Percolation was introduced by Broadbent and Hammersley in 1957. The simplest version to describe is on the Euclidean lattice Zd. Let p be a fixed probability between 0 and 1. Each bond in Zd is retained with probability p, and removed with probability 1-p, independently of all the others. The percolation cluster containing a point x, denoted C(x), consists of those points which can be reached from x by a path of retained bonds. There is a critical value pc∈(0,1) such that if p < pc then all clusters are finite, while for p > pc there is an infinite cluster.
Random walks on percolation clusters were introduced by De Gennes in 1976: he called this the problem of 'the ant in the labyrinth'. If p = p(n,x,y) is the probability that a random walker ('the ant'), starting at x, is at y at time n, then p describes diffusion of heat on the cluster.
For the supercritical phase (p > pc) this problem is now quite well understood, and p(n,x,y) converges to a Gaussian distribution as n→∞. PDE techniques introduced by Nash in the 1950s, play an important role in some of the arguments.
The critical case p = pc is much harder, since the clusters have fractal properties. One expects that p(n,x,x) ∼ x- ds/2, where ds is called the spectral dimension of the cluster. Alexander and Orbach conjectured in 1982 that ds = 4/3 in all dimensions: this has recently been proved in some high dimensional cases.
Note for Attendees
There will be tea and cookies in the math lounge at approximately 2:45pm.
We will discuss funding opportunities in pure and applied mathematics, and strategies for obtaining grants, fellowships, and so on. We will focus on NSERC grants in Canada and National Science Foundation grants in the U.S.
Breathing is an interesting and essential life-sustaining behavior for humans and all mammals. Like many rhythmic motor behaviors, breathing movements originate due to neural rhythms that emanate from a central pattern generator (CPG) network. CPGs produce neural-motor rhythms that often depend on specialized pacemaker neurons or alternating synaptic inhibition. But conventional models cannot explain rhythmogenesis in the respiratory preBotzinger Complex (preBotzC), the principal central pattern generator for inspiratory breathing movements, in which rhythms persist under experimental blockade of synaptic inhibition and of intrinsic pacemaker currents. Using mathematical models and experimental tests, here we demonstrate an unconventional mechanism in which metabotropic synapses and synaptic disfacilitation play key rhythmogenic roles: recurrent excitation triggers Ca2+-activated nonspecific cation current (ICAN), which initiates the inspiratory burst. Robust depolarization due to ICAN also causes voltage-dependent spike inactivation, which diminishes recurrent excitation, allowing outward currents such as Na/K ATPase pumps and K+ channels to terminate the burst and cause a transient quiescent state in the network. After a recovery period, sporadic spiking activity rekindles excitatory interactions and thus starts a new cycle. Because synaptic inputs gate postsynaptic burst-generating conductances, this rhythm-generating mechanism represents a new paradigm in which the basic rhythmogenic unit encompasses a fully inter-dependent ensemble of synaptic and intrinsic components.
The aim of this talk is to highlight the usefulness of continuous time branching process theory in understanding refined asymptotics about various random network models. We shall exhibit their usefulness in two different contexts:
(1) First passage percolation: Consider a connected network and suppose each edge in the network has a random positive edge weight. Understanding the structure and weight of the shortest path between nodes in the network is one of the most fundamental problems studied in modern probability theory. In the modern context these problems take an additional significance with the minimal weight measuring the cost of sending information while the number of edges on the optimal path (hopcount) representing the actual time for messages to get between vertices in the network. In the context of the configuration model of random networks we shall show how branching processes allow us to find the limiting distribution of the minimal weight path as well as establishing a general central limit theorem for the hopcount with matching means and variances.
(2) Spectral distribution of random trees: Many models of random trees (including general models embedded in continuous time branching processes) satisfy a general form of convergence locally to limiting infinite objects. In this context we find via soft arguments, the convergence of the spectral distribution of the adjacency matrix to a limiting (model dependent) non random distribution. For any \gamma we also find a sufficient condition for there to be a positive mass at \gamma in the limit.
Joint work with Remco van der Hoftsad, Gerard Hooghiemstra, Steve Evans and Arnab Sen.
Social learning is an important ability seen in a wide range of animals. Especially, humans developed the advanced social learning ability such as language, which triggered rapid cultural evolution. On the other hand, many species, such as viruses, rely on genetic evolution to adapt to environmental fluctuations. Here we propose an evolutionary game model of competition among three strategies; social learning, individual learning, and genetic determination of behavior. We identify the condition for learning strategies to evolve.
This is joint work with Eric Friedlander, Julia Pevtsova and Andrei Suslin. We consider modules over an elementary abelian group on which every element in the radical, but not the square of the radical, has the same Jordan canonical form. Such modules can be used to define bundles on projective spaces and Grassmanians. They have many interesting properties. We can get them as submodules of any module of the group algebra. In this talk I will discuss some of the constructions and their generalizations.
Note for Attendees
There will be tea and cookies in the PIMS 1st floor lounge at approximately 2:45pm.
The basic reproductive number, R_0, which is generally defined as the expected number of secondary infections per primary case in a totally susceptible population, is an important epidemiological quantity. It helps us to understand the possible outcome of an initial infection seeding in a social setting: whether it leads to a small outbreak, or it evolves into a large-scale epidemic. The basic reproductive number encapsulates the information about the biology of disease transmission as well as the structure of human social contacts. We use concepts from network theory to present a novel method for estimating the value of the basic reproductive number during the early stage of an outbreak. This approach will greatly enhance our ability to reliably estimate the level of threat caused by an emerging infectious disease.
In this talk, designed for a broad mathematical audience, we will describe what is known (and unknown) about the classification of smooth 4-manifolds. In particular we will uncover a new mechanism that makes progress towards the conjecture that every simply-connected closed 4-manifold has either zero or infinitely many distinct smooth structures.
Note for Attendees
There will be tea and cookies in the PIMS 1st floor lounge at approximately 2:45pm.
Coffee and cookies will be served at 2:30 p.m. There will be a 30 minute talk at 3 p.m., followed by a 20 minute forum. All members of the department are encouraged to meet the candidate for the joint Skylight/Mathematics position.
We introduce and discuss some new stochastic models of optimal portfolio selection with reduced impact of forecast errors. In particular, we found some new examples of optimal myopic strategies, including some discrete time models with serial correlations. In addition, we found some new cases when the strategies that don't leave the efficient frontier even if there is an error in the forecast. It may happen for non-myopic strategies if the required information about the future is limited.
This talk presents joint work with Kee Lam of UBC and Nancy Cardim, Maria Herminia de Mello, and Mario Olivero da Silva in Rio de Janeiro, Brazil. Given the total space of any orthogonal sphere bundle over a sphere, we determine its stable span and span.
Abstract: We will discuss the recent developments in the asymptotic theory of finite groups and their applications to theoretical computer science, number theory and topology.
Speaker's Biography: Efim Zelmanov is one of the world's leading algebraists. He is best known for his ground breaking work in Jordan algebras, Lie algebras and combinatorial group theory. In 1994 he was awarded a Field's medal for his solution of the restricted Burnside problem.
About the Niven Lectures: Ivan Niven was a famous number theorist and expositor; his textbooks won numerous awards, have been translated into many languages and are widely used to this day. Niven was born in Vancouver in 1915, earned his Bachelor's and Master's degrees at UBC in 1934 and 1936 and his Ph.D. at the University of Chicago in 1938. He was a faculty member at the University of Oregon since 1947 until his retirement in 1982. The annual Niven lecture series, held at UBC since 2005, is funded in part through a generous bequest from Ivan and Betty Niven to the UBC Mathematics Department.
Note for Attendees
The Niven lecture is meant to be broadly accessible.
Probabilities and randomness arise whenever we're not sure what will happen next. They apply to everything from lottery jackpots to airplane crashes; casino gambling to homicide rates; medical studies to election polls to surprising coincidences. This talk will explain how a Probability Perspective can shed new light on many familiar situations. It will also discuss "Monte Carlo" computer algorithms which use randomness to solve problems in many branches of science.
Note for Attendees
Professor Rosenthal is the author of the bestseller "Struck by Lightning: The Curious World of Probabilities".
In the first part of the mini-course I shall discuss how the entropy production observable can be defined for any classical or quantum dynamical system as a derivative of the Radon-Nikodym cocycle. For the so-called open systems, which describe the interaction of several thermal reservoirs, this definition coincides with the standard thermodynamical definition in terms of the fluxes (heat, charge, mass...) across the system. After reviewing some basic properties of the entropy production observable and non-equilibrium steady states, I shall describe the large deviation theory of the entropy production observable. The main topic will be certain symmetries (Evans-Searls and Gallavotti-Cohen) of the moment generating functionals which can be interpreted as an extension of the Green-Kubo linear response formula to far from equilibrium steady states. The emphasis of the course will be on the mathematical structure of the theory. One novelty of the exposition is that the classical nd quantum case will be treated in parallel. The mini-course is based on some very recent joint work with Claude-Alain Pillet, Yan Pautrat, Yoshiko Ogata and Luc Rey-Bellet.
Note for Attendees
If there is sufficient interest, this talk will be expanded to a small mini-course.
Employing an affine version of the plactic algebra (which arises in theRobinson-Schensted-Knuth correspondence) one can define non-commutative Schur polynomials. The latter can be employed to construct a combinatorial ring with integer structure constants. This combinatorial ring turns out to be isomorphic to what is called the su(n) WZNW fusion ring in the physics and the su(n) Verlinde algebra (extension over C) in the mathematics literature. There is a simple physical description of this ring in terms of quantum particles hoping on the affine su(n) Dynkin diagram. Many of the known complicated results concerning the fusion ring can be derived in a novel and elementary way. Using the particle picture one also arrives at new recursion formulae for the structure constants which are dimensions of moduli spaces of generalized theta functions. I explain the close connection with the small quantum cohomology ring of the Grassmannian and present a simple reduction formula which allows to relate the structure constants of the su(n) Verlinde algebra with Gromov-Witten invariants.
Given any two uniformly convex regions in Euclidean space, we show that there exists a unique diffeomorphism between them, such that the graph of the diffeomorphism is a special Lagrangian submanifold in the product space. This is joint work with Simon Brendle.
Weakly electric fish are fascinating animals that have evolved an electric sense that blends aspects of our senses of touch, vision and audition. Much is known about the relatively simple (compared to higher mammals) circuitry of their brains, the kinds of stimuli they respond to and their social communications/interactions. They are particularly well-suited to study principles of neural encoding and decoding because of the availability of electrophysiological recordings at many successive processing stations, enabling mathematical modeling of information transfer between stations. This talk will review past and current research on this topic from the experimental-theoretical collaboration of Len Maler, John Lewis and Andre Longtin at the University of Ottawa. We will focus especially on the role of feedback and how it interacts with stochastic spatio-temporal stimuli to induce oscillatory neural activity.
Abstract: Given a reasonable filtration of spaces F_0, F_1,..., F_n, and a homology theory H, we define what it means to split the filtration with respect to H and then give a criterion for when this is possible. We strongly refine this criterion
to decide when the filtration splits stably (and hence splits with respect to any homology theory). Many examples will be
discussed (like Steenrod splitting, snaith splitting, configuration spaces, commuting tuples in lie groups, Miller
splitting of unitary groups, etc). This work is by and with Stylian Zanos (Lille).
Starting from the standard contact structure of the supercircle, S^{1|1}, one considers the subgroups E(1|1), Aff(1|1), and SpO(2|1) of the group, K(1), of its contactomorphisms that respectively define its Euclidean, affine, and projective geometries. The notion of p|q-transitivity allows one to systematically construct the characteristic invariants of each geometry, in particular the super cross-ratio. One deduces the nontrivial associated 1-cocycles of K(1), e.g., the superschwarzian. The case of the supercircle S^{1|2} is also studied. The aim of this talk is to present in a synthetic fashion these geometric objects which are somewhat scattered in the literature. This is joint work with J.-P. Michel.
Abstract: Cocycle categories give a simple, flexible way to describe morphisms in a homotopy category, provided that the underlying model structure is sufficiently well behaved. "Well behaved" model structures include simplicial sets, spectra, simplicial presheaves and presheaves of spectra, together with all good localizations such as the motivic model structure of Morel and Voevodsky
Abstract: Suppose that H is an algebraic group which is defined over a field k, and let L be the algebraic closure of k. The canonical stalk for the etale topology on k induces a simplicial set map from the classifying space B(H-tors) of the groupoid of H-torsors (aka. principal H-bundles) to the space BH(L). The homotopy fibres of this map are groupoids of pointed torsors, suitably defined. These fibres can be analyzed with cocycle techniques: their path components are representations of the "absolute Galois groupoid" in H, and each path component is contractible. The arguments for these results are relatively simple, and applications will be displayed.
Abstract: Conjecture: For any map f: E \to S^1 from a closed 4-manifold to a circle whose homotopy fiber has the homotopy type of a 3-manifold, there exists a fiber bundle \bar f : \bar E \to S^1 where \bar E is a 4-manifold homotopy equivalent to E.
Theorem (joint with Shmuel Weinberger) The conjecture is true when the 3-manifold is a lens space with odd order fundamental group.
The proof involves a surgery theoretic argument which involves a lemma of Gauss used in his third proof of the law of quadratic reciprocity.
This theorem answers a question of Jonathan Hillman, asked in the context of 4-dimensional geometries:
Theorem: Any 4-manifold with Euler characteristic zero and fundamental group a semidirect product where Z acts on Z/odd is homotopy equivalent to a self isometry of a lens space.
Nonlinear dynamical systems are prevalent in systems biology, where they are often used to represent a biological system. Its dynamical behaviour is often impossible to understand by intuition alone without such mathematical models. Ideas and methods from systems and control engineering can help us to understand how the pathway architecture and parameter choices produce the desired performance and robustness in the observed dynamics. In this talk, we first show the direct interaction of a theoretical analysis with efficiently setting up experiments. We present the application of tools from engineering for designing biological experiments to elucidate the signalling pathway in the chemotactic system of /Rhodobacter sphaeroides/. In the second part, we focus on the problem of finding experimental setups that allow for full state observability and parameter identifiability of a nonlinear dynamical system; that is, whether the values of system states and parameters can be deduced from output data (experimental observations). This is an important question to answer as often observability and identifiability are assumed, which might lead to costly repetitions of experiments. We present a novel approach to check a priori for parameter identifiability and use new, state of the art computational tools for the implementation. Examples from biology are used to illustrate our method.
Many foods are complex fluids and chocolate is a prime example.
We infer the quality of chocolate by taste, which is our body's response
to the chemical and rheological processes that occur when we eat it.
Although we usually consume chocolate in the plastic state (plastic in
the engineering sense), it is usually - not always - processed in the
molten state, in which case it belongs to the rheological family
of 'granular suspensions' or 'pastes'. These materials inhabit the
rheological no-man's land between yield stress fluids and multiphase
flows. Even there it exhibits non-standard rheological behaviour,
arising from the nature of the components and the processing routes
used to make the chocolate. This presentation will outline some of the
challenges inherent in characterising and modelling this familiar
foodstuff, with a liberal sprinkling of facts from the world of
chocolate.
A basic question in probabilistic potential theory is the following: Consider a random subset $K\subset \mathbb{R}^d$, for what nonrandom sets $A$ is $P\{K\cap A\neq\emptyset\}>0$? In this lecture we will give some abstract results when $K$ is the range of a random field $\{v(x), x\in I\}$, $I\subset \mathbb{R}^k$. More specifically, we will establish upper and lower bounds of the hitting probabilities in terms of the Hausdorff measure and the Bessel-Riesz capacity of $A$, respectively, and highlight the role of the dimensions $d$ and $k$. Application to systems of stochastic wave equations will be discussed.
I will first describe a result on the uniqueness of invariant distributions for a certain process of coagulation and fragmentation. This result was first proved by Diaconis, Mayer-Wolf, Zeitouni and Zerner (2004) using representation theory, but subsequently Oded Schramm (2005) found a completely different and probabilistic proof. I will then explain how ideas from this approach can be used to give a new and probabilistic proof of the famous Diaconis-Shahshahani (1981) result about mixing times of random transpositions. In fact, this readily extends to much more general random walks on the permutation group (for which the increment is at each step uniformly selected from a given conjugacy class). This proves a conjecture of Roichman (1996). Joint work with Oded Schramm and Ofer Zeitouni.
Let G be a connected linear algebraic group over a number field K,
let H be a connected K-subgroup of G, and set X=H\G. We give a
convenient criterion to check, whether the set of rational orbits
X(K)/G(K) is finite, in terms of the Galois actions on pi1(H) and
on pi1(G). Using this criterion, we classify symmetric homogeneous
spaces of absolutely simple K-groups with finitely many rational
orbits.
The starting point of this talk is a theorem of Serre's on the
classification of compact p-adic manifolds. Every such manifold is a
disjoint union of n closed unit balls, for a unique value of n in
{0,...,p-1}. Using motivic integration, the ideas of Serre's proof can be
generalized to algebraic varieties X over a complete discretely valued
field K (for instance, the field of complex Laurent series). In this way,
one can define the motivic Serre invariant S(X) of X, which is an element
of a certain ring of virtual varieties over the residue field of K. We
will explain how one can consider S(X) as a measure for the set of
rational points on X, and how this measure admits a cohomological
interpretation by means of a trace formula.
In the first part of the talk, we recall the definitions of p-adic
numbers and p-adic manifolds, and we explain the elementary but elegant
proof of Serre's theorem. In the second part, we develop the basic
notions of motivic integration, and we illustrate the construction of the
motivic Serre invariant by some examples. In the third part, we explain
the statement of the trace formula, and we give some applications to
complex singularity theory and to arithmetic geometry.
This talk presents some coherent though incomplete conjectures for the emergence, replication and abundance of some chemical structures found in each prokaryote, with special emphasis on the tRNAs and the rRNA filaments that constitute a large part of the ribosomes.
In addition to the consideration of the data, two guiding principles for the formulation of these conjectures are Occam's razor, and the idea of uniformitarianism introduced with great success by the geologists of the 19th century. These ideas, aided by the empirical data, suggest that the abundance of the relevant cell structures should be regarded as a clue for their emergence. Also, in this talk, the distinction between the purines and the pyridines is emphasized, while distinguishing each purine (or each pyrimidine) from the others is often ignored; and the conjectures advanced in this talk also suggest some experiments that may justify or falsify their ideas.
We formulate a global form of Denef and Loeser's motivic
monodromy conjecture for complex hypersurface singularities, and we prove
it for tamely ramified abelian varieties A over a discretely valued field.
More precisely, we show that the motivic zeta function of A has a unique
pole, which coincides with Chai's base change conductor c(A), and we show
that this pole corresponds to a monodromy eigenvalue on the tame ell-adic
cohomology of A of degree dim(A). This is joint work with Lars Halvard
Halle (Hannover).
Abstract: We give a model for the moduli space of Riemann surfaces with one or more boundary curves using harmonic functions and canonical tesselations. The resulting simplicial complex is homeomorphic
to a flat vector bundle over the moduli space.
The simplest models of population genetics, useful as they are in analyzing data, often have obvious shortcomings. Such models might ignore the effects of natural selection, mutation, or, as we will be concerned with in this talk, geography and migration. We will briefly look at the Wright-Fisher model of evolution of a single population; then, we will look at a so-called stepping stone model, where instead of a single population living all in one place, we model several populations living on discrete islands, with migration between the islands. It is often useful to consider these models' associated dual processes, which correspond to tracing the lineages of a current-day sample backwards through history. We will discuss these dual processes as well.
We will then discuss two models of evolution with *continuous* geography. Unlike the previous models, which describe directly the dynamics of a population evolving as time moves forward, the continuous geography models are instead defined in terms of prescribed dual processes. Time permitting, we will also discuss some properties of these models, such as continuity.
I discuss recent results on the spatial dimensional reduction of the effective mean-field dynamics of many-body bosonic systems in strongly anisotropic harmonic potentials. In particular, the dynamics in the limit of strong anisotropy is effectively described by the nonlinear Hartree equation that is restricted to a submanifold of the original configuration space. Time permitting, I will discuss open problems regrading the mean-field dynamics of many-body constraint quantum systems.
PIMS/WMAX Postdoctoral Colloquium Abstract: This talk is based on a joint work with St\'ephane Kirsch. Pulsating traveling fronts are solutions of heterogeneous reaction-advection-diffusion equations that model some population dynamics. Fixing a unitary direction e, it is a well-known fact that for nonlinearities of KPP type (after Kolmogorov, Petrovsky and Piskunov, f(u)=u(1-u) is a typical homogeneous KPP nonlinearity), there exists a minimal speed c* such that a pulsating traveling front with a speed c in the direction of e exists if and only if c\geq c^*. In a periodic heterogeneous framework we have the formula of Berestycki, Hamel and Nadirashvili (2005) for the minimal speed of propagation. This formula involves elliptic eigenvalue problems whose coefficients are expressed in terms of the geometry of the domain, the direction of propagation, and the coefficients of reaction, diffusion and advection of our equation. In this talk, I will describe the asymptotic behaviors of the minimal speed of propagation within either a large drift, a mixture of large drift and small reaction, or a mixture of large drift and large diffusion. These ``large drift limits'' are expressed as maxima of certain variational quantities over the family of ``first integrals'' of the advection field. I will give more details about the limit and a necessary and sufficient condition for which the limit is equal to zero in the 2-d case.
The metric Mahler measure $M_1:A\to[1,\infty)$ is a modification of the classical Mahler measure $M$ that satisfies the triangle inequality $M_1(\alpha\beta)\leq M_1(\alpha)M_1(\beta)$. This function was first studied by Dubickas and Smyth in 2001, where they suggested a certain weakened version of Lehmer's conjecture. We establish this conjecture as well as give some applications showing that the value of $M_1$ cannot be too mysterious. We further examine a collection of other metric Mahler measures that give rise to new problems.
We prove a trace formula ŕ la Grothendieck-Lefschetz-Verdier for varieties X over a henselian discretely valued field with algebraically closed residue field. To the variety X, one can associate a motivic Serre invariant S(X), which measures the set of rational points on X. The trace formula expresses this measure in terms of the Galois action on the ell-adic cohomology of X, if X satisfies a certain tameness condition. If X is a curve, we relate the trace formula to Saito's criterion for tame ramification of the cohomology of X. If X is an abelian variety, we show how the trace formula gives a cohomological expression for the number of components of the Néron model of X.
The Min system in E.coli -- a group of three interacting proteins playing a role in cell division -- has attracted a lot of attention by modellers, some claiming it to be the 'measurement stick' in the rod-shaped bacterium. Different models have been proposed to explain the observed dynamical patterns -- oscillations, standing and travelling waves. Here, we will focus on a simple polymerisation/depolymerisation model. The model provides an interesting example of a stochastic hybrid dynamical system and we use probabilistic maps to compute probability distributions of experimentally accessible quantities. As a step towards model discrimination I will report on experiments we conducted on GFP-labelled E.coli.
We study the controllability of a shape changing body immersed in a perfect fluid. The shape changes are prescribed as functions of time and satisfy constraints ensuring that they are due to the work of body's internal forces only. The net locomotion of the body results from the exchange of momentum between the shape changes and the fluid. We consider the control problem that associates to any given shape changes the trajectory of the body in the fluid and we will show how this non-standard control problem can be solved within the framework of geometric control theory.
We formulate a global form of Denef and Loeser's motivic
monodromy conjecture for complex hypersurface singularities, and we prove
it for tamely ramified abelian varieties A over a discretely valued field.
More precisely, we show that the motivic zeta function of A has a unique
pole, which coincides with Chai's base change conductor c(A), and we show
that this pole corresponds to a monodromy eigenvalue on the tame ell-adic
cohomology of A of degree dim(A). This is joint work with Lars Halvard
Halle (Hannover).
In this presentation, I describe three projects aimed at exploring interfacial dynamics of viscoelastic polymeric liquids: drop deformation in converging pipe flow, experiment on selective withdrawal, and finally numerical simulation on selective withdrawal. The first project consists of finite-element simulations of drop deformation in converging flows in an axisymmetric conical geometry. The moving interface is captured using a diffuse-interface model and accurate interfacial resolution is ensured by adaptive refinement of the grid. The second and third projects concern the same process of selective withdrawal, in which stratified layers of immiscible fluids are withdrawn from a tube placed a certain distance from the interface. For experiment, we used video recording and imaging processing to analyze how the interfacial deformation is influenced by the non- Newtonian rheology of the liquid. For simulation, we used a sharp- interface, moving-grid method to explicitly track the moving interface. The work of this presentation has lead to two main outcomes. The first is a detailed understanding of how viscoelastic stress can lead to unusual and sometimes counter-intuitive effects on interfacial deformation. The second is a potentially important new method for measuring elongational viscosity of low-viscosity liquids. This is worth further investigation considering the poor performance of existing methods.
Abstract: The simpicial complex of talk I consists of pieces of the classifying spaces of symmetric groups. We use this to investigate the homology of moduli spaces. At the end, we discuss generalizations, where the symmetric groups are replaced by other families of Coxeter groups
In this talk a finite dimensional approximation of the recently constructed Wasserstein diffusion on the unit interval is presented. More precisely, the empirical measure process associated to a system of interacting, two-sided Bessel processes with dimension $0 < \delta < 1$ converges in distribution to the Wasserstein diffusion under the equilibrium fluctuation scaling. The passage to the limit is based on Mosco convergence of the associated Dirichlet forms in the generalized sense of Kuwae/Shioya. This is joint work with Max von Renesse.
This is the first talk of the new undergraduate mathematics colloquium
at UBC, or UBC/UMC. These biweekly talks will be centred on research
in math, and accessible to undergrads.
The first talk will be given by Adam Clay.
Title: Introduction to Knot Theory
Abstract: I will begin by describing some of the basic tools used in knot theory,
namely knot diagrams and Reidemeister moves. Next, we'll talk about
knot invariants, and I will give examples of some of the classical knot
invariants which are founded on the notion of knot diagrams. A good
example of such an invariant is "knot tricolourability". Time permitting,
I will also touch on some of the more powerful knot invariants, such as
the Alexander polynomial.
For more information, check out the UBC/UMC page at
http://www.math.ubc.ca/~fsl/UMC.html
This weekly seminar series (Wednesdays, 3:30-4:30pm, Math Annex 1102, except for October 7th will alternate between talks given by George Bluman and other speakers (to-date: Raouf Dridi, Zhengzheng Yang, Andy Wan, Mark Gotay, Alexandre Munnier). George Bluman will give a series of talks on the forthcoming Springer book (expected publication date: November 2009) "Applications of Symmetry Methods of Partial Differential Equations" by G. Bluman, A. Cheviakov and S. Anco. Topics include: local conservation laws and externsions of Noether's theorem, local symmetries, higher order symmmetries, invertible and noninvertible local mappings (including linearizations through symmetries and CLs), nonlocally related PDE systems, nonlocal symmetries, nonlocal CLs, nonlocal mappings, and the
nonclassical method to obtain solutions of PDEs. The new book is a sequel to the Springer book "Symmetry Methods for Differential Equations" by G. Bluman
Bluman and S. Anco. The main emphasis of Bluman's lectures will be on how to find systematically symmetries (local and nonlocal) of a given PDE system
and how to use systematically symmetries and conservation laws for related applications.
The first seminar will present an overview of the new book's topics. Participants can request a pdf file of the manuscript for private use.
A theorem of Hasse-Brauer-Noether states that every central simple
algebra over a number field is cyclic.
This does not hold for arbitrary fields. However, we have the
following result:
For any given field F there exists a regular field extension E/F such
that
i) any central simple E-algebra is cyclic,
ii) for any central simple F-algebra, index and exponent over
E (after field extension) are the same as over F,
iii) the restriction homomorphism res Br(F) --> Br(E) is injective.
In this talk I will give a brief introduction to the mean curvature flow and review some classic theory and results in the area. I will then introduce the Lagrangian mean curvature flow and discuss recent results which can be viewed as Lagrangian versions of classic results of Ecker-Huisken on the mean curvature flow of entire hypersurfaces. In particular, we prove existence of longtime smooth solutions to mean curvature flow of entire Lipschitz Lagrangian graphs. As an application of our estimates we classify all self-similar entire solutions to Lagrangian mean curvautre flow satisfying certain conditions. The results are from joint work with Jingyi Chen and Weiyong He.
Abstract: We give an example of two finitely generated quasi-isometric
groups that are not bilipschitz equivalent. The proof involves
structure of quasi-isometries from rigidity theorems, analysis of
bilipschitz maps of the n-adics and uniformly finite homology
One particularly easy way to compute generating functions for 3D Young diagrams, and for "pyramid partitions", is to use the commutation properties of vertex operators. In fact, the vertex operator method turns out to apply to a broader class of box-counting / dimer cover problems.
We will describe this more general class of problems, and explicitly give their generating functions. All of these generating functions can be readily turned into Donaldson-Thomas partition functions for the associated quivers (modulo a superpotential) by introducing signs on certain variables.
Wood is nature’s solution to the structural and hydraulic problems faced by the World’s largest living organisms-trees. The solutions to these problems are both elegant and energy efficient and are starting to inspire new (biomimetic) approaches to the design of novel materials. Wood in its native form, however, is still a very important material and sustains numerous industries that are vital to the economies of many countries, including Canada’s. The utilization of wood by these industries is constantly throwing up complex and important problems whose solution often benefits from an interdisciplinary approach involving the collaboration of biologists, physicists, chemists and mathematicians. This seminar will describe a range of fundamental and applied problems involving the utilization of wood that were solved by an interdisciplinary approach involving material scientists and mathematicians at The Australian National University. I will also introduce a new problem on the quantification of penetration of coatings into the porous microstructure of wood that could benefit from a similar interdisciplinary approach.
Biography.
Professor Philip Evans is the BC Leadership Chair in Advanced Forest Products Manufacturing Technology in the Faculty of Forestry at UBC. He is also an Adjunct Professor at The Australian National University and a Visiting Fellow in the Department of Applied Mathematics at The ANU. His research focuses on the surface properties of wood and the development of new wood-based materials that can compete with plastics, metals and concrete.
Let \gamma be a smooth closed curve of length 2\pi in R^3, and let \kappa(s) be its curvature regarded as a function of arc length s. We associate with this curve the one-dimensional Schroedinger operator H_\gamma = -d^2/ds^2 + \kappa^2(s) acting on the space of square integrable 2\pi-periodic functions. A natural conjecture is that the lowest eigenvalue e_0(\gamma) of H_\gamma is bounded below by 1 for any \gamma (this value is assumed when \gamma is a circle). We study a family of curves which includes the circle and for which e_0(\gamma)=1 as well, and show that the curves in this family are local minimizers; i.e., e_0(\gamma) does not decrease under small perturbations. A connection between the inequality and a dynamical elastica will be described. The conjecture remains open.
Recently, Y. Prokhorov classified all finite simple groups with faithful
actions on rationally connected threefolds. Using this classification,
I show that the essential dimensions of the alternating group, A7, and the
symmetric group, S7, are 4. In particular, the essential dimension of S7
is a measure of how much one can simplify a ``general polynomial'' of degree
7 by Tschirnhaus transformations. This was a long-standing open problem and
is related to algebraic forms of Hilbert's 13th problem.
Let F be a kxl (0,1)-matrix. We say that a (0,1)-matrix A has F as a
`configuration' if some row and column permutation of F is a
submatrix of A.
We are interested in `simple' matrices, namely (0,1)-matrices with no
repeated columns. If A is a simple matrix and has no configuration F then
what can we deduce about A? Our extremal problem is given m,F to
determine the maximum number of columns forb(m,F) in an m-rowed simple matrix A
which has no configuration F.
A `critical substructure' of F is a configuration F’ which is
contained in F and such that forb(m,F’)=forb(m,F). We give some examples to
demonstrate how this idea often helps in determining forb(m,F).
This represents joint work with Steven Karp and also Miguel Raggi.
Water striders and fishing spiders are creatures living on water. The special structure of their legs renders them highly nonwetting or superhydrophobic so that these creatures can stand effortlessly and walk quickly over the free surface of water. While the hydrostatics has been well understood, the propulsion mechanism of water walkers is still not fully resolved. We have performed finite-element simulations of the interfacial flow induced by the stoke motion of a leg, which is modeled as a cylinder. The free interface and the moving contact line are handled by using a diffuse-interface method. Results show that it is primary the curvature force pushing the water walkers forward, while the pressure force is of secondary role and the viscous force can be neglected. The superhydrophobicity doesn’t play an important role in the driving force, but it can decrease the resistance during the recovery stroke and increase the safety margin to delay the surface penetration. The relative importance of resulted waves and vortices is also discussed.
Define $e_{n}(t)=\{t/n\}$. Let $d_N$ denote the distance in $L^2(0,\infty ; t^{-2}dt)$ between the indicator function of $[1,\infty[$ and the vector space generated by $e_1, \dots, e_N$. A theorem of B\'aez-Duarte (2003) states that the Riemann hypothesis (RH) holds if and only if $d_N \rightarrow 0$ when $N \rightarrow \infty$. Assuming RH, we prove the estimate $$d_N^2 \leq (\log \log N)^{5/2+o(1)}(\log N)^{-1/2}.$$ I shall put this result in its historical context, from Nyman's criterion (1950) and its beautiful proof to a sketch of our proof. I shall focus on the main ingredient we used, a method of Maier and Montgomery, recently sharpened by Soundararajan, to get some upper bound for partial sums of the M\"obius function. (joint work with Michel Balazard)
In this hastily prepared talk, I will describe some preliminary results of Nathan Ng and myself that concern linear dependencies (with integer coefficients) among zeros of Dirichlet L-functions. We can show, for example, that given a Dirichlet L-function and an arithmetic progression of points on the critical line Re(s) = 1/2, a large number of points in the arithmetic progression are not zeros of the L-function. Furthermore, given a fixed linear form F in n variables, we show (assuming the Riemann hypothesis) that a large number of points of the form 1/2 + iF(gamma_1, ..., gamma_n) are not zeros of the Riemann zeta function, where the gamma_j are imaginary parts of such zeros. We also describe a theorem about prime number races that is linked to linear (in)dependence of zeros of Dirichlet L-functions.
In mean curvature flow (or MCF), a surface evolves to minimize its surface area as quickly as possible. One of the challenges of MCF is that the flow starting from a closed surface (like a sphere) always becomes singular and one of the most important problems is understanding these singularities. The simplest example comes from a round sphere, which evolves by staying round but having the radius shrink until it hits zero and then just disappears. Matt Grayson proved that this is the only type of singularity that occurs for simple closed curves in the plane. However, many other examples were discovered in higher dimensions (most of them by applied mathematicians doing numerical simulations). I will describe recent work with Toby Colding, MIT, where we classified the generic singularities of MCF of closed embedded hypersurfaces. The thrust of our result is that, in all dimensions, every singularity other than shrinking spheres and cylinders can be perturbed away.
In this talk I will overview some results about derived categories of toric stacks. In particular the problem of existense of strong exceptional collections of line bundles. Some connections of this problem to Mirror symmetry and combinatorics of polytopes will be mentioned.
We address the task of reconstructing images corrupted by Poisson noise, which is important in various applications, such as fluorescence microscopy, positronemission- tomography (PET) or astronomical imaging. We focus on reconstruction strategies, combining Bregman concepts, expectation maximization (EM) and total variation (TV) based regularization, and present analytical as well as numerical achievements. Recently extensions of the well-known EM/Richardson-Lucy algorithm received increasing attention for inverse problems with Poisson data. However, most algorithms for regularizations like TV produce images suffering from blurred edges due to lagged diffusivity, and neither can guarantee positivity nor provide analytical investigations including convergence. The first goal of this talk is to provide an accurate, robust and fast EM-TV method for computing cartoon reconstructions facilitating post-segmentation. The method can be reinterpreted as a modified forward-backward (FB) splitting strategy known from convex optimization. We establish the well-posedness of the basic variational problem and can prove the positivity preserving property of our method. A damped variant of the FB-EM-TV algorithm with modified time steps, is the key step towards convergence. Typically, TV-based reconstruction methods deliver reconstructions suffering from contrast reduction. Hence, as the second goal of this talk, we propose extensions to EM-TV, based on Bregman iterations and inverse scale space methods, in order to obtain improved imaging results by simultaneous contrast enhancement. We illustrate the performance of our techniques by 2D and 3D synthetic and real-world results in microscopy and tomography. Proceeding to 4D video reconstruction yields interesting challenges. Due to natural patient motion in medical imaging (e.g. heart or lung) or cell migration in microscopy, naive reconstructions can suffer from undesired blurring effects at object boundaries. Finally, we touch on combinations of reconstruction techniques and optimal transport strategies.
Subcellular oscillations of Min proteins within individual cells of E. coli serve to localize division to midcell. While significant progress has been made to understand the Min oscillation both experimentally and in modeling, I will present three outstanding Min mysteries. I will also present our ongoing work to develop generic submodels of the Min oscillation, and to systematically manipulate the Min oscillation experimentally. In particular, we find that the period of the Min oscillation responds dramatically to temperature and to the concentration of extracellular multivalent cations (including antimicrobial peptides).
The minimal heat kernel on a Riemannian manifold is conservative if it integrates to 1. If this is the case, the manifold is said to be stochastically complete. Since the heat kernel is the transition density function of Brownian motion, a manifold is stochastically complete if and only if Brownian motion does not explode. This interpretation opens a way for investigating conservation of the heat kernel by probability theory. To find a proper geometric condition for heat kernel conservation is an old geometric problem. The first result in this direction was due to S. T. Yau, who proved that a Riemannian manifold is stochastically complete if its Ricci curvature is bounded from below by a constant. However, it has been known for quite some time that the heat kernel conservation property is intimately related to the volume growth of a Riemannian manifold. We study this problem by looking at the more refined question of how fast Brownian motion escapes to infinity, for the existence of a deterministic upper bound for the escaping rate implies heat kernel conservation. We show how the Neumann heat kernel, time reversal of reflecting Brownian motion, and volumes of geodesic balls all come together in this problem and give an elegant and often sharp upper bound of the escaping rate solely in terms of the volume growth function without any extra geometric restriction besides geodesic completeness. The talk should be interesting and accessible to differential geometers, people in partial differential equations (pde-ers), and probabilists.
Let G be a nilpotent group and let g be its Lie algebra. I
will explain how Kirillov's orbit method associates an isomorphism
class of an irreducible representation of G to a point f in the dual
of g. I will then outline my joint work with T. Thomas, in which we
sharpen the Orbit Method and associate an irreducible representation
to f.
The arXiv code for our paper is: 0909.5670
We introduce the notion of a fat staircase and define when a skew diagram D is a sum of fat staircases. We give a collection of Schur-positivity results that may be obtained from each sum of fat staircases. Further, we determine conditions on when a diagram may be a sums of fat staircases.
The next talk for UBC/UMC, the undergraduate mathematics colloquium, will be given by George Bluman.
Title: Systematic Methods for Solving Ordinary Differential Equations (ODEs)
Abstract:
In the latter part of the 19th century, Sophus Lie initiated his studies on continuous groups (“Lie” groups) in order to put order to, and thereby extend systematically, the hodgepodge of heuristic techniques for solving ODEs. Lie showed how to find one-parameter Lie groups of transformations (point symmetries) of a given ODE and showed how a point symmetry reduces the order of an ODE. If an nth order ODE has n point symmetries that yield an n-parameter solvable group of point transformations (with good reason, solvable groups were initially called integrable groups), then the ODE is completely integrable, i.e., its solution reduces to n integrations (n quadratures).
Another systematic method for solving ODEs is based on finding and using integrating factors (seeking “first integrals” or “conservation laws”). This technique was fully developed in the 1990s. Lie showed that the symmetry and integrating factor methods are directly related for first order ODEs. This turns out not to be the case for higher-order ODEs..It turns out that, in general, the integrating factor method is complementary to Lie’s symmetry reduction method.
Symmetry and integrating factor methods are highly algorithmic and hence amenable to symbolic computation. These methods systematically unify and extend well-known ad-hoc techniques, learned in undergraduate ODE courses, to construct explicit solutions for differential equations, especially for nonlinear ODEs. The interplay between these two systematic methods is especially interesting.
This lecture will give an elementary presentation of symmetry and integration factor methods for solving ODEs (which today are the basis for the MAPLE software package DSOLVE as well as other software packages for solving ODEs).
The three dimensional incompressible Navier-Stokes system has two different structures: The antisymmetry of the nonlinearity, which provides an a priori bound for L^2 norm, and its natural scaling, which preserves L^3 norm. After a brief survey, I will talk about two problems. First the regularity problem, with a focus on current results assuming scaling invariant bounds (Type I singularity). Second the asymptotics problem, both temporal and spatial.
In this talk, I will introduce a new approach to rationality problem of Fano varieites using derive category, proposed by Kuznetsov. The idea is to construct a subcategory of the Fano variety, which is unchanged under birational tranforms. I will focus on the example where the Fano varieity is complete intersection of quadrics and explain the link between this new approach with some classical approaches of rationality problem.
In *classical mechanics of fluids* which is based on classical continuum field theories, the effect of forces is only considered on an infinitesimal fluid element. Hence, distribution and transmission of forces in the medium is described by the concept of *stress*. Such a simple model, which obtained with neglecting some physical aspects of system, can predict flows with small disturbances and smooth variations. These models fail when used for description of some complex phenomena like *turbulence*.
One of the ways to attack to the problems like turbulence is the use of non-classical continuum theories. We can use a non-local continuum model and the equations of gradient materials or a Cosserat continuum model. Another way is to use statistical mechanics and thermodynamics.
The aim of this work is to generalize the concept of continua by use of principles of *rational mechanics* such as *determinism* and *material objectivity*.
After derivation of equations of motion of fluids of second and third grade and of a Cosserat fluid, they have been solved for some simple problems of turbulent flows and the results have been compared with experimental data and the comparison showed that our model can describe turbulence with a acceptable precision.
Abstract: We study and determine a homotopy type of
the moduli space of all generalized Morse functions on d-manifolds for given d.
This moduli space is closely connected to the moduli space of all Morse functions
studied by Madsen and Weiss and classifying space of the corresponding cobordism
category.
This talk will survey recent progress on clarifying the connection between enumerative combinatorics and cluster expansions. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. Cluster expansions, on the other hand, are supposed to give convergent expressions for measures on infinite dimensional spaces, such as those that occur in statistical mechanics. There is a kind of dictionary between these two subjects that sheds light on each of them. In particular, it gives insight into new convergence results for cluster expansions.
This is the second part of ''New classification techniques for ordinary differential equations" in which I describe how one can use Cartan's equivalence method to compute the change of coordinates that maps two differential equations. In this talk I will present a new ordinary differential equation solver.
The Liouville function is defined by $\lambda(n):=(-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime divisors of the positive integer $n$ counting multiplicity. Let $m \geq 2$ be an integer and $\zeta_m$ be a primitive $m$--th root of unity. As a generalization of Liouville's function, we study the function $\lambda_m (n):=\zeta_m^{\Omega(n)}$. Using properties of this function, we will show how, for any integer $j$, properties about the set of all positive integers $n$ with $\Omega(n) \equiv j \pmod{m}$ can be obtained. In particular, we will show that this set has (natural) density $1/m$. In fact, we will also obtain much information about error terms and will illustrate how the case $m=2$ is very different from the case $m>2$. This is joint work with Michael Coons.
Note for Attendees
Cookies and tea will be served between the two talks.
Abstract:Equivariant Lefschetz invariants have already appeared in algebraic topology. Here I will show how to approach them using the so-called equivariant KK-theory of Kasparov — the main tool of the new field of noncommutative geometry. I will sketch the construction of Lefschetz invariants for equivariant self-maps of a G-space, where G is a
discrete group, and then define them for more general objects than just
self-maps, called correspondences. There are always many interesting
equivariant self-correspondences of a space with a group action, even if the
group is not discrete. The case of compact connected groups seems in particular quite interesting. We state an equivariant version of the Lefschetz fixed-point formula for this situation. In the resulting formula, the geometric side is based on equivariant index theory of elliptic operators, while the global algebraic side involves the module trace of Hattori and Stallings. Computing the relevant traces seems to be a problem belonging properly to algebraic geometry
Apart from sharing some interesting touristic and climatological observations, I will report on the computational number theoretic improvements I have included in Magma in June 2009.
While explicit p-adic analytic methods for solving diophantine equations based on Chabauty's ideas have been available for around 10 years now, there has been a recent shift to concentrate computational effort on an additional phase that can combine p-adic information at several primes. Heuristically, the method commonly referred to as "Mordell-Weil sieving" should yield arbitrarily detailed information on the location of possible solutions. In practice, however, there are severe combinatorial obstructions to exploiting that information.
In joint work with Michael Stoll, we have developed good ways of avoiding the intermediate combinatorial explosion. This strategy has now been implemented in Magma and yields an almost completely automatic procedure to determine the rational points on a considerable class of algebraic curves.
Studying non-smooth geometric objects is a very important
and modern research topic in differential geometry and geometric
analysis. In particular it is interesting to know to which extent
these objects can be approximated by smooth ones.
In this talk I want to indicate how this problem can be related to the
study of systems of parabolic equations with irregular initial data.
Moreover I want to discuss various situations in which existence results
for these initial value problems have been obtained.
If H is a variation of Hodge structure over a variety S, then there is a family of complex tori J(H) over S associated to H. Admissible normal functions are certain sections of J(H) over S. Roughly speaking, they are the ones that have the possibility of coming from algebraic geometry.
I will explain recent work with Gregory Pearlstein proving that the locus where a section of J(H) vanishes is an algebraic subvariety of S. This answers a conjecture of Griffiths and Green.
Gauss-Lobatto spectral elements, based on hexahedral meshes, provide a very efficient way to solve transient wave equations in terms of storage and of computational time. Unfortunately, it is very difficult and almost impossible in some cases to produce pure hexahedral meshes for complex geometries. Until now, we have remedied this shortcoming by using tetrahedral meshes in which tetrahedral were split into four hexahedra, but this technique provides very distorted meshes which imply tu use about three times more unknown than pure hexahedral meshes to get the same accuracy. For this reason, we developed a strategy of solvers based on hybrid meshes containing mainly hexahedra and some tetrahedral, pyramids and wedges. First results show a dramatical gain in performance versus split tetrahedra.
I will explain why formulas for compatible intertwiners for
representation of nilpotent groups require the notion of the
determinant of a finite abelian p-group. I will then outline an
approach of Deligne for defining determinant using K-theory.
This talk is a continuation of my previous talk, but I will try to
assume very little.
Superhydrophobicity and water repellency is important not only for outdoor apparel, but also for applications in the fields of corrosion resistance, micro-fluidics, anti-fouling, bio-compatibility, low drag and low friction surfaces. This talk will give an introduction to superhydrophobicity and the underlying physics. Also it will show a way of using a femtosecond laser to modify the wetting behavior of common engineering materials, namely metals, and their subsequent use as low friction materials.
Group field theory is the higher-dimensional generalization of random matrix models. As it has built-in scales and automatically sums over metrics and discretizations, it provides a combinatoric origin for space time. Its graphs facilitate a new approach to algebraic topology. I exemplify this approach by introducing a graph's cellular structure and associated homology.
The next talk for UBC/UMC, the undergraduate mathematics colloquium, will be given by Richard Anstee.
Title: If you can't square the circle, then at least you can square the square
Abstract:
I will talk partly about W.T. Tutte, a Professor at the University of Waterloo. I will partly talk about the problem of the squared square, which is a dissection of a square into smaller squares, all of different sizes.
There may even be a little linear algebra and graph theory in the talk.
This seminar continues with an overview of the topics of the forthcoming Springer book "Applications of Symmetry Methods to Partial Differential Equations" by Bluman, Cheviakov and Anco.. In Part II, it will be shown how to obtain systematically nonlocal symmetries of PDE systems. In turn this leads to the question of how to find systematically conservation laws of PDE systems. In particular, it will be shown how to generalize the classical Noether's theorem to find conservation laws of PDE systems that are not variational.
For millenia, many misguided mathematicians moved to prove that Euclid's parallel postulate was a consequence of his other axioms. In this talk I will discuss hyperbolic geometry, and draw a picture or two proving all of these valiant efforts wrong.
I will also discuss relationships between complex analysis and the classification of all two-dimensional geometries, why hyperbolic geometry can be described (somewhat tongue-in-cheek) as "God's Geometry", and given time, a discussion of extant three dimensional geometries.
Biological aggregations such as insect swarms, bird flocks, and fish schools are arguably some of the most common and least understood patterns in nature. In this talk, I will discuss recent work on swarming models, focusing on the connection between inter-organism social interactions and properties of macroscopic swarm patterns. The first model is a conservation-type partial integrodifferential equation (PIDE). Social interactions of incompressible form lead to vortex-like swarms. The second model is a high-dimensional ODE description of locust groups. The statistical-mechanical properties of the attractive-repulsive social interaction potential control whether or not individuals form a rolling migratory swarm pattern similar to those observed in nature. For the third model, we again return to a conservation-type PIDE and, via long- and short-wave analysis, determine general conditions that social interactions must satisfy for the population to asymptotically spread, contract, or reach steady state.
We review the idea of using lattice paths as models of phase boundaries rather than as models of polymers. We also present the solution of a particular model that has demonstrated some novel mathematical features. Additionally this model may describe the steady state of a non-equilibrium model of molecular motors.
Superlattice patterns and quasipatterns, while well-studied in waves on the surface of vertically vibrated viscous fluids (Faraday waves), have found little attention in forced oscillatory systems. We study such patterns, comprised of 4 or more Fourier modes at different orientations, by applying multi-frequency forcing to systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. For weak forcing composed of 3 frequencies near the 1:2- and 1:3-resonance, such systems can be described by a suitably extended complex Ginzburg-Landau equation with time periodic coefficients. Using Floquet theory and weakly nonlinear analysis we obtain the amplitude equations for simple patterns (comprised of 1, 2, or 3 modes) and superlattice patterns. We stabilize these patterns via spatiotemporal resonance and find stable subharmonic 4- and 5-mode patterns through judicious choice of the forcing function. For system parameters reported for experiments on the oscillatory Belousov–Zhabotinsky reaction we explicitly show that the forcing parameters can be tuned such that 4-mode patterns are the preferred patterns. We confirm our analysis through numerical simulation. This work was done jointly with Dr. Hermann Riecke at Northwestern University.
I will discuss some deformation properties of Fano
varieties. The general methods rely on the investigation of the
variation of the cone of effective curves and, more generally, of the
Mori chamber decomposition, which, according to Mori theory, encode
information on the geometry of the variety. The talk is based on
joint work with C. Hacon.
Computer algebra systems like Maple and Mathematica spend most of their time doing either polynomial arithmetic or linear algebra. For polynomials in one variable of degree n, the Fast Fourier Transform gives us an O(n log n) multiplication algorithm. But for polynomials in several variables, which are usually sparse (most of the coefficients are zero) there are no O(n log n) algorithms. So what's the best way to multiply and divide them?
In the talk I will present
Johnson's algorithm from 1974 for multiplying two sparse polynomials using a heap.
Our division algorithm which also uses a heap.
Some benchmarks showing that these algorithms are 100 times faster than Maple and Mathematica, and
a parallel algorithm for multiplication using a heap.
I'd also like to present an optimization which we call "immediate monomials" where we reduce multiplication of monomials ( x^i y^j z^k ) to one machine instruction and describe a new project we are doing with Maple to redesign the basic polynomial representation in Maple to hopefully get a good overall speedup.
We present a general variational method, involving self-dual variational calculus, for recovering non-linearities from prescribed solutions for certain types of PDEs which are not necessarily of Euler-Lagrange type, including parabolic equations. The approach can also be used for optimal control problems. The topological aspects involved, for the space of self-dual Lagrangians, and the space of maximal monotone vector fields on a reflexive Bancah space will be discussed.
We give a characterization of all degree 6 del Pezzo surfaces over an arbitrary
field. This characterization is derived by studying the toric structure of
the surface. Such a surface is determined, up to isomorphism, by a pair of
separable algebras, subject to some constraints. These algebras determine
geometric, arithmetic, and topological information about the variety, which
we will discuss. Some of this work is joint with Paul Smith and Sue Sierra.
In symbolic dynamics, a Z^d shift of finite type (or SFT) is the set of all ways to assign elements from a finite alphabet A to all sites of Z^d, subject to local rules about which elements of A are allowed to appear next to each other. A fundamental number associated to any SFT is its (topological) entropy, which is, roughly speaking, the exponential growth rate of the number of allowed patterns of size n.
The entropy of any Z SFT is easily computable (it is the log of an algebraic number). However, for d > 1, the situation becomes more complex. There are in fact only a few nontrivial examples of Z^2 SFTs whose entropies have explicit closed forms.
It is natural then to try to at least estimate these entropies. We will discuss some of the difficulties involved in doing this, and present a way of approximating entropy for a class of Z^2 SFTs by way of some easier-to-compute entropies of associated Z SFTs.
As a corollary of this technique, we can show that the entropy of any Z^2 SFT in this class is computable in polynomial time.
The potential catastrophic failure of a dam and the resultant widespread downstream flooding and damage is a scenario that is of great concern. A brief overview of the numerical methods most commonly used in the simulation of dam-break floods, namely Godunov-type finite-volume schemes solving the two-dimensional shallow water equations, will be given. Smoothed particle hydrodynamics will then be introduced and its potential as an alternative numerical scheme will be addressed. Results of the simulation of an actual historic dam-break event using both methods will be presented.
Abstract: In the 1960s, Wall developed a theory of finiteness obstructions for CW-complexes. We extend this result to diagrams of spaces and use it to investigate which homotopy G-spheres can be realized on a finite complex. We will conclude with some new examples of finite (non-linear, non-free) G spheres for some small groups G
We will describe a current approach to understanding stationary states in non-equilibrium statistical mechanics. We will then consider two examples: a system of stochastic differential equations for coupled oscillators, and a stochastic wave equation. Stationary states for these examples exhibit steady state energy flow.
Micro-Electro-Mechanical Systems (MEMS) and Nano-Electro-Mechanical Systems (NEMS), which combine electronics with miniature-size mechanical devices, are basic ingredients of contemporary technology. A key component of such systems is the simple idealized electrostatic device consisting of a thin and deformable plate, consisting of a dielectric material with a negligibly thin conducting film on its lower surface, that is held fixed along its boundary in the two dimensional plane. Above the deformable plate lies a rigid grounded plate. As one applies a positive voltage to the thin conducting film the deformable plate deflects upwards towards the ground plate. If the voltage is increased beyond a certain critical value then the deformable plate touches the ground plate, in finite time, and we have the so-called "pull in instability".
Unfortunately, models for electro-statically actuated micro-plates that account for moderately large deflections are quite complicated and not yet amenable to rigorous mathematical analysis. In the last 5 years, my students (Cowan, Esposito, Guo, Moradifam) and I, dealt with much simplified models that still lead to interesting second and fourth order nonlinear elliptic equations (in the stationary case) and to nonlinear parabolic equations (in the dynamic case). The non-linearity is of an inverse square type, which -- until recently – has not received much attention as a mathematical problem. It was therefore rewarding to see, besides the above practical considerations, that the model is actually a very rich source of interesting mathematical phenomena. Numerics and formal asymptotic analysis give lots of information and point to many conjectures, but even in the most simple idealized versions of electrostatic MEMS, one essentially needs the full available arsenal of modern nonlinear analysis and PDE techniques “to do" the required mathematics.
In this talk, we will look at the exciting, recent results showing that most images and other signals can be reconstructed from much less information than previously thought possible, using simple, efficient algorithms. A consequence has been the explosive growth of the new field known as compressive sensing, so called because the results show how a small number of measurements of a signal can be regarded as tantamount to a compression of that signal. The many potential applications include reducing exposure time in medical imaging, sensing devices that can collect much less data in the first place instead of collecting and then compressing, and getting reconstructions from what seems like insufficient data (such as EEG). We will see how replacing the convex optimization problem typically used in this field with a nonconvex variant has the effect of reducing still further the number of measurements needed to reconstruct a signal. A very surprising result is that a simple algorithm, designed only for finding one of the many local minima of the optimization problem, typically finds the global minimum. Understanding this is an interesting and challenging theoretical problem. We will see examples, and discuss algorithms, theory, and applications.
Kostant's remarkable formula (which I will recall) generalizes to smooth Schubert varieties in the flag variety G/B of an algebraic group G. On the other hand, Sara Billey noticed that rationally smooth Schubert varieties in G/B give an analogous formula, though it often doesn't agree with the remarkable formula in the singular case. This motivates the question of which rationally smooth Schubert varieties are smooth. I will show that there is a neat answer hinted at in the title.
I will discuss the scattering problem of mass-critical generalized KdV equation. We will see if the scattering of gKdV fails, then a minimal mass blow-up solution exist on the condition that scattering of mass-critical 1D NLS is true. We use concentration compactness argument in addition to an observation that a certain modulated, rescaled version of NLS solution is approximately gKdV solution for highly oscillatory profile.
Combinatorial games are typically played using a disjunctive sum of distinct components. On each player's turn, they choose one of the available components and make a legal move in that component. Traditionally, a player cannot choose a component where a legal move does not exist. We examine the case where this is allowed and call it the short disjunctive sum of games. In particular, we show that both interpretations are equivalent under normal play rules but differ under mis\`ere play rules. Finally, we show how this interpretation can be extended to the analysis of both normal and mis\`ere play games which have non-standard ending conditions.
We will show that a Del Pezzo surface of degree 6 has two explicitly
defined locally free sheaves, whose endomorphism rings generated the
K-theory of the surface. Time permitting, we will also show that a similar
result holds for the derived category of the surface. Some of this work is
joint with Paul Smith and Sue Sierra.
Essentially all cellular processes depend on spatially localized proteins. Some proteins localize to cell poles, others to the particular cell membranes, and yet others to specific cytoplasmic regions. This localization is often dynamic, with proteins shuttling between different regions. The Smoldyn biochemical simulator helps researchers study this intracellular organization; Smoldyn represents each protein as an individual point-like particle that diffuses, reacts, and interacts with membranes, all in continuous space. It was surprisingly difficult to make these processes quantitative, such as for finding the "binding radius" for bimolecular reactions and the adsorption probability for molecules that adsorb to membranes. Smoldyn has enabled a variety of research projects over the last several years. In one example, Smoldyn simulations showed that yeast cells appear to secrete a protease (called Bar1) which degrades extracellular pheromone so that, paradoxically, they can sense the pheromone gradient more accurately. This helps cells improve their mating success.
Start a totally asymmetric simple exclusion process with a second class particle at 0, particles to its left and holes to its right. If Xt is the location at time t of the second class particle, then Xt / t converges a.s. to a uniform [-1,1] random variable.
I will prove an analogous result for partially asymmetric exclusion process (with Balázs and Seppäläinen), and explain why this is interesting (with Amir and Valkó).
Abstract: In stable homotopy theory the thick subcategory theorem of Hopkins and Smith classifies thick subcategories of the triangulated category of p-torsion finite spectra. Unstably Bousfield classified nullity classes of p-torsion finite suspensions. We look at analagous results in the derived category of a commutative noetherian ring D(R), and some of its subcategories satisfying suitable finiteness conditions. Hopkins and later Neeman proved that thick subcategories of D_{perf}(R) can be classified by their supports, which are subsets of Spec(R). In analogy to Bousfield's result, we show that nullity classes in D^b_{fg}(R) can be classified by certain increasing functions from Z into Spec(R). By an observation of Keller and Vossieck, it turns out that t-structures are just nullity classes together with a right adjoint of the inclusion. From this we derive an extra condition that the increasing function must satisfy to correspond to a t-structure. When R has a dualizing complex, applying a construction of Deligne and Bezrukavnikov thengives a classification of all the t-sctructures in D^b_{fg}(R)
The next talk for UBC/UMC, the undergraduate mathematics colloquium, will be given by Greg Martin.
Title: Prime numbers: What we know, and what we know we think
Abstract:
Questions about the distribution of prime numbers, and about the existence of prime numbers of special forms, have been stymieing mathematicians for over two thousand years. It's almost necessary to study two different subjects: the theorems about prime numbers that we have been able to prove, and the (vastly more numerous) conjectures about prime numbers that we haven't yet succeeded at proving. In this talk I'll describe many of the open problems (and a few closed ones) concerning the distribution of primes, mentioning when I can some techniques that have been used to attack them.
The Graduate Colloquium will feature two speakers this Thursday, November 5. We'll have first Kael that will talk about "Knots, tensors, and statistical mechanics" and then Terry who will be speaking about "Nonmeasurable sets".
So join us from 12:30 to 1:30 in LSK 462 with the usual free pizza and pop! Abstracts below,
Cheers,
David
------------------------------------------------
Speaker: Kael Dixon
Title: Knots, tensors, and statistical mechanics
In this short talk, I will breifly introduce knot theory and abstract tensor diagrams. Then I will show how combining these ideas gives the Yang Baxter Equation, which originated in statistical mechanics. Don't worry, there's not actually going to be any statistical mechanics in the talk.
------------------------------------------------
Speaker: Terry Soo
Title: Nonmeasurable sets
We will discuss the nonmeasurable sets defined by Vitali, Ulam, and Shelaha-Soifer.
The Graduate Colloquium will feature two speakers this Thursday, November 5. We'll have first Kael that will talk about "Knots, tensors, and statistical mechanics" and then Terry who will be speaking about "Nonmeasurable sets".
So join us from 12:30 to 1:30 in LSK 462 with the usual free pizza and pop! Abstracts below,
Cheers,
David
------------------------------------------------
Speaker: Kael Dixon
Title: Knots, tensors, and statistical mechanics
In this short talk, I will breifly introduce knot theory and abstract tensor diagrams. Then I will show how combining these ideas gives the Yang Baxter Equation, which originated in statistical mechanics. Don't worry, there's not actually going to be any statistical mechanics in the talk.
------------------------------------------------
Speaker: Terry Soo
Title: Nonmeasurable sets
We will discuss the nonmeasurable sets defined by Vitali, Ulam, and Shelaha-Soifer.
While univariate polynomial interpolation has been a basic tool of scientific computing for hundreds of years, multivariate polynomial
interpolation is much less understood. Already the question from which polynomial space to choose an interpolant to given data has no obvious answer.
The talk presents, in some detail, one answer to this basic question, namely the ``least interpolant'' of Amos Ron and the speaker which, among other nice properties, is degree-reducing, then seeks some remedy for the resulting discontinuity of the interpolant as a function of the interpolation sites, then addresses the problem of a suitable representation of the interpolation error and the nature of possible limits of interpolants as some of the interpolation sites coalesce.
The last part of the talk is devoted to a more traditional setting, the complementary problem of finding correct interpolation sites for a given
polynomial space, chiefly the space of polynomials of degree le k for some k, and ends with a particular recipe for good interpolation sites in the square, the Padua points.
Carl de Boor is a Professor Emeritus in Mathematics and Computer Science at the University of Wisconsin-Madison. He won the 2003 U.S. National Medal of Science. An expert in numerical analysis, Dr. de Boor is the author of more than 150 papers and four books. He has earned world recognition for his work on spline functions, mathematical expressions that describe free-form curves and surfaces. In particular, Dr. de Boor developed simpler approaches to complex spline calculations, a contribution that revolutionized computer-aided geometric design. His work is now routinely applied in a range of fields that rely on precise geometry, including the use of special effects in films, and in the aircraft and automotive industries. Dr. de Boor grew up in East Germany and came to the United States in 1959. He received a doctorate from the University of Michigan in 1966 and joined the UW-Madison faculty in 1972. Until 2003, Dr. de Boor was the Steenbock Professor of Mathematical Sciences and the P.L. Chebyshev Professor of Mathematics and Computer Sciences. He was awarded the John von Neumann Prize by SIAM in 1996. In 1993 he was elected to the National Academy of Engineering, and in 1997 to the National Academy of Sciences. Dr. de Boor also is a member of the Deutsche Akademie der Naturforscher (1998) and a foreign member of the Polish Academy of Sciences (2000). He holds honorary doctorates from Purdue University (1993) and the Technion in Israel (2002)
I will describe some recent work in inverse problems from seismology for the acoustic wave equation. Caustics (or multi-pathing) present in the underlying ray geometry force one to consider certain Lagrangian manifolds with singularities and associate classes of Fourier integral operators, in analogy with the classical Hormander theory for smooth Lagrangians.
In this first talk we'll give an advertisement for the rigorous and formal tools of optimal transportation, highlighting their contribution to diffusion equations, simple proofs of Sobolev and isoperimetric inequalities, generalizing the Ricci-bounded-below condition beyond smooth manifolds, and geometrically reinterpreting the Schroedinger equation. We'll then learn about two ideas at the center of these applications: 1) that probability measures can be formally seen as a Riemannian manifold (F. Otto '01) and 2) certain entropy functionals are convex in this geometry (R. M cCann '94). We'll fill out the hour by reviewing the formal Riemannian structure (local geometry) and rigorous aspects of (global) Wasserstein distance.
n the first part of the talk, I will explain why we may need to go beyond the standard framework of transition state theory (TST) to describe activated processes and reactive events, and I will present another framework, termed transition path theory (TPT), that permits to do that. Unlike TST, which gives mainly an expression for the rate of the reactive event, TPT describes more fully the statistical properties of the reactive trajectories (i.e. those trajectories by which the reactive event occurs), in particular in terms of their probability density function and their probability current. In the second part of the talk, I will describe how TPT can be used to design and/or improve numerical methods for computing the pathways and rate of reactive events. I will focus in particular on the string method and milestoning. These techniques will be illustrated via examples from molecular dynamics.
In the first of a two lecture series (to be completed by Dave Anderson immediately following), we present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex $G$-variety $X$ by its associated arc space $J_{\infty} X$, with its induced $G$-action. If $X$ admits an `equivariant affine paving', then we deduce an explicit geometric basis for the equivariant cohomology ring. Moreover, under appropriate hypotheses, we obtain explicit bijections between bases for the equivariant cohomology rings of smooth varieties related by an equivariant, properbirational map. As an initial application, we present a geometric basis for the equivariant cohomology ring of a smooth toric variety.
Let G be an algebraic group acting on a smooth complex variety X. In joint
work with Alan Stapledon, we present a new perspective on the G-equivariant
cohomology of X, which replaces the action of G on X with the induced action
of the respective arc spaces. I will explain how this point of view allows
one to interpret the cup product of classes of subvarieties geometrically
via contact loci in the arc space, at least under suitable hypotheses on the
singularities. As an explicit example, I'll discuss GL_n acting on the space
of matrices.
In this talk the Meshing program Netgen and the Finite Element library NGSolve are introduced. The possibilities of this software package will be highlighted with the help of two modern examples for Advanced Higher Order Finite Element Methods: A Discontinuous Galerkin Formulation for the Time Domain Maxwell Equations and a Hybrid Discontinuous Galerkin Formulation for the incompressible Navier Stokes Equations. Both methods will be shown from a theoretical point of view as well as implementation aspects with Netgen/NGSolve.
In this second talk we'll see how commonly studied PDEs like the heat equation, nonlinear diffusion, thin film equation, and Schroedinger equation can be formally seen as geometric evolutions in the Riemannian geometry of probability measures. The work on Schroedinger equation is due to Max-K. von Renesse ('09).
Given a morphism of vector bundles on a variety, one is often interested in
the locus of points where the morphism drops rank. When the map is
sufficiently general, the classical Giambelli-Thom-Porteous formula
expresses the class of this locus in terms of the Chern classes of the
vector bundles. I will explain how this is related to a more general
problem of computing the T-equivariant cohomology classes of B-orbit
closures in g/p, and motivated by this perspective, describe a new symmetry
for morphisms coming from triality.
In this talk on joint work with Shahab Shahabi, I would like to describe how algebraic parts of periods of cycles on Shimura curves are interpolated by certain p-adic L-functions. In appropriate situations, derivatives of these p-adic L-functions are related to Heegner and Stark-Heegner points on elliptic curves. This generalizes results of Bertolini-Darmon and Shahabi concerning the analogous situations for classical modular curves.
Note for Attendees
Cookies and tea will be served between the two talks.
WMAX 110 (PIMS mini-symposium in PDE); time changed
Thu 12 Nov 2009, 3:00pm-4:00pm
Abstract
I will speak on certain kinetic and macroscopic models of traffic flow. After a review of the concept of a fundamental diagram the high-density regime will be considered, and the emergence of macroscopic models with nolocalities will be discussed. Numerical evidence (and real traffic data) suggest that travelling "braking" waves form and propagate in response to trigger events. A traveling wave ansatz for solutions of the macroscopic models leads to an unusual functional differential equation, for which preliminary studies will be shown.
Let E be an elliptic curve defined over Q. A pair of primes (p,q) is called an amicable pair for E if #E(F_p) = q and #E(F_q) = p. Although rare for non-CM curves, such pairs are relatively abundant in the CM case. I will explain the difference, present conjectures and experimental data for their frequency, discuss some generalisations and related questions, and spend some time on the still-mysterious j=0 case. This talk will afford an opportunity to use cubic reciprocity. This is joint work-in-progress with Joseph H. Silverman.
We describe a fourth order family generalizing the linear-mobility thin film equation on R^n. In joint work with R. McCann we derive formally sharp converence rates to self-similarity, using a link to Denzler-McCann's analysis of a second order diffusion. We then show (joint with Matthes, McCann, Savare) that a certain range of nonlinearity allows the obtaining of rigorous results for the fourth-order evolution in 1 dimension.
Imagine some commodity being produced at various locations and consumed at others. Given the cost per unit mass transported, the optimal transportation problem is to pair consumers with producers so as to minimize total transportation costs. Despite much study, surprisingly little is understood about this problem when the producers and consumers are continuously distributed over smooth manifolds, and optimality is measured against a cost function encoding some geometry of the product space.
This talk will be an introduction to the optimal transportation, its relation to Birkhoff's problem of characterizing of extremality among doubly stochastic measures, and recent progress linking the two. It culminates in the presentation of a criterion for uniqueness of solutions which subsumes all previous criteria, yet which is among the very first to apply to smooth costs on compact manifolds, and only then when the topological type of one of the two underlying manifolds is the sphere.
How plants and animals achieve their forms has been an enduring question in the history of biology, from early descriptions to modern genetic manipulations. Plant shapes are especially challenging, since spatial chemical patterns determine cell type, but also drive (and respond to) tissue growth, a major determinant of overall plant architecture. Increasingly, physical and mathematical scientists are becoming involved in the unique problems of mechanics, transport, and pattern formation in plants. My work uses Turing-type reaction-diffusion models to drive localized surface growth, in 3D. I have been able to generate many of the shapes seen in plants, fitting results to data from single-celled algae and more recently to conifer embryos. These shapes can be understood in terms of transitions between solutions to the reaction-diffusion equations in response to domain change. I will describe some of the open computational problems to achieving stable, accurate model solutions with large domain growth and arbitrary shape change, and some of the directions we are taking experimentally and analytically to further characterize the chemical control of plant shape.
Goresky and MacPherson observed that certain pairs of
algebraic varieties with torus actions have equivariant cohomology
rings that are "dual" in a sense that I will define. Examples of
such pairs come up naturally in both representation theory and
combinatorics. I will explain how this duality is in fact a shadow
of a much deeper relationship, in which certain categories of sheaves
on the varieties are Koszul dual to each other.
We present a new algorithm for the evaluation of the periodized Green's function for Helmholtz equation in two and three dimensions. A variety of classical algorithms (based on spatial and spectral representations, Ewald transformation, etc.) have been implemented in the past to evaluate such acoustic fields. As we show however, these methods become unstable and/or impractically expensive as the frequency of use of the sources increases. Here we introduce a new numerical scheme that overcomes some of these limitations allowing for simulations at unprecedented frequencies. The method is based on a new integral representation derived from the classic spatial form, and on suitable further manipulations of the relevant integrands to render the integrals amenable to efficient and accurate approximations through standard quadrature formulas. We include a variety of numerical results that demonstrate that our algorithm compares favorably with every classical method both for points close to the line where the poles are located and at high-frequencies while remaining competitive with them in every other instance.
We derive a general Hardy inequality and show most Hardy inequalities can be seen as special cases of this inequality. In addition we characterize the improvements of this inequality and (time permitting) we show an application of this inequality to the regularity of stable solutions to a nonvariational equation.
The representation dimension of a finite group G is the minimal dimension
of a faithful complex representation of G. I will discuss the maximal
representation dimension of groups of order p^n for fixed prime p and
integer n, and show that p-groups of maximal representation dimension can
be found (non-uniquely) in a certain class of groups of nilpotence class 2.
This talk is based on joint work with Zinovy Reichstein and Masoud
Kamgarpour.
Cluster expansions give convergent expressions for measures on infinite dimensional spaces, such as those that occur in statistical mechanics. One approach to these convergence questions is via fixed points for infinite dimensional increasing functions. Sometimes it is not so difficult to show that there is a fixed point, indeed, a unique least fixed point.
Abstract: Motivated by basic questions from robotics and molecular biology, we consider certain configuration spaces, and some associated maps to two dimensional Euclidian space. We are able to understand the singular sets for a natural subset of these maps.
For this subset, we apply variants of Bott-Morse theory to determine the structure of inverse images - in the configuration spaces - of curves and points in the plane.
In turn, these results answer key questions about the structure of spaces of length preserving immersions of polygons into Euclidean space and provide insights into the process of protein folding
The next talk for UBC/UMC, the undergraduate mathematics colloquium, will be given by David Kohler.
Title: The Missing Region Problem
Abstract:
In this talk, we will study the missing region problem and its solution. This will allow us to discuss various interesting mathematical ideas and concepts along the way, such as the strong law of small numbers and the Euler characteristic of a polyhedron ... some neat thinking outside the box, and we'll be back to Pascal's triangle in the end.
I will give a brief introduction to the development of soliton theory. Then I will present the constructive theory of nonlinear wave equations and its application to the (N + 1)-dimensional generalized nonlinear Klein-Gordon equation. Finally, the systematic method to compute the point symmetries of a PDE and its application will be introduced.
In geometry finite reflection groups (subgroups of the orthogonal group of an Euclidean space that are generated by reflections) appear, e.g., as groups of symmetries of certain regular polytopes. We will establish a correspondence between those groups and root systems, finite subsets of R^n that were first mentioned in Lie theory but have since then been observed in many different contexts like combinatorics or cluster algebras to name just a few. With help of those sets we will manage to give a nice description of finite reflection groups as well as a complete classification.
In the 1950's, reaction-diffusion equations were proposed as a model for the symmetry-breaking process that organisms undergo as they transition from single cells to embryos. There are plenty of other applications that involve them, most notably as a mechanism responsible for pattern formation in organisms. Perhaps the main reason for the popularity of the model is that linear stability analysis provides considerable insight into the types of patterns that the stationary solutions will have. The talk will serve as an introduction to reaction-diffusion equations and Turing analysis. For the people already familiar with the subject, we will also discuss the limitations that occur when considering domains in 2 or more dimensions.
A discussion of symmetry is good for the fluff section of any grant proposal. In this talk I will discuss finite symmetry groups of spheres. These questions have been part of topology from the beginning. Poincare showed that the binary icosahedral group acts freely on the 3-sphere and that the quotient is a homology 3-sphere. This led to his famous question about simply connected three manifolds.
Traditionally, three kinds of group actions on spheres have been studied, linear actions, smooth actions and continuous actions. The linear actions of a group are its orthogonal representations. A smooth action of a group is a homomorphism to Diff S^n and a topological actions is a homomorphism to Homeo S^n (the topological group of homeomorphisms from the sphere to itself). These correspond to three of the geometries of the sphere. Homotopy theory studies the most fundamental geometry, the one where only the toughest invariants are left.
There are several ways to define the homotopy actions of a group. In the easiest definition, a homotopy action of G on S^n is an action of G on a space X that is homotopy equivalent to S^n. But X can be complicated; it need not be a manifold or even finite dimensional.
The linear actions of G are the orthogonal representations of G. The smooth and topological actions are only completely understood when the group is acting freely, the so called spherical space form problem, and only a few finite groups can act freely on a sphere. Much to our surprise, Grodal-Smith have completely classified the homotopy actions of a finite group on a sphere. The surprise arises because the classification of homotopy actions is equivalent to computing
[BG,B Aut S^n]
where Aut S^n is the topological monoid of self equivalences of S^n; B Aut S^n and BG are the classifying spaces. Both are infinite dimensional and classical techniques are useless. The entry into computation is the modern technology for studying homotopy fixed points.
I will report on joint work with E. Macri on the space of stability
conditions for the derived category of the total space of the canonical
bundle on the projective plane. It is a 3–dimensional manifold, with
many chamber decompositions coming from the behaviour of moduli spaces
of stable objects under change of stability conditions.
I will explain how this space is related to classical results by Drezet
and Le Potier on stable vector bundles on the projective plane. Using
the space helps to determine the group of auto-equivalences, which
includes a subgroup isomorphic to \Gamma_1(3). Finally, via mirror symmetry,
it contains a universal cover of the moduli space of elliptic curves
with \Gamma_1(3)–level structure.
(Huang) We consolidate an unorganized point cloud with noise, outliers, non-uniformities, and in particular interference between close-by surface sheets as a preprocess to surface generation, focusing on reliable normal estimation. Our algorithm includes two new developments. First, a weighted locally optimal projection operator produces a set of denoised, outlier-free and evenly distributed particles over the original dense point cloud, so as to improve the reliability of local PCA for initial estimate of normals. Next, an iterative framework for robust normal estimation is introduced, where a priority-driven normal propagation scheme based on a new priority measure and an orientation-aware PCA work complementarily and iteratively to consolidate particle normals. The priority setting is reinforced with front stopping at thin surface features and normal flipping to enable robust handling of the close-by surface sheet problem.
We demonstrate how a point cloud that is well-consolidated by our method steers conventional surface generation schemes towards a proper interpretation of the input data.
(van den Berg) The use of l1 regularization in optimization problems to promote sparsity in the solution has increasingly become a standard technique. More recently, related types of regularization have been used in different contexts to obtain similar sparsifying effects. In this talk we present a framework that solves problem formulations in these new areas by applying root-finding on the Pareto curve.
The theory of optimal transport is concerned with phenomena arising when one matches two mass distributions in a most economic way, minimizing transportation cost of moving mass from one location to another. We consider an optimal transportation problem with costs satisfying certain type of degenerate curvature condition. This condition is a slightly stronger but still degenerate version of the Ma-Trudinger- Wang condition for regularity of optimal transport maps. We explain a continuity result of optimal maps with rough data on local and global domains. If time permits, we will also explain a connection to Principal- Agent problem in microeconomics. These reflect joint work in progress with Alessio Figalli and Robert McCann.
We study experimentally the effect of a mean flow, imposed on a buoyant exchange flow of two miscible fluids of equal viscosity, in a long tube oriented close to horizontal. We measure the evolution of the front velocity, Vf, as a function of the imposed velocity, V0. At low V0 an exchange flow dominated regime is found, as expected, and is characterized here by Kelvin-Helmholtz-like instabilities. With increasing V0 we unexpectedly observed that the flow becomes stable. Here also Vf increases linearly with V0, with slope > 1. At large V0 we find Vf ~ V0.
A simple random walk (SRW) on a graph is a Markov chain whose state space is the vertex set and the next state distribution is uniform among the neighbors of the current state. A graph is called recurrent if a SRW on it returns to the starting vertex with probability 1, and called transient otherwise. The path of a walk on a graph is simply the set of edges this walk has traversed. Our main result is that the path of a SRW on any graph is a recurrent graph. The proof uses the electrical network interpretation of random walks. We will give a sketch of the proof, including the necessary background, and discuss related questions and conjectures.
I will give a brief introduction to the development of soliton theory. Then I will present the constructive theory of nonlinear wave equations and its application to the (N + 1)-dimensional generalized nonlinear Klein-Gordon equation. Finally, the systematic method to compute the point symmetries of a PDE and its application will be introduced.
In this talk, through basic examples, I will try to illustrate how various classical problems of analysis arise when modeling the interactive motion of a fluid and a rigid structure: well-posedness, stability, asymptotic analysis, control and inverse problems...
We show how the method of Green and Tao can be used to obtain the correct order of magnitude for the number of k-term arithmetic progressions in the integers representable as the sum of two squares, with a similar Roth-like theorem for subsets of positive relative density. We'll discuss the possibility of sharpening this to an asymptotic formula.
This talk is based on a series of five lectures given by Doug Ulmer at the Park City Mathematics Institute in 2009. I will present Ulmer's proof that elliptic curves over function fields can have arbitrarily high Mordell-Weil rank.
Coin tosses are the very essence of probability theory. Yet, there are a number of simply stated problems that have resisted solution. These may involve Peter Winkler's clairvoyant demon, and they may sometimes be cast in the language of so-called dependent percolation. Three such problems and their extensions are discussed in this talk, together with some recent progress in collaboration with Ander Holroyd. The only prerequisite for the audience is an affinity for tossing a coin.
I will summarise how our quantitative description of the Universe as a whole has been dramatically improving, through advances in instrumentation which allow us to gather higher fidelity data which, combined with simple physical models and computationally-intensive statistical approach, enable us to determine a few cosmological parameters to high precision, leading (of course) to new questions.
I will present work-in-progress on the construction of compactifications of the moduli space of curves with A_k-singularities. These spaces conjecturally give moduli interpretations of certain log canonical models of the moduli space of curves. This is joint work with David Smyth and Fred van der Wyck.
We propose a hybrid method for solving large sparse linear least squares problems. The method is iterative in nature, as it is based on preconditioned LSQR. However, the preconditioner comes from an orthogonal factorization of a submatrix of the original matrix associated with the least squares problem; the construction of the preconditioner is based on well-known techniques in sparse direct methods. In this talk, we will discuss the quality of the preconditioner and the choices of the submatrix.
Consider a nonlinear Schr\"{o}dinger equation in $\mathbb{R}^3$ with a short-range potential. The linear Hamiltonian is assumed to have three or more eigenvalues satisfying some resonance conditions. We study the asymptotic behavior at time infinity of solutions with small initial data in $H^1 \cap L^1(\mathbb{R}^3)$. The results include the case that all of the eigenvalues are simple and also the case that the second eigenvalues are degenerate. These are joint works with Stephen Gustafson, Kenji Nakanishi and Tai-Peng Tsai.
This talk will be about the collection of theories that started from a
lecture by M. Kontsevich in 1995, and which are now known
by the name of "motivic integration". One of these theories
introduces integration-like techniques in order to prove results in
algebraic geometry (this was the original motivation behind it), and
the other one provides a geometric interpretation of p-adic integrals.
I will try to survey the main ideas behind the construction of motivic
measures, and mention some of the most spectacular applications.
Given a homogeneous Poisson point process it is well known that selecting each point independently with some fixed probability gives a homogeneous Poisson process of lower intensity. This is often referred to as thinning. Can thinning be achieved without additional randomization; that is, is it possible to choose a subset of the Poisson points as a deterministic function of the Poisson process so that the chosen points form a Poisson process of any given lower intensity?
On an infinite volume, it is always possible. Furthermore, on R^d , it is possible to define the deterministic function to be a translation-equivariant factor (that is, if a translation is applied to the original process, the chosen points are translated by the same vector). On a finite volume, the answer depends on both the intensities of the original and resulting Poisson processes. We will discuss joint work with Omer Angel, Alexander Holroyd, and Russell Lyons.
The last talk this term for UBC/UMC, the undergraduate mathematics colloquium, will be given by Eric Cytrynbaum.
Title: How cells get by without a compass and ruler
Abstract:
Cells need to be able to read their own geometries for various purposes. For example, cell division requires that a division plane through the "middle" of a cell be determined. Cell motility requires that various structures within the cell be properly aligned with the direction of "desired" motion. In this talk, I will discuss a few mechanisms by which cells carry out these space-sensing calculations.
This seminar continues with an overview of the topics in the just published (November 2009) Springer book "Applications of Symmetry Methods to Partial Differential Equations" by Bluman, Cheviakov and Anco. In Part IV, it will be shown how to find directly the multipliers and then the corresponding fluxes for conservation laws of PDE systems. This will be followed by a discussion of connections between symmetries and conservation laws.In particular, it will be shown how to use symmetries to find new conservation laws from known conservation laws.Furthermore relationships will be established between symmetries, solutions of adjoint equations and conservation laws.
Elliptic curves are very old mathematical objects but they are still element of current research. In my talk I will define the notion of an elliptic curve E motivated by some beautiful connection to meromorphic functions on complex tori. We will illustrate (but not prove) that an elliptic curve E carries an abelian group structure and will define its group of rational points, which leads to the famous Mordell-Weil Theorem. Time permits I will say something about the conjecture of Birch and Swinnerton-Dyer.
It is expected that a general K3 surface does not admit
self rational maps of degree > 1. I'll give a proof of this conjecture for
K3 surfaces of genus at least 4.
The outer automorphism group of a free group of finite rank shares many properties with linear groups and mapping class groups of surfaces. However the techniques for studying these three families of groups are generally quite different; problems which are difficult for one class may be easier and more intuitive for another. I will describe some algebraic, topological, and dynamical methods for studying such problems, thereby highlighting some of the interesting features which distinguish these groups.
Abstract: Fully irreducible outer automorphisms of a free group are analogous to loxodromic isometries of hyperbolic space, or to pseudo-Anosov elements of the mapping class group of a surface. We develop methods for constructing customized fully irreducible elements of a free group F of rank k. For example, there exists for any matrix A in GL(k,Z) a non-geometric fully irreducible element inducing the action of A on the non-abelian free group of rank k. This is an analogue of a well-known theorem for the mapping class group. This is joint work with Matt Clay.
By Mostow's rigidity theorem, geometric invariants of hyperbolic 3-manifolds are in fact topological invariants. On the other hand, it follows from the work of Thurston and Perelman that a 3-manifold is hyperbolic if and only if it satisfies some rather mild conditions. In light of these results, it is an interesting question to try to understand how topological conditions on a 3-manifold $M$ which admits a hyperbolic metric affect the geometry of the hyperbolic metric. This question is rather imprecise. In other words, it has many different incarnations. In this talk I will describe a few results on different concrete formulations of the question above.
John Mighton will speak on his work with JUMP Math.
John Mighton is a mathematician, author, playwright, and the founder of JUMP Math. He tirelessly volunteers his time and expertise at JUMP as the lead curriculum developer for the JUMP Math Student Workbooks and Teacher's Manuals. He also donates all proceeds from publications to JUMP.
Dr. Mighton completed a Ph.D. in mathematics at the University of Toronto and was awarded an NSERC fellowship for postdoctoral research in knot and graph theory. He is currently a Fellow of the Fields Institute for Research in Mathematical Sciences and has also taught mathematics at the University of Toronto.
Capture-recapture methods are widely used to monitor endangered wildlife populations. A requirement of simple capture-recapture models is that all individuals alive on one sampling occasion have the same probability of capture. While this assumption may be reasonable in small, isolated populations, there are many variables that might influence an individual's catchability and estimates of survival rates or the abundance will be biased if these differences are
ignored. However, covariates of the capture probability which vary both between individuals and over time, like body mass, present a challenge in the analysis of capture-recapture data because 1) their values can only be measured for the individuals captured on each sampling occasion and 2) the unknown values are not missing at random and cannot be ignored. I will present Bayesian methods to incorporate the effects of such covariates in the Cormack-Jolly-Seber and Jolly-Seber models -- the two most common models for open-population capture-recapture data.
My talk will begin with an introduction to Bayesian statistics, capture-recapture methods, and the problems associated with time-dependent covariates. I will then describe my method for including such covariates in the Cormack-Jolly-Seber model to estimate survival rates and how this method can be extended to the Jolly-Seber model to obtain estimates of abundance. I will illustrate my methods by application to data from the study of Soay sheep on the Isle of Hirta, Scotland, and conclude by discussing applications to more complicated models and comparisons with other approaches.
Abstract: Suppose that \Sigma is a hyperbolic surface of finite type and let \Map(\Sigma) be its mapping class group. It is due to Morita that the canonical homomorphism \Diff(\Sigma)\to\Map(\Sigma) does not split. A first goal of this talk is to give a very simple proof of this fact showing in fact that this also remains true when restricted to some rather small subgroups of the mapping class group. After having proved this, I will show that while \Map(\Sigma) admits a natural Lipschitz action on the unit tangent bundle T^1\Sigma of \Sigma, this action is not homotopic to any smooth action. This is partly joint work with Mladen Bestvina and Tom Church.
The talk will address the issues of numerical approximations of dynamical systems in presence of high oscillation. For the systems of highly oscillatory ordinary differential equations given in the vector form y' = A_w y + f, where A_w is a constant nonsingular matrix, ||A_w|| >> 1, \sigma(A_w) in iR, f is a smooth vectorvalued function and w is an oscillatory parameter, we show how an appropriate choice of quadrature rule improves the accuracy of numerical approximation as w -> \infty. We present a Filon-type method to solve highly oscillatory linear systems and WRF method, a special combination of the Filon-type method and the waveform methods, for nonlinear systems. The work is accompanied by numerical examples.
MATH ANNEX 1102 (relocated because of PIMS closure)
Mon 21 Dec 2009, 1:30pm-3:00pm
Abstract
In his influential works, A. Okounkov showed how to associate a convex body to a very ample G-line bundle L on a projective G-variety X such that it projects to the moment polytope of X and the push-forward of the Lebesgue measure on it gives the Duistermaat-Heckamnn measure for the correspoding Hamiltonian action. He used this to prove the log-concavity of multiplicities in this case. Motivated by his work, recently Lazarsfeld-Mustata and Kaveh-Khovanskii developed a general theory of Newton-Okounkov bodies (without presence of a G-action).
In this talk, I will go back to the case where X has a G-action. I discuss how to associate different convex bodies to a graded G-algebra which in particular encode information about the multiplicities of the G-action. Using this I will define the Duistermaat-Heckmann measure for a graded G-algebra and prove a Brunn-Minkowski inequality for it. Also I will prove a Fujita approximation type result (from the theory of line bundles) for this Duistermaat-Heckmann measure. This talk is based on a preprint in preparation joint with A. G. Khovanskii.
In classical Lie theory a homomorphism of Lie groups f : H--> G, with H simply connected, is uniquely given by its effect on
the Lie algebras Lie(f) : Lie(H) --> Lie(G). When f : H --> G is a weak
morphism of Lie 2-groups, with H 2-connected (i.e., \pi_iH vanish
for i=0,1,2), we prove that f is uniquely given by Lie(f), where
Lie(f) : Lie(H) --> Lie(G) is the induced morphism in the derived
category of 2-terms diff. graded Lie algebras. We also exhibit a
functorial construction of the 2-connected cover H<2> of a Lie
2-group H.
The theory of geometric quantization is one way of producing a "quantum system" from a "classical system," and has been studied a great deal over the past several decades. It also has surprising ties to representation theory. However, despite this, there still does not exist a satisfactory theory of quantization for systems with singularities.
Geometric quantization requires the choice of a polarization; when using a real polarization to quantize a regular enough manifold, a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld fibres. However, there are many types of systems to which this result does not apply. One such type is the class of completely integrable systems, which are examples coming from mechanics that have many nice properties, but which are nontheless too singular for Sniatycki's theorem to apply.
In this talk we will explore one approach to the quantization of integrable systems, and show a Sniatycki-type relationship to Bohr-Sommerfeld fibres. However, some surprising features appear, including infinite-dimensional contributions and strong dependence on the polarization.
I will give at least a brief explanation of both geometric quantization and integrable systems, and hope to make the talk accessible to a general differential geometric audience.
When molten rock rises from deep in the earth, invades the earth's crust and cools, this forms new rock structures called igneous intrusions. These widespread features are often associated with the formation of mineral deposits, and there is hope that understanding their emplacement mechanisms will benefit engineering applications by giving insight into how rock behaves at large scale. Particular attention is given here to so-called large mafic sills and laccoliths, which can be proposed as natural analogues to hydraulic fractures that grow relatively close to a free surface.
The use of elastic plate theory to model the growth of shallow igneous intrusions has been debated for over 40 years. Investigation has typically resulted in the elastic plate model being heavily questioned or abandoned because it fails to predict the characteristic flat-topped, steep-sided thickness profiles of laccoliths or the strikingly uniform thickness of large mafic sills. However, upon coupling elastic plate theory with a fracture mechanics based propagation criterion and, crucially, the driving force due to the weight of the magma, the predicted thickness profiles and thickness to length relationships for both laccoliths and large mafic sills are consistent with an extensive collection of field data. Furthermore, analysis of the large time asymptotics predict that large mafic sills will attain a thickness that is not only spatially uniform, but also constant in time, depending only on physical properties of the magma and host rock. While a number of questions remain open, it is an exciting prospect that a single, basic model could provide a unifying framework to understanding what controls the first-order behaviour of the growth of laccoliths and large mafic sills.
This talk will present work that results from collaboration with Professor Alexander Cruden of the University of Toronto, Department of Geology.
Let G be a finite group. A G-gerbe over a space X may be
intuitively thought of as a fiber bundle over X with fibers being the
classifying space (stack) BG. In particular BG itself is the G-gerbe
over a point. A more interesting class of examples consist of G-gerbes
over BQ, which are equivalent to extensions of the finite group Q by G.
Considerations from physics have led to conjectures asserting that
the geometry of a G-gerbe Y over X is equivalent to certain "twisted"
geometry of a "dual" space Y'. A lot of progresses have be made recently
towards proving these conjectures in general. In this talk we'll try to
explain these conjectures in the elementary concrete examples of G-gerbes
over a point or BQ.
At the heart of Thurston's proof of Geometrization for Haken manifolds is a family of analytic functions between Teichmuller spaces called "skinning maps." These maps carry geometric information about their associated hyperbolic manifolds, and I'll discuss what is presently known
about their behavior. The ideas involved form a mix of geometry, algebra, and analysis.
In this talk I will describe the coupled integro-partial differential equations that model the evolution of a fluid-driven fracture propagating in a state of plain strain. I will discuss the use of the Mellin Transform and matched asymptotics to establish the asymptotic behavior of the solution in the vicinity of the fracture tip for a number of regimes of propagation. I also describe a novel cubic Hermite collocation scheme to solve these coupled equations. This algorithm involves special blended cubic Hermite-power law basis functions, with an arbitrary index 0<1, which are developed to treat the singular behavior of the solution that typically occurs at the tips of a hydraulic fracture. I also discuss the implementation of blended infinite elements to model semi-infinite crack problems. The cubic Hermite collocation algorithm is used to solve a number of different test problems and the results are compared to published similarity, asymptotic, and numerical solutions.
It is well know that for p = 2, the K(1)-localization of KO is EO_1,
and for p = 2; 3, the K(2)-localization of TMF is
EO_2. When does the K(n)-localization of TAF contain a factor of EO_n?
We will provide a complete answer. This is joint work with Mike
Hopkins.
complexity of PGLn-torsors or, alternatively, central simple algebras. It was first raised by Procesi in the 1960 and the exact value is still
mostly unknown. We will discuss some recent developments which have led to both new lower
and upper bounds. These are obtained in part by studying classes of algebras with additional
structure, e.g. crossed-products or simple algebras split by a distinguished field extension.
Abstract: A Bers slice is a naturally embedded copy of the Teichmuller
space in the SL(2,C) character variety of a surface. We prove that Bers
slices are never algebraic. A corollary is that Thurston's skinning map
is never constant. The proof involves the theory of complex projective
structures and a little algebraic geometry.
We present the uniform distribution of roots of quadratic congruences on the unit circle. Several proofs have been rediscovered over the years. I will review some of the very different methods involved -- ergodic theory, exponential sums, automorphic forms. It constitutes an excellent introduction to the more delicate Linnik problems. I will then proceed to describe new generalizations. These have far reaching applications to arithmetic geometry and independence of Heegner points on rational elliptic curves.
In this talk, we discuss the behavior of heat kernel for symmetric jump-type process with jumping kernels comparable to radially symmetric function on the spaces. Parabolic Harnack principle and sharp two-sided heat kernel estimates for both small and large time will be discussed. This is a joint work with Zhen-Qing Chen and Takashi Kumagai.
This talk will be a friendly introduction to the basic concepts of game theory, starting with the Nash solution concept (more appropriately, the “Cournot-Nash” concept – you’ll see why). My goal is to highlight some of the central results, illustrate some interesting examples, and to give an idea why game theory has become so important in economic analysis (and other disciplines, too). Technicalities will be kept to a minimum, so it should be easy for everyone to walk away with a good idea of what game theory is all about.
This short talk aims at describing expander graphs and some of their fascinating applications to other fields of mathematics and computer science. Wether you are interested in coding theory, complexity theory, probability theory, number theory or group theory, (with or without a flavour of geometry and linear algebra on the side) there will be something for you. And if none of these really speak to you, at least you'll get a nice promenade in the mathematical landscape.
We discuss the question of quantitative bounds on the sup-norm of automorphic cusp forms. We present an improvement on a recent result by Blomer-Holowinski on Hecke-Maass forms on $X_0(N)$ with large level $N$. Analogous results are then established for all compact arithmetic surfaces by a geometric approach.
The stack of stable maps parameterizes maps from a complete curves having at worst nodal singularities into a smooth scheme. Generally this stack is not smooth, but we will explain how it can be made smooth by relaxing the condition that the source curves be complete. Although the resulting stack is not fibered in groupoids, and therefore may not be easily accessible to geometric intuition, it is a natural setting in which to construct the virtual fundamental class. We will discuss how this generalization can be used to prove a conjecture of Abramovich and Fantechi relating the virtual fundamental classes of two different moduli spaces parameterizing stable maps into mildly singular schemes.
The Langlands program relates complex representations of GL_n(Q_p) to Galois representations. For n = 1 this is explained by class field theory and for n = 2 this is closely related to the theory of modular forms. For general n, this is now understood by the work of Harris-Taylor and Henniart. In the last decade, a mod-p (as well as a p-adic) version of the Langlands program have been emerging, and they have already played an important role in some recent progress in number theory. But so far understanding has been limited to n = 1 and 2. We survey some of the known story in the classical and in the mod p case, and then discuss some recent progress on the classification of mod p representations of GL_n(Q_p), as time permits.
Kalman smoothing is an important topic in control theory, with a myriad of applications. We will discuss some of these applications, present the modeling framework amenable to solution by smoothing, and discuss two related approaches to making the smoother robust against errors in the measurement data. Specifically, we will consider two heavy tailed models for observation noise, discuss the merits of these models from a statistical point of view, show how in each case the statistical model gives rise to an optimization problem with special structure, and then solve these problems to find the a posteriori maximum likelihood (MAP) solution for each model. We will then compare the smoothers' performance on simulated data contaminated with different types of outliers and on real data in an underwater tracking experiment.
Let F be a finite extension of the p-adic numbers. We describe the classification of irreducible admissible smooth representations of GL_n(F) over an algebraically closed field of characteristic p, in terms of "supersingular" representations. This generalizes results of Barthel-Livne for n = 2. Our motivation is the hypothetical mod p Langlands correspondence for GL_n, which is supposed to relate smooth mod p representations to Galois representations.
Note for Attendees
This is a joint Number Theory/Algebraic Groups Seminar.
I will describe some background and recent results on singularity formation (and non-formation) for some simple, physical, and popular geometric PDE describing dynamics of maps into spheres -- the heat-flow, wave map, and Schroedinger map -- in the energy-critical 2D case. I'll try to keep it simple and accessible by illustrating the methods on a symmetric reduction of the heat-flow, leading to a single scalar PDE.
In recent work of Baas-Dundas-Richter-Rognes, the authors prove that the classifying space of 2-vector bundles, K(Vect) is equivalent to the algebraic K-theory of the connective K-theory spectrum ku. In this talk we will show that K(Vect) is the group completion of the classifying space of the 2-category of 2-vector spaces, which is a symmetric monoidal 2-category. We will explain how to use the symmetric monoidal structure to produce a $\Gamma$-2-category, which will give an infinite loop space structure on K(Vect). Then we will show that the equivalence of BDRR is a map of infinite loop spaces.
The first talk this term for UBC/UMC, the undergraduate mathematics colloquium, will be given by Mike Bennett.
Title: Diophantine equations for fun (and profit?)
Diophantine equations are one of the oldest, frequently celebrated and most abstract objects in mathematics. In this talk, I'll attempt to show some of the roles these equations play in modern mathematics and maybe even reveal how they can be used to make a (not particularly) fast buck.
This is an expository talk about the distributive law of algebra and its role in combinatorics, probability, and physics. The main idea is that one can run the distributive law either way, a process sometimes called re-summation. This extraordinarily simple idea underlies a fair amount of current research in probability and mathematical physics.
Let $M(\alpha)$ denote the Mahler measure of the algebraic number $\alpha$. Dubickas and Smyth constructed a modified version $M_1$ of $M$ having the triangle inequality. $M_1$ is called the metric Mahler measure. We produce an entire parametrized family $\{M_t\}$ of metric Mahler measures which gives rise to a new reformulation of Lehmer's problem. We further examine the functions $t\mapsto M_t(\alpha)$, for fixed $\alpha$, showing that they are constructed piecewise from certain simpler functions.
Note for Attendees
Cookies and tea will be served between the two talks.
This is the first talk in the teaching seminar associated with the TA Accreditation Program. (All are welcome!)
Title: Engaging students in the classroom: what is all this 'clicker' nonsense?
We will explore ways to engage students to think actively about course material in class using "clickers." We will make use of iClickers throughout this session.
Starting from a classical result of Skolem, Mahler and Lech for linear recurrence sequences, we present an algebraic geometric generalization of it. Then we interpret our result from the point of view of dynamics, linking it with the Mordell-Lang conjecture from Diophantine Geometry. We conclude by studying another dynamical question which generalizes the Manin-Mumford conjecture.
Motivated by the classical Mordell-Lang problem we formulate a dynamical generalization, which we show that it doesn't always hold. Then we discuss the cyclic case of our question, which we call the Dynamical Mordell-Lang Conjecture. We present several positive results which support our conjecture, and discuss the difficulties one has for proving the full conjecture. In particular, our work answers a basic question from complex dynamics.
In the first half of the talk, I will explain the notion of PT stability, as defined by Bayer. I will also explain how it is related to classical stability conditions on sheaves, and other Bridgeland-type stability conditions. In the second half of the talk, I will discuss results on the moduli space of PT-stable objects from my thesis. In particular, I will explain how to use semistable reduction to obtain the valuative criterion of completeness for PT-stable objects.
Temporal cycles in natural populations have long been observed. More recent field studies have shown that in some situations, the population cycles have different phases in different locations, consistent with the existence of a travelling wavetrain of population density, passing through the region. Numerical simulations of predator-prey models with spatial dependence show that travelling wavetrains and other patterns can form behind invading fronts of predators. For reaction-diffusion and similar models, mathematical analysis can be used to predict in some cases what spatiotemporal patterns will form behind an invading front. In this talk I will describe some of the analysis that is possible using dynamical systems methods, especially for predicting the properties of wavetrains that form behind an invading front.
The problem of determining the electrical conductivity of a body by making voltage and current measurements on the object's surface has various applications in fields such as oil exploration and early detection of malignant breast tumour. This classical problem posed by Calder\'on remained open until the late '80s when it was finally solved in a breakthrough paper by Sylvester-Uhlmann. In the recent years, geometry has played an important role in this problem. We will look at the connection between this analysis problem with seemingly unrelated fields such as symplectic geometry and differential topology as well as geometric scattering theory.
The speaker is partially supported by NSF Grant No. DMS-0807502}
This SCAIM lecture is more of a tutorial than a talk. I will present a simple Finite Element Method (FEM) code as a model for the more complex codes in common modern use. Code components include: mesh generation, matrix assembly, a linear solver, and a post-processor. I will discuss these components with reference to their object-oriented (C++) implementations. This tutorial is aimed at those needing a framework for modifying and using existing codes. As a model, the example computes a piecewise-linear approximation of the solution to the Poisson problem on a circular domain with Neumann or Dirichlet boundary conditions. Focus is on implementation issues rather than on issues of accuracy and convergence. The example C++ code will be provided on-line following the talk for use as a teaching and reference tool.
We show that on a smooth compact Riemann surface with boundary (M_0, g) the Dirichlet-to-Neumann map of the Schr\"odinger operator \Delta_g + V determines uniquely the potential V. This seemingly analytical problem turns out to have connections with ideas in symplectic geometry and differential topology. We will discuss how these geometrical features arise and the techniques we use to treat them.
This is joint work with Colin Guillarmou of CNRS Nice.
The speaker is partially supported by NSF Grant No. DMS-0807502 during this work.
We wish to understand the boundary between forbidden configurations on
4 rows that yield a quadratic bound and those that have cubic
constructions. The result is joint with my supervisor and Attila Sali.
The bounds we are concerned with are the following: For a (0,1)-matrix
F, we define forb(m,F) to be the maximum number of columns in an
m-rowed (0,1)-matrix which has no repeated columns and has no
submatrix which is a row and column permutation of F. The asymptotics
of forb(m,F) for arbitrary F have been conjectured by Anstee and Sali.
Abstract
The "colored Tverberg problem" asks for a smallest size of the color
classes in a (d+1)-colored point set C in R^d that forces
the existence of an intersecting family of r "rainbow" simplices with
disjoint, multicolored vertex sets from C. Using equivariant topology
applied to a modified problem, we prove the optimal lower bound
conjectured by Barany and Larman (1992) for the case of partition into
r parts, if r+1 is a prime.
The modified problem has a "unifying" Tverberg-Vrecica type
generalization, which implies Tverberg's theorem as well as the ham
sandwich theorem.
This is joint work with Pavle V. Blagojevic and Gunter M. Ziegler.
In the early 60s Kesten showed that self-avoiding walk in the upper half plane has a decomposition into an i.i.d. sequence of "irreducible bridges". Loosely defined, a bridge is a self-avoiding path that achieves its minimum and maximum heights at the start and end of the path (respectively), and it is irreducible if it contains no smaller bridges. Considering only the 2-dimensional case, one can ask if the (likely) scaling limit of self-avoiding walk, the SLE(8/3) process, also has such a decomposition. I will talk about recent work with Hugo Duminil from Ecole Normale Superieure that provides a positive answer, using only the restriction property of SLE(8/3). In the end we are able to decompose the SLE(8/3) path as a Poisson Point Process on the space of irreducible bridges, in a way that is similar to Ito's excursion decomposition of a Brownian motion according to its zeros. Our decomposition can actually be generalized beyond SLE(8/3) and applied to an entire family of "restriction measures", hence the title of the talk. If time permits I will also talk about the natural time parameterization for SLE(8/3), which has immediate applications towards the bridge decomposition.
A vector bundle is a continuously varying family of vector spaces; for example, the set of lines that are tangent to a smooth curve is a vector bundle. In this talk, we will draw pictures, give examples, state applications, and (time permitting) learn how vector bundles can mend a heart broken by Liouville's theorem. Our emphasis will be on intuition, and all technical details will be suppressed.
In statistical physics we study models for interacting particles. One of the main open questions is to find models which exhibit a phase transition (think about water turning into gas at 100 Celcius). First I will introduce all the big words people use in statistical physics. Then I will show a result of Lebowitz and Penrose on the existence of a phase transition for a certain class of models. If there is some time left I can talk about some extensions of the Lebowitz and Penrose result.
Dissipative nonlinear systems such as fluid dynamical systems can reach a chaotic state when the parameter measuring the nonlinearity is large. For instance, parallel shear flows of Newtonian fluids are turbulent when the ratio of the nonlinear inertial term and the viscous dissipation term, defined by the Reynolds number is sufficiently important. In non-Newtonian fluid flows, an additional nonlinearity is introduced via the constitutive equation. For viscoelastic fluids, this nonlinearity can give rise to turbulent flow at low Reynolds number (Larson Nature 2000 and Groisman and Steinberg Nature 2000). The degree of nonlinearity is expressed by the Weissenberg number which is a product of a characteristic rate of deformation and the relaxation time of the polymer. The shear-thinning behaviour, non linear decrease of the effective viscosity with the shear rate, is the most common property of non Newtonian fluids. It is reasonable to inquire, whether an interplay between this nonlinearity and inertia can lead to a chaotic flow. This point has been addressed in (Ashrafi and Khayat PRE 2000) using low order dynamical system (generalized Lorenz system) in the Taylor-Couette flow of weakly-thinning fluid. It is shown that the additional nonlinearity gives rise to a Hopf bifurcation otherwise non existent for Newtonian fluid. In the previous talk dealing with the transition to turbulence for a yield-stress shear-thinning fluid in a pipe, a new state with a robust coherent structure characterized by two weakly modulated counter-rotating longitudinal vortices was described. In this nonlinear asyrnmetric state, time-averaged axial velocity profiles exhibited increasing asymmetry with increasing Reynolds number. In the present talk, velocity fluctuations are analysed, and it is shown that this state displays the salient feature of chaos, namely, randomly fluctuating motion excited in a broad range of spatial and temporal scales. Beside the experimental part, a spectral Petrov-Galerkin method is used to study the nonlinear stability of Hagen-Poiseuille flow of shear-thinning fluid. In the first step and as suggest the experimental results, the perturbation is assumed homogeneous in the axial direction. In this situation, the numerical results show that travelling waves with an azimuthal wave number m=1, are not sustained.
Traveling fronts are special solutions of reaction-diffusion equations which model phenomena such as propagation of species in an environment or spreading of flames in combustible media. In this talk we will address questions of existence, uniqueness, and stability of traveling fronts in general inhomogeneous media. We will show that in certain circumstances a unique front exists and it is a global attractor of the corresponding parabolic evolution, thus describing long time dynamics for very general solutions of the PDE. In contrast to this, we will also present examples of media where no traveling front solutions exist.
Reaction-diffusion equations are parabolic partial differential equations used in the modeling of phenomena such as propagation of species in an environment or spreading of flames in combustible media. Their general solutions exhibit two basic behaviors, extinction (quenching) and spreading. In this talk we will review recent progress in our understanding of how the motion of the underlying medium, modelled by a fluid flow, affects both the occurence of quenching and the speed of spreading of reaction. The problem turns out to have fruitful connections to questions about mixing effixiency of flows and homogenization of advection-diffusion operators.
I will present a global construction of the Neron model for degenerating families of intermediate Jacobians; a classical case would be families of abelian varieties. The construction is based on Saito's theory of mixed Hodge modules; a nice feature is that it works in any dimension, and does not require normal crossing or unipotent monodromy assumptions. As a corollary, we obtain a new proof for the theorem of Brosnan-Pearlstein that, on an algebraic variety, the zero locus of an admissible normal function is an algebraic subvariety.
The Picard group of an algebraic variety $X$ is the set of complex line bundles over $X$. In this talk, we will describe the Picard groups of certain finite covers of the moduli space of curves. The methods we use combine ideas from algebraic geometry, finite group theory, and algebraic/geometric topology.
For the efficient solution of large, sparse, linear systems of equations, Ax = b, we usually need a preconditioning matrix P, in an appropriate sense close to the inverse of A, such that solving PAx = Pb converges fast. If we need to solve a sequence of problems in which the matrix A changes slowly (and the right hand side b arbitrarily), we would like to adapt the preconditioner rather than compute a new one from scratch for each problem.
After a brief introduction to iterative linear solvers, we discuss adaptive preconditioners for time-dependent simulations and nonlinear optimization problems (topology optimization) with dynamic mesh adaptation. Adaptive meshing greatly reduces the computational cost of simulations and optimization. Unfortunately, it also carries a number of problems for preconditioning in iterative linear solvers, as changes in the mesh lead to structural changes in the linear systems we must solve. As a result, a new preconditioner must be computed after every change in the mesh, which might be prohibitively expensive. Here, we propose preconditioners that are cheap to update for dynamic changes to the mesh as well as for changes in the matrix due to nonlinearity of the underlying problem; more specifically, we propose preconditioners that require only local changes to the preconditioner for local changes in the mesh and nonlinear terms. Our preconditioners combine sparse approximate inverses with multilevel correction. For further information see [1,2].
[1] Shun Wang and Eric de Sturler, Multilevel sparse approximate inverse preconditioners for adaptive mesh refinement. Linear Algebra Appl., 431:409-426, 2009.
[2] Shun Wang, Krylov subspace methods for topology optimization on adaptive meshes. PhD thesis, University of Illinois at Urbana-Champaign, Department of Computer Science, September 2007. Advisor: Eric de Sturler, Co-Advisor: Glaucio H. Paulino.
The Torelli group is the subgroup of the mapping class group of a surface
which acts trivially on the surface's first homology group. Despite the
pioneering work of Birman, Johnson, and many others numerous basic
questions about it remain open. I will begin by describing some history
and background, and then I will discuss a new (infinite) presentation of
the Torelli group whose generators and relations have simple topological
interpretations.
In this talk, we discuss how probabilistic ideas are applied to study PDEs. First, we briefly go over the basic theory of Gaussian Hilbert spaces and abstract Wiener spaces to determine function spaces which capture the regularity of the Brownian motion and the white noise. Next, we go over Bourgain's idea to establish the invariance of Gibbs
measures for PDEs. We then establish local well-posedness (LWP) of KdV with the white noise as initial data via the second iteration introduced by Bourgain. This in turn provides almost sure global well-posedness (GWP) of KdV as well as the invariance of the white noise. Then, we discuss how one can use the same idea to obtain LWP of the stochastic KdV with additive space-time (non-smoothed) white noise in the periodic setting.
We also consider the weak convergence problem of the grand canonical ensemble (i.e. the interpolation measure of the usual Gibbs measure and the white noise) with a small parameter (tending to 0) to the white noise. This result, combined with the GWP in $H^{-1}$ by Kappeler and Topalov, provides another proof of the invariance of the white noise for KdV. In this talk, we discuss the same weak convergence problem for mKdV and cubic NLS, which provides the ``formal'' invariance of the white noise. This part is a joint work with J. Quastel and B. Valk\'o.
Lastly, if time permits, we discuss the well-posedness of the Wick ordered cubic NLS on the Gaussian ensembles below $L^2$. The main ingredient is nonlinear smoothing under randomization of initial data. For GWP, we also use the invariance (of the Gaussian ensemble) under the linear flow. This part is a joint work with J. Colliander.
The next UBC/UMC talk is by Bud Homsy, Deputy Director of PIMS.
Title: Fluid motion and the Navier-Stokes Equations: Why is F=ma so tough for fluids and why haven't we solved these equations yet?
The differential equations governing the flow of fluids like air and water have been known since the 1800’s. Yet they have proven to be nearly impenetrable to mathematical analysis and to solutions using supercomputers. This talk will show many examples (in the form of movies) of physical flows from science, technology and everyday life that one would like to be able to describe. I will then give the highlights of how the Navier-Stokes equations are derived and what makes them so tough to solve.
In invasion percolation, the edges of a graph are assigned i.i.d. edge weights, and an infinite cluster is grown by recursively adding the boundary edge of minimal weight. By considering the edges whose weight is larger than all subsequently accepted weights, the invasion cluster is divided into a chain of ponds linked by outlets.
Working on the regular tree, we show that the sizes of the ponds grow exponentially, with law of large numbers, central limit theorem and large deviation results, and also give asymptotics for the size of a fixed pond.
We compare with known results for Z^2 and explore why these results should be expected on more general graphs.
In 1953, K. Roth proved that any subset of positive integers of positive density contains infinitely many non-trivial three-term arithmetic progressions. (By a non-trivial arithmetic progression we mean one of the form (a, a+d, a+2d) with d > 0.) First, I shall explain the main ideas of the proof of Roth's theorem. The second part of my talk will be devoted to Roth's theorem in the primes. I shall explain how B. Green proved that a subset of primes of positive relative density must contain some non-trivial 3-term arithmetic progressions and how H. Helfgott and I sharpened his quantitative result.
We present the results of a combined experimental and theoretical investigation of droplets falling onto a horizontal soap film. Both static and vertically vibrated soap films are considered. A quasi-static description of the soap film shape yields a force-displacement relation that allows us to model the film as a nonlinear spring, and yields an accurate criterion for the transition between droplet bouncing and crossing. On the vibrating film, a variety of bouncing behaviours were observed, including simple and complex periodic states, multiperiodicity and chaos. A simple theoretical model is developed that captures the essential physics of the bouncing process, reproducing all observed bouncing states. The system is among the very simplest fluid mechanical chaotic oscillators. The relevance of our model to a seemingly unlikely biological system is discussed.
In this talk, we first discuss the second iteration argument introduced by Bourgain to establish LWP of KdV with measures as initial data. Then, we establish LWP of the stochastic KdV (SKdV) with additive space-time white noise by estimating the stochastic convolution via Ito calculus and showing its continuity via the factorization method. Next, we discuss
well-posedness of SKdV with multiplicative noise in $L^2$. In order to treat the non-zero mean case, we derive a coupled system of a SDE and a SPDE.
Lastly, as a toy model to study KPZ equation and stochastic Burgers equation, we study stochastic KdV-Burgers equation (SKdVB). We discuss how Fourier analytic technique can be applied to show LWP. If time permits, we discuss how one can obtain global well-posedness of these equations via (1) analogue of conservation laws, (2) Applying Bourgain's argument for invariant measures (for deterministic PDEs) to SPDEs.
This is the second talk in the teaching seminar associated with the TA Accreditation Program. (All are welcome!) Graduate students will have their attendance credited toward their eventual accreditation.
Title: Difficult Matters
We will address some of the conceptual aspects of teaching mathematics by raising various pedagogical and management issues. We will explore these issues through group consideration and analysis of case studies developed by Solomon Friedberg and his team in the Mathematics Department of Boston College. These case studies have been used at universities including Boston University, Brown, Cornell, Harvard, Stanford and Dartmouth, as a tool in TA training programs for mathematics graduate teaching assistants.
We hope to have a fun and vibrant discussion, and to draw upon each others ideas, perspectives and experiences. Most importantly, this session will give us an opportunity to think in advance about these complicated situations so that we can handle them efficiently and decisively as they develop in practice.
Geometrization is a process of replacing finite sets by algebraic varieties over finite field and functions on such sets by sheaves on the corresponding variety. I will explain the meaning of the above sentence and state some applications.
PELICANS is a C++ framework with a set of integrated reusable components, designed to simplify the task of developing applications of numerical mathematics and scientific computing. The program is developed at IRSN (France) and available under an open source license.
In this talk I will give an introduction to PELICANS starting with the Laplace equation solved by finite elements. This example is used to demonstrate how to implement your own code by choosing appropriate components of PELICANS and wiring them together. I also show that it is fairly simple to compare a given analytic solution with the numerical one for verification purposes. As another more detailed example I present the advection-diffusion equation solved by the finite volume method. Finally, some results of more complicated problems as multi-layer visco-plastic flows will be shown.
The goal of this talk is to show that PELICANS can provide you with a C++ framework which allows focusing on the set up of the mathematical description and numerical scheme rather than on the implementation. PELICANS also provides lots of examples (e.g. Navier-Stokes), it is well documented and coupled with external libraries like PETSc, SPARSKIT, and UMFPACK.
PELICANS can be downloaded from https://gforge.irsn.fr/gf/project/pelicans
In joint work with Adam Clay, we establish a necessary condition that an automorphism of an orderable group can preserve an
ordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications will be given to knot theory and to the fundamental groups of fibred spaces. An example: if surgery on a fibred knot in $S^3$ (or in a homology 3-sphere) produces a 3-manifold whose fundamental group is orderable, then the surgery must be longitudinal (0-framed) and the Alexander polynomial of the knot must have a positive real root.
We present a new method of constructing a stress-energy-momentum tensor for a classical field theory based on covariance considerations and Noether theory. The stress-energy-momentum tensor T^\mu_\nu that we construct is defined using the (multi)momentum map associated to the spacetime diffeomorphism group. The tensor T^\mu_\nu is uniquely determined as well as gauge-covariant, and depends only upon the divergence equivalence class of the Lagrangian. It satisfies a generalized version of the classical Belinfante-Rosenfeld formula, and hence naturally incorporates both the canonical stress-energy-momentum tensor and the ``correction terms'' that are necessary to make the latter well behaved. Furthermore, in the presence of a metric on spacetime, our T^\mu_\nu coincides with the Hilbert tensor and hence is automatically symmetric.
Among the open problems for SLE suggested by Oded Schramm in his 2006 ICM talk is that of obtaining \reasonable estimates for the speed of convergence of the discrete processes which are known to converge to SLE." In this talk we derive a rate for the convergence of the Loewner driving function for loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2). This talk is based on joint work with Christian Benes (CUNY) and Fredrik Johansson (KTH).
In this talk I will present several ideas ranging from Euclid to Conway about how to put our intuitive and also sometimes not so intuitive ideas about what a number is on a rigorous foundation. Some questions you might have that I'm going to answer are: How to create something out of nothing? Why is a proof by induction actually a proof? What is a surreal number?
Computing zeta functions of curves over finite fields is an important problem in computer algebra with connections to cryptography and coding theory, among others. In this talk, I first want to highlight how rigid cohomology can be used to construct explicit algorithms and why their runtime is usually linear in the characteristic p. In a second part, I will restrict the problem to superelliptic curves and show how the complexity can be reduced to be linear in the squareroot of p.
Note for Attendees
Refreshments will be served between the two talks.
Given a positive integer $N$, an old problem of D.J. Newman is to bound the number of ways to express $N$ as
N = 2^a 3^b + 2^c + 3^d
in nonnegative integers $a, b, c$ and $d$. That this number is finite is a consequence of a result of Evertse on $S$-units equations. That it is at most 9 requires some new ideas. I will sketch a proof of this and attempt to show how such an odd question fits into a more general framework.
Canonical dimension is a numerical birational invariant of
algebraic varieties defined over a field k, which is zero for
varieties with a k-rational point. I will review the connection with
incompressibility, and discuss what is known about the canonical
dimensions of projective homogeneous varieties, such as Severi-Brauer
varieties, and quadrics. Then I will use a new birational equivalence
to deduce the canonical dimensions of some more projective
G-homogeneous varieties, where G is an algebraic group of classical
type, or type F4. In particular, we will see that a variety of type
F4/P4, which is not split by a cubic extension, is incompressible.
I will explain recent joint work with Xiaoyi Zhang on threshold solutions to critical nonlinear Schrodinger equations. These results are analogues of Liouville-type theorems in the dispersive setting. I will cover mainly the mass-critical case. Time permitting the energy-critical case will also be discussed.
In recent work, Haglund, Mason, van Willigenburg, and this author introduced a family of quasisymmetric functions which we call quasisymmetric Schur (QS) functions. These naturally refine the (symmetric) Schur functions and form a Z-basis of QSym, the quasisymmetric function algebra. We showed that this basis has interesting properties such as a Littlewood-Richardson rule for the product of a symmetric Schur with a QS function.
We extend the definition of QS functions to skew QS functions, which are counterparts to the classical skew Schur functions. Intimately related to these are the duals of the QS functions, which form a Z-basis of NSym, the graded Hopf algebra which is dual to QSym. The dual QS functions are noncommutative analogs of the classical Schur functions, having properties such as a Littlewood-Richardson rule and relationship to a poset of compositions which is analogous to Young's lattice of partitions. We discuss how the duals of the QS functions arise in the study the Poirier-Reutenauer tableaux algebra.
This is joint work with Christine Bessenrodt and Stephanie van Willigenburg.
The braid group has a left-invariant total ordering, called the Dehornoy ordering. In this talk, I introduce the Dehornoy floor of braids, which measures a complexity of braids by using the Dehornoy ordering.
I will give a new lower bound of knot genus by using the Dehornoy floor.
I also discuss other applications of the Dehornoy floor to knot theory.
Title: Mathematics at work in geomechanics: why miners and oilmen should learn PDEs
In this talk I will introduce some of the basic concepts used to develop mathematical models for fracture processes that occur around mining excavations. In particular I will demonstrate on a simplified model problem how a local analysis of the governing PDE enables one to predict the modes of fracture propagation observed around deep gold mines. While miners are intent on avoiding the unstable formation of fractures around mining excavations, petroleum engineers have found it profitable to deliberately create fractures by injecting high-pressure viscous fluids into oil reservoirs to enhance production. I will demonstrate how to formulate mathematical models of these hydraulically driven fractures and attempt to demonstrate some of the subtleties involved in their solution.
We present a new method of constructing a stress-energy-momentum tensor for a classical field theory based on covariance considerations and Noether theory. The stress-energy-momentum tensor T^\mu_\nu that we construct is defined using the (multi)momentum map associated to the spacetime diffeomorphism group. The tensor T^\mu_\nu is uniquely determined as well as gauge-covariant, and depends only upon the divergence equivalence class of the Lagrangian. It satisfies a generalized version of the classical Belinfante-Rosenfeld formula, and hence naturally incorporates both the canonical stress-energy-momentum tensor and the ``correction terms'' that are necessary to make the latter well behaved. Furthermore, in the presence of a metric on spacetime, our T^\mu_\nu coincides with the Hilbert tensor and hence is automatically symmetric.
Regarding his 1920 paper proving recurrence of random walks in Z^2, Polya wrote that his motivation was to determine whether 2 independent random walks in Z^2 meet infinitely often. Of course, in this case, the problem reduces to the recurrence of a single random walk in Z^2 , by taking differences. Perhaps surprisingly, however, there exist graphs G where a single random walk is recurrent, yet G has the finite collision property: two independent random walks in G collide only finitely many times almost surely. Some examples were constructed by Krishnapur and Peres (2004), who asked whether critical Galton-Watson trees conditioned on nonextinction also have this property. In this talk I will answer this question as part of a systematic study of the finite collision property. In particular, for two classes of graphs, wedge combs and spherically symmetric trees, we exhibit a phase transition for the finite collision property when growth parameters are varied. I will state the main theorems and give some ideas of the proofs.
This is joint work with Martin Barlow and Yuval Peres.
Abstract: I will describe a new operad (the "splicing operad") that acts
on a fairly broad class of embedding spaces. Previously I constructed an
action of the operad of little (j+1)-cubes on the space of framed long
embeddings of R^j in R^n. This operad action can be seen an extension of
the cubes action that allows for a general type of splicing operation. The
space of long embeddings of R into R^3 was described as a free 2-cubes
object over the subspace of prime long knots. With respect to the
splicing operad, long knots in R^3 are again free, but rather than being
free on the prime long knot subspace, the generating subspace is the (much
smaller) torus and hyperbolic knot subspace. Moreover, the splicing
operad has a particularly pleasant homotopy-type from the point of view of
its structure maps.
In this talk we outline the main ideas that come upon the computation of quotients of tori by actions of Z/p induced by linear
representations of Z/p. These are called toroidal orbifolds.
The talk will be aimed to a general audience.
This is joint work with Alejandro Adem and Ali Duman.
I will speak about the cohomology of the middle degree of graph hypersurfaces of some special Feynman graphs, the "generalised zigzag" graphs, and about how to connect this to counting rational points of this hypersurfaces. Then I will explain, why "wheel with spokes" WSn are polynomially countable (in the spirit of Kontsevich conjecture), and will give an example of graph which is not polynomially countable. I will say some words about the number of points of gluing of graphs.
Note for Attendees
Cookies and tea will be served between the first two talks.
At certain special points, the values of the Riemann zeta function and many other L-functions are algebraic, up to a well-determined transcendental factor. G. Shimura, H. Maass, and M. Harris extensively studied a class of differential operators on automorphic forms; these differential operators play an important role in proofs of algebraicity properties of many $L$-functions.
Building on work of N. Katz, we introduce a p-adic analogue of these differential operators, which should be similarly significant in the study of many p-adic L-functions, in particular p-adic L-functions attached to families of p-adic automorphic forms on unitary groups.
According to the Birch and Swinnerton-Dyer conjecture, the order of the Tate-Shafarevich group can be expressed as the ratio of several invariants of the curve. However, this ratio involves two numbers which are expected to be irrational; namely $L^{(r)}(E,1)$ and $\Omega_E$. In the case $r >= 2$, this ratio has never been proven to be rational. Recently, we have provably computed the ratio (1.000...) to 10kbits of precision for such a curve. We present a provable algorithm to compute $L''(E,1)$ with $O(p^3)$ runtime, where $p$ is the desired precision.
Let A be a finite subset of a group G. How rapidly does A grow?
More precisely: let |S| be the number of elements of a finite set S. In 2005, I proved that, for G = SL_2(Z/pZ), p a prime, A\subset G such that A generates G and
|A|<=|G|^{1-epsilon}, epsilon>0, we have
|A A A| >> |A|^{1+delta},
(*)
where A A A = {x y z: x,y,z\in A}, and delta>0 and the implied constant depend only on epsilon.
This implies directly that the diameter of any Cayley graph of G is polylogarithmic (Babai's conjecture). Further implications on expander graphs were derived by Bourgain and Gamburd (and used by Bourgain, Gamburd and Sarnak in their work on the {\em affine sieve}).
In 2008, I proved the same result for SL_3(Z/pZ). Half of the proof had become fully general in the process, but much work remained to be done. Nick Gill and I extended the result to small subsets of SL_n in 2009.
This January, two different teams (Pyber and Szabo; Breuillard, Green and Tao) announced proofs of (*) valid for all finite simple groups of Lie type. Their success is based in part on a strengthening of some of my intermediate results from my paper in SL_3, apparently inspired by papers by Larsen-Pink and Hrushovski-Pillay. The process of making ideas of growth ("pivoting" or "bootstrapping") independent from the context of the sum-product theorem has also reached its natural conclusion.
I will introduce the notion of categorical resolution of singularities which is based on the concept of
a smooth DG algebra. Then I will compare this notion with the traditional resolution in algebraic geometry and
give some examples.
semisimple split algebraic group is, and how to compute its essential p-dimension. This computation only requires us to understand how a Sylow p-supgroup of the Weyl group acts on the weight lattice of a split maximal torus. I will give some explicit examples of this computation for classical groups as well as exceptional groups. The group G2 will provide a good example, since its root system can be easily drawn on a blackboard.
A symmetry of a PDE is a transformation that maps any solution of the PDE to another solution. In this lecture, we will focus on determining whether there exists an invertible mapping of a given PDE into a member of a target class of PDEs that can be completely characterized by its admitted point or contact symmetries and how to construct such a mapping when it exists. Examples include: (1) Invertible mappings of nonlinear PDEs to linear PDEs through symmetries. (2) Invertible mappings of linear PDEs to linear PDEs with constant coefficients. (3) Invertible mappings of nonlinear PDEs to linear PDEs through conservation law multipliers.
This seminar is based on Chapter 2 Applications of Symmetry Methods to Partial Differential Equations by Bluman, Cheviakov and Anco.
Nowadays, mathematicians plays a more and more important role in the financial world. In this talk, I will briefly introduce the development of mathematical tools in finance. Then I will focus on the most famous formula in mathematical finance field, namely the Black-Scholes Formula (a SDE problem mathematically), and explain how it works in pricing financial derivatives.
One of the most ubiquitous operations in mathematics is the solution of polynomials in one variable. Unlike the situation for quadratics, cubics and quartics, Abel's Theorem tells us that we cannot find a general formula in radicals for polynomials of degrees greater than 4. Of course, one can always find roots numerically but this is unsatisfying. There are relatively nice solutions to quintic and sextic equations using functions more general than radicals. Motivated by these constructions, Hilbert's thirteenth problem asks if there is a general solution to a seventh degree polynomial of a particular form. Arnold and Kolmogorov found that the answer was yes, and furthermore, their solution could be extended to polynomials of arbitrary degree. However, we shall see why their answer was probably not what Hilbert was seeking.
This is the third talk in the teaching seminar associated with the TA Accreditation Program. (All are welcome!) Graduate students will have their attendance credited toward their eventual accreditation.
Title: Introduction to the First-Year Calculus Workshops
The first-year calculus workshop programs in MATH 180 and 184 provide an activity where students meet once a week outside of lecture time to work on math problems in small groups. This may sound simple enough, by in fact the design and delivery of the program is a complex process, and makes for quite a different (though rewarding!) job compared to other TA-ships. We aim to address the following questions:
What does a workshop session look like?
What does it mean for a TA to run a workshop week to week?
What do students actually do?
What do students and TAs think about the experience?
Why do we put in all of this effort?
Past workshop TAs are encouraged to attend and share their experiences as well, and we hope to have some time for discussion.
Who we are: Costanza Piccolo has been involved with content development as well as data collection associated with the workshops for the past two years as part of the Carl Wieman Science Education Initiative in the Math Department. Warren Code has been involved as a workshop TA for the past four years, including work as Head TA for MATH 184 this past fall.
Description: Djun Kim, UBC Math's resident expert in computer-based educational tools, will demonstrate the capabilities of WeBWorK, an online homework system currently in use in the department. Features include quick student feedback, straightforward management tools for the instructor, as well as a large existing problem base coupled with authoring tools for new custom questions. Students at UBC and elsewhere have responded well to this system; some results will be presented. Others who are currently using WeBWorK in the department will be available to share their recent experiences in implementation.
One of the most important open questions in climate research is the value of the climate sensitivity, defined as the equilibrium response of the annually and globally averaged surface air temperature to a doubling of atmospheric carbon dioxide. Climate models, all of which give reasonably accurate representations of current climate, differ in their estimates of the climate sensitivity by about a factor of two, with a standard deviation range between 2.5 - 4.5 degrees C. This range has narrowed only slightly over the last two decades, despite significant progress in representing fundamental physical processes in global climate models. We know that differences in the parameterization of boundary layer clouds explain about 60% of the spread in these sensitivity estimates, making low clouds the single biggest uncertainty in climate modeling at 10-100 year time scales. In this talk I will give an overview of how a feedback analysis is done using a global climate model, and discuss some of the challenges that need to be overcome to accurately represent the impact of sub-grid scale cloudiness on resolved-scale energy and moisture transport. Large eddy simulations (LES), run on horizontal domains of 10 x 10 km and grid resolutions of 10-50 meters, are an extremely useful tool for understanding the processes governing sub-grid scale statistics of these boundary layer clouds. I will conclude with recent work that Jordan Dawe and I have done on the direct calculation of cloud-environment mixing for these climatologically important shallow cloud layers.
I will discuss joint work in progress with Rajesh Kulkarni
on the moduli of maximal orders on surfaces. In contrast to the
"classical" case of Azumaya algebras, ramified maximal orders have
several potentially interesting moduli spaces. I will discuss three
different scheme structures on the same set of points: a naive
structure, a structure arising from a non-commutative version of
Koll\'ar's condition on moduli of stable surfaces, and a structure
that comes from hidden Azumaya algebras on stacky models of the
underlying surface. Only (?) the third admits a natural
compactification carrying a virtual fundamental class, giving rise to
potentially new numerical invariants of division algebras over
function fields of surfaces.
The conjugate gradient (CG) algorithm is usually the method of choice for the solution of large symmetric positive definite linear systems Ax=b. If however the matrix-vector products Av required at each iteration can not be calculated accurately, the delicate mechanisms on which CG is built can be easily disturbed and cause disaster. In such cases we may consider gradient descent methods, which are more robust against such effects. The classical steepest descent (SD) method, which takes the best possible (greedy) step in terms of reducing the error at each iteration, is well-known to wiggle agonizingly slowly to the solution. Fortunately its behaviour improves dramatically (by orders of magnitude) by some tinkering with the step size. This has given rise to a zoo of fast gradient descent methods known as BB, LSD(s), HLSD(k), SDOM, SD(\omega) etc. These methods are in practice much closer to CG in performance than to SD (though nobody has been able to prove this) and can outperform CG under certain conditions. I will present numerical experiments to establish which of the methods perform best on average, then show that the fast gradient descent methods generate chaotic dynamical systems. Very little is required to generate chaos here: simply damping steepest descent by a constant factor close to 1 will do. Some insight will be given into how chaos speeds up these methods, and I will show beautiful animations of the chaotic dynamics.
as a and b range over k. All representations of Cf have dimensions that are multiples of d, and occur in families. In this lecture we will discuss a construction of a fine moduli spaces U=Uf,r for the irreducible rd-dimensional representations of Cf for each r >= 2. Our construction starts with the projective curve C in P2k defined by the equation wd=f(u,v), and produces Uf,r as a quasiprojective variety in the moduli space
M(r, D) of stable vector bundles over C of rank r and degree D=r(d+g-1), where g denotes the genus of C.
In this talk, we will discuss several optimal (global) conditions for the existence of a smooth solution to the mean curvature flow. Our focus will be on quantities involving only the mean curvature. We will also discuss several applications of a local curvature estimate which is a parabolic analogue of Choi-Schoen estimate for minimal submanifolds. This is joint work with Natasa Sesum.
I discuss algorithms and complexity results for two game theoretic extensions of integer programming: integer programming games and bilevel integer programming. In the case of integer programming games, I discuss an algorithm which computes pure Nash equilibria using rational generating functions which runs in polynomial time when certain parameters are fixed. In the case of bilevel integer programming, I describe an algorithm which decides the existence of and computes ``optimistic" optimal solutions using parametric integer programming and binary search. I show that this algorithm runs in polynomial time when the number of integer variables are fixed, extending a result by Lenstra on integer programming in fixed dimension to the bilevel setting.
This is joint work with Matthias Koeppe and Maurice Queyranne.
Abstract: I discuss a scheme of fault-tolerant quantum computation which is driven by local projective measurements on an entangled quantum state of many qubits, a so-called cluster state. There are two
fundamentally different ways of evolving quantum states, namely unitary evolution and projective measurement. Both can be used to realize quantum computation. The approach discussed in this talk uses the
latter. The constructions involved in making cluster state quantum computation robust against decoherence (=quantum-mechanical error) are in large part topological. In particular, Z_2 relative homology plays an
important role.
I begin with a short introduction to quantum computation, and explain the notions of "universality" and "fault-tolerance" in quantum computation. A very brief introduction to the field of quantum error-correction
will be included. Then I turn to the main subject of my talk, cluster state quantum computation. After a brief discussion its universality, I will turn to the question of how to make cluster state quantum computation
fault-tolerant. At that point, elements of topology will come into the picture.
The next UBC/UMC talk is by Jennifer Johnson-Leung, visiting from the University of Idaho.
Title: What's modularity got to do with it?
Modular forms seem to crop up all over the place in modern number theory and beyond -- from Fermat's Last Theorem to the theory of partitions to the Langlands Program. We'll introduce the notion of a modular form, look at some of its properties, and try to understand why it is such a useful and powerful mathematical object.
We will show that the dynamics of a mechanical system consisting of a rigid structure (with a finite number of degrees of freedom) interacting with a fluid can sometimes be governed by a system of ODEs. In the literature, we can usually find two different ways allowing one to obtain equations of motion for these systems: the first one is based on Newton's laws (Classical Mechanics) and the second on Hamilton's principle (Analytic Mechanics). However, the resulting equations are far from being obviously identical in both cases. Quite surprisingly, we will prove in this talk that it is not always the case.
This is joint work with Brooks Roberts. Let E be a real quadratic field and let P be a cuspidal, irreducible, automorphic representation of GL(2) of the adeles of E with trivial central character and infinity type (2, 2n+2). We show that there exists a Siegel paramodular newform F with weight, level, epsilon factor, Hecke eigenvalues and L-function determined explicitly by P. These invariants are tabulated for all choices of P. I will also discuss some applications of this result.
Note for Attendees
Refreshments will be served between the two talks.
In my thesis, I give a new construction of the tame local Langlands correspondence for GL(n,F), n a prime. The Local Langlands Correspondence for GL(n,F) has been proven recently by Henniart, Harris/Taylor. In the tame case, supercuspidal representations correspond to characters of elliptic tori, but the local Langlands correspondence is unnatural because it involves a twist by some character of the torus. Taking the cue from the theory of real groups, supercuspidal representations should instead be parameterized by characters of covers of tori. DeBacker has calculated the distribution characters of supercuspidal representations for GL(n,F), n prime, and they are written in terms of functions on elliptic tori. Over the reals, Harish-Chandra parameterized discrete series representations of real groups by describing their distribution characters restricted to compact tori. Those distribution characters are written down in terms of functions on a canonical double cover of real tori. I have succeeded in showing that if one writes down a natural analogue of Harish-Chandra's distribution character for p-adic groups, it is the character of a unique supercuspidal representation of GL(n,F), where n is prime, far away from the identity. These results pave the way for a new construction of the local Langlands correspondence for GL(n,F), n prime. In particular, there is no need to introduce any character twists.
Anders Bj\"orner characterized which finite, graded partially ordered sets (posets) are closure posets of finite, regular CW complexes, and he also observed that a finite, regular CW complex is homeomorphic to the order complex of its closure poset. One might therefore hope to use combinatorics to determine topological structure of stratified spaces by studying their closure posets; however, it is possible for two different CW complexes with very different topological structure to have the same closure poset if one of them is not regular. I will talk about a new criterion for determining whether a finite CW complex is regular (with respect to a choice of characteristic functions); this will involve a mixture of combinatorics and topology. Along the way, I will review the notions from topology and combinatorics we will need. Finally I will discuss an application: the proof of a conjecture of Fomin and Shapiro, a special case of which says that the Schubert cell decomposition of the totally nonnegative part of the space of upper triangular matrices with 1's on the diagonal is a regular CW complex homeomorphic to a ball.
Abstract: Recently Kaledin proved a non-commutative generalization of
the Deligne-Illusie theorem about the de Rham complex of an algebraic
variety in characteristic p.
I will explain how his approach can be used to prove new results in
commutative algebraic geometry.
The objective of this Ph.D work is the analytical, numerical and experimental investigation of selectedflows of viscoplastic materials. The focus is restricted to slow, inertialess flow situations.The first test case is the flow through a plane channel with sinusoidal walls. Analytic results show the existence of an unbroken, central plug which breaks under certain conditions. Asymptotic results for the broken plug solution are derived, and an extensive comparison with numerical simulations is conducted. The difference between the usual regularized approach, in contrast to an augmented Lagrangian approach that is used throughout the thesis, is illustrated through example calculations. The second test case concerns the settling of a solid particle in a visco-plastic fluid. Due to the yield stress, it is possible for the particles to remain trapped in the material, even if they are not neutrally buoyant. The variational form of this problem allows the estimation of the leading order order terms in the weak functionals close to this stopping situation, and to formally derive a stopping condition. Applying slip line analysis, it is possible to derive an analytic stopping criterion for ellipsoid particles. The augmented Lagrangian method is used to calculate numerical solution to this flow problem. A series of experiments on the settling of spheres in a \trademark{Carbopol} solution, a typical visco-plastic fluid, is conducted. Particle image velocimetry (PIV) is used to visualize the flow fields. In contrast to the theoretical results, symmetry breaking in the flow fields is observed, and a link to carefully established rheological measurements is found. Guided by the experiments, and an analogous problem in magnetization, a new set of constitutive equations is derived. These models are tested against a large set of different rheological measurements.
We will discuss some results, both recent and ancient, on Weyl and Coxeter
groups. These results impinge on topics related to the theory of complex
algebraic groups and representation theory. We will also mention some open
problems.
Given n points in the real plane, the unit distance problem asks for
an asymptotic upper bound on the number of unit distances between
pairs of the points. We consider this problem under the restriction
that the line segments between the points make a rational angle (in
degrees) with the x-axis. In the complex plane, that allows us to
think of such segments of length 1 as roots of unity. Given a point
set with many such segments, we deduce simple linear equations with
many solutions in roots of unity. Using an algebraic theorem of Mann
from 1965, we can give a uniform bound on the number of such
solutions, which will give us a tight asymptotic bound on the number
of unit distances with rational angles. These results can then be
extended to rational distances. This is joint work with Jozsef
Solymosi.
For a complex number $\lambda$ in the open unit disk, consider the
attractor $A_\lambda$ of the
iterated function system $\{\lambda z, 1+\lambda z\}$ in the complex plane,
which can be represented explicitly
as the set of all power series with coefficients 0,1 evaluated at $\lambda$.
M. Barnsley and A. Harrington (1985)
defined the "Mandelbrot set" for pairs of linear maps $M$ as the set of
$\lambda$ for which $A_\lambda$ is
connected, by analogy with the classical Mandelbrot set defined as the set of
complex numbers $c$ for which the
Julia set of the quadratic polynomial $z^2+c$ is connected. There are both
similarities and differences between our piecewise-linear and the classical
quadratic setting. I will discuss some recent results and open problems
concerning the topological,
measure-theoretic, and dimension properties of $M$ and the family
$A_\lambda$.
Abstract: I will talk about work with Dan Isaksen on the motivic
homotopy groups of spheres, focusing on the story of the Hopf
maps. In classical algebraic topology the Hopf maps generate a
very small and easily computed subring of the stable homotopy ring.
The motivic Hopf maps generate a larger ring, and we don't yet
know explicity what it is. In the talk I'll describe some of what we do know.
A symmetry of a PDE is a transformation that maps any solution of the PDE to another solution. In the first lecture, we focussed on determining whether there exists an invertible mapping of a given PDE into a member of a target class of PDEs that can be completely characterized by its admitted point or contact symmetries. In this second lecture, we show how to construct such a mapping when it exists. Examples include: (1) Invertible mappings of nonlinear PDEs to linear PDEs through symmetries. (2) Invertible mappings of linear PDEs to linear PDEs with constant coefficients.
This seminar is based on Chapter 2 Applications of Symmetry Methods to Partial Differential Equations by Bluman, Cheviakov and Anco.
We consider a gradient Gibbs measure with non convex potential and show that it behaves at high temperature like a gaussian free field. The proof is based on the fact that the marginal distribution of the even sites has a strictly convex Hamiltonian for which we can apply the random walk representation.
This is a joint work with Codina Cotar.
We all know some probability, if nothing more than what we help first year students with in the tutorial center. But what makes a concept probabilistic as opposed to measure-theoretic? I'll try to answer this question, which sheds some light on the strange language and notation used by Probabilists.
This talk will be an introduction to the theory of stochastic differential equations (SDEs), with an emphasis on the connections to statistics. In any case, I could look at SDEs from two different perspectives: applied mathematics and statistics. The differences between the two perspectives will quickly be explained. I will also briefly mention some of the results of a directed studies project that I am doing and highlight the connection to SDEs. My project is about modelling diffusion of one particle into a two‐dimensional system of spatially homogeneous particles. The eventual goal is to have a working molecular dynamics simulation representative of the system in question, as well as to validate one of two theoretical predictions for the diffusion coefficient and its dependence on spacing between particles.
Coisotropic submanifolds of symplectic manifolds are canonically foliated. Their deformation is described by an L-infinity algebra with a known geometric description. I will describe some calculations with the obstruction theory of this L-infinity algebra, and my attempts to relate this to models for the leaf space of the foliation.
The next talk in the TAAP seminar series is by Dr. Joanne Nakonechny, Director of Skylight, UBC's Science Centre for Teaching and Learning. Graduate students will have their attendance credited toward their eventual accreditation.
Title: Appraisal = Assessment + Feedback Some essential questions to consider:
What is the difference between assessment and evaluation?
What is appraisal?
When should I start to think about appraisal?
What factors influence appraisal?
What forms of appraisal can I use to achieve my course goals?
Good appraisal and implementation provide appropriate steps to facilitate and measure students' learning. Without suitable formative and summative appraisal tasks throughout the course, students and instructors lack valid reference points from which to develop, track, and evaluate the learning process.
Simple random walk is well understood. However, if we condition a random walk not to intersect itself, so that it is a self-avoiding walk, then it is much more difficult to analyze and
many of the important mathematical problems remain unsolved. This lecture will give an overview of some of what is known about the self-avoiding walk, including some old and some more recent results, using methods that touch on combinatorics, probability, and statistical mechanics.
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There will be a special reception at 2:30 pm in Math 125.
In this thesis we exhibit an explicit non-trivial example of the twisted fusion algebra for a particular finite group. The product is defined for the third power of modulo two group via the pairing of projective representations where the three cocycles are chosen using the inverse transgression map. We find the rank of the fusion algebra as well as the relation between its basis elements. We also give some applications to topological gauge theories.We next show that the twisted fusion algebra of the third power of modulo p group is isomorphic to the non-twisted fusion algebra of the extraspecial p-group of order pł and exponent p.
The final point of my thesis is to explicitly compute the cohomology groups of toroidal orbifolds which are the quotient spaces obtained by the action of modulo p group on the k-dimensional torus. We compute the particular case where the action is induced by the n-th power of augmentation ideal.
Turbulence and other complex flows are often characterized by their transport and mixing properties. Turbulence can greatly enhance molecular viscosity for the transport of momentum, and molecular diffusion coefficients for the transport of heat, chemical concentrations, etc. Such flows are usually too complicated to study exactly, and they are often extremely challenging for direct numerical computations. Nevertheless, rigorous analysis of the fundamental equations of motion can yield qualitatively and quantitatively precise estimates of turbulent transport properties that can be compared with theories, experiments, and simulations.
Given a system of polynomial equations in some variables
and
depending on one parameter, when can every solution which is a power
series in the parameter be approximated to arbitrary order by
solutions
which are polynomial in the parameter? Hassett observed that a
necessary
condition is that the generic fiber is "rationally connected", i.e.,
for a
general choice of the parameter, every pair of solutions are
interpolated
by a family of solutions which are the output of a polynomial
function in
one variable. Hassett and Tschinkel conjecture the converse holds:
if a
general fiber is rationally connected, then "weak approximation"
holds.
I will review progress by Hassett -- Tschinkel, Colliot-Th\'el\`ene
--
Gille, A. Knecht, Hassett, de Jong -- Starr, and Chenyang Xu. Then I
will
present a new perspective by Mike Roth and myself using "pseudo ideal
sheaves", a higher codimension analogue of Fulton's effective pseudo
divisors. I will also mention a theorem of Zhiyu Tian, who used this
perspective to relate weak approximation to equivariant rational
connectedness, thereby proving many new cases of weak approximation.
Percolation is concerned with the existence of an infinite path in a random subgraph of the lattice Z^D. We can rephrase this as the existence of a Lipschitz embedding of the infinite line Z into the random subgraph. What happens if we replace the line Z with another lattice Z^d? I'll answer this for all values of the two dimensions d and D, and the Lipschitz constant. Based on joint works with Dirr, Dondl, Grimmett and Scheutzow.
In the context of optimal control, the Hamilton-Jacobi Partial Differential Equation (HJ PDE) is a continuous analogue to the discrete Bellman dynamic-programming equation. Both of these equations satisfy a causal property: the solution value at a state is independent of equal or greater solution values. Dijkstra's algorithm is a dynamic-programming method that exploits this causal property to compute the minimal path costs from a source node to all nodes in a discrete graph in a single pass. The Fast Marching Method (FMM) is an analogous single-pass method for approximating the continuous solution to the Eikonal equation, an HJ PDE for which the speed of motion is the same in all directions. We present a generalization of FMM, a single-pass method that approximates the solution to a static HJ PDE for which the speed of motion may depend on the direction of travel. We use examples drawn from robot path planning and seismology to demonstrate the benefits over competing methods.
I will discuss recent results concerning the regularity of the extremal solution associated with fourth order nonlinear eigenvalue problems on general domains. We show that the extremal solution is bounded under various assumptions on the nonlinearity and/or the space dimension. This is a joint work with Pierpaolo Esposito and Nassif Ghoussoub.
A group is left orderable if there exists a strict total ordering of its elements that is invariant under multiplication from the left.The set of all left orderings of a group comes equipped with a natural topological structure and group action, and is called the space of left orderings.My thesis investigates the topology of the space of left orderings for a given group, by analyzing those left orderings that correspond to isolated points and by characterizing the orbits of the natural group action using categorical notions.We also present an application of the space of left orderings in the field of 3-manifold topology, by using compactness to show that the fundamental groups of certain manifolds obtained from Dehn surgery are not left orderable.
ABSTRACT: Let f be the obvious covering map from Euclidean n-space to the n-torus. It is well known that if L is any straight line in n-space, then the closure of f(L) is a very nice submanifold of the n-torus. In 1990, Marina Ratner proved a beautiful generalization of this observation that replaces Euclidean space with any Lie group G, and allows L to be any subgroup of G that is "unipotent." We will discuss the statement of Ratner's Theorem, and a few of its important consequences. Topological and geometric aspects will be emphasized, while algebraic technicalities will be pushed to the background.
A positive rational number is said to be congruent if it is the area of some right triangle with rational sides. The question of determining whether a given number is congruent is called the congruent number problem. This is a thousand-year-old unsolved problem in number theory. In this talk, we present a detailed description of this problem and discuss its connection with the theory of elliptic curves.
We will show that the dynamics of a mechanical system consisting of a rigid structure (with a finite number of degrees of freedom) interacting with a fluid can sometimes be governed by a system of ODEs. In the literature, we can usually find two different ways allowing one to obtain equations of motion for these systems: the first one is based on Newton's laws (Classical Mechanics) and the second on Hamilton's principle (Analytic Mechanics). However, the resulting equations are far from being obviously identical in both cases. Quite surprisingly, we will prove in this second talk that it is not always the case.
We will discuss continuous-time, multi-type branching problems to model aspects of HIV-virus and T-cell dynamics in the blood stream. We are motivated by observations of viral load in HIV+ patients on anti-retroviral treatment (ART). While on ART for HIV, an infected individual’s viral load remains non-zero, though it is very low and undetectable by routine testing. Further, blood tests show occasional viral blips: very short periods of detectable viral load. We hypothesize that this very low viral load can be explained principally by the activation of cells latently infected by HIV before the initiation of treatment, which constitute a reservoir that has been observed to decay in time. Viral blips then represent large deviations from the mean. Modeling this system as a sub-critical 3-type branching process (latently infected cells, activated cells, virus), we derive equations for the probability generating function. Using a novel numerical approach we extract probability distributions for viral load yielding blip amplitudes consistent with patient data. This technique also allows us to calculate extinction probability distributions in time, which we relate to extinction of the latent reservoir. We also consider related problems including a 2-type super-critical branching process (virus and target cells only) with small initial numbers representing early HIV infection dynamics, to assess probabilities of infection initiation and early-time viral loads.
The present thesis is concerned with the development and practical implementation of robust a-posteriori error estimators for discontinuous Galerkin (DG) methods for convection-diffusion problems.It is well known that solutions to convection-diffusion problems may have boundary and internal layers of small width where their gradients change rapidly. A powerful approach to numerically resolve these layers is using hp-adaptive finite element methods, which control and minimize the discretization errors by locally adapting the mesh sizes and the approximation orders to the features of the problems. In this work, we choose DG methods to realize adaptive algorithms. As compared to standard finite element discretizations, DG methods make use of approximating spaces that are discontinuous over inter-elemental boundaries. As a result, these methods yield stable and robust discretization schemes for convection-dominated problems, and are naturally suited for hp-adaptive algorithms.At the heart of adaptive finite element methods are a-posteriori error estimators. They provide information on the errors on each element and indicate where local refinement/derefinement should be applied. An efficient error estimator should always yield an upper and lower bound of the discretization error in a suitable norm. For convection-diffusion problems, it is desirable that the estimator is also robust, meaning that the upper and lower bounds differ by a factor that is independent of the Péclet number of the problem.We develop a new approach to obtain robust a-posteriori error estimates for convection-diffusion problems. As a starting point, we consider the h-version DG method. Then we extend our techniques to hp-version methods, both on isotropically and anisotropically refined meshes. The main technical tools in our analysis are new hp-version approximation results of an averaging operator, which are derived for irregular hexahedral meshes in three dimensions, as well as for irregular anisotropic rectangular meshes in two dimensions. Our numerical results indicate that the error estimator is effective in resolving layers. For the hp-adaptive algorithms, once the local mesh size is of the same order as the width of layers, both the energy error and the error estimator are observed to converge exponentially fast in the number of degrees of freedom.
Motivated by geometry, we consider a `less discrete' way of counting lattice points in polytopes, in which one assigns a certain `weight' to each lattice point. On the combinatorial side, this approach reveals some `hidden symmetry' which improves upon and makes transparent some classical results in Ehrhart theory. On the geometric side, the combinatorial invariants count orbifold Betti numbers of toric stacks. If time permits, we will discuss a generalization involving motivic integration.
We will discuss our motivation for understanding what the analogue of DeRham
cohomology should be for E_{\infty}-algebras in connection with algebraic
K-theory and topological cyclic homology
The Schur functions $s_\lambda$ and ubiquitous
Littlewood-Richardson coefficients $c_{\mu \nu}^{\lambda}$ are
instrumental in describing representation theory, symmetric functions,
and even certain areas of algebraic topology.
Determining when two skew diagrams $D_1$, $D_2$ have the same skew
Schur function or determining when the difference of two such skew
Schur functions $s_{D_1}-s_{D_2}$ is Schur-positive reveals
information about the structures corresponding to these functions.
By defining a set of staircase diagrams that we can augment with other
diagrams, we discover collections of skew diagrams for which the
question of Schur-positivity among each difference can be resolved.
Furthermore, for certain Schur-positive differences we give explicit
formulas for computing the coefficients of the Schur functions in the
difference.
We extend from simple staircases to fat staircases, and carry on to
diagrams called sums of fat staircases. These sums of fat staircases
can also be augmented with other diagrams to obtain many instances of
Schur-positivity.
We establish explicit criteria of solvability for the quasilinear Riccati type equation $-\Delta_p u =|\nabla u|^q + \omega$ in a bounded $\mathcal{C}^1$ domain $\Omega\subset\mathbb{R}^n$, $n\geq 2$. Here $\Delta_p$, $p>1$, is the $p$-Laplacian, $q$ is critical $q=p$, or super critical $q>p$, and the datum $\omega$ is a measure. Our existence criteria are given in the form of potential theoretic or geometric (capacitary) estimates that are sharp when $\omega$ is compactly supported in the ground domain $\Omega$. A key in our approach to this problem is capacitary inequalities for certain nonlinear singular operators arising from the $p$-Laplacian.
The next talk in the TAAP seminar series is by Fok-Shuen Leung. Graduate students will have their attendance credited toward their eventual accreditation.
Title: What's next for 599? MATH 599 is the department's main tool for helping graduate instructors get off the ground. In this seminar, we'll discuss how well it does that. What did you expect, and what did it deliver? What works, and
what doesn't? And most importantly, where does the course go from here? The goal is to produce a road map for future iterations.
The Schur functions and ubiquitous Littlewood-Richardson coefficients are instrumental in describing representation theory, symmetric functions, and even certain areas of algebraic topology. Determining when two skew diagrams have the same skew Schur function or determining when the difference of two such skew Schur functions is Schur-positive reveals information about the structures corresponding to these functions. By defining a set of staircase diagrams that we can augment with other diagrams, we discover collections of skew diagrams for which the question of Schur-positivity among each difference can be resolved. Furthermore, for certain Schur-positive differences we give explicit formulas for computing the coefficients of the Schur functions in the difference.We extend from simple staircases to fat staircases, and carry on to diagrams called sums of fat staircases.These sums of fat staircases can also be augmented with other diagrams to obtain many instances of Schur-positivity. We note that several of our Schur-positive differences become equalities of skew Schur functions when the number of variables is reduced. Finally, we give a factoring identity which allows one to obtain many of the non-trivial finite-variable equalities of skew Schur functions.
The calculus of functors provides a framework for analyzing
functors of spaces, and more generally, model categories, in
terms of polynomial-like approximations to the functors. In this
talk, I will give an introduction to the calculus of functors
and discuss recent work with Kristine Bauer and Randy McCarthy
aimed at using calculus to better understand DeRham cohomology
in new contexts.
From knowledge of the conservation law multipliers of a given PDE system, one can determine whether it can be mapped invertibly to a linear PDE system and explictly find such a mapping when it exists. This method will be compared with the symmetry approach to this problem and it will be explained why the conservation law approach should be simpler computationally. Several examples comparing these approaches will be given. A systematic way of extensions to non-invertible mappings of nonlinear PDEs to linear PDEs will be presented. If time permits, it will also be shown how to systematically find non-invertible mappings of linear PDEs with variable coefficients to linear PDES with constant coefficients.
Tree polymers are simplifications of 1+1 dimensional lattice polymers made up of polygonal paths of a (nonrecombining) binary tree having random path probabilities. As in the case of lattice polymers, the path probabilities are (normalized) products of i.i.d. positive random weights. The a.s. probability laws of these paths are of interest under weak and strong types of disorder. The case of no disorder provides a benchmark since the polymers are simple symmetric random walk paths where all of the probability laws are known. We will discuss some recent results, speculation and open problems for this class of models. This is largely based on joint work with Stan Williams.
Basics of Mathematical Finance will be presented using concepts from probability. Risk measures and option replication strategies will be introduced, which allow the introduction of fundamental theorems of asset pricing. Brief results from the author’s research concerning regularity of profits under technical trading rules will be also presented.
I will present this open problem in complex geometry via the example of a Hopf surface. This is (roughly) the only compact Hermitian surface that admits infinitesimal isometries which are not holomorphic, and it exhibits some interesting behaviour which is indicative of some known results in higher dimensions. This talk will not require very much knowledge of differential geometry, with most of the discussion on the more intuitive level of group actions by Lie groups.
A celebrated result of Halasz characterizes the multiplicative functions taking values in the complex unit disc which have a non-zero mean value; recent work of Granville and Soundararajan characterizes the Dirichlet characters which have large character sums. I'll describe how one can prove a hybrid of these two, and show how this leads to improvements over Granville and Soundararajan's bounds. In particular, on the assumption of the Generalized Riemann Hypothesis the method yields a sharp bound on cubic character sums.
Note for Attendees
Refreshments will be served between the two talks.
I will talk about work in progress on certain adelic period integrals of modular forms on SL_2 and GL_2. it turns out that the situation for SL_2 is quite different from that of GL_2 and we'll try to explain what some of the differences mean for non-vanishing of L-functions.
Research on mathematical cognition indicates that learning the basics – that is, mastering the facts and procedures of the discipline – is only a small part of what learning to think mathematically is all about. Other vitally important aspects of mathematical thinking and problem solving are:
·heuristic problem solving strategies (rules of thumb for making progress when you're “stuck”);
·“control” skills (having a degree of self-awareness during problem solving that keeps you on the right track, and keeps you from squandering problem solving resources on wild goose chases); and
·“having a sense of what mathematics is all about” – developing a mathematician's point of view and being able to engage in mathematics rather than merely knowing about it.
There is nothing special about mathematics, at least in this regard: I argue that the same is true of all problem-solving domains, including the physical sciences, engineering, and even writing! A mistaken focus on subject matter mastery alone can have some disastrous consequences. This talk outlines the story in mathematics, with a few examples from other fields. I describe what can go wrong, and provide a few examples of what can go right if we attend to the broad spectrum of problem solving competencies in all of our instruction.
Biography: Alan Schoenfeld is the Elizabeth and Edward Conner Professor of Education and Affiliated Professor of Mathematics at the University of California at Berkeley. He is a Fellow of the American Association for the Advancement of Science an inaugural Fellow of the American Educational Research Association, and a Laureate of the education honor society Kappa Delta Pi. He has served as President of the American Educational Research Association and as the vice president of the National Academy of Education. In 2008 he was given the Senior Scholar Award by AERA’s Special Interest Group for Research in Mathematics Education.
After obtaining his Ph.D. in mathematics from Stanford in 1973, Schoenfeld turned his attention to issues of mathematical thinking, teaching, and learning. His work has focused on problem solving (what makes people good problem solvers, and how can people get better at it?), assessment, teachers’ decision-making, and issues of equity and diversity, with the goal of making meaningful mathematics truly accessible to all students.
Schoenfeld was lead author for grades 9-12 of the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics. He was one of the founding editors of Research in Collegiate Mathematics Education, and has served as associate editor of Cognition and Instruction. He has written, edited, or co-edited twenty-two books and nearly two hundred articles on thinking and learning. He has an ongoing interest in the development of productive mechanisms for systemic change and for deepening the connections between educational research and practice. His most recent book, How we Think, provides detailed models of human decision making in complex situations such as teaching.
The recently developed graph limit theory is part of a bigger picture in which limits of axiomatizable structures are studied. We present new results about the case when the underlying structures are functions on groups. The corresponding limit theory is deeply connected to a theory called "higher order Fourier analysis" which was founded by Gowers to generalize Roth's approach to three term arithmetic progressions in integer sets.
Rigid cohomology is one flavor of Weil cohomology. This entails for instance that one can asociate to a scheme X over F_p a collection of finite dimensional Q_p-vector spaces H^i(X) (and variants with supports in a closed subscheme or compact support), which enjoy lots and lots of nice properties (e.g. functorality, excision, Gysin, duality, a trace formula -- basically everything one needs to give a proof of the Weil conjectures).
Classically, the construction of rigid cohomology is a bit complicated and requires many choices, so that proving things like functorality (or even that it is well defined) are theorems in their own right. An important recent advance is the construction by le Stum of an `Overconvergent site' which computes the rigid cohomology of X. This site involves no choices and so it trivially well defined, and many things (like functorality) become transparent.
In this talk I'll explain a bit about classical rigid cohomology and the overconvergent site, and explain some new work generalizing rigid cohomology to algebraic stacks (as well as why one would want to do such a thing).
We will describe the large scale behaviour that is conjectured to be universal for a large class of one dimensional systems, and recent progress on the continuum versions of these models.
Thus [xr, xs] = Nr,s xr+s for some constant Nr,s whenever r+s is also a root. One of Chevalley's remarkable results from about 1950-1955 was that a basis can be chosen so that Nr,s is an integer, and more particularly |Nr,s| = pr,s where pr,s is the greatest p such that s-p.r is a root. This was the first, crucial step in constructing reductive groups over arbitrary fields in terms of root data.The derivation of this equation remains mildly mysterious even after fifty years, as does the derivation of the signs of the constants. In 1966 Jacques Tits discussed these matters in a paper published in the `Publications de l'IHES', but this work has the reputation of being extremely difficult, and I imagine few have understood it. I hope to make Tits' presentation digestible.
The omega transformation takes a Schur function indexed by a partition to the Schur function indexed by the partition's transpose. In this joint work with Jeff Remmel, we explore a refinement of the omega transformation defined on the quasisymmetric Schur functions. The resulting polynomials are called row-strict quasisymmetric Schur functions since they are described combinatorially as generating functions for row-strict composition tableaux. The interaction between row-strict quasisymmetric Schur functions and quasisymmetric Schur functions provides a natural method for interpolating between the compositions that rearrange a given partition and those that rearrange the partition's transpose. This allows us to define an operation on compositions which is similar to the transposition operation on partitions.
The last UBC/UMC talk of the year is by Alex Duncan.
Title: Solving equations with Groebner bases
The theory of Groebner bases is fundamental to modern computational algebra. One of their most useful applications is the solution of systems of polynomials in multiple variables. I will give a gentle introduction to Groebner bases emphasizing this application.
A systematic way of obtaining non-invertible mappings of nonlinear PDEs to linear PDEs will be presented. It will also be shown how to systematically find non-invertible mappings of linear PDEs with variable coefficients to linear PDES with constant coefficients. In particular, an extension to non-invertible mappings of the problem posed by Kolmogorov in his original 1931 paper will be shown (i.e, for which coefficients can a Kolomogorov equation be mapped into the heat equation).
If time permits, Raouf Dridi will further discuss the Cartan equivalence problem.
Let T be the return time to the origin of a simple random walk on an infinite recurrent graph. We show that T is heavy tailed and non-concentrated. More precisely, we have
i) P(T>t) > c/sqrt(t)
ii) P(T=t|T>=t) < C log(t)/t
Inequality i) is attained on Z, and we construct an example demonstrating the sharpness of ii). We use this example to answer negatively a question of Peres and Krishnapur about recurrent graphs with the finite collision property (that is, two independent SRW on them collide only finitely many times, almost surely).
In the 1990's, Greg Arone gave a description of the Snaith splitting of spaces of the form $\Omega^m \Sigma^m X$. His method extended to give a kind of functorial filtration of any space of the form Maps(K, X) where K is a finite complex. In good cases, this leads to a stable splitting of these mapping spaces. We will describe Arone's result with an eye toward applying this to other mapping space functors.
In this talk we will present our recent work on ‘’Closed geodesics in Alexandrov spaces of curvature bounded from above’’. This is an extension of Colding and Minicozzi’s width-sweepout construction of closed geodesics on closed Riemannian manifold to the Alexandrov setting, which provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics. We will explain how the width-sweepout construction works and discuss some future work in this direction.
This is the last talk this term in the teaching seminar associated with the TA Accreditation Program. All are welcome. Graduate students will have their attendance credited toward their eventual accreditation.
Title: The TAAP meta-seminar
The goal of this last session is to address what direction we would like the seminar to take in the Fall. What are the concerns, wishes for improvement or techniques related to your activity as a TA that you would like to discuss? What formats would best serve these themes? Take this opportunity to help shape your seminar.
Let $a,b,c \geq 2$ be integers satisfying $1/a + 1/b + 1/c > 1$. Darmon and Granville proved that the generalized Fermat equation $x^a + y^b = z^c$ has only finitely many coprime integer solutions; conjecturally something stronger is true: for $a,b,c \geq 3$ there are no non-trivial solutions and for $(a,b,c) = (2,3,n)$ with $n \geq 10$ the only solutions are the trivial solutions and $(\pm 3,-2,1)$ (or $(\pm 3,-2,\pm 1)$ when n is even).
I'll explain how the modular method used to prove Fermat's last theorem adapts to generalized Fermat equations and use it to solve the equation $x^2 + y^3 = z^{10}$. One new ingredient is the use of number field enumeration techniques to classify Galois representations associated to hypothetical solutions; classically one uses Ribet's level lowering theorem, but here the representations are wildly ramified and his method does not apply.
A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic manifolds arising as a quotient of the upper half-plane by a discrete ``arithmetic" subgroup of SL_2(R) (for example, SL_2(Z), and in this case the corresponding eigenfunctions are called Maass cusp forms). In this case, Rudnick and Sarnak have conjectured that the high energy eigenfunctions become equi-distributed. I will discuss some recent progress which has led to a resolution of this conjecture, and also on a holomorphic analog for classical modular forms. I will not assume any familiarity with these topics, and the talk should be accessible to graduate students.
This talk will discuss mathematical problems which are challenged by the fact they involve functions of a very large number of variables. Such problems arise naturally in learning theory, partial differential equations or numerical models depending on parametric or stochastic variables. They typically result in numerical difficulties due to the so-called ''curse of dimensionality''. We shall explain how these difficulties may be handled in various contexts, based on two important concepts: (i) variable reduction and (ii) sparse approximation.
Tropicalization is a technique that transforms algebraic geometric objects to combinatorial objects. Specifically, it associates polyhedral complex to subvarieties of an algebraic torus. One may ask which polyhedral complexes arise in this fashion. We focus on curves which are transformed by tropicalization to immersed graphs. By applying toric geometry and Baker's specializing of linear systems from curves to graphs, we give a new necessary condition for a graph to come from an algebraic curve. In genus 1 and in certain geometric situation, this condition specializes to the well-spacedness condition discovered by Speyer and generalized by Nishinou and Brugalle-Mikhalkin. The techniques in this talk give a combinatorial way of thinking about deformation theory which we hope will have further applications.
This thesis consists of four research papers and one expository note that study factors of point processes in the contexts of thinning and matching.In “Poisson Splitting by Factors,” we prove that given a Poisson point process on Rd, with intensity l, as a deterministic function of the process, we can colour the points red and blue, so that each colour class forms a Poisson point process on Rd, with any given pair of intensities summing l; furthermore, the function can be chosen as a isometry-equivariant finitary factor (that is, if a isometry is applied to the points of the original process the points are still coloured the same way). Thus using only local information, without a central authority or additional randomization, points of a Poisson process can split into two groups, each of which are still Poisson.In “Deterministic Thinning of Finite Poisson Processes,” we investigate similar questions for Poisson point processes on a finite volume. In this setting we find that even without considerations of equivariance, thinning can not always be achieved as a deterministic function of the Poisson process and the existence of such a function depends on the intensities of the original and resulting Poisson processes. In “Insertion and Deletion Tolerance of Point Processes,” we define for point processes a version of the concept of finite-energy. This simple concept has many interesting consequences. We explore the consequences of having finite-energy in the contexts of the Boolean continuum percolation model, Palm theory and stable matchings of point processes. In “Translation-Equivariant Matchings of Coin-Flips on Zd,” as a factor of i.i.d. fair coin-flips on Zd, we construct perfect matchings of heads and tails and prove power law upper bounds on the expected distance between matched pairs.
In the expository note “A Nonmeasurable Set from Coin-Flips,” using the notion of an equivariant function, we give an example of a nonmeasurable set in the probability space for an infinite sequence of coin-flips.
I aim to explain a recent paper of my collaborator Bai-Ling Wang in which he proves that there is a generalisation of the Baum-Douglas geometric cycles which realise ordinary K-homology classes to the case of twisted K-homology. We propose that these twisted geometric cycles are D-branes in string theory. There is an analogous picture for manifolds that are not string.
We classify bifurcations of the asymmetric states from a family of symmetric states in the focusing (attractive) Gross-Pitaevskii equation with a symmetric double-well potential. Depending on the shape of the potential, both supercritical and subcritical pitchfork bifurcations may occur. We also consider the limit of large energies and show that the asymmetric states always exist near a non-degenerate extremum of the symmetric potential. These states are stable (unstable) in the case of subcritical nonlinearity if the extremum is a minimum (a maximum). All states are unstable for large energy in the case of supercritical nonlinearity. This is a joint work with E. Kirr and P. Kevrekidis.
Abstract: This will be a mostly expository talk on the new 3-manifold invariant, Heegaard-Floer homology, developed in the last decade in a series of remarkable papers by Ozsvath and Szabo. Emphasis will be on the remarkable applications of the theory.
We discuss channels for which the input is constrained to be from a given set of D-dimensional arrays over a finite alphabet. Such a set is called a constraint. An encoder for such a channel transforms arbitrary arrays over the alphabet into constrained arrays in a decipherable manner. The rate of the encoder is the ratio of the size of its input to the size of its output. The capacity of the channel or constraint is the highest achievable rate of any encoder for the channel.
Given a binary D-dimensional constraint, a D-dimensional array with entries in {0,1, } is called “valid”, for the purpose of this abstract, if any “filling” of the ‘ ’s in the array with ‘0’s and ‘1’s, independently, results in an array that belongs to the constraint. The density of ‘ ’s in the array is called the insertion rate. The largest achievable insertion rate in arbitrary large arrays is called the maximum insertion rate. An unconstrained encoder for a given insertion rate transforms arbitrary binary arrays into valid arrays having the specified insertion rate. The tradeoff function essentially specifies for a given insertion rate the maximum rate of an unconstrained encoder for that insertion rate.
Given a 1-dimensional constraint, one can consider the D-dimensional constraint formed by collecting all the D-dimensional arrays for which the original 1-dimensional constraint is satisfied on every row in a direction along an axis. The sequence of capacities of these D-dimensional constraints has a limit as D approaches infinity, sometimes called the infinite-dimensional capacity.
As time permits, we will present some of our results: we computed the exact capacity of 2 families of multidimensional constraints; we generalized a known method for obtaining lower bounds on the capacity, for a certain class of 2-dimensional constraints, and improved the best known bounds for a few constraints of this class; we determined the tradeoff function for a certain family of 1-dimensional constraints; and finally, we partially answer a question of Poo et al., by proving that for a large class of 1-dimensional constraints with 0 maximum insertion rate, the infinite dimensional capacity equals 0 as well.
Our primary goal is to describe a strong quantitative form of the Shannon-McMillan-Breiman theorem for log-concave probability measures on linear spaces, even in the absence of stationarity. The main technical result is a concentration of measure inequality for the “information content” of certain random vectors. We will also discuss other related work. For instance, by combining this concentration result with ideas from information theory and convex geometry, we obtain a reverse entropy power inequality for convex measures that generalizes the reverse Brunn-Minkowski inequality of V. Milman. This is joint work with Sergey Bobkov (Minnesota).
We consider optimal control problems with control appearing linearly. The evaluation of the Pontryagin Minimum Principle shows that optimal controls are composed of bang-bang and singular arcs. Values of bang-bang controls switch discontinuously between their upper and lower bounds, whereas singular controls can take values in the interior of the control region. The optimal control problem induces a finite-dimensional optimization problem with respect to the switching times between bang-bang and singular arcs. The arc-parametrization is an efficient method for solving the induced optimization problems and allows to check second-order sufficient conditions (SSC). It can be shown that SSC for the induced optimization problem and a regularity property imply SSC for bang-bang control problem. SSC for singular control problems require stronger conditions which are currently under investigation. SSC constitute the basis for a parametric sensitivity analysis and the development of real-time control techniques. Several examples illustrate theory and numerics: optimal control of (1) a Van-der-Pol oscillator, (2) a semiconductor laser, (3) a batch fermentation process and (4) chemotherapy of HIV.
Special Undergraduate Awards will be presented at the beginning of the reception and refreshments will be served. This will be followed by the Niven lecture at 4:00 pm in MATX 1100.
I will examine some of our most primitive images of random phenomenon: tossing coins, spinning a roulette wheel and shuffling cards. In each case, practical experiments, combined with a bit of mathematics shows that while randomness is possible, usually we are lazy and things are quite far from random. Connections problems with large scale modelling are developed.
We apply self-dual variational calculus to inverse problems, optimal
control problems and homogenization problems in partial differential
equations.
Self-dual variational calculus allows for the variational formulation of
equations which are not necessarily of Euler-Lagrange type. Instead, a
monotonicity condition permits the construction of a so called self-dual
Lagrangian. This Lagrangian then permits the construction of a non-negative
functional whose minimum value is zero, and its minimizer is a solution to
the corresponding equation.
In the case of inverse and optimal control problems, we use the functional
given by the self-dual Lagrangian as a penalization function, which
naturally possesses the ideal qualities for such a role. This allows for
the application of standard variational techniques in a convex setting, as
opposed to working with more complex constrained optimization problems.
This extends work pioneered by Barbu and Kunisch.
In the case of homogenization problems, we develop variational
counterparts to results by dal Maso, Piat, Murat and Tartar with the use of
simpler machinery. In this context self-dual variational calculus permits
one to study the asymptotic properties of the potential functional using
classical Gamma-convergence techniques which are simpler to handle than the
direct techniques required to study the asymptotic properties of the
equation itself. The approach also allows for the seamless handling of
multivalued equations.
The study of such problems leads to the introduction of suitable
topological structures of the spaces of maximal monotone operators and
their corresponding self-dual potentials, which we develop in detail
throughout this thesis.
In joint work with Tom Salisbury, we study a class of random directed graphs on Z^d that includes site percolation and oriented percolation. Motivated by the study of random walks in these random environments, we focus on those models in two dimensions for which the set C_o (sites reachable from the origin) is infinite almost surely. We describe phase transitions in the geometry of C_o and other relevant clusters in some of the more interesting cases.
We construct a supercritical interacting measure-valued diffusion with
representative particles that interact with the center of mass by using
the historical stochastic calculus of Perkins to modify a super
Ornstein-Uhlenbeck process.
In doing so we prove continuum analogues of
results of Englander (2010) for binary branching Brownian motion.
On the survival set it is shown, in the attractive case, that the mass
normalized process converges almost surely in the Vasherstein metric
to the stationary distribution of the Ornstein-Uhlenbeck process, centered
at the limiting value of its center of mass. In the repulsive case it is
shown that it converges in probability, provided the repulsion is not too
strong, by appealing to a result of Englander and Winter (2006).
A version of a result of Tribe (1992) is proven on the extinction set;
that is, as it approaches the extinction time, the normalized process
in both the attractive and repulsive cases converges to a random point a.s.
It is a well-known property of Monte Carlo methods that quadrupling the sample size halves the error. In the case of simulations of a stochastic partial differential equations, this implies that the total work is the sample size times the discretization costs of the equation. This leads to a convergence rate which is impractical for many simulations, namely in finance, physics and geosciences. With the Multi-level Monte Carlo method introduced herein, the overall work can be reduced to that of the discretization of the equation, which results in the same convergence rate as for the standard Monte Carlo method. The model problem is an elliptic equation with stochastic coefficients. Multi-Level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. Numerical examples complete the theoretical analysis.
Abstract: In this talk, we'll use classic definitions from calculus of real variables to motivate what it means to be a polynomial functor of degree n, the construction of the derivative of a functor and the basics of the `Taylor Series' of a functor as well as do a few simple calculations.
In Applied Mathematics, linear and nonlinear eigenvalue problems arise frequently when characterizing the equilibria of various physical systems. In this thesis, two specific problems are studied, the first of which has its roots in micro engineering and concerns Micro-Electro Mechanical Systems (MEMS). A MEMS device consists of an elastic beam deflecting in the presence of an electric field. Modelling such devices leads to nonlinear eigenvalue problems of second and fourth order whose solution properties are investigated by a variety of asymptotic and numerical techniques.
The second problem studied in this thesis considers the optimal strategy for distributing a fixed quantity of resources in a bounded two dimensional domain so as to minimize the probability of extinction of some species evolving in the domain. Mathematically, this involves the study of an indefinite weight eigenvalue problem on an arbitrary two dimensional domain with homogeneous Neumann boundary conditions, and the optimization of the principal eigenvalue of this problem.
Under the assumption that resources are placed on small patches whose area relative to that of the entire domain is small, the underlying eigenvalue problem is solved explicitly using the method of matched asymptotic expansions and several important qualitative results are established.
Geodynamic flows, such as the convection within the Earth's mantle, are characterized by the extremely viscous nature of the flow, as well as the dependence of the viscosity on temperature. As such, a PDE-based approach, coupling the (stationary) Stokes Equations for viscous flow with a time-dependent energy equation, offers an accurate mathematical model of these flows. While the theory and practice of solving the Stokes Equations is well-understood in the case of an isoviscous fluid, many open questions remain in the variable-viscosity case that is relevant to mantle convection, where large jumps occur in the fluid viscosity over short spatial scales. I will discuss recent progress on developing efficient parallel solvers for geodynamic flows, using algebraic multigrid methods within block-factorization preconditioners.
The branched transport problem is the minimization of a concave functional on vecror measures with prescribed divergence. The only admissible measures are those concentrated on 1-rectifiable sets and the energy is the integral of a power $\theta^\alpha$ of their multiplicity $\theta$. I'll present an approximation by Gamma-convergence, through elliptic functionals defined on more regular functions : the idea is minimizing fucntionals such as $\frac 1 \varepsilon \int |v|^\alpha + \varepsilon |Dv|^2$ under constraints on the divergence of the $H^1$ function $v$. Obviously the exponents on the $\varepsilon$ and on the power of $|v|$ are to be changed if the result wants to be true. This approximation result recalls those of Modica-Mortola for the perimeter functional, where a double-well potential $W$, minimal on $0$ and $1$, is considered, and the energies $\frac 1 \varepsilon \int W(v)+ \varepsilon |Dv|^2$ converge to the perimeter of the interface between $\{v=0\}$ and $\{v=0\}$. Here the double-well is replaced with a concave power, so that there is a sort of double-well at $0$ and $\infty$. In ths case as well, the energy at the limit concentrates on a lower dimensional structure. Besides the link with the theory of elliptic approximations, the interest of this convergence lies in its applications for numerics. Actually, we built (in collaboration with E. Oudet, Chambéry) a quite efficient method, which allows to find reasonable local minima of the limit problem, avoiding the NP complications of the usual combinatorial approaches. The Steiner problem of minimal connection may be approached in this way as well.
Abstract: Two G-spaces X and Y are said to be hG-equivalent if their Borel constructions are equivalent over BG. In this talk we introduce an obstruction which determines when a G-sphere X is hG-equivalent to a finite G-sphere Y, at a prime p dividing the order of G. We will also discuss how to generalize this to a global finiteness obstruction, and, in the case of G-spheres, relate the finiteness obstruction to the dimension function of a G-sphere.
A review of several applications of Lie groups of transformations to difference equations, meshes (lattices) and difference functionals is presented. Examples of difference models (i.e. difference equations and appropriate meshes) which admit the same symmetry group as there continuous counterparts are presented.
For integrable cases of ODEs, discrete representations of ODEs ("an exact finite-difference scheme") are developed. Invariant variational problems for difference equations are considered. Lagrangian and Hamiltonian formalisms and Noether-type constructions for difference functionals, meshes and difference equations are illustrated by several examples.
A Galois descent theorem for n-types will be displayed and explained. This result is used, together with an appropriate version of the homotopy theory of pro objects, to give a descent criterion for diagrams of spaces which are defined on the etale site of a field. The need for such a criterion first arose in connection with the algebraic K-theory of fields.
Usually, a nonlinear functional involving the Sobolev exponent does not satisfy the Palais-Smale condition. However, Brezis and Nirenberg showed that under a threshold , the minimax value is in fact a critical value. In this talk, we should extend this idea to the equation involving with the Hardy singular potential.
The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation. We will discuss the application of Lie group variational integrators to the optimal control of mechanical systems. These methods are based on a discretization of Hamilton's principle that preserves the Lie group structure of the configuration space, and yield minimum-dimensional global representations of the dynamics. Recent extensions to homogeneous spaces yield intrinsic methods for Hamiltonian flows on the sphere, and have potential applications to the simulation of geometrically exact rods, structures and mechanisms. Extensions to Hamiltonian PDEs and uncertainty propagation on Lie groups using noncommutative harmonic analysis techniques will also be discussed. In particular, this allows one to develop attitude state estimation methods for satellite dynamics without assuming that the measurements are frequent, and that the uncertainty distribution is localized.
Somitogenesis is a process for the development of somites which are transient, segmental structures that lie along the anterior-posterior axis of vertebrate embryos. The pattern of somites is traced out by the ``segmentation clock genes" which undergo synchronous oscillation over adjacent cells. In this presentation, we analyze the dynamics for a model on zebrafish segmentation clock-genes which are subject to direct autorepression by their own products under time delay, and cell-to-cell interaction through Delta-Notch signaling. For this system of delayed equations, we present an ingenious iteration approach to derive the global synchronization and global convergence to the unique synchronous equilibrium. On the other hand, by applying the delay Hopf bifurcation theory and the method of normal form, we derive the criteria for the existence of stable synchronous oscillations. Our analysis provides the basic range of parameters and delay magnitudes for stable synchronous, asynchronous oscillation, and oscillation-arrested dynamics. Based on the derived criteria, further numerical findings on the dynamics which are linked to the biological phenomena are explored for the considered system.
Note for Attendees
Coffee and cookies will be served at around 1:45 pm.
Gedrich and Gruenberg(1987)and Ikenaga(1984)studied the algebraic
invariants silpZG,the supremum of the injective lengths of the projective
ZG-modules,and spli,the supremum of the projective lengths of the
injective ZG-modules,in connection with the existence of complete
cohomological functors for the group G and showed that these are related
to the virtual cohomological dimension of G. We will show that silpZG and
spliZG are related to the Gorenstein dimension of ZG and to the existence
of a finite dimensional model for E_G, the classifying space for proper
actions
All instructors teaching a graduate course for 2010-11 should attend this "trade show" style graduate orientation and be prepared to talk about their course.
A catered lunch will be provided to faculty and students in Math 125.
We will begin with some basic background on shifts of finite type (SFT) and then talk about the recent paper by K. McGoff with the title above. Given an SFT X, fixed n, and alphat in [0,1], one defines a random sub-SFT of X by selecting words of length n, independently with probability alpha. McGoff shows that the probability that the resulting SFT has various properties as n -----> infty can be described explicitly in terms of alpha. (This will be the first talk in an informal weekly seminar on various topics within Symbolic Dynamics and Ergodic Theory).
Cooperation represents a key organizing principle in genetic and cultural evolution. Yet cooperation is a conundrum because cooperators make a sacrifice to benefit others at some cost to themselves. Exploiters or defectors reap the benefits and forgo costs. Despite the fact that groups of cooperators outperform groups of defectors, Darwinian selection or utilitarian principles based on rational choice should favor defectors. Nevertheless, cooperation is ubiquitous in biological and social systems. Public goods games have established as the leading mathematical and game theoretical metaphor to study such social dilemmas, which are characterized by the conflict of interest between the group and the individual. An analysis based on evolutionary game theory shows that cooperation can be stabilized by punishing defectors. Punishment is also ubiquitous in animal and human societies - ranging from toxin producing microorganisms to law enforcement institutions. However, it remains unresolved how initially rare, costly punishment behavior can gain a foothold and spread throughout a population. In nature, animals and humans often carefully select their interaction partners or adjust their behavioral patterns. In the simplest case individuals simply refuse to participate in risky enterprises. Such voluntary participation in social endeavors describes a simple yet efficient mechanism to prevent deadlocks in states of mutual defection and thus represents a potent promoter of cooperation - but fails to stabilize it. However, the combined efforts of punishment and volunteering are capable of changing the odds in favor of cooperation - but only in finite populations. Under the stochastic dynamics of finite populations the freedom to withdraw leads to prosocial coercion. This implements Garret Hardin’s principle to overcome the 'Tragedy of the Commune': 'mutual coercion mutually [and voluntarily] agreed upon'. To date, theory and experiments emphasize the role of such peer-punishment, which is of crucial importance in various animals, but at least in most human societies peer-punishment has been largely superseded by sanctioning institutions and vigilantism deemed illegal. This can be modeled by introducing pool-punishment, which represents a precursor of executive power and echoes Elinor Ostroms self-governing principles in her work on 'Governing the Commons'. Pool-punishment always incurs costs to those committed to it even if no one requires reprimanding. Interestingly, our model predicts that individuals still trade the higher efficiency of peer-punishment for the increased stability of pool-punishment to maintain cooperation.
Note for Attendees
Tea and cookies will be served in the Math Lounge (MATH 125) at approximately 2:45pm.
I will present a joint work with Behrend.
For any smooth projective variety, we construct a differential graded scheme (stack) structure on the moduli space of
complexes of coherent sheaves. The construction uses the Hochschild cochain complex of A infinty bi-modules.
As an application, we show that the DT/PT wall crossing can be intepreted as change of stability conditions on
dg schemes.
We give a criterion under which a solution g(t) of the Kahler-Ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. This is a joint work with Jian Song.
Implicit time stepping applied to discretized incompressible fluid flow problems results in discrete Stokes-like (saddle-point, KKT) problems. There is a class of so-called Projection Methods in which these Stokes-like problems are efficiently approximated by the solutions of standard, second order elliptic problems. Over the last 20 years, these methods have been increasingly well understood. In this talk, a review of the methods and the analytic results are given. Leading order errors (both character and order) from the methods can be precisely described using asymptotic techniques. This asymptotic error analysis is applied to a number of commonly used projection methods. The results are confirmed by careful numerical computations using a spectral method on a representative model problem.
In these talks we describe our proofs of the Hanna Neumann Conjecture. This
conjecture of the late 1950's can be described both as a problem in group
theory or as one in graph theory. Our first proof is longer and interprets the
problem using homology of sheaves on graphs; our second proof is very short and
uses only simple graph theory, but was inspired from the type of induction used
in our first proof. Both proofs demonstrate the strengthened form of the
conjecture formulated by Walter Neumann, and both proofs use earlier resolved
cases of the conjecture.
A crucial idea of the proof is to express the seemingly awkward notion of
"reduced cyclicity" of a graph in simpler terms, involving limits over covering
maps. This "limit homology theory" may be of independent interest, and is
related to the Atiyah Conjecture; our theory requires some curious linear
algebra that also may be of independent interest.
Our talks will not assume any previous knowledge of sheaf theory.
We consider a system of bosons in a box, under the influence of a mutually
repellent pair potential. The particle density is positive and is kept
fixed while the volume of the box is increased. We discuss the following
result: the identification of the limiting free energy, at positive and sufficiently
small temperature, in terms of an explicit variational formula. We
use a large deviations approach combined with the representation of the
system of bosons through a system of interacting Brownian bridges.
This is a joint work with S. Adams (Warwick) andW. Konig (T.U. Berlin
and Weierstrass Institute).
This is the first talk this term for UBC/UMC, the undergraduate mathematics colloquium. The speaker is Professor Brian Wetton.
Title: Approximating the arctan function
Calculators can perform the same basic operations that humans can do by hand: add, subtract, multiply and divide. How these basic operations can be used to approximate more complicated functions (square roots and the arctan function are used as examples) to arbitrary precision will be discussed. Some history is given as to how these functions were computed routinely before calculators were available.
Graphs naturally pop up everywhere e.g. the Internet, networks of social contacts etc. Random graphs (and thus probability theory) are used to study these networks which are too large to study analytically. I will talk a bit about the questions which random graph theory hopefully can resolve. Also I will show introduce random graph models and discuss some results.
----
Simon Rose on "Why are generating functions cool?"
A generating function is a formal power series (that is, an infinitely long polynomial) where the coefficients are chosen to be some numbers we find interesting. In this talk I will show a few surprising results that can be proven quite easily within this theory, and encourage you to no longer care about whether or not something converges.
We will continue with background on symbolic dynamics, in particular the dynamical zeta function. Then we will state and prove some of McGoff's results on random shifts of finite type.
We consider the complex Monge-Amp\`ere equation on a compact K\"ahler manifold $(M, g)$ when the right hand side $F$ has rather weak regularity. In particular we prove that estimate of second order and the gradient estimate hold when $F$ is in $W^{1, p_0}$ for any $p_0>2n$. As an application, we show that there exists a classical solution in $W^{3, p_0}$ for the complex Monge-Amp\`ere equation when $F$ is in $W^{1, p_0}$.
Polarization occurs when cells segregate specific proteins and other factors to opposite ends of the cell in response to some signal. A cell with a symmetric distribution of proteins must have a symmetry breaking event in order to become polarized, resulting in a stable asymmetric protein distribution. In this informal talk, I will discuss possible mechanisms used by embryos of the nematode worm C. elegans to initiate the process of polarization, including new experimental evidence produced this summer.
In these talks we describe our proofs of the Hanna Neumann Conjecture. This
conjecture of the late 1950's can be described both as a problem in group
theory or as one in graph theory. Our first proof is longer and interprets the
problem using homology of sheaves on graphs; our second proof is very short and
uses only simple graph theory, but was inspired from the type of induction used
in our first proof. Both proofs demonstrate the strengthened form of the
conjecture formulated by Walter Neumann, and both proofs use earlier resolved
cases of the conjecture.
A crucial idea of the proof is to express the seemingly awkward notion of
"reduced cyclicity" of a graph in simpler terms, involving limits over covering
maps. This "limit homology theory" may be of independent interest, and is
related to the Atiyah Conjecture; our theory requires some curious linear
algebra that also may be of independent interest.
Our talks will not assume any previous knowledge of sheaf theory.
Gersgorin theorem says that the eigenvalues of finite matrices lie on
the so called Gersgorin region in the complex plane. This theorem implies
in particular that if a matrix is strictly diagonally dominant, then it
is invertible. Taussky improves: if a matrix is irreducibly diagonally dominant,
then it is invertible (quite long time ago, in 1949!).
In this talk, we discuss this problem in probabilistic way. We introduce
a Markov process from the given diagonally dominant matrix. Then by
using Feynman-Kac formula and large deviation principle, we can bound
the components of the inverse.
In this first talk we will introduce the notion of a symmetry of a PDE system. A symmetry of a PDE system is any transformation that maps all solutions into other solutions of the same PDE system. Lie’s algorithm provides an effective way to find directly the local symmetries of a given PDE system. Since the continuous symmetries of a PDE system are defined topologically, they are not restricted to local symmetries. It turns out that a natural way to find such nonlocal symmetries is through the construction of nonlocally related systems arising from the conservation laws of the given PDE system. A local symmetry of such a nonlocally related system can yield a nonlocal symmetry of the given PDE system. In this series of talks, it will be shown how to construct further nonlocally related systems (and consequently further nonlocal symmetries) through various means.
This is the first talk this term for the TAAP Seminar Series. Our speaker is Isabeau Iqbal, from the Department of Educational Studies and the Centre for Teaching, Learning and Technology.
Title: What is the value of the peer review of teaching?
Abstract: The literature suggests that the peer review of teaching process can foster a culture of teaching and learning and can enhance the level of collegiality in departments. In this session, I will present a snapshot of the peer review of teaching at UBC, highlighting distinctions between "formative" and "summative" peer review, and making links to the improvement of teaching. I will draw from the higher education literature and my own research to feature how departmental culture can support or hinder the peer review of teaching process.
I will begin by discussing some recent theoretical and numerical joint work with Liang and Sujatha on the Tate-Shafarevich group of elliptic curves over Q with complex multiplication and rank at least 2. In the latter part of the lecture, I will explain some rather wild general speculation which grew out of these computations.
Note for Attendees
Refreshments will be served between the two talks.
The Yoshida lift is a theta lift that takes a pair of automorphic forms on a definite quaternion algebra to a holomorphic Siegel modular form. We show that a natural refinement of the classical Yoshida lift in fact preserves p-integrality in a suitable sense. Since the result is rather technical, we will motivate it by casting it as the first step in a program sketched out by Harris-Skinner-Li aimed at relating the p-divisibility of the special values of certain automorphic L-functions to the existence of non-trivial elements of a certain Selmer group. Time permitting, we will also discuss a non-vanishing modulo p result of the integral Yoshida lift.
MATH ANNEX 1100 (PIMS/UBC distinguished colloquium)
Fri 24 Sep 2010, 3:00pm-4:00pm
Abstract
I will try to explain the general background to the subject, and discuss the formulation of the main conjectures of non-commutative Iwasawa theory, on which there has been important recent progress.
The Ricci flow is by now well-known as the technique used to prove the Poincare conjecture and several other remarkable mathematical results. It is natural to search for applications in physics. I will describe two, both related to mass-energy.
The first is that the Ricci flow is an approximation to a renormalization group flow in bosonic string theory. In the RG flow, a natural question to ask is how does the spacetime mass behave at different scales. In the Ricci flow approximation, we can sometimes answer this question.
The second is that, while mass-energy is a global property of spacetime, nonetheless various proposals for quasilocal mass exist and have sometimes proved useful. Bartnik has suggested a quasilocal mass which has many desirable properties, including both nonnegativity and manifest monotonicity as the quasilocal region is expanded. Nonetheless, his definition presents practical difficulties. This led Bartnik to make certain geometrical conjectures which, if true, would improve the situation. One of these is the Static Minimization Conjecture, much of which is now a theorem due to recent work of Anderson and Khuri. The minimization conjecture leads to the study of a version of Ricci flow. I will explain Bartnik's quasilocal mass and give the results of early attempts to study the minimization conjecture via this method.
In this talk we will describe the underlying construction of the main
theorem of Galatius' paper. This main theorem is a generalization of the
Pontryagin-Thom construction and establishes a weak homotopy equivalence
between the classifying space of a cobordism category and the infinite loop
space of the spectrum MTSO(n). We then compute the rational cohomology of
this infinite loop space and describe how the generalized MMM-classes come
from it.
Let G be a simple Lie group or Kac-Moody group and P a parabolic
subgroup.
One of the goals Schubert calculus is to understand the product
structure
of the cohomology ring H^*(G/P) with respect to its basis of Schubert
classes. If G/P is the Grassmannian, then the structure constants
corresponding to the Schubert basis are the classical
Littlewood-Richardson
coefficients which appear in various topics such as enumerative
geometry,
algebraic combinatorics and representation theory.
In this talk, I will discuss joint work with A. Berenstein in which
we give
a combinatorial formula for these coefficients in terms of the Cartan
matrix corresponding to G. In particular, our formula implies
positivity
of the “generalized” Littlewood-Richardson coefficients in the
case
where the off diagonal Cartan matrix entries are not equal to -1. Moreover, this positivity result does not rely on the geometry
of
the flag variety G/P.
Vortex solitons are standing wave solutions with complex phase that is an (integer) multiple of the angular polar coordinate. This multiple we call the 'spin', and indexes a family of solutions with increasing L2 norm. In the case of no spin, Merle and Raphael have shown that there exists a range of data that blowup with the Townes profile (the regular soliton) and whose H1 norm grows at a precise 'log-log' rate. We prove that in the case of spin 1, there is comparable data that blows up with the vortex profile and the log-log rate. The case of spin 2 and 3 will be discussed. This is joint work with Gideon Simpson (Toronto)
Title: A mathematical model of antigen bonds on immune cells
The bond between an antigen and a T-cell is one of the most important elements of our adaptive immune system. How the T-cell has such high sensitivity to slight differences in antigen remains a mystery. It is also a mystery why the antigen bond's lifetime is different depending on what the antigen is attached to. I will introduce a model of a single antigen bond. The model uses diffusion-advection equations, elasticity mechanics, and stochastic processes. As with many physical processes, the entire model can be understood in terms of energy minimization. We developed this model to explain experimental data on bond lifetimes, but it ended up generating a new hypothesis about T-cell sensitivity.
Motivated by the classical Mordell-Lang conjecture, we formulate a natural dynamical analogue in the context of semiabelian varieties, which we prove under a stronger hypothesis. We also present counterexamples which
occur in the Dynamical Mordell-Lang problem once we relax our hypothesis.
Self-avoiding polygons are simple closed curves embedded in a regular lattice. They belong to a family of objects, known collectively as lattice animals, that lie at the heart of lattice models of polymers, magnets and other phenomena. This large family of discrete structures has been a source of combinatorial problems for over 50 years and many
basic questions remain stubbornly unsolved. For example, the simple enumerative question - "How many self-avoiding polygons are there of length n?" is currently best answered by an algorithm that requires exponential time and memory.
The question of counting polygons on the cubic lattice is enriched by topology - self-avoiding polygons have well defined knot types. The exact computation of the number of polygons of length n and fixed knot type K is extremely difficult - indeed the current best algorithms can barely touch the first knotted polygons. In this talk I will discuss a
different approach to the problem. Instead of of exact methods, we have used an approximate enumeration method - which we call the GAS algorithm. This is a generalisation of the famous Rosenbluth method for sampling self-avoiding walks.
Using this algorithm we have estimated the number of polygons of different lengths and knot types on three different cubic lattices. These give direct evidence that the asymptotic growth of the number of polygons of a fixed knot type K is simply related to the growth of the number of unknotted polygons and the number of prime components in K. We have also studied the relative frequencies of different knots - for example, the ratio of the number of trefoils to the number of figure-eights. We see strong evidence that these ratios are universal suggesting that a very long closed curve is about 27 times more likely to be a trefoil than a figure-eight.
Markley & Paul, Engel, Calkin-Wilf and Friedland show a sequence of computable and very good lower and upper bounds
on the entropy of 2-D symmetric SFT's. I'll describe the method and show Engel's observation that these bounds monotonically converge to the entropy. I'll also discuss some of Engel's related conjectures and disprove some of them."
Motivated by applications to Brill-Noether theory and higher-rank Brill-Noether theory, we discuss several variations on Grassmannians. These include "doubly symplectic Grassmannians", which parametrize subspaces which are simultaneously isotropic for a pair of symplectic forms, "linked Grassmannians", which parametrize tuples of subspaces of a chain of vector spaces linked via linear maps, and "symplectic linked Grassmannians", which is an amalgamation of the linked Grassmannian and symplectic Grassmannian.
We introduce cohomological dimension following Serre's Galois Cohomology book. With the goal of proving some theorems on fields of cohomological dimension less or equal to 1.
I will describe new and old conjectures and some recent results
concerning typical maximal words in the group of signed permutations,
called signed sorting networks. These mirror similar results and
conjectures, concerning sorting networks, and suggest that these
models are in a common universality class.
The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers, particle systems, matrices and stochastic PDEs, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian.
The recent solution of M. Hill, M. Hopkins and D. Ravanel of the Kervaire invariant problem is likely to lead to a new area in the study of homotopy groups of spheres. In this talk I will explain that problem, and indicate what the above three authors have achieved. I’ll discuss the impact of their result on the study of vector fields and mention some problems which now open up in this “Post Kervaire-Milnor” era
The next talk in the TAAP Seminar Series will be by David Kohler and Desiree Mou. David Kohler is a senior graduate student in our department, and a recent winner of the UBC Graduate Teaching Award. Desiree Mou is a facilitator and instructional developer.
Title: How to assess your teaching skills
Abstract: This session will discuss, in a very honest and down-to-earth fashion, how to improve your ability to assess your own teaching skills. The focus will be on two specific items: getting better feedback and using goal-setting techniques.
I'll describe a joint result with Michal Hochman characterizing entropy
numbers of multidimensional subshifts of finite type (SFTs) in
recursive-theoretic terms. This result relates to Berger's theorem which
is the solution to Wang's problem: a non-empty 2 dimensional SFT may not
have periodic points, and (thus) the emptiness problem is undecidable.
The first part of this talk will be devoted to background on Berger's
theorem, apriodic SFTs, undecidability, and substitution systems.
Just as Cheeger's inequality relating spectral gap and
isoperimetric constants for manifolds has an analogue for graphs, so it has
an analogue for an action of a group preserving a probability measure. In
the case of Bernoulli actions, one can determine the spectral gap. This
allows one to analyze when certain combinatorial constructions are possible
as factors of IID random variables. (Joint work with Fedja Nazarov.)
We study the space M_g of isometry classes (or conformal equivalence classes) of smooth manifolds, diffeomorphic to #^g(S^d \times S^d), the connected sum of g copies of S^d \times S^d. For 2d=2, this is essentially the moduli space of Riemann surfaces. There is a variant M_{g,1} where we consider moduli of manifolds with an embedded D^{2d}; connected sum with S^d \times S^d gives a map M_{g,1} \to M_{g+1,1}, and we can form the direct limit M_{\infty,1}. The work of Madsen and Weiss on Mumford's conjecture determines the homology of M_{\infty,1} in the case 2d=2. We give a similar description of the homology of M_{\infty,1} in higher dimensions (2d \geq 6). This is joint work with Oscar Randal-Williams.
Speaker: Tyler Helmuth Title: What is Percolation?
Percolation, in the mathematical sense, is an abstraction of the physical phenomena of percolation: the movement of fluids through porous materials. I'll introduce the most classical percolation models, along with some basic questions and results. I'll also describe some more sophisticated questions that are related to the Fields Medal winning work of Werner (2006) and Smirnov (2010).
Speaker: Maxim Stykow Title: What is Cohomology?
We've all heard this "too-abstract-for-it's-own-good" word float around the math department but what actually does it mean? I will describe the motivation and intuitive ideas behind cohomology and provide insight into different flavors of cohomology and their applications. Finally, I will discuss the surprising fact that homotopy theory is the mother of all cohomology theories - great and small.
The Rallis inner product formula provides a precise relationship between the inner product of a theta lift and the special value of a $L$-function. We hope to give a flavor of the ingredients involved and a rough description of the recipe for cooking up this formula. We also hope to discuss/speculate on possible applications.
Location Changed to IAM Lounge - Room 306 Leonard S. Klinck Bldg.
Thu 14 Oct 2010, 2:00pm-3:00pm
Abstract
Animal cells crawl on surfaces using the lamellipod, a flat dynamic network of actin polymers enveloped by the cell membrane. Recent experiments showed that the cell geometry is correlated with speed and with actin dynamics. I will present mathematical models of actin network self-organization and viscoelastic flow explaining these observations. According to this model, a force balance between membrane tension, pushing actin network and centripetal myosin-powered contraction of this network can explain the cell shape and motility. In addition, I will discuss Darci flow of cytoplasm and its role in the cell movements.
After an introduction to the classical category of motives, we shall explain the construction of the category of birational motives which was carried out in joint work with B. Kahn, along with some applications.
Microtubules confined to the two-dimensional cortex of elongating plant cells must form a parallel yet dispersed array transverse to the elongation axis for proper cell wall expansion. Collisions between microtubules, which migrate via hybrid treadmilling, can result in plus-end entrainment (“zippering”) or catastrophe. Here, I present (1) a cell-scale computational model of cortical microtubule organization and (2) a molecular-scale model for microtubule-cortex anchoring and collision-based interactions between microtubules. The first model treats interactions phenomenologically while the second addresses interactions by considering energetic competition between crosslinker binding, microtubule bending and microtubule polymerization. From the cell-scale model, we find that plus-end entrainment leads to self-organization of microtubules into parallel arrays, while collision-induced catastrophe does not. Catastrophe-inducing boundaries can tune the dominant orientation. Changes in dynamic-instability parameters, such as in mor1-1 mutants in Arabidopsis thaliana, can impede self-organization, in agreement with experiment. Increased entrainment, as seen in clasp-1 mutants, conserves self-organization, but delays its onset. Modulating the ability of cell edges to induce catastrophe, as the CLASP protein may do, can tune the dominant direction and regulate organization. The molecular-scale model predicts a higher probability of entrainment at lower collision angles and at longer unanchored lengths of plus-ends. The models lead to several testable predictions, including the effects of reduced microtubule severing in katanin mutants and variable membrane-anchor densities in different plants, including Arabidopsis cells and Tobacco cells.
In finite fields and Euclidean space, we shall discuss criteria under which the set $A\cdot A+A \cdot A+\dots+A \cdot A$ is substantially larger than the set $A$ itself. We shall also discuss connections between this problem and the question of distribution of simplexes in vector spaces over finite fields and Euclidean space.
Given a projective variety X of dimension d, a "flag" of subvarieties Y_i, and a big divisor D, Okounkov showed how to construct a convex body in R^d, and this construction has recently been developed further in work of Kaveh-Khovanskii and Lazarsfeld-Mustata. In general, this Okounkov body is quite hard to understand, but when X is a toric variety, it is just the polytope associated to D via the standard yoga of toric geometry. I'll describe a more general situation where the Okounkov body is still a polytope, and show that in this case X admits a flat degeneration to the corresponding toric variety. This project was motivated by examples, and as an application, I'll describe some toric degenerations of flag varieties and Schubert varieties. There will be pictures of polytopes.
We prove the unique existence of solutions of the 3D incompressible Navier-Stokes equations in an exterior domain with small non-decaying boundary data, for all $t \in R$ or $t \in (0,\infty)$. In the case $t \in (0,\infty)$ it is coupled with a small initial data in weak $L^{3}$. If the boundary data is time-periodic, the spatial asymptotics of the time-entire solution is given by a Landau solution which is the same for all time. If the boundary data is time-periodic and the initial data is asymptotically discretely self-similar, the solution is asymptotically the sum of a time-periodic vector field and a forward discretely self-similar vector field as time goes to infinity. This is a joint work with Kyungkuen Kang and Hideyuki Miura.
Let R be an associative algebra with linear basis {x_w}. The problem of
determining the multiplicative structure coefficients of R with respect to
the basis {x_w} has been an interesting problem in combinatorics, algebra
and geometry. For example, the algebra of Symmetric polynomials has a
basis of Schur functions whose structure coefficients are the classical
Littlewood-Richardson numbers.
In this talk, I will discuss a new formula for computing
Littlewood-Richardson type coefficients for a large class of algebras
including cohomology rings of flag varieties of a Kac-Moody group and
coinvariant algebras of finite Coxeter groups. This formula relies only on
the data given in a Cartan matrix. This talk will be similar to the one I
gave in the AG seminar a few weeks ago, but with an emphasis on the
combinatorics involved.
There is a general cohomology defined by Sweedler for
co-commutative Hopf algebras, generalizing the usual cohomology of a
group or a Lie algebra. Recently it was discovered that
low-dimensional groups could be defined without the co-commutativity
requirement. In joint work with Christian Kassel, we have given the
first few examples of computations with these, in the case of algebras
of functions on groups. These turn out to be related to torsors in
algebraic geometry, and Drinfeld twists in quantum groups theory
I will review some recent (and also not-so-recent) progress
on the understanding of the behavior of the random walk on the
hypercubic lattice among bounded i.i.d. random, nearest neighbor
conductances that have a heavy lower tail at zero. The center of
focus of my talk will be the large-$n$ behavior of the probability
that the random walk in this random medium returns back to the
starting point after $2n$ steps. First I will show that there are
universal upper bounds on this probability, which enforce the
standard (diffusive) decay in spatial dimensions $d=2,3$ but permit
subdiffusive decay in higher dimensions. Then I will show how one
constructs examples for which the decay is actually provably
subdiffusive in all $d\ge4$. In $d=4$ this is particularly subtle
because there one has to control the walk over a whole range of
spatial scales. Based on various joint papers with N. Berger, O.
Boukhadra, C. Hoffman , G. Kozma and T. Prescott.
The mean ergodic theorem says that when T is a measure preserving transformation of a probability space (X,m), averages of foT^n converge to a limit L when the parameter n is averaged over longer and longer intervals, assuming f is in L2(m). We consider averages the form foT^(a(n)), where a(n) is some sequence growing polynomially. For a large class of sequences a(n), the averages of foT^(a(n)) still converge to the same limit L as above.
We will introduce some of the relavent background, prove some special cases of these results, and survey the situation in general. A subsequent talk will apply these results to additive combinatorics.
I will describe work with Dubi Kelmer on the first Laplace eigenvalue in towers of manifolds covered by real or complex hyperbolic space. All congruence quotients in a given dimension have a uniform spectral gap; we show how to deduce from this a uniform spectral gap for the family of congruence covers of a fixed arithmetic (non-congruence) manifold.
Abstract:
The exclusion process is one of the basic particle processes. In this process, particles move randomly, with the constraint that two particles may not occupy the same position. Of particular interest -- both mathematical and as a model for flow along a path -- is the process where the individual motions are along a line, with a drift in one direction.
Certain instances of this process are closely related to many other seemingly different models. These include a random growth model for a pile, and a certain card shuffling process. These connections allow us to prove certain properties of the exclusion processes. It is commonly believed that many large scale properties of such models, do not depend on the microscopic parameters of the model. This is referred to as universality and verifying it is a major open problem.
I will describe several aspects of the exclusion process, including its so called hydrodynamic limit, its stationary distributions and the asymptotic speed of particles. I will discuss some progress on extending these results to less specific exclusion processes.
Note for Attendees
The speaker aims to keep the talk accessible to a general mathematical audience:
Familiarity with continuous time Markov chains and stationary distributions may be useful, but is NOT crucial, since very concrete examples will be discussed. Some of the finer points alluded to depend on other notions from an undergraduate probability course.
In this talk we use a combination of the methods developed by Gowers and
refined by Green and Tao in the proof of Szemeredi's Theorem on long
arithmetic progressions in sets and the Hardy-Littlewood circle method to
obtain a bound for the upper density of a subset A of the integers Z such
that if d>=7, then the set A^d contains no triangles which are similar to a
given triangle. The bound is independent of the given triangle.
Yau conjectured that the first eigenvalue of the Laplacian on compact embedded minimal surfaces in $\mathbb S^3$ should be equal to 2. We prove that Yau's conjecture is true for all minimal surfaces that are known to exist so far: the minimal surfaces constructed by Lawson, by Karcher-Pinkall-Sterling, and by Kapouleas-Yang. (Joint work with M. Soret)
We are interested here in interacting particle systems. Such systems
are used to approximate probability measures. The two examples I will talk about
are the genetic particle systems and Bird/Nanbu-like particle systems.
Propagation of chaos is a property shared by these systems, it is an asymptotic
property in term of the number of particles. I will show how one can do precise
computation concerning this propagation of chaos and deduce convergence results
for these systems.
I am going to explain how representations of the braid
groups endowed with a compatible symplectic form give rise to link
invariants with values in the Witt ring of the field considered. The
construction makes use of Maslov indices. In the end, using the Burau
representation, we get one invariant which "contains" many others:
signatures, Jones metaplectic invariants, and a polynomial which is
almost the one by Alexander-Conway.
A wide range of questions in discrete mathematics begin with ``How many...''. The main focus of this talk will be to discuss the use of generating functions as a means of constructing, counting, and encoding information about combinatorial classes. In particular, the question of how to construct bigger combinatorial classes using smaller ones will be discussed. Examples of classes that we can count and those where counting is not so easy will be used throughout to illustrate the concepts and ideas presented.
The theory of complex multiplication provides algorithms for obtaining elliptic curves over finite fields with a number of points known in advance, which finds applications in cryptography and primality proving. The main ingredient is the construction of Hilbert class fields of imaginary-quadratic number fields. While these are of exponential size with respect to the input, several approaches have been described that are quasi-linear in the output. I will give a self-contained overview of the algorithms and the latest record computations.
Note for Attendees
Refreshments will be served between the two talks.
In applications ranging from energy harvesting to toxin detection, nanocrystalline materials promise to yield revolutionary new technologies. The ability to produce nanomaterials with controlled morphologies remains a difficulty, however. Producing nanomaterials using self-assembly promises to be a low-cost, high-yield approach. However a fundamental understanding of growth instabilities that occur and lead to the natural formation of nanostructures and patterns is still needed. In this talk, we will discuss mathematical models and simulation methods for such problems that account for the influence of strongly anisotropic surface tension forces and elastic stresses that arise during growth of nanocrystalline materials. In the context of epitaxial growth of thin films, we demonstrate the dependence of nanostructure dynamics on the growth conditions, such as the deposition rate and the temperature.
We introduce a collection of polynomials G_N in Z[z] having the following property: the Nth cyclotomic polynomial divides G_N if and only if N cannot be represented as a sum of two odd primes. Numerical evidence suggests that, in fact, G_N is irreducible and has no roots on the unit circle. We proceed to discuss some basic properties of G_N, including giving asymptotic estimates on the size of their coefficients. At various stages, this work is joint with P. Borwein, K.K. Choi, and G. Martin.
Most tissues are hierarchically organized into lineages, which are sets of progenitor-progeny relationships where the cells differ progressively in their character due to differentiation. It is increasingly recognized that lineage progression occurs in solid tumors. In this talk, we develop a multispecies continuum model to simulate the dynamics of cell lineages in solid tumors. The model accounts for spatiotemporally varying cell proliferation and death mediated by the heterogeneous distribution of oxygen and soluble chemical factors. Together, these regulate the rates of self-renewal and differentiation of the different cells within the lineages and lead to the development of heterogenous cell distributions and formation of niche-like environments for stem cells. As demonstrated in the talk, the feedback processes are found to play a critical role in tumor progression, the development of morphological instability, and response to treatment.
The multiplicative group of units in the ring Z/nZ is one of the first groups an undergraduate student of algebra encounters. As n varies over the positive integers, these multiplicative groups form a naturally occurring family of finite abelian groups, whose structure encodes various interesting arithmetic invariants of the modulus n (for example, the number of prime factors of n). Their algebraic structure is already somewhat complicated, but the real fun begins when we want to understand the statistical distribution of these invariants. What distribution do we get, for example, if we choose positive integers "at random" and record how many prime factors they have? The answer, in a sense that analytic number theory has made rigorous, is surprisingly a Gaussian distribution.
This combination of an algebraic setting, analytic number theory questions, and probabilistic answers provides a thriving ecosystem in which to do field research (well, group research, anyway). Many of the speaker's current projects involve dissecting multiplicative groups and seeing what structures there are to discover. In this talk we take a tour of several interesting arithmetic invariants connected to the multiplicative group (mod n) and the investigation of their distribution.
Note for Attendees
Background knowledge that will help:
The quotient ring Z/nZ and its multiplicative group of units (Z/nZ)^x
The Euler phi function
The structure theorem for finite abelian groups
The Gaussian (normal) distribution
Refreshments will be served in MATH 125 at 2:45 p.m.
A 0-dimensional scheme is said to be "smoothable" if it deforms to a disjoint union of points. Determining if a given 0-dimensional scheme is smoothable seems to be quite a difficult problem in general, and I will survey some of the main results in this area of research, including some recent progress that is joint work with David Eisenbud and Mauricio Velasco. In particular, I will explain how Gale duality provides a geometric obstruction to smoothability.
This is part of our ongoing Lunch Series for Teaching & Learning.
Description: This session will feature a panel of six UBC Math instructors of various stripes (three faculty and three grad students, one of whom is teaching for the first time) who are using clickers this term or did last year. There will be a brief introduction on instructional use, driven in part by clickers in the audience, so you will be able to see a sample of the technology in action. This will not be a technical how-to, however; the bulk of the time will be devoted to hearing from the panel members about their classes:
Did the technology help to achieve instructional goals?
What were the challenges in terms of lesson design and technology use?
What was surprising?
After experiencing the pain of the tech bubble crash Sunny Wong decided to break all the rules and throw in the towel to lose the rest of his money. Instead of losing any money he makes money.
He combines all 3 markets into one. He is somewhat of a “Jack of all Markets” and a “Master of None”.But he still manages to consistently make profits every day.He also calls his trading style a business and not investing.Sunny is not a fundamental or technical trader. Sunny will now discuss his Active Neutral Trading as a Business in a nutshell for all of us to hear.
Dr. Calvin Winter CFA will introduce Sunny Wong and frame his trading in the context of the Behavioral Finance Cognitive Biases which Sunny is exploiting. Ample time will be available for questions and answers.
(Joint work with C. Procesi and M.Vergne) Let G be a compact
Lie group with Lie algebra g. Given a G-manifold M with a G equivari-
ant one form w we consider the zeroes M^{0} of the corresponding moment
map and de fine a map, called in nitesimal index, of S[g*]^{G}-modules
from the equivariant cohomology of M^{0} with compact support to the
space of invariant distributions on g*.
In the case in which G is a torus, N is a linear complex representation
of G, M = T*N with tautological one form we are going to explain
how this is used to compute the equivariant cohomology of M^{0} with
compact support using certain spaces of polynomial which appear in
approximation theory.
This colloquium will examine the broad themes of nonlinear dispersive PDE, with the nonlinear Schrodinger and wave equations as examples. We'll discuss the underlying linear theory (Strichartz estimates + conservation laws), weakly nonlinear phenomena (scattering), full nonlinear behaviour (standing waves), and instability (blowup). The emphasis will be on motivation and not proof. This talk is intended for mathematically mature non-experts.
We study sumsets A+B:={a+b: a in A, b in B} where A and B are sets of integers, and B has positive density. We show that under an equidistribution condition, A+B must be highly structured, and in particular must contain all the finite configurations from some Bohr set. This generalizes work of Renling Jin, and Bergelson, Furstenberg, and Weiss. We will see how a description of sumsets follows from a description of ergodic averages, and how embeddings of a set of integers A into compact abelian groups influence the structure of summands A+B where B is a dense set of integers. This raises questions, some new and some old.
We consider the situation in which a weak solution of the Navier-Stokes equations fails to be continuous in the strong L^2 topology at some singular time t=T. We identify a closed set S_T in space on which the L^2 norm concentrates at this time T. The famous Caffarelli, Kohn Nirenberg theorem on partial regularity gives an upper bound on the Hausdorff dimension of this set. We study microlocal properties of the Fourier transform of the solution in the cotangent bundle T*(R^3) above this set. Our main result is a lower bound on the L^2 concentration set. Namely, that L^2 concentration can only occur on subsets of T*(R^3) which are sufficiently large. An element of the proof is a new global estimate on weak solutions of the Navier-Stokes equations which
have sufficiently smooth initial data.
The chordal Schramm-Loewner evolution (SLE) is a measure on curves
connected boundary points of a domain. Schramm defined the
process for simply connected domains, but it is not immediate
how to extend the definition to multiply conneceted domains.
I will discuss an approach using the Brownian loop measure
and some work in progress showing that the measure is well
defined. The recent work uses ideas from a recent paper of
Dapeng Zhan on reversibility of whole plane SLE.
I present an inverse function theorem for differentiable maps between Banach spaces, the proof of which relies on Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C^{1}, or even Fr\'{e}chet-differentiable. I then state a inverse function theorem for differentiable maps between Fr\'{e}chet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C^{2}, or even C^{1}, or even Fr\'{e}chet-differentiable.
Note for Attendees
Please plan to attend a Special Tea to be held at 2:15 pm prior to Ivar Ekeland's colloquium. This will take place in MATH 125 and special cakes will be served.
When a thin rectangular sheet is clamped along two opposing edges and stretched, its inability to accommodate the Poisson contraction near the clamps may lead to the formation of wrinkles with crests and troughs parallel to the axis of stretch. A variational model for this phenomenon is proposed. The underlying energy functional includes bending and membranal contributions. Motivated by work of Cerda, Ravi-Chandar, and Mahadevan, the functional is minimized subject to a global constraint on the area of the mid-surface of the sheet. Analysis of a boundary-value problem for the ensuing Euler–Lagrange equation shows that wrinkled solutions exist only above a threshold of the applied stretch. A sequence of critical values of the applied stretch, each element of which corresponds to a discrete number of wrinkles, is determined. Whenever the applied stretch is sufficiently large to induce more than one wrinkle, previously proposed scaling relations for the wrinkle wavelength and root-mean-square amplitude are confirmed. Comparisons with experimental measurements and numerical results indicate that the analytical results are remarkably robust.
I state and prove an inverse function theorem between Fr\'{e}chet spaces, which does not require that the function to be inverted is C^{2}, or even C^{1}, or even Fr\'{e}chet-differentiable.
The problem of finding a combinatorial rule for computing Kronecker
coefficients i.e. the multiplicities appearing when the tensor product
of two irreducible representations of the symmetric group is
decomposed into irreducibles, is a long standing open problem in
combinatorial
representation theory.
I'll start with a crash course in Symmetric functions and give an
account of the work done on the aforementioned problem and the
motivation behind it. I'll point out the cases in which answers were
obtained and end by mentioning the special cases that I've tackled.
The Navier–Stokes-alpha-beta equation regularizes the Navier–Stokes equation by the addition of dispersive and dissipative terms. The dispersive term is proportional to the divergence of the corotational rate of the symmetric part of the velocity gradient. The dissipative term is proportional to the bi-Laplacian of the velocity. The coefficients of these terms involve factors alpha and beta, respectively, both having dimensions of length. Calculating the energy spectrum for an assembly of stretched spiral vortices reveals an inertial range where Kolmogorov's -5/3 law holds and shows that choosing beta less than alpha yields a better approximation of the inertial range of the Navier–Stokes equation. Direct numerical simulations of three-dimensional periodic turbulent flow confirm this and also show that vorticity structures behave more realistically when beta is less than alpha. However, the simulations indicate that optimal choices of alpha and beta are resolution dependent. This suggests the possibility of developing multigrid methods that capitalize on resolution dependence by using the Navier–Stokes-alpha-beta equation at coarse grid levels, with different choices of alpha and beta at each level, to accelerate convergence to solutions of the Navier–Stokes equation at the finest grid level. Results obtained from a two-dimensional spectral multigrid algorithm of this type show promise.
Note for Attendees
Refreshments and cookies will be served at 1:15pm in the PIMS Lounge downstairs near WMAX 110.
Littlewood-Paley functions are used to characterize some function spaces such as Hardy space, Sobolev space and Lipschitz space. In the last 30 years, Brownian motion has been widely used to give such characterizations in terms of Lp-norm. Recently, there has been an increasing interest in discontinuous processes, particularly in stable processes. An interesting question is if there is an analogue of the Lp characterization in the case of discontinuous processes. In this talk, first I will discuss the Dirichlet problem which forms the core of this problem and then I will talk about its solution, corresponding L-P functions, and my recent results in this topic.
To each finite group G one associates an algebraic invariant rk(G), called
the rank of G, and a topological invariant hrk(G), called the homotopy rank
of G. The number hrk(G) is defined in terms of free G-actions on products
of spheres. In this talk we define the two invariants and discuss the rank
conjecture, which states that rk(G)=hrk(G).
The speaker for UBC/UMC this week is Josh Zukewich.
Title: Evolution of cooperation in structured populations
A quick introduction to evolutionary game theory as a powerful tool to model frequency dependent selection. We'll develop a spatially structured model where individuals are represented by nodes in a graph (or network). Analysis of the model reveals simple and surprising implications for the evolution of cooperation.
I will define and discuss some basic properties of algebraic groups over a field, including the classification of the split simple (connected) algebraic groups (i.e. Dynkin diagrams). Then I will discuss G-torsors, give examples of their related algebraic structures, and define the essential dimension and essential p-dimension of algebraic groups. Finally, for some (all, if there is time) split simple groups I will survey what is known about these invariants.
Since Riemann's 1859 monograph, the study of the distribution of prime numbers has been dominated by the study of the zeros of the Riemann zeta function and Dirichlet L-functions. Although there have been ad hoc elementary proofs of some of the key results, there has been no coherent alternative approach to that of Riemann. In this talk we will introduce a new way to develop analytic number theory, without zeros, stemming from the concept of "pretentiousness". This is joint work with Soundararajan.
Note for Attendees
---A note from the speaker about preliminaries
There are not really any notes since this is an introduction to a new way of thinking about an old topic. If anything, a student should look at Davenport's book "Multiplicative Number Theory" for the best version of the classical approach, but I doubt someone will do that within the next week! I will explain things without assuming much background (from profs as well as students!)
To any polynomial over a perfect field of positive characteristic, one may associate an invariant called the F-pure threshold. This invariant, defined using the Frobenius morphism on the ambient space, can be thought of as a positive characteristic analog of the well-known log canonical threshold in characteristic zero. In this talk, we will present some examples of F-pure thresholds, and discuss the relationship between F-pure thresholds and log canonical thresholds. We also point out how these results are related to the longstanding open problem regarding the equivalence of (dense) F-pure type and log canonical singularities for hypersurfaces in complex affine space.
The modulation spaces are spaces of distributions that behave like
the Besov spaces, by essentially replacing the dilation in their
definition with a time-frequency shift (modulation).
The main goal of the talk is to convey the idea that modulation space
estimates arise as natural alternatives when estimates on other
classical function spaces fail. We will discuss this idea through two
examples: wave or Schr"odinger multipliers, and a class of bilinear
pseudodifferential operators.
Abstract: When identical information has to be delivered from a source to multiple destinations, we would like to use a transmission scheme known as multicast. Multicast delivers the information simultaneously to all receivers in a single transmission over the network, and thus, in general, uses less transmission energy and time than the traditional scheme consisting of multiple independent deliveries to different destinations. Mathematically, the network is modeled as a directed graph, and information as streams of numbers in some finite field. Often, when network resources are limited, it is beneficial to implement multicast with network coding, that is, to allow network nodes to linearly combine the incoming information streams and forward the resulting merged streams to their neighbors. We show to which extent, how, and at what cost, the information can be hidden from a wiretapper who can observe all (possibly merged) information streams on a fixed number of network edges of his choice, and knows how the information streams are linearly combined throughout the network. We are interested in perfect (unconditional) secrecy rather than encryption which relies on the limited computational power of the wiretapper.
A linear space is an incidence structure of points and lines such that
every line is on at least two points and any two distinct points are
on exactly one line. This is also known as a pairwise balanced
design. The dimension of such a structure is the maximum positive
integer D such that any D of its points generate a proper subspace. In
more detail, let's consider linear spaces on n points such that any d
points generate subspaces of size at most s.
The main result I will present is the construction of arbitrarily
large linear spaces having an upper bound on s (the largest generated
subspace size) depending only on d (the number of generating points).
A somewhat surprising consequence in extremal graph theory will be
discussed as well.
Delay is one of the most important performance measures of a communication system. In scenarios as different as mobile, ad hoc, wireless networks and content distribution networks, delay is caused by seemingly very different random phenomena, which are often statistically similar. Thus, in a wide range of applications, the delay can be reduced by following the same idea. To help understand this idea, we consider the following scenario: Imagine an agent having a piece of information that he wants to communicate to his partner. The agent knows that his partner resides in an occupied city but does not want to be seen talking with him, or with anyone else on the street for a long time. The agent and his partner have a number of friends walking in the same city, who are willing to relay small pieces of information between the secret couple. Because of that, the agent decides to split his data in small chunks which he can then inconspicuously pass to his friends. To increase the data lifetime among his friends who are moving in an adverse environment, the agent also decides to make these data chunks redundant by erasure coding, that is, by generating a larger number of chunks s.t. the original data can be recovered based on any sufficiently large subset of the redundant chunks. Assuming that all participants in this multi-agent information transfer perform simple random walks over a finite, random, regular graph, and can exchange information only when they are on the same node of the graph, we describe how coding, at the expense of introducing redundancy and processing complexity, not only increases the lifetime of the data, but also reduces the average time necessary for the transfer of information between the agent and his partner.
Multiplicative differential forms are relevant whenever considering an object with a smooth groupoid of symmetries. One can ask what is the corresponding infinitesimal object, and in fact some of the most important examples arise from this direction. The geometric structures of Hamiltonian mechanics - Poisson manifolds, Dirac structures, etc. - can be viewed as infinitesimal data which, when integrated, yield multiplicative 2-forms on Lie groupoids.
I will explore the relationship between multiplicative structures on Lie groupoids and their infinitesimal counterparts on Lie algebroids.
Let T:X \to X be a measure preserving transformation of some infinite-measure space (X,\mathcal{B},\mu) with \mu(X)=\infty.
Associated with T is a natural probability-preserving map T_* which acts on discrete countable subsets of X, with respect to the probability measure defined by Poisson processes on X. This map is called the Poisson suspension of T.
I will review some basic properties of Poisson suspensions.
Under the assumption that the transformation T is recurrent and ergodic, I will prove ergodicity of the map T \times T_*, which acts on Poisson processes with one ``marked particle''.
Given a nonzero integer d and a positive integer n we know, by Hermite's Theorem, that there exist only finitely many degree n number fields of discriminant d. It is thus natural to ask whether there are refinements of the discriminant which completely determine the isomorphism class of a number field. In this talk we will consider the integral trace form as such refinement. By using one of Bhargava's composition of cubes, we show that the integral trace form is a complete invariant for cubic fields with positive fundamental discriminant. If time allows we will discuss some further results for higher degree number fields.
Note for Attendees
Refreshments will be served between the two talks.
The boundary rigidity problem consists in determining the Riemannian metric of a compact Riemannian manifold with boundary by measuring the lengths of geodesics joining points of the boundary. The lens rigidity problem consists in determining the Riemannian metric of a compact Riemannian manifold with boundary by measuring the scattering relation or lens relation: We know the point of exit and direction of exit of a geodesic if we know its point of entrance and direction of entrance.
These two problems arise in travel time tomography in which one attempts to determine the index of refraction of a medium by measuring the travel times of waves going through the medium.
We will survey what is known about this problem and some recent results.
In 1980 A. P. Calder\'on wrote a short paper entitled "On an inverse boundary value problem". In this seminal contribution he initiated the mathematical study of the following inverse problem: Can one determine the electrical conductivity of a medium by making current and voltage measurements at the boundary of the medium? There has been substantial progress in understanding this inverse problem in the last 30 years or so.
In this lecture we will survey some of the most important developments.
The goal of this talk is to describe the sheaf model for the infinite loop space of the Thom spectrum. Since the main theorem of Galatius, Madsen, Tillmann, Weiss is that this sheaf model is homotopy equivalent to the classifying space of the cobordism category understanding this sheaf model is an important step in proving the Mumford conjecture.
We construct new smooth CY 3-folds with 1-dimensional Kaehler moduli by pfaffian method and determine their fundamental topological invariants. The existence of CY 3-folds with the computed invariants was previously conjectured by C. van Enckevort and D. van Straten. We then report mirror symmetry for these non-complete intersection CY 3-folds. We explicitly build their mirror candidates, some of which have 2 LCSLs, and check the mirror phenomenon.
Cooperation has been often studied in the framework of evolutionary game theory. Usually each player adopts a single strategy against everyone: cooperation or defection. But humans can discriminate and adopt different strategies against different opponents. In this talk I am going to present some analytical and simulational results for the case where the players can distinguish the opponents and, in the second part, I am going to talk about the extension of these ideas that has been developed jointly with prof. Christoph Hauert.
We consider the problem of existence and uniqueness of multi-soliton solutions for the L²-supercritical generalized Korteweg-de Vries equation. We recall that a multi-soliton is a solution which behaves as a sum of N solitons in large time. After a survey of existing results in the subcritical and critical cases, and also in the 1-soliton case, we will state the theorem of existence and uniqueness of an N-parameter family of N-solitons in the supercritical case. Finally, we will sketch a proof of the classification part of this theorem.
The Schramm-Loewner evolution is a one-parameter family of random growth
processes in the complex plane introduced by Oded Schramm in 1999. In the
past decade, SLE has been successfully used to describe the scaling limits
of various two-dimensional lattice models. One of the first proofs of
convergence was due to Greg Lawler, Oded Schramm, and Wendelin Werner who
gave a precise statement that the scaling limit of loop-erased random walk
is SLE with parameter 2. However, their result was only for curves up to
reparameterization. There is reason to believe that the scaling
limit of loop-erased random walk is SLE(2) with the very specific natural
time parameterization that was recently introduced by Greg Lawler and
Scott Sheffield, and further studied by Greg Lawler and Wang Zhou. I will
describe several possible choices for the parameterization of the discrete
curve that should all give the natural time parameterization in the limit,
but with the key difference being that some of these discrete time
parameterizations are easier to analyze than the others. This talk is
based on joint work in progress with Tom Alberts and Robert Masson.
Toric schemes admit a combinatorial description which, in turn, allows
one to describe sheaves of modules by certain diagram categories. I
will explain these basic constructions in some detail, and then give a
non-standard approach to constructing the derived category of a
regular toric scheme; in technical language, the derived category will
appear as the homotpy category of a colocalisation of a simple diagram
category.
The speaker for UBC/UMC this week is Professor Leah Keshet.
Title: Getting Together: Flocks and the single bird
In this talk, I will present a few examples of the
phenomena that accompany the formation and dynamics
of swarms, schools, bird flocks and other collective social groups. I
then describe some of the common questions that scientist are
interested in addressing. How do such flocks stay together?
how do individuals keep from colliding or getting too crowded?
what exactly are the individuals doing inside that flock?
I'll describe some recent work that combines mathematics and
the real world. Thanks to the work of Ryan Lukeman, my former
PhD student, we were able to answer some of these questions
for a local flock of ducks (a.k.a. surf scooters).
The quadratic reciprocity law (dating back to Gauss) is a cute result of algebra letting you decide almost instantly whether a given integer is a square modulo a prime. This theorem is famous for having about 100 different proofs, and counting. I'll explain what it really says and give one proof which is particularly elementary.
I shall present an energetic variational phase field model for multiphase incompressible flows which leads to a set of coupled nonlinear system consisting a phase equation and the Navier-Stokes equations. We shall pay particular attention to situations with large density ratios as they lead to formidable challenges in both analysis and simulation.
I shall present efficient and accurate numerical schemes for solving this coupled nonlinear system, in many case prove that they are energy stable, and show ample numerical results which not only demonstrate the effectiveness of the numerical schemes, but also validate the flexibility and robustness of the phase-field model.
I'll explain a new method for computing the structure constants according to a method implicit in an old paper of Jacques Tits. This will be an expansion of the talk I gave last Spring. Then I'll recall how this allows one to compute in reductive groups, following Chevalley, Carter, and Cohen-Murray-Taylor.
Special Seminar Abstract: Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality".
We shall construct a new sparse spectral method for high-dimensional problems, and present, in particular, rigorous error estimates as well as efficient numerical algorithms for elliptic equations.
We shall also propose a new weighted weak formulation for the Fokker-Planck equation of FENE dumbbell model, and prove its well-posedness in weighted Sobolev spaces. Based on the new
formulation, we are able to design simple, efficient, and unconditionally stable semi-implicit Fourier-Jacobi schemes for the Fokker-Planck equation of the FENE dumbbell model.
It is hoped that the combination of the two new approaches would make it possible to directly simulate the five or six dimensional Navier-Stokes Fokker-Planck system.
MATX 1100. Special PIMS Distinguished Lecture --CANCELED
Fri 26 Nov 2010, 3:00pm-4:00pm
Abstract
At the International Congress of Mathematicians held in Hyderabad, India this past August, Louis Nirenberg, from the Courant Institute for the Mathematical Sciences at New York University, was awarded the inaugural Chern Medal for his role in the formulation of the modern theory of non-linear elliptic partial differential equations and formentoring numerous students and post-docs in this area.
In addition to this prestigious prize, Professor Nirenberg has received many
other awards and honours, including: the American Mathematical Society’s Bôcher Prize in 1959, the Jeffrey-Williams Prize of the Canadian Mathematical Society in 1987, the Steele Prize of the AMS in 1994 for Lifetime Achievement, the Crafoord Prize in 1982and the U.S. National Medal of Science in 1995. Please seewww.icm2010.in/prize-winners-2010/chern-medal-louis-nirenberg for a full citation.
Professor Nirenberg was born in Hamilton, Ontario and obtained his undergraduate degree from McGill University before emigrating to the United States. He has shown a longstanding interest in the Canadian mathematical community, mentoring and supporting many of our colleagues. This fact and his enormous contributions tothe mathematical sciences will be recognized by the University of British Columbia by awarding him an honorary degree during its 2010 Fall Congregation (November 24-26, 2010). On November 26 he will deliver a special PIMS Distinguished Lecture at 3pm at UBC.
Everyone is invited to attend and celebrate Louis Nirenberg's wonderful career in mathematics.
After talking about how a sheaf of categories gives rise to a topological category, I will finish the proof of the main theorem of Galatius, Madsen, Weiss and Tillmann which states that the classifying space of the cobordism category is weakly homotopy equivalent to the infinite loop space of the Thom spectrum.
Log canonical thresholds are invariants of singularities that play an important role in birational geometry. After an introduction to these invariants, I will describe recent progress on a conjecture of Shokurov predicting the Ascending Chain Condition for such invariants in any fixed dimension. This is based on joint work with Tommaso de Fernex and Lawrence Ein.
Abstract: Mark has been experimenting with the Livescribe Echo, a pen that records what you write along with audio commentary you may make while writing. By producing "pencasts" using this pen, he has been able to post responses to students' questions and solutions to problems that include discussion in the natural way that we do when working with students during our office hours or presenting ideas in class. The technology is simple to use: you just write and speak as you normally would. There are many possibilities for using this technology in mathematics classes.
Invariants of singularities are defined in birational geometry via divisorial valuations, and are computed by resolution of singularities. In positive characteristic, one defines similar invariants via the action of the Frobenius morphism. The talk will give an overview of the known results and conjectures relating the two sets of invariants.
This presentation starts with a quick epidemiological overview that puts emphasis on neglected diseases and health disparities in the context of developing and/or poor nations. The primary emphasis is however on Tuberculosis (TB). A review of mathematical models and results on issues related to the transmission dynamics and control of TB, under various degrees of complexity is provided. The presentation continues with a discussion on the relationship between urban growth and TB decline in the USA. The observations are supported using demographic and TB epidemiological time series that capture the observed patterns of disease prevalence in growing urban centers in the States of Massachusetts and a large aggregate of cities in the USA, over a long window in time.
We will show that the probability that a simple random walk will cover a finite, bounded degree graph in linear time is exponentially small.
More precisely, for every D and C, there exists a=a(D,C)>0 such that for any graph G, with n vertices and maximal degree D, the probability that a simple random walk, started anywhere in G, will visit every vertex of G in its first Cn steps is at most exp(-an).
The Uniform Infinite Planar Quadrangulation (UIPQ) is an infinite planar graph which is the local limit of uniform planar quadrangulations of size n. It has been introduced by Krikun in 2005 following a pioneer work of Angel and Schramm on triangulations. The geometry of this object is particularly interesting, for example the volume growth exponent is 4. In this talk we introduce a new construction of the UIPQ and deduce several geometric properties. For example an essential uniqueness of infinite geodesics reminiscent of the work of Le Gall on the Brownian Map.
joint work in progress with L. Ménard and G. Miermont.
The local Langlands correspondence seeks to parameterize the smooth irreducible representations of a reductive group G over a local field F in terms of Langlands parameters, objects closely related to representations of the Galois group of F. Each parameter is supposed to correspond to a finite set of representations of G, called an L-packet. The broad principle of Langlands functoriality suggests that often such an L-packet \Pi_G transfers to an L-packet \Pi_H on an endoscopic group H. The transfer is encoded in identities between the characters of representations in \Pi_G and those in \Pi_H. Endoscopic character identities play an important role not only in representation theory, but also in number theory, via the stabilization of the Arthur-Selberg trace formula.
In this talk, we will motivate the problem that the theory of endoscopy addresses, and then formulate the precise statement of the endoscopic character identities, after recalling the necessary notions from the local Langlands correspondence. If time permits, we will then discuss their proof for the depth-zero supercuspidal L-packets recently constructed by DeBacker-Reeder. The main technical tool involved is Waldspurger's work on endoscopy for p-adic Lie algebras, which ultimately rests on the fundamental lemma.
Note for Attendees
Refreshments will be served between the two talks.
Let f \in S_2(\Gamma_0(N)) be a modular newform of weight 2 and level N. Then given a prime ideal \lambda , in certain favourable cases, we can say if there is a modular newform g of weight 2 at level M such f \equiv g \pmod \lambda . When M|N this is a level lowering result, while when N |M this is a level raising result. In this talk I will discuss what happens when \lambda is not a prime ideal, but rather a power of a prime ideal.
This is a seminar in the WENTS ("West End Number Theory Seminar") series.
Abstract:
In this talk we shall survey the theory of complex multiplication from its inception to the present and indicate some of the current outstanding challenges. The talk is in "colloquium style" and as such should be accessible to mathematicians that are not necessarily number theorists.
A major branch of modern combinatorics, usually called Ramsey theory, studies properties of structures which are preserved under partitions. Its guiding philosophy can be neatly summarized by the statement, "Complete disorder is impossible". In this talk I will survey what is known and what is still unknown from this perspective for solution sets of linear equations over the integers.
In the first half of the talk we will provide a global picture of how all the pieces (that we constructed throughout this series) fit together to prove Mumford's conjecture following the approach in Galatius, Madsen, Tillman and Weiss. In the second half we outline the proof of the last piece of information needed to complete the proof.
The first of a series of lectures on Severi-Brauer varieties. Starting from their definition and the geometry (line bundles, etc) they hold. And more importantly, the connection between Central Simple Algebra and Brauer-Severi varieties, such as how to relate the ground field, the Brauer group function field of the variety and the Picard group of the variety.
We show that the catenoid is the unique surface of least area (suitably understood) within a geometrically natural class of minimal surfaces. The proof relies on a techniques involving the Weierstrass representation used by Osserman and Schiffer to show the sharp isoperimetric inequality for minimal annuli. An alternate approach that avoids the Weierstrass representation will also be discussed. This latter approach depends on a conjectural sharp eigenvalue estimate for a geometric operater and has interesting connections with spectral theory. This is joint work with J. Bernstein
This is the weekly study seminar MATH 620A. If you are interested, please come to this meeting (or send an e-mail to Julia Gordon,
if you have not done so already). The future meetings of this seminar will not be announced on this page.
In 1960, Feit and Fine were interested in the question posed by the title and to answer it, they found a beautiful formula for the number of pairs of commuting n by n matrices with entries in the field F_q. Their method amounted to finding a stratification of the variety of commuting pairs of matrices into strata each of which is isomorphic to an affine space (of various dimensions). Consequently, their computation can be interpreted as giving a formula for the motivic class of the commuting variety, that is, its class in the Grothendieck group of varieties. We give a simple, new proof of their formula and we generalize it to various other settings. This is joint work with Andrew Morrison.
The second of a series of lectures on Severi-Brauer varieties. Starting from their definition and the geometry (line bundles, etc) they hold. And more importantly, the connection between Central Simple Algebra and Brauer-Severi varieties, such as how to relate the ground field, the Brauer group function field of the variety and the Picard group of the variety.
Consider percolation on the Hamming cube {0,1}^n at the critical probability p_c
(at which the expected cluster size is 2^{n/3}). It is known that if
p=p_c(1+O(2^{-n/3}), then the largest component is of size roughly 2^{2n/3} with
high probability and that this random variable is not-concentrated. We show that for any
sequence eps(n) such that eps(n)>>2^{-n/3} and eps(n)=o(1) percolation at
p_c(1+eps(n)) yields a largest cluster of size (2+o(1))eps(n)2^n.
This completes the description of the phase transition on the Hamming cube and settles a
conjecture of Borgs, Chayes, van der Hofstad, Slade and Spencer.
Our approach is to show that large percolation clusters have inherent randomness
causing them to clump together and form a giant cluster. The behavior
of the random walker on the Hamming cube plays a key role in the proofs of such statements.
We study the probability that the origin is connected to the sphere of radius R in critical percolation in high dimensions, namely when the dimension d is a large fixed constant, or when d>6 and the lattice is sufficiently spread out. I will present highlights of the proof that this probability decays like R^{-2}. No knowledge of percolation will be assumed.
Professor George Bluman will be kicking off UBC/UMC this term.
Title: Dimensional Analysis, Modelling and Invariance
In a new field of research, for a given quantity of interest, called an unknown or "dependent variable", the first step of an expert is to determine the essential independent quantities that it can depend upon (including constants/parameters and variables ("independent variables") as well as the fundamental dimensions of all quantities. Often, the expert needs to conduct intelligent experiments to determine quantities and their dimensions. The only mathematics that can be used at this stage is "dimensional analysis".
The application of dimensional analysis reduces the number of essential independent quantities. For a real problem, any formula (equation, differential equation, etc.) relating quantities is invariant under any change of systems of units where each fundamental dimension is scaled by a positive factor. It follows that each formula can be made dimensionless. As a consequence, the expert can design proper (and safe) model experiments at more reasonable cost to predict the values of a quantity of interest in terms of relevant values of essential independent quantities in an actual situation.
The talk will give the mathematical background of dimensional analysis. Applications will include a proof of the Pythagoras Theorem and the deduction of the amount of energy released in the first atomic explosion of 1945 from motion picture data released in 1947 even though the amount of energy released was classified "top secret" until 1950! The talk will be accessible to all students and should serve as a motivation to learn different topics in linear algebra.
In this talk we present an algebraic context for knot theory. Knotted
trivalent graphs (KTGs) along with standard operations defined on them
form a finitely presented algebraic structure which includes knots, and in
which many topological knot properties are defineable using simple
formulas. Thus, a homomorphic invariant of KTGs provides an algebraic way
to study knots. We present a construction for such an invariant: the
starting point is extending the Kontsevich integral of knots to KTGs. This
was first done in a series of papers by Le, Murakami, Murakami and Ohtsuki
in the late 90's using the theory of associators. We present an elementary
construction building on Kontsevich's original definition, and discuss the
homomorphic properties of the invariant, which, as it turns out,
intertwines all the standard KTG operations except for one, called the
edge unzip. We prove that in fact no universal finite type invariant of
KTGs can intertwine all the standard operations at once, and present an
alternative construction of the space of KTGs on which a homomorphic
universal finite type invariant exists. This space retains all the good
properties of the original KTGs: it is finitely presented, includes knots,
and is closely related to Drinfel'd associators. Partly joint work with
Dror Bar-Natan.
How do simple local interactions combine to produce complex large-scale structure and patterns? The abelian sandpile model provides a beautiful test case. I'll discuss a pair of conjectures about the scale invariance and dimensional reduction of the patterns formed. A new perspective on sandpiles views them as free boundary problems for the discrete Laplacian with an extra integrality condition. Joint work with Anne Fey and Yuval Peres.
Let $K$ be a field of characteristic $p>0$ and let $f(t_1,\ldots ,t_d)$ be a power series in $d$ variables with coefficients in $K$. We discuss a recent generalization of both Derksen's recent analogue of the Skolem-Mahler-Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices $(n_1,\ldots ,n_d)\in \mathbb{N}^d$ for which the coefficient of $t_1^{n_1}\cdots t_d^{n_d}$ in $f(t_1,\ldots ,t_d)$ is zero is generated by a finite-state automaton that accepts the base $p$ expansions of $d$-tuples of natural numbers as input. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to $S$-unit equations and more generally to the Mordell--Lang Theorem over fields of positive characteristic. (joint with Boris Adamczewski)
Note for Attendees
Refreshments will be served between the two talks.
Abstract: In a previous lecture I waved my hands at the subject of period integrals for automorphic forms on SL_2. If that lecture had a point, it was that the phenomenon of non-vanishing of quadratic twists of L-functions was somehow related to the non-vanishing of the symmetric square L-function. In this lecture, I will try to substantiate that claim by describing the relative trace formula of Jacquet, and by sketching the general shape of this formula for the group SL_2. In the end, I'll try to explain how one might get information about quadratic twists directly from the symmetric square if only one had a stable form of Jacquet's formula, and if one could explicitly evaluate the stable and kappa-stable portions.
A substantial part of extremal combinatorics studies relations existing between densities with which given combinatorial structures (fixed size ``templates'') may appear in unknown (and presumably very large) structures of the same type. Using basic tools and concepts from algebra, analysis and measure theory, we develop a general framework that allows to treat all problems of this sort in an uniform way and reveal mathematical structure that is common for most known arguments in the area. The backbone of this structure is made by commutative algebras defined in terms of finite models of the associated first-order theory.
In this talk I will give a general impression of how things work in this framework, and we will pay a special attention to concrete applications of our methods.
Start with n particles at the origin in the square grid Z2, and let each particle in turn perform simple random walk until reaching an unoccupied site. Lawler, Bramson and Griffeath proved that with high probability the resulting random set of n occupied sites is close to a disk. We show that its fluctuations from circularity are, with high probability, at most logarithmic in the radius of the disk, answering a question posed by Lawler in 1995. These logarithmic fluctuations were predicted numerically by chemical physicists in the 1980's. Joint work with David Jerison and Scott Sheffield.
This will be a survey of what we do and don't know about what happens to potential Mathematics, Science, and Engineering majors as they make the transition from high school to university mathematics, highlighting where the need for more information is most pressing and how university programs are adapting to meet the needs of entering students.
This talk is presented by the CWSEI and the Mathematics Department.
A quantum cluster algebra is a subalgebra of an ambient skew field of rational functions in finitely many indeterminates. The quantum cluster algebra is generated by a (usually infinite) recursively defined collection called the cluster variables. Explicit expressions for the cluster variables are difficult to compute on their own as the recursion describing them involves division inside this skew field. In this talk I will describe the rank 2 cluster variables explicitly by relating them to varieties associated to valued representations of a quiver with 2 vertices. I will also indicate to what extent the theory. I present is applicable to higher rank quantum cluster algebras.
Informally, the essential dimension of a finite group is the minimal number of parameters required to describe any of its actions. It has connections to Galois cohomology and several open problems in algebra. I will discuss how one can use techniques from birational geometry to compute this invariant and indicate some of its applications to the Noether Problem, inverse Galois theory, and the simplification of polynomials.
Building on I. Dolgachev and V. Iskovskikh's recent work classifying finite subgroups of the plane Cremona group, I will classify all finite groups of essential dimension 2. In addition, I show that the symmetric group of degree 7 has essential dimension 4 using Yu. Prokhorov's classification of all finite simple groups with faithful actions on rationally connected threefolds.
We re-describe what a Severi-Brauer variety is by displaying a particularly good embedding into projective space. We also present a theorem by Chatelet, that uniquely describes the projective space among Severi-Brauer varieties.
Efforts to control the Mountain Pine Beetle infestation in British Columbia and Alberta include large-scale landscape manipulations such as clearcutting, and cost-intensive techniques such as green attack tree removal. Unfortunately, it is unclear just how effective these techniques are in practice. In order to determine and predict the effectiveness of various management strategies, we need to understand how MPB disperse through heterogeneous habitat, where heterogeneity is measured in terms of species composition and tree density on the landscape. In this talk I will present a spatially-explicit hybrid model for the Mountain Pine Beetle (MPB) dispersal and reproduction. The model is composed of reaction-diffusion-chemotaxis PDEs for the beetle flight period and discrete equations for the overwintering stage. Forest management activities are also included in the model. I will discuss the formation of beetle attack patterns and the impacts of management in the PDE model.
Introduced in 1963, the Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. I will survey a number of new results on the mixing of the dynamics at high, critical and low temperatures. At high temperatures the dynamics exhibits the cutoff phenomena, a sharp transition in the convergence to equilibrium. We show a polynomial upper bound on the mixing time at the critical temperature establishing the conjectured critical slowdown behaviour. Finally at low temperatures with all plus boundary conditions we prove a quasi-polynomial time upper bound on the mixing time.
Joint work with Eyal Lubetzky, Fabio Martinelli, and Fabio Toninelli.
In this talk I will introduce some of the combinatorics
behind affine permutations. These are a generalization of classical
permutations that, when viewed as a reflection group, amount to adding
an affine reflection to the set of generators (hence the name). We
now end up with an infinite group, however, many of the same results
from classical permutations still hold. Classifying families of
permutations in terms of the patterns they avoid has been studied for
a long time. However, applying pattern avoidance to affine
permutations is fairly new. I will discuss pattern avoidance, and in
particular, the role it plays in the geometry of the corresponding
affine Schubert varieties. Part of this talk is joint work with Sara
Billey.
The reconstruction problem on the tree concerns the propagation of information in Markov processes on trees and has been studied in probability, statistical physics, computational biology, information theory and theoretical computer science. I will discuss progress in establishing thresholds for the reconstruction problem and give an overview of its applications to phylogenetic reconstruction, mixing times of Markov chains, random constraint satisfaction problems and the computational complexity of counting problems.
I will first give a gentle introduction to LS category and
then review some of the progress that has been made over the last 10
years in connection with Iwase's counterexample to Ganea's conjecture.
This will include many problems that are still open. At the end I will
describe some recent work with Rodriguez.
This talk is directed to non-specialists and students. The Metropolis is a very famous algorithm which can used to simulate random variables. The Ising model is a model coming from statistical physics. After defining the notions needed, I will present the Metropolis algorithm and apply it to the simulation of the Ising model.
Abstract: The Langlands correspondence relates global Galois
representations with automorphic representations; the local
correspondence works at each prime. For any reductive group $G$ over
a local field $K$ we construct a complex reductive group $^LG$. For
any homomorphism from the Galois group of $K$ to $^LG$ (called a
Langlands parameter) we then construct a set of representations of
$G(K)$ (called an L-packet). I will make these constructions explicit
in the case that the Langlands parameter is discrete, tamely ramified
and regular and that $G$ is the unitary group associated to a tame
extension of
$K$.
We will prove that solutions to the defocusing energy-critical Schrodinger equations are global in the hyperbolic space H^3. The relevance of the energy-critical case is that in this case, one needs to understand how to take into account the scaling limits of the equation. In particular, one needs to see how to connect solutions to the corresponding equation on a Euclidian space to solutions of the original equation which concentrate as they evolve. To try and understand the influence of the geometry, we will also look at some results on other spaces in the other directions (the volume of balls grows slowly as the radius goes to infinity). This is a joint work with A. Ionescu and G. Staffilani.
Motivated by Hochman's notion of subdynamics of a Z^d subshift, we define and examine projective subdynamics of Z^d shifts of finite type (SFTs) where we restrict not only the action but also the phase space. In analogy with the notion of stable and unstable limit sets in cellular automata we distinguish between stable and unstable projective subdynamics.
First we review the classification of Z sofic shifts which can (not) appear as projective subdynamics of Z^2 (Z^d) SFTs both in the stable and unstable regime - these are results obtained jointly with Ronnie Pavlov.
In a second part of the talk we present results on the projective subdynamics of Z^d SFTs with some uniform mixing condition. In particular there is a compatibility condition assuring the projective subdynamics of Z^d SFTs has to be sofic and if time permits we explain a construction that allows to realize any mixing Z sofic as stable projective subdynamics of some strongly irreducible Z^2 (Z^d) SFT.
Dispersive equations are evolution equations in which different components move with different velocities. This way the solution is mixed and dispersed without being damped. Examples include the wave equation, Schrodinger equation, Beam equation...
Focusing on the fourth order case, we will show how one can understand and control the global dynamics of some nonlinear equations when there is a lot of volume to disperse, especially in the critical case, which corresponds to situations where there is no preferred scale.
I will explain an interpretation of Illusie's results on the deformation theory of commutative rings in terms of the cohomological classification of torsors and gerbes. Then I'll show how this point of view can be used to solve some other deformation problems. I'll also indicate some deformation problems that I don't know how to approach this way.
This will be an expository talk on some modern variants of the Hilbert transform. The main viewpoint will be from an L^p-theory perspective, but connections with other fields will also be discussed. I will focus on the double Hilbert transform along polynomial surfaces in \mathbb{R}^3, and outline the proof of a result by Carbery, Wainger and Wright that states such operators are bounded on all L^p, 1 < p < \infty, if and only if the polynomial surface is "well-behaved" enough with respect to its Newton diagram.
This is a first talk of a series about Hochschild (co)homology and cyclic (co)homology.
We will start with the basic definitions and later we will see some applications of these
theories to stringy topology
We define the Picard group of a variety using (Weil) divisors and show that the Picard group of projective space P^n is the integers. We also describe the two generators of the Picard group of P^n as line bundles.
The real trees form a class of metric spaces that extends the class of
trees with edge lengths by allowing behavior such as infinite total
edge length and vertices with infinite branching degree. We use
Dirichlet form methods to construct Brownian motion on a given locally
compact R-tree equipped with a Radon measure. We then characterize
recurrence versus transience.
This is joint work with Anita Winter and Michael Eckhoff.
Humans are currently jeopardizing the other species in life's fabric and potentially our own future due to our overuse of common resources. Over the last two decades, a large effort has focused on trying to persuade individuals to consume differently. These conservation efforts largely appeal to guilt and an individual's willingness to do the right thing. What about the role of shame in solving the tragedy of the commons? I will explore the differences between guilt and shame and then present results from a recent public goods experiment conducted with Christoph Hauert and others that tests the effects of shame on cooperation. I will also examine our findings in the context of shame's real world applications and concerns.
In modern language, Fermat's Descent Infini establishes that an elliptic curve has a Mordell-Weil group of rank 0. Since then, the method has been generalized to provide an upper bound on the rank of any elliptic curve and further work also allows the analysis of the Mordell-Weil group of Jacobians of many hyperelliptic curves. Reformulating work of Schaefer, we present a general framework, in principle applicable to any curve, which allows us, under certain technical conditions, to provide an upper bound on the rank of the Jacobian of any curve. In particular, we have been able to compute some rank bounds on Jacobians of smooth plane quartic curves. This is joint work with Bjorn Poonen and Michael Stoll.
Note for Attendees
Refreshments will be served between the two talks.
Piecewise isometries are a class of mappings of low complexity
and a test case for discontinuous dynamics. We analyze the dynamical
behaviour of a simple family of piecewise isometries and use methods from
uniform distribution to give a quantify the behaviour.
The Skyrme model is one of the important nonlinear sigma models in quantum field theory. In this talk I will report some recent progress on he dynamics of Skyrmions, focusing on the (3+1) space-time case.
The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. By using the dispersive Klein-Gordon effect, Guo first constructed a global smooth irrotational solution in the three dimensional case. It has been conjectured that same results should hold in the two-dimensional case. The main difficulty in 2D comes from the slow dispersion of the linear flow and certain nonlocal resonant obstructions in the nonlinearity. I will discuss a new method to overcome these difficulties and construct smooth global solutions for the 2D Euler-Poisson system.
We study the category of sheaves on the étale site of a scheme. In particular we show an equivalence between sheaves on (Spec(k))_ét and the category of discrete Gal(k_sep/k)-module.
I will describe recent work on motivic DT invariants for 3-manifolds, which are expected to provide a refinement of Chern-Simons theory. The conclusion will be that these should be possible to define and work with, but there will be some interesting problems along the way. There will be a discussion of the problem of upgrading the description of the moduli space of flat connections as a critical locus to the problem of describing the fundamental group algebra of a 3-fold as a "noncommutative critical locus," including a result regarding topological obstructions for this problem. I will also address the question of whether a motivic DT invariant may be expected to pick up a finer invariant of 3-manifolds than just the fundamental group.
I will present a time-discretization algorithm which allows to build Leray weak solutions of the Navier-Stokes equations. Such algorithm has a genuine variational structure and has been inspired by the study of gradient flows. From a collaboration with S.Mosconi.
We define the Picard group of a variety using (Weil) divisors and show that the Picard group of projective space P^n is the integers. We also describe the two generators of the Picard group of P^n as line bundles. (A continuation of last week's talk)
Aim of the talk is to make a survey on some recent results concerning analysis over spaces with Ricci curvature bounded from below. I will show that the heat flow in such setting can be equivalently built either as gradient flow of the natural Dirichlet energy in L^2 or as gradient flow if the relative entropy in the Wasserstein space. I will also show how such identification can lead to interesting analytic and geometric insights on the structures of the spaces themselves. From a collaboration with L.Ambrosio and G.Savare'.
appeared along with Jannsen's theorem that the category of motives is abelian semi-simple if and only if the equivalence relation is numerical equivalence. The proof of this theorem is beautiful, simple and short. (The main tool is the double-centralizer theorem.) I will try to do it justice in my lectures.
An edge-weighting vertex colouring of a (di)graph is an edge-weight
assignment such that the accumulated weights at the vertices yields
a proper vertex colouring. If such an assignment from a set S exists,
we say the graph is S-weight colourable. Using the Combinatorial
Nullstellensatz and a classical theorem of Schur, we show that every
digraph is S-weight colourable for any set S of size 2. It is
conjectured that every graph with no isolated edge is {1,2,3}-weight
colourable. We explore the problem of classifying those graphs which
are {1,2}-weight colourable.
This is joint work with Reza Naserasr, Mike Newman, Ben Seamone,
and Brett Stevens.
As part of Smale's program for smooth dynamics, David Ruelle gave a definition of a Smale space as (roughly) a topological dynamical system which has a local product structure of contracting and expanding directions for the dynamics. A special case is certain symbolic dynamical systems called shifts of finite type. In the late 1970's, Wolfgang Krieger, motivated by ideas from C*-algebra theory and K-theory, provided a beautiful algebraic invariant for shifts of finite type. The aim of this talk is to show how this invariant may be extended to the class of all Smale spaces as a kind of homology theory which provides a Lefschetz formula. Such a theory was conjectured by Bowen. (I will attempt to define and give examples of all the dynamical concepts: Smale space, shifts of finite type, etc.)
A type A_{n-1} root polytope is the convex hull in R^n of the origin and a subset of the points e_i-e_j, 1\leq i< j \leq n. A collection of triangulations of these polytopes can be described by reduced forms of monomials in an algebra generated by n^2 variables x_{ij}, for 1\leq i< j \leq n. In a closely related noncommutative algebra, the reduced forms of monomials are unique, and correspond to shellable triangulations whose simplices are indexed by noncrossing alternating trees. Using these triangulations we compute Ehrhart polynomials of a family of root polytopes. We extend the above results to more general families of polytopes and algebras of types C_n and D_n. Special cases of our results prove several conjectures of Kirillov regarding these algebras.
While in the theory of one-dimensional subshifts we have a fairly complete picture about when a factor map between two shifts of finite type exist. The situation for Z^d shifts of finite type is much more complicated and so far even the case of factoring onto the full Z^d shift has only a partial answer (which we will present in the talk).
Moreover we will introduce a class of Z^d matrix shifts whose entropy is constructed to be exactly log N and which come with a natural factor map onto the corresponding full N shift. We will give some examples of those equal-entropy extensions of Z^d full shifts realizing various mixing and periodic point conditions.
This will be a rather informal talk on some things I am working on and some related open questions.
Branched polymers are certain configurations of nonoverlapping disks in the plane. In 2003 Brydges and Imbie discovered some remarkable formulas for the volumes of configuration spaces of branched polymers. These formulas mysteriously involve combinatorial numbers like (n-1)!. We introduce branched polymers arising from any central hyperplane arrangement A and express the volume of the resulting configuration space through the characteristic polynomial of A. This is joint work with Alex Postnikov.
We describe Galois covers in the context of the étale site of a scheme, seen as a generalization of Galois field extensions. In particular, we use this idea to show that the étale cohomology of a field agrees with the standard galois cohomology.
The Hilbert scheme of projective space is a fundamental moduli space in algebraic geometry. The naive generalization of the Hilbert scheme can fail to exist for some spaces of interest, however. D. Rydh and I have generalized the Hilbert scheme, to the Hilbert stack, and have shown that the Hilbert stack is always algebraic. I will describe the Hilbert stack and some of the ideas behind the proof.
The one-factorizations of the complete 3-uniform hypergraph with 9 vertices are classified by means of an exhaustive computer search. It is shown that the number of isomorphism classes of such one-factorizations is 103000.
We're pleased to have Bud Homsy back to give a UBC/UMC talk.
Title: Viscous Fingering in Porous Media: Modelling Unstable Growth Processes
Q: What do oil production, fractal aggregates, electrochemical deposition, and lightening have in common?
A: The underlying mathematics, of course! This talk will focus on modeling unstable flow in porous media, which important in enhanced oil production.It is also a good example of unstable growth processes and of free boundary problems in mathematics.I’ll start by reviewing the basic partial differential equations of fluid flow in porous materials and then discuss the stability analysis of the uniform flow state, with some emphasis on physical mechanisms.Then I’ll discuss the nonlinear regime of this instability which is only accessible through numerical simulation.This will help understand the connection between this instability and other unstable growth processes.
Determinantal (also called fermion) point processes are point processes (on discrete or continuous spaces) whose correlation functions are given by determinants of matrices coming from certain kernel functions. In this talk we start with a couple of examples where DPP's appear. Focusing on the discrete DPP's, we discuss the Gibbianness of them. We benefit from this property to construct some dynamics that leave a DPP invariant. We will typically construct Glauber and Kawasaki dynamics for DPP's.
The well understood phenomenon of Anderson localization says (in its dynamical formulation) that adding random fluctuations to the potential of a Schrodinger operator will lead to the absence of wave transport for the solution of the time-dependent Schrodinger equation. Several years ago it was argued by Burrell and Osborne that a corresponding phenomenon should hold in quantum spin systems. As an example they used the xy-spin chain to show on the physical level of rigor that the introduction of disorder will lead to zero-velocity Lieb-Robinson bounds. We will show
how recent results on Anderson localization can be used to make this result rigorous and, in fact, to improve on the conclusions reached by Burrell and Osborne.
There is a general principle for algebraic dynamical systems
that the growth rate of periodic points should be the entropy.
This has to be suitably interpreted, and I will formulate a
conjecture for which there are no known counterexamples. For
a single toral automorphism this is equivalent to a deep
result in diophantine analysis. For several commuting group
automorphisms the corresponding diophantine result is not
known, but I will describe recent work with Schmidt and
Verbitskiy using homoclinic points which provides an
alternative approach in many cases.
Let x be an algebraic number and let M(x) denote its Mahler measure. If x = x1...xN the t-metric
Mahler measure Mt(x) is a convenient way to study the smallest possible values of M(xn) in
terms of x. In joint work with J. Jankauskas, we resolve an earlier conjecture
regarding Mt(x) for rational x. This result suggests a generalization to higher degree x,
which turns out, however, to be false. We provide an infinite family of quadratic
counterexamples and discuss how the conjecture should be modified.
Consider the negative Dirichlet or Neumann Laplacian on a square. Add a potential perturbation
which is supported on a small disk. How should the potential be placed in the square in order
to minimize the lowest eigenvalue of the resulting Schrodinger operator? The answer to this
question for the case of Neumann conditions is very different from the answer for the
Dirichlet case. In particular, for the Neumann case the answer is independent of the
sign of the potential. We will discuss how the solution of this problem ultimately
led to a proof of localization near the spectral boundary of the random displacement
model. The latter is a model for a random Schrodinger operator which is used to
model structural disorder in a crystal. A proof of localization for this model was long
considered challenging due to the non-monotone dependence of the operator on the random parameters.
We continue to describe sheaves of abelian groups on an arbitrary étale site. In particular, we prove that the category has enough injective; allowing us to define étale sheaf cohomology.
As in most cohomology theories over a topological space, calculations can be tricky. To solve this problem, we talk about Cech cohomology on an étale site, and show the equivalence of this cohomology to the étale cohomology whenever the base space satisfies some mild conditions.
I will talk about the sheaf associated to a group scheme, focusing on the case of the multiplicative group. I will discuss connections between its cohomology and the Picard and Brauer groups of a scheme.
This is the first of a series of talks in which, subject to time and interests, we will discuss some or all of the following:
-Definitions and basic results from the theory of Polynomial Identity (PI) rings.
-Major results from PI-theory: The Amitsur-Levitzki Theorem, Kaplansky’s Theorem, Posner’s Theorem.
-Construction of the ring of generic matrices and the universal division algebra.
-The centre of the universal division algebra.
-“Universal” properties of the universal division algebra: Galois subfields and rational specialization.
-The universal division algebra is an object of maximal essential dimension.
We continue on with the study of H^1(X_et, G) and its interpretation the set of principal homogenous spaces for G, when G is a sheaf of groups on X_et.
This is the second of a series of talks in which we will discuss some of the following:
-Major results from PI-theory: The Amitsur-Levitzki Theorem, Kaplansky’s Theorem, Posner’s Theorem.
-Construction of the ring of generic matrices and the universal division algebra.
-The centre of the universal division algebra.
-“Universal” properties of the universal division algebra: Galois subfields and rational specialization.
-The universal division algebra is an object of maximal essential dimension.
Let G be a compact connected Lie group act on a topological space X
in such a way that all isotropy subgroups are connected and of maximal
rank. In this talk we provide a criterion to determine when K_{G}^{*}(X) is
free over the representation ring R(G).
The Mayer expansion is a power series expansion that has a central
place in statistical mechanics. It is also full of combinatorial miracles that
relate it to graphs, forests and branched polymers. I will discuss the background,
the results, and some open problems.
Do your students (or other people you meet) say anything about math or learning math that makes you cringe?
Student perceptions of mathematics play a role in their motivation and approaches to learning in their math courses. Last term, we adapted an existing survey for Physics (*) and surveyed students at the beginning and end of a range of Math courses (including several first-year calculus courses). This has allowed us to assess student attitudes and perceptions, and to track how they shift over time.
In this Lunch Series, we will present our development of the survey and some of these initial results. We would also like to gather input from members of the department about the content of the survey: what perceptions or attitudes do mathematicians have about their own subject, and which would you hope students develop as they pursue their undergraduate degree? We hope to see you there for discussion and pizza.
Entropy" is a key notion in the study in of dynamical systems. This quantity reflects the "uncertainty", or "randomness" of a system. Subshifts are topological dynamical systems whose elements are sequences over a given finite alphabet. A translation-invariant measure on a subshift corresponds to a finite-valued stationary stochastic process. Measures obtaining maximal entropy are in some sense "most random" or "most uniform" among those with a given support. In this talk, I will present older and newer results of various authors regarding measures of maximal entropy.
Background:
The Canadian mathematics and statistics communities are engaged in a long range planning (LRP) exercise for 2012-2017/22. This exercise is funded by NSERC, and draws on the expertise of the societies, institutes and the communities of researchers in mathematics and statistics. The site longrangeplan.ca serves as the primary repository for information about the development of the plan.
Members of the steering committee are visiting universities to as part of their community consultations.
In this capacity, Rachel Kuske and Alejandro Adem will give a short presentation on the process, and the progress to date, which will be followed by general discussion.
A call for submission of written documents has also been issued, and we welcome input from interested parties. More details and guidelines for submission are provided on the home page of the website
The call for abstracts for Frontiers in Biophysics 2011 is now open! Please send your name, talk title, abstract, and affiliation (UBC, SFU, etc) to frontiersinbiophysics2011@gmail.com. This is a one-day conference highlighting interdisciplinary research within the areas of biophysics and computational and mathematical biology in the greater Vancouver area. Participants from undergraduates to emeritus are welcome and encouraged to attend and/or present a talk or poster. The fifth annual conference of its kind, Frontiers in Biophysics 2011 will be held on the UBC campus in Vancouver on Saturday February 26, 2011. The schedule will include student presentations, a poster session, and a keynote address by Professor Tim Elston from the University of North Carolina.
The structure of the zero set of a multivariate polynomial is a topic of
wide interest, in view of its ubiquity in problems of analysis, algebra,
partial differential equations, probability and geometry. The study of
such sets originated in the pioneering work of Jung, Abhyankar and
Hironaka and has seen substantial recent advances in an algebraic setting.
In this talk, I will mention a few situations in analysis where the study of
polynomial zero sets plays a critical role, and discuss prior work in this
analytical framework in two dimensions. Our main result (joint with
Tristan Collins and Allan Greenleaf) is a formulation of an algorithm for
resolving singularities of a real-analytic function in any dimension with
a view to applying it to a class of problems in harmonic analysis.
This event will feature a panel discussion involving four of the department's winners of the Killam Teaching Award. All are welcome to this event in the TA Accreditation Program Seminar Series. Refreshments will be served.
We explore the question of whether there are nontrivial solutions to the two-valued minimal surface (2MSE) equation defined over the punctured plane. The 2MSE is a non-uniformly elliptic PDE, degenerate at the origin, originally introduced by N.Wickramasekera and L.Simon to produce examples of stable branched minimal immersions.
I will explain how to use Rost's theory of "Chow groups with coeffients" in order to effectively compute the cohomology invariants of (some) algebraic groups. The method is quite geometric and requires understanding the orbits of the group in at least one representation. One can also derive a useful spectral sequence from this approach.
Motivated by the analysis of the singularity of the Bergman kernel on strictly pseudoconvex complex domains, Fefferman launched in the late 70s the program of determining all local biholomorphic invariants of a strictly pseudoconvex complex domain. This program has since been extended to other "parabolic" geometries such as conformal geometry. After a review of Fefferman's program, we shall explain how to compute explicitly the logarithmic singularities of the Green kernels of the conformal powers of the Laplacian, including the Yamabe and Paneitz operators. As applications we obtain a new characterization of locally conformally flat manifolds and a spectral-theoretic characterization of the conformal class of the round sphere.
In this talk a quenched invariance principle for the random conductance model is presented.
More precisely, we consider a continuous time random walk X in an environment
of i.i.d. non-negative random conductances. In recent years quenched invariance
principles have been proven for X under various assumptions on the law of the conductances,
while we present here the result for general i.i.d. environments.
This is joint work with Martin Barlow, Jean-Dominique Deuschel and Ben Hambly.
Myxococcus xanthus is a model bacteria famous for its coordinated multicellular behavior resulting in formation of various dynamical patterns. Examples of these patterns include fruiting bodies - aggregates in which tens of thousands of bacteria self-organize to sporulate under starvation conditions and ripples - dynamical bacterial density waves propagating through the colony during predation. Relating these complex self-organization patterns in M. xanthus swarms to motility of individual cells is a complex-reverse engineering problem that cannot be solved solely by traditional experimental research. Our group addresses this problem with a complementary approach - a combination of biostatistical image quantification of the experimental data with agent-based modeling. To illustrate our approach we discuss our methods of modeling predatory traveling waves - ripples, quantifying emergent order in developmental aggregation under starvation conditions and discovering features that affect the aggregation dynamics.
Note for Attendees
This seminar will be held on a Thursday rather than our regular Tuesday.
Many arithmetic geometric results have an arithmetic dynamic analogue. For instance, Siegel's theorem that an elliptic curve has only finitely many integer points is analogous to the fact that any orbit under a rational function whose second iterate has non-constant denominator has only finitely many distinct integer values.
A conjecture of Lang states that the number of integer points on a minimal Weierstrass model of an elliptic curve is uniformly bounded. In order to translate this conjecture, one needs a dynamic concept of minimality. We present one such notion, affine minimality, an algorithm to compute affine minimal forms of rational functions and some recent results pertaining to the dynamical analogue of Lang's conjecture.
Note for Attendees
Refreshments will be served between the two talks.
We summarize the current meager state of knowledge concerning how often values of polynomials have only small prime factors (that is, the values are "friable" or "smooth"). We also present some evidence, in the form of a theorem conditional upon a suitably explicit hypothesis on prime values of polynomials, to support a conjectured asymptotic formula for the number of friable values of any polynomial.
Hilbert functions are fundamental objects in algebraic geometry, commutative algebra, and combinatorics. After recalling the basic definitions and motivating examples, we will discuss Macaulay's characterization for the collection of all Hilbert functions. We'll then contrast this with a newer viewpoint and look at potential applications.
The Heisenberg algebra is defined by the relations;
[x_i,p_j] = \delta_{i,j} id.
Grojnowski and Nakajima constructed a geometric representation of this
algebra using correspondences between the collection of all the Hilbert
schemes of C^2.
This is a report on my joint work with Prakash Belkale and Nicolas Ressayre. We prove a generalization of Fulton’s conjecture which relates intersection theory on an arbitrary flag variety to invariant theory.
Given a sequence of roots, one can construct a corresponding Bott-Samelson variety. These varieties are basic tools in representation theory and
geometry of G/P's; for instance, the Bott-Samelson varieties corresponding to reduced sequences resolve singularities of Schubert varieties. Partly because of these connections, a good understanding of line bundles and divisors, including descriptions of the nef and effective cones, is of interest. In this talk, I'll explain what is known about these questions, what I would like to know, and how much I know of what I'd like to know.
We discuss research conducted last summer by Dr. Anstee and the
speaker, with help from Miguel Raggi, in the extremal set theory
problem area of forbidden configurations. For an integer m and a given
(0,1)-matrix F, we define forb(m, F) (as you may remember!) as the
maximum number of columns that an m-rowed (0,1)-matrix A can have
subject to the conditions that A has no repeated columns and no
submatrix that is a row and column permutation of F. Let [G|H]
represent the concatenation of the matrices G and H, and let K(k)
denote the matrix of all possible (0,1)-columns on k rows. Fundamental
results determine forb(m, K(k)) and forb(m, [K(k)|K(k)]) exactly for
all m and k. Our research extends these results by identifying
matrices B, C such that forb(m, [K(k)|B]) = forb(m, K(k)) and forb(m,
[K(k)|K(k)|C]) = forb(m, [K(k)|K(k)]). Analysis of base cases tends to
be the most difficult part in applying standard induction techniques!
We also explain the motivation to consider a new function that is
somewhat opposite in spirit to forb (and appropriately named "req"!)
and its relation to the research problems.
Abstract: A set in the plane K is called a Kakeya set if a needle (i.e. a unit segment) can be continuously rotated around within K, returning to its original position with its ends reversed. In 1917, Soichi Kakeya asked for the minimal area for Kakeya sets, initially for convex sets but then expanded to include all sets. This talk will cover some of the ideas considered for this problem, including the "3-point turn", and a clever construction known as Besicovitch sets which answers a related problem. Kakeya and Besicovitch sets are useful in a surprising number of areas in mathematics; we will discuss their influence in one problem in harmonic analysis - convergence of partial Fourier integrals. No background is necessary for the discussion on construction of Kakeya sets, although knowledge of some Fourier analysis would help in the later portion of the talk.
The Mayer expansion is a power series expansion that has a central
place in statistical mechanics. It is also full of combinatorial miracles that
relate it to graphs, forests and branched polymers. I will discuss the background,
the results, and some open problems.
(Special PIMS Lecture: the inaugural Hugh C. Morris Lecture). CANCELLED
Fri 11 Mar 2011, 3:00pm-4:00pm
Abstract
The quantification of uncertainty in large-scale scientific and engineering computations is rapidly emerging as a research area that poses some very challenging fundamental problems which go well
beyond sensitivity analysis and associated small fluctuation theories. We want to understand complex systems that operate in regimes where small changes in parameters can lead to very different solutions. How are these regimes characterized? Can the small probabilities of large (possibly catastrophic) changes be calculated? These questions lead us into systemic risk analysis, that is, the calculation of probabilities that a large number of components in a complex, interconnected system will fail simultaneously.
I will give a brief overview of these problems and then discuss in some detail two model problems. One is a mean field model of interacting diffusions and the other a large deviation problem for conservation laws. The first is motivated by financial systems and the second by problems in combustion, but they are considerably simplified so as to carry out a mathematical analysis. The results do, however, give us insight into how to design numerical methods where detailed analysis is impossible.
Note for Attendees
Due to illness, PIMS regrets that the Friday, March 11 Hugh C. Morris Lecture with speaker George Papanicolaou has been cancelled. Our apologies for this late notice; we will reschedule this lecture for a later date.
As the very first step of talking about spectral sequences in the context of étale cohomology, we review the basic theory of spectral sequence starting with spectral sequences associated to double complex. The goal with this series of talk is to describe the spectral sequence associated to Galois coverings.
Using the theory of polynomial identity rings and the construction of the
universal division algebras, we will prove the following theorem:
Let k be a field, UD_n(k) the universal division ring of degree n over k,
and D a division ring of degree n whose centre contains k. If UD_n(k)
contains a (not necessarily maximal) subfield which is Galois over Z(UD_n)
with Galois group G, then D contains a subfield Galois over Z(D) with
Galois group G.
This was an important step in Amitsur's proof that the universal division
algebra is not a crossed product for certain values of n.
Order of information plays a crucial role in the process of updating beliefs across time. In fact, the presence of order effects makes a classical or Bayesian approach to inference difficult. As a result, the existing models of inference, such as the belief-adjustment model, merely provide an ad hoc explanation for these effects. We postulate a quantum inference model for order effects based on the axiomatic principles of quantum probability theory. The quantum inference model explains order effects by transforming a state vector with different sequences of operators for different orderings of information. We demonstrate this process by fitting the quantum model to data collected in a medical diagnostic task and a jury decision-making task. To further test the quantum inference model, new jury decision-making experiments are developed. The results of these experiments are used to compare the quantum model to the belief-adjustment model and suggest that the belief-adjustment model faces limitations whereas the quantum inference model does not.
Symmetric functions have applications in the real world as well as
many branches of mathematics. The most important basis for the
algebra of symmetric functions is that of the Schur functions. As
such they are much studied and have spawned many analogs and
generalizations. A combinatorial rule for the way they multiply is
given by the Littlewood-Richardson Rule.
In recent work, Haglund, Mason, van Willigenburg, and this author
introduced a family of quasisymmetric functions which we call
quasisymmetric Schur (QS) functions. These naturally refine the
ordinary Schur functions and form a basis, over the integers, of QSym,
the quasisymmetric function algebra.
Dual to the QS functions are noncommutative analogs of the ordinary
Schur functions, having analogous properties such as a
Littlewood-Richardson rule and relationship to a poset of compositions
which is analogous to Young's lattice of partitions. This is joint
work with Christine Bessenrodt and Stephanie van Willigenburg.
I will give a brief review of the combinatorics of ordinary Schur
functions in terms of tableaux. Some familiarity with symmetric
functions is helpful, but is not necessary. I will explain what
quasisymmetric functions are, then introduce you to these
quasisymmetric and noncommutative Schur analogs.
In this talk we will discuss properties of spaces of homomorphisms
Hom(Q,G) where Q is a discrete group and G a Lie group. The example
given by the ordered commuting n-tuples in a compact Lie group will be
explained in some detail. We will discuss how spaces of homomorphisms
and the descending central series of the free groups can be used to
construct a filtration of the classifying space BG. Homotopy properties
of these constructions will be given for finite groups, and cohomology
calculations provided for compact Lie groups. We will also describe
results on understanding both the number and stable homotopy type of
the components of related spaces of representations.
A real-valued stationary time series is said to produce 1/f noise
if the power spectral density is like 1/f on a long frequency interval. A
question of interest since shot noise was noticed in the 1920's is the title
of this talk. I will present some progress toward a general construction to
explain such data and indicate some of the remaining questions.
Thanks to the work of Howe, Freitag, J.-S. Li, and most others, we can identify the holomorphic Siegel modular forms of degree n that arise as theta liftings associated with quadratic forms in at most n variables in terms of their Fourier coefficients. We shall review this work (using the language of theta correspondence) and discuss some ideas for studying the case when the number of variables in the quadratic form is greater than n.
Note for Attendees
Refreshments will be served between the two talks.
Let f be a Holder potential on the full one-sided shift {0,1}^N, N={0,1,2,...}, and let mu_beta denote the Gibbs measure for f at inverse temperature beta (existence and uniqueness are classical, as is the smooth dependence on beta). I will discuss the question of existence of the limit of these measures as beta tends to infinity, i.e. as the temperature tends to zero. Although over infinite state spaces it was known, due to work of Van Enter and Ruszel, that the limit may not exist, it was believed that over finite state spaces (e.g. the case {0,1} above) the measures should converge. In particular, for finite alphabets Bremont proved that convergence occurs when f takes only finitely many values. I will present joint work with Jean-Rene Chazottes in which we construct a counterexample, and discuss some of its features. If time permits I will also discuss results in the multidimensional case.
I will talk about a class of functions on local fields that can be defined by means of logic (the so-called constructible exponential functions). These functions are in a sense built from charactersitic functions of balls, and additive characters of the field. It turns out that they have unexpectedly simple "integration theory". It also turns out that Harish-Chandra characters of representations of p-adic groups near the identity element belong to this class of functions. This allows to transfer Harish-Chandra's theorem about local integrability of characters that was known in general only for local fields of characteristic zero, to large positive characteristic. All of the notions mentioned in this abstract will be defined in the talk.
MATH ANNEX 1100 (PIMS/UBC distinguished colloquium)
Fri 18 Mar 2011, 3:00pm-4:00pm
Abstract
A group of scientists at the University of North Carolina, from theorists to clinicians, have coalesced over the past decade on an effort called the Virtual Lung Project. There is a parallel VLP at the Pacific Northwest Laboratory, focused on environmental health, but I will focus on our effort. We come from mathematics, chemistry, computer science, physics, lung biology, biophysics and medicine. The goal is to engineer lung health through combined experimental-theoretical-computational tools to measure, assess, and predict lung function and dysfunction. Now one might ask, with all due respect to Tina Turner: what's math got to do with it? My lecture is devoted to many responses, including some progress yet more open problems.
We continue our survey about basic spectral sequences. In this talk we describe convergence of a first quadrant spectral sequence, and as examples the spectral sequence associated with a Serre fibration and a complete proof of the Wang sequence.
We will continue with the historical development of systematic procedures to find nonlocally related but equivalent systems of PDEs and to use such systems to find nonlocal symmetries and nonlocal conservation laws of a given PDE system. As a consequence, one is able to find and use symmetries for PDE systems which exhibit no Lie point symmetries and exhibit no local symmetries.
Vanishing theorems and rational singularities are closely related and play important roles in classification theory as well as other areas of algebraic geometry. In this talk I will discuss these roles their interrelations and a new notion that helps understand these singularities and their connections to the singularities of the minimal model program and moduli theory of higher dimensional varieties.
We start a series of talk on Amitsur's conjecture. In the first talk, we re-introduce Brauer-Severi varieties and discuss some constructions associated to projective representations of the group PGL_n.
The hierarchical spherical model is investigated at the critical point
by solving the underlying evolution equation, corresponding to the
renormalization group transformation in the limit as the block size
goes to 1. Some limit theorems can be obtained following the
trajectories of this PDE equation. Starting far away from the
stationary Gaussian fixed point, an interpretation of the critical
trajectory is given in terms of the geometric theory of functions
which describes the motion of the Lee-Yang zeros under the scale
transformation.
We consider the space L of long knots, i.e. of smooth embedding of the real line into a d-dimensional vector space R^d with a fixed behaviour at infinity. This space is almost the same as the space of all smooth embeddings of the circle into a d-dimensional sphere, Emb(S^1,S^d).
The fact that there is no knot in codimension > 2 implies that L is connected when d>3, but actually this space L is not contractible. In this talk I will explain how we can actually describe the homotopy type of that space. I will in particular explain that the rational homology of that space L is computable as the homology of some explicit combinatorial chain complex of generalized chord diagrams. Another way of saying it is that the Vassiliev spectral sequence computing the rational homology of L collapses at the E2-page. This fact is closely related to the fact that the
operad of little disks is formal.
Joint work with Victor Turchin and Ismar Volic
Title: Building a diagnostic test for proof skills. Can we predict and increase success in an "introduction to proof" course?
One of our department's Carl Wieman Science Education Initiative projects is to assess and improve learning in our introductory proof course, Math 220. As part of this project, we have been developing a short (<=20 min) test to administer at the start of the term to assess incoming students' basic logic, computational skills and mathematics reading comprehension. Such a diagnostic test could then be used to predict individual students' performance in the course and to inform us of gaps in student knowledge and skills. In addition, by giving the test again at the end of term we can track learning gains and assess the effectiveness of any changes we are considering.
In this talk we will discuss our "first draft" and our first results from Math 220 students in Fall 2010. Already this uncovers some interesting results and we will discuss how the predictive power of the test compares with using results from Calculus II. We also hope to gain some feedback from department members on the test itself. Are the skills we test the most important for learning mathematical proof? Are there important skills missing?
I will explain a method for constructing particle preserving cellular automata with no anticipation and velocity one, by showing that they must preserve a certain partial order. Some of this dynamical systems may be discrete models for the movement of drops of rain in a window, subject to a force that can be regarded as gravity or wind. I will also expose some related probability problems. E.g. What is the probability that a raindrop will continue to move forever (in an infinite window) subject to a sequence of wind directions (a random walk) ?
We consider moduli spaces of semistable sheaves on an elliptically fibered K3 surface, so that the first Chern class of the sheaves is a numerical section. For pairs of complementary such moduli spaces subject to numerical restrictions, we establish the strange duality isomorphism on sections of theta line bundles. We will also present applications to Brill-Noether theory for sheaves on a K3.
We will continue with the historical development of systematic procedures to find nonlocally related but equivalent systems of PDEs and to use such systems to find nonlocal symmetries and nonlocal conservation laws of a given PDE system. As a consequence, one is able to find and use symmetries for PDE systems which exhibit no Lie point symmetries and exhibit no local symmetries.
Dr. Sinha's talk will cover both genome analysis of pathogens (HIV-1 in particular), SIR type models, and statistical modelling of disease prevalence data (of Malaria).
Let G be a connected reductive group over F_q((t)) and let B be a Borel subgroup. Principal series representations are realized on the space of twisted functions (equivalently, sections of line bundles) on G/B. To geometrize this, in the frame work of geometric Langlands program, we would like to replace functions by perverse sheaves. The trouble is that G/B is infinite dimensional (as a variety over F_q) and so it is not clear how to define perverse sheaves on such objects. I will explain how to overcome this difficulty by employing methods of Howe-Bushnell-Kutzko-Roche to realize (families of) principal series representations via compact subgroups.
A tight fusion frame is a sequence of orthogonal projection matrices which
sum to a scalar multiple of the identity. To any such sequence, we can
associate a weakly decreasing sequence of positive integers given by the
ranks of these projections. The question we address is the following:
For which sequences of positive integers do tight fusion frames exist?
In this talk, I will discuss ongoing work with K. Luoto and M. Bownik where
we explore this problem. One of our main tools is the Knuston-Tao
Honeycomb Model. In particular, we use the Honeycomb model to prove
majorization of tight fusion frames. Majorization gives a partial order
structure on sequences for which tight fusion frames exist.
Tuberculosis (TB) granulomas are organized collections of immune cells that form in the lung as a result of immune response to Mycobacterium tuberculosis (Mtb) infection. Formation and maintenance of granulomas are essential for control of Mtb infection and are regulated in part by a pro-inflammatory cytokine, tumor necrosis factorâ€Î± (TNF). We have developed a multi-scale computational model that includes molecular, cellular and tissue scale events that occur during TB granuloma formation. At the molecular scale, we focus on TNF. TNF receptor internalization kinetics are predicted to play a critical role in infection outcome, controlling whether there is clearance of bacteria, excessive inflammation, containment of bacteria in a stable granuloma, or uncontrolled growth of bacteria. Our results suggest that there is an inter-play between TNF and bacterial levels in a granuloma that is controlled by the combined effects of both molecular and cellular scale processes. We also use the model to explain what mechanisms lead to differential effects of TNF-neutralizing drugs (generally used to treat anti-inflammatory diseases) on reactivation of TB. Ultimately, these results can help to elaborate relevant features of the immune response to Mtb infection, identifying new strategies for therapy and prevention.
The Mayer expansion is a power series expansion that has a central
place in statistical mechanics. It is also full of combinatorial miracles that
relate it to graphs, forests and branched polymers. I will discuss the background,
the results, and some open problems.
We will present new conjectures on polynomial equations with constant coefficients over a function field of arbitrary characteristic (joint work with Ghioca). These statements are inspired by previous conjectures from Zilber, Pink and Bombieri, Masser and Zannier. We will try to explain how known results on the latter may give some information on the former in the case of characteristic zero.
Note for Attendees
Refreshments will be served between the two talks.
Hammersley Clifford Theorem gives us the equivalence of Markov random fields and measures with a nearest neighbour Gibb's potential on finite graphs provided that the measure is fully supported. This generalises naturally when the graph is infinite. However it fails to hold when the measure is not fully supported. We will discuss the theorem and some examples for which it fails.
We consider a system of k diagonal polynomials of degrees 1, 2,..., k. Using methods developed by W.T. Gowers and refined by Green and Tao to obtain bounds in the 4-term case of Szemeredi's Theorem on long arithmetic progressions, we show that if a subset A of the natural numbers up to N of size d_N*N exhibits sufficiently small local polynomial bias, then it furnishes roughly the expected number of solutions to the given system. If A furnishes no non-trivial solutions to the system, then we show via an energy incrementing argument that there is a concentration in a Bohr set of pure degree k, and consequently in a long arithmetic progression. We show that this leads to a bound on the density d_N of the set A of the form d_N << exp(-c*sqrt(log log N)), where c>0 is a constant dependent at most on k.
The most important family of statistical mechanics models on graphs are the Fortuin-Kasteleyn random cluster measures FK(p,q), giving a random subset of the edges of the graph. E.g., the q=1 case
is independent bond percolation, and the q=2 case gives rise to the Ising model of magnetization. For every q, there is a critical value p_c(q) for p, where the connectivity properties of the system suddenly change completely. On planar lattices, the system at p_c(q) is conjectured to be conformally invariant, proved for the q=1,2 cases on some lattices by Stas Smirnov.
There is a natural Markov chain with stationary measure FK(p,q), updating locally the states of the edges as time goes. For the q=1 case, in several joint projects with Christophe Garban and Oded
Schramm, we understood exactly how the macroscopic connectivity properties are changing with this dynamics. We also found that by doing the updates asymmetrically, hence pushing the system out of criticality, we can describe well the near-critical behavior. In particular, we constructed conformally covariant continuum dynamical and near-critical percolation processes, the scaling limits of the
discrete systems, using the same space-time scaling in the two cases.
In recent work with C. Garban, we have extended the dynamical scaling limit construction to the q=2 case. However, we have found that the near-critical window is now governed by very different mechanisms. In particular, understanding Onsager's classical results on the near-critical Ising model via the conformally invariant critical system remains a huge challenge, which I will try to explain. I will
also present some results and conjectures on the noise and dynamical sensitivity of critical FK(q) models.
I will describe an action of a quantized Heisenberg algebra on the (derived) categories of coherent sheaves on Hilbert schemes of ALE spaces (crepant resolutions of C^2/G). This action essentially lifts the actions of Nakajima and Grojnowski on the cohomology of these spaces. (Joint with Tony Licata.)
In the last of this series of seminars on nonlocally related systems and nonlocal symmetries, we will discuss the interesting situation for linear PDE systems and the situation for PDE systems with three or more independent variables. In the case of three or more independent variables, it will be seen that it is necessary to introduce gauge constraints to obtain potential symmetries. Open problems will be discussed as usual.
We prove results about the L^p-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L^p, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,...,N} of sizes alpha N and beta N then A+B contains an arithmetic progression of length at least about exp(c (alpha beta log N)^{1/2}). Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least about exp(c (alpha log N/(log(beta^{-1}))^3)^{1/2}). (Joint work with Ernie Croot and Olof Sisask.)
We will state some theorems in arithmetic combinatorics, and give outlines of the proofs.
One such theorem bounds the absolute density in the integers of sums of relatively dense sets of prime numbers in terms of the relative density. In discussing the proof of this result, we will recall the concept pseudorandomness of a set previously developed by Green and Green-Tao. These ideas are used to reduce the problem to an analogous statement for relatively dense subsets of the multiplicative subgroup of integers modulo a large integer N, and this statement may be proved using a combinatorial argument. This result is joint work with Mariah Hamel.
We will also present some variants of the Erdös-Szemerédi sum-product problem which have the following flavor: A set of complex numbers whose elements produce few products will necessarily produce many sums. To demonstrate the proofs, we will introduce the concept of the multiplicative dimension of a finite set of complex numbers.
We will view these results in the context of connections between a set's structure and arithmetic.
A permutation $\pi$ is realized by the shift on $N$ letters if there is
an infinite word on an $N$-letter alphabet whose successive shifts by
one position are lexicographically in the same relative order as
$\pi$.
Understanding the set of permutations realized by shifts, as well as
other one-dimensional dynamical systems, is important because
it provides tests to distinguish deterministic sequences from random ones.
In this talk I will give a characterization of permutations realized
by shifts, and also by a natural generalization of them, where instead
of $N$ we have a real number $\beta$.
In the talk I will survey some results, techniques, and open problems, concerning finite group actions on (i) spheres, (ii) products of spheres and (iii) non-compact space forms modelled on indefinite quadratic forms.
Given a finite graph we wish to order to its vertices in some random way that depends on the graph. If we impose some natural constraints on the order, it turns out that there is only one possible distribution on
the order. I will describe connections to newly developed theory of limits of graphs, and a surprising new proof of a result about embedding metric spaces.
I will introduce the concept of p-local compact groups, which
are combinatorial structures that generalize p-local finite groups and p-compact
groups, along with some examples. Then I will describe some partial results
towards showing the existence of embeddings of these objects in p-completed
classifying spaces of unitary groups.
We show that the threshold for a subset sampled uniformly from the
range of a random walk on Z^d_n (the d dimensional torus of size n) d>=3, to become indistinguishable from a
uniformly chosen subset of Z^d_n is 1/2 the cover time. As a consequence of
our methods, we show that the total variation mixing time of the random
walk on the lamplighter graph of Z^d_n d>=3, has a cutoff with threshold at
1/2 the cover time. We give a general criterion under which both of these
results hold; other examples for which this applies include bounded degree
expander families, the intersection of an infinite super-critical percolation
cluster with an increasing family of balls, the hypercube, and the
Cayley graph of the symmetric group generated by transpositions. The
proof also yields precise asymptotics for the decay of correlation in the
uncovered set. This is joint work with Yuval Peres.
We will discuss a surprising relationship between stationary Markov chains, stationary Markov random fields and measures with nearest neighbour Gibbs potential on the integer lattice. We will also see why this does not extend to higher dimensions.
In this work we will consider several questions concerning the asymptotic nature of arithmetic functions. First, we conduct an asymptotic analysis of the behavior of the composition of Carmichael's lambda-function and Euler's totient function. Second, we establish an asymptotic formula for a sum of a generalized divisor function on the Gaussian numbers. And third, for complex-valued multiplicative functions that are sufficiently close to 1 on the primes and bounded on prime powers, we determine the average value over a short interval.
MATX 1118 (Note the special location and special time)
Fri 8 Apr 2011, 1:00pm-2:00pm
Abstract
In recent years, there has been an increasing interest in studying constrained variational problems with a fractional diffusion. One of the motivations comes from mathematical finance: jump-diffusion processes where incorporated by Merton into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock's prices, in contrast with the classical lognormal diffusion model of Black and Scholes. These models allow to take into account large price changes, and they have become increasingly popular for modeling market fluctuations, both for risk management and option pricing purposes.
In a joint paper with Luis Caffarelli we study the parabolic version of the fractional obstacle problem, i.e. where the elliptic part of the operator is given (at least at the leading order) by a fractional laplacian. We prove optimal spatial regularity and almost optimal time regularity of the solution, recovering in particular the optimal regularity for the stationary case. To obtain this result, we crucially exploit the fact that the solution coincides with the obstacle at the initial time, which corresponds to the fact that (for the backward operator) the stock's price coincides with the payoff at the final time.
The research solves the problem of Lie group analysis on the Buckley-Leverett equation
and provides the construction of some invariant solutions.
We also view the analytical solutions of a diffusion-convection equation (Rapport-L model)
with capillary effects.
Abstract:
What are the basic skills in Mathematics that can affect success in first-year Calculus and beyond?
We will give an overview of some of the results gathered from the Basic Skills Test and other similar
tests. We will show what questions had the highest degree of correlation with performance in
differential Calculus and discuss common students' errors. Data from a variety of courses will
be presented, ranging from Math 110 to Math 180, 104, and 184, but also including a few
examples from Science One. Finally, we will show that some of the weaknesses in
basics skills continue beyond first-year courses by discussing recent data from Math 220.
After reviewing some basic constructions with multiplier ideals on complex algebraic varieties, we recall the definition of multiplier ideals in positive characteristic and highlight the failure of some desirable properties to carry over in this setting. This leads us to a related measure of singularities coming from commutative algebra -- the test ideal -- which seems to exhibit better behavior than the multiplier ideal in positive characteristic. While test ideals were first introduced in the theory of tight closure, our goal in this talk will be to describe a new and algebro-geometric characterization of test ideals using regular alterations. This characterization is also holds for multiplier ideals in characteristic zero (but not in positive characteristic!!!), providing a kind of uniform description with new insight and intuition. Time permitting, we will use this result to give an analogue of Nadel Vanishing in positive characteristic.
We develop and analyze mixed discontinuous Galerkin finite element methods for the numerical approximation of incompressible magnetohydrodynamics problems.Incompressible magnetohydrodynamics is the area of physics that is concerned with the behaviour of electrically conducting, resistive, incompressible and viscous fluids in the presence of electromagnetic fields. It is modelled by a system of nonlinear partial differential equations, which couples the Navier-Stokes equations with the Maxwell equations.
In the first part of this thesis, we introduce an interior penalty discontinuous Galerkin method for the numerical approximation of a linearized incompressible magnetohydrodynamics problem. All the variables are discretized using completely discontinuous finite element spaces. Under minimal regularity assumptions, we carry out a complete a-priori error analysis and prove that the energy norm error is optimally convergent in the mesh size in general polyhedral domains, thus guaranteeing the numerical resolution of the strongest magnetic singularities in non-convex domains.
In the second part of this thesis, we propose and analyze a new mixed discontinuous Galerkin finite element method for the approximation of a fully nonlinear incompressible magnetohydrodynamics model. The velocity and the magnetic field are now discretized by partially continuous elements. In addition to correctly capturing magnetic singularities, the method yields exactly divergence-free velocity approximations, and is thus energy-stable. We show that the energy norm error is convergent in the mesh size in possibly non-convex polyhedra, and derive slightly suboptimal a-priori error estimates under minimal regularity and small data assumptions.
Finally, in the third part of this thesis, we present two extensions of our discretization techniques to time-dependent incompressible magnetohydrodynamics problems and to Stokes problems with non-standard boundary conditions.
All our discretizations and theoretical results are computationally validated through comprehensive sets of numerical experiments.
Given an infinite stack of arrows (each pointing left or right) at each
vertex of Z, we can define a walk on Z that moves by following and consuming
arrows. If we switch a left arrow to a right arrow, what happens to
the walk? The answer to this question gives some interesting results when
applied to 1-dimensional random walks (such as multi-excited random
walks), and projections of higher-dimensional random walks.
(Joint work with Tom Salisbury)
The overlapping cycles shuffle mixes a deck of n cards by moving either the nth card or (n-k)th card to the top of the deck, with probability half each. Angel, Peres and Wilson determined the spectral gap for the location of a single card and found the following surprising behaviour. Supppose that k is the closest integer to cn for a fixed c in (0,1). Then for rational c, the spectral gap is on the order of n^{-2}, while for poorly approximable irrational numbers c, such as the reciprocal of the golden ratio, the spectral gap is on the order of n^{-3/2}. We show that these bounds also apply, up to logarithmic factors, to the mixing time for all the cards. The talk is based on work in progress with Olena Bormashenko and Sukhada Fadnavis.
Using a group of automorphisms of the free group F_2 isomorphic to the braid group B_3, we construct a non-abelian one-cocycle P: F_2 --> F_2, which turns out to take palindromic values and to be continuous for the profinite topology on F_2. We characterize the elements of the image of the map P and use B_3 to express the relation between elements having the same image. (Joint work with Christophe Reutenauer, UQAM.)
The corner transfer matrix renormalization group method is a powerful
algorithm for evaluating physical quantities in statistical mechanical
models. It was developed by Nishino and Okunishi in 1996, and
originates from Baxter's corner transfer matrix equations and method,
which were developed in 1978. The method has been applied to numerous
models since its development to great success, but has not been used
to calculate series expansions (which Baxter did with his original
method).
In this talk, we discuss the CTMRG method, our work on this method
which has enabled us to calculate series expansions from it, and the
models we have applied it to.
Abstract:
The focus of this presentation is a class of random processes known as
interacting measure-valued stochastic processes. These processes are
related to another class of measure-valued stochastic processes known as
superprocesses. Both superprocesses and interacting measure-valued
stochastic processes arise naturally from branching particle systems as
scaling limits. A branching particle system is a collection of particles
that propagate randomly through space, and that upon death give birth to a
random number of particles (children). Therefore when the populations of
the particle system and branching rate are large one can often use a
superprocess to approximate it and carry out calculations that would be
very difficult otherwise.
There are many branching particle systems which do not satisfy the strong
independence assumptions underlying superprocesses and thus are more
difficult to study mathematically. We attempt to address two
measure-valued processes with different types of dependencies
(interactions) that the associated particles exhibit. In both cases, the
method used to carry out this work is called Perkins' historical stochastic
calculus, and has never before been used to investigate interacting
measure-valued processes of these types. That is, we construct the
measure-valued stochastic process associated with an interacting branching
particle system directly without taking a scaling limit.
The first type of interaction we consider is when all particles share a
common chaotic drift from being immersed in the same medium, as well as
having other types of individual interactions. The second interaction
involves particles that attract to or repel from the center of mass of the
entire population. For measure-valued processes with this latter
interaction, we study the long-term behavior of the process and show that
it displays some types of equilibria.
We consider a uniformly chosen proper 3-coloring of the discrete torus Z_n^d. We show that in sufficiently high dimensions, such a coloring is very rigid, taking a single color on most of either the even or odd sublattice. We will further explain the relation between proper 3-colorings and a certain class of Lipschitz functions, and how it implies that high-dimensional Lipschitz functions are typically flat. Our main tool is a detailed study of the combinatorics of certain cutsets in Z^d. Along the way, we will mention many related conjectures and open questions concerning proper colorings and random functions.
All concepts used will be introduced and the talk will be illustrated by many pictures.
Doodling has many mathematical aspects: patterns, shapes, numbers, and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. I'll begin by doodling, and see where it takes us. It looks like play, but it reflects what mathematics is really about: finding patterns in nature, explaining them, and extending them. By the end, we'll have seen some important notions in geometry, topology, physics, and elsewhere; some fundamental ideas guiding the development of mathematics over the course of the last century; and ongoing work continuing today.
This is a meeting of instructors from UBC and Vancouver high schools. The purpose of the meeting is to discuss issues of transition from high school to university, particularly in the context of mathematics.An agenda of speakers and panels may be found at www.math.ubc.ca/~fsl/meeting.html
Consider the Poisson point process in Euclidean space. We are interested in functions on this random point set whose value in each configuration point is given by some ``local" rule (no ``central planning"). One example is the so-called allocation problem,
where we want to partition $\R^d$ to sets of volume 1 and match them with the point process, in a translation equivariant way. We want to make the allocated set optimal in some sense (e.g., the distribution of the diameter shows fast decay). We will present some allocation schemes, among them one with an optimal tail, which is our recent construction with R. Marko.
In the first part of the thesis we discuss the rank conjecture of Benson and Carlson. In particular, we prove that if G is a finite p-group of rank 3 and with p odd, or if G is a central extension of abelian p-groups, then there is a free finite G-CW-complex homotopy equivalent to the product of rk(G) spheres; where rk(G) is the rank of G.
We also treat an extension of the rank conjecture to groups of finite virtual cohomological dimension. In this context, for p a fixed odd prime, we show that there is an infinite group L satisfying the two following properties: every finite subgroup G<L is a p-group with rk(G)<3 and for every finite dimensional L-CW-complex homotopy equivalent to a sphere, there is at least one isotropy subgroup H<L with rk(H)=2.
In the second part of the thesis we discuss the study of homotopy G-spheres up to Borel equivalence. In particular, we provide a new approach to the construction of finite homotopy G-spheres up to Borel equivalence, and we apply it to give some new examples for some semi-direct products.
I plan to review several applications described by delay differential equations (DDEs) starting from familiar examples such as car following models to physiology and industrial problems. DDEs have the reputation to be mathematically difficult but there is a renewed interest for both old and new problems. I’ll emphasize the need for analytical tools in order to guide our numerical simulations and identify key physical phenomena. These ideas will be illustrated by problems in nonlinear optics and neurobiology.
There is a growing body of work that supports a connection between L-spaces and 3-manifolds with non-left-orderable fundamental group, in fact a Seifert fibred manifold is an L-space if and only if its fundamental group is not left-orderable. In this talk I'll provide evidence for a connection that extends beyond the class of Seifert fibred manifolds, by showing that L-spaces behave similarly to non-left-orderability with respect to the operation of Dehn surgery on a manifold. It is with this goal in mind one is led to define a decayed knot; decayed knots have the property that sufficiently large surgery always yields a manifold with non-left-orderable fundamental group. Moreover, cables of decayed knots are also decayed, as long as the ratio of the cabling coefficients is chosen to be large enough. I'll show how both of these properties mirror the behaviour of knots which admit L-space surgeries, and outline some questions for future research. This is joint work with Liam Watson.
We explore the field of Forbidden Configurations, a problem in
Extremal Set Theory. We consider a family of subsets of {1,2,...,m} as
the corresponding {0,1}-incidence matrix. For {0,1}-matrices F, A, we
say F is a *subconfiguration* of A if A has a submatrix which is a row
and column permutation of F. We say a {0,1}-matrix is *simple* if it
has no repeated columns. Let ||A|| denote the number of columns of A.
A {0,1}-matrix F with row and column order stripped is a
*configuration*. Given m and a family of configurations G, our main
function of study is forb(m,G) := max{||A|| : A simple and for all F
in G, we have F not a subconfiguration of A }.
We give a general introduction to the main ideas and previous work
done in the topic. We develop a new more computational approach that
allows us to tackle larger problems. Then we present an array of new
results, many of which were solved in part thanks to the new
computational approach. We use both new ideas and new spins on old
ideas to tackle the problems. The new results include finding exact
bounds on small configurations that were previously unknown, and
proving some previously conjectured asymptotic bounds for "boundary''
configurations. We also develop a relationship between Forbidden
Configurations and Patterns, which we use to prove some results.
In the first part of the talk we discuss some new observations on the blow-up problem in the 3D Euler equations. We consider the scenarios of the self-similar blow-ups and the axisymmetric blow-up. For the self-similar blow-up we prove a Liouville type theorem for the self-similar Euler equations. For the axisymmetric case we show that some uniformity condition for the pressure is not consistent with the global regularity. In the second part we present Liouville type theorems for the steady Navier-Stokes equations for both of the incompressible and the compressible cases. In the time dependent case we prove that some pressure integrals have definite sign unless the solution is trivial.
In this talk, I will explain SYZ proposal on describing mirror symmetry as a Fourier-Mukia transformation along special Lagrangian torus fibration. By computing certain open Gromov-Witten invariants, we show that the mirror map is the same as the SYZ map for certain toric Calabi-Yau manifolds.
In this talk, we will discuss the edit distance function, a function of a hereditary property mathcal{H} and of p, which measures the maximum proportion of edges in a density-p graph that need to be inserted/deleted in order to transform it into a member of mathcal{H}. We will describe a method of computing this function and give some results that have been attained using it. The edit distance problem has applications in property testing and evolutionary biology and is closely related to well-studied Tur'an-type problems. The results we address will involve constructions used on the problem of Zarankiewicz as well as strongly regular graphs. This is joint work with Tracy McKay, Iowa State University.
In this talk, the symmetry group direct method and some applications for nonlinear systems are discussed. The direct method by Clarkson and Kruskal can derive symmetry reductions of a nonlinear system without using any group theory. Recently, the direct method has been modified to find the generalized Lie and non-Lie symmetry groups for some nonlinear systems. The Lie symmetry groups obtained via traditional Lie approaches are only special cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches. Now, the direct method can be extended to construct some finite transformations between some variable-coefficient PDEs and their related constant-coefficient PDEs or simpler variable-coefficient PDEs. These applications will be illustrated by some examples.
This talk will be the first in a sequence of five talks on recent developments in Symmetries and Differential Equations. All talks will take place from 2-3pm each Wednesday. Future talks will take place in Math Annex 1102 and expected speakers include ZZ Yang, Ian Lisle, Jeremy Hoskins, Raouf Dridi and Shoufu Tian.
In the procedure of finding nonlocally related systems, subsystems are obtained by excluding dependent variables of a given PDE system, or after ``hodograph'' type transformations. More generally, one can further extend this procedure to more general invertible point transformations. As an example, I have investigated the nonlinear wave equation and have shown that a more general point transformation of the potential system of the nonlinear wave equation does yield a subsystem that can yield previously unknown nonlocal symmetries of the nonlinear wave equation. Moreover, some open problems will be posed.
Department of Chemical Engineering and Minerva Center for Nonlinear Physics of Complex Systems, Technion - Israel Institute of Technology, Haifa 32000, Israel
Cells and tissues rearrange under the action of chemical signals. Numerous examples are found in eggshell development, wing disc remodeling, dorsal closure, wound healing, etc. In many cases, this can be attributed to changing local mechanical properties of cytoskeleton due to motor attachment/detachment and rearrangement of the actin network triggered by signaling. I consider in more detail the action of myosin motors on nonlinear viscoelastic properties of cytoskeleton. It turns out that motors activity may either stiffen the network due to stronger prestress or soften it due to motor agitation, in accordance with experimental data. Prestress anisotropy, which may be induced by redistribution of motors triggered by either external force or a chemical signal, causes anisotropy of elastic moduli. Based on this assumption, we developed a cellular mechano-diffusive model cell that describes reshaping of the Drosophila wing disc. Similar models may be applicable to other processes where mechanics is influenced by chemical signals through the action of myosin motors.
Symmetry analysis of differential equations or other geometric objects typically gives an overdetermined system of linear 'determining equations' for the components of the symmetry vector field. Most methods assume the vector fields are explicitly available, i.e. that the determining system has been solved, but this is not algorithmically possible in general. Greg Reid and others have shown how to get the dimension and structure constants of the Lie symmetry algebra algorithmically from the determining system, but considerably more is possible. We describe a calculus that uses differential reduction and completion to derive determining equations for structural components of the Lie algebra such as the centre, derived algebra, radical, etc.
In classical physics, especially electrodynamics, often one must consider the behaviour of waves as they move between two media in which the speed of the waves differs. In the past this has often been approached through studying a system in which this change occurs instantaneously. Using symmetry methods it is possible to extend this result to obtain an exact solution for the wave behaviour in a system in which there is a smooth transition between the two media. The details of this method will be discussed, as well as the use of the solution in obtaining relevant physical quantities, such as the reflection coefficient, in these new systems.
A striking feature of symmetries of DEs or other geometric objects is that the group action may project to subsets of the variables. For example, the 3-d Navier-Stokes equations, with independent variables (x,y,z,t) and dependent variables (u,v,w,p) (velocity components and pressure), has a symmetry group that projects naturally to actions on t, or on (x,y,z,t) or on (t,u,v,w) or various other subsets of the variables. These projections can be characterised automatically from the determining equations by constructing and condensing a certain directed graph. The digraph suggests a block elimination ranking to assist in solving the determining system. The methods work in both finite- and infinite-dimensional cases.
I will discuss Elie Cartan's paper "Les sous-groupes des groupes continus de transformations. Annales scientifiques de l'Ecole Normale Suprieure, Sr. 3, 25 (1908), p. 57-194" and relate it to the equivalence problem of partial differential equations completely characterized by their symmetries.
The electromagnetic radio frequency spectrum is a scarce and valuable resource. Its utilization can be improved by allowing multi-layer communications, in which several signals are simultaneously transmitted and received in the same frequency band. In this talk I will describe some algorithms for the transmission and reception of multi-layer signals. These algorithms are compatible with commonly used transmitter and receiver equipment that was not designed for multi-layer communications. Such algorithms have great practical importance because they yield increased network capacity and at the same time allow telecom equipment manufacturers (including both handset and network vendors) and wireless operators to obtain additional returns on their multi-billion dollar investments.
WMAX 110 (PIMS) (Schedule and location subject to change)
Thu 8 Sep 2011, 3:30pm-4:30pm
Abstract
The Aharonov-Bohm effect is a quantum mechanical phenomenon where electrons passing through a region of vanishing magnetic field gets scattered due to topological effects. It turns
out that this phenomenon is closely related to the cohomology of forms with integer coefficients. We study this relationship from the point of view of the Calder´n problem and see that it can be captured in how Cauchy data of the connection laplacian determines uniquely the holonomy representation of the connection.
The work was partially supported by Finnish Academy of Science and by NSF Grant No.DMS-0807502.
Schur functions were introduced early in the last century with respect to representation theory, and since then have become important functions in other areas such as combinatorics and algebraic geometry. They have a beautiful combinatorial description in terms of diagrams, which allows many of their properties to be determined.
These symmetric functions form a subalgebra of the algebra of quasisymmetric functions, which date from the 1980s. Despite this connection, the existence of a natural quasisymmetric refinement of Schur functions has been considered unlikely.
In this talk we introduce quasisymmetric Schur functions, which partition Schur functions in an intuitive way. Furthermore, we show how these quasisymmetric Schur functions refine many well-known Schur function properties with combinatorics that strongly reflect the classical case. This is joint work with Christine Bessenrodt, Jim Haglund, Kurt Luoto and Sarah Mason.
The talk will require no prior knowledge of any of the above terms.
Note for Attendees
Tea and cookies will be served in the Math 125 Lounge at approximately 2:45 p.m. prior to the colloquium.
In this talk I will give an introduction to Donaldson-Thomas invariants, and then their motivic incarnation. I'll discuss motivic vanishing cycles and lambda rings, before moving to the main example of the talk - the one loop quiver with potential. It turns out that the motivic DT invariants in this simple example have a neat presentation, and in a break with other worked out examples these invariants really involve the mondromy of the motivic vanishing cycle.
We show that any non-degenerate vector field u in L^{\infty}(\Omega, \R^N), where \Omega is a bounded domain in \R^N, can be written as {equation} \hbox{u(x)= \nabla_1 H(S(x), x) for a.e. x \in \Omega}, {equation} where S is a measure preserving point transformation on \Omega such that S^2=I a.e (an involution), and H: \R^N \times \R^N \to \R is a globally Lipschitz anti-symmetric convex-concave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be the identity, which suggests that our result is a self-dual version of Brenier's polar decomposition for the vector field u as u(x)=\nabla \phi (S(x)), where \phi is convex and S is a measure preserving transformation. We also describe how our polar decomposition can be reformulated as a self-dual mass transport problem.
We describe some recent results about isometry groups of aspherical Riemannian manifolds, and also isometry groups of their universal covers. For instance, we show that on an irreducible locally symmetric space of dimension > 2, no metric has more symmetry than the locally symmetric metric.
We consider a certain class of non-symmetric Markov chains and obtain heat kernel bounds and parabolic Harnack inequalities. Using the heat kernel estimates, we establish a sufficient condition for the family of Markov chains to converge to non-symmetric diffusions. As an application, we approximate non-symmetric divergence forms with bounded coefficients by non-symmetric Markov chains. This extends the results by Stroock-Zheng to the non-symmetric divergence forms.
This is the academic year's first UBC Undergraduate Mathematics Colloquium talk. UBC/UMC talks are meant for undergraduate students interested in mathematics beyond the curriculum. They are accessible at all levels. Our speakers are dynamic professional mathematicians with a reputation for interesting research and strong teaching.
Eric Cytrynbaum is our first speaker. Professor Cytrynbaum will talk about mathematical modelling. He will present two case studies from everyday life, including one describing the rise and fall of an air-breathing mammal attempting to maintain neutral buoyancy in the water.
Given a polynomial f with complex coefficients, and a complex number z, we call the orbit of z under f the set of all images of z under the iterates of f. If the orbit of z under f is finite, we call the number z preperiodic for f. We study the following two basic questions regarding orbits of complex numbers under polynomials.
1) For two polynomials f and g, and for two complex numbers a and b, when does the orbit of a under f intersect the orbit of b under g in infinitely many points?
2) For two polynomials f and g, when there exists an infinite set of complex numbers z which are preperiodic both for f and for g?
We investigate the probability that "Buffon's Needle" lands near a one-dimensional self-similar product set in the complex plane, where the similarity maps have rational centers and identical scalings. If the factors A and B are defined by at most 6 similarities, then the likelihood that the needle intersects an e^{-n}-neighborhood of such a set is at most Cn^{-p/\log\log n} for some p>0.
We introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for frozen triples given by the data O_X^{r}(-n)-->F where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zero-dimensional virtual fundamental class by hand, by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of X. This yields the first deformation-theoretic construction of a higher-rank enumerative theory for Calabi-Yau threefolds. We calculate this enumerative theory for local P^1 using the Graber-Pandharipande virtual localization technique. In a sequel to this project (arXiv:1101.2251), we show how to compute similar invariants associated to frozen triples using Kontsevich Soibelman, Joyce-Song wall-crossing techniques.
I will talk about the backward uniqueness of the heat equation in unbounded domains. It is known that a bounded solution of the heat equation in a half-space which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the half-space is replaced by cones with opening angle smaller than 90 degrees. In a joint work with Vladimir Sverak we show the result remains true for cones with opening angle larger than 110 degrees. Our proof covers heat equations having lower-order terms with bounded measurable coefficients.
We define and discuss representable functors and groups schemes, as a first step towards describing the various group schemes associated to central simple algebras with involution.
We prove that every closed, smooth manifold of at least dimension 3 admits a sequence of Riemannian metrics with pinched curvature, volume tending to infinity but whose first eigenvalue of the Laplacian remains bounded away from 0. As a consequence we construct sequences of hyperbolic knots whose complements have again volume tending to infinity and whose Cheeger constant is uniformly bounded away from 0. This is joint work with Marc Lackenby.
Joint work with Thomas Mountford and Qiang Yao. We consider
the contact process on a random graph chosen with a fixed degree,
power law distribution, according to a model proposed by Newman,
Strogatz and Watts (2001). We follow the work of Chatterjee and
Durrett (2009) who showed that for arbitrarily small infection
parameter $\lambda > 0$, the limiting metastable density does not tend
to zero as the graph size becomes large. We show three distinct
regimes for this density depending on the tail of the degree law.
References:
- Charterjee,S. and Durrett, R. (2009): Contact process on random
graphs with degree power law distribution have critical value zero.
Annals of Probability 37 (2009)
- Newman, M.E.J., Strogatz, S.H. and Watts, D.J. (2001): Random graphs
with arbitrary degree distributions and their applications. Physical
Review E. 64, paper 026118
Let s(n) denote the sum of the proper divisors of n, so, e.g., s(4)=1+2=3. A natural number n is called *perfect* if s(n)=n and *amicable* if s(n) =/= n but s(s(n))=n. For example, 6 is a perfect number, and 220 is an amicable number. Questions about perfect and amicable numbers constitute some of the oldest unsolved problems in mathematics. I will talk about old and new theorems concerning these numbers and their generalizations. Some of this is joint work with Mits Kobayashi (Cal Poly Pomona), Florian Luca (Universidad Nacional Autónoma de México), and Carl Pomerance (Dartmouth College).
Note for Attendees
Refreshments will be served between the two talks.
We consider two classical number theoretic problems that may seem quite
unrelated:
* What is the power of 2 dividing the class number of Q(sqrt(-n))
* Which n are congruent numbers (n called congruent if it occurs as the
area of a right-angled triangle with rational length sides)
The second question is equivalent to determining whether the elliptic curve E_n: y^2=x^3-n^2*x has positive rank. This observation suggest we might want to consider:
* What is the power of 2 in the order of Sha(E_n).
If we restrict to prime values n=p, it is already known that partial answers to these questions can be related to the splitting of p in the quartic number field Q(sqrt(1+i)).
In this talk we will discuss the next step in the classification.
If X is a Riemannian manifold, the Laplacian is a second order elliptic operator on X. The hypoelliptic Laplacian L_b is an operator acting on the total space of the tangent bundle of X, that is supposed to interpolate between the elliptic Laplacian (when b -> 0) and the geodesic flow (when b -> \infty). Up to lower order terms, L_b is a weighted sum of the harmonic oscillator along the fibre TX and of the generator of the geodesic flow. In the talk, we will explain the underlying algebraic, analytic and probabilistic aspects of its construction, and outline some of the applications obtained so far.
Other Information:Jean-Michel Bismut was born in 1948. He is a Professor of Mathematics at University Paris-Sud (Orsay), and a member of the Academie des Sciences, of the Academia Europaea, and of the Deutsche Akademie Leopoldina. He received his 'Doctorat d'Etat' from Universite Paris VI in 1973 for his work in the control of stochastic processes. His interests in probability theory led him to study refinements of the index theorem of Atiyah-Singer. Through his work on Quillen metrics, he participated to the proof of a Riemann-Roch theorem in arithmetic geometry. He constructed an exotic Hodge theory, whose corresponding Laplacian is a hypoelliptic operator on the cotangent bundle of a Riemannian manifold. Recently, he used the hypoelliptic Laplacian to give a new approach to the evaluation of orbital integrals. Jean-Michel Bismut was a plenary speaker at the International Congress of Mathematics in Berlin in 1998, and a vice-president of International Mathematical Union (I.M.U.) from 2002 to 2006.
Projectivized toric vector bundles are a large class of rational varieties that share some of the pleasant properties of toric varieties and other Mori dream spaces. Hering, Mustata and Payne proved that the Mori cones of these varieties are polyhedral and asked if their Cox rings are indeed finitely generated. We present the complete answer to this question. There are several proofs of a positive answer in the rank two case [Hausen-Suss, Gonzalez]. One of these proofs relies on the simple structure of the Okounkov body of these varieties with respect to a special flag of subvarieties. For higher ranks we study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles, we give generators for the cone of effective divisors and a presentation of the Cox ring as a polynomial algebra over the Cox ring of a blowup of a projective space along a sequence of linear subspaces [Gonzalez-Hering-Payne-Suss]. As applications, we show that the projectivized cotangent bundles of some toric varieties are not Mori dream spaces and give examples of projectivized toric vector bundles whose Cox rings are isomorphic to that of M_{0,n}.
I will explain the probabilistic interpretation of the hypoelliptic Laplacian L_b . To L_b, one can associate the diffusion on the manifold X that is a solution of the differential equation b^2 x'' = −x' + w'. For b = 0, we get x' = w', the equation of Brownian motion, and for b = +∞, we obtain the equation of geodesics x'' = 0. I will explain the rigorous results one can derive on the corresponding heat kernels via the Malliavin calculus. These will include uniform Gaussian decay of the hypoelliptic heat kernel over a symmetric space.
The Kompaneets equation describes the spectra of photons interacting with a rarefied electron gas and includes three parameters. In 2010, Ibragimov obtained some time-dependent exact solutions for several restrictions of the parameters of this equation. In this talk, a tree of equivalent nonlocally related PDE systems is constructed for the nonlinear Kompaneets (NLK) equation. The tree includes some nonlocally related equivalent subsystems. For a two-parameter class of NLK equations, a point symmetry classification is given of these nonlocally related PDE systems and shown to yield previosuly unknown nontrivial nonlocal symmetries of the NLK equation. Invariant solutions arising from these nonlocal symmetries are shown to yield wider classes of time-dependent exact solutions for the NLK equation beyond those previously obtained by Ibragimov. In particular, for five classes of initial conditions, each involving two parameters, previously unknown explicit solutions are obtained. Interestingly, each of these solutions is expressed in terms of elementary functions. Three of the classes exhibit quiescent behaviour, and the other two classes exhibit blow up behaviour in finite time. As a consequence, it is shown that the corresponding nontrivial stationary solutions, obtained by Dubinov in 2009, are unstable. In particular, it is shown that only the stationary quiescent solution is stable.
Third talk in the series. If G is a reductive Lie group with Lie algebra g, orbital integrals are key ingredient in Selberg’s trace formula. I will explain how one can think of the evaluation of orbital integrals as the computation of a Lefschetz trace. Using in particular the Dirac operator of Kostant, the standard Casimir operator of X = G/K is deformed to a hypoelliptic operator L_b acting on the total space of a canonically flat vector bundle on X, that contains TX as a subbundle. The symbol of this hypoelliptic operator is exactly the one described in the previous talks. When descending the situation to a locally symmetric space, the spectrum of the original Casimir remains rigidly embedded in the spectrum of the hypoelliptic deformation. Making b → +∞ gives an explicit evaluation of semisimple orbital integrals.
A rational distance set is a subset of the real plane such that all
pairwise distances are rational numbers. It's not too hard to
construct an infinite rational distance set contained in a line or in
a circle, but if you do not allow 3 points on a line or 4 on a circle,
the current record is a set of 7 points, found a few years ago with a
computer. On the other hand, no one knows if a rational distance set,
no 3 points on a line or 4 on a circle, could be infinite. Erdős
conjectured that it would have to have a very special form, like an
algebraic curve.
In a paper with Jozsef Solymosi we showed that the only algebraic
curves that contain infinite rational distance sets are lines and
circles. In my talk I will explain the ideas involved and give an
outline of our proof.
We develop path integral representations for Quantum Ising models. To
connect with classical Fortuin-Kasteleyn (FK) representation, we begin by
presenting the FK representation of the classical Curie-Weiss model (the
Ising model on complete graph) via the language of Poisson Point Processes.
We then show how to derive a general FK representation for Quantum Ising
model. This representation was originally developed by M. Campanino, A.
Klein, J.F. Perez (1991) and M. Aizenman, A. Klein, C.M. Newman (1993).
We apply the above to the quantum Curie-Weiss model in transversal field.
First, we present the full FK representation of this model. Examining the
form of the resulting measure and dropping the weight component from it
leads to the natural extension of the Erd\"os - Rényi random graphs. Finally,
we consider the ground state of the quantum Curie-Weiss model via partial FK
representation. We prove the existence of a phase transition in the ground
state when the strength of the transversal field equals one.
A central result in discrete geometry is the Szemeredi-Trotter theorem which gives a sharp bound on the number of point-line incidences in the Euclidean plane. The result has various generalizations and applications.
In this talk we prove an extension of the Szemeredi-Trotter theorem and we show some new applications. For example, using incidence bounds, we show that if M is an n-element set of k x k matrices with real coefficients such that det(A - B) is not zero for any distinct A,B elements of M and V,W are n-element sets of k dimensional vectors, then |V +W| + |MW| >> n^{5/4}.
Part of the talk is based on joint work with Terry Tao.
Note for Attendees
Refreshments will be served in MATH 125 at 2:45 p.m.
I will talk on joint work in progress with Behrang Noohi. We study the GIT problem given by the differential graded Lie algebra of Hochschild cochains of a finite graded algebra. This will lead to a definition of stability for non-commutative polarized projective schemes, and to the construction of quasi-projective moduli spaces for them. These moduli spaces are differential graded schemes. There may be new moduli spaces with symmetric obstruction theories coming out of this.
This talk is a continuation of Matt Bond's talk from Sept 19: We investigate the probability that "Buffon's Needle" lands near a one-dimensional self-similar product set in the complex plane, where the similarity maps have rational centers and identical scalings. If the factors A and B are defined by at most 6 similarities, then the likelihood that the needle intersects an e^{-n}-neighborhood of such a set is at most Cn^{-p/\log\log n} for some p>0. (Joint work with M. Bond and A. Volberg.)
Are you teaching a course for the first time and would like to see old midterm exams? Are you looking for old course materials from when you taught the same course a few years ago but can't find them? Would you like to have access to old homework assignments and worked solutions for the course you are teaching? Have you ever had to spend time collecting and compressing files to share course materials with your colleagues only to find out that your zip file is too big to send by email?
If any of these applies to you, then the Course Archive Project (CAP) is what you need. Now available on a secure department site, CAP allows you to store, search through, and easily access commonly-used course materials. We will give a short demo of the site and discuss effective ways to organize materials, including strategies to upload documents and enter information. Pizza and pops will be available.
We consider regularity for Monge solutions to the optimal transport problem when the initial and target measures are supported on the embedded sphere, and the cost function is the Euclidean distance squared. Gangbo and McCann have shown that when the initial and target measures are supported on boundaries of strictly convex domains in $\mathbb{R}^n$, there is a unique Kantorovich solution, but it can fail to be a Monge solution. By using PDE methods, in the case when we are dealing with the sphere with measures absolutely continuous with respect to surface measure, we present a condition on the densities of the measures to ensure that the solution given by Gangbo and McCann is indeed a Monge solution, and obtain higher regularity as well. This talk is based on joint work with Micah Warren.
The computation of the joint probability distribution
of the eigenvalues of random matrices originating in
statistical mechanics often leads to the evaluation
of a Selberg-type integral which can be reduced to a
problem in algebraic combinatorics. Such a problem
was first identified by Dyson in 1962, leading to
far reaching generalizations. We present a simple
algebraic method which among others leads to a very
short proof of Andrews' q-Dyson conjecture as well as
to the solution of a long-standing open problem of
Forrester.
It seemed quite difficult to determined crossing number of a link until Jones discovered a new polynomial invariant. Important results about crossing number of a link were shown after Kauffman gave a method for calculating Jones polynomial.
I will survey the relation between the reduced degree of Jones polynomial of a link and its crossing number and discuss Jones polynomial of a pretzel link. I will introduce an adequate diagram and discuss minimality of a diagram from the viewpoint of it. I will discuss the difference between the reduced degrees of Jones polynomials of pretzel links and their crossing numbers, and whether Jones polynomial is a complete invariant on alternating pretzel knots.
Consider a Poisson point process of intensity \lambda in R^d and let each
point move as an independent Brownian motion. Consider a target particle
that is initially placed at the origin at time 0 and can move according to
any continuous function. We say that the target is detected at time t if
there exists at least one point of the point process within distance 1 of
the target at time t. We show that if \lambda is sufficiently large, then
the target will eventually be detected even if its motion can depend on the
past, present and future positions of the points. In the proof we use
coupling and multi-scale analysis to show that some good events percolate in
space and time.
In early 1960's, Lang proved that if for a given polynomial G(X,Y) with complex coefficients, there exist infinitely many pairs (x,y) where both x and y are roots of unity such that G(x,y) = 0, then essentially G(X,Y) = X^mY^n - c, for some integers m and n, and a root of unity c. In 2009, Masser and Zannier proved a result (similar in the spirit of Lang's result) for torsion points on a family of elliptic curves. In our talk we explain how both results come from the same general principle in arithmetic geometry, and at the same time we present a partial result to a more general conjecture which subsumes both Lang and Masser-Zannier theorems.
Note for Attendees
Refreshments will be served between the two talks.
Representations of the Galois group of the rationals abound in nature and are important in understanding the structure of the Galois group of the rational numbers. Iwasawa theory studies certain arithmetic modules arising from such representations. Hida theory provides a technique to package some of these representations in a family and study them simultaneously. In this talk I will demonstrate the usefulness of this technique in obtaining information on certain fundamental invariants arising in Iwasawa theory for such families and studying their variation within a family.
The Schroedinger map PDE is a simple, natural, geometric (hence nonlinear) generalization of the classical (linear) Schroedinger equation of quantum mechanics. Remarkably, it is also a basic model in ferromagnetism.
Without assuming a PDE background, I will discuss the key questions one asks about the qualitative properties of solutions of such equations -- which concern singularity (non)-formation and asymptotic behaviour -- leading up to results of mine (joint with Eva Koo, and with Kenji Nakanishi and Tai-Peng Tsai) and others which address these questions, and along the way touching on recent developments in the field of nonlinear dispersive equations more generally.
This thesis includes three papers and one expository chapter as background for one of the papers. They have in common that they combine algebra with discrete geometry, mostly by using algebraic tools to prove statements from discrete geometry. Algebraic curves and number theory also recur throughout the proofs and results.
In the first paper, we prove that an infinite set of points in the real plane such that all pairwise distances are rational cannot be contained in an algebraic curve, except if that curve is a line or a circle, in which case at most 4 resp. 3 points of the set can be outside the line or circle. In the proof we use the classi cation of curves by their genus, and Faltings' Theorem.
In the second paper, we prove a bound on the number of unit distances that can occur between points of a finite set in the real plane, under the restriction that the line segments corresponding to these distances make a rational angle with the horizontal axis. In the proof we use graph theory and an algebraic theorem of Mann.
In third paper, we give an upper bound on the length of a simultaneous arithmetic progression (a two-dimensional generalization of an arithmetic progression) on an elliptic curve, as well as for more general curves. We give a simple proof using a theorem of Jarnik, and another proof using the Crossing Inequality and some bounds from elementary algebraic geometry, which gives better explicit bounds.
A new class of Inverse Problems seeks to significantly improve both the quantitative accuracy and the resolution of traditional inverse boundary value problems by using data which can be determined in the interior of the object. I will briefly explain how such measurements of current density can be obtained
using Magnetic Resonance Imaging in a non-standard way. Imaging electric conductivity then leads to beautiful mathematics involving minimal surfaces in a conformal metric determined by the measured data, and a corresponding variable coefficient least gradient problem.
This series of lectures will be concerned with some recent developments on how to find and use symmetries for PDEs; how to find and use conservation laws for PDEs; connections between symmetries and conservation laws for PDEs. Much of the material appears in Applications of Symmetry Methods to Partial Differential Equations (Bluman/Cheviakov/Anco), Springer (2010). Background material appears in Symmetry and Integration Methods for Differential Equations (Bluman/Cheviakov), Springer (2002) on Lie groups of transformations and their applications to solving ODEs, the construction of conservation laws (integrating factors) for ODEs, and finding invariant solutions of PDEs.Both of these books are available online through the UBC library.
No background beyond undergraduate mathematics is necessary as a prerequisite for this series. These lectures were presented at Moscow State University in June 2011 and will flow as follows.
General introduction giving an extensive overview of topics to be covered.
Review of Lie’s work on point symmetries and invariant solutions for PDEs.
Local symmetries—point, contact, higher-order.How to find them for a given PDE system.
Construction of conservation laws—direct method to find them, connections with Noether’s theorem.
Use of symmetries to construct new conservation laws from known CLs.
Invertible mappings of nonlinear PDEs to linear PDEs through symmetry analysis and through conservation law multiplier analysis.
Invertible mappings of linear PDEs with variable coefficients to linear PDEs with constant coefficients.
The nonclassical method to find solutions of PDEs.
How to systematically find trees of equivalent but nonlocally related PDE systems for a given PDE system.
How to systematically find nonlocal symmetries and nonlocal conservation laws for a given PDE system.
The multidimensional situation--gauge constraints.
In this talk, we will discuss about Li-Yau-Hamilton type differential Harnack inequalities, heat kernel estimates and their applications to study type I ancient solutions of the Ricci flow. Some of this is joint work with Q. S. Zhang.
We produce a derived equivalence for some examples of Projective Homogeneous Varieties (Severi-Brauer Varieties, Quadrics, and Involution Varieties) by constructing a Tilting Bundle in each case.
A fairly recent thesis by John Feldman (MIT 2002) presents
maximum-likelihood decoding of binary linear codes as an integer
linear program. For LDPC and turbo-like codes, certain relaxations
allow for a good decoding algorithm. Since then, there have been a
number of results on improved versions of the LP decoder, suggesting
that it may become a more useful paradigm than more commonly used
message-passing algorithms.
Following a 2008 by Navin Kashyap (Queen's), we discuss an old but
previously neglected connection between codes and matroids. We give a
survey of results obtained by matroid decomposition techniques, and
their implications for LP decoding.
In this talk I will prove that in the presence of bounds for the rank of the fundamental group and the injectivity radius, the kth eigenvalue of the Laplacian of a closed hyperbolic 3-manifold M is bounded from above and below by a multiple of vol(M)^{-2}.
I will present recent work on modelling treated HIV infection using branching process models. We use simulations and novel numerical methods to calculate probability distribution functions for virus and cell concentrations. An overview of relevant HIV biology will be included in the talk. Most of this work is joint with Jessica Conway (UBC).
Let L be a reductive Lie algebra. The strong Macdonald theorems of Fishel, Grojnowski, and Teleman state that the cohomology algebras of L[z]/z^N and L[z,s] (where s is an odd variable) are free skew-commutative algebras with generators in certain degrees. The theorems were originally conjectured by Hanlon and Feigin as Lie algebra cohomology extensions of Macdonald's constant term identity in algebraic combinatorics. The proof uses ideas from the Kahler geometry of the loop Grassmannian.
I will explain how to extend Fishel, Grojnowski, and Teleman's ideas to generalized flag varieties of (twisted) loop groups, and consequently get strong Macdonald theorems for p[s] and p/z^N p when p is a parahoric. When p has a non-trivial parabolic component the cohomology of p/z^N p is no longer free, as it contains a factor which is isomorphic to the cohomology algebra of the flag variety of the corresponding parabolic.
The finite element method is a widely accepted tool for the numerical solution of partial differential equations. Nowadays a posteriori error estimation is an expected and assessed feature in scientific computing. It is used for adaptively creating approximation spaces and to assess the accuracy of numerical solutions. The performance of the method can be improved by mesh refinement (h-refinement) or the use of higher oder ansatz spaces (p-refinement). Taking a combination of both (hp-refinement) can lead to exponentially fast convergence with respect to the number of degrees of freedom. Especially for hp-FEM there have been proposed several strategies for adaptively creating problem-dependent meshes, e.g. estimating the analyticity of the solution, solving local boundary value problems and minimize the global interpolation error can be minimized.
In this talk we present a fully automatic hp-adaptive refinement strategy, which is based on the solution of local boundary value problems. We present the strategy for the Poisson and the Maxwell boundary value problem and show convergence of the algorithm. The talk is concluded by some numerical examples.
In this talk, we present the 4th chapter of Bain's book. The problematic is to determine what college teachers should expect from their students. More info under : wiki.ubc.ca/Sandbox:MathTeachingSeminar
Tutoring is one of the most common TA assignments for new graduate students, and is a skill all TAs can benefit from improving. But what makes a tutor effective? What are some good practices to use while tutoring, and do we follow them? In this session, we will examine questions such as these in a reflective process and discussion in an attempt to better our abilities as tutors.
This is the first talk in the TA Accreditation Program Seminar Series; more information can be found at
The relativistic Vlasov-Darwin (RVD) system is a kinetic model that describes the evolution of a collisionless plasma whose particles interact through the self-induced electromagnetic field. In contrast with the Vlasov-Maxwell system, the particle interaction is assumed to be a low-order relativistic correction (i.e., the Darwin approximation) to the full Maxwell case. A consequence of this assumption is that instead of the less tractable hyperbolic Maxwell equations, the resulting system has elliptic features even though there is a fully coupled magnetic field. We use optimal transportation techniques to show uniqueness of the compactly supported weak solutions of the RVD system. Our proof extends the method used by Loeper in [J. Math. Pures Appl., 86 (2006), pp. 68-79 ] to obtain uniqueness results for the Vlasov-Poisson system. This is a joint work with Martial Agueh.
This work studies buoyant displacement flows with two miscible fluids in pipes and 2D channels that are inclined at angles close to horizontal. Detailed experimental, analytical and computational approaches are employed in an integrated fashion. The displacements are at low Atwood numbers and high Peclet numbers, so that miscibility effects are mostly observable after instability and via dispersive mixing. For iso-viscous Newtonian displacements, studying the front velocity variation as a function of the imposed flow velocity allows us to identify 3 distinct flow regimes: an exchange flow dominated regime characterized by Kelvin-Helmholtz-like instabilities, a laminarised viscous displacement regime with the front velocity linearly increasing with the mean imposed flow rate, and a fully mixed displacement regime. The transition between the first and the second regimes is found to be marked by a stationary layer of displaced fluid. In the stationary layer the displaced fluid moves in counter-current motion with zero net volumetric flux. Different lubrication/thin-film models have been used to predict the flow behaviour. We also succeed in characterising displacements as viscous or inertial, according to the absence/presence of interfacial instability and mixing. This dual characterisation allows us to define 5-6 distinct flow regimes, which we show collapse onto regions in a two-dimensional parameter plane. In each regime we have been able to offer a leading order approximation to the leading front velocity. A weighted residual method has also been used to include the effect of inertia within the lubrication modelling approach, which allows us to predict long-wave instabilities.
The Torelli group is the subgroup of the mapping class group which acts trivially on the homology of the surface. It is the first term of the Johnson filtration, the sequence of subgroups which act trivially on the surface group modulo some term of its lower central series. We prove that the abstract commensurator of each of these subgroups is the full mapping class group. This is joint work with Martin Bridson and Juan Souto.
The Carmichael lambda function \lambda(n) is defined to be the smallest positive integer m such that a^m \equiv 1 \pmod{n} for all (a,n)=1. \lambda_k(n) is defined to be the k th iterate of \lambda(n). We will discuss some previous known results about k=1,2 as well as sketch a proof of a normal order for n/\lambda_k(n) for all k.
Note for Attendees
Refreshments will be served between the two talks.
We will survey results on, and techniques for, the generalized Fermat equation x^p + y^q = z^r, where p, q, and r satisfy 1/p + 1/q + 1/r < 1. This is joint work with Imin Chen, Sander Dahmen, and Soroosh Yazdani.
We develop hp-adaptive finite element schemes for the numerical approximation of linear second-order elliptic boundary-value problems. We begin by reviewing some classical results from the late eighties. Then we show that on geometrically and anisotropically refined meshes in three dimensional polyhedral domains, hp-methods achieve exponential rates of convergence in the number of degrees of freedom, thereby proving a longstanding conjecture in the theory of hp-finite element methods. In the last part of the talk, we discuss some recent results related to fully automated hp-adaptivity and illustrate numerically that our proposed adaptive refinement strategies lead to exponential rates of convergence. The talk includes work done with my former Ph.D. student Liang Zhu.
We will introduce and study the notion of an equivariant pretheory. Basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes of the respective Tits algebras.
This talk presents the development of BEM and XFEM frameworks for modelling hydraulic fractures, which arise in a wide range of geoengineering applications. The mathematical formulation of the problem involves a system of coupled nonlinear partial differential equations with a moving boundary, arising from the coupling between the fluid flow in the evolving fracture and the deformation of the parent material. Each of the discussed approaches has its own advantages: the BEM can efficiently simulate a propagating crack in linear homogeneous domains, while the XFEM is able to model complex settings such as multiple fractures in porous and layered rocks or plastic material deformation. The first part of the talk presents a BEM algorithm coupled with the finite-volume fluid flow model. An example of a near-surface radial crack is investigated, for which the required Green’s functions, that represent the crack as a distribution of material discontinuities, are derived. A comparison of the numerical results generated by this numerical model with data from laboratory experiments identifies particular physical phenomena that have to be accounted for in the mathematical formulation for accurately capturing the complex fracture propagation process.
In the second part of the talk, an XFEM approach to this problem is discussed. The development includes derivation of shape functions that enrich the underlying finite element formulation by representing discontinuities and singularities associated with the hydraulically driven crack. An example is presented in which a coupled XFEM model simulates a crack driven by a viscous fluid through a layered material.
It will be shown how to find directly all conservation laws for a given PDE system. A consequence is the extension of the classical Noether theorem to non-variational systems. Connections to Noether's theorem and connections between symmetries and conservation laws will be discussed.
In this talk, I will first recall the notions of superlinearity and subcriticality for strongly coupled elliptic systems. I will present various functional frameworks and their limitations. I will then discuss two reduction methods that allow to get rid of the indefiniteness of the energy functional. These reductions to a single equation are powerful to treat basic questions for superlinear systems. For instance, I will discuss the notion of ground states, in bounded domains and in R^N, show how to get the information on the symmetry and the sign of the ground states through the definition of a convenient Nehari manifold or constrained minimization problem. I will also discuss the classical question of existence of infinitely many critical points of perturbed indefinite symmetric functionals and how one of the reduction method allow to use the notion of Morse index. Finally, I will show how these reduction methods can help in proving partial symmetry and symmetry breaking. As a paradigm, I will illustrate the ideas on the Lane-Emden system with Hénon weights.
Consider a crystallographic root system together with its Weyl group W acting on the weight lattice M. Let Z[M]^W and S^*(M)^W be the W-invariant subrings of the integral group ring Z[M] and the symmetric algebra S^*(M) respectively. A celebrated theorem of Chevalley says that Z[M]^W is a polynomial ring over Z in classes of fundamental representations w_1,...,w_n and S^*(M)^{W} over rational numbers is a polynomial ring in basic polynomial invariants q_1,...,q_n, where n is the rank. In this talk we will establish and investigate the relationship between w_i's and q_i's over the integers.
We show that the time for a random walk to cover all
the edges of a graph with m edges is bounded by 2m^2;
if all edges must be covered in both directions, 3m^2.
These results generalize to graphs with edge-lengths
(even with infinitely many vertices) and to Brownian
motion.
Joint work with Agelos Georgakopoulos.
I will try to give a short survey of some of the major results in the study of Kleinian groups in recent years. We will concentrate on the proof of the Bers' Density Conjecture and I will try to give an outline of the proof based on a joint work with J. Souto.
The curve shortening flow has been much studied in differential geometry over the past 25 years. In this talk we will introduce the problem,
which involves the flow of a plane curve in the normal direction with speed equal to the curvature. It was shown by M. Grayson in 1987 that
the flow converts an arbitrary "nice" closed curve into a circle! Over the next 10 years Grayson's proof was clarified and simplified by
R. Hamilton and G. Huisken.
We will describe the main ingredient in these simplifications, which involves finding a geometric quantity that is "improved" under the flow
and prevents the formation of certain types of singularities. Although this is a hard theorem, its proof uses only two variable calculus
together with a lot of cleverness. The curve shortening flow is the simplest of several geometric flows whose study is a topic of current
research interest in differential geometry. These flows have certain features and methods in common, particularly with regard to the
formation of singularities.
We show that in any submodular system there is a maximal chain that is minimal, in a very strong sense, among all paths from 0 to 1. The consequence is a set of general conditions under which optimal
scheduling can be done without backward steps. Among the applications are a fast algorithm for scheduling multiple processes without overusing a resource; a theorem about searching for a lost child in a forest; and a closed-form expression for the probability of escaping from the origin in a form of coordinate percolation. Joint work in part with Graham Brightwell (LSE) and in part with Lizz Moseman (NIST).
We construct an isomorphism from Somekawa's K-group associated to a finite collection of semi-abelian varieties (or more general sheaves) over a perfect field to a corresponding Hom group in Voevodsky's triangulated category of effective motivic complexes.
There is a continuing interest in developing numerical methods for treating problems that are characterized by multiple time scale features. Traditional finite element method (FEM) based on semi-discrete schemes, however, is not well suited for these classes of problems due to their lack of flexibility in establishing multiscale approximations in the temporal domain. In this presentation, we show that a multiscale method that is capable of incorporating both multiple spatial and temporal features can be established based on the space-time
discontinuous Galerkin method which was originally developed in the context of linear elastodynamics. After an initial assessment of the convergence and its connection to the various time stepping algorithms, we show that space-time FEM is a stable, high-order convergent numerical method. We further explore the incorporation of fine scale features based on the extended finite element method. The nonlinear formulation incorporating enriched space-time FEM with stabilization least-square term is further developed and numerical solution based on GMRES is proposed. Through numerical examples, it is shown that multiscale space-time FEM enjoys superior convergence properties over the traditional space-time FEM and the proposed method represent a new paradigm towards resolving structural and solid mechanics problems with strong temporal nonlinearity.
What do we know already? We present some recent data from a selection of first-year Math courses to look for indicators of poor study habits, and to give a sense of how widespread this issue can be. There are indicators from other departments that some students need a boost even later on in their programs.
How do we measure them? We will present the instruments we have used, and suggestions for collecting the same sort of data in your own class.
What are some ways to address study skill issues? We suggest some easy-to-implement strategies to help get students on track.
We will also have time for discussion about study skills - what have you noticed among your own students, and what have you tried?
In this talk, we consider the problem of non-existence of solutions for some basic elliptic equations and systems with weights. Starting with Henon-Lane-Emden system, we present a Liouville-type theorem for bounded solutions in dimension N=3 as well as the full Henon-Lane-Emden conjecture in higher dimensions. Since systems are normally much more complicated than equations, in higher dimensions we back to single equations (both second order and fourth order) to prove such theorems under some additional assumptions on solutions.
Also, during the talk we will see many open problems.
This work has been done under supervision of N. Ghoussoub.
It is a long standing open problem in algebraic combinatorics to compute the Kronecker coefficients of the symmetric group using a combinatorial rule. In this talk I will discuss explicit combinatorial formulae (for 5 new cases) for Kronecker coefficients arising in the inner tensor product of two Schur functions indexed by near-rectangular partitions of small height. As an application of the description of Kronecker coefficients thus obtained, I will also also describe an enumerative formula for a specific case of counting Standard Young tableaux of bounded height in terms of Catalan and Motzkin numbers.
Interfacial instabilities of multi-layer shear flows may be eliminated by
astute positioning of yield stress fluid layers that remain unyielded at
the interface(s). This lecture consists of three parts.
Firstly, I present a computational study of these flows in the setting of
a Newtonian core fluid surrounded by a Bingham lubricating fluid, within
pipe and channel configurations. The simulations include an inlet geometry
in the computational model and study the multi-layer flows, both as the
fluids are initially injected (start up) and later the established steady
flows (development lengths). Nonlinear perturbations are also studied,
showing in particular that during energy decay of stable perturbations the
initial rapid decay of the perturbation kinetic energy relates to
reforming/breaking of the unyielded plug and is followed by slower viscous
decay. For axisymmetric perturbations these flows can be stable to order
unity initial perturbation amplitudes and for Re<100. The channel geometry
allows for symmetry breaking and appears to be less stable. A number of
interesting effects are explored using the channel geometry.
Secondly, I focus on demonstrating whether the stable core annular flow can
be achieved when lubricating a visco-elastic core fluid with a yield stress
fluid. Over 100 experiments have been performed using Carbopol solutions as
the lubricating yield stress fluid and Polyethylene Oxide solutions as the
visco-elastic fluid.
Thirdly, I will briefly explain application of energy stability method to
study nonlinear stability of a core-annular flow of an Oldroyd-B fluid
surrounded by a Bingham fluid. Together with the experimental study, this
shows that visco-elasticity is not a barrier to use of this methodology.
Contributors: S. Hormozi(1,2), I.A. Frigaard(1,2), K. Wielage-Burchard(2) &
D.M. Martinez(3)
Let G denote a compact connected Lie group with torsion-free fundamental group acting on a compact space X such that all the isotropy subgroups are connected and of maximal rank. Let T be a maximal torus with Weyl group W. We derive conditions on the induced action of W on the fixed-point set of T which imply that the equivariant K-theory of X is a free module over the representation ring of G. This can be applied to compute the equivariant K-theory of spaces of ordered commuting elements in certain compact Lie groups. This is joint work with J.M.Gomez.
We show that crossing probabilities in 2D critical
percolation converge at a polynomial rate in the mesh size to their
limit given by the Cardy-Smirnov formula. We then use this to obtain
that in the half plane the probability that the cluster at the origin
has diameter R decays like R^{-1/3} with polylogarithmic corrections,
improving the previously known estimate of R^{-1/3+o(1)}.
Joint work with Dana Mendelson, Scott Sheffield and Sam Watson.
A primitive set is a set of positive integers with the property that no element of the set divides another. We review a 1934 result of Besicovitch, showing that certain primitive sets can have large upper density even though their counting function is usually small, and a 1935 result of Erdős, showing that the sum of 1/(n log n) over all elements n of a primitive set is bounded by an absolute constant. We go on to describe two new theorems on primitive sets. First, in joint work with Carl Pomerance, we construct primitive sets with consistently large counting functions (as opposed to occasionally large as in Besicovitch's example), essentially providing a converse to Erdős's theorem. Second, the optimal absolute constant in Erdős's theorem is conjectured to be the sum of 1/(p log p) over all primes p, but this conjecture is still open; we describe current joint work with Bill Banks that makes progress towards this conjecture.
Note for Attendees
Refreshments will be served between the two talks.
Deligne and Kazhdan used the notion of close local fields to establish a way of analyzing representations of Galois groups and Algebraic groups over fields of characteristic p, using information about corresponding representations in characteristic 0. I will first explain this theory and some related results. Then I will state our results on the extension of Kazhdan's and Howe's work on Hecke Algebra isomorphims. I will also explain how this extension has enabled us to generalize Lemaire's work on generic representations of Gln(F) over close local fields.
As has been known since the time of Gromov's Nonsqueezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. In the past few years we have gained significant new insight into the question of when there is a symplectic embedding of one basic geometric shape (such as a ball or ellipsoid) into another (such as an ellipsoid or torus). After a brief introduction to symplectic geometry, this talk will describe some of this progress, with particular emphasis on results in dimension four.
Diffeomorphisms that preserve a symplectic structure have unexpected rigidity properties. In particular, many manifold have subsets that cannot be displaced (i.e. moved to a disjoint position) by a symplectic isotopy though they can be smoothly displaced. Toric manifolds provide a good setting in which to study these questions because they have a purely combinatorial description.
This talk will describe some recent progress in understanding which toric fibers can be displaced. I will try to make the subject accessible to those who do not know toric or symplectic geometry.
The quantification of uncertainty in large-scale scientific and engineering computations is rapidly emerging as a research area that poses some very challenging fundamental problems which go well beyond sensitivity analysis and associated small fluctuation theories. We want to understand complex systems that operate in regimes where small changes in parameters can lead to very different solutions. How are these regimes characterized? Can the small probabilities of large (possibly catastrophic) changes be calculated? These questions lead us into systemic risk analysis, that is, the calculation of probabilities that a large number of components in a complex, interconnected system will fail simultaneously.
I will give a brief overview of these problems and then discuss in some detail two model problems. One is a mean field model of interacting diffusions and the other a large deviation problem for conservation laws. The first is motivated by financial systems and the second by problems in combustion, but they are considerably simplified so as to carry out a mathematical analysis. The results do, however, give us insight into how to design numerical methods where detailed analysis is impossible.
Note for Attendees
This talk will be preceded by a reception at 2:00 pm in Math 125.
Let X be a smooth projective scheme over a field F. We say that Rost nilpotence is true for X in the category of Chow motives with integral coefficients if for any field extension E/F the kernel of
CH_2(S x S) --> CH_2(S_E x S_E)
consists of nilpotent correspondences. In my talk I will present a proof of Rost nilpotence for surfaces over fields of characteristic zero which uses Rost's theory of cycle modules.
To fit a model of a process described by a system of Partial Differential Algebraic Equations to a given set of experimental data we have to solve constrained, nonlinear parameter estimation problems. Since the data usually contains statistical errors the parameters are random variables too. The uncertainty of a parameter estimation can be quantified by the variance-covariance matrix of the estimator.
For minimizing the confidence region of the parameter estimation an optimized experimental setup is needed. We present our approach for the minimization of quality criteria on the variance-covariance matrix of the parameters. Thereby process controls and the layout of measurements are the optimization variables.
Our approach are derivative based optimization strategies. We introduce the general Optimum Experimental Design optimization problem and the methods implemented in the software package VPLAN, such as Quasi-Newton methods, tailored derivative evaluation by Internal Numerical Differentiation and Automatic Differentiation and exploitation of multiple experiment structures. To use experimental design for practical problems, we have developed strategies including robustification, multiple experiment formulations, a sequential strategy and an online-approach.
In the second part of the talk we give an overview of parameter estimation methods to fit the parameters to the data. We treat this kind of problems by (reduced) Gauss-Newton-Type methods and multiple-shooting. Furthermore, we will give a short outlook on what is next to come (multiple shooting for OED, Proper Orthogonal Decomposition (POD) and reduced approach).
In Part V, we show how one can use a symmetry (discrete or continuous) of a given PDE system to map a given conservation law into another conservation law--in particular, a recently found simple and explicit formula will be presented. In addition, we show how one can use symmetries and conservation laws to determine whether a given nonlinear PDE system can be invertibly mapped to a linear PDE system and also show how to find such a mapping when it exists.
In this talk I will discuss the notion of Rost nilpotent and point out some of its applications to quadratic forms or projective homogeneous varieties.
Associated to a tree T in the boundary of Outer space is a symbolic dynamical system called the dual lamination of T, denoted L^2(T). We develop a two-part inductive procedure for studying L^2(T). One part is known: it is a slight generalization of the Rips machine as developed by Coulbois-Hilion; the other part is new: it is a generalization of the classical Rauzy-Veech induction. As an application we characterize trees T for which L^2(T) is minimal. As a further application we give a description of the Gromov boundary of the complex of free factors: it is the space of measure classes of arational trees. (Jointly with T. Coulbois and A. Hilion.)
Given a Markov chain, when is it possible to construct two or more copies of the chain such that they never coincide? For random walks on the complete graph we show that it is possible to couple roughly $n^{log2/log5}$ walkers for all $n$, and $n^{1/2}$ for some $n$.
Joint with Alexander E. Holroyd, James Martin, David B. Wilson and Peter Winkler.
I will discuss some recent results (joint with Zev Klagsbrun and Barry Mazur) on the distribution of 2-Selmer ranks in families of quadratic twists of elliptic curves over arbitrary number fields. We study the density of twists with a given 2-Selmer rank, and obtain some surprising results on the fraction of twists with 2-Selmer rank of given parity. Since the 2-Selmer rank is an upper bound for the Mordell-Weil rank, this work has consequences for Mordell-Weil ranks in families of quadratic twists.
This talk will explore a technique of using equivariant cohomology to say something about the action of a group on the cohomology of a space. In particular, we will look at examples of cohomology of flag varieties and configuration spaces. Also, we will look at a family of algebras with an algebro-geometric interpretation that admits an S_n action, and use the results we developed to make progress toward a result about these algebras.
This will be the first of two expository lectures on the Guth-Katz solution of the Erdos distinct distance problem in the plane. The first talk will introduce the polynomial method and focus on the "joints problem".
Compressed sensing has been one of the great successes of applied mathematics in the last decade. It allows one to reconstruct sparse signals from seemingly incomplete collections of measurements, and thereby circumvent the classical Nyquist barrier. However, compressed sensing is currently a finite-dimensional theory: it concerns the recovery of vectors in finite-dimensional vector spaces. With this in mind, the purpose of this talk is to introduce a new framework that extends the current theory and techniques to infinite-dimensional problems.
This new framework originates from recent developments in classical (i.e. Nyquist rate) signal recovery, known as generalized sampling. Generalized sampling, which I will introduce in the first part of the talk, allows for signal reconstruction in arbitrary bases in a manner which is both numerically stable and, in a certain sense, optimal. The infinite-dimensional compressed sensing framework builds on this approach by allowing one to take advantage of sparsity to obtain significant subsampling.
In Part VI, we will show how to solve two important mapping problems.
1, Given a nonlinear PDE system, does their exist an invertible mapping to a linear PDE system? Find such a mapping when it exists.
2. Given a linear PDE with variable coefficients, does their exist an invertible mapping to a linear PDE system with constant coefficients? Find the most general such mapping when it exists.
For any smooth compact Riemannian 3-manifold with boundary, we prove that there always exists a smooth, embedded minimal surface with (possibly empty) free boundary. We also obtain a priori upper bound on the genus of such minimal surfaces in terms of the topology of the ambient compact 3-manifold. An interesting note is that no convexity assumption on the boundary is required. In this talk, we will describe the min-max construction for the free boundary problem, and then we will sketch a proof of the existence part of the theory.
Number conserving cellular automata (NCCA) are discrete
models of particles in a grid of cells that move according to a local
rule. I will present some results for surjective one-dimensional NCCA.
For example, we will find what to expect of a reversible traffic model
with no mean drivers.
I will describe recent experiments in which an angled plate is dragged horizontally over a fluid surface; the plate is attached by a pivot in such a way as to be able to move freely up and down in response to the lift force from the fluid. We find that the steady planing state, in which the plate's vertical position remains constant, becomes unstable if it is towed above a threshold speed. Instead, the plate oscillates up and down, leaving a rippled imprint on the fluid layer. The same instability occurs on a granular layer, and is responsible for the troublesome phenomenon of 'washboard' or 'corrugated' roads.
After describing the experiments I will discuss attempts to rationalize these observations using simple theoretical models for the dynamics of the plate and the lift force provided by the deforming fluid. Special attention will be given to yield stress fluids, for which the imprinted pattern remains 'frozen' into the surface.
We consider continuous time simple random walks with arbitrary speed measure \theta on infinite weighted graphs. Write p_t(x,y) for the heat kernel of this process. Given on-diagonal upper bounds for the heat kernel at two points x_1,x_2, we obtain a Gaussian upper bound for p_t(x_1,x_2). The distance function which appears in this estimate is not in general the graph metric, but a new metric which is adapted to the random walk. Long-range non-Gaussian bounds in this new metric are also established. Applications to heat kernel bounds for various models of random walks in random environments are discussed.
I will first review the notion of Galois averages of Rankin-Selberg L-functions, in particular those of Rankin-Selberg L-functions of weight-two cusp forms times theta series associated to Hecke characters of imaginary quadratic fields. I will then present a conjecture about the behaviour of these averages with the conductor of the character, of which the nonvanishing theorems of Rohrlich, Vatsal and Cornut-Vatsal are special cases. Finally, I will explain a strategy of proof, at least in the setting where the class number is equal to one.
Note for Attendees
Refreshments will be served between the two talks.
Ergodic optimization is the process of finding invariant probability measures that maximize the integral of a given function. It has been conjectured that ‘most’ functions are optimized by measures supported on a periodic orbit, and it has been proved in several separable spaces that an open and dense subset of functions is optimized by measures supported on a periodic orbit. All known positive results have been for separable spaces. We give in this paper the first positive result for a non-separable space, the space of super-continuous functions on the full shift, where the set of functions optimized by periodic orbit measures contains an open dense subset.
Part of the Langlands Program is to find a meaningful correspondence between representations of Galois groups and representations of reductive algebraic groups. I will attempt to motivate this through an example and then concentrate on what happens at a (real) Archimedean place of the global picture. In this context the idea of endoscopy arises in a natural fashion and suggests identities between representations of different Lie groups. These identities have been proven by Shelstad. I will sketch the theory of endoscopy under twisting by a group automorphism and describe character identities between discrete series representations.
Topological essential dimension of a covering is the minimal dimension of a base-space such that the original covering can be induced from some covering over this base-space.
We will see how to compute the topological essential dimension for coverings over tori.
Surprisingly this question turns out to be useful in obtaining estimates in Klein's resolvent problem: what is the minimal number k such that the equation z^n+a_1z^n+...+a_n=0 with complex coefficients a_1,...,a_n can be reduced by means of a rational substitution y=R(z,a_1,...,a_n) to an equation on y depending on k algebraically independent parameters.
We will also obtain some bounds in the analogue of this question for other algebraic functions and get a sharp result for functions on C^n unramified outside of coordinate hyperplanes.
This talk discusses algorithms to solve systems of linear equations where the matrix is the Laplacian matrix of a graph. These systems arise in many applications: in scientific computing, when using the finite difference method to approximately solve Poisson's equation; in machine learning, in some methods for semi-supervised learning on graphs; and in theoretical computer science, in fast algorithms for network flow problems.
For two decades, theoretical computer scientists have been developing algorithms with provable running-time bounds for solving such systems of equations. The current state-of-the-art algorithm computes a solution with relative error epsilon in the energy norm in running time O(m log n (log log n)^2 log(1/epsilon)) for any Laplacian matrix of size n x n with m non-zero entries.
These algorithms use several sophisticated tools, including low-stretch trees and concentration of random matrices. In this talk we give a survey of these results.
This informal presentation provides an overview of current technologies that are being used at UBC to enhance student learning through online technologies, including screen casting, WeBWorK, Wolfram Alpha, UBC Blogs, and the UBC Wiki.
1. Given a linear PDE with variable coefficients, find all forms of the variable coefficients so that the PDE can be mapped invertibly to a linear PDE with constant coefficients. This problem was originally posed by Kolmogorov in a limited way in his celebrated paper introducing "Kolmogorov's equation". In particular, he posed the problem of when can a diffusion process be mapped into a Wiener process. The complete solution to this problem was found in the early 1980's using group methods.
2. The so-called nonclassical method to find systematically solutions of nonlinear PDEs. This significantly extends Lie's work on finding invariant solutions for PDEs based on invariance under point (Lie) symmetries. In Lie's approach, one looks for symmetries (that are invertible transformations) that map any solution of a PDE to another solution of the PDE, i.e, leave invariant the solution manifold and then seeks corresponding invariant solutions. In the "nonclassical" approach, one seeks symmetries that leave invariant a submanifold of solutions of a given PDE (but maps other solutions of the PDE to solutions of another PDE) and then seeks corresponding invariant solutions. This approach has proven very fruitful to discover solutions of many well-known nonlinear PDEs.
In 1922 J.F. Ritt classified rational functions of prime degree invertible in radicals. Since then exactly the same family of rational functions or families of rational functions closely related to it have appeared in many different parts of mathematics, for instance as commuting rational functions (with interpretation as integrable discrete dynamical systems) or as rational functions that appear in rational function analogue of Schur's conjecture in number theory.
We will see several characterizations of these rational functions and discuss a possible generalization of them to many dimensions. This proposed generalization is related to actions of groups generated by generalized complex reflections. For instance the polynomials among these rational functions have been previously studied under the name of "folding polynomials" because of the way they act on a fundamental domain of a real group generated by reflections.
I will present the ideas of a version of a theorem of Hedlund for compact laminations (or foliations). More precisely: If L is a compact minimal Riemannian lamination by surfaces of negative curvature, we give a sufficient condition for the horocycle flow on its unit tangent bundle to be minimal, in other words every orbit of the flow is dense.
We investigate the survival probability, \theta_n, in high-dimensional statistical physical models, where \theta_n denotes the probability that the model survives up to time n. Models to which our results apply are oriented percolation above 4+1 dimensions, the contact process above 4+1 dimensions, and lattice trees above 10 dimensions. We show that, similarly to branching processes, for these models, Kolmogorov's result that n\theta_n converges holds, as well as Yaglom's theorem stating that, conditionally on survival to time n, the number of particles is approximately n times an exponential random variable.
In more detail, we prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if a certain self-repellence conditions is satisfied, then n\theta_n->2/(AV), where a) A is the asymptotic expected number of particles alive at time n, and b) V is the vertex factor of the model. Our proofs are based on simple weak convergence arguments.
In the case of oriented percolation, this reproves a result with den Hollander and Slade (that was proved using heavy lace expansion arguments), at the cost of losing explicit error estimates.
[This is joint work with Mark Holmes, building on work with Gordon Slade, Frank den Hollander and Akira Sakai.]
The reflector antenna problem is the problem of constructing a reflective surface which directs a specified energy distribution (i.e. light, or radio signals) emanating from the center of a sphere to another specified energy distribution on the so-called far-field sphere. I will discuss some of the basic analytic and geometric background of this problem, and a connection that has been discovered in recent years to Monge's optimal mass transportation problem.
Note for Attendees
Tea & cookies PRECEDING the talk at 2:45 in the WMAX upstairs lounge!
The physicists Bershadsky, Cecotti, Ooguri and Vafa argued that the mirror to the theory of Gromov-Witten invariants is provided by a certain quantum field theory on Calabi-Yau varieties. I'll describe joint work in progress with Si Li, which gives a rigorous construction of the BCOV quantum field theory. In the case of the elliptic curve, Li has shown that our theory recovers the Gromov-Witten invariants of the mirror curve, and so proving mirror symmetry in this example.
We'll present the sixth chapter of Bain's book on the relationship between teachers and students. All info under : http://wiki.ubc.ca/Sandbox:MathTeachingSeminar
I will first discuss actions of finite groups on metric spaces by Lipschitz maps (maps which preserve distance up to a constant), starting with an open problem about the group with two elements.
I will then discuss the passage to infinite groups, leading to recent results on the fixed-point property for random groups in Gromov's density model.
A vortex column supports oscillations known as Kelvin modes. These
modes only exist for vortices with a compact core. For smooth vortices, the
Kelvin modes are replaced by exponentially damped quasi-modes, a collective
response of the continuous spectrum eigenfunctions. In the first part of
this talk, we discuss the inviscid response of a 2D vortex to external
disturbances. We show that for certain initial conditions, vorticity
gradient can arrest the Landau damping process by a screening mechanism. In
the second part of the talk, we discuss a novel way to understand the
continuous spectrum of smooth 3D vortices. This is facilitated by a well
known analogy between 3D disturbances to a vortex column and disturbances in
a stratified shear flow.
We show the convergence of large random quadrangulations, i.e. random decompositions of the sphere into a large number of quadrangles, towards the so-called Brownian map, which is a universal model for a continuum random surface. Proving this convergence, which holds in the Gromov-Hausdorff topology after proper rescaling of distances in the random map, requires a precise study of geodesics in large quadrangulations and in the limiting space, and in particular, of the locus where geodesics tend to separate.
There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems, especially for young researchers. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both ``bottom up'' and ``top down'' approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.
The study of finite-sheeted covering spaces of 3-manifolds has been invigorated in recent years by the resolution of several long-standing conjectures by Kahn-Markovic, Agol and Wise. In this talk, I will discuss how using this work one can reformulate some of the central open questions in the field in terms of objects called solenoids. These objects are formed by taking inverse limits of families of finite-sheeted covering spaces of a compact manifold M, and they can be thought of as pro-finite analogues of covering spaces of M. While such an object can in general be quite complicated, I will show in this talk that if M is a compact aspherical 3-manifold, then the solenoid given by taking the inverse limit of the family of all finite-sheeted connected covering spaces of M looks like a disk from the perspective of Cech cohomology with coefficients in any finite module. I will then talk about the relevance of this result to elementary questions about finite-sheeted covers.
Let (b_n) = (b_1, b_2, \dots) be a sequence of integers. A primitive prime divisor of the k-th term is a prime which divides b_k but does not divide any previous term in the sequence. A zero orbit of a polynomial f(x) is a sequence of integers (c_n) where the n-th term is the n-th iterate of f at 0. We consider primitive prime divisors of zero orbits of polynomials and restrict to polynomials with zero linear term. In this talk, we show that if zero has an infinite orbit, then b_n will have primitive prime divisors for all n \geq 3.
We then consider a generalized notion of primitive prime divisors as it applies to families of commutative polynomials. We consider families of commutative polynomials which all have zero linear term and give an effective bound on the number of terms in the forward orbit of 0 which do not have primitive prime divisors.
This talk is based on joint work with Jason Bell and builds on previous joint work with Anna Haensch.
Note for Attendees
Refreshments will be served between the two talks.
Last year Francis Brown and Oliver Schnetz defined the c_2 invariant of a graph. Let p be prime, take the Kirchhoff polynomial of a graph, and count points on the variety of this polynomial over the finite field with p elements. For the graphs of interest to us, this point count will be divisible by p^2 and the result modulo p is the c_2 invariant at p.
This invariant has important things to say about the Feynman integrals of scalar Feynman graphs, and links together the combinatorial and algebro-geometric approaches to understanding Feynman integrals.
In this talk I will describe this setup and then explain some joint results with Francis Brown and Oliver Schnetz concerning the c_2 invariant of graphs with subdivergences.
Using techniques from the theory of Banach algebras, Kuranishi constructed an analytic germ of the moduli space of deformations of a holomorphic vector bundle on a compact complex manifold. In order to extend his results to deformation theory of perfect complexes, we introduce higher analytic stacks, which are simplicial Banach analytic varieties satisfying a horn filler condition modeled on that satisfied by Kan complexes. We show that there is a natural way to attach a higher analytic stack to a Banach algebra, and apply this to the deformation theory of perfect complexes. This is a joint work with Kai Behrend.
The coherent constructible correspondence matches coherent sheaves on a toric variety to constructible sheaves on a compact torus T^n. Microlocal sheaf theory allows one to view the latter sort of object as a Lagrangian submanifold in the symplectic manifold T^n x R^n, making this a form of mirror symmetry. I will discuss this correspondence, and an extension of of it to hypersurfaces in toric varieties, which in some sense matches coherent sheaves to Legendrian submanifolds of the contact manifold T^n x S^{n-1}.
Diagnostic test is a useful tool that you can build to measure and determine many aspects of your course and your incoming students: background, existing misconceptions, knowledge and beliefs about your subject - in short, anything you may wish to know and that will influence student learning in in your course. In the talk we will look at the diagnostic tests that have been built in our department for various courses, and discuss some practical issues of putting such a tool together.
If you are interested in creating a diagnostic test for a course you teach, please contact Mathematics Teaching and Learning Initiative - and we will build one for you, based on your needs and requirements. http://www.math.ubc.ca/~cwsei/
This week's TAAP session will be a debate: be it resolved that blackboards and whiteboards should not be used in the classroom as a medium to transfer information for students. David Kohler will argue for the affirmative, and Fok-Shuen Leung will argue for the negative.
This is the fifth talk in the TA Accreditation Program Seminar Series; more information can be found at http://www.math.ubc.ca/~fsl/TAAP%20Seminar%20Series.html
Domino tilings on the Aztec Diamond have been extensively studied over the past twenty years. Tilings of large Aztec diamonds exhibit four frozen regions and an elliptic liquid region. In this talk, we focus on the interface between the liquid and frozen regions, explaining some of the new behavior we have observed. This is ongoing work with Benjamin Young and Kurt Johansson.
To a logician diophantine geometry, the study of geometric relations on points of arithmetic origin, is almost a contradiction in terms since geometrical reasoning has long been known to be tame and decidable (from the 1929 work of Tarski on elementary geometry) while by Gödel's incompleteness theorems arithmetic is known to be infected by inherent undecidability. However, this tension has not prevented geometers and number theorists from investigating a subject they know to be hard but one in which ideas from geometry explain the apparent regularity of the solutions in rational numbers, roots of unity, or other arithmetically interesting to certain geometrically constrained systems of equations. In many important cases, model theorists can explain or prove such theorems by showing that the structure on the arithmetic points simply reflects the tameness of the class of definable sets in some structure intermediate between algebraic geometry and arithmetic.
With this lecture I will broadly survey the project of applying the model theory of enriched, though still tractable, geometries such as those coming from differential or difference algebra, to number theoretic problems such as those around the conjectures of Mordell-Lang and André-Oort or arising from arithmetic dynamics.
A way of quantifying is to impose integrability conditions on powers of the polynomial. This leads to the definition of the complex singularity exponent, an analytic invariant of the singularity that appears naturally in several branches of mathematics and has become known in algebraic geometry with the name of log canonical threshold. In this talk I will overview general properties of this invariant and present some applications to the study of algebraic varieties.
Wahl’s characteristic number is a topological invariant of the link of a normal surface singularity: its behavior under finite morphisms captures information about surface singularities that carry finite non-invertible endomorphisms. In this talk I will discuss how this is in fact a completely general invariant of an isolated normal singularity of any dimension. This fact has implications for instance regarding polarized endomorphisms of projective manifolds. The talk is based on joint work with S. Boucksom and C. Favre.
The "group" a knot in 3-space is by definition the fundamental group of its complement; it is one of the oldest algebraic tools used to study knots. Only recently was it discovered that all knot groups can be endowed with a left-invariant ordering. Some even have two-sided invariant orderings, while others do not. This talk will discuss the current state of the art on this subject, and why it is interesting. It will be accessible to grad students.
The so-called Harris criterion (A. B. Harris '74), is a non-rigorous argument, frequently used in theoretical physics literature to predict whether the critical properties of a statistical mechanics system will be qualitatively modified by a small concentration of impurities ("relevance of disorder"). Giving mathematical bases to this argument is an old challenge in the theory of disordered systems and rigorous results are very scarce. In recent years, the so-called polymer pinning model has proved to be the ideal context to attack this problem. In particular, we have given a rigorous justification of the Harris criterion for this class of models and we have solved a long controversy about the "relevance" of disorder for the pinning model in (1+1) dimensions. I will give a non-technical overview of related results.
Based on joint works with: B. Derrida, G. Giacomin, H. Lacoin.
Given a PDE system, it will be shown how to construct systematically a tree of nonlocally related systems with the properties that
*a solution of any PDE system within the tree yields, through a specific formula, a solution of any other system in the tree.
*there is not a one-to-one correspondence between two systems in the tree.
It will be shown that such systems within a tree can arise constructively as follows:
*through conservation laws.
*through symmetries.
*through subsystems.
As a consequence, one is able to seek systematically nonlocal symmetries and nonlocal conservation laws for a given PDE system. Many examples will be given including nonlinear wave equations, nonlinear diffusion equations, and gas dynamics equations. It will be shown that Eulerian and Lagrangian descriptions of continuum mechanics arise systematically as nonlocally related systems in a tree of nonlocally related systems
The situation is especially interesting in multidimensions where gauge constraints can play a central role.
The usual Mordell-Lang conjecture, a theorem of Faltings, asserts that if A is an abelian variety over C, \Gamma < A(C) is a finitely generated subgroup, and X \subseteq A is a closed subvariety, then X(C) \cap \Gamma is a finite union of cosets of subgroups of \Gamma. If one were to ask instead that A be defined over a field K of positive characteristic, then such a conclusion cannot hold in general as if A were defined over a finite field, F: A \to A were the associate Frobenius morphism, X \subseteq A were defined over the same finite field, and P \in Y(K) \cap \Gamma, then { Fn(P): n \in N } \subseteqY(K) \cap \Gamma. Other anomalous intersections may arise as sums of such orbits. Some years ago, in joint work with Moosa, I showed that these are essentially the only counterexamples to a naïve translation of the Mordell-Lang conjecture to semiabelian varieties defined over a finite field. Our proof which was long but elementary yields bounds which explicitly depend on the characteristic. In these lectures, I shall explain how to deduce characteristic independent bounds from a differential algebraic argument.
Note for Attendees
Refreshments will be served between the two talks.
I will begin from two different motivations: 1) (the mathematical physics motivation) trying to derive a macroscopic, deterministic "mean-curvature-type" evolution for the boundary between two coexisting phases, starting from a microscopic stochastic dynamics and 2) (the computer science motivation) analyzing the running time of Monte Carlo Markov Chain (MCMC) algorithms that sample uniformly among lozenge tilings of a large region of the plane. I will discuss the link between these two problems and try to emphasize their connections with dimer covering problems and with interacting particle systems. Our results include: a) bounds on the disappearence time of droplets for the three-dimensional Ising model at zero temperature b) derivation of mean-curvature evolution for the two-dimensional Ising model at zero temperature and c) (in some situations) "almost-optimal" bounds on the running time of the lozenge tiling MCMC.
Based on joint works with: P. Caputo, H. Lacoin, E. Lubetzky, F. Martinelli, F. Simenhaus, A. Sly.
Metallic gas-liquid foams are precursors to porous solid materials that are lightweight and useful in transportation, energy, medical and other applications. The crowded gas bubbles in a gas-liquid metal foam are highly unstable, because there is capillary-suction flow of liquid from the thin lamellar regions into the Plateau borders. The liquid drainage leads to lamellar rupture, gas bubble coalescence and rapid foam coarsening, so that successful solidification of a metal foam is not possible without liquid additives such as particles. Our research focuses on the microscale flow and interface evolution of pure liquid foam films, in order to glean knowledge that can be used to improve the processing of bulk metal foam. In this talk, our calculations of the thinning and the onset of instability of foam lamella and of the evolving unstable foam films with Plateau borders in a pure gas-liquid metallic foam will be presented. The linear stability results show that a draining lamella with Plateau borders is more stable than a lamellar film without flow and without Plateau borders. Numerical calculations track film evolution to a time just prior to rupture. The effects of variations in the Plateau border radii of curvature and of different initial conditions on rupture phenomena will be shown.
In practice, gas-liquid metal foams are stabilized against coarsening by the addition of particles. One effect of the addition of particles to thin draining films is the development of an oscillatory structural component of the disjoining pressure. Our research involving the analysis of a model of a particle-laden free film that includes the oscillatory structural component of the disjoining pressure will also be presented. By studying the effect of this component of the disjoining pressure on film dynamics we hope to gain insight into its role as a possible mechanism for foam stabilization by particles, to explain unique behaviors observed in particle-laden films such as stepwise thinning, and to identify potential novel film or foam structures. To date, the results of our analysis show that for certain ranges of film thickness to particle diameter ratios, a uniform film can spontaneously evolve into a multilayered film as a result of the structural oscillatory effect. The results reveal an analogy between the layering behavior of a particle-laden free film and classical first-order phase transformation kinetics observed in multicomponent solutions. A "phase" diagram delineating regions of stable layered and uniform films, constructed on a plot of the ratio of structural oscillatory to van der Waals components of the disjoining pressure versus the ratio of film thickness to particle diameter, will also be discussed.
Abstract: Recent advances in DNA sequencing technology have led to new
mathematical challenges in the analysis of the massive datasets
produced in current evolutionary studies. In particular, much progress
has been made in the design and analysis of computationally efficient
algorithms for assembling the Tree of Life from present-day molecular
sequences. In the first half of the talk, I will briefly review some
of the mathematical techniques that have led to our current
understanding of large-scale tree-building algorithms. Prior rigorous
results, however, have typically relied on models of molecular
evolution that are too simplistic. In the second half, I will discuss
recent work on the probabilistic modeling and analysis of more complex
settings, including mutation-rate variation across sites,
insertion-deletion events and lateral genetic transfer. No biology
background will be assumed.
In this talk we a introduce mixed finite element method for the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We consider mixed formulations in both the Stokes domain and the Darcy region, which yields the introduction of the traces of the porous media pressure and the fluid velocity as suitable Lagrange multipliers. Then, considering generic finite element spaces with some technical conditions, we apply the Babuska-Brezzi theory together with a classical result on projection methods for Fredholm operators of index zero to show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, we generalize the previous results and introduce a mixed finite element method for the coupling of fluid flow with nonlinear porous media flow.
This is joint work with G.N. Gatica (University of Concepcion) and F.-J. Sayas (University of Delaware).
I will describe recent results on a connection between the so-called reconstruction problem on Markov random fields on trees and two important problems in computational evolutionary biology: the inference of ancestral states and the estimation of phylogenies using maximum likelihood. This is joint work with Allan Sly. No biology background will be assumed.
Stratified flops show up in the birational geometry of symplectic varieties such as moduli spaces of sheaves. Varieties related by such flops are often derived equivalent (meaning that there is an equivalence between their derived categories of coherent sheaves). After recalling a bit about the geometry of flops I will discuss a general method for constructing such equivalences and illustrate with some examples and applications.
After recalling a few properties of sl(2), we will discover that it acts in a very elementary way via counting points. I will then sketch how this simple example is the starting point of many constructions in algebraic geometry with applications to representation theory and even low dimensional topology.
We show that on bounded degree graphs and general trees the cover time of the simple random walk is asymptotically equal to the product of the number of edges and the square of the supremum of the Gaussian free field on the graph, assuming that the maximal hitting time is significantly smaller than the cover time. Furthermore, for general trees, we derive exponential concentration for the cover time, which implies that the standard deviation of the cover time is bounded by the geometric mean of the cover time and the maximal hitting time.
In the talk, I will try to explain main steps for the proof of bounded degree graphs, including an application of sprinkling method, a detection argument for Gaussian free field, as well as a reconstruction of the embedded walk from local times which uses a connection with enumeration of Eulerian circuits.
More examples of nonlocally related systems and their use to obtain nonlocal symmetries. Focus on equations of gas dynamics--Eulerian, Lagrangian and other nonlocally related systems and their use to obtain nonlocal symmetries for both Eulerian and Lagrangian descriptions.
This is joint work in progress with Alan Haynes and Jeffrey Vaaler. Let A be a finite, nonempty set of positive integers. For x in R/Z, we study Delta(A,x) := min { ||ax|| : a \in A }, where || y || = \min { |y-n| : n \in Z } is the distance from y to the nearest integer. If each element of A is odd, then it is obvious that Delta(A,1/2)=1/2. However, in this talk, we will show that for most points x in R/Z the value of Delta(A,x) is not much bigger than |A|-1/2.
Note for Attendees
Refreshments will be served between the two talks.
For this first session, we will have a short overview of the compilation of survey papers "Making the connection: Research and Teaching in Undergraduate Mathematics Education". Following from this, we will decide in which order to discuss the survey papers.
To give you a teaser, the topics covered by these surveys is organized by themes, each of which is then detailed in three to four survey papers. The themes are:
- Foundations for beginning calculus
- Infinity, limits and divisibility
- Proving theorems
- Interacting with students
- Using definitions, examples and technology
- Knowledge, assumptions and problem solving behaviours for teaching
I will explain how to apply the modular method to resolve cases of this family of generalized Fermat equations. (joint with M. Bennett, S. Dahmen, S. Yazdani)
The main theme of this talk is that studying implicit tree structures of random processes is of significance in understanding their extreme values. I will illustrate this by several examples including cover times for random walks, maxima for two-dimensional discrete Gaussian free fields, and stochastic distance models. Our main results include:
(1) An approximation of the cover time on any graph up to a multiplicative constant by the maximum of the Gaussian free field, which yields a deterministic polynomial-time approximation algorithm for the cover time (D.-Lee-Peres 2010); the asymptotics for the cover time on a bounded-degree graph by the maximum of the GFF (D. 2011); a bound on the cover time fluctuations on the 2D lattice (D. 2011).
(2) Exponential and doubly exponential tails for the maximum of the 2DGFF (D. 2011); some results on the extreme process of the 2D GFF (D.-Zeitouni, in preparation).
(3) Critical and near-critical behavior for the mean-field stochastic distance model (D. 2011).
We consider the ferromagnetic Potts on typical d-regular graphs, and the independent set model on typical bipartite d-regular graphs, with graph size tending to infinity. We show that the replica symmetric (Bethe) prediction applies for all parameter values in these two models. In this talk I will describe some of the proof techniques, which will give an indication of the contrast with the anti-ferromagnetic Potts model and the independent set model at high fugacity on non-bipartite graphs, where the Bethe prediction is known to fail.
This is joint work with Amir Dembo, Andrea Montanari, and Allan Sly.
Many important problems in representation theory have analogues in geometry. For example, decomposing tensor products of representations of GL(n) into irreducible representations is very closely tied to the geometry of Grassmannians. Similarly, studying the restriction of a representation of GL(n) to subgroups such as SO(n) or SP(n) has geometric analogues in terms of the geometry of flag varieties. In this talk, I will show you how drawing pictures can help elucidate these problems. I will specifically concentrate on Littlewood-Richardson rules and geometric branching rules. I intend to make the talk accessible to anyone who is willing to be seduced by pictures.
High contrast in electrical conductivity motivates the investigation of electromagnetic methods in geophysics, for instance, for hydrocarbon and mineral exploration. A straightforward approach to modelling the spatial distribution of this parameter within the earth is the assumption of piecewise constant values, defined on a moderately fine tessellation of the volume under investigation by hexahedra or tetrahedra. We study here the solution of the 3-D forward problem for time-harmonic electromagnetic fields using finite elements, based on the above mentioned tessellation. Furthermore, we seek to reconstruct the spatial distribution of conductivity of an overparameterized model by a regularised output least squares approach. Our model assumption, a piecewise constant coefficient, allows for simplifications of the forward solver which eventually lead to an overall faster imaging algorithm. The model assumption also requires special care when the regularisation operator is derived for unstructured meshes within the finite element framework.
In 2009 and 2010 the Math 152 computer labs underwent major changes both in content and structure. In-depth assessments of the effectiveness of the new lab activities showed that learning of basic Matlab skills did occur in the labs and was not due to previous programming experience. More importantly, students who had attended the new labs performed better than the previous year's cohort when tested on basic Matlab programming concepts a few months after the end of the labs. We will discuss in what ways the new lab activities may have contributed to improving student learning and what challenges instructors face when they develop activities that support the learning of Matlab for novice programmers.
The Hilbert scheme of n points in the plane P2[n] is a compactification of n unordered tuples of points in the plane. It is a smooth, irreducible variety of dimension 2n and plays a central role in algebraic geometry, combinatorics, representation theory and mathematical physics. In this talk, I will describe the birational geometry of P2[n]. I will explain how to run the minimal model program on P2[n] and how to interpret the resulting models as moduli spaces of Bridgeland semi-stable objects. This is joint work with Daniele Arcara, Aaron Bertram and Jack Huizenga.
We prove that solutions to a fully nonlinear elliptic equation F(D^2u)=0 are classical outside a set of dimension at most n-epsilon, where n is the dimension and epsilon is a small constant depending on the ellipticity bounds of F and dimension. We do not make any convexity assumption on the equation F, but we assume that it is differentiable. We will also discuss the relationship of the partial regularity result with the question of unique continuation of solutions. This is a joint work with Scott Armstrong and Charles Smart.
This talk presents the concept of obstacle number--a recently defined characteristic of graphs, arising from how a graph can be drawn in the plane with 0, 1, or 2-dimensional "obstacles" blocking the non-edges of the graph. We will define obstacle number and then present several results, including characterization of graphs with obstacle number 1 for 1 and 2 dimensional obstacles, as well as constructions of graphs with arbitrarily large obstacle number.
For spin systems on a tree, roughly, the reconstruction problem
is to determine whether correlations persist between vertices
deep inside the tree and the root. Reconstruction on trees plays
an important role in explaining threshold phenomena in random
constraint satisfaction problems on sparse random graphs as well
as the efficiency of finding and sampling solutions for these
problems.
In this talk, I will speak about the following results:
(1) Bounds on the reconstruction threshold for colorings and
rapid mixing of the block dynamics for sampling colorings. (with
Vera, Vigoda, and Weitz '11)
(2) An algorithm to compute the reconstruction threshold, with
an application showing bounds on the threshold for the Potts
model on small-degree trees. (with Maneva '11)
I will speak about the connections of the reconstruction problem on trees with constraint satisfaction on random graphs and efficiency of algorithms. I will describe our bounds on the reconstruction threshold for the hardcore model on trees. (with Sly and Tetali '11)
Time permitting, I will speak about some unrelated results on non-uniform models for random permutations and bounds on the length of the longest increasing subsequences. (with Peled, in progress)
It will be shown that each point symmetry of a given PDE system systematically yields a nonlocally related PDE system. Examples will be exhibited that yield nonlocal symmetries for a given PDE system not obtainable systematically through previously known methods.
We discuss Alpern theorems (a generalization of Rokhlin's lemma) for R^d actions
in which the `towers' are rectangular boxes of prescribed sizes, proving both
sufficient and necessary conditions on the number of boxes. (joint work with
Bryna Kra and Ayşe Şahin).
This week Greg Mayer will conduct the discussion. We will study the section title "Using definitions, examples and technology". This section is composed of three papers, our discussion will focus on the third one:
The role of mathematical definitions in mathematics and in undergraduate mathematics courses (pdf)
Compute-based technologies and plausible reasoning (pdf)
Worked examples and concept example usage in understanding mathematical concepts and proofs (pdf)
I shall consider a rhombus tiling model (equivalently, a dimer model on a hexagonal graph) with a free boundary. The correlation of a small triangular hole in this model will be determined. As I shall explain, this kind of problem features phenomena which are parallel to phenomena in electrostatics. This is joint work with Mihai Ciucu.
The acoustic wave equation is a mathematical model commonly used for the propagation of seismic waves through the Earth's subsurface, where the wavespeed is determined by the geological composition of the Earth. The seismic imaging problem is to recover this geological composition based on surface data, and is typically formulated as an inverse problem to recover the wavespeed in the acoustic wave-equation model. To linearize this inverse problem, it is typically assumed that waves reflect only once in the subsurface. This assumption, combined with the data acquisition geometry which restricts sources and receivers to lie on the Earth's surface, makes these algorithms most effective at imaging structures that are nearly flat. The inclusion of multiply scattered waves mitigates this to some extent, but at the cost that the inverse problem becomes nonlinear. I will describe a series framework for including multiply scattered waves in the seismic inverse problem allowing us to approach this nonlinear problem as a series of linear problems. Within this approach, we use techniques of seismic interferometry, a method based on Green's theorem. I will briefly describe how we use these techniques in other applications in particular for locating micro earthquakes relative to one another.
I will survey the constructions of holomorphic Chern-Simons functionals for the moduli spaces of sheaves on CY 3-folds and several applications to the theoretical and computational aspects of Donaldson-Thomas theory.
Loosely speaking, a stochastic system exhibits the correlation decay property if correlations between components of the system decay as a function of the distance between the components. The notion appears primarily in the context of uniqueness of Gibbs measures in statistical physics, but also appears to have interesting algorithmic applications. We illustrate how the correlation decay property can be used for designing a polynomial time deterministic approximation algorithm, which we call the cavity expansion algorithm, for the problem of computing the partition function for Gibbs distributions on general graphs. Prior algorithms for this problem rely primarily on the Monte Carlo simulation technique and thus suffer from the sampling error. In contrast, the cavity expansion algorithm is deterministic and takes advantage of some structural property of Gibbs measures on graphs. We will illustrate our method on the problem of counting partial matchings in a graph - a well known #P hard algorithmic problem. Then we will demonstrate the practicality of our approach for the problem of computing the counting entropy for some lattice models. Specifically, we obtain an estimate of the entropy of the monomer-dimer coverings of a lattice, improving earlier estimates by several orders of magnitude.
Let W be a Coxeter group. For any w in W, let P_w denote the Poincare polynomial of w (i.e. the generating function of the principle order ideal of w with respect to length). If W is the Weyl group of some Kac-Moody group G, then P_w is the usual Poincare polynomial of the corresponding Schubert varitey X_w.
In this talk, I will discuss work in progress with W. Slofstra on detecting when the sequence of coefficients of a Poincare polynomial are the same read forwards and backwards (i.e. palindromic). The polynomial P_w satisfies this property precisely when the Schubert variety X_w is rationally smooth. It turns out that for many Coxeter groups, this property is relatively easy to detect.
High-order surface integral methods are powerful modeling tools for problems involving isolated variations in sound speed. The challenge in these methods lies in designing discretizations that appropriately account for the singularities in the Green's functions. I will discuss a method for doing this and will illustrate its applicability with two examples: (i) Modelling ultrasound vibro-acoustography, a nonlinear medical imaging technique that uses two different frequencies of ultrasound, combining the resolution of high-frequency ultrasound with the clean images possible when imaging at low frequencies. (ii) Understanding wavefront healing, a phenomena observed in seismic tomography where traveltime delays are reduced as waves propagate further from an isolated anomaly. I will also briefly discuss a related method for modeling seismic data near topography; our method allows us to quickly estimate a rough structure that can then be used as a starting model for full-waveform inversion techniques.
A classical problem in probability is to determine the length of the
longest increasing subsequence in a random permutation. Geometrically,
the question can be formulated as follows: given n independent,
uniform random points in the unit square, find the longest increasing
chain (polygonal path through the given points) connecting two
diagonally opposite corner of the square, where "length" means the
number of points on the chain. The variant of the problem I am going
to talk about asks for the length of the longest convex chain
connecting two vertices. We determine the asymptotic expectation up to
a constant factor, and derive strong concentration and limit shape
results. We also prove an ergodic result as well as giving a heuristic
argument for the exact asymptotics of the expectation. Some of these
results are joint with Imre Barany.
I will give a survey of results concerning the actions of a mapping class on the homology of various finite covers to which it lifts. I will draw connections to 3-manifold theory, especially largeness, growth of torsion homology and Alexander polynomials.
Lattice trees and lattice animals are used to model branched polymers. They are of interest in combinatorics and in the study of critical phenomena in statistical mechanics. A lattice animal is a connected
subgraph of the d dimensional integer lattice. Lattice trees are lattice animals without cycles. We consider the number of lattice trees and animals with n bonds
that contain the origin and form the corresponding generating functions. We are mainly interested in the radii of convergence of these functions, which are the
critical points. In this talk we focus on the calculation of the first three terms of the critical points for both models as the dimension goes to infinity.
This is ongoing work with Gordon Slade.
While the combination of standard finite-element discretizations and fast multigrid-based solvers is now well-established and well-understood for uniformly elliptic systems of PDEs, much work remains for problems that fall outside this class. This talk focuses on recent developments in the theory and practice of multigrid solvers for two important classes of non-uniformly elliptic systems: singular-perturbation and saddle-point problems. For singular-perturbation problems, boundary and interior layers are generated as the perturbation parameter goes to zero, leading to the need for strongly locally refined meshes to efficiently resolve the solutions to these systems. I will show how layer-aware multigrid methods can be developed to achieve optimal solver efficiency for these meshes. For saddle-point problems, difficulties arise due to the indefiniteness of the discretized linear systems, leading to many approaches based on block factorization treatment of two smaller but definite matrices. I will discuss an alternative monolithic approach, based on the constrained optimization viewpoint, to developing optimal multigrid solvers for these problems.
Whether you are an experienced clicker user or a pure novice who barely knows what clickers are, this session is for you! Clickers allow us to ask multiple-choice questions to a large crowd and gather global information about the state of your learners. Everyone will agree that creating effective clicker questions is a challenge. In this session we will offer some activities to allow everybody to improve their skills crafting such questions.
Note: Despite the fact that clickers themselves are mainly used in the classroom, this session isn't designed solely for instructors. Since designing effective questions which challenge the conceptual understanding of students is at the core of our activities, every TA will improve their skills of crafting effective questions, understand common misconceptions and reflect on what type of feedback informs you of the state of understanding of your learners.
Cookies and tea will be offered by our generous sponsors.
L-spaces are 3-manifolds with simplest possible Heegaard Floer homology. These arise naturally in many applications of Heegaard Floer theory, and as a result it has been asked if there is an alternate characterization of this class of 3-manifolds. A recent conjecture suggests the following: A 3-manifold is an L-space if and only if its fundamental group is not left-orderable. This talk will attempt to put this conjecture in context and describe some of the evidence supporting it.
Computational simulation is an important tool in many fields of science and engineering, providing crucial insight into a wide range of real-world problems. At the core of many simulations lies the solution of large linear systems of equations that can, in principle, be solved using simple techniques, such as Gaussian elimination. The high computational cost of Gaussian elimination, however, often makes high-fidelity simulations intractable. The modelling of convective flows in the Earth's mantle, for example, are limited to simulations with a few hundred thousands of degrees of freedom using this approach. In this talk, I will discuss both the theory and practice of algebraic multigrid methods, which can be used to overcome this barrier in many cases. I will show both theoretical analysis, leading to a rigorous framework for motivating the choice of components within a multigrid algorithm, and practical results, enabling efficient simulations of mantle convection with up to 100 million degrees of freedom.
We will discuss how to modify an approach of Solymosi and Tao, based on the so-called relative triangle removal lemma, to show that subsets of positive relative density of P^2 contain infinitely many corners, that is sets of the form {(a,b), (a+d,b),(a,b+d)}. This is ongoing joint work with T. Titichetrakun.
We characterize Schubert varieties (for GLn) which are local complete intersections (lci) by the combinatorial notion of pattern avoidance. For the Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighborhoods at the identity. Although the statement only requires ordinary pattern avoidance, showing the other Schubert varieties are not lci appears to require more complicated combinatorial ideas which have their own geometric underpinnings. The Schubert varieties defined by inclusions, originally introduced by Reiner and Gasharov, turn out to be an important subclass of lci Schubert varieties. Using the explicit equations at the identity for the lci Schubert varieties, we can recover formulas for some of their local singularity invariants at the identity as well as explicit presentations for their cohomology rings.
This is joint work with Henning Ulfarsson (Reykjavik U.).
T cells of the immune system, upon maturation, differentiate into either Th1 or Th2 cells that have different functions. The decision to which cell type to differentiate depends on the concentrations of transcription factors T-bet (x_1) and GATA-3 (x_2). These factors are translated by the mRNA whose levels of expression, y_1 and y_2, depend, respectively, on x_1 and x_2 in a nonlinear nonlocal way. The population density of T cells, \phi(t,x_1,x_2, y_1, y_2), satisfies a hyperbolic conservation law with coefficients depending nonlinearly and nonlocally on (t, x_1,x_2, y_1, y_2), while the x_i, y_i satisfy a system of ordinary differential equations. We study the long time behavior of \phi and show, under some conditions on the parameters of the system of differential equations, that the gene expressions in the T-cell population aggregate at one, two or four points, which connect to various cell differentiation scenarios.
In this talk we prove Roth type theorems in finite groups. Our main
tool is the Triangle Removal Lemma of Ruzsa and Szemeredi. For example
we show that for every c>0 there is a bound n such that if a finite
group G has order at least n, then any set S which is at least c-dense
in GxG contains four elements (a, b), (a, c), (e,c), (e,f) such that
ab = ec and ac = ef.
Over the past few years there has been considerable activity in exploiting the power of algebraic topology to investigate areas outside of mathematics. The phrase 'applied algebraic topology'
is no longer an oxymoron!
Even more recently the intrinsically random nature of the world is beginning to bring statistical and probabilistic tools to these problems, leading to the birth
of a new area of 'random algebraic topology'.
In this talk I will discuss some of the few results in random algebraic topology, including the persistence homology of the sub-level sets of Gaussian processes
over manifolds, and limit theorems for the Betti numbers of random complexes built over random point processes.
Since this is to be a joint Probability/Topology seminar, I shall assume no prior knowledge in either area.
Feynman's famous quote is of particular relevance for research at the interface of mathematics and physics in recent decades. A striking example is the impressive number of fascinating results (many of them still conjectural) in various areas of mathematics, such as geometry, topology and number theory, that have been obtained via string theory. In this vein, mirror symmetry and topological string theory have been particularly fruitful. In this talk I will focus on a number of mathematical conjectures and theorems that we have obtained through careful study of mirror symmetry. I will discuss what string theory tells us about (quasi-)modularity of the generating functions of Gromov-Witten invariants of Calabi-Yau threefolds, and what it implies for the crepant resolution conjecture relating Gromov-Witten invariants of an orbifold to the invariants of its crepant resolution. I will also talk about a new mysterious recursive structure conjecturally satisfied by the Gromov-Witten generating functions for toric Calabi-Yau threefolds, with far-reaching and still mostly unexplored consequences. By the end of the talk, you should hopefully be convinced of "the unreasonable effectiveness of string theory in mathematics"!
I will present a biologically-based mathematical model that accounts for several features of human sleep and demonstrate how particular features depend on interactions between a circadian pacemaker and a sleep homeostat. The model is made up of regions of cells that interact with each other to cause transitions between sleep and wake as well as between REM and NREM sleep. Analysis of the mathematical mechanisms in the model yields insights into potential biological mechanisms underlying sleep and sleep disorders including stress-induced insomnia and fatal familial insomnia.
Let S be an ergodic measure-preserving automorphism on a non-atomic
probability space, and let T be the time-one map of a topologically weak
mixing suspension flow over an irreducible subshift of finite type under a
Holder ceiling function. We show that if the measure-theoretic entropy of
the S is strictly less than the topological entropy of T, then there
exists an embedding from the measure-preserving automorphism into the
suspension flow. As a corollary of this result and the symbolic dynamics
for geodesic flows on compact surfaces of negative curvature developed by
Bowen and Ratner, we also obtain an embedding from the measure-preserving
automorphism into a geodesic flow, whenever the measure-theoretic entropy
of S is strictly less than the topological entropy of the time-one map of
the geodesic flow.
In recent years, a unifying theme has been found for a surprising number of counting problems. It appears that in many seemingly unrelated contexts, generating functions for enumerative invariants satisfy a particular topological recursion, based on the geometry of a complex curve. As examples, Hurwitz theory,
Gromov-Witten theory of the complex line, and open/closed Gromov-Witten theory of the three-dimensional complex plane and other toric Calabi-Yau threefolds are all encoded by the same topological recursion on certain complex curves. In this talk, I will first review applications of the topological recursion on genus zero curves, and then report on work in progress on studying the topological recursion for families of elliptic curves. In this context, the recursion
produces an infinite tower of quasi-modular forms. The question is: to what counting problem should these quasi-modular forms be related?
This week we will study the section titled "Foundations for beginning calculus". This section is composed of four papers: 1) On developing a rich conception of variable 2) Rethinking change 3) Foundation reasoning abilities that promote coherence in students' function understanding and 4) The concept of accumulation in calculus. Please see the Seminar's wiki page available at: http://wiki.ubc.ca/Sandbox:Math_Teaching_Seminar
You will find access to the pdf version of these papers.
At a previous seminar, we proposed a "strong normality" test, to exclude numbers like Champernowne's number. Now we give a sharp version of this test. Almost all numbers are strongly normal, and every strongly normal number is normal. We use a method of Sierpinski to construct an absolutely normal number satisfying the new criterion. (This is joint work with Peter Borwein.)
This talks introduces a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we present some novel theory showing that our entire approach may be provably surprisingly effective.
If X is a finite CW complex of small dimension, information about the homotopy groups of unitary groups can be translated into cohomological classification results for complex vector bundles on X. I will explain how A^1-homotopy theory can be used in an analogous fashion in the classification of vector bundles of on smooth affine varieties of small dimension. In particular, I will explain some joint work (in progress) with J. Fasel which shows how to give a complete classification of vector bundles on smooth affine 3-folds over certain fields. No knowledge of A^1-homotopy theory will be assumed.
We explain how the computation of motivic Donaldson-Thomas invariants associated to a quiver with potential reduces to the computation of the motivic classes of simpler quiver varieties. This has led to the calculation of these invariants for some interesting Calabi-Yau geometries derived equivalent to a quiver with potential. Here we observe q-deformations of the classical generating series.
For many inverse problems, accuracy of data is required by standard methods, yet rarely achievable in practice. In many applications, data may contain large artifacts, such as outliers caused by measurement errors, or physical phenomena not explained by the predictive model. In this setting, robust methods, i.e. methods that find reasonable results even in the face of gross errors, are an appealing alternative to pre-processing, outlier removal, or very complex modeling.
In this talk, we will discuss two applications: geophysical imaging and inference for dynamical systems. In both cases, we will show how to design robust methods by modifying the statistical error models. We can then get robust solutions by finding the maximum likelihood estimates for parameters in these modified models. In order to solve these problems quickly, optimization techniques must exploit the underlying problem structure. We will highlight this structure for both classes of applications, and present numerical results to show how
the methods work in practice.
We continue to introduce basic ideas in Cohomological invariant theory, with a particular focus on unramified cohomology and on invariants of étale algebras.
I'll give a historical overview of computations of the zeta function on the critical line and then describe some recent computations from the past year or so. Mostly this will focus on an implementation of Ghaith Hiary's O(t1/3) method for computing ζ(1/2+it) and some computations that Hiary and I have run using it. Highlights will include the 1032nd zero, values of |ζ(1/2+it)| larger than 14000, and some record observations of irregularities in the distribution of zeros.
Note for Attendees
Refreshments will be served between the two talks.
Abstract: I will describe some mathematical issues related to the analysis of localized patterns (spikes and interfaces) in reaction diffusion systems. For spikes, the analysis of the spectrum of various classes of nonlocal eigenvalue problems (NLEP) is essential. I will discuss some new and interesting NLEPs arising in cross diffusion systems and crime models. For interfaces, a nonlocal geometric problem involving mean curvature and Newtonian potential is derived and analyzed. Our goal is to give mathematically rigorous proofs of existence and stability of various classes of patterns that have been observed in experiments and simulations in the physics literature. Our further goal is to predict the existence of some new patterns which have not yet been found in experiments. (Joint works with T. Kolokolnikov, X. Ren, M. Ward, and M. Winter.)
Note for Attendees
Refreshments will be served in the PIMS Lounge from 3:30-3:45 p.m.
This week Djun Kim will conduct the discussion. We will study the section titled "Knowledge, assumptions and problem solving behaviours for teaching". This section is composed of four papers:
From concept images to pedagogic structure for a mathematical topic (pdf)
Promoting effective mathematical practices in students: insights from problem solving research (pdf)
When students don't apply the knowledge you think they have, rethink your assumptions about transfer (pdf)
How do mathematicians learn to teach? Implications from research on teachers and teaching for graduate student professional development (pdf)
The study of numerical stability of algorithms for computing with polynomials and matrices over archimedian fields like R and C constitutes an entire branch of numerical analysis. The analogous problems over Qp have received far less attention. In this talk I will outline some number theoretic questions in which these problems arise, and describe joint work with Xavier Caruso in which we propose a general methodology for approaching them in practice.
It is well-known that the self dual non-Abelian Chern-Simons gauge equations coupled with a Schrodinger matter field can be reduced to Toda systems with Cartan matrix of rank r and singular sources. In this talk, using purely PDE methods, we give a complete classification of SU(n+1) Toda system with a single source. Then we apply this classification result to construct non-topological solutions for the SU(3) Chern-Simons system and obtain sharp estimates for the blow-up rates of SU(3) Toda system on a surface. (Joint work with Ao, CS Lin and D. Ye.)
Note for Attendees
Tea and cookies will be served from 2:45-3:00 p.m.
The goal of the talk is to survey recent progress in understanding
statistics of certain exactly solvable growth models, particle systems,
directed polymers in one space dimension, and stochastic PDEs. A
remarkable connection to representation theory and integrable systems is
at the heart of Macdonald processes, which provide an overarching theory
for this solvability. This is based off of joint work with Alexei Borodin.
In this talk we will present a formula to count the number of hyperelliptic curves on a polarized Abelian surface, up to translation. This formula is obtained using orbifold Gromov-Witten theory, the crepant resolution conjection and the Yau-Zaslow formula to related hyperelliptic curves to rational curves on the Kummer surface Km(A). We will show how this formula can be described in terms of certain generating functions studied by P. A. MacMahon, which turn out to be quasimodular forms.
Multi-scale modelling has become a paradigm that transcends most scientific disciplines. A key challenge that arises in many scientific problems is the connection between discrete atomistic and continuum descriptions of matter. The Cauchy-Born rule postulates such a connection for crystal elasticity, which seems almost naive at first glance. Nevertheless, the Cauchy-Born model has been found to provide an accurate description of crystal elasticity even at the sub-grain scale. In this talk, I will derive the Cauchy-Born model from a formal perspective and then present rigorous approximation results for 0T statics and dynamics. New ideas that have arisen from my work on this problem include a novel localisation mechanism, an atomistic notion of stress, and a surprising symmetry of certain multi-lattices.
Continuum mechanics models generally have an astonishing range of validity. For example, nonlinear elasticity can accurately describe the elastic behaviour of crystals even at the sub-grain scale (see talk on Monday). However, crystal defects such as cracks and dislocations are difficult to model quantitatively within CM. By coupling atomistic models of defects with continuum models of elastic far fields one can, in principle, obtain models with near-atomistic accuracy at significantly reduced computational cost. That said, various pitfalls must be overcome in the construction of efficient and reliable coupling mechanisms. In this talk, I will take a numerical
analysis approach to this problem and analyze a sequence of coupling mechanisms in terms of their accuracy relative to their computational cost, which requires a thorough understanding of the consistency errors committed in the coupled models.
We will review the foundations on the Cohomology of k(t) and it's various completions. By doing so we will be able to define the notion of unramified elements, and we'll see how these fit on the calculation of Inv_k(C_2,C_2).
We study two of the simple rules on finite graphs under the death-birth updating and the imitation updating discovered by Ohtsuki, Hauert, Lieberman, and
Nowak [\emph{Nature} {\bf 441} (2006) 502-505]. Each rule specifies a payoff-ratio cutoff point for the magnitude of fixation probabilities of the underlying
evolutionary game between cooperators and defectors. We view the Markov chains associated with the two updating mechanisms as voter model perturbations. Then
we present a first-order approximation for fixation probabilities of general voter model perturbations on finite graphs subject to small perturbation in terms of
the voter model fixation probabilities. In the context of regular graphs, we obtain algebraically explicit first-order approximations for the fixation
probabilities of cooperators distributed as certain uniform distributions. These approximations lead to a rigorous proof that both of the rules of Ohtsuki et al. are valid and are sharp.
Given a lie group G and a finitely generated group P we can give a topology to the set of group homomorphisms Hom(P,G) as a subset of G^k. There has
been an increasing interest in understanding these spaces, and particularly their connected components, for their relevance in bundle theory. In particular when P=Z^k the space Hom(Z^k,G) is identified with the set of commuting k-tuples in G. We will present some of the generalities of these spaces and a possible systematic approach to their study. Then we will use that approach applied to the particular case when G=O(n), the group of orthogonal matrices, and compute the number of components of Hom(Z^k,O(n)).
Compressed sensing has brought the use of sparsity- and compressibility-based signal models to the forefront of data acquisition and inverse problems.The well-known analyses of compressed sensing are indirect and hold pointwise over the possible signals of interest.Inspired by the conservatism of these analyses, we developed a Bayesian analysis framework.Under the assumption of replica symmetry, we prove convergence in distribution as problem size grows of the joint marginal of one variable of interest and its estimate to the joint distribution in a simple scalar equivalent problem.This gives a simple mechanism for asymptotically-exact performance predictions that applies to a large class of estimators applied to a large class of problems.For example, it shows that l1-regularized least squares estimation typically performs much better than predicted by previous analyses.It can also be applied to l0-regularized least squares and various other estimators.Taken together, these analyses are significantly more optimistic than the traditional analyses.
Compressed sensing is far from the only reason to look at inverse problems with linear forward models.I will also present a cross section of work in magnetic resonance imaging where we have exploited sparsity-based regularization.This includes excitation design, GRAPPA kernel calibration, and image reconstruction.
Biography:
Vivek Goyal received the B.S. degree in mathematics and the B.S.E. degree in electrical engineering from the University of Iowa, where he received the John Briggs Memorial Award for the top undergraduate across all colleges.He received the M.S. and Ph.D. degrees in electrical engineering from the University of California, Berkeley, where he received the Eliahu Jury Award for outstanding achievement in systems, communications, control, or signal processing.
His previous positions include Member of Technical Staff in the Mathematics of Communications Research Department of Bell Laboratories, Lucent Technologies; and Senior Research Engineer for Digital Fountain, Inc., Fremont, CA.His research interests include source coding theory, quantization, sampling, and computational imaging.
Professor Goyal was awarded the IEEE Signal Processing Society Magazine Award and an NSF CAREER Award.He served on the IEEE Signal Processing Society’s Image and Multiple Dimensional Signal Processing Technical Committee, is a permanent Co-chair of the SPIE Wavelets and Sparsity conference series, and is a TPC Co-Chair of the IEEE International Conference on Image Processing 2016.He is a co-author of a forthcoming textbook available for download at FourierAndWavelets.org, and he will present tutorials on teaching signal processing at IEEE ICASSP 2012 and ICIP 2012.
This thesis develops a method (dimensional reduction) to compute motivic Donaldson-Thomas invariants. The method is then employed to compute these invariants in several different cases.
LIDAR systems and time-of-flight cameras use time elapsed from transmitting a pulse and receiving a reflected response, along with scanning by the illumination source or a 2D sensor array, to acquire depth maps. We introduce a method for compressive acquisition of scene depth with high spatial and range resolution using a single, omnidirectional, time-resolved photodetector and no scanning components. This opens up possibilities for accurate and high-resolution 3D sensing using compact and mobile devices.
In contrast to compressive photography, the information of interest -- scene depths -- is nonlinearly mixed in the measured data. To overcome this aspect of the inverse problem, the depth map reconstruction relies on parametric signal modeling of the impulse response of piecewise-planar scenes. Through the use of parametric deconvolution, we achieve much finer depth resolution than dictated by the illumination pulse width and detector bandwidth alone. Spatial resolution in our framework is rooted in patterned illumination or patterned reception followed by decoupling the inverse problems of range estimation and spatial resolution recovery during computational processing.
Our compressive depth acquisition camera (CoDAC) framework is an example of broader research themes of exploiting time resolution in optical imaging and identifying and exploiting structure in inverse problems.
This session will discuss what are learning objectives precisely and how to create effective ones using the Qualifying Exam for practice. Coffee, tea and cookies will be provided by the MGC and there will be a board games event just after this session.
In this talk, I introduce the restricted normal cone, which is a novel generalization of the Mordukhovich (also known as basic or limiting) normal cone. Basic properties are presented. In the case of subspaces, we make a connection to the Friedrichs angle between the subspaces. Restricted normal cones are useful in extending work by Lewis, Luke and Malick on the method of alternating projections for two (possibly nonconvex) sets.
Based on joint work with: Heinz Bauschke (UBC Kelowna), Russell Luke (Goettingen, Germany), and Shawn Wang (UBC Kelowna).
Homogeneous dynamics is another name for the theory of flows on homogeneous spaces, or homogeneous flows. The study of homogeoeus flows has been attracting considerable attention for the last 40-50 years. During the last three decades, it has been realized that some problems in number theory and, in particular in Diophantine approximation, can be solved using mathods from the theory of homogeneous flows. The purpose of the talk is to give examples of interactions between number theory and homogeneous dynamics; mostly only formulations will be given, but there will be also very brief description of some proofs.
The Mori cone of curves of the Grothendieck-Knudsen moduli space of stable rational curves with n markings, is conjecturally generated by the one-dimensional strata (the so-called F-curves). A result of Keel and McKernan states that a hypothetical counterexample must come from rigid curves that intersect the interior. In this talk I will show several ways of constructing rigid curves. In all the examples a reduction mod p argument shows that the classes of the rigid curves that we construct can be decomposed as sums of F-curves. This is joint work with Jenia Tevelev.
Radar imaging is a technology that has been developed, very successfully, within the engineering community during the last 50 years. Radar systems on satellites now make beautiful images of regions of our earth and of other planets such as Venus. One of the key components of this impressive technology is mathematics, and many of the open problems are mathematical ones.
This lecture will explain, from first principles, some of the basics of radar and the mathematics involved in producing high-resolution radar images.
A common misconception is that raindrops take the form of teardrops. In fact,
they tend to be nearly spherical due to surface tension forces. This is an example
of how at small scales fluid the tendency of molecules to adhere to each other
is the dominate effect driving a fluid's motion. In this talk we will explain
how surface tension arises from intermolecular forces. We will also examine some
examples of the behavior that can occur at small scales due to the balance between
fluid-fluid and fluid-solid forces, with applications as varied as understanding
how detergents help clean clothes to the design of fuel tanks in zero gravity
environments.
This talk explores the spatial, temporal, and spectral attributes of radar data and the associated target information. We see that existing signal-processing approaches combine these attributes, two at a time, to form images.
The question of combining all three attributes arises in the problem of using fixed, distributed sensors to image multiple moving targets. This talk explains how a comprehensive theory can be developed that incorporates sensor positions and the (possibly different) waveforms transmitted by the various transmitters. The theory leads to a formula for producing an image and a formula for the imaging point-spread function. Included are plots of the point-spread function for various geometries and waveforms.
I will present a recent result on the $W^2_p$-solvability of elliptic equations in convex wedge domains or in convex polygonal domains with discontinuous coefficients. A corresponding result for parabolic equations in polyhedrons with time-irregular coefficients will also be discussed.
For a smooth compact toric variety X, the equivariant cohomology of X is identified with the ring of piecewise polynomials on the associated fan. For singular toric varieties, this correspondence breaks down: the equivariant cohomology ring is not well understood. However, Payne identified the ring of piecewise polynomials with the operational equivariant Chow cohomology of X; this agrees with the usual equivariant cohomology when X is smooth. It turns out that a similar story holds for K-theory: when X is smooth and compact, the equivariant K-theory of algebraic vector bundles on X can be identified with the ring of "piecewise Laurent polynomials" on the associated fan. So, what is this ring for a general toric variety? In this talk, I will describe joint work with Sam Payne: for an arbitrary toric variety X, we identify the ring of piecewise Laurent polynomials on the fan with the operational equivariant K-theory of X. The proof requires us to develop some foundational aspects of operational K-theory, as well as the usual equivariant K-theory of coherent sheaves. Our point of view leads to the curious result that the abstract operational theory is tractable and computable on varieties where the usual K-theory (of algebraic vector bundles) is completely unknown.
Abstract: There are many operators in harmonic analysis for which the
curvature of some underlying manifold plays a significant role. We
will discuss recent efforts to establish uniform estimates for such
operators by compensating for degeneracies of curvature with an
appropriate measure. We will focus on the case when the underlying
manifolds are polynomial curves.
I will discuss recent pedagogical efforts which involve moving lecture-based elements of courses to a pre-class online format. These efforts aim to encourage students' mental effort outside of class and to leverage in-class meeting time to meet particular learning goals. The course design is built upon three technological tools. First, the LiveScribe pen enables pre-recording of lecture material. Second, Google Moderator is an online environment that aggregates and sorts user-submitted questions. Finally, PRS clickers facilitate a dialogue about material during class meeting time, and facilitate ongoing student assessment, feedback, and metacognitive reflective practice. Through a course design case study and some demonstrations, we will explore pedagogical and technical aspects of this learning ecosystem. I will leave ample time for discussion and questions.
I will discuss a few examples of analysis of stochastic models of social and economic behaviors on networks. The models presented are coming from statistics, computer science, marketing, learning and network games.
I will talk on work in progress (with Albert Ruiz) on Kac-Moody groups over finite fields from a topological point of view, including some explicit cohomological computations (at non-characteristic primes) as well as some (conjectural) general properties.
Over the last decade, generalization of the central limit theorem titled Non-Linear Invariance Principles, have played a major role in the theory of approximation algorithms in computer science and in the theory of voting schemes in theoretical economics. The talk will provide a broad overview of non-linear invariance, Gaussian geometry and their connection to hardness of approximation and social choice theory.
A fundamental question in the study of integral quadratic forms is the representation problem which asks for an effective determination of the set of integers represented by a given quadratic form. A slightly different, but equally interesting problem, is the representation problem for inhomogeneous quadratic forms. In this talk, we will discuss a characterization of positive definite almost universal ternary inhomogeneous quadratic forms which satisfy some mild arithmetic conditions. Using these general results, we will then characterize almost universal ternary sums of polygonal numbers.
Note for Attendees
Refreshments will be served between the two talks.
This week Katya Yurasovskaya will conduct the discussion. We will study the section titled "Proving theorems". This section is composed of five papers:
Overcoming students' difficulties in learning to understand and construct proofs (pdf)
Mathematical induction: cognitive and instructional considerations (pdf)
Proving starting from informal notions of symmetry and transformations (pdf)
In 2010 Duminil-Copin and Smirnov (DCS) proved rigorously that the
growth constant for self-avoiding walks on the honeycomb lattice is
equal to $\sqrt{2+\sqrt{2}}$, a value conjectured by Nienhuis in 1982.
One of the main ingredients in DCS's proof is a finite lattice
identity for generating functions obtained from a discretely
holomorphic observable. I will discuss an extension of their proof to
include boundary weights, and hence a method for establishing the
critical fugacity for the adsorption transition for self avoiding
walks.
Many questions in analysis and geometry lead to problems of quasiconformal geometry on non-smooth or fractal spaces. For example, there is a close relation of this subject to the problem of characterizing fundamental groups of hyperbolic 3-orbifolds or to Thurston's characterization of rational functions with nite post-critical set. In recent years, the classical theory of quasiconformal maps between Euclidean spaces has been successfully extended to more general settings and powerful tools have become available. Fractal 2-spheres or Sierpinski carpets are typical spaces for which this deeper understanding of their quasiconformal geometry is particularly relevant and interesting. In my talk I will give a survey on some recent developments in this area.
Edge-flip distance between polygonal triangulations measures the degree of similarity between two rooted triangulations and has applications in balancing rooted binary trees. Currently, there are no known poly-time algorithm for computing edge-flip distances. However, the existence of matched edges makes computing geodesic distances more manageable. In this talk, we will describe a method of reducing the problem by partitioning it into smaller components. We will present some results related the asymptotic distribution of these components as they are central to determining the effectiveness of this method.
Studying Gromov-Hausdorff limits of sequences of Riemannian manifolds (M_i) satisfying suitable conditions on their local geometry is an extremely fruitful idea. However, in the most interesting case that the diameter of M_i grows without bounds, one is forced to choose base points p_i\in M_i and consider limits of the pointed spaces (M_i,p_i) in the pointed Gromov-Hausdorff topology. The choice of the base points p_i influences enormously the obtained limits. Benjamini and Schramm introduced the notion of distributional limit of a sequence of graphs; this basically amounts to "choosing the base point by random". In this talk I will describe the distributional limits of sequences (M_i) of manifolds with uniformly pinched curvature and satisfying a certain condition of quasi-conformal nature. I will also explain how these results yield a modest extension of Benjamini's and Schramm's original result. This is joint work with Hossein Namazi and Pekka Pankka.
Let X be a variety over a field k, with a fixed rational point x_0 in X(k). Nori defined a profinite group scheme N(X,x_0), usually called Nori's fundamental group, with the property that homomorphisms N(X,x_0) to a fixed finite group scheme G correspond to G-torsors P --> X, with a fixed rational point in the inverse image of x_0 in P. If k is algebraically closed this coincides with Grothendieck's fundamental group, but is in general very different. Nori's main theorem is that if X is complete, the category of finite-dimensional representations of N(X,x_0) is equivalent to an abelian subcategory of the category of vector bundles on X, the category of essentially finite bundles.
After describing Nori's results, I will explain my work in collaboration with Niels Borne, from the University of Lille, in which we extend them by removing the dependence on the base point, substituting Nori's fundamental group with a gerbe (in characteristic 0 this had already been done by Deligne), and give a simpler definition of essentially finite bundle, and a more direct and general proof of the correspondence between representations and essentially finite bundles.
In 1965, Hironaka showed resolution of singularities for all algebraic varieties over fields of characteristic 0. Since then, the analogue in positive characteristic has remained an open problem. When one cannot solve a problem globally, one tries to solve it locally. Here this means getting rid of singularities one at a time. Since Zariski it is known that the local version of resolution of singularities, called ``local uniformization'', is of valuation theoretical nature. It was proved by Zariski in 1939 in the case of characteristic 0. Also for local uniformization, the positive characteristic case is still unsolved. The only known partial solutions so far are:
- resolution up to dimension 3 (Abhyankar; Cossart and Piltant)
- resolution by alteration (de Jong)
- local uniformization for Abhyankar places (Knaf & Kuhlmann)
- local uniformization by alteration with Galois extensions (Knaf &
Kuhlmann)
- inseparable local uniformization (Temkin)
Alteration takes an extension of the function field of the algebraic variety into the bargain. The version for local uniformization gives better information about the extension than can be deduced from de Jong's result, and the proof by Knaf and Kuhlmann is purely valuation theoretical. Temkin arrived at a complementary result which uses alteration by purely inseparable extensions.
In our talk we will discuss the various known results about local uniformization and how they relate to the structure theory of valued function fields. Can the complementary results of Knaf & Kuhlmann and Temkin be put together to avoid extensions of the function field altogether? To hear our opinion, you will have to wait until the end of our talk...
Reaction-diffusion processes occur in many materials with microstructure such as biological cells, steel or concrete. The main difficulty in modelling and simulating accurately such processes is to account for the fine microstructure of the material. One method of upscaling multi-scale problems, which has proven reliable for obtaining feasible macroscopic models, is the method of periodic homogenisation.
The talk will give an introduction to multi-scale modelling of chemical mechanisms in domains with microstructure as well as to the method of periodic homogenisation. Moreover, certain aspects particularly relevant in upscaling reaction-diffusion processes in biological cells will be discussed.
The Ising model is a discrete mathematical model for ferromagnetism. A classical argument shows that to study the Ising model on a finite graph G it is equivalent to study the generating function of even subgraphs of G. In the 60s Sherman showed that when G is planar this generating function can be rewritten as a generating function of closed non-backtracking walks on G, and recent work of Cimasoni and Loebl has removed the restriction that G is planar. I'll outline an alternative proof of these facts and explain what this means for the Ising model.
The Abelian sandpile model on the d-dimensional integer lattice is a particle system that is critical, in the sense well-known from lattice models of statistical physics. That is, several observables follow power law distributions, at least numerically, and occasionally this can be proved. Here we study a natural one parameter family of models called dissipative sandpiles, where a small amount gamma of mass can be lost (dissipated) on each toppling. As gamma approaches 0, the critical model is recovered, while for any gamma>0, the model has exponential decay of correlations. After discussing some basic properties, I will present estimates in d = 2 and 3, on how fast the stationary measure of the dissipative model approaches the critical sandpile measure. (Partly joint work with F. Redig and E. Saada.)
Let G be a finitely presented group. If the process of iteratively passing to vertex groups in a maximal graph of groups decomposition of G over finite subgroups, and then to vertex groups in maximal decompositions of the factors over two-ended subgroups, terminates, we say that G is strongly accessible. Delzant and Potyagailo argue that this process always terminates for certain types of splittings of finitely presented groups, in particular hyperbolic groups without two-torsion. I will give an example showing that their proof cannot be correct, and sketch a new proof that (relatively) hyperbolic groups without two-torsion are strongly accessible. This is joint work with N. Touikan.
We investigate a one-dimensional model describing the motion of liquid drops sliding down an inclined plane (the so-called quasi-static approximation model). We prove existence and uniqueness of a solution and investigate its long time behavior for both homogeneous and inhomogeneous medium (i.e. constant and non-constant contact angle). This is joint work with Antoine Mellet (U.Maryland).
Associated to a newform F in S2(Γ0(N)) and an anticyclotomic finite-order Hecke character χ of an imaginary quadratic field K, one defines the twisted L-function L(F,χ,s). Prof. Vatsal obtained various results pertaining to the non-vanishing modulo a prime λ in the algebraic closure of Q of the special values of these L-fuctions in the anticyclotomic tower of conductor p∞ over K when the sign in the functional equation is +1. In this talk, we report on a work in progress to establish some generalizations of such results.
Note for Attendees
Refreshments will be served between the two talks.
This week Costanza Piccolo will conduct the discussion. We will study the section titled "Interacting with students". This section is composed of four papers:
Meeting new teaching challenges: teaching strategies that mediate between all lecture and all student discovery (pdf)
Examining interaction patterns in college-level mathematics classes: a case study (pdf)
Mathematics as a constructive activity: exploiting dimensions of possible variation (pdf)
Supporting high achievement in introductory mathematics courses: what we have learned from 30 years of the emerging scholars program (pdf)
Please see the Seminar's wiki page available at: http: // wiki.ubc.ca/Sandbox: Math Teaching Seminar
You will find access to the pdf version of these papers.
I will give a complete classification of the isogeny classes of supersingular abelian varieties for all dimensions by explicitly finding all possible characteristic polynomial of Frobenius endomorphism up to dimension 7 and giving an algorithm to find for all dimensions using Honda-Tate Theory. This is joint work with Gary McGuire and Alexey Zaytsev.
Sea ice is a leading indicator of climate change. It also hosts extensive microbial communities which support life in the polar oceans. The precipitous decline of the summer Arctic sea ice pack is probably the most visible, large scale change on Earth's surface in recent years. Most global climate models, however, have significantly underestimated these losses. We will discuss how mathematical models of composite materials and statistical physics are being used to study key sea ice processes such as melt pond evolution, snow-ice formation, and nutrient replenishment for algal communities. These processes must be better understood to improve projections of the fate of sea ice, and the response of polar ecosystems. Video from recent Antarctic expeditions where we measured sea ice properties will be shown.
Note for Attendees
Reception will be held at 2:30 pm in the WMAX 110 at PIMS.
In the past two years, Church, Farb and others have developed the concept of 'representation stability', an analogue of homological stability for a sequence of groups or spaces admitting group actions. In this talk, I will give an overview of this new theory, using the pure string motion group P\Sigma_n as a motivating example. The pure string motion group, which is closely related to the pure braid group, is not cohomologically stable in the classical sense -- for each k>0, the dimension of the degree k rational cohomology of P\Sigma_n tends to infinity as n grows. The groups H^k(P\Sigma_n, \Q) are, however, representation stable with respect to a natural action of the hyperoctahedral group W_n -- that is, in some precise sense, the description of the decomposition of these cohomology groups into irreducible W_n-representations stabilizes for n>>k. I will outline a proof of this result, verifying a conjecture by Church and Farb.
The Kodaira vanishing theorem and its generalizations are extremely important tools in higher dimensional geometry and the failure of these theorems in positive characteristic causes great difficulties in extending the existing theories to that realm. In this talk I will discuss new results about cases where an appropriate vanishing theorem holds and cases where the expected one fails even in characteristic zero. These results are joint works (separately) with Christopher Hacon and with János Kollár.
High contrast in electrical conductivity motivates the investigation of electromagnetic methods in geophysics, for instance, for hydrocarbon and mineral exploration. A straightforward approach to modelling the spatial distribution of this parameter within the earth is the assumption of piecewise constant values, defined on a moderately fine tessellation of the volume under investigation by hexahedra or tetrahedra. We study here the solution of the 3-D forward problem for time-harmonic electromagnetic fields using finite elements, based on the above mentioned tessellation. Furthermore, we seek to reconstruct the spatial distribution of conductivity of an overparameterized model by a regularised output least squares approach. Our model assumption, a piecewise constant coefficient, allows for simplifications of the forward solver which eventually lead to an overall faster imaging algorithm. The model assumption also requires special care when the regularisation operator is derived for unstructured meshes within the finite element framework.
We consider a nonlinear Schrödinger equation with a general nonlinearity. In space dimension 2 or higher, this equation admits solitons (standing/traveling waves) with a fixed profile which is not a ground state. These types of profiles are called excited states. Due to instability, excited solitons are singular objects for the dynamics of NLS. Nevertheless, we will show in this talk how to exhibit solutions of NLS behaving in large time like a sum of excited solitons with high relative speeds.
We compute the exact value of essential dimension for split spinor groups and split even Clifford groups. I will also discuss some applications of these results to quadratic forms.
This talk is based on joint work with A. Merkurjev; see www.math.uni-bielefeld.de/lag/man/455.html
The notions of covering maps, Galois theory and representations, Laplacians, L^2 Betti numbers, and sheaves are well known in many areas of mathematics. Despite the fact that these notions are easy to describe in the context of graph theory, these tools seem to be greatly underutilized there. Furthermore, such ideas can be used to prove theorems that have no obvious connection to sheaf theory, or even to graph theory.
We describe the above notions, and briefly describe the applications of sheaves and covering maps to (1) a solution of the Hanna Neumann Conjecture of the 1950's, and (2) an equivalence of two notions of "2-independence," which are generalizations of ordinary linear independence (which has no obvious connection to graph theory). We also describe our original motivation to study sheaves on discrete structures that arose from complexity theory.
Application (2) seems to make essential use of a result regarding covering maps of graphs and "scaled Abelian limits" of Betti numbers, that may be described as a "baby version" of l-adic cohomology.
We give an overview of the most basic Ramsey theoretic principles such as the Hales-Jewett theorem and the Halpern-Läuchli theorem and the corresponding Ramsey space theories that they lead to. The theory has natural counterparts both in the sense of dimensions and cardinalities of the structures. For example, we explain the close relationships between the finite and the infinite-dimensional theory and we also explain the Ramsey theory of finite structures and its close relationship to the Ramsey theory of infinite structures. If time permits we will also explain some of the most recent advances in the density Ramsey theory with a particular emphasis on the new phenomena that show up in the context of infinite structures. We point out some of the successes in applying this theory to, for example, topological dynamics and functional analysis. Professor Todorcevic obtained his Ph.D. in Belgrade and currently holds a Canada Research Chair at the University of Toronto.
Note for Attendees
Following PIMS Tea (2:45pm) the Lecture takes place from 3-4pm in WMAX 110.
Every transverse knot in a contact 3-manifold is represented as a closed braid in an open book. In this talk, based on a new technique called an open book foliation, we give a formula of self-linking number in terms of braids and open books. Surprisingly, our self-linking number formula essentially uses Johnson's homomorphism. This is a joint work with Keiko Kawamuro (Univ. Iowa).
A famous result by Steven Smale states that we can turn the sphere inside-out through immersions: this is called the eversion of the sphere. We will explain this result and the strategy of its proof which is a "cut-and-paste" strategy quite standard in algebraic topology. This approach allows us to understand globally the space of all immersions of a given manifold in another one, like the space of all immersion of the sphere in R^3 in the case of Smale's eversion. This theory has been enhanced by Goodwillie in the 1990's to understand spaces of embeddings. We will explain how this can be applied to understand spaces of knots, that is the spaces of all embeddings of a circle into a fixed euclidean space.
A richly illustrated talk about Goedel's time in Vienna. These fifteen years cover his brilliant start at the University and his discovery of the incompleteness theorems, the confused period between Hitler's rise to power in Germany and the annexation of Austria, and Goedel's desperate struggle to leave national-socialist Vienna.
This week Kyle Hambrook will conduct the discussion. We will study the section titled "Infinity, limits and divisibility". This section is composed of three papers:
Evolutionary game theory helps to investigate the role of incentives in promoting cooperative behavior in joint enterprises. In particular, this lecture deals with the surprising effects of optional participation in collaborative enterprises. Coercion works better for voluntary rather than compulsory collaboration. A social contract need not be based on rational deliberation or the command of a higher authority. It can emerge spontaneously through social learning of individuals guided by no more than their myopic self-interest.
In this talk I explain a promising and previously unnoticed link between electronic structure of molecules and optimal transportation (OT), and I give some
first results. The `exact' mathematical model for electronic structure, the many-electron Schroedinger equation, becomes computationally unfeasible for more than a dozen or so electrons. For larger systems, the standard model underlying a huge literature in computational physics/chemistry/materials science is density functional theory (DFT). In DFT, one only computes the single-particle density instead of the full many-particle wave function. In order to obtain a closed equation, one needs a closure assumption which expresses the pair density in terms of the single-particle density rho.
We show that in the semiclassical Hohenberg-Kohn limit, there holds an exact closure relation, namely the pair density is the solution to a optimal transport problem with Coulomb cost. We prove that for the case with $N=2$ electrons this problem has a unique solution given by an optimal map; moreover we derive an explicit formula for the optimal map in the case when $\rho$ is radially symmetric (note: atomic ground state densities are radially symmetric for many atoms such as He, Li, N, Ne, Na, Mg, Cu).
In my talk I focus on how to deal with its main mathematical novelties (cost decreases with distance; cost has a singularity on the diagonal). I also discus the derivation of the Coulombic OT problem from the many-electron Schroedinger equation for the case with $N\ge 3$ electrons, and give some results and explicit solutions for the many-marginals OT problem.
Joint works with Gero Friesecke (TU Munich) and Claudia Klueppelberg (TU Munich).
We provide results related to the study of prime points on level sets of homogeneous integral forms which are linear or quadratic. In the linear case we present an extension of the Green-Tao Theorem, which finds affine copies of finite intervals in relatively dense subsets of the primes, to a higher dimensional setting in which one finds affine copies of suitably generic point configurations in relatively dense subsets of the Cartesian product of the primes. For general integral quadratic forms we present a result which is a Birch-Goldbach type theorem for a single quadratic form with sufficient rank. This guarantees solubility among the primes on the level set of a quadratic form subject to local conditions. This is an extension of a well known result due to Hua.
From flapping birds to space telescopes: The modern science of Origami.
The last decade of this past century has been witness to a revolution in the development
and application of mathematical techniques to origami, the centuries-old Japanese art
of paper-folding. The techniques used in mathematical origami design range
from the abstruse to the highly approachable. In this talk, I will describe how geometric
concepts led to the solution of a broad class of origami folding problems - specifically,
the problem of efficiently folding a shape with an arbitrary number and arrangement of
flaps, and along the way, enabled origami designs of mind-blowing complexity and
realism, some of which you'll see too. As often happens in mathematics, theory originally
developed for its own sake has led to some surprising practical applications. The
algorithms and theorems of origami design have shed light on long-standing
mathematical questions and have solved practical engineering problems. I will discuss
examples of how origami has enabled safer airbags, robdingnagian space telescopes,
and more.
Note for Attendees
6:30 p.m. - cakes and refreshments will be served. To ensure we have enough refreshments, please email events@pims.math.ca and specify how many will be attending. Seating is limited.
We formulate an optimal stopping problem for a geometric Brownian motion where the probability scale is distorted by a general non-linear function. The problem is inherently time inconsistent due to the Choquet integration involved. We develop a new approach, based on a reformulation of the problem where one optimally chooses the probability distribution or quantile function of the stopped state. An optimal stopping time can then be recovered from the obtained distribution/quantile function, either in a straightforward way for several important cases or in general via the Skorokhod embedding. This approach enables us to solve the problem in a fairly general manner with different shapes of the payoff and probability distortion functions. We also discuss economical interpretations of the results. In particular, we justify several liquidation strategies widely adopted in stock trading, including those of “buy and hold”, “cut loss or take profit”, “cut loss and let profit run”, and “sell on a percentage of historical high”.
A polynomial is a product of distinct cyclotomic polynomials if and only if it is a divisor over Z[x] of xn–1 for some positive integer n. In this talk, we will examine two natural questions concerning the divisors of xn–1: "For a given n, how large can the coefficients of divisors of xn–1 be?" and "How often does xn–1 have a divisor of every degree between 1 and n?" We will consider the latter question when xn–1 is factored in both Z[x] and Fp[x], using sieve methods and other techniques from analytic number theory in order to obtain our results.
Note for Attendees
Refreshments will be served between the two talks.
Quantum Unique Ergodicity has been a widely studied conjecture of Rudnick and Sarnak (1994), concerning the distribution of large frequency eigenstates on a negatively curved manifold. Arithmetic Quantum Unique Ergodicity (AQUE) restricts the problem to arithmetic manifolds, such as SL(2,Z) \ H, the classical modular surface. Work of Lindenstrauss (2006) combined with the elimination of escape of mass proved by Soundararajan (2010) confirmed AQUE for the classical modular surface. This talk is concerned with AQUE for Hilbert modular surfaces, and in particular, my thesis work involving the elimination of escape of mass in this case.
We will give an overview of Ruelle's book, Thermodynamic Formalism. On May 1 and 3, we plan to cover Chapter 0, interspersing examples with Ruelle's abstract treatment We will relate the subject to material covered in the Entropy course, Math 601D (Fall, 2011):
http://www.math.ubc.ca/~marcus/math601D.
Familiarity with entropy will be helpful, but concepts will be reviewed. This will be the first of 10 informal seminars on the subject, through the month of May, given by the participants.
This is the first talk in a series where we'll define fibered categories, quotient stacks, and gerbes banded by commutative groups. We'll also discuss essential and canonical dimension of fibered categories and some applications to the essential dimension of algebraic groups. In this first talk, we'll define fibered categories and categories fibered in groupoids (CFGs). We'll also discuss examples and introduce quotient stacks as a CFG. The only prerequisites are basic category theory and basic algebraic geometry. No prior knowledge of stacks, essential dimension, etc. is needed.
We consider a symmetrised functional of Brownian bridges, which is related to the partition function of an interacting many-particle system. The aim is to calculate the associated free energy in the thermodynamic limit where temperature and particle density remain positive. More precisely, we give a lower and upper bound on the free energy in terms of certain variational principles. The bounds coincide if either temperature is fixed and the particle density is small or conversely if the density remains fixed and the temperature is high. The novel idea is a representation of the partition function in terms of a marked point process, where the marks are Brownian bridges starting and ending at the corresponding points of the point process. Based on the large deviations results for marked point processes, we employ an argument analogously to Varadhan's lemma to eventually obtain the bounds on the partition function. The difference in upper and lower bounds, however, is not merely technical, but hints at the emergence of Bose-Einstein condensation, where infinitely long cycles appear and whose description remains a major challenge of the field. In a second part we outline the connection to random permutation and random partitions models.
This is the second talk in a series where we'll define fibered categories, quotient stacks, and gerbes banded by commutative groups. We'll also discuss essential and canonical dimension of fibered categories and some applications to the essential dimension of algebraic groups.
In this second talk, we'll define and describe the quotient stack X/G for an algebraic group G and a G-scheme X.
The only prerequisites are basic category theory and basic algebraic geometry. No prior knowledge of stacks, essential dimension, etc. is needed.
Given a partial difference equation (PDiffE), how can one find its conservation laws? Indeed, what is a conservation law? These questions have fairly simple answers for partial differential equations (PDEs), but are harder for PDiffEs. However, recent work has shown that similarities in the key algebraic structures for PDEs and PDiffEs can be exploited. In particular, there are difference analogues of Noether's first and second theorems. Furthermore, each class of conservation laws is characterized by a function, from which the components of the conservation law may be reconstructed. This function is used: a) to establish the converse of Noether's Theorem, b) to prove that infinite hierarchies of inequivalent conservation laws exist for certain integrable PDiffEs, c) to construct finite difference schemes that preserve multiple (non-hyperbolic) conservation laws.
This talk is suitable for a general mathematical audience.
Peter Hydon is well known for his 2000 Cambridge Text in Applied Mathematics: Symmetry Methods for Differential Equations: A Beginner's Guide
The impetus arose from the fact that complete analytical solutions for the metric of a covariant N-body system have proven elusive in General Relativity. To simplify the problem, the number of dimensions was lowered to (1+1) namely one spatial dimension and one-time dimension. This model problem, known as R=T theory (as opposed to the general G=T theory) is amenable to exact solutions in terms of a generalization of the Lambert W function. It was also found that the field equation governing the dilaton (derived from differential geometry) was none other than the Schrodinger equation and consequently amenable to quantization. Thus, one had a theory which combined gravity, quantization and even the electromagnetic interaction. The outcome revealed a previously unknown and already existing natural link between general relativity and quantum mechanics.
We will define a few Grothendieck topologies and what is a torsor is in each of these topologies. This is in preparation to talk about the various cohomology theories these topologies define. We shall also define the Grothendieck-Brauer group and the cohomological Brauer group and go over their basic properties.
The diversity of form in living beings led Darwin to state that it is "enough to drive the sanest man mad". How can we describe this variety? How can we predict it? Motivated by biological observations on different scales from molecules to tissues, I will show how a combination of biological and physical experiments, mathematical models and simple computations allow us to begin to unravel the physical basis for morphogenesis.
About the Niven Lectures: Ivan Niven was a famous number theorist and expositor; his textbooks have won numerous awards and have been
translated into many languages. They are widely used to this day. Niven was born in Vancouver in 1915, earned his Bachelor's and Master's
degrees at UBC in 1934 and 1936 and his Ph.D. at the University of Chicago in 1938. He was a faculty member at the University of Oregon since 1947 until his retirement in 1982. The annual Niven Lecture, held at UBC since 2005, is funded in part through a generous bequest from
Ivan and Betty Niven to the UBC Mathematics Department.
The Graduation Reception is held at 12:30-2:00 pm in the MATH building, room MATH 125.
The lecture begins at 2:00-3:00 pm in the Math Annex building, room MATX 1100. Everyone is welcome to attend both events.
Special Lecture by Malabika Pramanik. Title: Needles, bushes, hairbrushes and trees
Abstract: Points, lines and circles are among the most primitive and fundamental of mathematical concepts, yet few geometric objects have generated more beautiful and nontrivial mathematics. Deceptively simple in their formulation, many classical problems involving sets of lines or circles remain open to this day. I will begin with a sample that has spearheaded much of modern research, and explore connections with analysis, geometry, combinatorics and probability. By the end, we will have seen applications in physics and computer science, and maybe in the not-too-distant future, parallel parking?
We will start with a quick exposition of the random cluster representation of the Ising model on general graphs and see that this representation has a particularly simple form on trees. Relying on this, we will then study two phase transitions for the Ising model on trees (as the inverse temperature increases). The first one is the change from a unique Gibbs state to multiple Gibbs states. In the second one, the Gibbs state obtained from free boundary conditions changes from being extremal to non-extremal. The latter transition is related to the reconstruction problem in information networks. The talk will be partly based on the paper "Broadcasting on Trees and the Ising Model", by Evans, Kenyon, Peres and Schulman.
Symmetries are among the most successfully employed concepts in science and mathematics. They form a cornerstone of various branches of physics, such as classical and quantum mechanics, particle physics and relativity. The governing equations of hydrodynamics generally possess wide symmetry
groups and therefore there is a great potential to exploit these symmetries so as to derive similarity solutions, conservation laws and invariants or to study the effects of symmetry breaking due to the
presence of boundaries or additional body-forces. However, to date,symmetries are often used in a non-explicit or indirect way in hydrodynamics and turbulence theory. On the other hand, there exist
powerful and general methods introduced in the field of group analysis of differential equations which, when suitably adapted, can be readily applied to the aforementioned fields. In this talk we will introduce an algorithmic method which allows associating with a given object its invariant counterpart. The object
under consideration can be, e.g., a turbulence closure model or a finite-difference discretization of a differential equation, which can then be invariantized to yield a turbulence model or a finite-difference
discretization that is invariant under the same Lie point symmetry group as admitted by the original governing equations of hydrodynamics. This method can therefore be used to correct artificial symmetry breaking due to non-appropriately designed turbulence models. As an example it is shown that classical hyperdiffusion as used in two-dimensional (decaying) turbulence simulations violates the symmetries of the incompressible Euler equations. Invariantization of these hyperdiffusion terms yields symmetry-preserving but nonlinear diffusion-like terms. Using the notion of differential invariants it is
demonstrated that the invariantized hyperdiffusion models can be modified with quite some flexibility while still preserving their desired invariance characteristics. First numerical tests show that the invariant hyperdiffusion schemes which can be obtained by this method might be able to reproduce the -3 slope of the energy spectrum in the enstrophy inertial range.
Abstract: The Donaldson-Thomas (DT) theory of a Calabi-Yau threefold X gives rise to subtle deformation invariants. They are considered to be the mathematical counterparts of BPS state counts in topological string theory compactified on X. Principles of physics indicate that the string theory of a singular Calabi-Yau threefold and that of its crepant resolution ought to be equivalent, so one might expect that the DT theory of a singular Calabi-Yau threefold ought to be
equivalent to that of its crepant resolution. There is some difficulty in defining DT when X is singular, but Bryan, Cadman, and Young have (in some generality) defined DT theory in the case where X is the coarse moduli space of an orbifold. The crepant resolution conjecture of Bryan, Cadman, and Young gives a formula determining the DT invariants of the orbifold in terms of the DT invariants of the crepant resolution. In this dissertation, we begin a program to prove the crepant resolution conjecture using Hall algebra techniques inspired by those of Bridgeland.
Let X=(X^{diff}, J) be a compact complex manifold. Kodaira and Spencer developed from 1957 a theory to address the problem:describe the set I of complex structures on X^{diff} close to J. The crowning piece of this theory is Kuranishi's Theorem(1962) which states that the (infinite-dimensional) analytic set I is isomorphic to the product of a (finite-dimensional) analytic set K by a vector space. Moreover, every structure of I is isomorphic to one of K, so that K contains all classes of isomorphisms of complex structures close to J.
Let (E,J) be a CR-structure on a smooth manifold X^{diff}. It is natural to ask the same question in that context that is: describe the set I of CR-structures close to (E,J). However, this is out of reach for a general (E,J).
In this talk, I will introduce the notion of polarized CR-structures, which are a very special type of CR-structures; and give a description of the set I in the case where (E,J) is polarized.
In the first part of this thesis, we study the existence and stability of multi-spot patterns on the surface of a sphere for a singularly perturbed Brusselator and Schnakenburg reaction-diffusion model. The method of matched asymptotic expansions, tailored to problems with logarithmic gauge functions, is used to construct both symmetric and asymmetric spot patterns. There are three distinct types of instabilities of these patterns that are analyzed: self-replication instabilities, amplitude oscillations of the spots, and competition instabilities. By using a combination of spectral theory for nonlocal eigenvalue problems together with numerical computations, parameter thresholds for these three different classes of instabilities are obtained. For the Brusselator model, our results point towards the existence of cycles of creation and destruction of spots, and possibly to chaotic dynamics. For the Schnakenburg model, a differential-algebraic ODE system for the motion of the spots on the surface of the sphere is derived.
In the second part of the thesis, we study the existence and stability of mesa solutions in one spatial dimension and the corresponding planar mesa stripe patterns in two spatial dimensions. An asymptotic analysis is used in the limit of a large diffusivity ratio to construct mesa patterns in one spatial dimension for a general class of two-component reaction-diffusion systems that includes the well-known Gierer Meinhardt activator-inhibitor model with saturation (GMS model), and a predator-prey model. For such one-dimensional patterns, we study oscillatory instabilities of the pattern by way of a Hopf bifurcation and from a reduction to a limiting ODE-PDE system. In addition, explicit thresholds are derived characterizing transverse instabilities of planar mesa-stripe patterns in two spatial dimensions. The results of our asymptotic theory as applied to the GMS and predator-prey systems are confirmed with full numerical results.
The Schrödinger equation, an equation central to quantum mechanics, is a dispersive equation which means, very roughly speaking, that its solutions have a wave-like nature, and spread out over time. We will consider global behaviour of solutions of two nonlinear variations of the Schrödinger equation.
In particular, we consider the nonlinear magnetic Schrödinger equation. We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in H^1, then the solution of the above equation decomposes uniquely into a standing wave part, which converges as t goes to infinity, and a dispersive part, which scatters.
We also consider the Schrödinger map equation into the 2-sphere. We obtain a global well-posedness result for this equation with radially symmetric initial data without any size restriction on the initial data.
We study lattice trees and lattice animals in high dimensions. Lattice trees and animals are interesting combinatorial objects used to model branched polymers in polymer science. They are also of interest in combinatorics and in the study of critical phenomena in statistical physics.
The nearest-neighbor and spread-out models on the d dimensional integer lattice \mathbb{Z}^d, have edge set consisting of pairs \set{x,y} with \|x-y\|_1=1 and \|x-y\|_\infty\leq L with L\geq1 fixed, respectively. On either graph, a \emph{lattice animal} is a finite connected subgraph, and a \emph{lattice tree} is an animal without cycles. Let t_n and a_n be the number of lattice trees and animals with n bonds that contain the origin, respectively. Standard subadditivity arguments provide the existence of the \emph{growth constants} \tau=\lim_{n\to\infty}t_n^{1/n} and \alpha=\lim_{n\to\infty}a_n^{1/n}. We are interested in the \emph{critical points}
of these models, which are the reciprocals of the corresponding growth constants.
We rigorously calculate the first three terms of a 1/d--expansion for the critical points of nearest-neighbor lattice trees and animals. The proof follows an inductive argument similar to the one used in \cite{HS95} and \cite{HS06}, to obtain analogous results for the critical points of self-avoiding walks and percolation. To provide the leading terms in the expansions, we use a mean-field model, related to the Galton-Watson branching process with critical Poisson offspring distribution, and results obtained with the lace expansion. The leading terms are also calculated in the spread-out model. Then we develop expansions for the nearest-neighbor generating functions and, together with the lace expansion, obtain the first and second correction terms.
Our result gives a rigorous proof for previous work on the subject \cite{GP00}, \cite{Harr82,PG95}. Given the algorithmic nature of the proof, it can be extended, with sufficient labor, to compute higher degree terms. It may provide the starting point for proving the existence of an asymptotic expansion with rational coefficients, for the critical point of nearest-neighbor lattice trees.
With the goal of understanding polymer entanglements, for over 20 years there has been interest in questions about knotting and linking of self-avoiding polygons on the simple cubic lattice. Notably, in 1988 Sumners and Whittington
proved that all but exponentially few sufficiently long self-avoiding polygons are knotted. This proved the long standing Frisch-Wasserman-Delbruck conjecture that sufficiently long ring polymers will be knotted with high probability.
Since then there has been progress both theoretically and numerically using lattice polygon models to investigate polymer entanglements. Much of this progress has been motivated by questions arising from the study of
DNA topology. For lattice models, these questions lie at the interface between statistical mechanics, enumerative combinatorics, topology, graph theory and
applied probability/Monte Carlo methods. I will review progress made and highlight new results and open problems, focusing on models of enzyme action on DNA and polymers in confined geometries.
This is joint work with Srikanth Iyengar on results that connects work of Mike Hopkins in homotopy theory and commutative algebra with a theorem of Dave Benson, Jeremy Rickard and myself on group representations. I will spend most of the lecture talking about what the words mean and why we are interested in the results.
Note for Attendees
Please note coffee and cookies will be served beforehand at 2:45 p.m.
We study questions in three arithmetic settings, each of which carries aspects of random-like behaviour.
In the setting of arithmetic functions, we establish mild conditions under which the tuple of multiplicative functions [f1(n), f2(n), ..., fd(n)] densely approximates points in Rd for a positive proportion of n; we obtain a further generalization allowing these functions to be composed with various arithmetic progressions.
Secondly, we examine the eigenvalues of random integer matrices, showing that most matrices have no rational eigenvalues; we also identify the precise distributions of both real and rational eigenvalues in the 2 × 2 case.
Finally, we consider the set S(k) of numbers represented by the quadratic form x2+ky2, showing that it contains infinitely many strings of five consecutive integers under many choices of k; we also characterize exactly which numbers can appear as the difference of two consecutive values in S(k).
We consider the heat equation with random coefficients on Z^d. The randomness of the coefficients models the inhomogeneous nature of the medium where heat propagates. We assume that the distribution of these coefficients is invariant under spatial translations, and has a finite range of dependence. It is known that if a solution to this equation is rescaled diffusively, then it converges to the solution of a heat equation with constant coefficients. In probabilistic terms, this convergence corresponds to the fact that the associated random walk satisfies a central limit theorem. I will present recent progress on the estimation of the speed of this convergence, based on the random walk representation.
When an algebraic group G acts linearly on a vector space V, the stabilizer subgroup G_x of elements fixing a point x is dependent on x. For example, the stabilizer at zero is all of G. But often there is a Zariski open subset of V on which all of the stabilizers G_x are the same (up to conjugation, and after extending scalars). I will discuss this phenomenon, and some consequences, paying particular attention to base fields of positive characteristic.
Consider a permutation \sigma\in S_n as a deck of cards numbered from 1 to n and laid out in a row, where \sigma_j denotes the number of the card that is in the j-th position from the left.\rm\ We study some probabilistic and combinatorial aspects of the shuffle on S_n defined by removing and then randomly reinserting each of the n cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be quite strong and possesses some surprising features.
The Monge-Kantorovich optimal transportation problem is to pair producers with consumers so as to minimize a given transportation cost. When the producers and consumers are modeled by probability densities on two given manifolds or subdomains, it is interesting to try to understand the structure of the optimal pairing as a subset of the product manifold. This subset may or may not be the graph of a map.
The talk will expose the differential topology and geometry underlying many basic phenomena in optimal transportation. It surveys questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It establishes new connections --- some heuristic and others rigorous ---based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.
The classical problem of optimal transportation can be formulated as a linear optimization problem on a convex domain: among all joint measures with fixed marginals find the optimal one, where optimality is measured against a given cost function. Here we consider a variation of this problem by imposing an upper bound constraining the joint measures, namely: among all joint measures with fixed marginals and dominated by a fixed measure, find the optimal one. After computing illustrative examples, we given conditions guaranteeing uniqueness of the optimizer and initiate a study of its properties. Based on a preprint arXived with Jonathan Korman.
Economists are interested in studying who matches with whom (and why) in the educational, labour, and marriage sectors. With Aloysius Siow, Xianwen Shi, and Ronald Wolthoff, we propose a toy model for this process, which is based on the assumption that production in any sector requires completion of two complementary tasks. Individuals are assumed to have both social and cognitive skills, which can be modified through education, and which determine what they choose to specialize in and with whom they choose to partner.
Our model predicts variable, endogenous, many-to-one matching. Given a fixed initial distribution of characteristics, the steady state equilibrium of this model is the solution to an (infinite dimensional) linear program, for which we develop a duality theory which exhibits a phase transition depending on the number of students who can be mentored. If this number is two or more, then a continuous distributions of skills leads to formation of a pyramid in the education market with a few gurus having unbounded wage gradients. One preprint is on the arXiv; a sequel is in progress.
The well-known fiber dimension theorem in algebraic geometry says that for every morphism f : X -> Y of integral schemes of finite type, the dimension of each nonempty fiber of f is at least dim X - dim Y. We will discuss two analogues of this theorem, where schemes are replaced by algebraic stacks (or categories fibered in groupoids) and dimension is replaced by essential or canonical dimension. These results will be applied in particular to derive new relations between the essential dimension and canonical dimension of group schemes.
In this talk we compare several different stratifications of parameter spaces of sheaves. The starting point is the infinite Yang-Mills stratification of the space of vector bundles on a compact Riemann surface, which is equal to the stratification by Harder-Narasimhan types. We then go on to look at finite stratifications of some quot schemes associated to a certain group action (the geometric invariant theory quotient for this action is a moduli space for sheaves) and relate this to a stratification of the quot scheme by Harder-Narasimhan types. Finally we discuss the limitations of the finite stratifications and how we could instead modify the set up to get infinite stratifications.
-2 curves are a favorite toy of birational geometers working in dimension 3 - they are slightly more complicated cousins of the resolved conifold. In this talk I'll try to give a reasonably self contained introduction to the theory of motivic DT invariants, and integrality, by explaining how this theory plays out in the case of "noncommutative" -2 curves. It turns out that, in common with the noncommutative conifold, the motivic DT partition function for -2 curves have a strikingly nice form, confirming the integrality conjecture in this case.
The critical sandpile model was introduced by Bak, Tang and Wiesenfeld in 1987--88 and attracted a lot of attention after the discovery of a somewhat elusive abelian group structure of the model by Dhar. In this lecture I will try to shed some light on this group property by showing that the sandpile model is an equal entropy symbolic cover of a Z2-action (or, in d dimensions, of a Z^d-action) by automorphisms of a compact abelian group, the 'harmonic model'. The construction involved goes back to Vershik, who used it to define Markov or sofic partitions for hyperbolic toral automorphisms (joint work with Evgeny Verbitskiy).
Lunch will be provided only if you are writing the Analysis exam. Lunch is served 12:00-1:00 pm in Math 125.
If you have food allergies, please let Lee know.
I will show a simple method to decompose the Gaussian free field associated to a (weighted) graph or manifold into a sum of finite range Gaussian fields, which are fields that are smoother than the original field and have spatially localized correlations.
We consider three counter-example: A non-smooth group which has a fppf-torsor that is not étale, similarly for a formally smooth group. We also consider how to differentiate between free and non-free actions for µ_p.
We construct noninvertible maps on every compact surface, and study their chaotic properties from both the measure theoretic and topological points of view (joint work with Jane Hawkins).
In this talk we describe a generalization of the result of Escauriaza-Seregin-Sverak on blow-up of the L^3 norm at a Navier-Stokes singularity by establishing the blow-up of any weaker critical Besov norm with finite third index as well. Following previous joint works with C. Kenig and with I. Gallagher and F. Planchon respectively, we use the "dispersive-type" method of concentration compactness and critical elements developed by C. Kenig and F. Merle. Joint work with I. Gallagher and F. Planchon
In this talk, I would like to discuss some recent progress in analysis and geometry of optimal transport maps that arise when mass distributions are matched in most cost efficient way. Especially, continuity of optimal maps will be addressed, which is related to nonlinear partial differential equations as well as Riemannian geometry.
Note for Attendees
Refreshments will be served in MATH 125 at 2:45 pm before the colloquium.
Let X be a Calabi-Yau threefold. We study the symmetric trilinear form on the integral second cohomology group of X defined by the cup product. Our study is motivated by C.P.C. Wall's classification theorem, which roughly says that the diffeomorphism class of a spin sixfold is determined by the trilinear form. We investigate the interplay between the Chern classes and the trilinear form of X, and demonstrate some numerical relations between them. If time permits, we also discuss some properties of the associated cubic form. This talk is based on a joint work with P.H.M. Wilson.
We analyze stochastic completeness, or non-explosiveness, of the variable-speed
random walk (VSRW) on weighted graphs. We prove a criterion relating volume
growth in an adapted metric to stochastic completeness of the VSRW. This
criterion is analogous to the optimal result for Riemannian manifolds and is
shown to be sharp. The proof is accomplished through the construction of a
Brownian motion on a metric graph which behaves similarly to the VSRW under
consideration. Results of Sturm on stochastic completeness for local Dirichlet
spaces are then applicable to this Brownian motion, and non-explosiveness of
the Brownian motion is shown to imply non-explosiveness of the VSRW.
It was a result of Greenlees and Sadofsky that classifying spaces of finite groups satisfy a Morava K-theory version of Poincare duality, which was proved by showing the contractibility of the corresponding Tate spectrum. In this series of two talks, I will explain the proof, discuss its generalization to quotient orbifolds and consequences with examples. Some background in equivariant stable homotopy theory will be given. If time permits, I will also explain why the duality map can be viewed as coming from a Spanier-Whitehead type construction for differentiable stacks.
We will discuss some results concerning the representability of torsors. In the case of an abelian scheme, we will explain how the properties "X is representable" and "the class of X in the first cohomology group is torsion" are related. This will lead us to the construction of an fppf sheaf torsor which is not representable.
A formula for the dimension of the space of cuspidal modular forms on Γ0(N) of even weight k≥2 has been known for several decades. More recent but still well-known is the Atkin-Lehner decomposition of this space of cusp forms into subspaces corresponding to newforms on Γ0(d) of weight k, as d runs over the divisors of N. A recursive algorithm for computing the dimensions of these spaces of newforms follows from the combination of these two results, but it is desirable to have a formula in closed form for these dimensions. In this talk we describe such a closed-form formula, not only for these dimensions, but also for the corresponding dimensions of spaces of newforms on Γ1(N) of weight k≥2. This formula is much more amenable to analysis and to computation. For example, we derive asymptotically sharp upper and lower bounds for these dimensions, and we compute their average orders. We also establish sharp inequalities for the special case of weight-2 newforms on Γ0(N), and we report on computations of these dimensions. For example, we can find the complete list of all N such that the dimension of the space of weight-2 newforms on Γ0(N) is less than or equal to 100; previous such results had only gone up to 3.
Beginning with the work of J. Hersch for the two sphere and that of P. Li and S. T. Yau for more general surfaces, the question of determining surfaces of fixed area that maximize the first eigenvalue has been actively studied. In this talk I will describe recent work with R. Schoen concerning extremal eigenvalue questions for surfaces with boundary. In both cases the eigenvalue problems are related to minimal surface questions. For closed surfaces these are minimal surfaces in spheres while for surfaces with boundary they are related to minimal surfaces in the ball satisfying a natural boundary condition. I will describe some results on determining optimal surfaces. I will also describe some recent work with Martin Li on compactness of the space of free boundary minimal surfaces with a fixed topological type.
Note for Attendees
Refreshments will be served at 2:45 p.m. in the Math Lounge (MATH 125).
The Gopakumar-Marino-Vafa formula, proven almost ten years ago, evaluates certain triple Hodge integrals on moduli spaces of curves in terms of Schur functions. It has since been realized that the GMV formula is a special case of the Gromov-Witten/Donaldson-Thomas correspondence for Calabi-Yau threefolds.
In this talk, I will introduce an orbifold generalization of the GMV formula which evaluates certain abelian Hodge integrals in terms of loop Schur functions. I will introduce local Z_n gerbes over the projective line and show how the gerby GMV formula can be used to prove the GW/DT correspondence for this class of orbifolds. With the remaining time, I will sketch the main ideas in the proof of the formula and discuss generalizations to other geometries.
A three-phase contact line forms when a gas-liquid interface intersects a solid substrate, and a moving contact line presents a well-known singularity that cannot be computed using the conventional Navier-Stokes formalism. I will discuss the use of a diffuse-interface model for computing moving contact lines. The Cahn-Hilliard diffusion is known to regularize the singularity and makes possible a continuum-level computation. But relating the results to physical reality is subtle. I will show numerical results that suggest a well-defined sharp-interface limit, with a finite contact line speed that can be related to measurements. Then I will discuss applications including enhanced slip on textured substrates and propulsion of water striders on the air-water interface.
Abstract: It is by now a classical result that sets of positive density of R^2 contain all large distances. It was however observed by Bourgain that the analogue result does not hold for 3 term progressions (3 equally spaced points along a line) in any dimensions. Our aim is to discuss an approach to show that such results are still possible if one changes the metric, for example using the l^4 metric. If time permits we'll mention other possible point configurations. This is ongoing joint work with Brian Cook and Malabika Pramanik.
During the last two decades, semidefinite optimization has grown into a significant field of research with applications in many diverse areas such as graph theory, distance geometry, combinatorial optimization, low-rank matrix completion, and polynomial optimization. In this talk, I will discuss my work in two of these areas, namely distance geometry and combinatorial optimization.
In distance geometry, Euclidean distance matrices (EDMs) have recently received revived interest due to their use in modern applications such as sensor network localization, protein structure determination, and machine learning. The second reason for this revived interest is the fact that we now have semidefinite optimization solvers that we can use to solve problems involving EDMs (however, we are limited in the size of problems we can solve efficiently due to the high complexity of these semidefinite solvers). I will discuss my theoretical contribution relating cliques in the graph of a partial EDM to identifying a reduced problem formulation, and how I used this result to develop numerical methods to solve large-scale instances in each of the three modern applications mentioned above.
In combinatorial optimization, semidefinite optimization is used to efficiently compute high-quality bounds to many difficult (in fact, NP-hard) problems, such as Max-Cut and binary quadratic optimization. This has led to the development of state-of-the-art branch-and-bound methods for solving such problems to optimality. I will discuss my work on a bounding procedure for Max-Cut which has been obtained by adding a regularization term to the standard semidefinite bound — this allows us to use basic numerical tools (a eigenvalue decomposition method, and a quasi-Newton optimization method) to compute high-quality semidefinite bounds efficiently. I will show how this new bounding procedure gives a significant improvement over the current state-of-the-art method.
We will discuss a chapter of a book on Delay differential equations by Hal Smith. This chapter can be downloaded from the internet at the following link provided. The chapter is called "Delayed Negative Feedback: A warm-up", and it will introduce some of the basics of delay differential equations and how to obtain some analytical results about their stability.
Since the famous work of V. Scheffer about 20 years ago, it has been known that the Cauchy problem for the incompressible Euler equations has non-unique weak solutions. Recently, De Lellis and Szekelyhidi demonstrated that this phenomenon can be viewed as an instance of the so-called h-principle, thereby providing a shorter and more general proof of the non-uniqueness. In this talk I will briefly review their method and then present some subsequent results, including global existence and non-uniqueness for 3D Euler, the approximation of measure-valued solutions by weak ones, and non-uniqueness for shear flow initial data.
The random interlacements (at level u) is a one parameter family of random subsets of Z^d (d>=3), introduced recently by A.-S. Sznitman, which arises as the
local limit of the trace of a simple random walk on a d-dimensional torus, when the size of the torus goes to infinity. The parameter u controls the density of
the interlacement. The vacant set at level u (i.e. the complement set of the random interlacement at level u) undergoes non-trivial percolation phase transition
as u varies. We study the supercritical phase and show that finite connected components of the vacant set are "small" for all d>=3 if u is small enough. Our
method is markedly different from that of A. Teixeira (2011), which gives the analogous result for d>=5.
This is joint work with Tetsuya Ito. Fractional Dehn twist coefficient (FDTC), defined by Honda-Kazez-Matic, is an invariant of mapping classes. In this talk we study FDTC by using open book foliation method, then obtain results in topology, geometry, and contact geometry of the open-book-manifold of a mapping class.
The theory of reductive groups has many applications in number theory, geometry, and representation theory. For some purposes it is natural to consider a more general notion of "pseudo-reductive" group, first studied by Borel & Tits. We will explain the motivation for this (with examples), and discuss the structure theory that has been established in recent years, and mention some applications. If time permits, we'll discuss some more recent developments. This is joint work with O. Gabber and G. Prasad.
Note for Attendees
Please note the nonstandard starting time for this seminar. PIMS will serve refreshments (coffee/tea/cookies) in the PIMS Lounge (ESB Room 4133) at 3:00 p.m.
Let p be a prime, r >= 3, and n_i = p^{a_i} for positive integers a_1,...,a_r. Set G = GL_{n_1} x ... x GL_{n_r}, and let \mu be a central subgroup of G. The Galois cohomology set H^1(K, G/\mu) classifies r-tuples of central simple algebras satisfying linear equations in the Brauer group Br(K). We study the essential dimension of G/\mu by constructing the 'code' associated to /mu.
It was a result of Greenlees and Sadofsky that classifying spaces of finite groups satisfy a Morava K-theory version of Poincare duality, which was proved by showing the contractibility of the corresponding Tate spectrum. In this series of two talks, I will explain the proof, discuss its generalization to quotient orbifolds and consequences with examples. Some background in equivariant stable homotopy theory will be given. If time permits, I will also explain why the duality map can be viewed as coming from a Spanier-Whitehead type construction for differentiable stacks.
Note for Attendees
The topology seminar is rescheduled to Tuesday for this week.
We study the regularity of solutions of parabolic equations of the form
u_t - Iu = f,
where I is a fully non linear non local operator. We prove C^\alpha regularity in space and time and, under different assumptions on the kernels,
C^{1,\alpha}; in space for translation invariant equations. The proofs rely on a weak parabolic ABP and the classic ideas of K. Tso and L. Wang. Our results remain uniform as \sigma goes to 2 allowing us to recover most of the regularity results of the local case.
This is a joint work with H ector Chang Lara.
We will explain the basic definitions of Chow groups, correspondences and Chow motives. This will be the first talk in a series to help us study the incompressibility of certain varieties.
UBC/UMC is the Undergraduate Mathematics Colloquium at UBC.
These talks are for undergraduates interested in mathematics and mathematical research. They are put on by professors, senior graduate students and visitors to the department. Graduate students are also invited to attend.
Title: Topological Data Analysis
Abstract: This is not an applied math talk. On the contrary: I will show how the abstract tools of Algebraic Topology can be used to assemble low-dimensional representations of data into high-dimensional structures. An example of such an assembly would be the brain's inference of a 3D environment from a 2D image from each eye.
The Sylvester-Gallai Theorem states that, given any set P of n points in the plane not all on one line, there is at leads one line through precisely two points of P. Such a line is called an ordinary line. How many ordinary lines must there be? The Sylvester-Gallai Theorem says that there must be at leads one but, in recent joint work with T. Tao, we have shown that there must be at least n/2 if n is even and at least 3n/4 - C if n is odd, provided that n is sufficiently large. These results are sharp. My plan in the talk is to give an overview of this problem and of our work towards its solution.
Note for Attendees
The lecture will be preceded by a reception in ESB 4133 at 2:30 pm.
The Euclidean division is a basic tool when dealing with the ordinary integers. It does not extend to rings of integers of algebraic number fields in general. It is natural to ask how to measure the "deviation" from the Euclidean property, and this leads to the notion of Euclidean minimum. The case of totally real number fields is of special interest, in particular because of a conjectured upper bound (conjecture attributed to Minkowski). The talk will present some recent results, obtained jointly with Piotr Maciak.
Note for Attendees
Please note the nonstandard starting time for this seminar.
The study of quadratic forms is a classical and important topic of algebra and number theory. A natural example is the trace form of a finite Galois extension. This form has the additional property of being invariant under the Galois group, leading to the notion of "self-dual nornal basis", introduced by Lenstra. The aim of this talk is to give a survey of this area, and to present some recent joint results with Parimala and Serre.
Note for Attendees
Refreshments will be served in Math Lounge Room 125 at 2:45 p.m.
The existence and stability of localized patterns of criminal activity
are studied for the reaction-diffusion model of urban crime that was
derived by Short et.~al.~[Math. Models. Meth. Appl. Sci., {\bf 18},
Suppl. (2008), pp.~1249--1267] as a continuum limit of an
agent-based model. Such patterns, characterized by the concentration
of criminal activity in localized spatial regions, are referred to as
hot-spot patterns and they occur in a parameter regime far from the
Turing point associated with the bifurcation of spatially uniform
solutions. Singular perturbation techniques are used to construct
steady-state hot-spot patterns in one and two-dimensional spatial
domains. Explicit stability thresholds for these patterns are obtained
by first deriving and then analyzing a new class of nonlocal
eigenvalue problems. In a certain parameter regime of this model,
analytical and numerical methods are used to analyze the phenomena of
hot-spot nucleation, whereby elevated regions of criminal activity
emerge ``spontaneously'' from a quiescent background. Finally, an
extended reaction-diffusion model that incorporates the effect of
police is analyzed, and optimal strategies to prevent the occurrence
of stable hot-spots are discussed.
Joint work with: Theodore Kolokolnikov (Dalhousie), Simon Tse (UBC), and
Juncheng Wei (CUHK, UBC).
The equivariant embeddings of a split torus have been well-known since the 70s. The isomorphism classes of such embeddings are classified by combinatorial objects called fans (after Demazure). In this talk, we address the classification of the embeddings of a non-necessary split torus and ask: Are the isomorphisms classes of such embeddings classified by Galois-stable fans? If time permits, we will discuss the analogous results in the setting of spherical homogeneous spaces.
This talk concerns applications of ergodic theory to Euclidean Ramsey theory. We shall focus mainly on the problem of finding triangular configurations in thickened sets of positive density in the plane (due to Furestenberg, Katzenelson, and Weiss). The generalization for arbitrary configurations in dimension at least 2 (due to Ziegler) is also to be discussed.
It is well known that an upper and lower bound on the Monge-Amp{\`e}re measure of a convex function u implies this function must actually be strictly convex. A lesser known result, also by Caffarelli, states that if the Monge-Amp{\`e}re of u has only a lower bound, the contact set between u and a supporting affine function must have affine dimension strictly less than n/2. By means of a careful geometric construction involving the subdifferential, we give an alternative proof of Caffarelli's result, and extend the result to optimal transportation problems with cost functions satisfying the weak Ma-Trudinger-Wang condition. This talk is based on a joint work with Young-Heon Kim.
A closed geodesic on a surface can also be viewed as a closed orbit of the geodesic flow on the unit tangent bundle of the surface. In this talk I will discuss the main tool for studying the knot-properties of closed orbits of (three dimensional) flows. This tool is called a template, first defined and used by Birman and Williams to study the well known Lorenzbutterfly.
The theory of templates was first used for geodesic flows by Ghys. I will discuss questions related to his extraordinary result, that the closed geodesics on the modular surface equal the closed orbits on the Lorenz butterfly, and will discuss some generalization of his methods to other hyperbolic surfaces.
Our goal will be to get a feeling for what Chow groups are and how do they "look like". We will discuss CH_0 of a smooth compact complex curve and see how the classical Abel-Jacobi theorem gives a very concrete description of it. We will then talk a little about the group of divisors on a smooth variety, for which the situation is not much worse. We will mention the group of zero-dimensional cycles on a surface, which is sometimes considerably more complicated. Finally we will see that while the Chow groups can be defined for non-smooth varieties, defining the intersection product is not a trivial matter at all.
Suppose S: X --> X and T: Y --> Y are continuous selfmaps of metric spaces X,Y (especially, with X,Y compact). They are topologically conjugate if there is a homeomorphism h: X --> Y such that hS = Th . Various equivalence relations coarser than topological conjuacy have been considered for such maps S,T including the following: entropy conjugacy (as defined by Bowen); entropy conjugacy (as defined by Buzzi); Borel conjugacy (h is only required to be a Borel-bimeasurable bijection); Borel isomorphism after restriction to a set of measure one for every invariant Borel probability. I'll discuss theorems and questions around these ideas, especially with regard to countable state Markov shifts, for which another relation ("almost isomorphism", in the spirit of Adler-Marcus) is relevant. I'll include new results joint with Buzzi and Gomez, and also a new universality theorem of Mike Hochman.
Berkovich's rigid analytic spaces are path-connected, Hausdorff, locally compact spaces that generalize non-archimedean fields in a way that allows conducting analysis. We use them to prove non-archimedean analogs of results in complex dynamics.
It is a classical result that over the complex numbers, whenever a rational function φ has a fixed point that is attracting but not superattracting, that is a fixed point z with 0 < |φ'(z)| < 1, there is a critical point of φ whose orbit is attracted to z. We show that a similar, but not identical, result holds over non-archimedean fields, with applications to both global and local non-archimedean dynamics.
Shift equivalence and strong shift equivalence of matrices over a semiring are matrix relations of great significance for symbolic dynamics; they are also interesting simply for matrices. The classification of nonnegative integral matrices up to strong shift equivalence is equivalent to the classification of shifts of finite type, the basic systems of symbolic dynamics. This classification problem has been open for 40 years.
I'll discuss the meaning of these relations and recent results (joint with Kim and Roush) for the case of positive matrices with entries from a dense subring of the reals. These results are obtained by extending the path methods approach developed by Kim and Roush in the early 1990s, which is based on a study of paths of conjugate positive real matrices.
Note for Attendees
Refreshments will be served at 2:45 p.m. in MATH 125 before the colloquium.
Extending the work of Jia and Sverak on self-similar solutions of the Navier-Stokes equations, we show the existence of large, forward, discrete self-similar solutions.
We consider the contact process on finite trees. We assume that
the infection rate is larger than the critical rate for the one-dimensional
process. We show that, for any sequence of trees with increasing number of
vertices and degree bounded by a universal constant, the expected
extinction time of the process grows exponentially. Additionally, the
extinction time divided by its expectation converges in distribution to the
unitary exponential distribution. This is joint work with Thomas Mountford,
Jean-Christophe Mourrat and Qiang Yao.
The talk will be an extended advertisement of a program to apply topological ideas to computation of essential dimension of groups.
The essential dimension of a group G measures to what extent every generically free action of G on an algebraic variety can be "compressed". For the symmetric group S_n, the essential dimension is directly related to the classical question how much an algebraic equation of degree n can be simplified by a rational change of variables.
I will introduce a topological approach to obtaining lower bounds on essential dimension. I will then survey some (non-topological) advances in theory of essential dimension and discuss some parallels in topology. Finally I will speculate on the possibility to relate results in the theory of topological group actions and results on essential dimension in a way that might benefit both fields.
We will continue the discussion from the previous talk on the basic ideas of Chow theory. We review the basic properties that make the Chow functor to be an oriented Borel-Moore homology theory, and then we will discuss some further topics including their equivariant version, their operational version and their universal version.
Let K be a field of characteristic zero and let k and l be two multiplicatively independent positive integers. We prove the following result: a power series F(x) in K[[x]] satisfies both a k- and a l-Mahler type functional equation if and only if it is a rational function. This proves a conjecture of Loxton and van der Poorten. This is joint work with Boris Adamczewski.
The Schubert subvarieties of a rational homogeneous variety X are distinguished by the fact that their homology classes form an additive basis of the integer homology of X. In general, the Schubert varieties are singular.
The cominuscule rational homogeneous varieties are those admitting the structure of a compact Hermitian symmetric space (eg. complex Grassmannians). In this case, type-dependent characterizations of the singular loci are known.
I will discuss a type-independent description, by representation theoretic data, of the singular loci. The result is based on a characterization (joint with D. The) of the Schubert varieties by an non-negative integer and a marked Dynkin diagram.
(If there is time left, I will discuss the project in which the integer-diagram characterization arose as a technical lemma. This work aims to determine whether or not the Schubert classes admit any algebraic representatives (other than the Schubert varieties). It is a remarkable consequence of Kostant's work that these algebraic representatives are solutions of a system of PDE; as a consequence, differential geometric techniques may be applied to this algebro-topological question.)
Asphaltene deposition is one of the important problems of
oil production that requires an accurate predictive modeling.
Uncontrolled deposition of asphaltenes may lead to a significant
reduction of oil production, or even to a total pipe plugging. We
developed an asphaltene deposition model in a pipeline. The model is
based on data, which are obtained by experiments performed in a
Couette device, where the inner cylinder rotates, and deposition on
the outer wall is studied. A detailed theoretical analysis of an
applicability of a Couette device for imitation of the asphaltene
deposition in a pipe flow is presented. A hypothesis stating that the
asphaltene deposition is a particle size controlled process is
accepted. A population balance model is employed for modeling the
particle size evolution. A concept of the critical particle size is
introduced; only particles that are smaller than the critical size
can deposit. The model developed contains only three parameters that
are determined experimentally by using a Couette device. The model of
asphaltene deposition in a Couette device allows accurate describing
the deposit mass growth in time. Performance of the deposition model
for a pipeline with the coefficients obtained by a laboratory Couette
device is also demonstrated.
Furstenberg-Katznelson's Theorem states that if "A" is a subset of Z^d with
positive upper density then for any finite set F of Z^d, A contains an affine image of F. We
wish to prove analogue theorem in prime tuples P^d where positive upper density is replaced by
relative upper density in P^d. This is partially done by Magyar and Cook in the case that no
two points in F have a common projection on some axis. When we count such configurations P^d
behaves like a random subset of Z^d with density(1/log N)^d but this is not true in general
since P^d has direct product structure. In this talk, we will discuss how to use hypergraph
approach and Gowers' Transference Principle to deal with the case that F is a corner (i,e,.
simplex of the form {(x_1,..,x_d),(x_1+s,x_2,..,x_d),...,(x_1,...,x_d+s)},s \neq 0). We expect
that the same method should also work for any finite set F. This is a joint work with Akos
Magyar.
I will describe recent work that characterizes the Schubert varieties that arise as variations of Hodge structure (VHS). I will also discuss the central role that these Schubert VHS play in our study of general VHS. In particular: (i) infinitesimally their orbits under the isotropy action `span' the space of all VHS, yielding a complete description of the infinitesimal VHS; (ii) the cohomology classes dual to the Schubert VHS form an (integral) basis of the invariant characteristic cohomology associated to the system of PDE characterizing VHS.
One of the biggest challenges in the field of random Schrödinger operators is to prove the existence of absolutely continuous spectrum for the Anderson model for small disorder in dimensions greater equal to 3. So far, the existence of absolutely continuous spectrum is only known for models on infinite-dimensional tree structures. The first proof, done by Abel Klein for a regular tree, dates back to 1994.
Recent developments considered trees of finite cone type and cross products of trees with finite graphs, so called tree-strips. I will present a proof for the existence of absolutely continuous spectrum for models on tree-strips of finite cone type. The proof uses a version of the Implicit Function Theorem in Banach spaces which are constructed by a supersymmetric formalism using Grassmann variables.
In 1997 Elekes showed the following. Suppose you have an $n\times n$ cartesian product $A \times B$ in the real plane. If $cn$ lines each contain at least $cn$ points of the cartesian product then $c'n$ of the lines are parallel or $c'n$ of them have the same intersection point. We show that this result can be extended by considering fewer lines. Specifically, if we have $cn^{2/3+\beta/3}$ lines each containing $c'n$ points of the cartesian product then $c''n^{\beta} of the lines are parallel or have the same intersection point. We used this to extend a theorem of Elekes and Ronyai regarding the structure of a surface containing many points of a cartesian product. These results give a proof of a conjecture of Purdy saying that if two collinear sets in the plane give too few distinct distances then the lines are parallel or orthogonal. This is joint work with Jozsef Solymosi and Frank de Zeeuw.
Let G be a finite group, we consider the homotopy colimit of classifying spaces of abelian subgroups of G. This space is a K(\pi,1) for certain finite groups, but there are examples when the space has non-vanishing higher homotopy groups. I will also talk about the complex K-theory of this space and give some examples.
Planar maps are embeddings of graphs on compact orientable
surfaces. We shall consider a special class of maps: triangulations with an
infinite simple boundary. We shall consider measures supported on this
space satisfying two natural properties: translation invariance and domain
Markov property. We shall show these two characters completely characterize
them. The methods reveal some standard maps and some mysterious maps of
hyperbolic nature.
I will discuss a conjecture which provides a relationship between Manin symbols in the homology of modular curves and cup products of cyclotomic units in Galois cohomology with restricted ramification. I hope to be able to give something of an idea of its meaning, with an eye towards generalization. I will also relate recent progress by Fukaya and Kato that essentially proves the conjecture under a mild but difficult to remove hypothesis.
One remarkable application of classical Schubert calculus on the cohomology of the Grassmannian is its close connection to the eigenvalue problem on sums of hermitian matrices. The eigenvalue problem asks: Given three sequences of real numbers, do there exist hermitian matrices A+B=C with eigenvalues given by the three sequences? This problem has a generalization to eigenvalues of majorized sums of hermitian matrices where we replace "A+B=C" with "A+B>C".
In this talk, I discuss joint work with D. Anderson and A. Yong where we show that the eigenvalue problem on majorized sums is related to the Schubert calculus on the torus-equivariant cohomology of the Grassmannian in the same way that classical Schubert calculus is related to eigenvalue problem on usual sums of Hermitian matrices. One consequence of this connection is a generalization of the celebrated saturation theorem to T-equivariant Schubert calculus.
I will present an upper bound on the (d-1)-volume and covering numbers of a filtration of the non-differentiability locus of the distance function of a compact set in R^d. A consequence of this upper bound is that the projection function to a compact subset K depends in a Hoelder way on the compact set, in the L^1 sense. This in turn implies that Federer's curvature measure of a compact set with positive reach can be reliably estimated from a Hausdorff approximation of this set, regardless of any regularity assumption on the approximation.
In the local Langlands program the (smooth) representation theory of p-adic reductive groups G in characteristic zero plays a key role. For any compact open subgroup K of G there is a so called Hecke algebra H(G,K). The representation theory of G is equivalent to the module theories over all these algebras H(G,K). Very important examples of such subgroups K are the Iwahori subgroup and the pro-p Iwahori subgroup. By a theorem of Bernstein the Hecke
algebras of these subgroups (and many others) have finite global dimension.
In recent years the same representation theory of G but over an algebraically closed field of characteristic p has become more and more important. But little is known yet. Again one can define analogous Hecke algebras. Their relation to the representation theory of G is still very
mysterious. Moreover they are no longer of finite global dimension. In joint work with R. Ollivier we prove that over any field the algebra H(G,K), for K the (pro-p) Iwahori subgroup, is Gorenstein.
Although directed random graph models are frequently used in modeling the electrical activity of networks of cortical neurons, experimental results consistently reveal that the actual network topology is complex, and tends to be clustered locally. This suggests that the random network assumption is unrealistic and that when analysing population dynamics in cortical networks, it is necessary to employ directed network models that incorporate clustering.
In this seminar I shall describe simulation results that demonstrate that replacing random connectivity with clustered connectivity can induce instability in subsets of neurons, in terms of significantly increased firing rates. Moreover, it is shown that one specific network topology gives rise to slow bistable switching between low and high states.
The aim of presenting this seminar is not to describe finished mathematical work, but rather to seek collaboration or assistance with finding mathematical explanations that predict that clustered connectivity can lead to the bistable or unstable states observed in simulations.
UBC/UMC is the Undergraduate Mathematics Colloquium at UBC.
These talks are for undergraduates interested in mathematics and mathematical research. They are put on by professors, senior graduate students and visitors to the department. Graduate students are also invited to attend.
Title: Coding Theory and Practice
Abstract: In this talk, we will introduce the framework of coding theory, including some classical results, open problems and applications to data transmission and storage.
In this thesis we present a reduction theory for the symmetrizable split maximal Kac-Moody groups. However there are many technical difficulties before one can even formulate a reduction theorem. Combining the two main approaches commonly seen in the literature we define a group, first over any field of characteristic zero and then on any commutative ring of characteristic zero. Then we prove a number of structural properties of the group such as representation in the highest weight modules, existence of a Tits system and an Iwasawa decomposition over $\RR$ and $\CC$. Finally we arrive at reduction theory which can only hold for part of the group.
We show that for each prime p congruent to 1 (mod 8), there exists a threefold Xp such that the existence of certain rational points on Xp produces families of generalized Mordell curves and of generalized Fermat curves that are counterexamples to the Hasse principle explained by the Brauer–Manin obstruction. We also introduce a notion of the descending chain condition (DCC) for sequences of curves, and prove that there are sequences of generalized Mordell curves and of generalized Fermat curves satisfying DCC.
Abstract: We will show how elementaryproduct decompositions of measures can detect directionality in sets, and show how this can be used to describe non-differentiability sets of Lipschitz functions on R^n, and to understand the phenomena that occur because of behaviour of Lipschitz functions around the points of null sets.
In order to prove this we will need to prove results about the geometry of sets of small Lebesgue measure: we show that sets of small measure are always contained in a "small" collection of Lipschitz surfaces.
The talk is based on a joint work with G. Alberti, P. Jones and D. Preiss.
Higher stacks arise in many contexts in algebraic geometry and differential topology. The simplest type are higher principal bundles, special cases of which include principal bundles and n-gerbes. Locally, these objects are presentable by higher cocycles on a hypercover. With ordinary principal bundles, we obtain a bundle from a cocycle by using the cocycle to construct a Lie groupoid over the trivial bundle on the cover, and then passing to its orbit space. We establish the existence of an analogous construction for arbitrary higher principal bundles. Unpacking this construction in examples, we recover the familiar definitions of principal bundles, bundle gerbes, multiplicative gerbes and their equivariant versions, now seen as instances of a single construction. Applications beyond this include establishing a representability criterion for connected simplicial presheaves, and a Lie's 3rd theorem for finite dimensional L_oo-algebras.
Sparse signal recovery has been dominated by the basis pursuit denoise (BPDN) problem formulation for over a decade. In this paper, we propose an algorithm that outperforms BPDN in finding sparse solutions to underdetermined linear systems of equations at no additional computational cost. Our algorithm, called WSPGL1, is a modification of the spectral projected gradient for minimization (SPGL1) algorithm in which the sequence of LASSO subproblems are replaced by a sequence of weighted LASSO subproblems with constant weights applied to a support estimate. The support estimate is derived from the data and is updated at every iteration. The algorithm also modifies the Pareto curve at every iteration to reflect the new weighted minimization problem that is being solved. We demonstrate through extensive simulations that the sparse recovery performance of our algorithm is superior to that of minimization and approaches the recovery performance of iterative re-weighted (IRWL1) minimization of Candčs, Wakin, and Boyd, although it does not match it in general. Moreover, our algorithm has the computational cost of a single BPDN problem.
In this talk we will discuss the integer distances problem: can we find n points in a general position on the plane such that all the distances between the points are integers? So far this question has been resolved up to 7 points. We will present results related to the problem and our recent efforts to find 8 points with integer distances.
The study of central simple algebras over a field is a venerable topic in ring theory. There is a generalization of central simple algebras to schemes in the étale topology (in fact to arbitrary ringed sites) due to Grothendieck. The group of equivalence classes of Azumaya algebras over X is known as the Brauer group of X. By comparing the étale topology on a smooth complex variety X with the classical topology, we are able to use results from classical obstruction theory in order to obstruct the existence of certain Azumaya algebras. After giving an introduction to Azumaya algebras and the Brauer group, we shall present one such result, which furnishes lower bounds on the ranks of Azumaya algebras on spaces of low cohomological dimension.
The Lie bracket of a Lie algebra L induces a linear map L^L → L. When can the kernel of this map be generated by indecomposable vectors of the form x ^ y with [x,y] = 0? This seemingly elementary question does not seem to be tractable by elementary methods. For semisimple Lie algebras over the complex numbers Kostant has given a positive answer by means of representation theory. I will explain why a number theorist is interested in this question over fields of positive characteristics. Here representation theoretic arguments become very difficult. I will sketch a solution for the Lie algebra of matrices.
I will consider two models arising in statistical physics, loop-erased random walk and the self-avoiding random walk, and discuss the scaling limit which is curve of fractal dimension. I will focus on two dimensions where we have a good description of this process and I will discuss recent progress in understanding the fine structure of the paths. I also plan to discuss open problems in three dimensions.
This talk is intended for a general mathematics audience.
Let K be a regular noetherian commutative ring. I consider finite type commutative K-algebras and K-schemes. I will begin by explaining the theory of rigid residue complexes on K-algebras, that was developed by J. Zhang and myself several years ago. Then I will talk about the geometrization of this theory: rigid residue complexes on K-schemes and their functorial properties. For any map between K-schemes there is a rigid trace homomorphism (that usually does not commute with the differentials). When the map of schemes is proper, the rigid trace does commute with the differentials (this is the Residue Theorem), and it induces Grothendieck Duality.
Then I will move to finite type Deligne-Mumford K-stacks. Any such stack has a rigid residue complex on it, and for any map between stacks there is a trace homomorphism. These facts are rather easy consequences of the corresponding facts for schemes, together with etale descent. I will finish by presenting two conjectures, that refer to Grothendieck Duality for proper maps between DM stacks. A key condition here is that of tame map of stacks.
Transient or unstable behavior is often ignored in considering long time dynamics in the deterministic world. However, stochastic effects can change the picture dramatically, so that the transients can dominate the long range behavior. This talk will contrast the effects of different noise sources in certain systems with delayed feedback, driving both detrimental and beneficial noise-stabilized behaviors that are composed of transients from the deterministic system. The analysis starts with a canonical model of delayed feedback in mechanical systems and is extended to consider questions about biological applications.
Isoperimetry is a well studied subject that have found many applications in geometric measure theory (e.g. concentration of measure, heat-kernal estimates, mixing time, etc.) Consider the super-critical bond percolation
on \mathbb{Z}^d (the d-dimensional square lattice), and \phi_n the Cheeger constant of the super-critical percolation cluster restricted to the finite box [-n,n]^d. Following several papers that proved that the leading
order asymptotics of \phi_n is of the order 1/n, Benjamini conjectured a limit to n\phi_n exists. As a step towards this goal, Rosenthal and myself have recently shown that Var(n\phi_n)< C n^{2-d}. This implies
concentration of n\phi_n around its mean for dimensions d>2.
Consider the super-critical bond percolation on \mathbb{Z}^2 (the square lattice). We prove the Cheeger constant of the super-critical percolation cluster restricted to finite boxes scale a.s to a deterministic quantity.
This quantity is given by the solution to the isoperimetric problem on \mathbb{R}^2 with respect to a specific norm. The unique set which gives the solution, is the normalized Wulff shape for the same norm.
Joint work with Marek Biskup, Oren Louidor and Ron Rosenthal.
UBC/UMC is the Undergraduate Mathematics Colloquium at UBC.
These talks are for undergraduates interested in mathematics and mathematical research. They are put on by professors, senior graduate students and visitors to the department. Graduate students are also invited to attend.
Title: The Transatlantic Cable
Abstract: A look at some fundamental partial differential equations through the lens of time.
In this talk, I plan to discuss Gaussian decay of the Box_b-heat kernel on polynomial models in C^2 and explore how to recover exponential decay via a Fourier transform.
We will continue past discussions on the basic ideas of Chow theory. We review the basic properties that make the Chow functor to be an oriented Borel-Moore homology theory, and then we will discuss some further topics including their equivariant version, their operational version and their universal version.
In this talk, I'll present the tools that I've been using to run one of our department's multi-section first year calculus courses (Math 102). I'll discuss the how and why of what we have set up including two wikis (one public, one private), WeBWorK, and a third party forum called Piazza.com.
Littlewood was interested in finding polynomials with ±1 coefficients (now called Littlewood polynomials) having a small ratio of L4 norm to L2 norm on the complex unit circle. This is equivalent to the problem of finding binary sequences with small mean-squared aperiodic autocorrelation, important in engineering and physics. The best known examples are polynomials whose coefficients are supplied by complex-valued characters of finite fields. As of 1988, the lowest known asymptotic ratio of norms was (7/6)1/4, and this was conjectured to be the lowest possible. In 2004, Borwein, Choi, and Jedwab analyzed a construction that reliably produces polynomials with a low ratio of norms, and they conjectured that these polynomials do indeed break the 1988 record. We prove that their conjecture is true, and the character sum methods we have devised settle further conjectures. (joint work with Jonathan Jedwab and Kai-Uwe Schmidt)
The Taylor-Goldstein (TG) equation governs the linear stability of internal gravity waves in an inviscid, stratified shear flow. It can be used to predict the most unstable modes in a flow, and is therefore useful in a wide range of geophysical applications. The TG equation is a singular eigenvalue problem, and attempting to solve it numerically for general velocity and stratification profiles is technically challenging. I will describe an ongoing project where the TG equation is used to understand processes in the atmospheric boundary layer. The focus of the talk will be on the numerical methods that are used to solve the TG equation, and some of the difficulties that arise in their application.
Codimension 2 complete intersections in P^N have a natural parameter space that is a projective bundle over a projective space given by the data of the two equations.
In this talk, we will be interested in the birational geometry of this parameter space. In particular, is it a Mori dream space? If this is the case, is it possible to describe its MMP explicitly? We will give motivations for these questions and answers in particular cases.
Let X and X' be two smooth Deligne-Mumford stacks. We call dash arrow X-->X' a Crepant Transformation if there exists a third smooth Deligne-Mumford stack Y and two morphisms \phi:Y-> X, \phi': Y-> X' such that the pullbacks of canonical divisors are equivalent, i.e. \phi^*K_{X}\cong \phi'^*K_{X'}. The crepant transformation conjecture says that the Gromov-Witten theory of X and X' is equivalent if X-->X' is a crepant transformation. This conjecture was well studied in two cases: the first one is the case when X and X' are both smooth varieties; the other is the case that there is a real morphism X-> |X'| to the coarse moduli space of X', resolving the singularities of X'. In this talk I will present some recent progress for this conjecture, especially in the case when both X and X' are smooth Deligne-Mumford stacks.
We discuss the general problem of computing the cohomology and topological K-theory for classifying spaces of crystallographic groups. Integral computations will be provided for groups with prime order holonomy.
I will talk about our recent progress on strong law of large numbers for some classes of superprocesses X corresponding to \partial_t u_t=A u_t+\beta u_t-\psi(u_t) in a domain D of {\bf R}^d, where A is the generator of a diffusion or a stable process, and the branching mechanism \psi(x,\lambda)=\beta\lambda+a\lambda^2+\int_0^\infty (e^{-\lambda r}-1+\lambda r)n(x, {\rm d}r) satisfies \sup_{x\in D}\int_0^\infty (r\wedge r^2) n(x,{\rm
d}r)<\infty .
Recently many people have established limit theorems for branching Markov processes or super-processes using the principal eigenvalue and ground state of the linear part of the characteristic equations. All the papers above assumed that the processes satisfy a second moment condition or a (1+\theta)-moment condition with \theta>0. Asmussen and Hering (1976) established a Kesten-Stigum L\log L type theorem for a class branching diffusion processes under a condition which is later called a positive regular property. We established Kesten-Stigum L\log L type theorems for superdiffusions and branching Hunt processes respectively.
Recently, we established strong law of large numbers for a class of superdiffusions in a domain D of {\bf R}^d with general branching mechanism, and for super-\alpha-stable processes in {\bf R}^d with \psi(x, \lambda)=-\beta\lambda+\eta\lambda^2, where \beta and \eta are positive constants. The main tool is the stochastic integral representation of superprocesses.
This informal talk will provide background material for graduate students and others who will be attending the seminar talk "Level-lowering for Galois representations" later today. Three main background items for that seminar will be:
Modular forms are analytic/geometric gadgets with lots of interesting arithmetic properties.
Modular forms have a weight and a level.
We can attach a Galois representation to an eigenform.
In this talk we will examine these three items in more detail.
Note for Attendees
Attendees are welcome to bring their lunch with them to this informal talk.
About 25 years ago, Ribet proved his famous level lowering result, which is an existence statement about congruences between modular forms of different levels. In this talk, I'll survey some recent progress towards giving a new proof of Ribet's result without any modularity assumptions. In place of a modular form, we start with a p-adic Galois representation, and in place of the level, we consider the conductor of this representation. We'll outline some ideas about how to show the existence of a second p-adic Galois representation of lower conductor which is congruent to the first.
An interesting problem in various areas involves examining the size of a set in a geometric way, namely if it contains a given pattern. If we focus on finite patterns in Euclidean space, the first non-trivial case is a 3 point pattern, and since a similar copy of a triangle preserves its angles, a natural question might be: For a fixed ambient dimension and fixed angle, what is the minimal dimension for which any set of higher dimension is guaranteed to contain three points which form the specified angle? Or conversely, what is the maximal dimension for which there exists a set of that dimension which does not "contain" the specified angle? In this talk, we discuss a few new bounds on this number. Time permitting, we will also discuss a generalization of this notion, where the set avoids all angles close to the specified angle.
I will explain how to associate a Satake-type isomorphism to certain characters of the compact torus of a split reductive group over a local field. I will then discuss the geometric analogue of this isomorphism. (Joint work with Travis Schedler).
In this talk, I will discuss some analytical aspects in the study of a finite dimensional Alexandrov space. Loosely speaking, the question I will consider is: to what extent does an Alexandrov space resemble a Riemannian manifold? In the first part of the talk, I will recall the background of Alexandrov's theory of metric spaces with curvature bounded from below, including results on the topology of these spaces.
One remarkable application of Schur functions and Littlewood-Richardson coefficients is their close connection to the eigenvalue problem on sums of hermitian matrices. The eigenvalue problem asks: Given three sequences of real numbers, do there exist hermitian matrices A+B=C with eigenvalues given by the three sequences? This problem has a generalization to eigenvalues of majorized sums of hermitian matrices where we replace "A+B=C" with "A+B>C".
In this talk, I discuss joint work with D. Anderson and A. Yong where we show that the eigenvalue problem on majorized sums is related to factorial Schur functions in the same way that classical Schur functions are related to eigenvalue problem on usual sums of Hermitian matrices. One consequence of this connection is a generalization of the celebrated saturation theorem to structure constants corresponding to factorial Schur functions.
We discuss the general problem of computing the cohomology and topological K-theory for classifying spaces of crystallographic groups. Integral computations will be provided for groups with prime order holonomy.
How small/large can be the connective constant of a regular graph? We give sharp inequalities for transitive graphs, and we explain how to prove strict inequalities as the graph varies. (joint work with Zhongyang Li)
The study of periods of automorphic forms, and their relations with L-functions, has for a long time been regarded as a field separate from the mainstream of the Langlands program, a collection of fortunate coincidences allowing us to get our hands on difficult arithmetic objects. The work of Jacquet and others, however, has continuously emphasized the relation of periods to functoriality: the nonvanishing of H-period integrals of G-automorphic forms (where H is a spherical subgroup of G) should detect functorial lifts from some other group to G. Building on the theory of spherical varieties developed by Brion, Knop, Luna, Vust and others, Gaitsgory and Nadler attached a dual group to every spherical variety H \ G. This can be used to recast periods in the language of the Langlands program, with many classical aspects of the program recovered as special cases when G=H x H. I will give an overview of this, mostly conjectural, program.
I'll discuss two results regarding how DT invariants (of smooth and projective Calabi-Yau threefolds) change under birational modifications. The first deals with flops and the second is related to the McKay correspondence and work of Jim Bryan and David Steinberg.
In this talk we will focus on some of the genome and transcriptome analysis software developed at the Lab for Computational Biology at SFU. These programs are all based on combinatorial optimization formulations for which exact or provably approximate polynomial time solutions exist - either for the original problem or its dual formulation. Some of the problems can be solved through standard integer linear program solvers while others require specialized (approximate) solutions to known NP-hard problems. The software we will cover include VariationHunter, a program to identify structural differences between a next generation sequenced genome and a reference genome, CommonLAW, a program to compare two or more next generation sequenced genomes through the help of a reference genome, deFuse, Comrad and nFuse, programs to detect gene fusions through the use of genomic or transcriptomic data or both, dissect, a program to associate assembled transcripts to a genome, and CLIIQ, a program to simultaneously identify and quantify novel splice variants.
Bio: S. Cenk Sahinalp is a Professor of Computing Science at Simon Fraser University, Canada. He received his B.Sc. degree in Electrical Engineering from Bilkent University and his Ph.D. in Computer Science from the University of Maryland at College Park. He did his postdoctoral work at Bell Labs, Murray Hill and spent the next two years on the faculty of U. Warwick, while holding a visiting position at U. Penn. Before moving to SFU he was on the faculty at Case Western Reserve. Sahinalp is an NSF Career Awardee, a Canada Research Chair, a Michael Smith Foundation Scholar and an NSERC DAS Awardee. He was recently named University of Maryland distinguished CS alumnus. Sahinalp has (co)chaired some of the leading conferences in computer science and bioinformatics such as RECOMB 2011 in Vancouver and serves on the editorial boards of several leading journals. His lab's recent research on computational genomics, in particular algorithms for high throughput sequence data and network biology have received several best paper awards, are used by major international research consortia, and have been highlighted by some leading scientific journals and media outlets.
The participants of the TA Accreditation Program will present some concrete tools they have gotten from participating at this course.
This presentation is intended for the whole department, both faculties curious about TAs professional development and grad students wanting to know more about this program. For more info : http://blogs.ubc.ca/mathtaap
Given two Gaussian vectors that are positively correlated, what is the probability that they both land in some fixed set A? Borell proved that this probability is maximized (over sets A with a given volume) when A is a
half-space. We will give a new and simple proof of this fact, which also gives some stronger results. In particular, we can show that half-spaces uniquely maximize the probability above, and that sets which almost
maximize this probability must be close to half-spaces.
We'll discuss how Karpenko and Merkurjev computed the canonical p-dimension of certain products of Brauer Severi varieties. They related these varieties to subgroups of the Brauer group, and used the theory of Grothendieck groups and Chow groups.
We study systems consisting of interacting spin particles which can have a positive or negative spin. We consider an Ising model and a type of Widom-Rowlinson (WR) model. The interactions between spin particles are regulated by Kac potentials which carry a parameter g. It is known that in the Kac limit as g tends to zero models with Kac potentials become mean field theory. Mean field theory is known to exhibit phase transitions. The focus of this work is to prove phase transitions not only in the Kac limit but also near the Kac limit i.e., for g small but strictly positive.
Placing the Ising and WR model in a rectangular box with side-length L and periodic boundary conditions defines finite volume Gibbs measures. The infinite volume Gibbs state n is the limit of the finite volume Gibbs measures as L tends to infinity. A particle system exhibits a phase transition if n is a mixture of ergodic states.
The main achievement of this thesis is the development of a new method to prove phase transitions. We first apply the Kac-Siegert transformation which reformulates the particle system by introducing an external field. The spin-spin interactions are replaced by interactions of the spin particles with the external field. The main idea of this dissertation is to study the mean field. In principle it should be easier to work with the mean field because, as we will show, it is approximately Gaussian. By a new expansion around mean field theory we prove that for g strictly positive but small the infinite volume Gibbs state for the external field, for both the Ising and the WR model, is a mixture of two ergodic states. It is shown that this implies that the infinite volume Gibbs state for both the Ising and WR model is a mixture of two ergodic states. One Gibbs state predominantly has positive spin particles, the other Gibbs state predominantly has negative spin particles. The new expansion is related to the Glimm Jaffe Spencer expansion around mean field theory.
In this thesis which is a compendium of seven papers, we explore the behaviour of solutions of various semilinear elliptic equations and systems on both bounded and unbounded domains of dimension N.
On unbounded domains which is the main part of our work, our motivation is a celebrated conjecture of De Giorgi (1978) stating that bounded and monotone solutions of the Allen-Cahn equation, up to dimension 8, must be one-dimensional. This conjecture is known to be true for N=<3 and with extra (natural) assumptions for 4=<N=<8.
Focusing on system of equations, we state a counterpart of the above conjecture for gradient systems introducing the concept of monotonicity for systems. Then, we prove this conjecture for dimensions up to three following ideas given for the scalar case but unfortunately for higher dimensions we are not able to give any (even partial) results. On the other hand, replacing the Laplacian operator by the divergence form operator for the Allen-Cahn equation, we ask under what conditions solutions of this new equation would be m-dimensional. This leads us to define the concept of “m-Liouville theorem” for PDEs. We say a PDE satisfies m-Liouville theorem for 0<=m<N if all solutions of the PDE are at most m-dimensional. The motivation to this definition is the Liouville theorem (or 0-Liouville theorem) that we have seen in elementary analysis stating that bounded harmonic functions on the whole space must be constant (0-dimensional). We present various 2- and higher- Liouville theorems, however, we are not sure whether or not any of these results are optimal. 0-Liouville theorem is at the heart of our work and this thesis includes various 0-Liouville theorems for the Henon-Lane-Emden system, Lane-Emden equation, Gelfand equation and gradient systems.
On bounded domains, following ideas observed for unbounded domains, we present regularity of solutions for gradient and twisted-gradient systems. The novelty here is a stability inequality that gives us the chance to adjust the known techniques and ideas to systems.
A highlight in the study of quantum physics was the work of Knizhnik, Polyakov and Zamolodchikov (1988), in which they proposed a relation (KPZ relation) between the scaling dimension of a statistical physics
model in Euclidean geometry and its counterpart in the random geometry. Recently, Duplantier and Sheffield used the 2D Gaussian free field to construct the Liouville quantum gravity measure on a planar domain, and
gave the first mathematically rigorous formulation and proof of the KPZ relation in that setting. We have applied a similar approach to generalize part of their results to R^4 (as well as to R^(2n) for
n>=2). To be specific, we construct a random Borel measure on R^4 which formally has the density (with respect to the Lebesgue measure) given by the exponential of an instance of the 4D Gaussian free field.
We also establish the KPZ relation corresponding to this random measure. This is joint work with Dmitry Jakobson.
At York, we have found first-year science majors coming to us from the Ontario high school system in general rather poorly prepared for first-year university mathematics. The result is very high drop-plus-fail rates in our first-year math courses and a resulting high attrition rate in the early years of our degree programs. A major source of the problem appears to be the widespread use in the schools of an approach heavily emphasizing the memorization of solution problem templates, an approach which leaves a majority of our incoming science majors with deficiencies in very basic algebra, trigonometry, and, even more problematic, their intuitive understanding of the basic operations of arithmetic. In this discussion, I will outline an approach I have developed involving 4-day, 4-hour-per-day intensive remediation sessions focused on changing the way such students approach mathematics. The program was begun in 2005 and significantly expanded in 2009, now handling between 15 and 20% of the incoming class each year. I will present statistics outlining the significant impact we have seen on student performance. The aim is to keep the presentation very informal, leaving lots of time for discussion, feedback and suggestions on possibilities for further improving the initiative.
I will discuss uniqueness and Monge solution results for multi-marginal optimal transportation problems with a certain class of cost functions; this class arises naturally in multi-agent matching problems in economics. This result generalizes a seminal result of Gangbo and \'Swi\c{e}ch on multi-marginal problems. I will also discuss some related observations about multi-marginal optimal transport on Riemannian manifolds.
Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis concerns large but finite dimensional matrices. Spectacular progress was done in the past two decades to prove universality of random matrices with orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4.
I will first review the many occurrences of these statistics from the random matrices universality class. I will then report on a joint work with L. Erd{\H o}s and H.-T. Yau, which yields universality for the log-gases at arbitrary temperature at the microscopic scale.
Note for Attendees
Refreshments will be served in MATH 125 at 2:45 p.m.
Fluctuations of random matrix theory type have been known to occur in analytic number theory since Montgomery's calculation of the pair correlation of the zeta zeros, in the microscopic regime. At the mesoscopic scale, the analogy holds, through a limiting Gaussian field, which present an ultrametric structure similar to log-gases. In particular we will consider an analogue of the strong Szeg{\H o} theorem for L-functions.
For a smooth projective variety over a finite field, Poonen’s Bertini Theorem computes the probability that a high degree hypersurface section of that variety will be smooth. We prove a semiample generalization of Poonen's result, where the probability of smoothness is computed as a product of local probabilities taken over the fibers of a corresponding morphism. This is joint with Melanie Matchett Wood.
Note for Attendees
ESB 4133 is the library room attached to the PIMS lounge.
For a system of polynomial equations, it has long been known that the relations (or syzygies) among the polynomials provide geometric information about the corresponding projective variety. I will describe a collection of new ideas about how to study syzygies,and how these lead to classification results and a duality between syzygies and vector bundles.
Consider a critical point of a function f(x,y) in two complex variables. Milnor showed that the number of non-degenerate critical points into which it splits under a general perturbation is encoded by the Alexander polynomial of the link of the singular fibre of f:C^2 --> C passing through the critical point.
Analogously, one can consider a (k-1)-parameter family of such functions f, and ask: into how many k-nodal curves does the fiber containing a critical point split under a general perturbation? As we will explain, they are counted by the coefficients of the HOMFLY polynomial of the link. The relation goes through the Hilbert schemes of points, which parameterize subschemes of the singular curve.
These latter spaces and their generalizations in fact contain contain enough information to recover all the coefficients of the HOMFLY polynomial, and, conjecturally, their cohomologies are the Khovanov-Rozansky homology of the link.
A divisor on a curve is called "special'' if its linear equivalence class is larger than expected. On a hyperelliptic curve, all such come from pullbacks of points from the line. But one can ask subtler questions. Fix a degree zero divisor Z; consider the space parameterizing divisors D where D and D+Z are both special. In other words, we wish to study the intersection of the theta divisor with a translate; the main goal is to understand its singularities and its cohomology.
The real motivation comes from number theory. Consider, in products of the moduli space of elliptic curves, points whose coordinates all correspond to curves with complex multiplication. The Andre-Oort conjecture controls the Zariski closure of sequences of such points (and in this case is a theorem of Pila) and a rather stronger equidistribution statement was conjectured by Zhang. The locus introduced above arises naturally in the consideration of a function field analogue of this conjecture. This talk presents joint work with Jacob Tsimerman.
De Giorgi in 1978 conjectured that bounded and monotone solutions of the Allen-Cahn equation must be one-dimensional up to dimension eight. This conjecture is known to be true for N=<3 and with extra (natural) assumptions for 4=<N=<8. We state a counterpart of the above conjecture for gradient systems introducing the concept of monotonicity for systems. Then, we prove this conjecture for dimensions up to three and applying a geometric Poincare inequality for stable solutions we show that the gradients of various components of the solutions are parallel.
On the other hand, we ask under what conditions we can prove solutions of a PDE are m-dimensional for 0=<m=<N-1. This leads us to define the concept of “m-Liouville theorem” for PDEs. The motivation to this definition is the Liouville theorem (or 0-Liouville theorem) that we have seen in elementary analysis stating that bounded harmonic functions on the whole space must be constant (0-dimensional).
This is the main part of my dissertation under the supervision of Nassif Ghoussoub.
Consider the moduli space M_{g,s} of Riemann surfaces of genus g with s marked points as an orbifold. In this talk I will determine all (non-constant) holomorphic maps M_{g,s}\to M_{g',s'} if g\ge 6 and g'\le 2g-2. This is joint work with Javier Aramayona.
We all "Google". You may even have found this talk by Googling. What you may not know is that behind Google and others' search engines is beautiful and elegant mathematics.
Margot Gerritsen will explain the workings of page ranking and search engines using only rusty calculus.
Dr. Margot Gerritsen is an Associate Professor and Director, Institute for Computational and Mathematical Engineering at Stanford University. Her research focuses on the design of highly accurate and efficient parallel computational methods to predict the performance of enhanced oil recovery methods with a particular interest in gas injection and in-situ combusion processes. Outside petroleum engineering, she is active in coastal ocean simulation, yacht research and pterosaur flight mechanics, and the design of search algorithms in collaboration witth the Library of Congress and colleagues from the Institute of Computational and Mathematical Engineering.
This is a joint CMS/PIMS/IAM public lecture, part of the Cross Canada Series of Lectures on Math of Planet Earth.
When studying sequences of convolution operators, the curvature or pseudorandomness of measures are reflected in the Fourier transform. But in cases where the Fourier transform is inconvenient or unavailable, it can be gleaned instead from convolution powers of the measure with itself. I will show how this tactic, pioneered by Fefferman and by Christ, helps to prove some endpoint results for nonstandard ergodic theorems and singular variants of the Lebesgue Differentiation Theorem.
By a theorem of Behrend Donaldson-Thomas invariants can be defined interns of a certain constructible function. We will compute this function at all points in the Hilbert scheme of points in three dimensions and see that it is constant. As a corollary we see that this Hilbert scheme of points is generically reduced and its components have the same dimension mod 2. This gives an application of the techniques of BPS state counting to a problem in Algebraic Geometry.
Large-scale production of very heavy oil is gaining momentum. Unfortunately, production of such reservoirs typically leads to large environmental impacts. One promising technique that may mitigate these impacts is in-situ combustion (ISC). In this process, (enriched) air is injected into the reservoir. After ignition a combustion front develops in situ that burns a small percentage of the oil in place and slowly moves through the reservoir producing steam along the way. The steam moves ahead of the front, heats up the oil, makes it runnier and hence easier to produce. A side benefit of this process is that the heat thus generated often cracks the oil into heavy, undesirable components that stay behind in the reservoir and lighter, more valuable components that can be brought up to the surface. In the last few years, my colleagues and I plunged into heavy oil recovery to see if computational mathematics could make a difference in pushing this process over less environmentally friendly processes in the industry. ISC processes are notoriously hard to predict. We developed a workflow involving laboratory experiments, various simulation tools and upscaling methods that increases the confidence of the oil reservoir engineer in ISC. We hope that this will lead to a wider acceptance and use of this technique.
A classical theorem in minimal surface theory says that any entire minimal graph in R^3 is a plane. We ask the same question for the Willmore equation which is of 4th order. We prove that an entire Willmore graph is a plane if its Willmore functional is finite (i.e. if the mean curvature of the graph is square integrable). This is joint work with Tobias Lamm.
Topological Complexity (TC) of a space is a concept motivated by the motion planning problem in Robotics. This turns out to be a homotopy invariant, closely related to Lusternik-Schnirelmann category (LS-cat). In fact, TC has proved more delicate than LS-cat given its relationship with some difficult problems in Algebraic Topology such as the Immersion Problem for Projective Spaces.
In this talk we will discuss basic properties and examples of TC. We will also discuss some recent progress on the computation of TC of some homogeneous spaces, and talk about some new techniques to compute TC based on Hopf-like invariants.
I will discuss a new proof of the boundedness of the Wolff circular maximal function. This maximal function helps us understand BRK sets---compact subsets of the plane that contain a circle of every radius, and it is also a stepping stone towards understanding the Kakeya problem in three dimensions. This new proof uses some modern ideas from combinatorial geometry, namely the "discrete polynomial partitioning" method developed by Guth and Katz to solve the Erdos distinct distances problem in the plane.
We begin by rediscovering the Lie algebra sl(2) in a very elementary way via counting points on Grassmannians. Natural generalizations of this construction lead to quantum groups and then "higher" representation theory. We illustrate how this theory can be used to understand the original geometry and then discuss various applications such as new constructions of derived equivalences and homological knot invariants.
Note for Attendees
Refreshments will be served in MATH 125 at 3:45 p.m.
We define an action of a Heisenberg algebra on categories of coherent sheaves on Hilbert schemes of points of C^2. This lifts the constructions of Nakajima and Grojnowski from cohomology to derived categories. Vertex operator techniques are then used to extend this to an action of sl_infty. We end with applications to knot homology and a discussion of future research directions.
Let F be a locally compact non archimedean field with residue characteristic p and G the group of F-points of a split connected reductive group. Let k be an algebraically closed field of characteristic p.
We are interested in the link between the k-representations of the spherical and affine Hecke algebras associated to G and the smooth k-representations of G. In particular, the so-called supersingular representations of G (and the corresponding supersingular Hecke modules) are still poorly understood. However, these are the representations which are expected to play a prominent role in a potential mod p local Langlands correspondence for G.
In the pro-p Iwahori-Hecke k-algebra H, we define a family of commutative subalgebras, each containing the center of H. We use these subalgebras to do the following: 1. Construct an inverse Satake isomorphism (one can subsequently show that this is the inverse of the Satake isomorphism defined by Herzig). 2. Prove that the center of H contains an affine semigroup algebra which is naturally isomorphic to the spherical Hecke algebra attached to an irreducible smooth k-representation of a given hyperspecial maximal compact subgroup of G. 3. Apply this to study the "supersingular block" of the category of finite length H-modules and relate it to supersingular representations of G.
The representation theory of finite groups is significantly more complicated and deeper if, instead of looking at representations with complex coefficients, we consider modular representations such as, for example, mod p representations of GL_n(F_p).
Likewise, the (smooth) mod p representation theory of the p-adic group GL_n(Q_p) is more complicated than its complex representation counterpart. Various subtleties can be illustrated by observing the simpler but still curious behavior of mod p representations of GL_n(F_p).
Using GL_n(F_p) a guideline, we will explain what can be said about GL_n(Q_p). More precisely, while trying to avoid number theoretic technicalities, we will describe a few results pointing towards a mod p local Langlands correspondence. While hopeful, we will also sketch some strange phenomena which add to the many mysteries in this field.
We expose and exploit a surprising relationship between elliptic gradient systems of PDEs and a multi-marginal Monge-Kantorovich optimal transport problem. We show that the notion of an "orientable" elliptic system (Fazly-Ghoussoub) conjectured to imply that stable solutions are essentially 1-dimensional, is equivalent to the definition of a "compatible" cost function (Carlier-Pass), known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems. We use this equivalence to show that solutions to these elliptic PDEs, with appropriate monotonicity properties, are related to optimal measures in the Monge-Kantorovich problem. We also prove a decoupling result for solutions to elliptic PDEs and show that under the orientability condition, the decoupling has additional properties, due to the connection to optimal transport.
A sequence (a_1, \dots, a_n) of positive integers is a {\it parking function} if its nondecreasing rearrangement (b_1 \leq \dots \leq b_n) satisfies b_i < i+1 for all i. Parking functions were introduced by Konheim and Weiss to study a hashing problem in computer science, but have since received a great deal of attention in algebraic combinatorics. We will define two new objects attached to any (finite, real, irreducible) reflection group which generalize parking functions and deserve to be called parking spaces. We present a conjecture (proved in some cases) which asserts a deep relationship between these constructions. This is joint work with Drew Armstrong at the University of Miami and Vic Reiner at the University of Minnesota.
The random-cluster model, introduced by Fortuin and Kasteleyn as a way to unify the study of percolation and the Ising and Potts models, provides a tool to extend geometric intuition to the derivation of properties of spin systems. Interest in its two-dimensionalversions has been revived with Smirnov's introduction of the parafermionic observable.
I will present our recent results with Duminil-Copin and Smirnov, focusing more specifically on two of them: the derivation of its critical point, and the study of the asymptotic behavior of its two-point function away from criticality.
UBC/UMC is the Undergraduate Mathematics Colloquium at UBC.
These talks are for undergraduates interested in mathematics and mathematical research. They are put on by professors, senior graduate students and visitors to the department. Graduate students are also invited to attend.
The first talk will be given by one of our most popular speakers, Professor George Bluman.
Title: Introduction to Similarity Methods for Partial Differential Equations
Abstract:
It will be shown how to find systematically solutions and the conservation laws for PDEs. Most systematic methods are symmetry-based (directly or through extensions). A PDE is a compact way of describing a family of surfaces. The surfaces are the solutions of the PDE. It turns out that one can find symmetries as well as conserved quantities of such a family of surfaces systematically without knowing specific surfaces. In turn, this allows one to find specific surfaces.
In general, a symmetry of a PDE is any transformation that maps its solutions to other solutions, i.e. a symmetry leaves invariant the family of surfaces that are the solutions of the PDE. Hence, in principle, any PDE has symmetries. Problems: How to use symmetries systematically, how to find symmetries systematically, and how to calculate symmetries efficiently for a given PDE, especially a nonlinear PDE.
Related to this talk, the following references are available online through the UBC library.
Bluman and Anco, Symmetry and Integration Methods for Differential Equations, Springer 2002. Bluman, Cheviakov and Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer 2010.
All Gibbs states of the supercritical q-state Potts model on Z^2 are convex combinations of the q pure phases; in particular, they are all translation invariant. We recently proved this theorem with Hugo Duminil-Copin (Geneva), Dima Ioffe (Haifa) and Yvan Velenik (Geneva). I will explain the basic concepts
underlying this result and present the heuristics of the proof, which consists of considering the model in large finite boxes with arbitrary boundary condition, and proving that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical
value of the inverse temperature beta > beta_c(q)=log(1+sqrt(q)).
We will discuss roughly the first half of Chapter 2 of Mézard-Montanari, which contains an elementary overview of Statistical Mechanics in the context of finite state spaces. We will introduce Boltzmann distributions, partition functions and thermodynamic potentials such as free energy, internal energy and entropy. Some properties of these potentials will be explored, including the fluctuation-dissipation relations. Next, we will have a somewhat heuristic discussion of thermodynamic limits and phase transitions. Finally, we will start studying the Ising model on finite graphs.
Two-dimensional models of statistical physics have long been studied by physicists, using tools such as quantum and conformal field theories and renormalization groups as well as through explicit computations in integrable cases. On the mathematics front, two objects were introduced over the last decade, shedding new light on their geometry: first, stochastic Loewner evolutions, proved by Schramm to be the unique possible scaling limits of models exhibiting conformal invariance; second, (para)fermionic observables, used by Smirnov to actually prove conformal invariance of several of them. I will present a panorama of these recent advances and some of the most puzzling open questions remaining to be solved.
Note for Attendees
Refreshments will be served at around 2:30 in the Math lounge.
A computational framework is presented for materials science models that come from energy gradient flows that lead to the evolution of structure involving two or more phases. The models are considered in periodic cells and standard Fourier spectral discretization in space is used. Implicit time stepping is used, and the resulting implicit systems are solved iteratively with the preconditioned conjugate gradient method. The dependence of the condition number of the preconditioned system on the size of the time step and the order parameter in the model (that represents the scaled width of transition layers between phases) is investigated. The framework is easily extended to higher order derivative models, higher dimensional settings, and vector problems. Several examples of its application are demonstrated, including a sixth order problem in three dimensions. Higher order time stepping and a GPU implementation is described briefly. A comparison to time-stepping with operator splitting (into convex and concave parts) is done.
We consider the Nelson model with variable coeffcients, which can be seen as a model describing a particle interacting with a scalar field on a static space time. We consider the problem of the existence of the ground state, showing that it depends on the decay rate of the coeffcients at infinity. We also show that it is possible to remove the ultraviolet cutoff, as it is in the flat case. We'll explain some open conjecture. (joint work with C.Gérard, F.Hiroshima, A.Suzuki)
We provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the
Euclidean distances, focusing primarily on models with long-range correlations. We also prove a shape theorem for balls in the chemical
distance under such conditions. Our general statements give novel results about the structure of the infinite connected component of the
vacant set of random interlacements and the level sets of the Gaussian free field. As a corollary, we obtain new results about the (chemical)
diameter of the largest connected component in the complement of the trace of the random walk on the torus. Joint work with Alexander Drewitz and Artem Sapozhnikov.
Consider the variety \Hom(\Gamma,G) where \Gamma is a finitely generated nilpotent group, and G a semisimple Lie group. I will discuss joint work with Juan Souto on homotopy retractions from this variety to the representation variety in K, a maximal compact subgroup of G.
We will continue our discussion of Chapter 2 of Mézard-Montanari. We will show the solution of the Ising model on the complete graph (also called the Curie-Weiss model) and discuss the finite-temperature phase transition. Time permitting, we will also briefly discuss spin glasses, by defining the Edwards-Anderson model and presenting some of the difficulties that arise in its study.
This is a continuation of part 1, the talk given on January 24.
In this talk, we'll survey the state-of-the-art on this and related topics, including recent techniques for solving such equations in more than two variables.
AKSZ Theory is a topological version of the Sigma Model in quantum field theory, and includes many of the most important topological field theories. I will present two generalizations of the usual AKSZ construction. The first is closely related to the generalization from symplectic to Poisson geometry. (AKSZ theory has already incorporated an analogous step from the geometry of cotangent bundles to the geometry of symplectic manifolds.) The second generalization is to phrase the construction in an algebrotopological language (rather than the usual language of infinite-dimensional smooth manifolds), which allows in particular for lattice versions of the theory to be proposed. From this new point of view, renormalization theory is easily recognized as the way one constructs strongly homotopy algebraic objects when their strict versions are unavailable. Time permitting, I will end by discussing an application of lattice Poisson AKSZ theory to the deformation quantization problem for Poisson manifolds: a _one_-dimensional version of the theory leads to a universal star-product in which all coefficients are rational numbers.
Finding geometrical correspondences between two images, called image registration, is one of the numerous challenging problems in image processing. Commonly, image registration is phrased as a variational problem that is known to be ill-posed. Thus, regularization is used to ensure existence of solutions, introduce prior knowledge about the expected solution, and/or increase the robustness against noise. This talk gives a comprehensive overview of theory, numerical methods, and applications of regularization energies based on hyperelasticity.
In this talk, we discuss the Landau damping -- the asymptotic stability of the linearly stable homogeneous states of the Vlasov-Possion system. It has been proved that solutions to the system linearized at stable homogeneous states decay algebraically in time. In such a Hamiltonian system, this decay is not caused by any dissipation. The nonlinear asymptotic stability is open until recently when Mouhot and Villani proved it of solutions in the Gevery class. The problem in Sobolev spaces remains open. We show that the nonlinear damping does not happen in Sobolev space with too low regularity by constructing BKG waves -- traveling waves -- arbitrarily close to stable homogeneous states. In the contrary, in Sobolev spaces with higher regularity, we show that there are no invariant structures -- including BGK waves -- near any stable homogeneous states and thus the same obstacle for the damping as in the rough Sobolev spaces does not appear. Similar results have also been proved for the Euler equation near Couette flow. These are joint works with Zhiwu Lin.
In this first of two parts, we will discuss the basic definitions necessary for the study of computability and computational complexity. We will begin by defining the Turing machine model and its limitations in terms of undecidability results. Then, we will introduce some complexity classes such as P and NP, and concepts such as polynomial-time reductions and completeness. Time permitting, some specific combinatorial problems and its relation with statistical physics will be discussed. This talk is a continuation in the informal learning seminar series on Information, Physics and Computation.
I will first review some ideas of algebraic number theory from a computational point of view, and then address the computational complexity of the problems of determining whether an ideal A in the ring of integers of a (fixed) algebraic number field K is prime, and of finding the prime factorization of A. Specifically, I will answer questions of Gil Kalai by giving polynomial-time reductions for the problems of determining whether A is prime and finding the prime factorization of A to the corresponding problems over the rational integers. I will then discuss the problem of factoring an algebraic integer into irreducibles and conclude with the problem of irreducibility testing.
I will begin this talk by recalling the classic inequalities of Alexandrov-Fenchel and Polya-Szego for convex surfaces of 3-dimensional Euclidean space.
Then, I will present my joint work with Freire, which generalizes the inequalities -with rigidity- to both a larger class of hypersurfaces and to arbitrary dimensions. I will conclude by mentioning some applications of the results, including a mass-capacity inequality for black holes.
The moduli space of Riemann surfaces M_g parametrizes bundles of genus g surfaces. A classical theorem of J. Harer implies that the homology H_k(M_g) is independent of g, as long as g is large compared to k. In joint work with Oscar Randal-Williams, we establish an analogue of this result for manifolds of higher dimension: The role of the genus g surface is played by the connected sum of g copies of S^n \times S^n.
UBC/UMC is the Undergraduate Mathematics Colloquium at UBC. These talks are for undergraduates interested in mathematics and mathematical research. They are put on by professors, senior graduate students and visitors to the department. Graduate students are also invited to attend.
The next talk is by Iain Moyles, a graduate student at the Institute for Applied Mathematics.
Title: Case studies in industrial mathematics
Abstract:
I will present three different projects that highlight a connection between mathematics and industry outside of academia. Each project utilizes a different type of modelling technique and mathematical tool to provide insight into the problem. The aim of my talk is to provide an overview into ways that math taught in the classroom is being used in the "real world" and to highlight important results without presenting too many technical details.
Non-commutative Iwasawa theory, especially the non-commutative main conjecture, predicts some congruences between p-adic L-values of elliptic curves, ordinary at a given odd prime p, and having isomorphic residual representations. Some of these results have been proved. On the algebraic side, this should have implications for the algebraic invariants. We shall discuss the theoretic results that can be proved in this framework.
Bootstrap percolation, one of the simplest cellular automata, can be viewed as an oversimplified model of the spread of an infection on a graph. In the past three decades, much work has been done on bootstrap percolation on finite grids of a given dimension in which the initially infected set A is obtained by selecting its vertices at random, with the same probability p, independently of all other choices. The focus has been on the critical probability, the value of p at which the probability of percolation (eventual full infection) is 1/2.
The first half of my talk will be a review of some of the fundamental results concerning critical probabilities proved by Aizenman, Lebowitz, Schonman, Cerf, Cirillo, Manzo, Holroyd and others, and by Balogh, Morris, Duminil-Copin and myself. The second half will about about the very recent results I have obtained with Holmgren, Smith, Uzzell and Balister on the time a random initial set takes to percolate.
Every knot can be presented on the union of finitely many half planes which have a common boundary line, so that each half plane contains a single arc of the knot. Such a presentation is called an arc presentation of the knot. The arc index of a knot is the minimal number of half planes needed in its arc presentations.
A grid diagram of a knot is a knot diagram constructed by finitely many vertical line segments and the same number of horizontal line segments such that at each crossing a vertical segment crosses over a horizontal segment. A grid diagram with n vertical segments is easily converted to an arc presentation on n half planes, and vice versa.
Grid diagrams are useful in several ways. A slight modification of a grid diagram gives a front projection of its Legendrian imbedding. Grid diagrams are used to compute Heegaard Floer homology and Khovanov homology.
In this work, we've tabulated prime knots by arc index up to arc index twelve. This is achieved by generating grid diagrams of prime knots up to arc index twelve. This tabulation contains all prime alternating knots up to ten crossings and all prime non alternating knots up to twelve crossings. The largest crossing number among prime knots with arc index twelve is twenty eight.
We will explore the L^p-boundedness of directional maximal operators in three dimensions. Given a curve on the surface of the sphere, we will describe how the associated set of directions gives rise to a directional maximal operator bounded on all L^p, 1<p<\infty . Extensions of this result to more arbitrary subsets of the sphere given by certain rooted trees will be discussed.
Suppose that a polarized self-morphism \phi of X dominates a polarized self-morphism \psi of Y. Szpiro and Tucker asked if, if \phi is isotrivial, then \psi also descends to an isotrivial morphism. We give an affirmative answer in a large set of cases, including the case Y = P^1. At heart is a result of Petsche, Szpiro, and Tepper on isotriviality and potential good reduction for self-maps of P^n, which we extend to more general polarized self-morphisms of projective varieties.
ABSTRACT
Despite a century of study, the macroscopic behaviour of quasistatic granular materials remains poorly understood. In particular, we lack a fundamental system of continuum equations, comparable to the Navier-Stokes equations for a Newtonian fluid. In this thesis, we derive continuum models for two-dimensional granular materials directly from the grain scale, using tools of discrete calculus, which we develop.
To make this objective precise, we pose the canonical isostatic problem: a marginally stable granular material in the plane has 4 components of the stress tensor σ, but only 3 continuum equations in Newton’s laws ∇·σ = 0 and σ = σT. At isostaticity, there is a missing stress-geometry equation, arising from Newton’s laws at the grain scale, which is not present in their conventional continuum form.
We first show that a discrete potential ψ can be defined such that the stress tensor is written as σ = ∇ × ∇ × ψ, where the derivatives are given an exact meaning at the grain scale, and converge to their continuum counterpart in an appropriate limit. The introduction of ψ allows us to understand how force and torque balance couple neighbouring grains, and thus to understand where the stress-geometry equation is hidden.
Using this formulation, we derive the missing stress-geometry equation Δ(F : ∇∇ψ) = 0, introducing a fabric tensor F which characterizes the geometry. We show that the equation imposes granularity in a literal sense, and that on a homogeneous fabric, the equation reduces to a particular form of anisotropic elasticity.
We then discuss the deformation of rigid granular materials, and derive the mean-field phase diagram for quasistatic flow. We find that isostatic states are fluid states, existing between solid and gaseous phases. The appearance of isostaticity is linked to the saturation of steric exclusion and Coulomb inequalities.
Finally, we present a model for the fluctuations of contact forces using tools of statistical mechanics. We find that force chains, the filamentary networks of con- tact forces ubiquitously observed in experiments, arise from an entropic instability which favours localization of contact forces.
We give a full description for the shape of the classical (2+1)-dimensional Solid-On-Solid model above a wall, introduced by Temperley (1952). On an L\times L box at a large inverse-temperature \beta the height of most sites concentrates on a single level h = \lfloor \frac1{4\beta}\log L\rfloor for most values of L. For a sequence of diverging boxes the ensemble of level lines of heights (h,h-1,\ldots) has a scaling limit in Hausdorff distance iff the fractional parts of \frac1{4\beta}\log L converge to a noncritical value. The scaling limit is explicitly given by nested distinct loops formed via translates of Wulff shapes. Finally, the h-level lines feature L^{1/3+o(1)} fluctuations from the side boundaries.
Based on joint works with Pietro Caputo, Fabio Martinelli, Allan Sly and Fabio Toninelli.
UBC/UMC is the Undergraduate Mathematics Colloquium at UBC. These talks are for undergraduates interested in mathematics and mathematical research. They are put on by professors, senior graduate students and visitors to the department. Graduate students are also invited to attend.
The next talk is by Ari Belenkiy, a mathematician and historian, and a visitor to the Mathematics Department.
Title: Norms of the coining at the Royal Mint and Newton's Revolution
Abstract:
Minting at the Royal Mint includes ten diverse operations organized by the Master to produce coins. Norms were introduced in mid-13th century by King Edward I to check on the Master’s overall performance.
From the onset, the remedies were designed to check the statistical mean, i.e., the weight and fineness of the average coin. The actual check was done by sampling from the daily production and estimating deficiency at the public trials, known as the trials of the Pyx. In hindsight, the remedies laid a check on the combination of the mean and a deviation from the mean. This ambiguity, coupled with technological imperfection in a coin’s production, defeated the original purpose of the checks since the variation of coins in weight could not be properly controlled in that era. As a result of inability to control the variation, the remedies were set unreasonably wide.
With the advent of a civil society in Britain in the 17th century, together with rise of transparency and accountability in state institutions, the remedies began serving another purpose – to ascertain the quality of coining. This became especially urgent at the turn of the 18th century, when a large variation in weight of the gold coins led bankers and goldsmiths to cull heavy pieces out of circulation and recoin them to their advantage.
In 1719, Sir Isaac Newton claimed to have stopped the practice of culling in Great Britain, thus “saving some thousands of pounds to the Crown.” By improving the minting procedure, he reduced the variation of coins in weight. Judging from the extant statistical data preserved in the Jury Verdicts written from the late 17th century on, he saved the Crown about Ł40,000.
Moreover, Newton tightened the norms of coins’ admissibility: beginning in 1707, a small sample of coins was tested at each trial, effectively reducing the margin (remedy in weight).
If time permits we shall discuss the post-Newton progress of coining in Great Britain till modern times, presenting the standard deviation of coins in weight as a measure of the technological advance of the country. We shall also pose some open questions.
Bone remodeling involves the coupled action of osteoblasts and osteoclasts. Osteocytes are believed to sense and respond to mechanical loading applied to bone at the tissue level and regulate bone remodeling process. However, how osteocytes regulate the action of osteoblasts and osteoclasts is unknown. To systematically investigate the cellular level mechanism underlying the osteocyte mechanoregulation of bone remodeling, we applied different types of mechanical stimuli to osteocytes. Cell-cell communications and bone formation and bone resorption markers at transcriptional level and protein level were analyzed. We found those osteocytes are highly responsive to dynamic fluid flow, dynamic hydraulic pressure, and low magnitude and high frequency vibration. Our data indicate that mechanically challenged osteocytes release soluble factors promote bone formation and inhibit bone resorption.
This is a continuation of previous seminars on Feb. 7 and 14. This talk will focus on classes of algorithms
and in particular on algorithms for approximate counting.
These talks are part of the informal learning seminar on Information, Physics and Computation.
I will recall the important features of the p-adic local Langlands correspondence for GL2(Qp). Extending this correspondence to other groups seems to require doing p-adic Hodge theory in a slightly different way. I will explain the new features that arise when one does this, in the simplest setting.
We explain the compatibility of determinant functors with spectral sequences and apply it to descent calculations in local Iwasawa theory related to the ε-isomorphism conjecture of Fukaya and Kato.
The Stein-Tomas restriction theorem says, essentially, that if \mu is the surface measure on the n-1-sphere in R^n (n \geq 2), then the Fourier transform is a bounded mapping from L^p(R^n) to L^2(\mu) for a certain range of exponents p. The range of exponents in the Stein-Tomas theorem is known to be sharp. Mockenhaupt's restriction theorem is a generalization of the Stein-Tomas theorem to more exotic measures on R^n. Izabella Laba and I have shown that the range of exponents in Mockenhaupt's theorem is sharp for the important case of Salem measures on R.
Flow in porous media is difficult to model due to the non-linearity of the equations but more importantly, due to the lack of knowledge about earth parameters. In this talk we show how geophysical imaging can be used in order to compensate for the lack of information about the subsurface, and improve our ability to image and forecast subsurface flow.
The notion of relative twisting of curves on a surface, the special case of subsurface projection to an annulus, is an important tool in the theory of mapping class groups. We develop an analogue for the outer automorphisms of a free group. This is joint work with Matt Clay.
The classical central limit theorem is one of the most important theorem in probability theory. The theorem states that if X_1, \cdots , X_n are independent identically distributed random variables and if F_n is the difference between the sample mean and the mean of the random variables properly normalized, then F_n converges to a normal distribution in distribution. Recent results extend this results to other random variables for example given by Wiener chaos (multiple It\^o-Wiener integrals). In this talk, we shall obtain some conditions on F_n such that the distributions of the random variables F_n have densities f_n(x) with respect to Lebesgue measure and f_n(x) converges to the normal density \phi(x)=\frac{1}{\sqrt{2\pi}}e^{-|x|^2/2}.
The tool that we use is the Malliavin calculus and a brief introduction will also be given.
This is an ongoing joint work with Fei Lu and David Nualart.
What protects your computer password when you log on, or your credit card number when you shop on-line, from hackers listening on the communication lines? Can two people who never met create a secret language in the presence of others, which no one but them can understand? Is it possible for a group of people to play a (card-less) game of Poker on the telephone, without anyone being able to cheat? Can you convince others that you can solve a tough math (or SudoKu) puzzle, without giving them the slightest hint of your solution? These questions (and their remarkable answers) are in the realm of modern cryptography. In this talk I plan to survey some of the mathematical and computational ideas, definitions and assumptions which underlie privacy and security of the Internet and electronic commerce. We shall see how these lead to solutions of the questions above and many others. I will also explain the fragility of the current foundations of modern cryptography, and the need for stronger ones. No special background will be assumed.
Bio:
DR. AVI WIGDERSON is a widely recognized authority in theoretical computer science. His main research area is computational complexity theory. This field studies the power and limits of efficient computation and is motivated by such fundamental scientific problems as: Does P=NP? Can every efficient process be efficiently reversed? Can randomness enhance efficient computation? Can quantum mechanics enhance efficient computation? He has received, among other awards, both the Nevanlinna Prize and the Gödel Prize.
Note for Attendees
A reception will be held in the PIMS Lounge, ESB 4133 at 2:30 p.m. Undergraduate and Graduate Students are welcome and invited to attend.
Man has grappled with the meaning and utility of randomness for centuries. Research in the Theory of Computation in the last thirty years has enriched this study considerably. I'll describe two main aspects of this research on randomness, demonstrating respectively its power and weakness for making algorithms faster. I will also address the role of randomness in other computational settings, such as space bounded computation and probabilistic and zero-knowledge proofs.
We consider a queuing system with multi-type customers and servers. When a server is available, each customer is assigned a score which depends on the customer's waiting time, type, and the server's type. The service is then provided to the customer with the highest score. We develop a fluid limit process to approximate the behavior of such a system. Our model has two important features: (1) the service rate in the transient state coincides with the max-flow of a parameterized network; (2) the service rate at the steady state coincides with the the minimal-cost max-flow of a capacitated network. Thanks to these properties, we may solve the transient and stationary behavior of the fluid limit process efficiently by combinatorial methods, and predict the performance of the system when a scoring policy has been implemented. By properly defining the performance metrics, we may solve the scoring formula that leads to the optimal efficiency-fairness tradeoff. We illustrate the application of our fluid model in the context of kidney allocation policy design. In particular, the fluid model we developed can be used to predict the steady-state allocation outcome of the scoring policy proposed by the United Network of Organ Sharing (UNOS) in 2008.
After a short presentation of the equation of harmonic heat flow and corotational solutions, I am presenting a result of finite time blow-up dynamics. We will have to see how similar results for wave maps and Schrödinger map allow us to conjecture the instability of this regime in the general case. Finally, I will give a strategy of the proof.
Algebra and topology are old friends. Many topological problems are solved by applying algebraic methods. But sometimes the relationship can work the other way. My talk will discuss how the topological viewpoint can be used to establish the basic facts regarding orderability of groups.
We study the behavior of the so-called quantum random graphs inside the "scaling window". The quantum random graphs were first introduced by Ioffe and Levit
in 2007 and they turn to be a certain generalization of the Erdős–Rényi random graphs. We show results for the quantum random graphs which are analogous
to those of Aldous (1997), who proved that inside the "critical window", the rescaled sizes of components of the Erdős–Rényi graphs converge to lengths
of excursions of a certain process related to Brownian motion.
Assuming GRH, I will give a proof for the asymptotic formula of the eighth moment of Dirichlet L-functions averaged over primitive characters χ modulo q, over moduli q<Q, with a short average in the t-aspect. We obtain the constant 24,024 as a factor in the leading-order term of the eighth moment, which is as predicted for the eighth moment of the Riemann zeta function. This talk is based on joint work with Xiannan Li.
Note for Attendees
Refreshments will be served between the two talks.
I will talk about the distribution of the values of the zeta function on points lying in an arithmetic progressions on the critical line. This research was originally motivated by questions about the primes and the linear independence conjecture. We discover some interesting correlations between such distributions along sparse discrete points and the usual distribution of values on the entire critical line. Among other applications, this allows us to prove that a positive proportion of such points are not zeros of zeta, improving a previous result of Martin and Ng. (based on joint work with M. Radziwill)
A common type of problem in additive number theory involves estimating the number of solutions to a system of Diophantine equations among the primes. We estimate the number of ways to embed a similar copy of a given triangle T in the primes by applying the Hardy-Littlewood circle method to count the number of prime solutions a system of quadratic Diophantine equations depending on the triangle T. The number of \log -weighted solutions to the system for which each coordinate is at most N is shown to be asymptotic to a constant depending on the singular series times N^{2n-4}, where n \geq 7 is the number of dimensions.
ESB Rm 4127, (PIMS video conference room). Note the date, time and place change.
Mon 18 Mar 2013, 3:30pm-4:30pm
Abstract
Questions related to the asymptotic behavior of nonlinear dispersive equations in the presence of a potential term are of great interest both for mathematical and physical reasons. Our main concern will be equations with low-degree nonlinearities, namely below the Strauss exponent threshold, for which classical energy and decay methods fail to suffice. For this, we we use the spectral theory of the operator H=-\Delta+V to develop a space-time resonance analysis adapted to the inhomogeneous setting. A key ingredient in this setup is the development of a sufficiently comprehensive multilinear harmonic analysis in the context of the corresponding distorted Fourier transform. This turns out to exhibit several intriguing differences in comparison to the unperturbed Euclidean setting (no matter how small V is). As a first application, we treat the case of a quadratic nonlinear Schrodinger equation on \R^3.
This is joint work with Pierre Germain and Samuel Walsh (Courant Institute, NYU).
In 1980 Henry Wente considered the variational stability of rotationally symmetric pendant drops and obtained a number of results for various problems.
We consider one version of one of those problems for cylindrical pendant drops trapped between parallel planes. The analysis is different in various ways, and leads to results of a different nature. Most notably, Wente's rotationally symmetric pendant drops form stable families which terminate at a maximum volume. We find stable families which terminate at a maximum volume, but are followed by (distinct disconnected) families of stable drops. As a result, we may have "large" stable pendant drops which become unstable and "drip" when the volume is decreased.
We will attempt to explain these results using the simpler zero gravity case as a model.
Consider a weak bialgebra H, is it possible to invert some elements and still have a weak bialgebra structure? If so, is there conditions on H or on the set of elements we want to invert? We thoroughly investigate these questions and establish sufficient conditions for the localization of a weak bialgebra to exist. We show that a monoid of group-like elements that is almost central, a condition we introduce here, forms a suitable set to be localized. We give constructive proofs of these results and then use them to produce interesting examples of bialgebras and weak bialgebras.
We also present a reformulation of Manin's Hopf envelope and use it to define the notion of weak Hopf envelope. We finally discuss the relationship between the localization of a weak bialgebra relative to the monoid of all group-like elements and the weak Hopf envelope.
For each of the sequences of groups in the title, the k-th rational cohomology is independent of n in a linear range n >= c*k, and this "stable cohomology" has been explicitly computed in each case. In contrast, very little is known about the unstable cohomology, which lies outside this range.
I will explain a conjecture on a new kind of stability in the unstable cohomology of these groups, in a range near the "top dimension" (the virtual cohomological dimension). For SL_n(Z) the conjecture implies that the unstable cohomology actually vanishes in that range. One key ingredient is a version of Poincare duality for these groups based on the topology of the curve complex and the algebra of modular symbols. Based on joint work with Benson Farb and Andrew Putman.
The local Shimura correspondence relates representations of PGL2 to those of the metaplectic cover of SL2, where both groups are p-adic and the representations are on C-vector spaces. Many pieces of the usual construction break down, due to non-semisimplicity, when the representations are instead taken over a field of positive characteristic. The mod p case is especially problematic. In the specific case of the pair (\tilde{SL}2, PGL2), I will discuss an alternate approach using a comparison of certain Hecke algebras of each group. The focus will be on the nonsupercuspical representations; I'll classify these for the cover of SL2, and relate them to the genuine spherical and Iwahori Hecke algebras.
Note for Attendees
Refreshments will be served between the two talks.
Let k be a positive integer that is congruent to 3 (mod 4), and let N be a positive square-free integer. In this talk, we show how to compute a basis for the two-dimensional subspace Sk/2(Γ0(4N),F) of half-integral weight modular forms associated, via the Shimura correspondence, to a newform F in Sk-1(Γ0(N)), which satisfies L(F,1/2) ≠ 0. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined.
If you choose a squarefree polynomial f(T) in F_q[T] uniformly at random, it will have slightly less than one linear factor on average; as deg f(T) goes to infinity, this expectation stabilizes and converges to 1 - 1/q + 1/q^2 - 1/q^3 + ... = q / (q+1). In joint work with J. Ellenberg and B. Farb, we proved that the stabilization of this combinatorial formula, and other statistics like it, is equivalent to a representation-theoretic stability in the cohomology of braid groups. I will describe how combinatorial stability for statistics of squarefree polynomials, of tori in GL_n(F_q), and other natural geometric counting problems can be converted to questions of representation stability in topology, and vice versa.
This talk will assume no background, and is intended for a general mathematical audience.
The last few years saw an explosion of interest in understanding the geometry of the projections and slices of random fractals (such as fractal percolation). We develop a framework that allows us to recover, improve and unify many of these results, for a large class of random measures (essentially Kahane's T-Martingales with an additional spatial independence assumption).
Our results have several applications, of which I will focus on two: sharp dimension results for tube-null sets, and the presence of patterns such as progressions, angles or distances in random fractals. These applications are motivated by problems in harmonic analysis and geometric measure theory.
I will discuss a combinatorial problem which comes from algebraic geometry. The problem, loosely, is to show that two theories for "counting" "curves" (Pandharipande-Thomas theory and reduced Donaldson-Thomas theory) give the same answer. I will prove a combinatorial version of this correspondence in a special case (X is toric Calabi-Yau), where the difficult geometry reduces to a study of the "topological vertex'' (a certain generating function) in these two theories. The combinatorial objects in question are plane partitions, perfect matchings on the honeycomb lattice and the double dimer model.
There will be many pictures. This is a combinatorics talk, so no algebraic geometry will be used, except as an oracle for predicting the answer.
We consider models or signals with simultaneous structure, for example a matrix that is simultaneously sparse and low-rank. Our goal is to find suitable convex penalties that allow us to reconstruct such signals given random measurements and noisy observations.
Often penalties that promote each individual structure are known and yield an order-wise optimal number of measurements (e.g., \ell 1 norm for sparsity, nuclear norm for matrix rank), so it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multi-objective optimization with the individual norms, then we can do no better (order-wise) in terms of required measurements than an algorithm that exploits only one of the structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, not one that is a function of convex relaxations used for each structures.
Bio: Maryam Fazel is an assistant professor in Electrical Engineering at the University of Washington since 2008. She received her MS and PhD in EE from Stanford University and her BS in EE from Sharif University in Iran, and was a Postdoctoral Scholar at Caltech prior to joining UW. Maryam is a recipient of the NSF Career Award (2009), and the UW EE Outstanding Teaching Award (2009).
We consider a quasi-static free boundary problem (the Hele-Shaw problem) in a randomly perforated domain with zero Neumann boundary conditions. A homogenization limit is obtained as the characteristic scale of the domain goes to zero. Specifically, we prove that the solutions as well as their free boundaries converge uniformly to those corresponding to a homogeneous and anisotropic Hele-Shaw problem set in $\mathbb{R}^d$. This is joint work with Nestor Guillen.
The classroom is a microcosm of the human social universe. In it, there are numerous opportunities for cooperation and for competition—over marks, reputation, status, alliances, and mates. How the tension between cooperation and competition is resolved both in and outside of the classroom may be dictated in part by the structural demands of the educational system (e.g. the distribution of competition in space) as well as information about the value of education arising from external sources (e.g. cues of life expectancy). Here, I present new research on the possible effects of these informational and structural demands on measures of personal “investment” in education: educational attainment, graduation rates, school attendance, and course performance. Following this, I discuss avenues for future investigation.
An n x n complex matrix is called pseudo-reflection if its eigenvalues are 1, ..., 1. t, where t \ne 1 is a root of unity. Finite groups generated by pseudo-reflections were classified by Shephard and Todd in the 1950s. This classification is one of the high points of invariant theory of finite groups. Research into the various aspects of the structure of pseudo-reflection groups continues to this day. In this talk, based on joint work with A. Duncan, I will present a simple formula for the local essential dimension of a pseudo-reflection group. I will also discuss global essential dimension and a related intermediate notion between local and global. Some of these results can be restated in purely representation-theoretic terms, without any reference to essential dimension.
For a weight two modular form f, the Gross-Zagier theorem is a formula relating two things: the central derivative of the convolution L-function of f with a weight-one theta series, and the Neron-Tate pairing of a Heegner point with itself. I'll discuss a generalization to higher weight modular forms, where the Heegner point is replaced by certain special cycles on a unitary Shimura variety. This is joint work with Jan Bruinier and Tonghai Yang.
We show that the electric resistance between the origin and the n-th generation of a critical oriented branching random walk in dimensions d<6 is at most n^(1-a) for some a>0. As a corollary, the spectral dimension of the trace is strictly larger than 4/3 (its value when d>6) answering a question of Barlow, Jarai , Kumagai and Slade.
This thesis consists of four chapters and deals with two different problems which are both related to the broad topic of special values of anticyclotomic L-functions.
In Chapter 3, we generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic Zp-extension. Let g be a cuspidal Hilbert modular form of parallel weight (2,...,2) and level N over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant D. We study the l-adic valuation of the special values L(g,χ,1/2) as χ varies over the ring class characters of K of P-power conductor, for some fixed prime ideal P. We prove our results under the only assumption that the prime to P part of N is relatively prime to D.
In Chapter 4, we compute a basis for the two-dimensional subspace Sk/2(Γ₀(4N),F) of half-integral weight modular forms associated, via the Shimura correspondence, to a newform F of level N and weight k-1, which satisfies L(F,1/2)≠0. Here we let k be a positive integer such that k ≡ 3 mod 4 and N be a positive square-free odd integer. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined. The squares of the Fourier coefficients of these forms are known to be essentially proportional to the central critical values of the L-function of F twisted by some quadratic characters.
Symplectic topology can be thought as the mathematical versant of String theory: they were born independently at the same time, the second one as a fantastic enterprise to unify large-scale and low-scale physics, and the first one to solve classical dynamical problems on periodic orbits of physical problems, the famous Arnold conjectures. In the 80's, Gromov's revolutionary work opened a new perspective by presenting symplectic topology as an almost Kähler geometry (a concept that he defined), and constructing the corresponding theory which is entirely covariant (whereas algebraic geometry is entirely contravariant). A few years later, Floer and Hofer established the bridge between the two interpretations of Symplectic topology, the one as a dynamical theory and the one as a Kähler theory. This bridge was confirmed for the first time by Lalonde-McDuff who related explicitly the first theory to the second by showing that Gromov's Non-Squeezing Theorem is equivalent to Hofer's energy-capacity inequality. Nowadays, Symplectic Topology is a very vibrant subject, and there is perhaps no other subject that produces new and deep moduli spaces at such a pace ! More recent results will also be presented.
Given a planar curve singularity, Oblomkov and Shende conjectured a precise relationship between the geometry of its Hilbert scheme of points and the HOMFLY polynomial of the associated link. I will explain a proof of this conjecture, as well as a generalization to colored invariants proposed by Diaconescu, Hua, and Soibelman.
The work of Misha Gromov has revolutionized geometry in many respects, but at the same time introduced a geometric point of view in many questions. His impact is very broad and one can say without exaggeration that many fields are not the same after the introduction of Gromov's ideas.
I will try and explain several avenues that Gromov has been pursuing, stressing the changes in points of view that he brought in non technical terms.
Here is a list of topics that the lecture will touch:
1. The h-Principle,
2. Distance and Riemannian Geometry,
3. Group Theory and Negative Curvature,
4. Symplectic Geometry,
5. A wealth of Geometric Invariants,
6. Interface with other Sciences,
7. Conceptualizing Concept Creation
Note for Attendees
There will be a reception at ESB immediately after the lecture, in conjunction with the opening of the IHES Exhibit "The Unravelers" in the Earth Sciences Building.
We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension n >=3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n >=3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. This is joint work with Yaiza Canzani, Rod Gover and Raphael Ponge.
Hessenberg varieties are closed subvarieties of the full flag variety. Examples of Hessenberg varieties include both Springer fibers and the flag variety. In this talk we will show that Hessenberg varieties corresponding to nilpotent elements which are regular in a Levi factor of the Lie algebra are paved by affines. We then provide a partial reduction from paving Hessenberg varieties for arbitrary elements to paving those corresponding to nilpotent elements, generalizing results of Tymoczko.
I will apply Martha Precup's theorem on affine pavings to describe the equivariant cohomology algebras of (regular) Springer fibres in terms of certain Weyl group orbits. This will also yield a simple description of Springer's representation of W on the cohomology of the above Springer fibres.
The T T-inverse process or equivalently "random walk in random scenery" is a family of stationary processes that exhibits an amazing amount of behavior. Each random walk yields such a process and as you vary the random walk, you obtain essentially all possible ergodic theoretic behaviors. There is also a phase transition that arises which we can only partially prove. I will give an overview of this area which contains work both old and (somewhat) new.
This work is done jointly with a number of people including Frank den Hollander, Mike Keane and Sebastien Blachere.
Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k 2 and of the same level N, both eigenfunctions of the Hecke operators, and both normalized so that a1 = 1. The main result we seek is
that when E and f are congruent mod a prime p (which may be a prime ideal lying over a rational prime p > 2), the algebraic parts of the special values L(E; ; j) and L(f; ; j) satisfy congruences mod the same prime. On the way to proving the congruence result, we construct the modular symbol attached to an Eisenstein series.
1
We study the number of solutions of diophantine equations f(x1,...,xn)=v when the variables xi are restricted to primes. It has been established by Birch and Schmidt that one has the expected number of integer solutions if f is a homogeneous integral polymomial of sufficiently large rank with respect to its degree. We show that the same phenomenon holds when the variables are restricted to primes, extending the results of Hua for diagonal forms. We illustrate some of the ideas on quadratic forms and discuss some elements of the proof of the general case.
This talk addresses the problem of how to analyze and discuss singularities of a variety X that "naturally'' sits inside a flag manifold. Our three main examples are Schubert varieties, Richardson varieties and Peterson varieties. The overarching theme is to use combinatorics and commutative algebra to study the "patch ideals", which encode local coordinates and equations of X. Thereby, we obtain formulas and conjectures about X's invariants. We will report on projects with (subsets of) Erik Insko (Florida Gulf Coast U.), Allen Knutson (Cornell), Li Li (Oakland University) and Alexander Woo (U. Idaho).
We consider convex nonsmooth optimization problems whose objective function is known through some expensive procedure. For example, this is the case in several problems that arise in electricity production management, where the objective function is itself the result of an optimization subproblem.
In this context, it often exists extra information - cheap but with unknown accuracy - that is not used by the algorithms. In this talk, we present a way to incorporate this coarse information into two classical nonsmooth optimization algorithms: Kelley method and level bundle method. We prove that the resulting methods are convergent and we present numerical illustrations showing that they speed up resolution.
I will describe a theory of jeu de taquin for increasing tableaux, extending Schutzenberger's work on standard Young tableaux. Our original motivation came from Schubert calculus. However, I'll also describe a specific connection to the study of longest increasing sequences (LIS) of words. This is joint work with Hugh Thomas (U. New Brunswick) and Ofer Zeitouni (U. Minnesota).
The reflected fractional Ornstein-Uhlenbeck (RFOU) process arises as the key approximating process for stochastic flow systems with reneging customers/jobs. Our aim is to statistically estimate the key parameters of the system based on the (partially) observed data. We derive the explicit formulas for the standard and sequential maximum likelihood estimators, and their asymptotic/nonasymptotic properties. Our analysis is based on the fractional Girsanov formulas and fundamental martingales for the fractional Brownian motions.
In Hilbert spaces, five classes of monotone operator of relevance to the theory of monotone operators, variational inequality problems, equilibrium problems, and differential inclusions are investigated. These are the classes of paramonotone, strictly monotone, 3-cyclic monotone, 3*-monotone (or rectangular, or *-monotone), and maximal monotone operators.
Examples of simple operators with all possible combinations of class inclusion are given, which together with some additional results lead to an exhaustive knowledge of monotone class relationships for linear operators, linear relations, and for monotone operators in general.
Many of the example operators considered are the sum of a subdifferential with a skew linear operator (and so are Borwein-Wiersma decomposable). Since for a single operator its Borwein-Wiersma decompositions are not unique, clean, essential, extended, and standardized decompositions are defined and the theory developed. In particular, every Borwein-Wiersma decomposable operator has an essential decomposition, and many sufficient conditions are given for the existence of a clean decomposition.
Various constructive methods are demonstrated together which, given any Borwein-Wiersma decomposable operator, are able to obtain a decomposition, as long as the operator has starshaped domain. These methods are more accurate if a clean decomposition exists. The techniques used apply a variant of Fitzpatrick's Last Function, the theory of which is developed here, where this function is shown to consist of a Riemann integration and be equivalent to Rockafellar's antiderivative when applied to subdifferentials. Furthermore, a different saddle function representation for monotone operators is created using this function which has theoretical and numerical advantages over more classical representations.
My thesis is dedicated to the study of various spatial stochastic processes from theoretical biology.
For finite interacting particle systems from evolutionary biology, we study two of the simple rules for the evolution of cooperation on finite graph in Ohtsuki, Hauert, Lieberman, and Nowak [Nature 441 (2006) 502-505] which were first discovered by clever, but non-rigorous, methods. We resort to the notion of voter model perturbations and give a rigorous proof, very different from the original arguments, that both of the rules of Ohtsuki et al. are valid and are sharp. Moreover, the generality of our method leads to a first-order approximation for fixation probabilities of general voter model perturbations on finite graphs in terms of the voter model fixation probabilities.
For spatial branching processes from population biology, we prove pathwise non-uniqueness in the stochastic partial differential equation (SPDE) of some one-dimensional super-Brownian motions with immigration and zero initial value. In contrast to a closely related case studied in a recent work by Mueller, Mytnik, and Perkins, the solutions of the present SPDE are assumed to be nonnegative and are unique in law. In proving possible separation of solutions, we use a novel method, called continuous decomposition, to validate natural immigrant-wise semimartingale calculations for the approximating solutions, which may be of independent interest in the study of superprocesses with immigration.
In this talk, I will describe construction and estimates for Green's function for elliptic and parabolic systems of second order in divergence form subject to various boundary conditions.
Here, we assume minimal regularity assumptions on the coefficients and domains.
The replication process is known to be strand asymmetric: it requires the opening of the DNA double helix and acts differently on the two DNA strands, which generates different mutational patterns and in turn different nucleotide compositions on the two DNA strands (compositional asymmetry). During my PhD thesis, we modeled the spatio-temporal program of DNA replication and its impact on the DNA sequence evolution. I will show how this model helps understand the relationship between compositional asymmetry and replication in eukaryotes and explains the patterns of compositional asymmetry observed in the human genome. During the last part of my talk, I will present our on-going project: inferring the spatio-temporal replication program from experimental replication kinetics data.
Note for Attendees
The Seminar will be preceded by refreshments at PIMS, ESB 4th floor.
Delay differential equation problems appear in all areas of science and engineering and are mostly investigated numerically. Analytical studies using asymptotic techniques are rare but are needed because some of the
dynamical phenomena caused by the delay have never been seen before. We review the method of multiple time scales for an oscillator described by a delay differential equation and admitting a Hopf bifurcation. We show that if the delay is large, the slow time amplitude equation is itself a delay differential equation. A specific example of a delayed optoelectronic oscillator is examined and a secondary bifurcation to quasiperiodic oscillations is predicted. Analytical and experimental bifurcation diagrams are compared quantitatively. Periodic square-wave oscillations of scalar delay differential equations exhibiting a large delay have been rigorously studied in the 1980’s. They result from a Hopf bifurcation and the plateau lengths are nearly equal to one delay. The total period is close to two delays. Recent experimental observations of delayed optoelectronic oscillators and lasers
subject to delayed feedbacks show more complex forms of square-wave oscillations. The square-waves may become asymmetric with two plateaus of different lengths and a total period of one delay or they may exhibit bursting oscillations on one of the two plateaus. Two specific examples motivated by experiments are investigated.
Dr. Thomas Erneux is currently a Research Director by the Fonds National de la Recherche Scientifique (Belgium) and he teaches asymptotic methods for nonlinear problems in physics and chemistry. In 2009, he received the Prix de La Recherche mention Sciences de la Communication – parrainage CNRS for his work on the applications of delay differential equations. He received his PhD in Chemistry in 1979 in the group of Ilya Prigogine (Nobel Prize 1977) at the Université Libre de Bruxelles (ULB). He then went to the U.S. to study applied mathematical techniques (1979-80 Caltech, 1980 Northwestern University). After his military duty at the Royal Military School in Brussels (1981), he came back to Northwestern University now as a professor (1982-1993). He worked on bifurcation problems mostly motivated by chemical instabilities and by laser stability problems. In 1993, he joined the newly formed Nonlinear Optics group at ULB. His current interests concentrate on nonlinear dynamical problems, and, in particular, on delay differential equation problems in all areas of science and engineering.
During this talk we will introduce a method, called 'soft local times', to couple the trace of two Markov chains running on the same state space. This method is general and can give non-trivial bounds for several models of interest. We then give a particularly useful example of application involving the model of random interlacements on Z^d. For that, we will first introduce this process in detail and show
how the method of soft local times can help decoupling the random interlacements in two separated sets. An interesting consequence of such result is the exponential decay of the connectivity function for d > 3.
ABSTRACT
We investigate some problems on the uniqueness of mean curvature flow and the
existence of minimal surfaces, by geometric and analytic methods. A summary of the
main results is as follows.
(1) The special Lagrangian submanifolds form a very important class of minimal
submanifolds, which can be constructed via the method of mean curvature flow. In
the graphical setting, the potential function for the Lagrangian mean curvature flow
satisfies a fully nonlinear parabolic equation.
We prove a uniqueness result for unbounded solutions for this equation without any
growth condition, via the method of viscosity solutions: for any continuous initial
function in , there exists a unique continuous viscosity solution in ×[o,∞). We
also study the Cauchy-Dirichlet problem for this equation.
(2) We prove an existence result for free boundary minimal surfaces of general
topological type. Let N be a complete, homogeneously regular Riemannian manifold
of dim(N) not less than 3 and let M be a compact submanifold of N. Let Σ be a
compact Riemann surface with boundary. A branched immersion u: (Σ, ∂Σ) →(N,M) is
a minimal surface with free boundary in M if u(Σ) has zero mean curvature and u(Σ) is
orthogonal to M along u(∂Σ). We prove that
• if Σ is not a disk, then there exists a minimal immersion of Σ with free boundary in M
that minimizes area in any given conjugacy class of a map in Cş(Σ,∂Σ;N,M) that is
incompressible;
• the kernel of the induced map of the inclusion M → N admits a generating set such
that each member is freely homotopic to the boundary of an area minimizing disk that
solves the free boundary problem.
(3) Under certain nonnegativity assumptions on the curvature of a 3-manifold N and
convexity assumptions on the boundary M=∂N, we investigate controlling topology for
index-1 or stable free boundary minimal surfaces:
• We derive bounds on the genus, number of boundary components;
• We prove a rigidity result;
• We give area estimates in term of the scalar curvature of N.
Delay problems appear in all scientific disciplines from biology to physics. As soon as there is a mechanical, physiological, or human control, there is a delay because time is needed to observe and react. If the delay is too important, oscillatory responses appear. But a properly used delayed feedback may also stabilize an unstable system. Our understanding of the positive and negative effects of a delay has progressed to the point that oscillatory outputs are used in applications. The presentation will review a series of problems and illustrate the different expectations of the researcher depending on his background.
The existence of a height pairing on the equivalence relation defining Bloch's higher cycle groups is a surprising consequence of some recent joint work by myself and Xi Chen on a nontrivial K_1-class on a self-product of a general K3 surface. I will explain how this pairing comes about.
I will discuss some earlier joint work with E. M. Stein concerning div-curl type inequalities
for the exterior derivative operator and its adjoint in Euclidean space.?I will then present various
higher-order generalizations of the notion of exterior derivative, and discuss some recent div-curl
type estimates ?for such operators (part of this work is joint with A. Raich).
In 1757, Euler presented to the Berlin Academy of Sciences the basic equations of fluid mechanics. As pointed out by V.I. Arnold in 1966, the Euler equations for incompressible fluids have a very simple geometric interpretation that combines the concept of geodesics and the concept of volume preserving maps. The later concept is very simple and nothing but a continuous version of the discrete and more elementary concept of permutation. Conversely, the Euler equations have a natural discrete counterpart in terms of permutation and combinatorial optimization, which establishes a direct link with the mathematical theory of "optimal transport". This theory, that goes back to Monge 1781 and has been renewed by Kantorovich since 1942, is nowadays a flourishing field with many applications, in natural sciences, economics, differential geometry and analysis.
About the Niven Lectures: Ivan Niven was a famous number theorist and expositor; his textbooks won numerous awards, have been translated into many languages and are widely used to this day. Niven was born in Vancouver in 1915, earned his Bachelor's and Master's degrees at UBC in 1934 and 1936 and his Ph.D. at the University of Chicago in 1938. He was a faculty member at the University of Oregon from 1947 until his retirement in 1982. The annual Niven Lecture Series, held at UBC since 2005, is funded in part through a generous bequest from Ivan and Betty Niven to the UBC Mathematics Department.
Motivated by seeking stationary solutions to the Euler equations with prescribed vortex topology, H.K. Moffatt has described in the 80s a diffusion process, called "magnetic relaxation", for 3D divergence-free vector fields that (formally) preserves the knot structure of their integral lines. (See also the book by V.I. Arnold and B. Khesin.)
The magnetic relaxation equation is a highly degenerate parabolic PDE which admits as equilibrium points all stationary solutions of the Euler equations. Combining ideas from P.-L. Lions for the Euler equations and Ambrosio-Gigli-Savar\'e for the scalar heat equation, we provide a concept of "dissipative solutions" that enforces first the "weak-strong" uniqueness principle in any space dimensions and, second, the existence of global solutions at least in two space dimensions.
Hoffman's theorem gives an upper bound on the independence ratio of regular graphs in terms of the minimum eigenvalue of the adjacency matrix. We use invariant Gaussian processes on graphs to get a lower bound in the vertex-transitive case. Joint work with Bálint Virág.
This conference brings together world-renowned researchers in areas of mathematical analysis and PDE such as optimal transportation, the calculus of variations, convex analysis, elliptic systems, and geometric analysis, which are grounded in applications to the natural and social sciences while generating exciting new directions for mathematical research. Its primary aims are to survey the state-of-the-art in these interrelated fields, expand the connections between them, identify key future directions, and encourage a new generation of scientists to advance this fundamental area of mathematics.
The conference begins Monday 8th morning and ends Friday 12th evening.
Uniqueness theory in stochastic partial differential equations (SPDE’s) concerns their completeness and can induce fundamental properties of solutions such as Markov property. Nonetheless, there remain no robust methods to determine uniqueness in general SPDE’s with non-Lipschitz diffusion coefficients. The most important problem, open for more than two decades, is whether pathwise uniqueness in the SPDE of one-dimensional super-Brownian motion holds. A recent work by Mueller, Mytnik, and Perkins sheds light on this difficult problem, proving, however, that pathwise uniqueness for some closely related SPDE’s fails. In contrast to these particular SPDE’s, the SPDE’s of one-dimensional super-Brownian motions with immigration share more properties with the SPDE of super-Brownian motion but, at the same time, raise additional difficulties in settling the question of pathwise uniqueness.
I will first review the SPDE of super-Brownian motion and some notions of uniqueness. I will then introduce the class of super-Brownian motions with immigration considered in our work and discuss our pathwise non-uniqueness result for their SPDE’s. In the rest of this talk, I will explain certain key arguments of our proof.
The theory of superconductivity of Bardeen, Cooper, and Schrie er (BCS) plays an important role in condensed-matter physics in understanding superconductors from fi rst principles. It is based on the idea that electrons in metals can form Cooper pairs, which then behave like bosons and become super fluid at very low temperatures. Mathematically, a proof of existence of a superconducting state as a charge-symmetry breaking state (in the quantum statistical sense of a positive linear functional on the observable algebra), remains a widely open problem.
In this talk I will briefly review the context of BCS theory, introduce BCS models as quantum many-fermion systems with an attractive interaction between Cooper pairs, and then focus on the so-called reduced BCS model, in which the interaction among the Cooper pairs does not decay with their distance. This is a vast simpli cation compared to the general situation, but the analysis of this model remains nontrivial because the underlying algebra is noncommutative. I will sketch a proof by functional integral methods that this model has a superconducting state at low enough temperatures in the thermodynamic limit
Over the years Balaban has developed a powerful renormalization group technique that is applicable to many problems in quantum field theory and statistical mechanics. In this expository talk we fi rst review his results for various gauge theories and the linear sigma model. Then we describe in some detail how the method can be applied to a simpler case, namely to the ultraviolet problem for the \phi^4_3 model. The treatment deals with the basic renormalization problem in a somewhat novel way which fits naturally with Balaban's scheme. This is a discrete dynamical systems method that makes no reference to perturbation theory.
The main results of this thesis concern the spatial decomposition of
Gaussian fields and the structural stability of a class of dynamical
systems near a non-hyperbolic fixed point. These are two problems that
arise in a renormalization group method for random fields and
self-avoiding walks developed by Brydges and Slade. This renormalization
group program is outlined in the introduction of this thesis with
emphasis on the relevance of the problems studied subsequently.
The first original result is a new and simple method to decompose the
Green functions corresponding to a large class of interesting symmetric
Dirichlet forms into integrals over symmetric positive semi-definite and
finite range (properly supported) forms that are smoother than the
original Green function. This result gives rise to multiscale
decompositions of the associated free fields into sums of independent
smoother Gaussian fields with spatially localized correlations. Such
decompositions are the point of departure for renormalization group
analysis. The novelty of the result is the use of the finite propagation
speed of the wave equation and a related property of Chebyshev
polynomials. The result improves several existing results and also gives
simpler proofs.
The second result concerns structural stability, with respect to
contractive third-order perturbations, of a certain class of dynamical
systems near a non-hyperbolic fixed point. We reformulate the stability
problem in terms of the well- posedness of an infinite-dimensional
nonlinear ordinary differential equation in a Banach space of carefully
weighted sequences. Using this, we prove the existence and regularity of
flows of the dynamical system which obey mixed initial and final
boundary conditions. This result can be applied to the renormalization
group map of Brydges and Slade, and is an ingredient in the analysis of
the long-distance behavior of four dimensional weakly self-avoiding
walks using this approach.
This thesis discusses various aspects of continuous-time simple random walks on measure weighted graphs, with a focus on behaviors related to large-scale geometric properties of the underlying graph. In contrast to previous work in this area, the majority of the results presented here are applicable to random walks with unbounded generators. A recurring theme in this research is the use of novel distance functions for graphs known as adapted metrics, which are demonstrated to be a powerful tool for studying random walks on graphs.
Using adapted metrics, we prove two off-diagonal estimates for heat kernels of continuous-time simple random walks; a Poisson-type long range estimate which is valid unconditionally, and a stronger, Gaussian-type estimate which is valid in a restricted space-time region.These results hold under mild geometric hypotheses and are applicable to many models of random walks in random environments.
Subsequently, we prove sharp upper bounds for the bottom of the essential spectrum of graph Laplacians in terms of the adapted volume growth. We prove two bounds, depending on whether the generator is bounded or not. Our estimate for the bounded case generalizes prior results of Fujiwara for graphs, and our estimate for the unbounded case is analogous to a result of Brooks for Riemannian manifolds.
Finally, we prove sharp criteria relating adapted volume growth to stochastic completeness of graphs. To do this, we construct a diffusion on a metric graph which behaves very similarly to the random walk under consideration, which makes it possible to use techniques from the theory of strongly local Dirichlet forms. This result is a significant improvement over the best previous results on this problem, and is analogous to the sharp result of Grigor'yan for stochastic completeness of Riemannian manifolds.
Run a Brownian motion on a torus for a long time. How large are the random gaps left behind when the path is removed? In three (or more) dimensions, we find that there is a deterministic spatial scale common to all the large gaps anywhere in the torus. Moreover, we can identify whether a gap of a given shape is likely to exist on this scale, in terms of a single parameter, the classical (Newtonian) capacity. I will describe why this allows us to identify a well-de fined "component" structure in our random porous set.
We discuss possible generalizations of the concept of Schubert and Grothendieck polynomials to the context of an arbitrary algebraic oriented cohomology theory. We apply these techniques to a rational formal group law and
obtain formulas for the respective polynomials in the A_n-cases. This is a joint project with C. Zhong.
We apply the techniques of formal Demazure operators to obtain an algebraic model for the T-equivariant oriented cohomology of the variety of Borel subgroups of a linear algebraic group. This is a joint project with B. Calm\`es and C. Zhong.
We investigate a random metric space on n points constrained to have all distances smaller than 2, or in other words, we take a random point from the Lebesgue measure on the intersection of the so-called metric polytope with the cube [0,2]^(n(n-1)/2). We find that, to a good precision, the distances behave simply like i.i.d. numbers between 1 and 2.
Our proof uses an interesting mix of entropy methods and the concept of "partial exchangeability".
Based on joint work with Gady Kozma Ron Peled and Wojciech Samtoij.
An important problem in Quantum Information is the transfer of states with high fidelity between locations. The devices performing this function are referred to as quantum wires. Spin chains can in principle be used to construct such wires. I shall discuss the design of spin chains that realize perfect and almost perfect transfer, that is that transport a state from one end of the chain to the other with probability one or almost one over some time.
Orthogonal Polynomial Theory and elements of Diophantine approximation will be called upon.
Room 203, Graduate Student Centre (6371 Crescent Road)
Thu 29 Aug 2013, 9:00am-11:30am
Details
Symmetry methods are important in the analysis of differential equation (DE)
systems. In this thesis, we focus on two significant topics in symmetry analysis:
nonlocally related partial differential equation (PDE) systems and the application
of the nonclassical method.
In particular, we introduce a new systematic symmetry-based method for constructing
nonlocally related PDE systems (inverse potential systems). It is shown
that each point symmetry of a given PDE system systematically yields a nonlocally
related PDE system. Examples include applications to nonlinear reactiondiffusion
equations, nonlinear diffusion equations and nonlinear wave equations.
Moreover, it turns out that from these example PDEs, one can obtain nonlocal
symmetries (including some previously unknown nonlocal symmetries) from
some corresponding constructed inverse potential systems.
In addition, we present new results on the correspondence between two potential
systems arising from two nontrivial and linearly independent conservation
laws (CLs) and the relationships between local symmetries of a PDE system and
those of its potential systems.
We apply the nonclassical method to obtain new exact solutions of the nonlinear
Kompaneets (NLK) equation
ut = x−2 x4 ux + u +
u2x
,
where > 0, 0 and
> 0 are arbitrary constants. New time-dependent exact
solutions for the NLK equation
ut = x−2 x4 ux +
u2x
,
for arbitrary constants > 0,
> 0 are obtained. Each of these solutions is
expressed in terms of elementary functions. We also consider the behaviours of
these new solutions for initial conditions of physical interest. More specifically,
three of these families of solutions exhibit quiescent behaviour and the other two
families of solutions exhibit blow-up behaviour in finite time. Consequently, it
turns out that the corresponding nontrivial stationary solutions are unstable.
Special Event: Fluid Lab Meetings
Abstract: Non-bearing materials currently used in building applications (walls or bulkheads) are generally formed by panels or blocks. Such materials require good thermal performances and minimal mechanical strength. Mineral foams are relevant solutions. We focus on calcium sulfate foams containing a large amount of air trapped in bubbles. The foam production starts with fluid suspension obtained by quick mixing of water, mineral binder and surfactants. Foaming is applied before the quick setting of the mineral matrix.
To study the interaction between foaming and rheological properties evolution, we use a transparent fluid model (dispersion of carbopol 676 in water). The air bubbles stability in yield stress polymer gel is analyzed by picture processing. Carbopol gel is placed in plexiglas container. Bubbles are inserted with a variable volume using a syringe for injection controlling gas pressure. Several phenomena are studied. The impact of the yield stress on a single bubble stability, growth, sphericity is first investigated. The contact conditions between is then studied. Experiments reveal surfactant role in foam stability. Results leads to a better understand of the foams stability at fresh state. It gives us tools to optimize the formulation regarding mechanical and thermal properties targets.
We study the stationary states of the semi-relativistic Schroedinger-Poisson system in the repulsive (plasma physics) Coulomb case. In particular, we establish the existence and the nonlinear stability of a wide class of stationary states by means of the energy-Casimir method. Moreover, we establish global well-posedness results for the semi-relativistic Schroedinger-Poisson system in appropriate functional spaces.
MATH 126 (previously listed as at 3 pm and in Henry Angus 254)
Thu 5 Sep 2013, 3:30pm-4:30pm
Abstract
Let G be an infinite group with a finite symmetric generating set S. The corresponding Cayley graph on G has an edge between x,y in G if their ratio xy^{-1} is in S. Kaimanovich-Vershik (1983), building on fundamental results of Furstenberg, Derrienic and Avez, showed that G admits non-constant bounded harmonic functions iff the entropy of simple random walk on G grows linearly in time; Varopoulos (1985) showed that this is equivalent to the random walk escaping with a positive asymptotic speed. Kaimanovich and Vershik (1983) also described the lamplighter groups (groups of exponential growth consisting of finite lattice configurations) where (in dimension at least 3) the simple random walk has positive speed, yet the probability of returning to the starting point does not decay exponentially. They conjectured a complete description of the bounded harmonic functions on these groups; In dimension 5 and above, their conjecture was proved by Anna Erschler (2011). In the talk, I will discuss the background and present a proof of the Kaimanovich-Vershik conjecture for all dimensions, obtained in joint work with Russ Lyons; the case of dimension 3 is the most delicate. No prior knowledge of group theory will be assumed.
Scalar curvature measures the asymptotic volume growth of small balls in Riemannian manifolds. In the case of positive scalar curvature the growth rate is smaller than in the flat, euclidean case. Typical examples are round spheres of dimension at least two.
We will discuss the interplay of analytic, geometric and topological methods for the investigation of manifolds of positive scalar curvature.
A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability. A hunter and a rabbit move on an n-vertex cycle without seeing each other until they meet. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such Kakeya sets. Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions. (Talk based on joint work withY. Babichenko, R. Peretz, P. Sousi and P. Winkler).
Note for Attendees
Coffee, tea and cookies served at 2:30pm in the PIMS Lounge.
Institute of Applied Mathematics (IAM) Students, Postdoctoral Fellows, Faculty and Staff gather to introduce and re-introduce themselves in an informal setting over refreshments. Even if you are not a member but interested in the IAM and its activities, you are welcome to join us.
CRG Geometry and Physics seminar.
This talk is hosted by U Alberta.
Abstract: For a toric manifold X and a Lagrangian torus fiber L in X, Fukaya-Oh-Ohta-Ono defined open Gromov-Witten invariants which are virtual enumerations of holomorphic disks in X with boundary conditions in L. Qualitative and quantitative properties of these open Gromov-Witten invariants play important roles in the symplectic geometry and mirror symmetry of X. Open Gromov-Witten invariants are difficult to compute because standard methods in Gromov-Witten theory (such as virtual localization) are not applicable. In this talk we explain a geometric method that leads to a complete calculation of these open Gromov-Witten invariants for compact semi-Fano toric manifolds. This is joint work with K. Chan, S.-C. Lau, N. C. Leung.
The area of sparse representation of signals is drawing tremendous attention in recent years in diverse fields of science and engineering. The idea behind the model is that a signal can be approximated as a linear combination of a few "atoms'' from a pre-specified and over-complete "dictionary'' (typically represented by columns from a matrix with more columns than rows). The sparse representation of a signal is often achieved by minimizing an L1 penalized least squares functional.
A well known conjecture in the theory of transformation groups states that if p is a prime and (Z/p)^r acts freely on a product of k spheres, then r ≤ k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non-simply connected spaces.
The ultimate success of many modeling applications depends on time to solution. I will illustrate the critical nature of time to solution by describing a joint project between my group at Rice University and Dr David Fuentes at the MD Anderson Cancer Center. The project goal is to evaluate the role and viability of using finite element modeling as part of the treatment planning process for MR Guided Laser Induced Thermal Therapy.
The commuting graph of a finite non-commutative semigroup S, denoted G(S), is a simple graph whose vertices are the non-central elements of S and two distinct vertices x,y are adjacent if xy=yx. This definition generalizes the corresponding concept of the commuting graph of a non-Abelian group.
Our work looks at the commuting graph of the symmetric inverse semigroup I(X). For a finite set X, let I(X) be the semigroup of all partial injective transformations on X under composition. The semigroup is universal for the class of inverse semigroups in the sense that every inverse semigroup can be embedded in I(X) for some finite set X, analog to the situation of the symmetric groups Sym(X) in group theory.
In 1989, Burns and Goldsmith classified the maximum order abelian subgroups of Sym(X). We extend this result to the semigroup I(X). As a consequence, we obtain a formula for the clique number of the commuting graph of I(X).We also calculate the diameter of I(X) when |X| is prime or even, and obtain tight bounds on it in the remaining cases.
This is joint work with Joao Araujo (CAUL) and Janusz Konieczny (Mary Washington).
In recent years large data sets have been collected and analyzed, typically, by using some machine learning algorithms. However, many types of data demand a much more in-depth analysis that requires simulation, that is, solving partial differential equations, and optimization to estimate parameters.
In this talk we discuss examples for such data sets in earth science. We describe the setting in which vast amounts of geophysical data is collected from the air. We present the large scale modeling that is required to simulate such a data set and the inverse problems that arise from these types of problems. We show that by using traditional techniques these problems cannot be solved in reasonable time on reasonable hardware.
We then discuss a new set of algorithms that enable us to solve such problems. These algorithms are based on a concept we call domain of interest computation for the forward coupled with stochastic programming for the inverse. We show that by using this combination we are able to solve very large scale inverse problems, using a rather modest hardware in reasonable time.
Many interesting and familiar varieties are collections of matrices. Two of the most familiar are the determinantal varieties and the matrix Schubert varieties. Often, these varieties are stable under the row operations of linear algebra. This talk will be about such varieties, specifically the equivariant K-theory classes of such varieties. Familiar combinatorial objects show up here, such as Schur polynomials and matroids. After giving a lengthy overview, I will move towards discussing the "shape" of the K-class of a variety. No background material will be assumed.
Our world is becoming increasingly complex: nanoscale processes need to be understood for the next generation of materials; global financial markets need to be understood to be able to optimize and stabilize the world's economy; long time environmental forecasts need to be done to understand the impact of global warming. Many of the underlying questions in these fields involve abstract models and require accurate computational approximation. The models may not involve new, research mathematics, but are often beyond the expertise of application experts. Mathematicians have the skills to contribute to these questions. Several examples of successful collaboration of mathematicians in industrial problems will be outlined, including the speaker's work in a larger group working with Ballard Power Systems on simulation tools for Hydrogen fuel cells. Some of the challenges to this collaboration will be discussed, including resistance of industrial engineers to working with mathematicians, but also resistance by our own community to recognizing the value of this activity. Some additional remarks will be made on the employment opportunities for mathematicians working in the Industry.
It is well-known that variational problems may fail to have a classical minimiser if the integrand is not convex. In the 1930s, L. C. Young suggested a relaxation of such problems, where the minimising map is allowed to be measure-valued. In physical applications (e.g. elasticity theory), one often looks at variational problems for gradients of vector fields. A crucial problem in the context of relaxation is to characterise those measure-valued maps that arise as limits of a sequence of gradients. While this was achieved by D. Kinderlehrer and P. Pedregal about 20 years ago, the question remained open whether a similar characterisation could be found under the additional constraint that the gradients have positive determinant, i.e. the underlying maps be orientation-preserving. I will present such a characterisation, recently obtained in joint work with K. Koumatos (Oxford) and F. Rindler (Warwick).
A length n parking function is a sequence of integers whose non-decreasing rearrangement (q_1,q_2,\dots,q_n) satisfies q_i \leq i. Any rearrangement of a parking function thus remains a parking function and one can consider the permutation representation P_n of the symmetric groups S_n generated by such sequences. In this talk I will describe a representation of S_{n+1} whose restriction to S_n is P_n. This will be viewed in the context of a result of Stanley which extends the regular representation of S_n to a representation of S_{n+1}. I'll include a review of the combinatorics of symmetric group representations. This is joint work with Brendon Rhoades.
We consider the Markov process given by products of i.i.d. random matrices that are perturbations of a fixed non-random matrix and the randomness is coupled with some small coupling constant. Such random products occur in terms of transfer matrices for random quasi-one dimensional Schrodinger operators with i.i.d. matrix potential. Letting the number of factors going to infi nity and the random disorder going to zero in a critical scaling we obtain a limit process for a certain Schur complement of the random products. This limit is described by an SDE. This allows us to obtain a limit SDE for the Markov processes given by the action of the random products on Grassmann and flag manifolds. Applied to random quasi-one dimensional Schrodinger operators we can describe the limiting eigenvalue process in a critical scaling by the zero process of a determinant of a matrix-valued function described by an SDE.
Homology stability for families of discrete groups such as the symmetric groups, linear groups, braid groups and mapping class groups are well-known. Extensions to diffeomorphism groups of manifolds more generally have only been proved recently in a few special cases. We will revisit some classical results on configuration spaces, extend them to the equivariant setting, and prove homology stability for so-called symmetric diffeomorphism groups for arbitrary manifolds.
It occasionally happens that, for a certain type of complete intersection, the associated complex moduli space is actually (open in) an arithmetic quotient of a complex ball. I will describe recent work, particularly involving the case of cubic surfaces, which suggests that this unexpected structure is the complex realization of a morphism of integral moduli spaces. A modest payoff of the general theory is the calculation, in characteristic at least five, of the Galois group of the 27 lines on a sufficiently general cubic surface.
Cobordisms have played an important role in the classification of manifolds through work of René Thom from the 1950s. In a different way, they are fundamental to the mathematical formulation for topological quantum field theory of Atiyah, Segal and Witten. For a long time these two theories did not seem to interact. We will explain how more recent results have unified the two.
Note for Attendees
Coffee, tea and cookies served at 2:30pm in the PIMS Lounge.
This talk will present the under utilized ideas of rigorous numerics as a means of solving problems in PDE analysis. The core concepts such as interval arithmetic, automatic differentiation and radii polynomial will be discussed and recent extensions using mesh adaptivity will be presented. A fully automated curve following code for a class of elliptic problems will be explained.
We define actions of certain Heisenberg algebras on the Hilbert schemes of
points on ALE spaces. This lifts constructions of Nakajima and Grojnowski
from cohomology to K-theory and derived categories of coherent sheaves.
This action can be used to define Lie algebra actions (using categorical
vertex operators) and subsequently braid group actions and knot
invariants.
Suppose you have a fixed number of boxes with a square base. These are transported in containers that also have a square base. In order to save space one wants to make containers small while still being able to hold all boxes. How large would the smallest such container be, and how would one need to pack the boxes into such a container? Questions such as these are called packing problems.
In mathematical language, this particular question would be considered as square packing, an area that traces its origin to a paper by Erdös and Graham. For small numbers of packed squares, tight (and conjectured optimal) packings have been obtained by a variety of methods, including computerized search. In contrast, only a surprisingly small number of non-trivial cases have actually been proved to be optimal, all of which involving relatively simple packings.
In this talk, which is accessible to undergraduate students, we give an overview on the subject and demonstrate the techniques used in proving square packings optimal.
A common experiment performed by web companies is the so-called A/B test, which compares two software designs assigning different interfaces or functionalities to two groups of randomly-selected users. Although it is usual to design these experiments as hypothesis tests, standard tests (such as t-tests) overestimate confidence, since web traffic is heavy-tailed. It is typical for a small number of individuals to radically affect the statistics of their test group.
In this talk, I discuss my work as a consultant for deviantART, an online art community, as we developed reliable models for their traffic data, as well as subsequent hypothesis tests.
This talk is both a review of properties and methods for handling heavy-tailed data, and an advertisement for the utility of probabilistic modeling in industry.
For any given integer N larger than 2, we show that every bounded measurable vector field is N-cyclically monotone up to a measure preserving N-involution. The proof involves the solution of a multidimensional symmetric Monge-Kantorovich problem, which we first study in the case of a general cost function on a product domain. The proof exploits a remarkable duality between measure preserving transformations that are N-involutions and those Hamiltonians that are N-cyclically antisymmetric.
The sandpile model was introduced by physicists Bak, Tang and Wiesenfeld in 1988. Then it was identified as a prototype of self organized criticality. This discrete model of diffusion is generally defined on any connected graph, and a configuration/state corresponds to a distribution of grains on vertices. Here are some of its general properties. It induces a Markov chain with uniform stationary distribution on some recurrent configurations in bijection with spanning trees. The distribution of external activity of Tutte polynomial corresponds to the distribution of the number of grains. Addition of recurrent configurations leads to a finite abelian group for each graph, sometimes called in other contexts the critical group or the Picard's group.
During this talk, I will focus on the case of complete and bipartite complete graphs. The symmetries of these graphs simplify a lot the analysis but leads, according to me, to not so completely trivial results. A study related to the evolution of the Markov chain to the recurrent configurations simply leads to an apparently not so well-known extension, due to Chottin (1975), of a classical cyclic lemma used to count Dyck words. The analysis of Dhar's criterion, testing recurrence of a given configuration, simply leads to bistatistics related to the q,t-Catalan numbers studied by Bergeron, Garsia, Haglund and Haiman in their study of the space of diagonal harmonics. On the complete graphs, we obtain an algorithm of linear arithmetic complexity for the rank parameter introduced by Baker and Norine in 2007 for their analogue of Riemann-Roch theorem for graphs.
Based on joint works with Mark Dukes, Michele D'adderio, Jean-Christophe Aval, Angela Hicks and Robert Cori.
[arXiv:1208.0024,arXiv:1301.4803, arXiv:1307.7740, arXiv:1308.5325]
We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity. Secondly, we characterize the growth of the variance of N(T). For the last part, we consider real stationary Gaussian functions on the real axis and discuss the "gap probability" (i.e., the probability that the function has no zeroes in [0,T]). This part is a joint work with Ohad Feldheim.
I will briefly Homological Projective Duality, an analog of projective duality in the setting of derived categories due to Kuznetsov. I will explain the relationship between this duality, GLSMs, and variation of geometric invariant theory quotients as part of joint work with M. Ballard, D. Deliu, U. Isik, and L. Katzarkov.
We prove that a uniformly chosen proper coloring of Z_{2n}^d with 3 colors has a very rigid structure when the dimension d is sufficiently high. The coloring almost surely takes one color on almost all of either the even or the odd sub-lattice. In particular, one color appears on nearly half of the lattice sites. This model is the zero temperature case of the 3-states anti-ferromagnetic Potts model, which has been studied extensively in statistical mechanics. The proof involves results about graph homomorphisms and various combinatorial methods, and follows a topological intuition. Joint work with Ron Peled.
Yield stress fluids encompass a large class of materials, from granular media and foams to dense assemblies of colloidal hard spheres and glassy suspensions of soft particles such as emulsions, microgels, etc. These soft glassy systems share the following feature: they display a solid-like behavior at rest, while they flow as viscous liquids when stressed above their yield stress. Such solid-fluid or yielding transition can be seen as a stress-induced unjamming scenario. The question of whether this shear-induced fluidization displays universal features, in a way similar to jamming driven by temperature or by volume fraction, has triggered much research effort in the recent years. Experimentally, difficulties arise from the need to measure deformations and flows close to yielding at vanishingly small shear rates with sufficient spatial and temporal resolutions.
In this talk, I will briefly review the current state of research on the steady state reached by a soft glassy system above yielding. I will then concentrate on the spatiotemporal fluidization dynamics of a "simple" yield stress material, namely a carbopol microgel, that presents negligible aging and thixotropy. Through long experiments combining standard rheology and ultrasonic velocimetry under imposed strain or stress, I will show that the material undergoes a transient regime characterized by (i) a short-time creep regime reminiscent of the primary, or Andrade, creep in solid materials followed by (ii) a long-lasting shear banding regime that progressively gives way to homogeneous flow. As a key result, the duration of the shear-banding regime decreases as power laws of the applied shear rate and of the applied viscous stress. These power laws nicely combine to recover the Herschel-Bulkley law characteristic of the steady-state rheology of our microgel, thus providing an interesting interpretation of this widely used phenomenological law.
In this talk, we survey past, present, and possible future results concerning the arithmetic nature of low complexity sequences. For example, what properties can be exhibited by numbers whose base expansion can be determined by a finite automaton? In the current context, this line of questioning was unknowingly initiated by Mahler, and later championed by Loxton and van der Poorten following the work of Cobham and Mendes France. In addition to describing some historical work, this talk will describe some of the the current advancements and generalisations concerning Mahler's method.
Note for Attendees
* People can attend the talk via videoconferencing in room MATH 126 at UBC.
This talk is about an application of methods originating in model theory to harmonic analysis on p-adic groups. It will start with the review of the definition of the p-adic numbers, and
then focus on a way to do integration over the field of p-adic numbers in a way that does not depend on p (this is where the model theory comes in). Then I will describe the application of these ideas to harmonic analysis, in particular, a way to get a uniform in p estimate for orbital integrals (which will be defined in the talk).
Note for Attendees
Please join your colleagues for refreshments in MATH 125 at 2:45 p.m. prior to today's colloquium.
The Mathematics Education Research Reading Group will meet monthly to review and discuss current papers about research in mathematics education. Anyone is welcome to participate. For this first meeting's paper we have chosen a broad overview of the purposes and methods in mathematics education research. Future meetings will focus on peer-reviewed studies.
The paper can be obtained at the following link: http://www.ams.org/notices/200006/fea-schoenfeld.pdf
Harmonic manifolds are those Riemannian manifolds whose harmonic functions satisfy the mean-value property, or equivalently, whose spheres have constant mean curvature. F. Ledrappier introduced an asymptotic version of harmonicity which was mainly studied in the cocompact and homogeneous cases. In this talk, I will review some classical facts on harmonic manifolds and prove some new results on asymptotically harmonic manifolds, including a characterization in term of the volume form . This is a joint work with Andrea Sambusetti.
The Schur-positivity order on skew shapes is denoted by B < A if the difference of their respective Schur functions is a positive linear combination of Schur functions. It is an open problem to determine those connected skew shapes that are maximal with respect to this ordering. In this talk we see that to determine the maximal connected skew shapes in the Schur-positivity order it is enough to consider a special class of ribbon shapes. We also explicitly determine the support for these ribbon shapes.
This is joint work with Peter McNamara and assumes no prior knowledge.
I will review connections, including the Dynkin isomorphism, between the Gaussian free fi eld on a lattice and local time of random walk. These connections give useful representations for walks with self-interactions.
The extended W-algebra of type sl_2 at positive rational level is a vertex operator algebra that is of great interest in logarithmic conformal field theory. In this talk I will give an overview of how it is constructed as a subvertex operator algebra of a lattice vertex operator algebra by means of so called screening operators. I will also explain how the screening operator formalism allows one to prove c_2 cofiniteness, compute relations in Zhu's algebra and classify all simple modules of the extended W-algebra of type sl_2 at positive rational level.
This talk presents analysis of the performance of multi-element guard-heated hot-film wall shear stress microsensors for turbulent flows. Previous studies of conventional, single-element sensors show that a significant portion of heat generated in the hot-film travels through the substrate before reaching the fluid, causing spectral and phase errors in the wall shear stress signal and drastically reducing the spatial resolution of the sensor. Earlier attempts to reduce these errors have focused on reducing the effective thermal conductivity of the substrate. New guard-heated microsensor designs proposed to overcome the severe deficiencies of the conventional design are investigated. Guard-heaters remove the errors associated with substrate heat conduction, by forcing zero temperature gradient at the edges and bottom face of the hot-film, and hence, block the indirect heat transfer to the flow. Air and water flow over the sensors are studied numerically to investigate design, performance and signal strength of the guard-heated sensors. Our results show, particularly for measurements in low-conductivity fluids such as air, that edge guard-heating needs to be supplemented by a sub-surface guard-heater, to make substrate conduction errors negligible. With this two-plane guard-heating, a strong non-linearity in the standard single-element designs can be corrected, and spectral and phase errors arising from substrate conduction can be eliminated.
Laurent:
The spiral wave is a pattern on a surface occurring in several natural phenomena, such as in chemical reactions an on the cardiac tissue. It is usually a by-product in several pathologies such as stated in Mathematical Physiology by Keener and Sneyd: "spirals on the heart are fatal, spirals on the cerebral cortex may lead to epileptic seizures, and spirals on the retina may cause hallucinations". In this talk, we will briefly go over the background research on spiral waves and will present results of a perturbed simple system, describing rigidly rotating waves and linearly travelling waves in the unperturbed case. The perturbation used must have lattice symmetry, that is symmetry on quarter rotations and unit translations in two orthogonal directions. We will first look at the derivation of the general form of the perturbation. Then, we will state findings obtained in both rotating and travelling waves, supported by numerical simulations.
Tom: A \emph{circle packing} $P$ is a collection $\{C_v : v \in V\}$ of discs of disjoint interior in the (extended) complex plane. Given a circle packing $P$, we may define its \emph{tangency graph} as $(V,E)$ where $\{u,v\} \in E$ if and only if $C_v$ and $C_u$ are tangent.
The Circle Packing Theorem states that any finite planar\footnote{A graph is said to be \emph{planar} if we can draw it in the plane such that no two distinct edges cross.} graph arises as the tangency graph of some circle packing. In fact, when the graph is a triangulation of the sphere, the circle packing is unique up to reflections and Mobius transformations of the sphere.
In this talk, we aim to provide an elementary, details-free introduction to the Circle Packing Theorem, its extensions and corollaries.
Note for Attendees
Pizza and pop will be provided during the break between the talks.
We compute the integral cohomology ring of configuration spaces of two distinct points on a given real projective space. As an application, we obtain the symmetric topological complexity of real projective spaces of dimensions 5 and 6.
We show how to realize the pro-p-Iwahori-Hecke algebra of SLn as a subalgebra of the pro-p-Iwahori-Hecke algebra of GLn. Using the interplay between these two algebras, we deduce a correspondence between "packets" of Hecke modules and mod-p projective Galois representations.
In this talk, we will review the literature on frame quantization and present a number of recent results. Motivated by compressed sensing we will focus on "sub-Gaussian random frames" and explain why classical quantization methods are bound to be substantially suboptimal. We will then show that by using the so-called sigma-delta quantizers along with reconstruction via ”Sobolev duals”, we can improve the quantization error substantially when we quantize sub-Gaussian frame expansions. Specifically, we prove that using an r-th order sigma-delta scheme, we get an accuracy of order (-r)-th power of the aspect ratio of the frame. Furthermore, if we optimize the order of the scheme depending on the aspect ratio, this yields root-exponential accuracy. Finally, we will discuss how these results can be used for constructing effective quantization methods for compressed sensing.
Note for Attendees
Refreshments will be served at 2:45 p.m. in the Math Lounge area, MATH 125.
A DC operating point is an equilibrium toward which a circuit will be drawn for sufficiently nearby initial conditions when any inputs are held fixed. DC operating points may or may not be desirable features in a circuit -- in an oscillator they represent lockup, but in a memory element they are the mechanism whereby discrete state is stored. Consequently, it is useful to identify a circuit's DC operating points. Because the circuit is naturally drawn towards them, the most common technique to identifying such equilibria is through simulation; however, it is quite possible for the domain of attraction of an equilibrium to be small enough that simulation is unlikely to find it, yet large enough to cause occasional problems.
In this joint work with Mohamed Zaki & Mark Greenstreet, we strung together a collection of public software from the formal verification and numerical analysis communities to rigourously identify and classify all potential DC operating points for surprisingly complex circuit models. Unfortunately, the resulting workflow has proved fragile, and significant effort would be required for reproduction and/or extension. In the second half of the talk I will discuss some tools and techniques that would have significantly improved the reproducibility of the results had they been adopted.
In this talk, I will review a few issues and results on compactness of equations of scalar curvature type. In particular, I will focus on the difficulty of the degeneracy of the kernel of the solutions to such equations. This is joint work with Jérôme Vétois (Nice).
Let M be a connected, orientable, piecewise linear manifold of dimension n and let B be a closed submanifold of M. Let PL(M, B) be the group of orientation preserving PL homeomorphisms of M which are pointwise fixed on B. The group operation is composition of functions.
In joint work with Danny Calegari we show that if B has codimension zero or one, the group PL(M,B) is locally indicable. This means that every finitely-generated subgroup has the integers as a quotient. It follows that PL(M,B) is left-orderable and therefore has no elements of finite order.
Let f be an irreducible polynomial over a number field. Under what conditions is it true that all iterates of f are also irreducible? We call polynomials with this property stable. Eventual stability is the weaker property that the number of factors of the nth iterate of f is bounded uniformly in n. We can extend this definition to rational functions. We conjecture that all rational functions are eventually stable when 0 is not periodic, and show that this is a case for a large class of functions using Newton polygon techniques. (joint with R. Jones)
We consider a inhibitory ternary system of threeconstituents, a model motivated by the triblock copolymertheory. The free energy of the system consists of two parts:the interfacial energy coming from the boundaries separatingthe three constituents, and the longer range interactionenergy that functions as an inhibitor to limit micro domaingrowth. One solution of this system, found by Lu Xie in herPhD thesis, is a core-shell pattern where the firstconstituent forms the core, the second forms the shell, andthe third fills the back ground. Another solution is shownin a joint work with Juncheng Wei: there is a perturbeddouble bubble that exists as a stable solution of the system.Each bubble is occupied by one constituent. The thirdconstituent fills the complement of the double bubble. Thissolution has two triple junction points.
I would like to report some advances made while on sabbatical in South Carolina working with Lincoln Lu. These are problems of extremal combinatorics in the area of Forbidden Configurations. We first crack an `old chestnut' that had been around since 1990. We use a new inductive approach. We then apply this new induction to a result of Balogh and Bollobas which is a kind of Ramsey Theorem for Forbidden Configurations. Not surprisingly Ramsey Theory shows up.
In the first lecture of this series the square of a Gaussian field was related to the local time of random walk and a Poisson process of random loops. In this lecture I will show how to "get rid" of the loops and end up with a representation for self-interacting walk as an almost Gaussian integral. This lecture will use the algebra of differential forms, but I will make it self-contained by reviewing what we need to know about differential forms.
I will discuss the size of the set of real numbers x that can be well approximated by rational numbers (in the sense that |x-p/q| < 1/q^s for infinitely many rational numbers p/q) using tools from number theory, harmonic analysis, and probability theory. The talk should be accessible to all math graduate students.
I will discuss the topology of the space Hom(N,G) of homomorphisms from a finitely generated group N into a reductive complex linear algebraic group G (e.g. a special linear group). When K is a maximal compact subgroup of G (e.g. the subgroup of special unitary matrices), Hom(N,K) is a subspace of Hom(N,G). Although in general these topological spaces are quite different, I will show that when N is nilpotent there is a strong deformation retraction of Hom(N,G) onto Hom(N,K).
We present analogues of the classical conjectures of Manin-Mumford, Bogomolov and Pink-Zilber for function fields (of arbitrary characteristic). We also present a function field analogue of the Bounded Height Conjecture (which appears in the study of the Pink-Zilber Conjecture).
Exquisitely structured patterns arise in many physical and biological systems as orderly outcomes of self-organization principles. Common in these pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its further increase. To prevent unlimited increase and spreading, pattern formation requires in addition a longer ranging confinement of the locally self-enhancing process. I will use the Ohta-Kawasaki diblock copolymer model and the Gierer-Meinhardt system as examples to demonstrate the roles played by the growth and inhibition properties, and show how they lead to a nonlocal geometric problem for which structures like discs, rings, balls, ovals, tori, etc, arise as solutions.
We study a scaling limit of the long range SIR epidemic model in which infected individuals cannot be reinfected. The limit, which exists in up to 3 dimensions, has been studied by Lalley and Zheng and is reminiscent of a one-dimensional model proposed by Durrett and studied by Mueller and Tribe. It is a measure-valued process similar to super-Brownian motion with drift \theta but with an additional killing term proportional to its local time. We show there is a non-trivial phase transition in \theta for dimension 2 and 3, above which the process survives and below which it goes extinct, and prove that in one dimension there is always extinction. Moreover we show that in any dimension there is always extinction on compact sets. The results suggest some conjectures for long range bond percolation. This is joint work with Steve Lalley and Xinghua Zheng.
I will give a brief introduction to high-resolution (shock-capturing) finite volume methods for hyperbolic PDEs that model wave propagation. These methods are based on solving Riemann problems at cell interfaces and using wave limiters to develop second-order accurate methods that avoid non-physical oscillations around discontinuities in the solution. Riemann solver methods are also well adapted to problems of wave propagation in heterogeneous media with discontinuities in the material parameters.
The Clawpack (Conservation Laws Package) software package implements these algorithms along with adaptive mesh refinement. I will give an overview of some of the recent developments in this project, including extensions to higher-order methods and to supercomputers through the PyClaw project. More about this open source software can be found at www.clawpack.org.
Theorem: If a finite group G acts freely and homologically trivially on a finite complex K which has the homotopy type of a product of k-spheres, then G acts freely and homologically trivially on a product of (k+1)-spheres.
Corollary: Any group acts freely and homologically trivially on a product of spheres.
In the early 70's Davenport and Heilbronn derived the leading term in the asymptotic formula for the number of cubic number fields with bounded discriminant. However, as algorithmic data became available, a large "gap" became evident between the actual number of cubic number fields of small discriminant and the asymptotic prediction. We will discuss this and the analogous situation in the function field setting. We will present methods for constructing and tabulating dihedral function fields (which includes non-Galois cubics) and prove the existence of a similar "gap" for cubic function fields of small discriminant and the leading term of the corresponding asymptotic.
Note for Attendees
* People can attend the talk via videoconferencing in room MATH 126 at UBC.
An object is rigid with respect to two properties if whenever it satisfies one property it automatically satisfies a stronger property. Many examples of rigidity will be given, focusing on examples involving group theory, topology, and geometry. The best example of topological rigidity is the Borel Conjecture, which states that two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic. The talk will conclude with a discussion of the equivariant rigidity problem and recent joint work with Connolly and Khan.
Note for Attendees
Refreshments will be served at 2:45 p.m. in MATH 125 lounge prior to the colloquium.
Chia will lead a discussion on "An Evaluative Calculus Project: Applying Bloom's Taxonomy to the Calculus Classroom", an article that can be found here: http://www.math.ubc.ca/~wes/bloomstaxonomy.pdf
The first part will consist of a tour through a sparse sample of the information theory literature - both classical and recent - on relations between information and estimation. Beyond aesthetic value, these relations underlie some of the main tools in Shannon theory. They also give considerable insight into and a quantitative understanding of several estimation theoretic objects, such as the costs of causality and of mismatch, as well as the performance and structure of minimax estimators. Further, they enable the transfer of analytic tools and algorithmic know-how from one domain to another. Examples will be given to illustrate these points. The second will begin by introducing Directed Information and making the case for caring about how to estimate it. I will then describe some approaches to this estimation problem which are based on universal sequential probability assignments. The estimators inherit much of the algorithmic simplicity and convergence properties of the latter. I will conclude by showcasing some experimental results illustrating the efficacy of directed information estimation as a tool for detecting and measuring causality.
Much of the talk is based on collaborations with Rami Atar, Jiantao Jiao, Young-Han Kim, Albert No, Haim Permuter, Kartik Venkat, and Lei Zhao.
Note for Attendees
Refreshments start 15 minutes before the talk in the IAM Lounge, Room 306 of the LSK building.
I will talk about my joint work with A. Sheshmani and Y. Toda. We study the stable pair theory of K3 fibrations over curves with possibly nodal fibers. We express the stable pair invariants of the fiberwise irreducible classes in terms of the famous Kawai-Yoshioka formula for the Euler characteristics of moduli space of stable pairs on K3 surfaces and Noether-Lefschetz numbers of the fibration. In the case that the K3 fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants of the fiberwise curve classes in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.
We consider fractional minimal surfaces introduced by Caffarelli, Roquejoffre and Savin (2009). Up to now the only examples of fractional minimal surfaces are hyperplanes. In this talk, we first prove the existence of the analog of fractional Lawson's minimal cones and establish their stability/instability in low dimensions. In particular we find that there are stable fractional minimal cones in dimension 7, in contrast with the case of classical minimal surfaces. Then we prove the existence of fractional catenoids and fractional Costa-Hoffman-Meeks surfaces. Interestingly the interaction of planes in fractional minimal surfaces is governed by an nonlinear elliptic equation with negative power which arises in the study of MEMS. (Joint work with J. Davila and M. del Pino.)
Motivated partly by previous work on the zero curvature representation (ZCR) of completely integrable chiral models and partly by the underlying Hamiltonian structures of ideal complex fluids, we derive systems of partial differential equations, called G-strands, that admit a quadratic zero curvature representation for an arbitrary real semisimple Lie algebra. Using the root space decomposition, the G-strand equations can be formulated explicitly for the compact real form and the normal real form of
any semisimple Lie algebra. We present several particular examples, including the exceptional group G_2. We also determine the general form of Hamilton's principles and Hamiltonians for these systems, and analyze the linear stability of their equilibrium solutions.
The discriminant and the Clifford invariant are classical invariants for quadratic forms. Milnor conjecture proposes successive higher invariants for quadratic forms and asserts that these invariants determine the isomorphism class of a quadratic form up to planes. Milnor conjecture is a theorem due to Voevodsky, Orlov and Vishik. We explain how the Galois cohomology invariants can be used to study the question whether a quadratic form in sufficiently many variables represents zero nontrivially over function fields of curves over number fields and local fields.
Each day we make many choices, often under time pressure and with poor information. How do we do this? The basic electro-chemistry of individual neurons and synapses in our brains is fairly well understood. The key problem is one of scale: how do almost a trillion neurons and many more synapses interact to sift noisy evidence and weigh it against prior knowledge? I will describe how mathematical models, coupled with human and animal experiments, illuminate the neural mechanisms responsible for some simple decisions and actions.
The talk will draw on joint work with Fuat Balci, Rafal Bogacz, Jonathan Cohen, Philip Eckhoff, Eric Shea-Brown, Patrick Simen, Marieke van Vugt, Kong Fatt Wong-Lin and Miriam Zacksenhouse. Research supported by NIMH and AFOSR.
Delay differential equations (DDEs) are used in a number of applications. Stability analysis of DDEs can be quite complex, particularly when multiple delays occur. We examine the scalar two-delay differential equation:
y'(t) = Ay(t) + B y(t - 1) + C y(t - R):
The stability region for this DDE has some very interesting features that this talk will explore. There are four parameters, A, B, C, and R, which can be varied. The stability region can be disconnected in the BC-space though for fixed R the 3D stability surface in the ABC-parameter space is connected. One of the most intriguing features is that when R is rational, this stability surface becomes larger. We demonstrate how certain rational values of R significantly increase the stability region, then show this significance in a nonlinear application. Understanding the details of this analysis can help mathematical modelers appreciate sensitivity in their stability analysis and the complexity of numerical solutions as delays vary in a model with multiple delays.
The Hilbert scheme of n points on C^2 is a smooth manifold of dimension 2n. The topology and geometry of Hilbert schemes have important connections to physics, representation theory, and combinatorics. Hilbert schemes of points on C^2/G, for G a finite group, are also smooth, and their topology is encoded in the combinatorics of partitions. When G is a subgroup of SL_2, the topology and combinatorics of the situation are well understood, but much less is known for general G. After outlining the well-understood situation, I will discuss some conjectures in the general case, and a combinatorial proof that their homology stabilizes.
Solutions of partial differential equations are often highly anisotropic and have strongly directional features. Examples include PDES which have shocks and interfaces in the solution. When calculating the solutions to these PDEs it is important to use computational meshes which align themselves with features in the solution. Many adaptive mesh methods explicitly and implicitly use equidistribution and alignment, and a metric tensor M is typically used to define the desired level of anisotropy. In this talk I will describe a mesh method which combines equidistribution with optimal transport that does not require the explicit construction of a metric tensor M, although such an M always exists. I will show that this method is very effective at aligning elements along solution features including linear shocks and radially symmetric structures. Furthermore, I will provide numerical results to show this method is cheap and robust to implement, and allows solutions to be very well approximated.
In this talk, I will give an introduction to aperiodic tilings. Usually, one studies a topological dynamical system associated to these tilings rather than one specific tiling (this is the analogue to studying a subshift rather that one single word in symbolic dynamics). It is a natural question to ask what happens to the underlying tilings when there is a homeomorphism between tiling spaces. The result I will present is the following: whenever two tiling spaces are homeomorphic, the complexity function is preserved up to some multiplicative constants and rescaling.
In 2007, Baker and Norine stated a theorem which they called a version for graphs of a Riemann-Roch theorem. In this theorem,the rank is an integer parameter defined by an optimisation among some compositions on labeling of vertices by integers. In the case of complete graphs, we provide a greedy algorithm to compute the rank in linear time. Then we study the joint distribution of the rank and the other parameter in Baker and Norine theorem. This involves objects like parking functions, numbers like the Catalan numbers, and bijections, in a framework close to the sandpile model studied by Dhar. The interest in these classical notions is renewed by the addition of seemingly new parameters (orappearing in other contexts) deduced from the analysis of our greedy algorithm.
Joint work with Robert Cori.
In the second lecture I reviewed differential forms and then related the square of a differential form to the local time of random walk. In this lecture I will prove this result in more detail and illustrate the idea by explaining why certain integrals concentrate on critical points even before any large deviation limit is taken. These are cases of a theorem called the Duistermaat Heckman theorem.
The concept of lattice polarization for a K3 surface was first introduced by Nikulin. I will discuss ways in which his definition can be extended to families of K3 surfaces over a (not necessarily simply connected) base curve, with the aim of gaining control over the action of monodromy upon the Néron-Severi lattice of a general fibre. I will then present an application of this to the study of Calabi-Yau threefolds that admit fibrations by Kummer surfaces.
The fragmentation of droplets is an essential stage of several natural and industrial applications such as fuel atomization and rain phenomena. In spite of its relatively long history, the mechanism of fragmentation is not clear yet. This is mainly due to small length and time scales as well as the non-linearity of the process. In the present study, two and three-dimensional numerical simulations have been performed to understand the early stages of the fragmentation of an initially spherical droplet. Simulations are performed in high Reynolds and a range of relatively high Weber numbers (shear breakup). To resolve the small-scale instabilities generated over the droplet, a second-order adaptive finite volume/volume of fluids (FV/VOF) method is employed, where the grid resolution is increased with the curvature of the gas-liquid interface as well as the vorticity magnitude. The study is focused on the onset and growth of interfacial instabilities. The role of Kelvin-Helmholtz instability (in surface wave formation) and Rayleigh-Taylor instability (in azimuthal transverse modulation) are shown and the obtained results are compared with the linear instability theories for zero and non-zero vorticity layers. Moreover, the analogy between the fragmentation of a single drop and a co-axial liquid jet is discussed. The current results can be used for the further development of the current secondary atomization models.
Darmon has developed a program to resolve the generalized Fermat equation xp + yp = zr using Galois representations and abelian varieties of GL2 type over a totally real field. I will survey some parts of his program and point out the key difficulties which remain. Recently, numerous irreducibility criteria for the mod p representations attached to elliptic curves over totally real fields have been developed (David, Billerey, Freitas-Dieulefait, Freitas-Siksek). These are based on a technique which first appeared in Serre's 1972 Inventiones paper. I will explain how this method can be adapted to Darmon's Frey abelian varieties of GL2 type over a totally real field and thereby show that the above equation does not have any non-trivial first case solutions for p large enough compared to r a regular prime ≥ 5.
Note for Attendees
* People can attend the talk via videoconferencing in room MATH 126 at UBC.
A great amount of work has been dedicated in the last years to understand
problems with integral diffusion for elliptic, parabolic or hyperbolic
equations and systems. In this talk, I will describe a new Yamabe problem
based on conformally covariant elliptic operators of fractional order. I
will describe some new results on existence of metrics for the regular and
singular problems.
Dyson-Schwinger equations are certain integral equations in quantum field theory which mirror the combinatorial decompositions of trees by subtrees, or of graphs by subgraphs. At the analytic level, many cases can still be interpreted combinatorially, as expansions indexed by rooted connected chord diagrams. I will explain this construction, talk about which cases we know and where we are going next.
The solvent casting process is a cost-effective technique with potential applications to the fabrication of microstructures such as microneedles. It involves the evaporation of a solvent from a polymer solution inside a micron scale cavity, and results in a polymeric coating on the walls of the cavity. This drying process represents a complex fluid mechanics problem that is generally associated with several flow phenomena contributing to complex flow patterns inside the fluid film. In addition to evaporation-driven flow and Marangoni flow, the velocity field also reveals single and multiple vortices generated by the creeping flow induced by evaporation. An experimental study including laser scanning microscopy and particle image velocimetry (PIV) are used to introduce and characterize the low Reynolds corner vortices that can occur during the evaporation of thin films inside microliter cavities. The observed corner vortices all show a similar unsteady behavior, in that they continuously shrink over time and finally disappear. The initial viscosity of the fluid leads to different vortex behavior during the evaporation process, as larger and more persistent vortices are generated in initially less viscous films. The presence, size and endurance of the corner vortex are highly affected by the geometry, in particular the depth of the cavity. These vortices exist at Reynolds numbers as low as Re<1e-5 indicating the absence of inertia forces; they are therefore driven by the viscous flow. However, high shear stresses caused by increasing viscosity finally destroy these vortices. Viscous flow therefore plays a curious role in this flow problem where it is necessary to generate these vortices, while it also makes them disappear.
The discovery of quanta began a revolution on how we think about the physical world and the mathematics we use to describe it. According to quantum mechanics, a particle that appears to have three co-ordinates (x, y, z) will in fact have a position described by a vector in an infinite-dimensional Hilbert space such as L^2(R^3). Quantum field theory is an attempt to apply the same principles to a system where the number of classical degrees of freedom is already infinite. We will see that this allows the description to be compatible with many phenomena from special relativity. Because of the need for a much larger Hilbert space, calculations in quantum field theory can be difficult to carry out, even numerically. This talk will work toward a derivation of the method of Feynman diagrams including a discussion of what these famous diagrams can, and cannot do. Nishant:
Nearest neighbour shifts of Finite Type are system of constrained configurations on the $\Z^d$ lattice which arise in numerous contexts,
e.g. probability, data storage, smooth dynamics and statistical mechanics.
After providing some motivation, we shall try to explore how things change when we move up in dimensions.
Note for Attendees
Pizza and pop will be provided during the break between the talks.
Given a finite group G, homotopy colimit of the classifying spaces of its abelian subgroups capture information about the commutativity in the group. For the class of extraspecial 2-groups of rank greater than 2 these colimits are not of the homotopy type of a K(\pi,1) space. The main ingreadient in the proof is the calculation of the fundamental group. Another natural question is the complex K-theory of these homotopy colimits, which can be computed modulo torsion.
Mazur observed that for a lot of elliptic curves E with non-trivial elements in Sha(E/Q)[n], one can find another elliptic curve E' that is n-congruent to E and for which the corresponding element in H^1(Q,E'[n]) lies in the image of the Mordell-Weil group of E'. Such an element in Sha is said to be made visible by E'.
It was since proved that for n=2,3, one can always find such an E', both when the n-congruence preserves Weil-pairing and when it inverts it. For given E and n=4, the question boils down to deciding if a certain K3 surface has a rational point. In joint work with Tom Fisher, we have been able to finally find equations for these K3 surfaces, which allows us to determine visibility computationally in specific cases.
Note for Attendees
* People can attend the talk via videoconferencing in room MATH 126 at UBC.
This seminar may be of interest to mathematicians and is included in the Mathematics listing of events on behalf of the Earth & Ocean Sciences Department.
Seismic imaging is an important geophysical tool for delineating and monitoring the earth’s subsurface structure and its oil, gas and mineral resources. Owing to the earth’s heterogeneity, such subsurface structures exist at different scales (sizes) with lateral and vertical variations in physical properties such as contrasts in bulk and shear moduli, and densities. Over the past decade, seismology entered a new era. Solving elastic and visco-elastic wave equations on large supercomputers, accurate and complete simulations for heterogeneous 3D earth models became a reality – replacing exact solutions for layered earth models. Seismic methods illuminate subsurface structures using compressional and shear waves. Recorded signal at surface and borehole seismic sensor locations arise from reflection, refraction, transmission, scattering and attenuation of elastic waves at lithological contacts, structural boundaries and the Earth's free surface, where abrupt and
Bernd Milkereit is the Teck Chair of Exploration Geophysics at the Department of Earth Sciences of the University of Toronto (since 2001). For more than 30 years, Bernd has worked at the intersection between mineral resources and exploration seismology. He carried out research projects on coal and groundwater before moving into hardrock seismic imaging and exploration for deep mineral deposits in the crystalline crust. He maintains a strong interest in international scientific drilling projects. Previous to his current position Bernd was a research scientist the Geological Survey of Canada (1985-1996) and professor of geophysics at Kiel University, Germany (1996-2001).
Note for Attendees
Anyone who wishes to meet with Dr. Milkereit during his visit here, please contact Miranda Joyce (mjoyce@eos.ubc.ca) or Brendan Smithyman (bsmithyman@eos.ubc.ca).
The reality of global warming has long been accepted within the scientific community, yet it remains a hotly debated topic at the political and social level. Why? Is it because the ultimate effects of global warming will not be felt within our lifetime? Do we feel little responsibility for future generations?
This talk will present historical foundations of the science of global warming. The range of projections of climate change over the next century will be summarized and the public confusion arising from the media portrayal of the science and its entry into the political arena will be discussed. A review will also be given as to how potential Canadian oil and gas production as well as international policy options fit within the framework of necessary actions required to reduce greenhouse gas emissions.
The technological solutions to mitigate global warming exist, although price and behavioural barriers to their introduction are present. As such, there has been a tendency in Canada, and more recently in BC, to entrench ourselves in the idea that our long-term economic prosperity lies within our continued export on fossil fuel extraction. I conclude the presentation with the suggestion that should society chose to deal with global warming, there is the potential to enter an age of creativity and innovation unlike any it has experienced before.
Note for Attendees
Refreshments start 15 minutes before the talk in the IAM Lounge, Room 306 of the LSK building.
The Verlinde bundles over the moduli space M_g of smooth curves have as fibers spaces of generalized theta functions i.e., spaces of global sections of determinant line bundles over moduli of parabolic bundles. I will discuss a formula for the Chern classes of the Verlinde bundles, as well as extensions over the compactification \overline M_g.
The problem of classifying off-shell representations of the $N$-extended one-dimensional super Poincar\'{e} algebra is closely related to the study of a class of decorated graphs known as Adinkras. We will discuss how these combinatorial objects possess a form of emergent supergeometry: Adinkras are equivalent to very special super Riemann surfaces with divisors. The method of proof critically involves Grothendieck's theory of "dessins d'enfants'', work of Cimasoni-Reshetikhin expressing spin structures on Riemann surfaces via dimer models, and an observation of Donagi-Witten on parabolic structure from ramified coverings of super Riemann surfaces.
This talk is concerned with Monte-Carlo methods for the estimation of the trace of an implicitly given matrix A, where the matrix information is only available through matrix-vector products. The need to estimate the trace of implicit matrices arises in many applications, where often A is symmetric positive semi-definite (SPSD). Thus, theoretical studies of accuracy and efficiency of these methods are very important. In order to set the scene, we initially present an application of such trace estimators which involves efficient stochastic methods for solving PDE-constrained inverse problems with many measurements.
The standard approach for estimating the trace of an implicit matrix involves averaging the quadratic forms of A with random vector realizations from a suitable probability distribution. We demonstrate the success of such stochastic methods in reducing the computational complexity of large scale inverse problems. We then derive new and improved theoretical results bounding the number of matrix-vector products required in order to guarantee a probabilistic bound on the relative error of the trace estimation. Bounds are derived for Rademacher (Hutchinson), Gaussian and uniform unit vector (with and without replacement) probability distributions. They provide some guidance in deciding which distribution to employ for a given application.
A common problem is to embed the complete graph on a surface so that every face is a triangle. To be perverse, suppose that we require that every triangle is a face. Let K^{(n-2)/2} denote the complete graph of order n where every pair of vertices are joined by (n-2)/2 parallel edges. For every even n at least 6 we construct a triangular embedding of this multigraph into both orientable and non-orientable surfaces such that any three vertices form a face. We give many other related results.
In the talk we discuss a recent common field in analysis and probability, the so-called Ginzburg-Landau interface models. In particular we
outline standard techniques as the Helffer-Sjoestrand PDE representation and relate them to the random field of gradients. An application of the
Helffer-Sjoestrand representation to non-convex energy functions leads to random walks having sparsely distributed negative conductances. The latter
problem is currently under intensive study and analysis and we will discuss different strategies. If time permits we will present a large deviation result
Budney recently constructed an operad which encodes splicing of knots and proved a theorem decomposing the space of (long) knots over this operad. Infection of knots (or links) by string links is a generalization of splicing from knots to links and is useful for studying concordance of knots. In joint work with John Burke, we have constructed a colored operad that encodes this infection operation. This operad captures all the relations in the 2-string link monoid. We can also show that a certain subspace of 2-string links is freely generated over a suboperad of our infection colored operad by its subspace of prime links.
In 1990s, in the process of proving the famous Moonshine conjecture, Borcherds discovered a new way to produce modular forms on a Shimura varieties of orthogonal type. Modular curves, Hilbert modular surfaces, Siegel 3-folds are all low dimensional Shimura varieties of this type. His modular form, typically called Borcherds product, has a very distinguished feature—its divisor is known. He later gave a more natural construction of these modular forms using "regularized theta lifting". This construction makes the computation of its values at CM points interesting, and gives some cool factorization formula for some very big integers. It can also be extended to prove high dimensional analogue of Gross–Zagier formula. In this talk, I will explain this story.
For the potential games there exists a potential function derived from the payoff variation of those players who modify their strategies. First we survey the general features of these games and discuss the evaluation of potential for some cases. Multi-agent spatial models will be constructed from pair interactions and studied for a specific evolutionary rule when these systems evolve into the Boltzmann statistics and become equivalent to the Ising type models used widely in solid state and statistical physics. Finally we briefly discuss the spatial version of a counter-example (matching pennies on chessboard).
Note for Attendees
Refreshments start 15 minutes before the talk in the IAM Lounge, Room 306 of the LSK building.
Title: A Proof of the Manickam-Miklos-Singhi Conjecture for Vector Spaces
Abstract: Let V be an n-dimensional vector space over a finite field.
Assign a real-valued weight to each 1-dimensional subspace in V so that
the sum of all weights is zero. Define the weight of a subspace S of V to
be the sum of the weights of all the 1-dimensional subspaces it contains.
We prove that if n >= 3k, then the number of k-dimensional subspaces in V
with nonnegative weight is at least the number of k-dimensional subspaces
in V that contain a fixed 1-dimensional subspace. This result verifies a
conjecture of Manickam and Singhi from 1988.
Joint work with Ghassan Sarkis (Pomona College) and Shahriar Shahriari
(Pomona College).
The Harvard Business Review recently called data scientist the 'sexiest job of the 21st century'. As a data scientist at Yammer, I'll give a brief overview of what data science is and what the day-to-day life of a data scientist looks like. I'll talk about some of the challenges present in our data, and try to give some insight into how we use data to better understand our users and make decisions at Yammer.
About the speaker:
Matthew is currently a data scientist at Yammer. Previously, he completed a Ph.D in mathematics at UBC, studying probability theory, and was a Fellow in the Insight Data Science Fellows Program.
About Yammer:
Yammer is an Enterprise Social Network that brings together people, conversations, content, and business data in a single location. Founded in 2008, Yammer was acquired by Microsoft Corporation in 2012 and is now part of the Microsoft Office Division.
There won't be any statistical background required, and Matthew will stay after the talk to answer any questions that people might have about data science/Yammer.
This is joint work with Tony Wong and Gjergji Zaimi. Relative BPS state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert and conjectured to be integers. For toric Del Pezzo surfaces, a proof of this conjecture will be presented.
Liquid crystals are known for their anisotropic characteristics, which lead to a preferred orientation of their molecules in the vicinity of solid surfaces. The ability of liquid crystalline materials to form ordered boundary layers with good load-carrying capacity and outstanding lubricating properties has been widely demonstrated. In order to study the advantages of implementing liquid crystals as lubricants, the steady state / time transient isothermal flow of nematic liquid crystals between two concentric / eccentric cylinders and in planar Couette geometries were studied numerically. To consider the influence of the microstructure formation / evolution on the macro-scale attributes of the flow; the Leslie-Ericksen and Landau-de Gennes theories were employed.
The simplicity of Leslie-Ericksen theory in capturing the orientational alignment angle of the molecules makes it a viable candidate for modelling the flow of flow-aligning nematic liquid crystals. On the other hand, the Landau-de Gennes nematodynamics equations are well suited for predicting texture formation since defects and disclinations are non-singular solutions of the governing equations. The Landau-de Gennes theory for the liquid crystalline microstructure along with continuity and momentum equations were solved simultaneously using General PDE and Laminar Flow modules of COMSOL Multiphysics. The investigation of flow characteristics and orientation of liquid crystalline molecules for different rotational shear rates and anchoring angles at the boundaries were presented. Furthermore, nucleation and evolution of singularities in texture of the liquid crystalline materials were tracked over the simulation time. Moreover, alterations in the macro-scale attributes of the flow such as velocity profile, pressure distribution and the first normal stress difference along with the evolution of defects were studied inside the liquid crystalline domain.
The implementation of Landau-de Gennes nematodynamic governing equations for LCs’ flow simulations offered an insight in application of these materials as lubricants. It was shown the LCs could provide protection against the wearing mechanism by forming a shielding layer in the vicinity of solid surfaces. Three-dimensional simulations of a simplified prosthetic hip joint suggested that liquid crystalline materials should be considered as potential bio-lubricants.
Nuclear fusion is a promising source of clean energy for the future, but designing an apparatus capable of fusing plasma and yielding a net energy gain has yet to be accomplished. This talk is aimed at providing an introduction to the basic physics of fusion, and outlining the methods and results obtained in analyzing the fusion reactor design of a local Canadian fusion research company.
This lunch series session will begin with a brief overview of the online homework system WeBWorK, and presentation and discussion of the large quantity of student feedback we have collected about WeBWorK through student surveys.This will be followed by an open discussion on the most effective ways to implement WeBWorK, and the benefits and challenges involved in using this system.We particularly encourage instructors who have past experience with WeBWorK to join us, so you can share your wisdom with your colleagues, as well as future instructors who are considering using this system next term.
Traditionally, we use mirror symmetry to map a difficult problem (A-model) to an easier problem (B-model). Recently, there is a great deal of activities in mathematics to understand the modularity properties of Gromov-Witten theory, a phenomenon suggested by BCOV almost twenty years ago. Mirror symmetry is again used in a crucial way. However, the new usage of mirror does not map a difficult problem to easy problem. Instead, we make both side of mirror symmetry to work together in a deep way. I will explain this interesting phenomenon in the talk. First we will give an overview of the entire story and then we will focus on the appearance of quasi-modularity.
Given U an nĂ—m matrix, the aim is to extract a large number of linearly independent columns of U and estimate the smallest and the largest singular value of the restricted matrix. For that, we give two deterministic algorithms: one for a normalized version of the restricted invertibility principle of Bourgain-Tzafriri, and one for the norm of coordinate restriction problem due to Kashin-Tzafriri. Merging the two algorithms, we are able to extract a well-conditioned block inside U, improving a previous result due to Vershynin. We give some applications of this result to the study of contact points of a convex body.
Let d ≥ 2 be an integer and let ft(z) := (zd+t)/z be a family of rational maps indexed by an algebraic number t. We study the variation of the canonical heights for this particular family. This is joint work with Dragos Ghioca.
During the last thirty years, there have been a great deal of interactions between mathematics and physics. During these interactions, mathematicians are often impressed by the magic formulas and conjectures physicists can come up with. At the same time, it is frustrating that we can not come up with similar formulas and conjectures on our own. In this talk, I will illustrate how to apply a simple physical principal (quantization principal) to discover several new areas of mathematics.
In their famous paper in 1994, Bershadsky, Cecotti, Ooguri and Vafa derived a set of equations called the BCOV holomorphic anomaly equations. The BCOV theory presents a generalization of the classical g=0 mirror symmetry (Hodge theory) and is capable of computing higher genus Gromov-Witten invariants. The key ingredient is the special Kaehler geometry of the moduli space of Calabi-Yau 3-folds. In this talk, I will explain the basic idea of the BCOV theory and show some interesting computations.
We discuss and show how to construct certain curves C of small genus satisfying the following conditions: (1) C is everywhere locally solvable; (2) the Jacobian of C is isogenous to the product of elliptic curves; (3) each elliptic curve factor in the Jacobian of C is of positive rank; (4) there is a Brauer-Manin obstruction for the failure of the Hasse principle for C. This is the joint work with Mike Bennett.
The deformation theory of holomorphic symplectic manifolds and their Lagrangian submanifolds is well known to be well behaved, thanks to the influence of Hodge theory. We will report on recent work extending some of these results to certain Poisson manifolds and their Lagrangian-like submanifolds, using mixed Hodge theory.
The numerical simulation results in the three sections are presented based on the finite element method as below:
1. Two particles interactions in Oldroyd-B fluid under confined shear are demonstrated showing the three behavior types "pass", "return" and "tumbling" in functions of initial separation and Weissenberg number. The single fibre dynamics is also presented with the slow motion showing the vorticity directional orientation in Oldroyd-B fluid and shear thinning fluid with Gisekus model.
2. The particle hydrodynamic effect on the mass transfer in the convection-diffusion problem are presented at Re=0 and Re=finitie. First, at Re=0, it is found that the disturbed streamlines on the convection enhance the mass transfer inducing the larger spectrum of the mass transfer area. Second, at Re>0, it is shown that the shear thickening phenomena by a inertial particle can promote the mass transfer on increasing effective viscosity. However, when Re_p is over critical value, it is also seen that inertial clustering particles being migrated in the region of low-vorticity by centifure-out can interrupt the mass transfer.
3. The mass transfer during the buoyant CO2 bubble absorption in a particle suspension is presented in terms of the experimental and computational studies. In both case, mass transfer enhancement (MTE) based on the mass transfer coefficient by the particle suspension is found. Especially, in numerical study, it is seen that the enhanced mass flux is highly influenced by the particle hydrodynamic interaction on the bubble surface. This can lead to declare a possible mechanism for MTE on a gas-absorption in a suspension.
In this talk, the briefly highlighted results are introduced in each section, not being included the numerical detail.
Natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. This conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from MRI to quantum state tomography. It is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? We explore this question under modern models of low-dimensionality and measurement acquisition.
Abstract: 1-bit compressed sensing combines the dimension reduction of compressed sensing with extreme quantization -- only the sign of each linear measurement is retained and thus the signal is truly compressed. Is this sufficient information to reconstruct the signal?
Behind this question lies a geometric question about random hyperplane tessellations. Picture a subset K of the unit sphere, as in the continents on our planet. Now slice the sphere in half with a hyperplane, and then slice it several times more, thus cutting the set K into a number of sections. How many random hyperplanes are needed to ensure that all sections have small diameter? How is the geodesic distance between two points in K related to the number of hyperplanes separating them? We show that a single geometric parameter, the mean width of K, governs the answers to these questions. This approach to binary data gives a new perspective on related statistical models such as logistic regression.
Global warming is one of the serious issues facing mankind today. In order to resolve the issue, numerous options have been proposed among which carbon dioxide (CO2) storage underground is the most promising method in hand. However, this method does not sound appealing to industry because of its non-profit nature. CO2 Enhanced Oil Recovery (CO2-EOR) is an oil extraction technique which can be combined with CO2 storage to not only increase the amount of recovered oil, but also store a huge amount of CO2 deep underground. In CO2-EOR, carbon dioxide acts as a solvent to reduce the oil viscosity and hence increase the oil production. However, the accurate impact of CO2 on the oil flow behavior needs to be investigated more thoroughly. In this presentation, the experimental and theoretical study of the reservoir oils flow properties in the absence and presence of CO2 will be discussed. Moreover, a novel methodology in measuring the diffusion coefficient in the gas-oil systems is presented.
An isoradial graph is a planar graph in which each face is inscribable into a circle of the same radius. We study perfect matchings on a bipartite isoradial graph, obtained from the union of an isoradial graph and its interior dual graph. Using the isoradial graph to approximate a simply-connected domain bounded by a simple closed curve, by letting the mesh size go to zero, we prove that in the scaling limit, the distribution of height is conformally invariant and converges to a Gaussian free field.
Abstract: We use stacky compactifications of ALE spaces and a theory of framed sheaves on them to give rigorous definitions of partition functions for supersymmetric gauge theories on ALE spaces. We use the notion of root stack to be able to incorporate instantons on the ALE spaces that have nontrivial holonomy at infinity. Joint work with F. Sala, and M. Pedrini and R. Szabo.
A lattice model is a probability measure on configurations of a graph parametrized by a continuous variable. The critical parameter is the parameter where the phase transition occurs, i.e., when the macroscopic properties of a lattice model change sharply with respect to the parameter. I will talk about three different lattice models including percolation, Ising model and self-avoiding walk, as well as recent progress on identifying the exact values of their critical parameters or bounding their critical parameters. Part of the talk is based on joint work with Geoffrey Grimmett.
We will discuss some topics on the theory of algebraic cobordism from joint work with Kalle Karu. We will explain the computation via envelopes and the descent exact sequence. We review the relation of algebraic cobordism with Chow and K-theory. We will also discuss an extension of the cobordism rings of smooth varieties to the singular setting via a well-behaved operational bivariant theory. As an example, we describe the operational equivariant cobordism of toric varieties.
Nature has solved the problem, and now engineers would like to -- how can we design small components to spontaneously form more complicated structures? To answer this question theoretically requires a way to describe the configuration space and assembly pathways of components with given interactions. For many systems of interest (e.g. colloids) these interactions are very short-ranged compared to the size of the components, so traditional approaches to energy landscapes struggle to capture the relevant dynamics. We propose a new framework to look at particles with short-ranged interactions and illustrate with several applications, such as computing the free energy landscape and transition rates for clusters of spheres, experimentally measuring the hydrodynamic interactions between colloids, and enumerating rigid packings of hard spheres.
Note for Attendees
Refreshments will be served before the colloquium in the IAM Lounge.
We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to hyperbolic partial differential equations, symmetric polynomials and the corresponding random growth models. The talk will be devoted to these recent developments which also shed new light on some beautiful old examples of intertwinings. Based on joint works with Vadim Gorin and Soumik Pal.
Note for Attendees
Refreshments will be served at 3:45 pm in the MATH 125 Lounge.
Advances in programming languages have changed the field of scientific computing in the past decades. Especially Mathworks' Matlab with its easy-to-use interface and huge potential for simulation, linear algebra, optimization, and visualization have dramatically changed the way we explore and teach numerical methods. While Mathworks has been dominating the market for scientific computing software almost unrivaled for quite a while, different open-source projects provide increasingly attractive alternatives today.
In a non-technical fashion, this talk will review Mathworks' contributions, discuss some of its limitations, and summarize first experiences with the open source software Python and Julia.
Systems of diffusion processes (particles) with rank-based interactions have been studied heavily due to their importance in stochastic portfolio theory and the intriguing relations with particle systems appearing in statistical physics. We study the behavior of this particle system as the number of particles gets large. By obtaining a large deviations principle (LDP), we will show that the limiting dynamics can be described by a porous medium equation with convection, whereas paths of finite rate are given by solutions of appropriately tilted versions of this equation. This is the first instance of an LDP for diffusions interacting both through the drift and the diffusion coefficients. Based on joint work with Amir Dembo, S. R. Srinivasa Varadhan and Ofer Zeitouni.
Hydraulic fracturing (HF) is a process where the material, such as rock, is cracked by a pressurized fluid. Among many applications of HF, the most common use is the stimulation of production from oil and gas wells. To prevent fracture from closing after the pressure is reduced, the propping agents, such as sand, are pumped together with the fracturing fluid. The problems of fluid-driven fracture propagation and flow of the suspensions have been studied extensively, but not many works combine both and address the fracturing caused by a slurry. To fill the gap, the aim of this study is to develop a computational model for calculating the propagation of a fracture induced by the mixture of the viscous fluid mixed with the spherical particles. First, the empirical constitutive law for the slurry is used to obtain the solution for the steady flow of the viscous fluid mixed with spherical particles in a channel. This solution is then used to formulate the conservation laws for the slurry and the particles, which govern the propagation of hydraulic fractures and the proppant transport inside them. The developments are applied to two fracture geometries -- Khristianovich-Zheltov-Geertsma-De Klerk (KGD) and pseudo-3D (P3D). Numerical simulations show that the proposed method allows to capture the proppant plug formation and growth, as well as the gravitational settling for both geometries. Dimensionless parameter, whose magnitude reflects the intensity of the settling, is introduced for the P3D fracture. Arising counterintuitive consequence, stating that the viscosity of the fluid can not directly affect the settling extent, is verified through the numerical examples.
Dr. Dontsov received his undergraduate degree in Physics from Novosibirsk State University in Russia, 2008. Then, in 2012, he graduated with PhD degree in Civil Engineering (and minor in Math) from the University of Minnesota, USA. During his PhD studies, he was doing nonlinear acoustic wave propagation from both theoretical and computational points of view with applications to medical imaging. After a short postdoc at the University of Minnesota, he joined Professor Anthony Peirce (Dept. of Math, UBC) in April 2013 and started to work on hydraulic fracturing.
Ecological competition between strains of a pathogen occurs when strains compete for hosts -- either for susceptible hosts, host resources during co-infection, or the ability to re-infect hosts. Competition is important because when strains compete with each other, intervening against only some of them can pave the way for rises in others. This has happened, for example, with the introduction of polyvalent vaccines against Streptococcus pneumoniae. However, detecting ecological competition between strains of an infection is very challenging, because competition is by its nature revealed over relatively long periods of time and is a population-level phenomenon which we would not expect to observe in small-scale studies. Even population-level dynamical (ODE) models, which are frequently used in such situations, are hard to formulate and calibrate. Indeed, such models often make hidden assumptions about competition, rather than aiding in its estimation. I have therefore been motivated to ask: can sequence data for pathogens allow us to detect ecological competition? Large and rich datasets of pathogen gene sequences are now available, due to the development of next-generation sequencing; perhaps they can be of assistance if appropriately linked to models with and without competition. Here, I present a dynamical model in which there is a competition parameter which ranges continuously from 0 (where pathogen strains are independent of each other) to 1 (where competition is complete, and strain dynamics show competitive exclusion). It predicts that the branching rates in phylogenies for competing strains should be anti-correlated. A stochastic implementation of the model gives rise to pathogen phylogenies that are quantitatively different, both in their structures and their branch lengths, from phylogenies without competition. This leads to a distinct profile for a phylogeny under ecological competition: such trees have high imbalance early in the tree, greater topological distances from the root to the tips, lower widths and a characteristic skew in inter-branch distances, among other properties. I analyse a phylogeny of within-host HIV sequences and show that it fits the profile of ecological competition. I conclude with a discussion of other organisms and future directions for this work.
Note for Attendees
Refreshments at PIMS (ESB 4th floor) from 2.40 or so.
Wolfgang Schmidt's Strong Subspace Theorem is a less well-known generalisation of his Subspace Theorem, and has not been studied much since its formulation in 1980. It is a result about integer points in parallelepipeds whose successive minima satisfy a certain condition. Thus, unlike the Subspace Theorem and its other generalisations, it falls within the field of the geometry of numbers.
This self-contained talk reintroduces the Strong Subspace Theorem and describes some new preliminary results, including a stronger reformulation of the theorem in dimension two.
Infections are an increasingly important challenge worldwide: new pathogens are emerging, and old ones are adapting to evade our control strategies. Recent dramatic improvements in sequencing technology have made many large and rich datasets of pathogen gene sequences available. In principle, these data can shed light on how pathogens are spreading, evolving, and interacting with each other -- information that is crucial for controlling infections. Such genomic data are usually interpreted with the aid of phylogenetic trees. Inferring a phylogeny from data is a mature field with decades of mathematics behind it. But comparing phylogenies from different sets of data, and linking the properties of phylogenies (and the associated data) to how pathogens spread, evolve and interact is a field in its infancy. In this talk I will briefly introduce the field, describing the data and the need for novel mathematical approaches. I will outline two recent contributions. In the first, we developed a new Bayesian method to infer a who-infected-whom tree from sequence data from an outbreak, and we applied it to whole-genome sequence data of tuberculosis. The method uses a novel mapping from lineages to hosts to find a nice decomposition of the key likelihood term into rapidly-computable simple parts, and is thus flexible and quick. In the second contribution, we develop novel summaries of phylogenetic trees and use them to classify trees according to the kind of transmission dynamics taking place. I will describe my research program in this emerging field, whose advancement calls for the engagement of several areas of mathematics including applied probability, discrete mathematics, geometry and topology.
Note for Attendees
Refreshments will be served at 2:45 pm in the MATH 125 Lounge.
The marriage of mathematics and epidemics has a long and distinguished history with a plethora of successes that go back to the work of Daniel Bernoulli (1700 – 1782) and Nobel Laureate and physician Sir Ronald Ross (1911) and associates. These individuals, mostly physicians, created the field of theoretical/mathematical epidemiology in their efforts to meet their commitment to diminish health disparities; the consequences of poverty and the lack of access to health services. The last four decades have seen deep and extensive computational and theoretical advances in the fields of computational, mathematical and theoretical epidemiology and the connections of this theoretical research to public health policy and security have had significant impact. These advances have been driven by the dynamics of specific emergent or re-emergent diseases including HIV, influenza, SARS and Tuberculosis as well as by bioterrorism concerns. Challenges and opportunities arise from the demands generated by the study of disease dynamics over multiple time scales and levels of organization and by the search for response to questions of importance to the fields of public health, homeland security and evolutionary biology. In this lecture, I will revisit some of the history of the field and discuss selected applications in the context of slow and fast diseases; highlight the differences between single and recurrent outbreaks and related issues.
A fundamental notion in statistical mechanics is phase transition: a microscopic system composed of a huge number of random particles depends on a thermodynamic parameter, and the system undergoes sudden changes in its large-scale structure as this parameter varies across a critical point.
Self-avoiding walk was introduced in the 1940s as a model in chemistry of a long chain of molecules, and is now viewed as a fundamental model in the rigorous theory of statistical mechancs. By introducing a positive parameter which provides a penalty to self-avoiding walk which is exponential in the walk's length, we obtain an example of phase transition. Recent work with Hugo Duminil-Copin shows that uniformly chosen self-avoiding walks of given high length move sub-ballistically, and this is related to the nature of this phase transition at the critical point. I will give an overview of the main elements of the proof.
Considering instead subcritical self-avoiding walk, and focussing on the planar case, we obtain a natural model for the problem of phase separation: when one substance is suspended in another, such as oil in water, a droplet forms, whose boundary approximates a smooth profile predicted by Wulff. Modelling the problem using a planar model such as subcritical self-avoiding walk, the fluctuation of the droplet boundary from its typical profile exhibits characteristic scaling exponents - 2/3 longitudinally and 1/3 latitudinally - which I derived a couple of years ago. The behaviour arises from a competition of local Gaussian randomness and global curvature constraints.
Phase separation in this guise is a static model. However, the Gaussian competition with curvature, and the two exponents, are shared by many dynamic models, of interfaces growing at random and subject to forces of surface tension. These models form the Kardar-Parisi-Zhang universality class. Resampling techniques from the phase separation papers find counterparts in more recent work, joint with Ivan Corwin, in which a natural Gibbs property is proved for the multi-line Airy process, which is a fundamental scaling limit encountered in KPZ universality. This Brownian-Gibbs property is valuable in, for example, improving regularity assertions about the Airy process. These ideas form the subject of the final part of the talk..
The KPZ equation, introduced by Kardar, Parisi and Zhang, is a stochastic PDE that models randomly evolving interfaces that are subject to constraining forces such as surface tension. It is anticipated to be a universal object, in the sense that many microscopic models will share the KPZ equation as an accurate asymptotic description of their late time behavior. This view is supported by extensive numerical evidence, recent experimental evidence involving liquid crystal instabilities, and a limited but growing body of mathematically rigorous work.
In recent work arXiv:1312.2600 with Ivan Corwin, we present a new technique for the analysis of the KPZ equation. The solution to the equation is represented as the lowest indexed curve in an N-indexed ensemble of curves, which we call a KPZ line ensemble. Curves within the ensemble enjoy a natural invariance under resampling, the H-Brownian Gibbs property, which property has the effect of energetically penalizing, but not absolutely forbidding, the crossing of adjacently indexed curves. This property is inherited from the O'Connell Yor semi-discrete continuum random polymer ensemble after a limiting procedure is applied.
The H-Brownian Gibbs property is an integrable one, in the sense that the precursor O'Connell-Yor ensemble is known to enjoy it by virtue of this ensemble's algebraic structure. However, it also offers a powerful probabilistic tool for the analysis of the KPZ equation. Since the solution of this equation is embedded in a KPZ line ensemble, we may analyse it using the H-Brownian Gibbs property, and in this way derive significant new estimates regarding the regularity and local structure of the KPZ solution. As I will aim to explain, these new estimates are valid uniformly in the time parameter for the KPZ equation, even after a natural rescaling of the equation is undertaken which accesses the fluctuation behavior of the KPZ evolution.
Let (M,g) be a compact Riemannian Manifold without boundry of dimension n \geq 3, x_0 \in M, and s \in (0,2). We let \crit: = \frac{2(n-s)}{n-2} be the critical Hardy-Sobolev exponent. I investigate the influence of geometry on the existence of positive distributional solutions u \in C^0(M) for the critical equation
\Delta_g u+a(x) u=\frac{u^{\crit-1}}{d_g(x,x_0)^s} \;\; \hbox{ in} \ M.
Via a minimization method, I prove existence in dimension n\geq 4 when the potential a is sufficiently below the scalar curvature at x_0. In dimension n=3, using a global argument, i prove existence when the mass of the linear operator \Delta_g + a is positive at x_0. On the other hand, by using a Blow-up around x_0, i prove that the sharp constant of the related Hardy-Sobolev inequalities on (M,g), which is equal to the one of the Euclidean Hardy-Sobolev inequalities, is achieved for all compact Riemannian Manifold of dimension n \geq 3 with or without boundary.
I will survey the general phenomenon of "integrable" probabilistic models, in which the presence of explicit formulas describing their distributions allow for an analysis by essentially algebraic methods.
Then I will discuss in detail an integrable probabilistic model of randomly tiling a hexagon drawn on the regular triangular lattice by lozenges of three types (equivalent formulations: dimer models on the honeycomb lattice, or random 3D stepped surfaces glued out of 1x1x1 boxes). This model has received a significant attention over the past 20 years (first results - the computation of the partition function - date back to P. MacMahon, 100+ years ago). Kenyon, Okounkov, and their co-authors (1998-2007) proved the law of large numbers: when the polygon is fixed and the mesh of the lattice goes to zero, the random 3D surface concentrates around a deterministic limit shape, which is algebraic. I will discuss finer asymptotics: local geometry, behavior of interfaces between phases (which manifests the Kardar-Parisi-Zhang universality), and global fluctuations of random surfaces (described by the Gaussian Free Field), as well as dynamical models associated with random tilings.
From a target space perspective, T-duality relates certain pairs of string theory backgrounds with a U(1) isometry. If we perform a Kaluza-Klein reduction on the corresponding circle, T-duality then acts as a symmetry of the reduced theory, and this symmetry can be argued to constrain the higher derivative couplings, which in turn constrains the couplings of the higher dimensional theory. I will explain an unsophisticated brute force implementation of this procedure and show how it can be used to completely fix the four-derivative action of type II O-planes coupling to NS-NS sector bulk fields.
In an Exposé published in 1968, Alexander Grothendieck generalized the notion of a central simple algebra over a field by defining Azumaya algebras in locally ringed topoi. Specific examples of Azumaya algebras in locally ringed topoi include (up to isomorphism) principal PUn or POn bundles on CW complexes, as well as Azumaya algebras over commutative rings. If the topos is connected, it is possible to define two invariants of an Azumaya algebra, the period & the index, which measure the nontriviality of the algebra. It is a classical theorem that the period & index have the same prime divisors in the case of central simple algebras over a field, and this is also known in the case of PUn bundles over CW complexes. In this talk, I will show that in the case of any locally ringed topos, the period & index have the same prime divisors.
Fouling control through air sparging in membrane systems is governed by the hydrodynamic conditions in the system and the resulting shear stress induced onto membranes. However, the relationship between hydrodynamic conditions and the extent of fouling control is not well understood. As a result, sparging approaches are designed using a capital and time intensive empirical trial-and-error approach that does not guarantee that optimal conditions are identified. To address this knowledge gap, the present research focused on characterizing the hydrodynamic conditions in a membrane system under different sparging conditions (bubble size and frequency) and on finding a correlation between the induced hydrodynamic conditions and fouling control efficiency.
New concepts of "zone of influence" of bubbles and "power transferred" were defined to characterize the hydrodynamic conditions in membrane systems. A non-homogenous fouling distribution was observed in the zone of influence of bubbles due to a non-homogenous distribution of velocity and shear stress in this zone.Fouling rates generally decreased with an increase in the area of the zone of influence, the root mean square of shear stress induced onto membranes and the rise velocity of bubbles. However, none of these parameters on their own could accurately describe the effect of the hydrodynamic conditions on fouling rate. On the other hand, power transferred onto fibers, which incorporates the effect of all the three parameters, could more effectively describe the effect of the hydrodynamic conditions on the rate of fouling.Power transfer efficiency into the system, defined as the ratio of power transferred onto membranes to the power input in the system, was used to identify optimal sparging approaches. For all cases investigated, the power transfer efficiency to the system was consistently much higher for pulse bubble than for coarse bubble sparging. The results from the present research are currently being used by the industry partners, GE Water and Processes Technologies, for the design of the air sparged membrane systems.
Sepideh Jankhah has a PhD in Civil and Environmental Engineering, She did her BSc and MSc in Chemical Engineering. She worked on developing a new catalyst for fuel cells during her MSc degree. Her PhD project focused on characterizing the hydrodynamic conditions in membrane filtration systems using high speed-high resolution camera/Particle Image Velocimetry (PIV) and Electrochemcial Diffusion (EDM) Shear probes.
Inspired by the invariant of a number field given by its Dedekind zeta function we define the notion of weak arithmetic equivalence, and we show that under certain ramification hypothesis this equivalence determines the local root numbers of the number field. This is analogous to a result Rohrlich on the local root numbers of a rational elliptic curve. Additionally we prove that for tame non-totally real number fields the integral trace form is invariant under weak arithmetic equivalence.
Since the end of 1990's there has been a significant progress in understanding the long time nonequilibrium behavior of certain integrable (1+1)-dimensional interacting particle systems and random growth models in the Kardar-Parisi-Zhang (KPZ) universality class. The miracle of integrability in most cases (with the notable exception of the partially asymmetric simple exclusion process) can be traced to an extension of the Markovian evolution to a suitable (2+1)-dimensional random growth model whose remarkable properties yield the solvability. So far, there have been two sources of such extensions. The first one originated from a classical combinatorial bijection known as the Robinson-Schensted-Knuth correspondence (RSK, for short) in the works of Johansson, O'Connell and their co-authors. The second approach introduced by Borodin-Ferrari was based on an idea of Diaconis-Fill of extending intertwined "univariate" Markov chains to a "bivariate" Markov chain that projects to either of the initial ones.
In a recent joint work with A. Borodin, we presented a way to unify these two approaches using a fairly general framework of Macdonald processes. This also provides new examples of integrable KPZ particle systems.
A theorem of Ryan and Wolper states that a type A Schubert variety is smooth if and only if it is an iterated fi bre bundle of Grassmannians. We extend this theorem to arbitrary finite type, showing that a Schubert variety in a generalized flag variety is rationally smooth if and only if it is an iterated fi bre bundle of rationally smooth Grassmannian Schubert varieties. The proof depends on deep combinatorial results in Coxeter groups. This is joint work with William Slofstra.
I will describe a certain class of hyperbolic dynamical systems called Smale spaces, which arose from Smale's program of smooth dynamics. I will give a number of different examples. The goal of the talk is to provide algebraic invariants for these systems, whose existence was first conjectured in the 1970's by Rufus Bowen. I will try to describe the motivation for these invariants and also how their construction parallels others in algebraic topology.
Nested dissection ordering and its graph partitioning generalization give rise to an ordered sequence of separators of (roughly) geometrically decreasing size. The fill-ins in the LU factors are confined in the parts of the matrix associated with the separators. In particular, the diagonal blocks associated with the separators are fully dense in the factors and they contribute to the dominant terms in the costs of the storage for the factors and the flops for computing the factors. Employing the data-sparse representations or compressions, such as low-rank approximation, for these separator blocks can drastically lower the overall factorization costs both in memory and flops. The low rankness can appear in many matrices from discretized PDEs.
Recently, we have been investigating fast and stable algorithms for one type of data-sparse representation called hierarchically semi-separable (HSS) structure, and using them in the sparse LU factorization. In this talk, we will show that both in theory and in practice, the HSS-embedded sparse LU factorization has much lower complexity than the traditional factorization algorithm. The complexity of this class of algorithms is closely related to the numerical ranks of the separator blocks which vary with the PDEs. For elliptic problems, we can achieve large amount of compression and the resulting factorization can be used as a nearly optimal-order direct solver. For wider classes of problems, the approximate factorization can be used as a preconditioner. We will show performance results from direct solvers as well as preconditioners for a wide range of problems. We will also illustrate the potential of such solvers/preconditioners being used for the extreme-scale computers and problem size.
This is joint work with A. Napov, F.-H. Rouet, S. Wang, and J. Xia.
This week's reading group will focus on the famous "Moore Method" of teaching mathematics, introduced by Robert Lee Moore. The discussion will centre around the following two (very short) articles:
Jones, F. Burton, 1977, "The Moore method," American Mathematical Monthly 84: 273-77.
Cohen, David W., 1982, "A modified Moore method for teaching undergraduate mathematics", American Mathematical Monthly 89(7): 473-474, 487-490.
We consider the nonlinear fourth-order problem \Delta^2 u=|u|^{p-1}u\ \ \mbox{in} \ \mathbb R^n, where p>1 and n\ge1. We give a complete classification of stable and finite Morse index solutions in the full exponent range. We also compute an upper bound of the Hausdorff dimension of the singular set of extremal solutions. A key tool is a new monotonicity formula.
Abstract: I will discuss homeomorphisms of the Cantor set with dense orbits. I will describe a simple algebraic invariant for such systems which leads to a complete classification of the systems up to orbit equivalence. The key ingredient for this result is a model, called the Bratteli-Vershik model. Finally, I will briefly described how these dynamical systems have been used to provide new examples in group theory.
The Gaussian Noise Stability of a set A in Euclidean space is the probability that for a Gaussian vector X conditioned to be in A, a small Gaussian perturbation of X will also be in A. Borel's celebrated Isoperimetric inequality states that a half-space maximizes noise stability among sets with the same Gaussian measure. We will present a novel short proof of this inequality, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: We show that the deficit between the noise stability of a set A and an equally probable half-space H can be controlled by a function of the distance between the corresponding centroids. As a consequence, we prove a conjecture by Mossel and Neeman, who used the total-variation distance.
In an Exposé published in 1968, Alexander Grothendieck generalized the notion of a central simple algebra over a field by defining Azumaya algebras in locally ringed topoi. Specific examples of Azumaya algebras in locally ringed topoi include (up to isomorphism) principal PUn or POn bundles on CW complexes, as well as Azumaya algebras over commutative rings. If the topos is connected, it is possible to define two invariants of an Azumaya algebra, the period & the index, which measure the nontriviality of the algebra. It is a classical theorem that the period & index have the same prime divisors in the case of central simple algebras over a field, and this is also known in the case of PUn bundles over CW complexes. In this talk, I will show that in the case of any locally ringed topos, the period & index have the same prime divisors.
Robust optimization is a paradigm for dealing with optimization problems whose input is uncertain. For instance, consider building a system (bridge, network, etc.) whose future demand (or stress) is unknown but bounded. Say we know that the set of possible demands will come from a given convex region called the "universe''. A solution/design is called robust if it supports any demand from the universe. Robust optimization asks for a minimum cost robust solution. We consider this model for the design of communication (data) networks where future demand can be both unknown and varying over time. We will discuss in detail one variant of robust network design known as the Virtual Private Network (VPN) problem. We will also mention several open (theoretical and practical) questions.
An instance of the edge-disjoint paths problem (EDP) consists of a graph G and some demand node pairs s_i,t_i, i=1,2, \ldots k. A subset I \subseteq [k] of the demand pairs is called routable if there exists edge-disjoint paths in G which satisfy (connect) each pair s_i,t_i with i \in I.
We consider two optimization problems associated with EDP. The maximum edge-disjoint paths problem (MEDP) asks one to find a maximum size routable subset. The minimum congestion problem asks for the minimum integer \alpha such that if we make \alpha copies of each edge, we can route all demands. We discuss the associated streams of work on approximating these problems, highlighting some recent advances and outstanding open problems. We will emphasize both the parallels and the key distinctions between the two streams.
The arrival of massive amounts of data from imaging, sensors, computation and the internet brought with it significant challenges for information science. New methods for analysis and manipulation of big data have come from many scientific disciplines. The first focus of this presentation is the application of ideas from PDEs, such as variational principles and numerical diffusion, to image and data analysis. Examples include denoising, segmentation, inpainting and texture extraction for images. The second focus is the development of new ideas in information science, such as wavelets, soft-thresholding, sparsity and compressed sensing. The subsequent application of these ideas to PDEs and numerical computation is the third focus of this talk. Examples include wavelet analysis for turbulent flows, the use of soft-thresholding in computation of PDEs with multi-scale features, and the construction of “compressed modes” (modes that are compactly supported in space) for density functional theory and other PDEs that come from variational principles.
Note for Attendees
Refreshments start 15 minutes before the talk in the IAM Lounge, Room 306 of the LSK building.
If X is a variety over a field, the Chow groups of X are defined in terms of closed subvarieties of X and form a kind of cohomology theory for X. Higher Chow groups, defined by Spencer Bloch in the 1980s in terms of subvarieties on X x A^m and now related to the theory of motivic cohomology, extend the theory of Chow groups, and may be nonzero even when the ordinary Chow groups vanish. I will explain how the higher Chow groups of GLn are related to a 'suspension' of the ordinary Chow groups of projective space. Time permitting, I will conjecture how this relationship might extend to 'suspensions' of the Chow groups of Grassmanians.
In this talk we present some joint work in collaboration with C. Castro (UPM, Madrid), R. Lecaros (CMM- Chile) and F. Palacios (Stanford) on flow control.
We address a classical optimal control problem of inverse design, aiming to identify the initial source leading to a desired final configuration.
First, in the one-dimensional case, we explain why classical strategies, based on linearization methods, fail, because of the lack of regularity of solutions. We then introduce an alternating descent method that exploits the generalized gradients that take into account the sensitivity of the smooth arcs of the solutions but also of shock locations.
We compare the performance of the method with classical purely discrete strategies through various numerical experiments.
We also address the multi-dimensional case and point towards perspectives of future development.
In this lecture we address the problem of the optimal placement of sensors and actuators for wave propagation problems.
Using Fourier series representation the problem can be recast as a spectral optimal design problem, involving all the spectrum of the Laplacian.
We show that, depending on the complexity of the data to be observed/controlled, several scenarios have to be distinguished. Those in which the solution is a classical set constituted by a finite number of simply connected subdomains, others in which the optimal set is of Cantor type and those leading to relaxation phenomena.
We also explain how closely this topic is related to the fine properties of the high frequency behavior of the eignefunctions of the Laplacian which is intimately linked to the ergodicity properties of the dynamical system generated by the corresponding billiard.
We shall also discuss the same problem for heat processes showing that, in that frame, according to intuition, the problem is governed by a finite number of Fourier modes.
These results will be illustrated by numerical simulations.
The lecture is conceived for a general audience and unnecessary technicalities will be avoided. It is based on recent joint work in collaboration with Y. Privat and E. Trélat from UMPC, Paris.
(Vector-valued) modular forms appear all over string theory. Often, one wants to know whether the Fourier coefficients are integers, or if it is already known they must be integral, one would like to know consequences. In my talk I'll describe the most effective tool for these sorts of questions: p-curvature.
We study interacting particle systems which can model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. The hydrodynamic limit and the fluctuation limit for the particle densities can be described, respectively, by a coupled Partial Differential Equation (PDE) and a Gaussian process solving a Stochastic Partial Differential Equation (SPDE). New tools of discrete approximations to reflected diffusions will be discussed. (Joint work with Zhen-Qing Chen)
The curve complex is a finite dimensional, locally infinite, unbounded and 17-hyperbolic simplicial
complex associated with surfaces. The intricate relationship between the curve complex of surfaces
embedded in 3-manifolds and the topology and geometry of the manifolds will be discussed in the talk.
Aerosols as a colloidal system of solid or liquid particles in a gas are significant two-phase systems which have an important rule in environmental, biological, health, industrial and other scientific studies. According to these important rules, some instruments namely “Aerosol Spectrometer” have been developed to measure aerosols’ characteristics and specifically size distribution of the sample particles. Optical Particle Counters (OPCs) are one of these instruments which are able to count and measure the particles’ size distribution in a specific aerosol sample. In the current study, an optical particle counter has been designed mechanically and optically. To validate our design, both optical and mechanical operation of the design was simulated. In this case the Mie theory was used for optical simulation and prescription of the scattered light by particles and the optical parts were redesigned using geometrical optic principals and simulations’ results to achieve an optimum sketch. Fluid flow was simulated by ANSYS Fluent and particles were tracked by means of a particle tracking UDF code in Lagrangian approach. Since the difference in particles’ refractive indices can cause wrong measured size, a novel method based on measurement of scattered light in different angular intervals was presented. This new method was checked by Mie theory simulation and its functionality was investigated. At last, a new sizing method by calculating intensity of collected light in CMOS sensors was studied experimentally under highly controlled conditions. This study showed that there is a good agreement between Mie theory and experimental study results for particles with bigger than 15 micron diameter.
Farzad received his undergraduate degree in Mechanical Engineering from Bu-Ali Sina University, (Iran), 2010. Then, in 2013, he graduated with MSc degree in Mechanical Engineering (Thermofluids) from Sharif University of Technology. He started his PhD career in UBC in January 2014 under supervision of Prof. Gwynn Elfring.
We calculate the logarithmic correction to the decay of the critical two-point function for networks of p mutually-avoiding weakly self-avoiding walks joining two distant points on the 4-dimensional integer lattice. While similar results have been obtained previously for dimensions d > 4 by lace expansion, our proof is based on a rigorous renormalisation group analysis of a representation of the self-avoiding walk as a supersymmetric field theory.
The talk is based on joint and ongoing work with Roland Bauerschmidt, David Brydges and my supervisor Gordon Slade.
The Masur criterion for Teichmuller geodesics relates the geometry of the Teichmuller space and random walks on the mapping class group of a surface to dynamical properties of vertical foliations of quadratic differentials. A major problem in the study of outer automorphism group OUT(F) of a nonabelian free group has been to find an analog for the Masur criterion. We discuss difficulties and explain our approach to this problem. We also mention applications of this result particularly in describing the space of random walks on the group of OUT(F). This is joint work with Alexandra Pettet and Patrick Reynolds.
In this talk, I'll introduce some basics of stochastic differential equations, used for modelling dynamical systems that are subject to stochastic effects. Typically, stochastic differential equations have a Gaussian white noise term that introduces the randomness into the dynamics for the purposes of capturing processes that are difficult (or undesirable) to quantify.
After this, I will discuss including a stochastic forcing known as \alpha stable noise into the dynamics. This noise process is used for incorporating stochastic effects with infinite variance into dynamical systems, however obtaining analytical results is significantly more challenging since moments of \alpha stable distributions do not exist beyond first order (in general). I will give some results from my stochastic averaging research project and try to offer some interesting questions to attendees.
This talk should be moderately accessible and will be on Beamer slides
In this talk, we will discuss the Math Educational Resource Wiki. Formerly called the Math Exam Resource, this wiki, located at http://wiki.ubc.ca/Science:Math_Education_Resources which redirects to http://wiki.ubc.ca/Science:Math_Exam_Resources has evolved into much more than just an exam solution database. We will first briefly demonstrate the wiki, then present usage statistics, recent enhancements, future ideas as well as advantages and limitations of the wiki platform. We will also discuss ways to supplement your current teaching with material from the wiki and where you, as an instructor, would like to see this resource go. We will also consider how this wiki can be enhanced to further facilitate teaching first year calculus courses.
I will outline a construction of an exotic solution of the nonlinear Schrödinger equation that exhibits a big frequency cascade. Recent advances related to this construction and some open questions will be surveyed.
Ideas from harmonic analysis and dynamical systems have led to spectacular advances in the understanding of Hamiltonian partial differential equations. This colloquium will survey some of these developments and highlight issues at the research frontier.
Note for Attendees
Refreshments will be served at 2:45 p.m. in the MATH 125 lounge.
The talk will describe a recent mathematical/computational model for understanding aspects of cancer metastasis, initiated when primary tumor cells enter the vasculature and lymphatic system becoming circulating tumor cells (CTC's). The model is based on a Markov chain dynamical system designed so that the transition matrix has as it's steady-state a `target' vector obtained from an autopsy data set. The target vector, chosen for a given type of primary cancer (i.e. lung), contains the distribution of metastatic tumors from an ensemble population. The transition matrix can be associated with a metastatic network (directed graph) with disease progression modeled as a random walker on the network. We focus on primary lung cancer, and using the metastatic network obtained from the transition matrix, we quantify (probabilistically) the most probable disease progression pathways, along with mean first-passage times of progression.
We highlight the multi-directional nature of the progression pathways that the model produces (self-seeding and re-seeding) which supports recent experimental observations carried out at Memorial Sloan-Kettering Cancer Center on the importance of primary tumor self-seeding. We also will describe how we use the concept of metastatic entropy to compare the complexity of different cancer types. The work is sponsored by the National Cancer Institute under the auspices of The Scripps Research Institute Physical Sciences Oncology Center `The Physics and Mathematics of Cancer Metastasis'. If time permits, we will finish with a brief overview of related projects modeling the fluid phase of cancer.
Note for Attendees
Refreshments start 15 minutes before the talk in the IAM Lounge, Room 306 of the LSK building.
Most inverse problems are ill-posed due to the fact that inputs such as parameters, physics and data are missing or inconsistent. This results in solution estimates that are not unique or unstable, i.e. small changes in the inputs result in large changes in the estimates. One common approach to resolving ill-posedness is to use regularization methods whereby information is added to the problem so that data are not over-fitted. Alternatively, one could take the Bayesian point of view and assign a probability distribution to the unknowns and estimate it by exploiting Monte Carlo techniques.
In this work we take the regularization approach and use uncertainties to weight added information and data in an optimization problem. This allows us to apply statistical tests with the null hypothesis that inputs are combined within their uncertainty ranges to produce estimates of the unknowns. For example, the Discrepancy Principle can be viewed as using a chi-squared test to determine the regularization parameter.
The chi-squared method developed by myself and colleagues uses a chi-squared test similar to the Discrepancy Principle, but differs in that the test is applied to the regularized residual rather than the data residual. This approach leads to a general methodology of using statistical tests to estimate regularization parameters or uncertainties in an inversion. I will give statistical tests for nonlinear algorithms and show results from benchmark problems in Geophysics. I will also describe how statistical tests can be used to find a regularization parameter for Total Variation and show results from Imaging.
We show how to lift positive Ricci and almost non-negative curvatures from an orbit space M/G to the corresponding G-manifold, M. We apply the results to get new examples of Riemannian manifolds that satisfy both curvature conditions simultaneously. This is joint work with Fred Wilhelm.
The non-commutative symmetric functions and quasi-symmetric functions
are the second and third examples of a combinatorial Hopf algebra that
one encounters (the first being the symmetric functions). In recent
years there have been at least two bases proposed as an analogues of
the Schur functions and they are in addition to the
"ribbon=funadmantal^*" basis. I'll list properties that we would want
these bases to have as analogues of the Schur functions and then
explain some computational results that tell us what is possible
(surprisingly, it is not possible to have it all!). I will also
discuss some symmetric function positivity open problems that we hope
these bases will resolve.
Given a finite dimensional unitary representation of a compact Lie group G one constructs the so-called symplectic quotient.This is given by the space of G-orbits in the zero fibre of the moment map. The symplectic quotient is stratified by symplectic manifolds and can be understood as a semi-algebraic set. We will present results related to the isomorphism problem of symplectic quotients. In particular, we will be concerned with the question when a symplectic quotient is symplectomorphic to a finite unitary quotient.
In order to construct moduli spaces in algebraic geometry, one typically must specify a notion of semi-stability for the objects one wishes to parameterize. To the objects that are omitted, the unstable objects, one can often associate a real number which measures "how unstable" that object is. In fact we can think of the moduli stack of all objects as stratified by locally closed substacks corresponding to objects of varying degrees of instability. The key examples of this phenomenon are the Kempf-Ness stratification of the unstable locus in GIT and the Shatz stratification of the moduli of G-bundles on a smooth projective curve. I will discuss a framework for describing stability conditions and stratifications of an arbitrary algebraic stack which provide a common generalization of these examples. Time permitting, I will discuss how some commonly studied moduli problems, such as the moduli of K-stable varieties and the moduli of Bridgeland-semistable complexes on a smooth projective variety, fit into this framework. One key construction assigns to any point in an algebraic stack a potentially large topological space parameterizing all possible `iso-trivial degenerations' of that point. When the stack is BG for a reductive G, this recovers the spherical building of G, and when the stack is X/T for a toric variety X, this recovers the support of the fan of X.
In this talk I will describe results obtained in two recent articles in collaboration with Xinyi Li concerning large deviations for the occupation-time profile in a large box of Z^d, d≥3, for random interlacements at a given level, and the probability that random interlacements disconnect a macroscopic body from infinity, when the vacant set is in the percolative regime.
I will attempt to give an overview of the Gross-Siebert program, emphasizing its guiding principles and their connection to the Strominger-Yau-Zaslow conjecture.
This talk will discuss a version in A^1 homotopy theory of the classical EHP sequence of James and Toda using the simplicial suspension map. This is joint work in progress with Ben Williams.
Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point in X. We show that the set containing all positive integers n such that the n-th iterate of x under f lands in Y is a union of at most finitely many arithmetic progressions along with a set of Banach density 0. This is joint work with Jason Bell and Tom Tucker.
This is the first in a series of talks on the papers:
(1) D. Weitz, Counting independent sets up to the tree threshold,
STOC'06, 2006
(2) A. Sly, Computational transition at the uniqueness threshold,
FOCS'10, 2010
Both papers study the hard-core model, a model for a lattice gas.
The hard-core model on a graph G=(V,E) is a probability measure on
the independent sets of G, that is, the subsets I of V with
the property that no two vertices of I are neighbours. The
probability of an independent set I is proportional to
lambda^|I|, where lambda is a positive parameter. The
problem of interest is computing the partition function Z of the
model, that is, the sum of lambda^|I| over all choices of I.
In the Computer Science literature, this is known as the problem of
counting (weighted) independent sets. Since this is well-known to be
computationally hard, one allows for algorithms that
compute Z only approximately, and also restrict attention to
graphs with degree bounded by a constant d. The combined results
of (1) and (2) show that there is a phase transition for the
computational hardness of this problem: Z can be approximated in
polynomial time (with respect to the size of G) if and only if
lambda is below a threshold lambda_c(d). Moreover, lambda_c
is also a threshold value for a Statistical Mechanical phase
transition, namely it separates the regimes of uniqueness and
non-uniqueness of Gibbs measures for the hard-core model on
d-regular trees. In this first, introductory talk, we will explain
the model, briefly survey the literature and cover the
pre-requisites concerning the hard-core model on trees.
The dichotomy between sparse and dense structures is one of the profound, yet fuzzy, features of contemporary mathematics and computer science.
We present a framework for this phenomenon, which equivalently defines sparsity and density of structures in many different yet equivalent forms, including effective decomposition properties. This has several applications to model theory, algorithm design and, more recently, to structural limits.
Note for Attendees
Coffee, tea and cookies served at 2:30pm in the PIMS Lounge.
Phase retrieval is the century-old problem of reconstructing a function, such as a signal or image, from intensity measurements, typically from the modulus of a diffracted wave. Phase retrieval problems - which arise in numerous areas including X-ray crystallography, astronomy, diffraction imaging and quantum physics - are notoriously difficult to solve numerically. They also pervade many areas of mathematics, such as numerical analysis, harmonic analysis, algebraic geometry, combinatorics, and differential geometry. I will introduce a novel framework for phase retrieval, which comprises tools from optimization, random matrix theory, and compressive sensing. In particular, we will see that for certain types of random measurements a function, such as a signal or image, can be recovered exactly with high probability by solving a convenient semidefinite program without any assumption about the function whatsoever and under a mild condition on the number of measurements. Our method, known as PhaseLift, is also provably stable vis-a-vis noise. I will describe how this approach carries over to the classical phase retrieval setting using structured random illuminations. I conclude with some open problems.
Note for Attendees
Refreshments start 15 minutes before the talk in the IAM Lounge, Room 306 of the LSK building.
A famous result of Springer says that the Weyl group of a reductive algebraic group G (over C) acts on the cohomology of the subvariety X_u of the flag variety G/B consisting of the flags fixed by a unipotent u in G. This result was unexpected since W does not act on X_u itself. Recently, Kumar - Procesi and Goresky - MacPherson showed that Springer's action lifts to the equivariant cohomology of X_u with respect to the maximal torus in C_G(u) for so called parabolic unipotents u with the proviso that the cohomology morphism j*: H*(G/B) \to H*(X_u) is surjective. In this talk we will describe the parabolic unipotents for which j* is surjective and indicate a direct proof of lifting.
Particle Image Velocimetry (PIV) is a technique using successive laser images of particles immersed in a fluid to measure the velocity field of the fluid flow. Traditionally, cross-correlation is employed to extract the field from each pair of recorded images. This talk will introduce a new approach based on Optimal Transport (OT) to approximate the velocity field. More specifically, we consider the solution of the L2 OT problem with initial and final densities given by successive images of tracers. We will first present a model for this situation and investigate the behaviour of the OT map with respect to the model's key parameters. Then, we will present some algorithms and numerical results applying this theory to synthetic and real examples. This is joint work with B.Khouider and M.Agueh.
Uniform infinite maps arise as local limits of uniformly chosen finite maps. Recently there has been keen interest in creating hyperbolic analogues of the uniform infinite planar maps. It is conjectured that uniformly distributed maps on surfaces with genus linear in the number of vertices should converge in the local topology to hyperbolic versions of uniform infinite maps. I will describe several models of such hyperbolic random maps which arise from several directions. Finally I will describe some recent results obtained for unicellular maps in high genus. Partly joint work with Omer Angel, Guillaume Chapuy and Nicolas Curien.
There are many analogies between the outer automorphism group of a free group Out(F) and the mapping class group of a surface Mod(S). I'll explain how each of these groups contains many right-angled Artin subgroups and how these subgroups can be used to understand the structure of both Mod(S) and Out(F). Interestingly, attempting to understand the properties of elements in right-angled Artin subgroups also reveals some major differences between Out(F) and Mod(S). I'll explain these differences and how they affect the study of Out(F).
Yield stress fluids have the property that they do not deform unless a given yield stress is exceeded. While in some flows this leads to unwanted features, this property can also be exploited in order to produce novel flow features. One example of such flows are visco-plastically lubricated (VPL) flows, in which a yield stress fluid is used to stabilize the interface in a multi-layer flow, far beyond what might be expected for a typical viscous-viscous interface. Here we extend this idea by considering the encapsulation of droplets within a visco-plastic fluid, for the purpose of transportation, e.g. in pipelines. The main advantage of this method, compared to others that involve capillary forces is that significantly larger droplets may be stably encapsulated, governed by the length scale of the flow and yield stress of the encapsulating fluid. We explore this setup both analytically and computationally. We show that sufficiently small droplets are held in the unyielded plug of the Poiseuille flow. As the length or radius of the droplets increase the carrier fluid eventually yields, potentially breaking the encapsulation. We study this process of breaking and give estimates for the limiting size of droplets that can be encapsulated.
Amir joined UBC for his master studies in the field of mechanical engineering in 2012. He is now working under supervision of Dr. Ian Frigaard in the complex fluids lab. He got his undergrad in 2011 from Sharif University of Technology.
We construct Weyl group multiple Dirichlet series associated to root systems of type C, through a combinatorial recipe involving Gelfand-Tsetlin patterns. These Dirichlet series are associated with an n-fold metaplectic cover of SO(2r+1) and we prove functional equations for them when n=1, via the Casselman-Shalika formula. We also prove that our description matches the so called "stable case," as described for general root systems by Brubaker, Bump and Friedberg. This is joing work with Jennifer Beineke and Ben Brubaker.
I will talk about one of the simplest and most important functors of algebraic topology, the fundamental group, which gives an algebraic object obtained from the loops on a space.
In this talk, we will discuss modular forms, what they are and why they are amazing. Topics we will discuss include equalities of values of \sigma_{n} (sums of powers of divisors), the sum of four squares theorem, a discussion on Ramanujan's constant, an aside on polynomials with many prime terms, and perhaps even a bit of moonshine. Little will be completely proved but this talk will leave you in awe and wonder of this powerful concept.
Network Science is a rapidly growing interdisciplinary area at the intersection of mathematics, computer science, and a multitude of disciplines from the natural and life sciences to the social sciences and even the humanities. Network analysis methods are now widely used in proteomics, in the study of social networks (both human and animal), in finance, in ecology, in bibliometric studies, in archeology, and in a host of other fields.
In this talk I will introduce the audience to some of the mathematical and computational problems and methods of complex networks, with an emphasis on the basic notions of centrality and communicability. More specifically, I will describe some of the problems in large-scale numerical linear algebra arising in this area, and how they differ from the corresponding problems encountered in more traditional applications of numerical analysis.
The talk will be accessible to students, requiring only a modest background in linear algebra and graph theory.
Note for Attendees
Refreshments start 15 minutes before the talk in the IAM Lounge, Room 306 of the LSK building.
I shall present results of work with Arinkin and Hablicsek which allow us to understand the Frobenius push-forward of the de Rham complex as the structure sheaf of a twisted derived intersection. Similar considerations also apply for twisted de Rham complexes, yielding results which have applications in singularity theory and in the study of matrix factorizations. Using our theorems we recover and strengthen earlier results of Deligne-Illusie, Barannikov-Kontsevich, and Sabbah. Our approach gives a new point of view on recent works of Joyce et al, generalizing results of Behrend, on understanding the holomorphic Fukaya-Floer homology.
We shall see how the positivity of some Schr\"odinger operator on a surface gives information on its topology and its conformal type. The potent of the operators considered here involve the curvature of the surface and appear naturally in the study of minimal and constant mean curvature surfaces. It is a joint work with Pierre B\'erard.
Based on some recent joint work of J. Lewis, and others, we formulate a variation of the Beilinson-Hodge conjecture pertaining to varieties defined over the complex numbers. In this talk, we explain the motivation for this conjecture, and some evidence in support of it.
We outline recent work uncovering intriguing connections between Otto's characterisation of diffusion as entropicgradientflowon one hand andlarge-deviationprinciples describing the microscopic picture (Brownian motion) on the other. Specifically, we connect macroscopicgradientflowswithlargedeviationprinciples, and point out the potential of a bigger picture emerging: we indicate that in some non-equilibrium situations, entropies and thermodynamic free energies can be derived vialargedeviationprinciples. The approach advocated in the talk is different from the established hydrodynamic limit passage but extends a link that is well known in the equilibrium situation.
The Discontinuous Galerkin (DG) method is very suitable for studying flows with mesh and accuracy adaptation due to weakly imposition of inter-element continuity. Moreover, DG provides sufficient stability for high-order finite-element schemes by direct implementation of flux upwinding. Here a robust, high-order and fast DG solver was developed for compressible flows.For increasing the accuracy, Hermitian super-parametric curved elements were designed for high-order boundary representation. For increasing the convergence rate, a Newton-Krylov algorithm was developed for pseudo-time advancement of the semi-discrete system. Local time-step adjustment based on flow history was introduced to stabilize the solution especially in startup iterations. To maximize the computational efficiency, the developed code was parallelized on the state-of-the-art Graphics Processing Unit (GPU) hardware with multiple layer of parallelism. According to compact nature of DG discretization, the fast block-tridiagonal solvers have been adapted recently both for structured and unstructured grids. A parallel Block Cyclic Reduction (BCR) linear solver was developed to work with DG on GPU. The developed solver has an improved speedup comparing with previous works due to optimizations in on-chip memory bandwidth and floating-point arithmetic rate. The proposed GPU-DG solver showed that it can be efficiently used in a wide range of high-order fluid flow simulations.
Dr. Baghapour finished his PhD at University of Tehran in the field of mechanical engineering in 2014. He worked under the supervision of Dr. Vahid Esfahanian in Vehicle, Fuel and Environment Research Institute (VFERI). He also got his MSc degree under the supervision of Dr. Mehdi Ashjaee in Laser Diagnostic laboratory of University of Tehran. He is now a research assistant at Simon Fraser University working on hydrodynamics.
This talk will retrace the main steps of the modern theory of prime numbers and in particular how the combinatorial sieve combined with the Dirichlet series theory to give birth to the modern representation of the primes via a linear combination of terms, some of which being "linear", while the other ones are "bilinear". This will lead us to the recent developments of Green & Tao, Mauduit & Rivat, Tao, Helfgott, and Bourgain, Sarnak & Ziegler.
Let G be a semisimple algebraic group over an algebraically closed field. A nonabelian free subgroup H is called strongly dense if every nonabelian subgroup of H is Zariski dense in G. We will discuss some recent results (with Breuillard, Green and Tao) regarding the existence of strongly dense subgroups and mention applications of these results to the Banach-Hausdorff-Tarski paradox, generation of finite simple groups of Lie type and the existence of families of expander graphs associated to finite groups of Lie type of fixed rank. If time permits, I will discuss some improvements with Breuillard and Larsen and the possibility of proving a generalizations of the Tits alternative.
The classification of finite simple groups is of fundamental importance in mathematics. It is also one of the longest and most complicated proofs in mathematics. We will very briefly discuss the result and a bit of history and then explain how it can and has been used to solve problems in many areas. We will end with mentioning some specific and perhaps surprising consequences in various fields.
Note for Attendees
Coffee, tea and cookies served at 2:30pm in the PIMS Lounge.
We will present a joint work with Young-Hoon Kiem on using family Chern-Simons charts to construct perverse sheaves on moduli of sheaves that gives a categorification of Donaldson-Thomas invariants.
Hydraulic fracturing (HF) is a process where the material, such as rock, is cracked by a pressurized fluid. Among many applications of HF, the most common use is the stimulation of production from oil and gas wells. To prevent fracture from closing after the pressure is reduced, the propping agents, such as sand, are pumped together with the fracturing fluid. The problems of fluid-driven fracture propagation and flow of the suspensions have been studied extensively, but not many works combine both and address the fracturing caused by a slurry. To fill the gap, the aim of this study is to develop a computational model for calculating the propagation of a fracture induced by the viscous fluid mixed with the spherical particles. First, the empirical constitutive law for the slurry is used to obtain the solution for the steady flow of the viscous fluid mixed with spherical particles in a channel. This solution is then used to formulate the conservation laws for the slurry and the particles, which govern the propagation of hydraulic fractures and the proppant transport inside them. The developments are applied to two fracture geometries -- one-dimensional Khristianovich-Zheltov-Geertsma- De Klerk (KGD) and pseudo-3D (P3D). Numerical simulations show that the proposed method allows to capture the proppant plug (the region of compacted sand) formation and growth, as well as the gravitational settling for both geometries. Calculations with different proppant size demonstrate that bigger particles tend to settle faster, but, at the same time, promote fluid filtration through the plug, which supports further fracture propagation. Another problem that is addressed in this presentation is generation of proppant schedule. Certain proppant distribution inside the fracture is often desired at the end of pumping, however, only special proppant pumping schedule may lead to this. A new methodology of proppant schedule generation is introduced. It avoids solving an inverse problem and, at the same time, is more accurate than the commonly used procedure. One of the biggest advantages of the proposed approach is its compatibility with numerous HF simulators, making it a universal tool for generating a proppant schedule.
The format of mathematics instruction in post-secondary contexts has remained problematic for undergraduate learners. Thus, the need to address the teaching practices of current and future teachers of post-secondary mathematics has become apparent. In preparing mathematics graduate students for their teaching responsibilities, many mathematics departments offer their new teaching assistants a one-time training session that addresses issues such as how to create quizzes, deal with disciplinary issues, and other practical matters. Beyond these shallow experiences, mathematics graduate students receive little support or mentorship for the development of their teaching practices. In this presentation, I will talk about three topics: my past research with mathematics graduate students, my current work on the creation of professional development experiences and support for mathematics graduate students, and my future work on the evolution of a deeper approach to cultivating mathematics graduate students’ teaching practices.
A preconditioned variant of the Golub-Kahan bidiagonalization process recently proposed by Arioli and Orban allows us to establish that SYMMLQ and MINRES applied to least-squares problems in symmetric saddle-point form perform redundant work and are combinations of methods such as LSQR and LSMR. A well-chosen preconditioner allows us to formulate a projected variant of the Golub-Kahan process that forms the basis of specialized numerical methods for linear least-squares problems with linear equality constraints. As before, full-space methods such as SYMMLQ and MINRES applied to the symmetric saddle-point system defining the optimality conditions of such problems perform redundant work and are combinations of projected variants of LSQR and LSMR. We establish connections between numerical methods for least-squares problems, full-space methods and the projected and constraint-preconditioned Krylov methods of Gould, Orban and Rees.
In the d-dimensional bin packing problem (VBP), one is given vectors x1, x2, …, xn in Rd and the goal is to partition them into a minimum number of "feasible" sets. A set is feasible if the sum of its vectors does not have a component exceeding 1. Online VBP refers to the case where the vectors arrive sequentially and an algorithm must try to create these feasible sets on the fly. This problem has received renewed interest due to its relevance to placing virtual machines in a cloud platform.
The competitive ratio for an online algorithm is an upper bound on its worst case performance against an adversary which tries to choose a difficult sequence of incoming vectors. It had been outstanding for almost 20 years to clarify the gap between the best lower bound W(1) on the competitive ratio for online VBP versus the best upper bound of O(d). We settle this by describing a W (d / log d) lower bound. We also present several remaining open questions in the area.
The n-component phi^4 model is a ferromagnetic continuous-spin model with interesting critical behaviour. In particular, the one-component model is predicted to be in the same universality class as the Ising model. We study the n-component model on the 4-dimensional integer lattice, for all n greater than or equal to 1, with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent (n+2)/(n+8) for the logarithm.
We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat hasfractional logarithmic scaling for n=1,2,3; double logarithmic scaling for n=4; and is bounded when n>4. In addition, for the model defined on the 4-dimensional discrete torus, we prove that the scaling limit near the critical point is a multiple of the Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity equal to the susceptibility.
The proofs are based on a rigorous renormalisation group method.
This is joint work with Roland Bauerschmidt and David Brydges.
I will discuss the recent work(-in-progress) on unifying various mirror constructions of various authors, such as Batyrev-Borisov and Berglund-Hübsch-Krawitz. This talk hopes to focus on questions, conjectures, and examples involved in this more generalized framework. This talk hopes not to focus on the difficulty of using a SmartBoard for seminars.
I will discuss some types of problems where techniques from algebraic topology have led to successful resolutions of open problems in algebraic geometry. Then, I will outline several future directions where better knowledge of the topology of classifying spaces of compact Lie groups could lead to more results in algebra.
Vector graphics software, such as Adobe Illustrator or Inkscape, typically use a data-structure made of paths independent from each other. This makes representing incidence relationships difficult.
With Rémi Ronfard and Michiel van de Panne, we developed a more topology-oriented data-structure, made of vertices, edges and faces, that I will first present. Then I will show how this combinatorial structure is in fact a presentation scheme whose geometric realization is (to the best of our computer scientist knowledge) a new kind of two-dimensional "cell" complex, that we call Point-Curve-Surface complex (PCS complex). We are currently trying to prove that every two-dimensional simplicial complex has a unique minimal PCS complex decomposition.
This short talk will focus on the interplay between number theory and dynamical systems by briefly describing some interesting yet simple to state questions.
Let l and p be distinct odd primes unramified in an imaginary quadratic extension K/Q. We outline the proof of the non-triviality of the p-adic formal group logarithm of Heegner points modulo p associated to the Rankin-Selberg convolution of an elliptic modular form of weight two and a theta series over the Zl-anticyclotomic extension of K. We also make remarks regarding the analogous non-triviality of generalised Heegner cycles.
ESB 2012 (PIMS building) note the time and location change
Fri 21 Mar 2014, 3:30pm-4:30pm
Abstract
Complex Monge-Ampere (CMA) equation is of fundamental importance in Kahler geometry. We will discuss regularity results for two versions of complex Monge-Ampere equation which are extensively studied in Kahler geometry. The first is the classical CMA equation solved by S.T. Yau in 1970s to prove the Calabi conjecture. The second is a homogenous complex Monge-Ampere, which is known as a geodesic equation of the space of Kahler metrics.
The goal of the talk is to present derived category techniques to study holomorphic symplectic varieties. In particular, we study and answer the following questions:
(1) the Hassett-Tschinkel Conjecture on the structure of the Mori cone of curves;
(2) the Bogomolov-Tyurin-Hassett-Tschinkel-Huybrechts-Sawon Conjecture on the existence of Lagrangian fibrations;
(3) the Kawamata-Morrison Cone Conjecture.
Irreducible Holomorphic Symplectic varieties (IHS for short) are simply connected projective manifolds endowed with a unique (up to scalars) holomorphic symplectic form; K3 surfaces are the lowest dimensional example. In this talk we concentrate on IHS of K3^[n]-type, namely IHS deformation equivalent to the punctual Hilbert scheme on a K3 surface. After giving a short introduction to the basics of IHS theory, we will present recent joint work with Arend Bayer on how to prove (1), (2), and (3) for moduli spaces of sheaves on K3 surfaces, by using derived categories and Bridgeland stability. If time permits, I will also sketch how to extend these results to all IHS of K3^[n]-type, as recently proven by Bayer-Hassett-Tschinkel, Mongardi, Matsushita, Markman-Yoshioka, and Amerik-Verbitsky.
We determine the stability threshold for a periodic arrangment of localized spots for some singularly perturbed two-component reaction-diffusion systems including the Gierer-Meihardt, Schnakenburg, and Gray-Scott models, in R^2. In the semi-strong interaction asymptotic limit where only one of the components has an asymptotically small diffusivity, the leading order stability threshold governing amplitude instabilities of the spots, as derived by Wei-Winter (2001, 2003), is independent of the arrangement of the spots in the lattice. By combining a spectral approach based on Floquet-Bloch theory together with the method of matched asymptotic expansions and appropriate Fredholm solvability conditions, we calculate the next order term in the expansion of the stability threshold in terms of the regular parts of certain Green's functions. In this way, we derive an asymptotic result for the location of a real-valued band of continuous spectrum for the linearized operator when a stability parameter is close to its critical value. This result depends on an objective function defined in terms of both the Bloch wavevector and the particular lattice arrangment. From a numerical min-max optimization of this objective function it is shown that a regular hexagonal lattice of localized spots is the most stable. Joint work with Juncheng Wei (UBC), David Iron (Dalhousie) and John Rumsey (Dalhousie)
The dimer method is a simple hessian-free algorithm for computing index-1 saddles. In this talk, I will review and analyze a few variants of this algorithm, focusing on some improvements to its efficiency, in particular adding preconditioning capabilities and line-search based on a local merit function. I will demonstrate the efficiency of the new variant on a range of applications from academic toy problems, an atomistic problem and a PDE problem.
Despite these new improvements, we can currently give no global convergence guarantee. Indeed, we can construct counterexamples to global convergence. I will conclude my talk by explaining some of the difficulties we encountered.
We will describe a class of two directed and friendly walks along the
square lattice that are restricted to the upper half-plane. One
motivation for studying this class is that it can be used to model an
idealised thermodynamic system of a DNA strand in a solvent near an
attractive surface. We will solve the combinatorial problem of finding
a closed-form expression for the number of $N$ paired walks that
contain $L$ lower-walk-to-$x$-axis and $M$ lower-walk-to-upper-walk
shared site contacts.
In this talk we will define a space built out of all the commuting n-tuples in a Lie group and discuss its role as a classifying space for commutativity. Applied to the unitary groups this gives rise to an infinite loop space and the notion of commutative K-theory. We will also provide computations for the rational cohomology in terms of mult-symmetric invariants. This is joint work with Jose Gomez.
Let F be a totally real number field. We consider the problem to show how congruences between the Fourier coefficients of a Hilbert eigenform over F and a Hilbert Eisenstein series over F (of the same parallel weight (k,...,k)) give rise to corresponding congruences between the algebraic parts of the special values of the associated L-functions. In the case F = Q, the congruences of L-functions were obtained by Vatsal (k=2), Heumann--Vatsal (k≥2), and the speaker (k≥2). In this talk, we generalize Vatsal's work to the case F is a real quadratic field and k=2.
In recent years it has become increasingly clear that stochasticity plays an important role in many biological processes.Examples include bistable genetic switches, noise enhanced robustness of oscillations, and fluctuation enhanced sensitivity or “stochastic focusing"..Numerous cellular systems rely on spatial stochastic noise for robust performance.We examine the need for stochastic models, report on the state of the art of algorithms and software for modeling and simulation of stochastic biochemical systems, and identify some computational challenges.
Note for Attendees
Coffee, tea and cookies served at 2:30pm in the PIMS Lounge.
The solution of vector-valued Cahn–Hilliard systems is of interest in many applications. We discuss strategies for the handling of smooth and nonsmooth potentials as well as for different types of constant mobilities. Whereas the use of smooth potentials leads to a system of parabolic partial differential equalities, the nonsmooth ones result in variational inequalities. Concerning the latter, we propose a Moreau–Yosida regularization technique that incorporates the necessary bound constraints. As a result, the variational inequalities are replaced by nonsmooth equations. Due to the use of fully implicit time discretizations, which are the most accurate, we have to solve in every time step nonlinear smooth or nonsmooth equations. This is done by standard Newton methods in the smooth case, and by semismooth Newton methods in the nonsmooth case. At the heart of both methods lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an effective Schur complement approximation. Numerical results illustrate the efficiency of our approach. In particular, we numerically show mesh and phase independence of the developed preconditioner in the smooth case. The results in the nonsmooth case are also satisfying and the preconditioned version always outperforms the unpreconditioned one. (Joint work with Martin Stoll.)
We will introduce a whole range of problems related to random permutations whose motivation goes back to the Bose-Einstein condensation in quantum statistical mechanics. After reviewing standard probabilistic approaches we will introduce a Gibbs formalism for random bijections of the planar integer lattice. Under certain energy assumptions we show the existence of Gibbs measures and discuss possible characterisation of different phases and address the problem of finite and infinite cycles in bijections of the planar integer lattice.
From the standpoint of basic fluid dynamics, katabatic winds are buoyantly driven boundary-layer-type flows along heated or cooled sloping surfaces in a stratified fluid. However, understanding their structure is of interest not only as a fundamental problem in itself, but also from a meteorological point of view, because of the broad range of areas and scales that they cover, influencing from local valleys micro-climate (e.g. over Salt Lake and Phoenix valleys) to synoptic scale motions (e.g. over ice sheets in coastal regions in Antarctica).
In katabatic flows turbulence is generated by shear and destroyed by negative buoyancy and viscosity. Because of this interplay between shear and buoyancy effects the strength of turbulence in the stable boundary layer that arise is much weaker, in comparison to the neutral and convective boundary layers, and this feature, together with the intrinsic complex dynamics of the system (e.g. occurrences of intermittency, Kelvin-Helmholtz instability, gravity waves, low-level jets and meandering motions) and the lack of any similarity theory, pose heavy burdens on numerical simulations.
The presentation will provide a brief overview of the state-of-the-art in numerical modeling of slope flows to then focus on recent numerical analyses, under idealized settings, which somehow resemble Prandtl's original model (1942) - an early milestone in the conceptual understanding of slope flows.
A modified set of filtered Boussinesq equations are solved on a regular domain relying on an operator-splitting technique to decouple the system. A mixed pseudo-spectral and finite difference approach is adopted in space and the fully explicit second-order accurate Adams-Bashforth scheme is used for time advancement. Closure of the equations is achieved through first order algebraic Smagorinsky models.
A statistical analysis of the initial oscillatory transient and on the properties of the steady state solution will be presented for a given subset of the parameter space, followed by an eduction of the coherent structures populating the flow. The behavior of Smagorinsky-type subgrid-scale models for such systems will also be discussed.
Here is a little intro about Marco:
- 2008 --> B.A. in Civil Engineering (University of Padua - Italy)
- 2010 --> M.A. in Civil Engineering (University of Padua - Italy), Thesis at the International Center for Numerical Methods in Engineering (CIMNE), Barcelona
- 2012 --> PhD candidate in Mechanical Engineering at École Polytechnique Fédérale de Lausanne (EPFL), at the EFLUM Lab. (Environmental Fluid Mechanics).
The unit spheres in orthogonal representations of finite groups give examples of group actions on spheres. We investigate non-linear actions by studying chain complexes over the orbit category, and constructing finite G-CW complexes. This leads to new examples of homotopy representations with isotropy of rank one. This project is joint with Ergun Yalcin (Bilkent University, Ankara).
The standard framework for studying representation theory of p-adic Lie groups is that of reductive groups over a p-adic field K. In this talk I will describe ongoing work with Clifton Cunningham and Takashi Suzuki where we instead work with limits of group schemes over the residue field of K. In particular, I will describe a sheaf-function dictionary for quasicharacters of tori over local fields, and early progress toward a definition for the affine Grassmannian for reductive groups over K.
In order to construct the moduli space of canonical polarized manifolds, three different stability conditions have been introduced, namely, KSBA-stability, K-stabilty and asymptotic GIT stability. In this talk, we try to explore the relations among them. In particular, any canonical polarized manifold is stable with respect to all three conditions above, however the compactifications they give are different. As a consequence, we answer a longstanding question by showing that asymptotically GIT Chow semistable varieties do not form a proper family.
We will first describe the conjecture of 'Quantum Unique Ergodicity' which is also known as 'Quantum Chaos'. This problem is in the intersection of Dynamical Systems, Harmonic Analysis, PDE, Differential Geometry and Mathematical Physics, and therefore attracts almost all branches of mathematics. Secondly, we will see why number theorists got immensely interested in this problem which is apparently coming from a different field of mathematics. We will also give a brief description of Lindenstrauss' ground-breaking work on this problem in a special case, for which he got Fields Medal in the ICM 2010.
Note for Attendees
Notice the special time and place. Last seminar of the academic year. Sushi will be served.
We will introduce the basic notions of first order model theory, the study of first order theories and their structures. Many familiar classes of mathematical objects are first order theories, such as groups, fields, ZFC, some theories of arithmetic, and more. I will present some basic theorems and present an interesting result, Skolem's Paradox.
Note for Attendees
Notice the special time and place. Last seminar of the academic year. Sushi will be served.
A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In a modern language, it can be shown that the special groups are those of essential dimension zero. In 1958 Grothendieck classified special groups in the case where the base field k is algebraically closed. In this talk I will explain some recent progress towards the classification of special reductive groups over an arbitrary field. In particular, I will give the classification of special semisimple groups, special reductive groups of inner type and special quasisplit reductive groups over an arbitrary field k.
Note for Attendees
Note the unusual time of the seminar. This is in order to avoid a clash with the PIMS-CRM Fields lecture at 3pm.
It is well known that Einstein's general theory of relativity provides a geometrical description of gravity in terms of space-time curvature. Einstein's theory poses some fascinating and difficult mathematical challenges that have stimulated a great deal of research in geometry and partial differential equations. Important questions include the well-posedness of the evolution problem, the definition of mass and angular momentum, the formation of black holes, the cosmic censorship hypothesis, the linear and non-linear stability of black holes and boundary value problems at conformal infinity arising in the analysis of the AdS/CFT correspondence. I will give a non-technical survey of some
significant advances and open problems pertaining to a number of these questions.
Note for Attendees
There will be a reception at 2:30pm in the PIMS Lounge.
I will describe a project to classify all smooth 4-dimensional manifolds triangulable with 6 or less 4-dimensional simplices. In the process we have found many simple triangulated 2-knot exteriors, forming a strong analogy with 3-manifold theory.
The traditional framework in the atmospheric boundary layer for relating turbulent fluxes of momentum, heat, and scalar quantities to their mean gradients, called Monin-Obukhov similarity theory (MOST) after its originators, can be viewed as an extension of law of the wall scaling to account for the effects of thermal stratification. Although MOST is the standard framework for interpreting atmospheric measurements and modeling turbulent fluxes in weather and climate models, a number of fundamental issues in MOST still are not well understood. Because MOST arises from dimensional analysis, the connections between the curves that relate turbulent fluxes to mean gradients and fundamental properties of turbulence (e.g. the spectra, integral scales, and TKE budget) are not well understood. Furthermore, although experimental data often indicate deviations from MOST, the cause of these deviations (experimental error vs. physical processes) remains an open question.
In this presentation, a theoretical framework to connect MOST curves to fundamental properties of turbulence will be introduced. Experimental data will be used to demonstrate the effects of buoyancy on the integral length scales and their linkage to the behavior of MOST curves. Asymptotic solutions for MOST curves will also be derived for slightly unstable and free convective conditions.
In the second part of the talk, error propagation analysis and atmospheric data will be used to quantify the extent to which deviations from MOST are due to experimental errors vs. physical processes that are not represented by MOST. Deviations from MOST are found to have a strong diurnal trend, which suggest that processes related to the growth of the unstable atmospheric boundary layer remain unaccounted for in MOST.
The final part of the talk will focus on the how buoyancy and mean shear together influence the large-scale organization of the unstable atmospheric boundary layer. For slightly unstable conditions, convective updrafts organize into longitudinal rolls, aligned with the mean wind; for highly convective conditions, updrafts organize into cells, similar to Rayleigh-Benard convection. Using large eddy simulation, the transition from roll- to cellular- type convection will be examined. A transitional state between rolls and cells is observed and is characterized by oscillatory behavior in velocity statistics and convective organization. The physical processes responsible for this transition will be discussed.
Bio statement written by Scott: "I received my B.S. in Science Education from Martin Luther College in New Ulm, MN in 2008. In 2010, I received my M.S. in Meteorology from Penn State University. I will complete my Ph.D. (also in Meteorology) from Penn State University in May 2014, working with Prof. Marcelo Chamecki. My research interests include turbulence, the atmospheric boundary layer, and environmental fluid mechanics"
I will discuss some recent developments in hyperbolic geometry and geometric group theory, namely Agol's proof of the virtual Haken conjecture and Wise's theory of special groups, together with their relationship with right-angled Artin groups and mapping class groups. I will then discuss a new result which shows that every hyperbolic 3-manifold admits a finite cover whose fundamental group embeds into a braid group, and into the group of diffeomorphisms of the circle. Finally, I will exhibit some higher dimensional closed hyperbolic manifold subgroups of braid groups and of the diffeomorphism group of the circle. The research in this talk represents work joint with Hyungryul Baik and Sang-hyun Kim.
We discuss a result which shows that every right-angled Artin group quasi-isometrically embeds in a planar pure braid group. As a consequence, we obtain examples of quasi-isometrically embedded closed hyperbolic manifold subgroups of pure braid groups in all dimensions. We also give some applications to decision problems in braid group theory. This represents joint work with Sang-hyun Kim.
Microlocal compactness forms (MCFs) are a new tool to study oscillations and concentrations in L^p-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending both the theory of (generalized) Young measures and the theory of H-measures. Since in L^p-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, MCFs allow to quantify the difference between these two notions of compactness. The definition involves a Fourier variable, whereby also differential constraints on the functions in the sequence can be investigated easily. Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar's framework of compensated compactness; consequently, we establish a new weak-to-strong compactness theorem in a "geometric" way. Moreover, the hierarchy of oscillations with regard to slow and fast scales can be investigated as well since this information is also is reflected in the generated MCF.
For any stationary mZ^d Gibbs measure that satisfies strong spatial mixing, we obtain sequences of upper and lower approximations that converge to its entropy. In the case d=2, these approximations are efficient in the sense that they are accurate to within epsilon and can be computed in time polynomial in 1/epsilon. The method is extended to approximate pressure of Gibbs interactions. Joint work with Ronnie Pavlov.
In this talk, we will describe a construction of compactly supported solutions to the three-dimensional incompressible Euler equations on R \times R^3 with Hölder regularity 1/5 -\epsilon in space and time. This work extends the earlier works of De Lellis-Székelyhidi, Buckmaster-De Lellis-Székelyhidi and Isett on construction of Hölder continuous dissipative Euler flows to the non-periodic setting. Our key technical innovation is a simple method for finding a compactly supported symmetric 2-tensor with a prescribed divergence, which obeys useful bounds.
The c-Plateau problem for surfaces in space asks, given c>0 and \gamma a closed curve in space, whether we can find M_{c} a smooth orientable surface-with-boundary, with \partial M_{c} = \sigma_{c}+\gamma where \sigma_{c} is a finite union of closed curves disjoint from \gamma, minimizing c-isoperimetric mass \mathbf{M}^{c}(M) := \text{area}(M)+c \cdot \text{length}(\partial M)^{2} amongst all M smooth orientable surfaces-with-boundary, with \partial M = \sigma+\gamma where \sigma is a finite union of closed curves disjoint from \gamma. In this talk we give several regularity results for solutions to the c-Plateau problem, formulated in the more general setting of integer two-rectifiable currents.
In the remarkable paper [Phys. Rev. Lett. 96, 100503 (2006)], Dimitri Gioev and Israel Klich conjecturedan explicit formula for the leading asymptotic growth of the spatially bi-partite von-Neumann entanglemententropy of non-interacting fermions in multi-dimensional Euclidean space at zero temperature. Based onrecent progress by one of us (A.V.S.) in semi-classical functional calculus for pseudo-differential operatorswith discontinuous symbols, we provide here a complete proof of that formula and of its generalization toR\'enyi entropies of all orders \alpha>0. The special case \alpha=1/2 is also known under the namelogarithmic negativity and often considered to be a particularly useful quantification of entanglement. Theseformulas, exhibiting a ``logarithmically enhanced area law'', have been used already in many publications.
This is joint work with Hajo Leschke and Alexander V. Sobolev which will be published in Phys. Rev. Lett.
The talk will be concerned with various optimization results for the principal eigenvalues of general second-order elliptic operators in divergence form with Dirichlet boundary condition in bounded domains of R^n. To each operator in a given domain, one can associate a radially symmetric operator in the ball with the same Lebesgue measure, with a smaller principal eigenvalue. The constraints on the coefficients are of integral, pointwise or geometric types. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new symmetrization technique, different from the Schwarz symmetrization. This talk is based on a joint work with N. Nadirashvili and E. Russ.
Hilbert's Tenth Problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would decide whether there exists a solution in integers. Around 1970, Matiyasevich, building on earlier work of Davis, Putnam, and Robinson, showed that no such algorithm exists. But the answer to the analogous question with integers replaced by rational numbers is still unknown, and there is not even agreement among experts as to what the answer should be.
Note for Attendees
There will be a departmental reception and student awards from 12:00pm-1:30pm in MATH 125.
I will discuss a family of fourth order PDE's and their corresponding second boundary value problem on a bounded strictly convex domain. Associated Monge-Amp\`ere functionals will be discussed as well. Special cases here include the equation for prescribed affine mean curvature of a graph, and also Abreu's equation for prescribed scalar curvature of certain toric varieties. The talk is based on joint work with Ben Weinkove.
I'm a recent math Ph.D from UBC who is now working as a data scientist for Yammer (Microsoft) in San Francisco. I'll give a brief overview of what data science is and what the day-to-day life of a data scientist at Yammer looks like. As well, I'll talk about why I chose industry over a postdoc, the major differences between academia and industry, and what skills and talents tech companies (and my team) are looking for.
Note for Attendees
Jointly with the PIMS Young Researcher's conference http://www.math.ubc.ca/~GradComm/YRC/
The classification of p-compact groups was a great achievement towards the understanding of finite H-spaces. In this talk we will talk about the history of finite H-spaces beginning from compact Lie groups. Then we will introduce the notions of p-compact group and finite p-local H-space and give some examples. We will finish with some open questions in the area.
The classical example of a log scheme is a variety X with a normal crossing divisor D. One can study differential forms on X with logarithmic (that is, order one) poles along D. Dual to these are log tangent vectors on (X, D), which have "zeroes along D." As ordinary jet schemes generalise tangent spaces, log jet schemes generalise log tangent spaces. We'll introduce the construction of log jet schemes for log schemes in the sense of K. Kato, which replace the divisor D with some combinatorial data, and some of their properties. This talk won't assume familiarity with jet schemes or log geometry.
Laminations and free group actions on R-trees are useful tools for studying the structure of elements of and subgroups of outer automorphisms groups of free groups, denoted Out(F). After introducing laminations and R-trees in the context of Out(F), we will mention some recent progress on various topics related to Out(F).
In this talk I will present some recent improvements in regularity estimates for the gradient of solutions to nonlinear elliptic and parabolic equations with nonstandard growth in irregular domains, In particular, when the nonhomogeneous terms belong to various function spaces including weighted Lebesgue spaces, Orlicz spaces and variable exponent spaces.
For a free group F, there is a natural action of Out(F) on the space of representations Hom(F,PSL(2,C)). Minsky introduced an open subset PS(F) (for primitive stable representations) on which Out(F) acts properly discontinuously which is expected to be maximal. I will explain the definition of this set PS(F) and introduce examples and properties of primitive stable representations with discrete image.
In this talk we give a new necessary and sufficient condition for a complex number to be an eigenvalue of a Cantor minimal systems of finite topological rank. This condition aims to be a good tool to understand eigenvalues of classical systems like interval exchange transformations. The case of Toeplitz systems where our new criteria is useful is deeply discussed.
In the context of stationary nearest neighbor Gibbs measures mu satisfying
strong spatial mixing, we present a new combinatorial property on the
support of mu sufficient for approximation of topological
pressure. We also give evidence that this property is in some cases
necessary for strong spatial mixing to hold. Most of these results are
extensions of results obtained by D. Gamarnik - D. Katz (2009) and B.
Marcus - R. Pavlov (2013), who gave a special representation of
topological pressure in terms of conditional probabilities.
Let S be a smooth Del Pezzo surface. We introduce three seemingly unrelated sets of invariants: local BPS state counts of S, relative BPS state counts of S relative to a smooth anticanonical divisor, and generalized DT invariants of the m-loop quiver. We explain how these invariants are connected and how they provide new insight into a conjecture by N. Takahashi on log mirror symmetry.
Room 203 Graduate Student Centre (6371 Crescent Road)
Fri 20 Jun 2014, 4:00pm-6:30pm
Details
Random planar maps have been an object of utmost interest over the last decade and half since the pioneering works of Benjamini and Schramm, Angel and Schramm and Chassaing and Schaeffer. These maps serve as models of random surfaces, the study of which are very important with motivations from physics, combinatorics and ramdom geometry.
Uniform infinite planar maps, introduced by Angel and Schramm which are obtained as local limits of uniform finite maps embedded in the sphere, serve as very important discrete model of infinite random surfaces. Recently, there has been growing interest to create and understand hyperbolic versions of such uniform infinite maps and several conjectures and proposed models have been around for some time. In this thesis, we mainly address these questions from several viewpoints and gather evidence of their existence and nature.
The thesis can be broadly divided into two parts. The first part is concerned with half planar maps or maps embedded in the upper half plane which enjoy a certain domain Markov property, a property which is reminiscent of that of the SLE curves.
The second part concerns investigating unicellular maps or maps with one face embedded in a high genus surface. The main motivation for studying the unicellular maps is the hope that this will shed some light into understanding the local limit of general maps via some well-known bijective techniques.
Discovered (or invented?) by Richard Laver in the 1990s, the tables that are now known as Laver tables are finite structures obeying the self-distributivity law x(yz)=(xy)(xz). Although their construction is totally explicit, some of their combinatorial properties are (so far) established only using unprovable set theoretical axioms, a quite unusual and paradoxical situation. We shall explain the construction of Laver tables, their connection with set theory, and their potential applications in low-dimensional topology via the recent computation of some associated cocycles.
This summer school is aimed at graduate students and post-docs, though all are welcome. The scientific goal is for participants to learn about the different aspects of the study of topological field theories, reaching the research frontier as much as possible. These different aspects include:
(1) the origins of topological quantum field theory in physics;
(2) the mathematical formulation and relation to bordism theory;
(3) extended theories and the cobordism hypothesis;
(4) examples and applications, such as string topology, factorization homology, and examples from representation theory.
Analogous to string topology of manifolds, string topology of classifying spaces studies the rich algebraic structure admitted by the homology groups of free loop spaces of classifying spaces of compact Lie groups. In this talk, I will discuss my recent joint work with Richard Hepworth where we extend the previously available structure in string topology of classifying spaces into a novel kind of field theory which includes operations parameterized by homology groups of automorphism groups of free groups with boundaries in addition to operations parameterized by homology groups of mapping class groups of surfaces. This work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds.
Preprint: http://arxiv.org/abs/1308.6169
This is an international, multi-disciplinary conference that has been held annually for 37 years -rotating between countries all around the world. This year’s theme is the connected classroom. The early-bird deadline for registration is May 16th and there is discounted registration for students and UBC delegates. If you are a graduate student and are interested in volunteering at this conference please contact Karen Smith, UBC site host and Lecturer, Dept. of Microbiology & Immunology, UBC karen.smith@ubc.ca
I'll give motivation for this topic; state some background on cross sections,
cocycles and the Parry Sullivan argument; outline the classifying invariants for
flow equivalence in the primitive and G-primitive SFT case; sketch the
proof strategy; discuss the mapping class group of an SFT (with many open
questions); and indicate the results and frontiers for understanding flow
equivalence in the sofic case.
This is taken out of joint works with Huang; M. Sullivan; and Carlsen and Eilers.
For the mapping class group, I will cite some results of Chuysurichay.
For more information on Qualifying Exams, please visit http://www.math.ubc.ca/Grad/QualifyingExams/index.shtml
Lunch will be provided in Math 125 for students writing the Analysis exam.
We consider - in uniformly strictly convex potentials case - two versions of random gradient models. In model A) the interface feels a bulk term of random fields while in model B) the disorder enters though the potential acting on the gradients itself. It is well known that without disorder there are no Gibbs measures in infinite volume in dimension d = 2, while there are gradient Gibbs measures describing an infinite-volume distribution for the increments of the field, as was shown by Funaki and Spohn. Van Enter and Kuelske proved that adding a disorder term as in model A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in d = 2. Cotar and Kuelske proved the existence of shift-covariant gradient Gibbs measures for model A) when d > 2 and the expectation with respect to the disorder is zero, and for model B) when d > 1.
In recent work, we prove uniqueness of shift-covariance gradient Gibbs measures with expected given tilt under the above assumptions. We also prove decay of covariances for both models.
We will also discuss in the talk the case of non-convex potentials with disorder.
This is partly based on joint work with Christof Kuelske.
The Langlands program, initiated in the 1960s, is a set of conjectures predicting a unification of number theory and the representation theory
of groups. More precisely, the Langlands correspondence provides a way to interpret results in number theory in terms of group theory, and vice versa.
In this talk we sketch a few aspects of the local Langlands correspondence using elementary examples. We then comment on some questions raised by the emerging "mod p" Langlands program.
Abstract: A variety of measurements may illuminate the reservoir with varying coverage and resolution such as: electromagnetic (EM); controlled-source EM (CSEM); magnetotelluric (MT), surface-to-borehole EM (STB-EM); crosswell EM; seismic (surface seismic, crosswell seismic, and VSP); gravity (surface and borehole); and production history/well testing data. The interpretation of each measurement on its own will provide incomplete information due to nonuniqueness and limited spatial resolution. However, when integrated and combined with other measurements such as near-wellbore data, they may provide considerable value such as, for example, to enable estimation of reservoir properties, to obtain an improved reservoir model, and to provide a physics-based reservoir upscaling. At the end, it will help us in making appropriate field management decisions with reduced uncertainty.
This presentation will review joint inversion algorithms and workflows for integrating EM, seismic, and production data. It will analyze challenges, advantages, and disadvantages of these approaches. In particular, for reservoir characterization applications, joint structural and petrophysical algorithms for integrating EM and seismic data (CSEM and surface seismic, and crosswell EM and crosswell seismic) will be presented. For reservoir monitoring applications, the talk will describe EM data (for single-well, crosswell and STB) inversion algorithms constrained by the fluid-flow simulator. In the inversion for both EM and seismic, a full nonlinear approach (the so-called full-waveform inversion) will be employed so that all the information in the data can be utilized. Some test cases will be discussed.
Biography: Aria Abubakar was born in Bandung, Indonesia, in 1974. He received a master’s degree (cum laude) in electrical engineering and a PhD (cum laude) in technical sciences from Delft University of Technology in 1997 and 2000, respectively. From September 2000 until February 2003 he was a researcher with the Laboratory of Electromagnetic Research and Section of Applied Geophysics of Delft University of Technology.
He joined Schlumberger-Doll Research in 2003 and his last position there was as a scientific advisor and the Research Program Manager of Multi-Physics Modeling and Inversion Program. While at Schlumberger-Doll Research, he worked on advanced modeling and inversion of electromagnetic and seismic waves. In addition, he has been developing algorithms and workflows for joint inversion of multiphysics data for both reservoir and wellbore-scale measurements. Since early 2013, he has been Interpretation Engineering Manager and Scientific Advisor at Schlumberger Houston Formation Evaluation in Sugar Land, Texas. He currently is leading Schlumberger Wireline and Logging-While-Drilling product development for electromagnetic, nuclear, and resistivity measurements and their joint inversion and interpretation. He is Associate Editor of Radio Science and Geophysics. He has published one book, four book chapters, over 75 scientific articles in refereed journals, 150 conference proceedings papers, and 50 conference abstracts. He holds four US patents and has five US patent applications under review.
The virtual Poincare polynomials of the stable pair moduli spaces of a Calabi-Yau threefold are conjecturally equivalent to the refined BPS numbers of Gopakumar and Vafa. As an application, stable pair invariants of the del Pezzo surfaces dP_n determine BPS Hilbert spaces which are observed to be representions of the exceptional Lie algebra E_n, consistent with expectations of string theory. In another direction, string theory on K3 x T^2 leads to a reduced DT theory on K3, hence
corresponding motivic and refined invariants. Work in progress on the rational elliptic surface dP_9 ("half K3") suggests that a blend of these two examples leads to a BPS Hilbert space with a representation of affine E_8. This talk includes separate joint works with Choi, Klemm, and Pandharipande.
Note for Attendees
Today the seminar has two talks. Sheldon 3-4pm (as usual) and Masoud 4:30-5:30pm. Coffee and cookies are served in between.
Local geometric Langlands program aims to establish a relationship between representations of the Galois group of a local field and irreducible representations of the dual group. It is expected that, under mild conditions, this correspondence preserves depths of representations. In this talk, I will explain the geometric analogue of this expectation, in the framework of Frenkel-Gaitsgory's local geometric Langlands correspondence. Based on a joint project with Tsao-Hsien Chen: http://arxiv.org/abs/1404.0598
Note for Attendees
Today the seminar has two talks. Sheldon 3-4pm (as usual) and Masoud 4:30-5:30pm. Coffee and cookies are served in between.
The (lattice) phi^4 model is a scalar field-theoretical model that exhibits a phase transition. It is believed to be in the same universality class as the Ising model. In fact, we can construct the phi^4 model as the N --> infinity limit of the sum of N Ising systems (with the right scaling of spin-spin couplings). Using this Griffiths-Simon construction and applying the lace expansion for the Ising model, we can prove mean-field asymptotic behavior for the critical phi^4 two-point function. In this talk, I will explain the key points of the proof, and discuss possible extensions of the results to the power-law coupling case.
I will discuss a new construction of finite random graphs motivated by the study of random walks on infinite groups, and show connections to the Poisson boundary and Sznitman's random interlacements.
Note for Attendees
This is an extra lecture for the Probability seminar, starting at 4pm rather than the usual 3pm.
In 1970, Manin showed that the Brauer group can obstruct the existence of rational points. Colliot-Thélčne and Sansuc have conjectured that this obstruction completely explains the failure of rational points on del Pezzo surfaces. We show that on degree 4 del Pezzo surfaces, this Brauer-Manin obstruction manifests itself through linear projections. As a consequence of the proof, we obtain a simple and efficient for computing the Brauer classes of a degree 4 del Pezzo surface. This is joint work with Anthony Várilly-Alvarado.
Convective flow in porous media can be found in numerous geophysical and industrial processes, and has recently been investigated in the context of geological CO_2 sequestration. I will present high-resolution numerical simulations of statistically steady high-Rayleigh-number convection in a porous medium that is heated at the base and cooled at the upper surface. I will discuss the relationship between the strength of convection, as described by the Rayleigh number, and the transfer of buoyancy, as described by the Nusselt number. I will examine the remarkable dynamical structure of high-Rayleigh-number porous convection: the structure is characterized in the interior by ordered vertical columnar exchange flow with a regular (Rayleigh-number dependent) horizontal wavenumber k(Ra), and near the boundaries by vigorous short-wavelength boundary-layer instabilities . I will investigate the physical control of this structure by performing a stability analysis of the interior columnar flow: the predictions of this analysis give good agreement with the measured horizontal wavenumber k(Ra). Numerical results will be presented in both two-dimensional and three-dimensional porous media.
I will then extend these results to consider the more physically relevant context of convection from below one boundary only, and consider the implications of these results for the long-term storage capacity of underground water-saturated porous rocks into which CO_2 has been injected.
We explore and take advantage of hidden structure behind the binomial identity above. This will lead us through some basic representation theory of sl(2), categorical equivalences and quantum knot invariants.
Adaptive dynamics is a general framework to study long-term evolutionary dynamics. It is typically used to study evolutionary scenarios in low-dimensional phenotype spaces, such as the important phenomenon of evolutionary branching (adaptive diversification). I will briefly recall the basic theory of evolutionary branching and present a well-studied empirical example. Because birth and death rates of individuals are likely to be determined by many different phenotypic properties, it is important to consider evolutionary dynamics in high-dimensional phenotype spaces. I will present some results about evolutionary branching in high-dimensional phenotype spaces, as well as results about more general types of non-equilibrium evolutionary dynamics, such as chaos. Finally, I will compare the deterministic adaptive dynamics in high-dimensional phenotype spaces to individual-based simulations of the underlying stochastic birth-death process.
Note for Attendees
Michael Doebeli is the winner of the 2014 CAIMS Research Prize.
Morrison defined canonical coordinates near a maximal degeneration point in the moduli of Calabi-Yau manifolds using Hodge theory. Gross and Siebert introduced a logarithmic-tropical algorithm to provide a canonically parametrized smoothing of a degenerate Calabi-Yau. We show that the Gross-Siebert coordinate is a canonical coordinate in the sense of Morrison. The coordinates are given by period integrals which we compute explicitly integrating over cycles constructed using tropical geometry. This is joint work with Siebert.
Self-assembly, a process in which a disordered system of preexisting components forms an organized structure or pattern, is both ubiquitous in nature and important for the synthesis of many designer materials. In this talk, we will address two variational paradigms for self-assembly from the point of view of analysis and computation. The first variational model is a nonlocal perturbation (of Coulombic-type) to the well-known Ginzburg-Landau/Cahn-Hilliard free energy. The functional has a rich and complex energy landscape with many metastable states. We present recent joint work with Dave Shirokoff (NJIT) and J.C. Nave (McGill) on developing a method for assessing whether or not a particular (computed) metastable state is a global minimizer. Our method is based upon a very simple idea of using a ``suitable" global convex envelope of the energy. We present full details for global minimality of the constant state, and then present a few partial results on the application to non-constant, computed metastable states.
The second variational model is purely geometric and finite-dimensional: Centroidal Voronoi Tessellations (CVT) of rigid bodies. Using a level set formulation, we a priori fix the geometry for the structures and consider self-assembly entirely dictated by distance functions. We introduce a novel fast algorithm for simulating CVTs of rigid bodies in any space dimension. The method allows us to empirically explore the CVT energy landscape. This is joint work with Lisa Larsson (Courant) and J.C. Nave (McGill).
Lagrangian Mean Curvature Flow (LMCF) is a geometric flow, aiming to deform a Lagrangian immersion to a minimal one. To understand the flow, it is important to understand the formation of singularity in LMCF. In this talk, I will introduce the concept of a self-shrinker (a local model for singularity), how it is formed in LMCF, and give some examples of Lagrangian self-shrinkers. Then I will discuss a recent work with Jingyi Chen concerning the space of all compact Lagrangian self-shrinkers in \mathbb C^2.
We propose an analog of the Little map for reduced expressions for signed permutations. We show that this map respects the transition equations derived from Chevellay's formula on Schubert classes. We discuss many nice properties of the signed Little map which generalize recent work of Hamaker and Young in type A where they proved Lam's conjecture. As a key step in this work, we define shifted dual equivalence graphs building on work of Assaf and Haiman and prove they can be characterized by axioms. These graphs are closely related to both the signed Little map and to the Coxeter-Knuth relations of type B due to Kraskiewicz.
This talk is based on joint work with Zach Hamaker, Austin Roberts and Ben Young.
Chern classes of complex manifolds play a key role in geometry and topology. In this talk we shall discuss how these classes extend to singular varieties. In fact there are various possible extensions, depending on which properties of Chern classes you want to preserve. This is closely related to asking who plays the role of the tangent bundle at the singular points (where there is no tangent bundle).
Ramanujan graphs are optimal expanders (from spectral point of view). Explicit constructions of such graphs were given in the 80's as quotients of the Bruhat-Tits tree associated with GL(2) over a local field F, by suitable congruence subgroups. The spectral bounds were proved using works of Hecke, Deligne and Drinfeld on the "Ramanujan conjecture" in the theory of automorphic forms. The work of Lafforgue, extending Drinfeld from GL(2) to GL(n), opened the door for the construction of Ramanujan complexes as quotients of the Bruhat-Tits buildings. This gives finite simplical complxes which on one hand are "random like" and at the same time have strong symmetries. Recently various applications have been found in combinatorics, coding theory and in relation to Gromov's overlapping properties. We will describe these developments and give some details on recent applications.
Expander graphs have played, in the last few decades, an important role in computer science, and in the last decade, also in pure mathematics. In recent years a theory of "high-dimensional expanders" is starting to emerge - i.e., simplical complexes which generalize various properties of expander graphs. This has some geometric motivations (led by Gromov) and combinatorial ones (started by Linial and Meshulam). The talk will survey the various directions of research and their applications, as well as potential applications in math and CS. Some of these lead to questions about buildings and representation theory of p-adic groups.
We will survey the work of a number of people. The works of the speaker in this direction are with various subsets of { S. Evra, K. Golubev, T. Kaufman, D. Kazhdan , R. Meshulam, S. Mozes }
Industrial mathematics is a field that spans a broad spectrum of activity ranging from applied R&D performed by mathematicians employed in industry, to purely academic research projects undertaken by university mathematics professors. In this talk, I will survey several research projects I have been involved with that fall under the heading of what I’ll call ”mathematics *for* industry”, which relates specifically to direct collaborations between university mathematicians and non-academic partner organizations. These projects encompass a diverse collection of mathematical techniques (ranging from simple algebra to partial differential equations, finite volume methods, inverse problems and homogenization theory) as well as applications from many scientific disciplines (such as fluid mechanics, image processing, atmospheric science and plant biology). In the process, I will attempt to characterize the job of an industrial mathematician and to identify the qualities and skills that are most desirable for anyone interested in making significant contributions to research at the interface between university and industry. I also hope to convince you that industrial collaborations can be a rich source of challenging and novel mathematical problems for academic mathematicians.
Note for Attendees
John Stockie is the winner of the 2014 CAIMS/Mprime Industrial Mathematics Prize.
Nakajima's quiver varieties are important geometric objects in representation theory that can be used to give geometric constructions of quantum groups. Very recently, graded quiver varieties also found application to monoidal categorification of cluster algebras. Nakajima's original construction uses geometric invariant theory. In my talk, I will give an alternative representation theoretical definition of graded quiver varieties. I will show that the geometry of graded quiver varieties is governed by the derived category of the quiver. This is joint work with Berhard Keller.
Given a general second-order, elliptic operator P on a general domain, we discuss the question of finding an "optimal", or "asymptotically optimal", Hardy inequality for P. Such an inequality can be considered as a gneralized spectral gap inequality of P. If time allows, we will also consider the $L^p$ case.
We study random walks on Z^d among random conductances {C_{xy}: x,y in Z^d} that permit jumps of arbitrary length. Apart from joint ergodicity with respect to shifts, we assume only that the nearest-neighbor conductances are uniformly positive and that sum_{x in Z^d} C_{0x} |x|^2 is integrable.
Our focus is on the Quenched Invariance Principle (QIP) which we establish in all d >= 3 by a combination of corrector methods and heat-kernel technology. We also show that our class contains examples where the corrector is not sublinear everywhere and yet the QIP holds. Thus, although the recent work of Andres, Slowik and Deuschel can be extended to long-range models, it cannot cover all cases for which the QIP is conjectured to hold. Notwithstanding, a combination of their methods with ours proves the QIP for random walks on long-range percolation graphs with exponents larger than d+2 in all d >= 2, provided all nearest-neighbor edges are present.
This is an ongoing joint work with Marek Biskup (UCLA).
In this talk I will mention some conjectures about group rings (Idempotent conjecture, unit conjecture) and mention their stable versions. Those involve algebraic K-theory. I will explain how the Farrell-Jones conjecture implies those stable versions. I will then finish with the status of the Farrell-Jones conjecture. The cheapest way to prove it for a certain group is to use its inheritance properties, but this also touches very interesting questions in (geometric) group theory.
Frames generalize the notion of bases and provide a useful tool for modeling the measurement (or sampling) process in several modern signal processing applications. In the digital era, such a measurement process is typically followed by quantization, or digitization. We discuss the quantization of frame coefficients using a scheme known as Sigma-Delta quantization. We show that a simple encoding via a discrete random Johnson-Lindenstrauss embedding of the integrated bit-stream yields near-optimal approximation error (as a function of the number of bits used). The result holds with high probability on the draw of the embedding, allows efficient reconstruction, and holds for a wide class of frames including random frames and deterministic smooth frames.
In addition, we show that if the same encoding scheme is applied to quantized compressed sensing measurements (albeit with a different reconstruction scheme), it also yields near-optimal approximation accuracy as a function of the bit-rate. Our results hold for Gaussian and sub-Gaussian compressed sensing matrices.
We discuss how the derived category of a smooth algebraic stack of finite type changes as one removes certain types of closed substacks. As an application, we show how wall-crossing in moduli of stable sheaves and Bridgeland stable objects yields semi-orthogonal decompositions of relating their derived categories.
In the Lego world, everything is built out of simple blocks fitting together in a prescribed way. In the mathematical world, you can just as easily think of cubes (or hypercubes) as Lego blocks and build away!
Note for Attendees
Notice the special time! This is to accommodate participation in the TA Union Orientation meeting: Date(for above event): Thursday, September 25 Time: 12:30-1:30 Location: Leonard S. Klinck Building (LSK) (6356 Agricultural Road) Room 200
One version of homological mirror symmetry relates the algebraic geometry of certain varieties to the symplectic topology of a Lefschetz pencil. Extracting symplectic (i.e. Floer theoretic) invariants from these fibrations is quite difficult, even in simple examples. I'll review some recent proposals (particularly from Kapranov-Kontsevich-Soibelman and Diemer-Kerr-Katzarkov) for deforming symplectic fibrations into more tractable components. The corresponding mirror theory appears to be closely related to birational geometry and the Mori program.
A space X is called Noetherian if there is no infinite descending chain of closed subsets of X. Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point in X. We show that the set containing all positive integers n such that the n-th iterate of x under f lands in Y is a union of at most finitely many arithmetic progressions along with a set of Banach density 0. This result has various consequences from the distribution of zeros in recurrence sequences to questions in arithmetic geometry.
Note for Attendees
Refreshments will be served at 2:45 p.m. in the Math Lounge area, MATH 125 before the colloquium.
We consider the ``limiting behavior'' of {\em discriminants}, by which we mean informally the closure of the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the "ring of motives", as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization can be described in terms of the motivic zeta values. The results extend parallel results in both arithmetic and topology. I will also
present a conjecture (on ``motivic stabilization of symmetric powers'') suggested by our work. Although it is true in important cases, Daniel Litt has shown that it contradicts other hoped-for statements. This is joint work with Melanie Wood.
Note for Attendees
Note the unusual place, ESB 2012, and the unusual time, 3:30-4:30pm.
We consider nonsingular saddle-point matrices whose (1,1) block is maximally rank deficient, and show that the inverse in this case has unique mathematical properties. We then develop a class of indefinite block preconditioners that rely on approximating the null space of the leading block. Under certain conditions, even though the preconditioned matrix is a product of two indefinite matrices, the conjugate gradient method can be applied and is rapidly convergent. Spectral properties of the preconditioners are observed, which are validated by numerical experiments.
This talk concerns the geometry of shrinking Ricci solitons, a class of self-similar solutions to the Ricci flows. We plan to provide some general background results and explain a recent work with Ovidiu Munteanu on the curvature estimates of four dimensional solitons.
We investigate various issues for stationary Kirchhoff systems in closed manifolds, such as the questions of existence, non-existence and compactness of solutions to the equations.
We consider the high-dimensional Markowitz optimization problem. A new approach combining sparse regression and estimation of optimal returns based on random matrix theory is proposed to solve the problem. We prove that under some sparsity assumptions on the underlying optimal portfolio, our novel approach asymptotically yields the theoretical optimal return, and in the meanwhile satisfies the risk constraint. To the best of our knowledge, this is the first method that can achieve these two goals simultaneously in the high-dimensional setting. We further conduct simulation and empirical studies to compare our method with some benchmark methods, including the equally weighted portfolio, the bootstrap-corrected method by Bai et al. (2009) and the covariance-shrinkage method by Ledoit and Wolf (2004). The results demonstrate substantial advantage of our method, which attains high level of returns while keeping the risk well controlled by the given constraint.
Based on joint work with Mengmeng Ao and Yingying Li.
In the beginning of this talk I will use the Farrell-Jones conjecture to express the K-theory of R[Z^2] in Terms of the K-theory of R. Geometric conditions on a Group that imply the conjecture will be mentioned . The class of Groups for which the conjecture is known is quite large I will define it and mention some interesting open cases.
An old conjecture of Lang (for elliptic curves) generalized by Silverman, asserts that the Néron-Tate height of a rational point of an abelian variety defined over a number field can be bounded below linearly in terms of the Faltings height of the underlying abelian variety. We shall explore the function field analogue of this problem.
The cubic Szegö equation is an Hamiltonian evolution on periodic functions with nonnegative Fourier modes, arising as a normal form for the large time behavior of a nonlinear wave equation on the circle. It defines a flow on every Sobolev space with enough regularity. In this talk, I will give the main arguments for the proof of the following theorem. The trajectories of the cubic Szegö equation are almost periodic in the Sobolev energy space, but
are generically unbounded in every more regular Sobolev space.This is a joint work with Sandrine Grellier and Zaher Hani.
The Ricci flow is one of the most important equations in geometric analysis, and has been used to solve deep problems in topology and geometry. Through a system of local parabolic PDE's, the flow governs the evolution of a Riemannian metric tensor in space, and it's general theory is fundamentally based on the assumption that the metric is complete with bounded sectional curvatures. I will give an overview of the general theory, then discuss the problem of flowing unbounded curvature metrics on non-compact manifolds. I will then discuss recent results for U(n) invariant Kahler metrics on C^n, and connections to Yau's uniformization conjecture. The talk is based in part on joint work with L.F. Tam and K.F Li.
Note for Attendees
Refreshments will be served at 2:45pm in the Math Lounge area, MATH 125 before the colloquium.
The arrival of massive amounts of data from imaging, sensors, computation and the internet brought with it significant challenges for information science. New methods for analysis and manipulation of big data have come from many scientific disciplines. The first focus of this presentation is the application of ideas from PDEs, such as variational principles and numerical diffusion, to image and data analysis. Examples include denoising, segmentation, inpainting and texture extraction for images. The second focus is the development of new ideas in information science, such as wavelets, soft-thresholding, sparsity and compressed sensing. The subsequent application of these ideas to PDEs and numerical computation is the third focus of this talk. Examples include wavelet analysis for turbulent flows, the use of soft-thresholding in computation of PDEs with multi-scale features, and the construction of “compressed modes” (modes that are compactly supported in space) for density functional theory and other PDEs that come from variational principles.
Russel Caflisch is a Professor in the Mathematics Department at UCLA and has a joint appointment in the Department of Materials Science and Engineering. He received his PhD from the Courant Institute at New York University in 1978 and has also held faculty positions at Stanford and NYU. He is currently the director of the Institute for Pure & Applied Mathematics (IPAM), and the Editor-In-Chief for the journal Multiscale Modeling and Simulation. He was a Sloan Foundation Research Fellow, and a fellow of the Society for Industrial and Applied Mathematics, the American Mathematical Society, and the American Academy of the Arts and Sciences. Caflisch’s expertise includes a wide range of topics in applied mathematics, including PDEs, fluid dynamics, plasma physics, materials science, Monte Carlo methods, and computational finance.
Note for Attendees
Refreshments start 15 minutes before the talk in the IAM Lounge, Room 306 of the LSK building.
Homological mirror symmetry initially concerned Calabi-Yau 3-folds and, from that point, rapidly expanded to incorporate local Calabi-Yau's and Fano varieties. In this talk, I will discuss joint work with Ludmil Katzarkov and Maxim Kontsevich on extending this correspondence further to include quasi-affine toric varieties, the most basic example of which is a punctured plane. The complex side of the correspondence, or B-model, remains the derived category of coherent sheaves of the variety. On the mirror side, the A-model is a partially wrapped Fukaya category on the cotangent bundle of the torus. The key ingredient is the wrapping Hamiltonian which is defined as a distance^2 function away from a mirror non-compact Lagrangian skeleton. I will explain the geometric intuition for the case of the punctured plane and discuss elements of the proof for the general case.
Note for Attendees
Note: Today we have a double-header again. Talk one: Gabriel Kerr 15:00-16:00. Talk two: Charlie Beil: 16:30-17:30. Coffee and cookies in between the two talks.
A dimer algebra is a type of quiver algebra whose quiver embeds in a torus, with homotopy-like relations. Dimer algebras with the cancellation property are Calabi-Yau algebras, and their centers are 3-dimensional Gorenstein singularities. Non-cancellative dimer algebras, on the other hand, are not Calabi-Yau, and their centers are nonnoetherian. In contrast to their cancellative counterparts, very little is known about these algebras, despite the fact that almost all dimer algebras are non-cancellative. I will describe how their centers are also 3-dimensional singularities, but with the strange property that they contain positive dimensional 'smeared-out' points. Furthermore, I will describe how this nonlocal geometry is reflected in the homology of certain vertex simple representations.
Note for Attendees
Note: Today we have a double-header again. Talk one: Gabriel Kerr 15:00-16:00. Talk two: Charlie Beil: 16:30-17:30. Coffee and cookies in between the two talks.
Strong solutions to the 3D Navier-Stokes equations are known to exist locally-in-time and are real analytic. Providing lower bounds for their analyticity radius is important as this length scale plays an important role in turbulent phenomenologies and can be used to establish blow-up criteria. In this talk we discuss one approach to estimating analyticity radii and a related conditional regularity criteria.
ESB 2012 Oct 7th update: This seminar has been canceled.
Wed 8 Oct 2014, 3:00pm-4:00pm
Abstract
We propose a framework for modelling stochastic systems with state-dependent diffusion coefficients. Rather than specifying dynamics through a state-dependent drift and diffusion coefficients, assuming detailed balance we specify an equilibrium probability density and a state-dependent diffusion coefficient. We argue that our framework is more natural from the modelling point of view and has a distinct advantage in situations where either the equilibrium probability density or the diffusion coefficient is discontinuous. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and elegantly handles discontinuities in the coefficients. This is joint work with Xin Yang.
A two-dimensional conformal field theory (CFT) is constructed from the representations of a vertex operator algebra. The Verlinde formula tells us much information about the fusion ring of modules in terms of the modular data of characters and this formula is known to be true if the vertex algebra is sufficiently nice, meaning essentially that the representation category is semi-simple and that there are only finitely many simple modules. Information about this relation is already included in the asymptotic behavior of characters. A conformal field theory/vertex algebra with modules that are not completely reducible is called logarithmic. In this case not much is known concerning the Verlinde formula. False theta functions appear frequently in characters of such theories. In this talk, I will introduce false theta functions, explain that they have a modular-like behavior and will relate their asymptotic behavior to fusion rings. Asymptotics are subtle and both wall-crossing and a relation to the D-brane fusion ring of minimal string theory will appear. The results presented are joint work with Antun Milas and Simon Wood.
This will be an introduction on recent developments in symmetries and PDEs by the speaker and collaborators. It will be based on the article with the same title as this talk by the speaker and Zhengzheng Yang, in Similarity and Symmetry Methods: Applications in Elasticity and Mechanics in Materials (J-F Ganghoffer, I Mladenov, Editors), Lecture Notes in Applied and Computational Mechanics 73, pp. 1-59 (2014).
I will talk about the concept of universality as it arises in critical phenomena from the point of view of probability. I will also briefly discuss why the Brownian motion is so cool.
I will define hyperelliptic curves, and describe three families of curves of each genus n ≥ 2. I will then discuss results of Bhargava, Poonen, Stoll, Shankar and Wang, which show that most curves in these families have the minimal number of rational points.
After quadratic equations in two variables come cubic equations, or elliptic curves. The set of rational points on an elliptic curve has the structure of a finitely generated abelian group. I will recall the basic theory of elliptic curves, then discuss the conjecture of Birch and Swinnerton-Dyer, which attempts to predict the rank of the group of rational points from the number of solutions (mod p) for all primes p. I will also discuss some recent results on the average rank, due to Manjul Bhargava and his collaborators.
Note for Attendees
Coffee, tea and cookies served at 2:30pm in the PIMS Lounge, ESB 4133.
So far, only two blow-up regimes have been studied for NLS equations: the pseudo-conformal regime, where the blow-up speed is like $|t| ^{-1}$ and the log-log regime where the blow-up speed is like $|t|^{-1/2}$ with a log-log correction.
In this talk, we consider the nonlinear Schrodinger with a double power nonlinearity where one of the power is L2 critical and the other one is L2-subcritical. We construct a minimal mass blowing up solution whose blow-up speed is neither the log-log speed nor the pseudo-conformal speed, but is of the type $|t|^{-s}$ with $s$ varying between $1/2$ and $1$ depending on the subcritical power. This is based on a joint work with Yvan Martel and Pierre Raphael.
We propose a framework for modelling stochastic systems with state-dependent diffusion coefficients. Rather than specifying dynamics through a state-dependent drift and diffusion coefficients, assuming detailed balance we specify an equilibrium probability density and a state-dependent diffusion coefficient. We argue that our framework is more natural from the modelling point of view and has a distinct advantage in situations where either the equilibrium probability density or the diffusion coefficient is discontinuous. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and elegantly handles discontinuities in the coefficients. This is joint work with Xin Yang.
Let G be a Lie group. The space Hom(Z^n,G) of commuting n-tuples in G was extensively studied over the last few years. Two related generalizations of this space are the space of almost commuting tuples, in the sense that the commutator of every pair of elements in each tuple is in the center of G, and the space of homomorphisms Hom(Gamma, G) from a central extension Gamma of a free abelian group by a free abelian group to G.
In this talk, I will describe the structures of these spaces and the relations between them in the case G=U(m). I will also discuss questions such as the number of path components and the rational homotopy type of these spaces.
We briefly review some methods for the construction of conservation laws for PDEs. These methods include the classical Nöther's theorem applicable to any variational system, symmetry action on known conservation laws, an auxiliary equations method, linearizing operator approach (use of a symmetry/adjoint symmetry pair) and, most generally, the direct method applicable essentially to any PDE system. In particular, we will exhibit examples for some of the above methods and applications of conservation laws.
The theory of compressed sensing considers the following problem: Let A be an m x n matrix and let x be an s-sparse vector in n dimensions, i.e., all but s of its entries vanish. One seeks to recover x uniquely and efficiently from linear measurements y = Ax, although m is much less than n. A sufficient condition to ensure that this is possible is the Restricted Isometry Property (RIP). A is said to have the RIP, if its restriction to any small subset of the columns acts almost like an isometry.
In this talk, we study two classes of matrices with respect to the RIP: First, we consider matrices A which represent the convolution with a random vector followed by a restriction to an arbitrary fixed set of entries. We focus on the scenario of a Rademacher vector, i.e., a vector whose entries are independent random signs, but also discuss the case of independent subgaussian entries. Second, we study Gabor synthesis matrices, that is, matrices consisting of time-frequency shifts of a such vectors.
In both cases, this question can be reduced to estimating a supremum of random variables taken over an indexing set of matrices. Random variables of this type are closely related to suprema of chaos processes. Using generic chaining techniques, we derive a bound for its moments in terms of concepts from the theory of empirical processes. As a consequence, we obtain that matrices from both classes under consideration have the RIP with high probability if the embedding dimension satisfies m > Cs log^4(n). This bound exhibits optimal dependence on s, while previous works had only obtained a suboptimal scaling of s^(3/2).
This is joint work with Shahar Mendelson and Holger Rauhut.
An Euler system is a collection of global arithmetic objects, most notably global cohomology classes arising from geometry, which are related to L-functions and can be made to vary in p-adic families. The talk will mainly focus on the Euler system of Hegneer points that played a key role in the seminal work of Gross–Zagier and Kolyvagin on the Birch and Swinnerton–Dyer Conjecture. We will present a construction of anticyclotomic p-adic Rankin–Selberg L-functions and discuss some related reciprocity laws in the spirit of Kato.
Knowledge of atmospheric carbon dioxide (CO2) concentrations in the past are important to provide an understanding of how the Earth's carbon cycle varies over time. This project combines ice core CO2 concentrations, from Law Dome, Antarctica and a physically based forward model to infer CO2 concentrations on an annual basis. Here the forward model connects concentrations at given time to their depth in the ice core sample and an interesting feature of this analysis is a more complete characterization of the uncertainty in "inverting" this relationship. In particular, Monte Carlo based ensembles are particularly useful for assessing the size of the decrease in CO2 around 1600 AD. This reconstruction problem, also known as an inverse problem, is used to illustrate a general statistical approach where observational information is limited and characterizing the uncertainty in the results is important. These methods, known as Bayesian hierarchical models, have become a mainstay of data analysis for complex problems and have wide application in the geosciences.
This work is in collaboration with Eugene Wahl (NOAA), David Anderson (NOAA) and Catherine Truding.
Note for Attendees
The Thursday talk is preceded by coffee, tea, cookies (15:30 pm -16:00 pm) in Room 101.
Donaldson-Thomas (DT) invariants of a Calabi-Yau threefold X are fundamental quantum invariants given by (weighted) Euler characteristics of the Hilbert schemes of X. We compute these invariants for the case where X is a so-called local elliptic surface --- it is the total space of the canonical line bundle over an elliptic surface. We find that the generating functions for the invariants admit a nice product structure. We introduce a new technique which allows us to use the topological vertex in this computation --- a tool which previously could only be used for toric threefolds. As a by product, we discover surprising new identities for the topological vertex. This is joint work with Martijn Kool, with an assist from Ben Young.
We propose the stochastic average gradient (SAG) method for optimizing the sum of a finite number of smooth convex functions. Like stochastic gradient (SG) methods, the SAG method's iteration cost is independent of the number of terms in the sum. However, by incorporating a memory of previous gradient values the SAG method achieves a faster convergence rate than black-box SG methods. Specifically, under standard assumptions the convergence rate is improved from O(1/k) to a linear convergence rate of the form O(p^k) for some p < 1. Further, in many cases the convergence rate of the new method is also faster than black-box deterministic gradient methods, in terms of the number of gradient evaluations. Beyond these theoretical results, the algorithm also has a variety of appealing practical properties: it supports regularization and sparse datasets, it allows an adaptive step-size and has a termination criterion, it allows mini-batches, and its performance can be further improved by non-uniform sampling. Numerical experiments indicate that the new algorithm often dramatically outperforms existing SG and deterministic gradient methods, and that the performance may be further improved through the use of non-uniform sampling strategies.
We show the nonexistence of traveling fronts in the combustion model with fractional Laplacian (-\Delta)^s when s\in(0,1/2]. Our method can be used to give a direct and simple proof of the nonexistence of traveling fronts for the usual Fisher-KPP nonlinearity. Also we prove the existence and nonexistence of traveling waves solutions for different ranges of the fractional power s for the generalized Fisher-KPP type model.
Sperner's theorem from 1928 states that the greatest number subsets of an n-element set such that no subset contains another (in other words the largest chain is length 1), is \binom{n}{\lfloor n/2\rfloor}. This result has many generalizations since: L-Sperner families are families where the largest chain is of length at most L, M-part families are families where there is no chain of length 2 where the increase of a chain is contained in a fixed M-partition of the underlying set, etc. Mixed orthogonal arrays are designs introduced by statisticians for designing experiments, so that factors potentially influential to the outcome occur simultaneously in a regular manner. We show that these distant topics have a strong connection (in particular mixed ortogonal arrays and homogeneous M-part (L_1,...,L_M)-Sperner families correspond to each other), and provide constructions for mixed orthogonal arrays. Joint work with H Aydinian and L.A. Szekely.
Given a Calabi-Yau complete intersection in a toric Fano variety, there are various ways to construct the mirror. Sometimes these mirrors are isomorphic and sometimes they are not. These distinct 'double' mirrors should be equivalent in some way if they all have a shot at being the 'correct' mirror in some setting of mirror symmetry. We will discuss the Batyrev-Borisov and Berglund-Hübsch-Krawitz construction and the double mirrors which arise, as well as their relationship through variation of geometric invariant theory quotients, Landau-Ginzburg models, and derived equivalence. This is joint work with Tyler Kelly.
For deterministic bounded degree triangulations, circle packing has proven a powerful tool for studying random walk via geometric arguments. In this talk, I will discuss extensions and analogues for random triangulations without the assumption of bounded degree. In particular, I will show that the circle packing type (hyperbolic or Euclidean) is determined by the expected degree at the root and that, in the hyperbolic case, the geometric boundary given by the circle packing coincides with the Poisson boundary of the random walk. No specialised knowledge will be assumed and I will review the main examples.
Joint work with Omer Angel, Asaf Nachmias and Gourab Ray.
The scope of what constitutes a math problem is far wider than "how many apples are left in the basket if..." or "prove the equation has at least one real root." Math problems can be seen in everything from the development of bone structures through the bending of light due to massive objects; math is everywhere. By going through a few projects I've recently had the fortune of working on, I want to highlight a few of the beautiful ways mathematical thinking finds its way into solving real-world problems including: using physical modelling in designing devices for water filtration by electrodialysis, implementing formal asymptotic analysis to predict the behaviour of a fusion reactor, and writing numerical methods to provide a proof-of-concept for a new method of mass spectrometry. Just as there is art in expressing the world through imagery and poetry, so there is in analyzing problems appropriately and making use of such analysis. No prior knowledge is expected: all the problems presented will include the relevant background information.
Classical knot groups, that is fundamental groups of knot complements in 3-space, are known to be torsion-free. However, we show that for many knots, their groups contain generalized torsion: a nontrivial element such that some product of conjugates of that element equals the identity. One example (the hyperbilic knot 5_2) was discovered with the aid of a Python program written by the USRA student Geoff Naylor. Other examples include torus knots, algebraic knots in the sense of Milnor (arising from singularities of complex curves) and satellites of knots whose groups contain generalized torsion. Although all knot groups are left-orderable, the existence of generalized torsion is an obstruction to their being bi-orderable.
We study the Noether and Lie symmetries that arise from the Euler-Lagrange equations, i.e., the ‘geodesic’ equations, related to manifolds that arise from a metric. In particular and as one of the examples, we present some peculiarities associated with the ASD Ricci-flat metric which depends on the `second heavenly equation'. It is noted, in general, that the Killing vectors are contained in the Noether symmetries generated by the Lagrangian of the geodesic equations. Specifically, a number of symmetries which are Noether and not Killing vectors are independent of the arc length variable ‘s’.
The probabilistic method, pioneered by Paul Erdős, is a means of proving the existence of a certain object. By describing a random process of choosing objects, if there is a nonzero probability of making a successful choice, then necessarily the desired kind of object exists. We present a problem in discrete math that employs this method.
Let X be a genus 2g curve defined over an arbitrary field of characteristic not equal to 2 and let J(X) the Jacobian variety of X. We say that a Jacobian variety is decomposable if it is isogenous to a product of abelian varieties. The type of decomposition can by characterized by the type of kernel of the isogeny and the dimensions of the varieties in the product. We consider isogenies with kernel type (Z/2Z)g and products of dimension g Jacobian varieties. Additionally, we insist that the isogeny is polarized. In this talk we describe a family of (non-hyperelliptic) genus 2g curves whose Jacobians are decomposable in this way. We prove that all genus 4 curves whose Jacobian has this decomposition type are either in this family or arise from a different construction considered by Legendre. Joint work with Nils Bruin.
The lopsided version of the Lovasz Local Lemma gives asymptotically tight lower boundsfor a number of enumeration problems. In the configuration model matching upper bounds are available. In this way a number of asymptotic enumeration results, mostly due to Wormald and McKay, can be proved in an alternative way. A new result is asymptotic enumeration of graphs with respect to degree sequence and girth. A classical probabilistic result of Paul Erdos showed the existence of graphs with arbitrary large girth and chromatic number. If the degree sequence satisfies some mild conditions, we show that almost all graphs with this degree sequence and prescribed girth have high chromatic number.
This is joint work with Lincoln Lu.
Note for Attendees
Refreshments will be served at 2:45pm in the Math Lounge area, MATH 125 before the colloquium.
In 1997, Rozansky and Witten built new finite-type invariants of 3-manifolds from hyperkahler manifolds. It was later shown by Kontsevich and Kapranov that those invariants only depend on the holomorphic symplectic structure of the hyperkahler manifolds. Indeed Kapranov proved that these invariants may be considered as an analogue of Chern-Simons type invariants, where the Atiyah class of the underlying complex manifold plays the role of Lie bracket. In this talk, we introduce symplectic structures on "Lie pairs" of (real or complex) algebroids, encompassing homogeneous symplectic spaces, symplectic manifolds with a $\mathfrak g$-action and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky-Witten-type invariants of three-manifolds. This is a joint work with Yannick Voglaire.
It has long been known that the way a student views the subject they study affects the approach they take to studying the subject. This, in turn, affects their performance in the subject. It seems, then, that the improvement of student outcomes not only requires addressing the approach a student takes to study, but also their view of the subject. In this presentation, I will present results from a series of surveys intended to explore two separate, but related questions: 1. Do math instructors actually know how their students view math? 2. What approaches to study do students take and how do these relate to their achievement? The crucial aspect of this work is that the data gathered was analysed by course year. It turns out that the answers to both questions above are different for lower and upper-year courses.
In this talk, we will discuss the existence of the traveling wave solution for the Allen-Cahn equation involving the fractional Laplacians. Based on the existence of the standing waves for the balanced Allen-Cahn equation, we will use the continuity method to obtain the existence of the traveling waves for unbalanced Allen-Cahn equation. The key ingredient is the the bound of the traveling speed in terms of the potential. Some open questions will be discussed.
In this talk, we investigate the degree distribution of shortest path trees of various weighted network models. The aim of many empirical studies is to determine the degree distribution of a network with unknown structure by using trace-route sampling. We derive the limiting degree distribution of the shortest path tree from a single source on various random network models with edge weights: the configuration model and r-regular graphs with i.i.d. power law degrees and i.i.d. edge weights, the complete graph with edge weights that are powers of i.i.d. exponential random variables. We use these results to shed light on an empirically observed bias in network sampling methods.
While an aspherical complex is determined up to homotopy by its
fundamental group, there are many geometrically different aspherical
manifolds with the same fundamental group. For instance, the punctured
torus and the pair of pants look quite different, but both have the same
fundamental group F_2. I will discuss constructions of aspherical
manifolds for a given fundamental group, talk about the smallest dimension
of such a manifold for a given group and describe some geometric
invariants that distinguish different aspherical manifolds with the same
fundamental group.
I will discuss this for right angled Artin groups (joint work with Mike
Davis, Boris Okun and Kevin Schreve) and possibly also for duality groups.
We study the Noether and Lie symmetries that arise from the Euler-Lagrange equations, i.e., the ‘geodesic’ equations, related to manifolds that arise from a metric. In particular and as one of the examples, we present some peculiarities associated with the ASD Ricci-flat metric which depends on the `second heavenly equation'. It is noted, in general, that the Killing vectors are contained in the Noether symmetries generated by the Lagrangian of the geodesic equations. Specifically, a number of symmetries which are Noether and not Killing vectors are independent of the arc length variable ‘s’.
Obstructions to the Hasse principle for the existence of rational points on principal homogeneous spaces under connected linear algebraic groups over a number fi eld are well understood. Similar questions of Hasse principle have been studied over more general fi elds, particularly, function fields of curves over complete discrete valued fields. There are several positive results in this direction for connected linear algebraic groups which are rational, thanks to the patching techniques developed by Harbater-Hartmann-Krashen. We shall explain some recent progress and open questions concerning Hasse principle for function fi elds of p-adic curves.
The talk is concerned with the application of sensor array imaging in complex environments. The goal of imaging is to estimate the support of remote sources or strong reflectors using time resolved measurements of waves at a collection of sensors (the array). This is a challenging problem when the imaging environment is complex, due to numerous small scale inhomogeneities and/or rough boundaries that scatter the waves. Mathematically we model such complexity (which is necessarily uncertain in applications) using random processes, and thus study imaging in random media. I will focus attention on the application of imaging in random waveguides, which exhibits all the challenges of imaging in random media. I will present a quantitative study of cumulative scattering effects in such waveguides and then explain how we can use such a study to design high fidelity imaging methods.
Note for Attendees
Coffee, tea and cookies served at 2:30pm in the PIMS Lounge, ESB 4133.
Bacterial biofilms are microbial depositions that form on immersed surfaces wherever environmental conditions sustain bacterial growth. They have been called the most successful life form on Earth and cities of microbes. Biofilms have important applications in environmental engineering, but are detrimental in a medical or industrial context. They have been characterised as both, spatially structured microbial populations, and as mechanical objects. Life in biofilm communities significantly differs from life in planktonic cultures. This is reflected in the complexity of mathematical models of biofilms that are essentially more involved than models of suspended microbial communities. In this talk I will focus on a class of highly degenerate diffusion-reaction biofilm models. In its simplest form this includes simultaneously two nonlinear diffusion effects: (i) a porous medium equation like degeneracy when the dependent variable biomass density vanishes, and (ii) a super-diffusion singularity when it attains its {\it a priori} known upper bound. I will summarize some analytical (well-posedness) results, and discuss applications of the model to answer questions about biofilms by numerical simulations. I will hereby focus on the contribution of mathematical models (this and others) to understand the formation of cluster-and-channel biofilm architectures, and I will illustrate how our model framework, extended by a model of bacterial communication by quorum sensing, can be used to shed light on the transition from an initial mode of biofilm colonization to a protected mode of biofilm growth.
We are interested in the problem of solving a translation-invariant linear equation in a dense subset of the squares. We focus on the quality of density bounds, and we explain how the efficient energy increment method developed by Heath-Brown and Szemeredi for Roth's theorem can be adapted to this problem. A key tool in the process is a restriction estimate of Bourgain for lattice sets, and we discuss its role in our density increment strategy.
Our goal is to develop an innovative, robust, intuitive and affordable system for guiding needle insertion into the lumbar spine.
Ultrasound imaging will be the basis for guidance before and during needle insertion. The main application is facet joints and epidural/steroid injection for the relief of chronic back pain, and labour/analgesia. The anesthesiologist is expected to use the system to choose a suitable puncture location, insert the needle at an appropriate angle and stop when the needle reaches the desired depth. The goal is to increase the confidence level by enabling an accurate needle placement on the first attempt, and hence reduce complications and pain for the patient. In current practice, the needle insertion is either done blindly, using palpation to choose a puncture site, or under fluoroscopy/CT guidance, which carry a high radiation dose risk to both patient andanesthesiologist. We propose to add real-time ultrasound capability to these procedures and display the ultrasound, pre-procedure CT and needle trajectory to the anesthesiologist. We also propose to incorporate spine statistical atlas information for enhanced interpretation of ultrasound images. I will finish the talk by also overviewing some of our emerging works in cancer interventions using machine learning techniques.
This is a survey on the joint works with Wilhelm Schlag, Joachim Krieger and Tristan Roy. We classify global behavior of all solutions with energy up to slightly more than the ground state for the nonlinear Klein-Gordon, Schrodinger, and wave equations. The dynamics include scatteing (to 0), blow-up, and scatttering to solitons. The solutions scattering to solitons form threshold hypersurfaces in the energy space, giving a complete classification under the energy constraint. It also describes how a solution can disperse in the past and blow up in the future.
Wilf-equivalence is one of the central concepts of pattern-avoiding permutations, and has been studied for more than thirty years. The two known infi nite families of Wilf-equivalent permutation pairs, due to Stankova-West and Backelin-West-Xin, both satisfy the stronger condition of shape-Wilf-equivalence. Dokos et al. recently studied a di fferent strengthening of Wilf-equivalence called inv-Wilf-equivalence, which takes account of the inversion number of a permutation. They conjectured that all inv-Wilf-equivalent permutation pairs arise from trivial symmetries. We disprove this conjecture by constructing an infi nite family of counterexamples derived from the permutation pair (231) and (312). The key to this construction is to generalize simultaneously the concepts of shape-Wilf-equivalence and inv-Wilf-equivalence. A further consequence is a proof of the recent Baxter-Jaggard conjecture on even-shape-Wilf-equivalent permutation pairs.
Much of the literature on random walk in random environment assumes uniformly ellipticity, i.e., that nearest neighbour steps have probabilities bounded away from zero. I’ll describe some work with Mark Holmes (Univ. of Auckland) in which we relax this assumption, and allow some such steps to be forbidden. This leads naturally to percolation models, using which one can in some cases prove ballisticity of the random walks (existence of non-zero asymptotic speeds).
We present a simple technique for estimating parts of the Singular Value Decomposition (SVD) of a data matrix from a small randomly compressed "sketch" of that matrix. In sensor network settings--where each column of the data matrix comes from a separate sensor--the sketch can be assembled using operations local to each sensor. As an application of this work, we consider the problem of Structural Health Monitoring (SHM). SHM systems are critical for monitoring aging infrastructure (such as buildings or bridges) in a cost-effective manner. Such systems typically involve collections of battery-operated wireless sensors that sample vibration data over time. After the data is transmitted to a central node, modal analysis can be used to detect damage in the structure. We propose and study three frameworks for Compressive Sensing (CS) in SHM systems; these methods are intended to minimize power consumption by allowing the data to be sampled and/or transmitted more efficiently. At the central node, all of these frameworks involve a very simple technique for estimating the structure's mode shapes without requiring a traditional CS reconstruction of the vibration signals; all that is needed is to compute a simple SVD. We support our proposed techniques theoretically and using simulations based on synthetic and real data. This project is joint work with Anna Gilbert and Jae Young Park.
We begin by defining the Grassmann integral of a function of both commuting ("bosonic") and anti-commuting ("fermionic") variables. An important example is the mixed bosonic-fermionic ("supersymmetric") Gaussian integral, which exhibits a surprising self-normalization property. Time permitting, we will mention applications of the Grassmann integral to the representation of self-avoiding walk as a supersymmetric quantum field theory.
ABSTRACT
We introduce a new class of parallel parameter learning algorithms for Markov random fields (MRFs) with untied parameters, which are efficient for a large class of practical models.
The algorithms parallelize naturally over cliques and, for graphs of bounded degree, have complexity that is linear in the number of cliques. We refer to these algorithms with the acronym LAP, which stands for Linear And Parallel. Unlike their competitors, the marginal versions of the proposed algorithms are fully parallel and for log-linear models they are also data efficient, requiring only the local sufficient statistics of the data to estimate parameters. LAP algorithms are ideal for parameter learning in big graphs and big data applications.
The correctness of the newly proposed algorithms relies heavily on the existence and uniqueness of the normalized potential representation of an MRF. We capitalize on this theoretical result to develop a new theory of correctness and consistency of LAP estimators corresponding to different local graph neighborhoods.
This theory also establishes a general condition on composite likelihood decompositions of MRFs that guarantees the global consistency of distributed estimators, provided the local estimators are consistent.
We introduce a conditional variant of LAP that enables us to attack parameter estimation of fully observed models of arbitrary connectivity, including fully connected Boltzmann distributions. We show consistency for this distributed estimator, and relate it to distributed pseudo-likelihood estimators.
Finally, for linear and non-linear inverse problems with a sparse forward operator, we present a new algorithm, named iLAP, which decomposes the inverse problem into a set of smaller dimensional inverse problems that can be solved independently.
Let R be a discrete valuation ring with fraction field F. Two algebraic objects (say, quadratic forms) defined over R are said to be rationally isomorphic if they become isomorphic after extending scalars to F. In the case of unimodular quadratic forms, it is a classical result that rational isomorphism is equivalent to isomorphism. This has been recently extended to "almost umimodular" forms by Auel, Parimala and Suresh. I will present further generalizations to related objects: hermitian forms over involutary R-algebras, quadratic spaces equipped with a group action ("G-forms"), and systems of quadratic forms. The results can be regarded as versions of the Grothendieck–Serre conjecture for certain non-reductive groups. (Joint work with Eva Bayer–Fluckiger.)
I will discuss: 1. A new problem concerning monotone subsequences in random data; 2. Several approaches towards its solution; 3. Relations to some old problems from analysis, Ramsey theory and even probability. Based on work with Louigi Adarrio-Berry, Guillaume Chapuy, Luc Devroye, Gabor Lugosi, Neil Olver, Yuval Peres and Richard Balka.
Note for Attendees
Refreshments will be served at 2:45pm in the Math Lounge area, MATH 125 before the colloquium.
Contact networks are graphs modeling population contact patterns. They can represent contact heterogeneity and fixed partners. On random contact networks without clustering (triangles), contact heterogeneity implies that the basic reproduction number (indicating the risk for disease invasion) is determined by the average degree of a node found by following a random edge. that partners are fixed imply that a node cannot reinfect its neighbors before its neighbours recover, and thus two diseases with identical transmissibility and duration of infection have different basic reproduction numbers on the same contact network if one induced lifetime immunity while the other does not. When the average degree of a neighbor becomes large, the two basic reproduction numbers become the same. Another major difference between disease dynamics on networks and in homogeneously mixed populations is that, on networks, the basic reproduction number does not scale linearly with the population size in a growing population, as predicted by classical models. Instead, it may reach maximum when the population is still growing.
Note for Attendees
tea and cookies available before the talk in the IAM lounge (LSK 306)
The term “versal” is best understood by subtracting “unique” from both sides of the formula
Universal = unique + versal.
In this talk based on joint work with Alex Duncan, I will discuss competing notions of versality for the action of an algebraic group G on an algebraic variety X and relate these notions to properties (such as existence and density) of rational points on twisted forms of X. I will then present examples, where this relationship can be used to prove that certain group actons are versal or, conversely, that certain varieties have rational points.
According to Merriam-Webster, “airy” means “having a light or careless quality that shows a lack of concern”. That describes pretty accurately quantum physics. But Airy was also a mathematician and physicist that did a lot of things, and somehow got his name attached to a very simple complex curve. It turns out that this so-called Airy curve encapsulates intersection numbers on the moduli space of curves via topological recursion. Moreover, the Airy curve can be quantized; the resulting Schrodinger differential operator recursively constructs intersection numbers through WKB analysis. A natural question then is to ask whether this circle of ideas holds in a much more general setting; given a complex curve that encapsulates some nice enumerative invariants via topological recursion, does there exist a (unique?) quantization of the complex curve that reconstructs the invariants recursively via WKB analysis? This question has closed connections with many fundamental conjectures in enumerative geometry and other areas of mathematics, such as the AJ conjecture in knot theory, and Witten’s conjecture for intersection numbers. In recent work with B. Eynard we construct such a quantization in a number of different settings; in this talk I will focus on the case of the r-Airy curve, which generates intersection numbers on the moduli space of r-spin curves.
Differentiable stacks are generalizations of smooth manifolds suitable for modelling poor quotients, such as quotients by non-free Lie group actions. In this talk, we will define differentiable stacks and explain how they can also be used to model the leaf space of a foliation. In the following week, we will explain some recent results of ours about a nice subclass of differentiable stacks, called etale differentiable stacks, and explain some applications to foliation theory.
In this talk, we consider the unsteady motion of the
conducting fluid in the rotating Cartesian coordinates system. By using Lie
symmetries, we successfully
reduced the system of partial differential equations to an ordinary
differential equation and thereby do the analysis. We firstly, obtained steady
state solutions which lead to infinite number of time-dependent solutions via
three arbitrary functions of time. Finally, the plots of the solutions along
with their physical interpretation are presented to understand the flow
behavior.
In this talk we will give a brief overview of stochastic gradient pursuit and the closely related Kaczmarz method for solving linear systems, or more generally convex optimization problems. We will present some new results which tie these methods together and prove the best known convergence rates for these methods under mild Lipschitz conditions. The methods empirically and theoretically rely on probability distributions to dictate the order of sampling in the algorithms. It turns out that the choice of distribution may drastically change the performance of the algorithm, and the theory has only begun to explain this phenomenon.
In the year 2003 I first heard of the notion of extremal valued fields when Yuri Ershov gave a talk at a conference in Teheran. He proved that algebraically complete discretely valued fields are extremal. However, the proof contained a mistake, and it turned out in 2009 through an observation by Sergej Starchenko that Ershov's original definition leads to all extremal fields being algebraically closed. In joint work with Salih Durhan (formerly Azgin) and Florian Pop, we chose a more appropriate definition and then characterized extremal valued fields in several important cases.
We call a valued field (K,v) extremal if for all natural numbers n and all polynomials f in K[X_1,...,X_n], the set {f(a_1,...,a_n) | a_1,...,a_n in the valuation ring} has a maximum (which is allowed to be infinity, which is the case if f has a zero in the valuation ring). This is such a natural property of valued fields that it is in fact surprising that it has apparently not been studied much earlier. It is also an important property because Ershov's original statement is true under the revised definition, which implies that in particular all Laurent series fields over finite fields are extremal. As it is a deep open problem whether these fields have a decidable elementary theory and as we are therefore looking for complete recursive axiomatizations, it is important to know the elementary properties of them well. That these fields are extremal seems to be an important ingredient in the determination of their structure theory, which in turn is an essential tool in the proof of model theoretic properties.
Further, it came to us as a surprise that extremality is closely connected with Pop's notion of "large fields". Also the notion of tame valued fields plays a crucial role in the characterization of extremal fields. A valued field K with algebraic closure K^ac is tame if it is henselian and the ramification field of the extension K^ac|K coincides with the algebraic closure.
In my talk I will introduce the above notions, try to explain their meaning and importance also to the non-expert, and discuss in detail what is known about extremal fields and how the properties of large and of tame fields appear in the proofs of the characterizations we give. Finally, I will present some challenging open problems, the solution of which may have an impact on the above mentioned problem for Laurent series fields over finite fields.
Inter-universal Teichmuller theory, as developed by Mochizuki in the past decade, is an analogue for number fields of the classical Teichmuller theory, and also of the p-adic Teichmuller theory of Mochizuki. In this theory, the ring structure of a number field is subject to non-ring theoretic deformation. Absolute anabelian geometry, a refinement of anabelian geometry, plays a crucial role in inter-universal Teichmuller theory. In this talk, we will try to give an introduction to these ideas.
A metric measure space (mms) is simply a complete, separable metric space equipped with a probability measure that has full support. A fundamental insight of Gromov is that the space of such objects is much ``tamer'' than the space of complete, separable metric spaces per se because mms carry within themselves a canonical family of approximations by finite structures: one takes the random mms that arises from picking some number of points independently at random and equipping it with the induced metric and uniform probability measure. A natural (commutative and associative) binary operation on the space of mms is defined by forming the Cartesian product of the two underlying sets equipped with the sum of the two metrics and the product of the two probability measures. There is a corresponding notion of a prime mms and an analogue of the fundamental theorem of arithmetic in the sense that any mms has a factorization into countably many prime mms which is unique up to the order of the factors. Moreover, a rich Fourier theory enables one to analyze convolutions of probability measures on the space of mms and obtain counterparts of classical results in the theory of infinitely divisible and stable probability measures on Euclidean spaces due to L\'evy, It\^o, Hin\u{c}in, and LePage. This is joint work with Ilya Molchanov (Bern).
Note for Attendees
Refreshments will be served at 2:45pm in the Math Lounge area, MATH 125 before the colloquium.
I will give the definition of birational motives and explain how they can beused in various areas of algebraic geometry: counting rational points over finite fields, defining the "Tate-Shafarevich motive" of an abelian variety over a function field, shedding a new light on Roitman's theorem on torsion 0-cycles.
In this talk, I will talk about the Chern-Simons equation arising from the study of physics of high critical temperature superconductivity. A long-standing open problem is the existence of non-topological solutions. We proved the existence of non-topological solutions for the rank 2 Chern-Simons system. This is joint work with Professor Changshou Lin and Juncheng Wei
We will discuss a sorting phenomenon for data in a rectangular array, called the "non-messing-up" property. Consider a collection of distinct numbers arranged in a rectangle. The non-messing-up property says that if you put the numbers of each row into increasing order, and then do the same thing to the (possibly new) numbers of each column, then the (possibly new) numbers in each row will still be in increasing order. In other words, the "in increasing order" feature of the rows doesn't get messed up, even though the values in the rows may change! We will explore what it means to generalize this property, and will look at some of these generalizations in detail.
Algorithms and computer algebra systems play an invaluable role in modern day mathematics in both pure and applied fields. In this talk I will discuss the open source mathematical software system called Sage. Started and led by William Stein currently at the University of Washington and released in 2005, Sage is a free alternative to commercial mathematical computing languages such as Maple and MATLAB. I will demonstrate the functionality of Sage by discussing a select number of problems in number theory such as the prime number theorem, sociable numbers, and counting unique entries in the multiplication table.
A metric space is a length space if the distance between two points equals the infimum of the lengths of curves joining them. For a measured length space, we characterize Gaussian estimates for iterated transition kernel for random walks and parabolic Harnack inequality for solutions of a corresponding discrete time version of heat equation by geometric assumptions (Poincaré inequality and Volume doubling property). Such a characterization is well known in the setting of diffusion over Riemannian manifolds (or more generally local Dirichlet spaces) and random walks over graphs (due to the works of A. Grigor'yan, L. Saloff-Coste, K. T. Sturm, T. Delmotte, E. Fabes & D. Stroock). However this random walk over a continuous space raises new difficulties. I will explain some of these difficulties and how to overcome them. We will discuss some motivating examples and applications.
This is joint work with Laurent Saloff-Coste. (in preparation)
We will introduce infinity-topoi as generalized topological spaces, and show how using this language unifies the notion of manifold with that of etale differentiable stacks (generalized orbifolds) and their higher-categorical analogues. We will then give a completely categorical description of etale stacks in terms of a representability theorem. This theorem gives a recipe for constructing moduli stacks of geometric structures, and we will explain some examples of how this produces moduli-stacks presented by Lie groupoids that have been well studied in the foliation theory literature. Finally, we will explain how a generalization of Segal's theorem follows which describes the homotopy type of certain classifying spaces, and will explain the connection to the classification of foliations with transverse structures.
The purpose of this talk is give some of the ideas behind Prohorov's metric and weak convergence of probability measures. Then, I will briefly discuss the most important basic result in the topic: Prohorov's theorem.
Geometric motivations will be used to explain both the definition and the theorem. Also, we will discuss random measures and the definition of weak convergence in measure (and if time permits, I will talk a bit about what I have been working during my stay at UBC). Finally, the version of the Portmanteau theorem for this context will be stated and I will talk about what was that motivated it.
Some times in algebra, we run across mysterious objects defined in terms of so-called "universal properties." What are they even? What do they mean? We explore this problem from the view of category theory, and the way it suggests we should look at objects.
Discovery, Structuring, and Justification are three diverse processes of mathematical thinking that might be brought out from subtle alterations in a problem statement. I will discuss some examples and explain why I think this is a valuable practice in undergraduate education.
In this talk I will discuss modularity of reducible mod l Galois representations. By analogy with the irreducible case, I will state several questions regarding characterization and optimization of the weights and levels of the various cuspidal forms attached to such representations. Finally I'll give an application of these results to the determination of an explicit lower bound for the highest degree of the coefficient fields of newforms of prime level and trivial Nebentypus. This is a joint work with Ricardo Menares.
In the Mathematics Department of The University of Auckland a major research project into undergraduate mathematics learning outcomes has required the development of three innovative ways to deliver undergraduate mathematics. One of these involves reducing lectures to less than one per week, handing responsibility for most of their mathematical learning to students using web or text resources. We then use the staff time saved to provide semi-authentic mathematical experiences in which students work in small groups for up to two hours at a time guided by a lecturer on open-ended mathematical situations. Such sessions require new teaching skills and new learning orientations. There is some evidence that we have made progress on the development of mathematical process skills.
Our research shows that, with our small trial groups, students perform at similar levels on the conventional assessments as do the students in the standard courses.
In this talk, I will attempt to introduce/discuss modules ofdifferentials for Lie algebras modelled after the corresponding notion for rings. This is relevant to the structure of certain infinite dimensional Lie algebras. This is joint work with Arturo Pianzola.
We will describe some recent work in the numerical simulation of problems of Geodynamics. The relevant partial differential equations share some of the features of the well-known Stokes equations, but there are significant differences. Our work has been to create rapid solvers for the large systems of equations arising from finite element approximation. We will briefly describe the relevant models and our preconditioned Krylov subspace iterative solvers which enable some of the first computations on these models.
This is joint work with Sander Rhebergen, Richard Katz, Garth Wells, John Rudge and Laura Alisic.
We prove the existence of a sequence of nondegenerate, in the sense of Duyckaerts-Kenig-Merle, nodal nonradial solutions to the critical Yamabe problem or stationary energy-critical wave equation.
Spin systems such as the Ising model are central topics in statistical mechanics and probability theory. In the late 1960s Symanzik made the important discovery that properties of spin systems could be expressed in terms of the behaviour of random walks. This thesis contributes to the understanding of these connections by developing and analyzing random walk representations of graphical models arising in statistical mechanics.
Concretely, the results of this thesis can be divided into two parts. The first part is a lace expansion analysis of a model called loop-weighted walk. Loop-weighted walk is a non-Markovian model of random walks that are discouraged (or encouraged), depending on the value of a parameter, from completing loops. The model arises naturally as a random walk representation of correlations in a statistical mechanics model called the cycle gas. A challenging aspect of this model is that it is not repulsive, meaning the weight of the future of a walk may either increase or decrease if the past is forgotten. Loop-weighted walk is the first finite range walk model with this property to be analyzed with lace expansion techniques.
The second part of this thesis is an essentially elementary derivation of a random walk representation for the partition function of the Ising model on any finite graph. Such representations have a long history for planar graphs. For non-planar graphs the additional ingredient needed is a way to compute the intersection numbers of curves on surfaces. The representations for non-planar graphs lead to random walk representations of spin-spin correlation functions that were previously unknown.
Given a simple graph $G$, a triangle decomposition of $G$ is a set of subgraphs isomorphic to $K_3$ whose edges partition the edge set of $G$. Further, a rational triangle decomposition of $G$ is a non-negative rational weighting of the copies of $K_3$ in $G$ such that the total weight on any edge of $G$ equals one. In this thesis, we will explore sufficient conditions for rational triangle decomposability. A famous conjecture in the area due to Nash-Williams states that any sufficiently large graph (satisfying some divisibility conditions) with minimum degree at least $3/4v$ is admits a triangle decomposition; the same conjecture stands for rational triangle decomposability (no divisibility conditions required). By perturbing and restricting the coverage matrix of a complete graph, we show that minimum degree of at least $22/23v$ is sufficient to guarantee that the given graph is rationally triangle decomposable. This density bound is a great improvement over the previously known results and is derived using estimates on the matrix norms and structures originating from association schemes.
We discuss recent results on phase transitions of Delaunay Potts models in dimension two where the interaction depends on Delaunay edges respectively Delaunay triangles.
This work is an extension of the Lebowitz & Lieb soft-core continuum Potts model to geometrically dependent interaction systems. The main tool is a FK (Fortuin-Kasteleyn) random cluster representation adapted to the Delaunay structure and percolation in the FK model. If time permits we discuss the Voronoi-Ising model where the interaction is function of the length of the common boundary and its connection to variants of the RSW (Russo-Seymour-Welsh) theorem in continuum percolation.
The formalism of six operations encodes the functorial behavior of (co)homology theories. It was first introduced by Grothendieck for the l-adic cohomology of schemes, and was later developed in a variety of other geometric contexts: D-modules on schemes, spectra parametrized by topological spaces, motivic spectra parametrized by schemes, etc. Equivariant homotopy theory is also best understood as a formalism of six operations for topological stacks.
In this talk I will discuss the basics and the significance of this formalism, and I will then describe an extension of motivic homotopy theory to algebraic stacks.
In this seminar I will discuss a function field analogue of a classical problem in analytic number theory, concerning the auto-correlations of divisor functions, in the limit of a large finite field.
We propose geometric flows to study the existence of a symplectic structure on an almost Hermitian manifold. We prove the short-time existence and uniqueness, and show some examples.
Magnetized target fusion reactors are a modern design to for hydrogen fusion energy on earth. The design entails confining a plasma with a magnetic field and crushing it in an imploding shell of molten metal. Such a design has many unresolved questions in terms of its feasibility as a power source and the most important elements in making it efficient. In this talk, we will look into two of the approaches undertaken to explore these questions. Firstly, through a coordinate transformation and implementing a novel flux-limited, split-step, finite volume scheme for nonlinear coupled hyperbolic partial differential equations, we do a parameter sensitivity analysis for the design performance. Secondly, by a careful series of asymptotic arguments, we establish a leading order asymptotic expression for the plasma compression. This expression is qualitatively consistent with the numerical work, but it also gives new insights into how the device operates. We will conclude with a look into the viability of magnetized target fusion and its future work.
Note for Attendees
Refreshments will be served at 2:45pm in the Math Lounge area, MATH 125 before the colloquium.
The Fukaya category of open symplectic manifolds is expected to have good local-to-global properties. Based on this idea several people have developed sheaf-theoretic models for the Fukaya category of punctured Riemann surfaces: the name topological Fukaya category appearing in the title refers to the (equivalent) constructions due to Dyckerhoff-Kapranov, Nadler and Sibilla-Treumann-Zaslow. In this talk I will introduce the topological Fukaya category and explain applications to Homological Mirror Symmetry for toric Calabi-Yau threefolds. This is work in progress joint with James Pascaleff.
The sparse inverse covariance estimation problem arises in many statistical applications in machine learning and signal processing. In this problem, the inverse of a covariance matrix of a multivariate normal distribution is estimated, assuming that it is sparse. An l-1 regularized log-determinant optimization problem is typically solved to approximate such matrices. Because of memory limitations, most existing algorithms are unable to handle large scale instances of this problem.
In this talk we present two contributions. First, we present a new block-coordinate descent (BCD) approach for solving the problem for large-scale data sets. Our method treats the sought matrix block-by-block using quadratic approximations, and we show that this approach has advantages over existing methods in several aspects. Next, we present an iterative multilevel framework for accelerating the solution of general convex optimization problems with sparsity promoting l-1 regularization. Taking advantage of the typical sparseness of the solution, we create a multilevel hierarchy of similar problems, which are traversed back and forth in order to accelerate the optimization process. We demonstrate this framework for solving the sparse inverse covariance estimation problem. Numerical experiments on both synthetic and real gene expression data sets demonstrate our BCD and multilevel approaches for solving both medium and large scale instances of this problem.
Collaborators:
Javier Turek & Irad Yavneh, CS dept. Technion Israel Institute of Technology.
In the standard approach to tropicalization, an algebraic subset X of a toric variety over a non-archimedean valued field k is sent to a weighted polyhedral set Trop(X) which we think of as a combinatorial shadow of X. The result depends only on the k-points of X. A system of polynomial equations often contains more information than the set of its solutions over a field, and the philosophy of scheme theory is that we should treat the system of equations itself as a fundamental geometric object from which the solution set is derived. Scheme-theoretic tropicalization is about realizing Trop(X) as the solution set to an underlying system of polynomial equations over the idempotent semiring of tropical numbers - a system that is constructed in a canonical way from the equations defining X. The theory involves the field with one element, and with these ideas the Berkovich analytification appears as the universal tropicalization of X and as the moduli space of valuations on X.
Note for Attendees
The seminar will start at 2:30, rather than the usual 3pm, in order to avoid a conflict with the 4pm colloquium.
Tea will be served at 2:15.
A man with a bad memory is given n bins into which he must place n balls, as uniformly as possible. Balls are given to him one at a time, and he must place the ball he has before he receives the next one. If for each ball he randomly chooses the bin, then after adding all n balls the most heavily loaded bin will have around log n/log log n balls. If instead, he chooses two bins at random, and then he places the ball in the less loaded bin, the most loaded bin after n steps will have around log log n balls. When n is large, this represents a dramatic improvement for little extra effort.
This phenomenon is referred to as the 'power of two choices.' In this talk, we will see how the power of two choices survives when we attempt to adapt it to two other settings: preferential attachment graphs and the spacings of randomly distributed points in an interval. In each of these settings, we'll see that adding two choices can have dramatic effects on the behavior of the random processes, though not always in the way one might hope.
Based on joint work with Yury Malyshkin (Tver) and Pascal Maillard (Orsay).
Note for Attendees
Refreshments will be served at 3:45pm in the Math Lounge area, MATH 125 before the colloquium.
Ribbon graphs provide a powerful combinatorial tool in the study of the moduli space of Riemann surfaces. The theory of quadratic differentials in complex analysis gives a cellular decomposition of the moduli space indexed by ribbon graphs, and this allowed the computation of the Euler characteristic and Kontsevich’s proof of Witten’s intersection number conjecture. Costello found a different ribbon graph model in his work constructing the B-model counterpart to Gromov-Witten theory in terms of topological field theories. In this talk I will review these ideas and describe how to produce Costello-type combinatorial models of moduli spaces of many related classes of objects, such as unoriented, spin and r-spin surfaces, surfaces with G-bundles, and 3-dimensional handlebodies.
We consider the hyperbolic Poisson Voronoi (HPV) tessellation, a triangulation of the hyperbolic plane whose vertices are given by a homogeneous Poisson point process. This triangulation fails to have a positive isoperimetric constant, however we show that it does have a positive "anchored" isoperimetric constant. HPV is an example of a stationary random graph, one which when viewed from the point of view of random walk, has the same law at all times. We review some of the theory of stationary random graphs and give some extensions that allow us to conclude random walk on HPV is ballistic and converges almost surely to a point on the boundary.
This is joint work with Itai Benjamini (Weizmann) and Josh Pfeffer (Harvard).
The central limit theorem of probability theory asserts under very general assumptions that properly renormalised sums of independent and identically random variables converge to a normal distribution. It can be viewed as a global stability result for a fixed point of a dynamical system. The dynamical system is given by sucessive convolution and the fixed point is the normal distribution. Here independence plays an important role by making the dynamical system autonomous. In statistical mechanics, collections of very strongly dependent random variables are at the heart of many problems. The renormalisation group is a grand extension of the dynamical view of the central limit theorem to systems with strong dependence and spatial structure, in which non-trivial phase portraits arise. I will discuss its background and some applications.
Spin systems generalise the Ising model to spins with a general number of components and general distributions. I will discuss a result with Brydges and Slade on the specific heat of the 4D |\varphi|^4 model, in which we obtain the precise asymptotic behaviour for the approach of the critical point for a general number of components.
Subword complexes are simplicial complexes introduced by A. Knutson and E. Miller as a tool to study Gröbner geometry of Schubert polynomials. In this talk, I will present some relevant results about of these objects in combinatorics, discrete geometry, and algebra. In particular, I will focus on:
- combinatorics of triangulations and multi-triangulations of convex polygons,
- two applications in cluster algebras and Hopf algebras, and
- geometric constructions of multi-associahedra.
This talk is based on joint works with Nantel Bergeron, Jean-Philippe Labbé, Vincent Pilaud, and Christian Stump.
Note for Attendees
Refreshments will be served at 2:45pm in the Math Lounge area, MATH 125 before the colloquium.
The usual Atiyah--Segal "functorial" description of quantum field theory corresponds to the "Schrodinger picture" in quantum mechanics. I will describe a slight modification that corresponds to the "Heisenberg picture", which I will argue is better physically motivated. The example I am most interested in is a version of quantum Chern--Simons theory that does not require the level to be quantized; it provides a neat packaging of pretty much all objects of skein theory.
The stochastic block model is a random graph model that was originally introduced 30 years ago tomodel community structure in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.
Note for Attendees
Refreshments will be served at 3:40pm in the Math Lounge area, MATH 125 before the colloquium.
Structure-preserving discretizations are numerical methods which can preserve important structures of differential equations at the discrete level. For systems with a Hamiltonian or variational structure, geometric integrators such as symplectic and variational integrators are a class of discretizations that can preserve symplectic structure, first integral, phase space volume or symmetry at the discrete level. In this talk, we present the multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations. The proposed discretization is shown to be consistent for any order of accuracy and that by construction, the discrete densities can be exactly conserved. In particular, the multiplier method does not require the system to possess a Hamiltonian or variational structure. Examples, including dissipative problems, are given to illustrate the method. This is joint work with Alexander Bihlo at Memorial University and Jean-Christophe Nave at McGill University.
Given two correlated Gaussian vectors, X and Y, the noise stability of a set A is the probability that both X and Y fall in A. In 1985, C. Borell proved that half-spaces maximize the noise stability among all sets of a given Gaussian measure. We will give a new, and simpler, proof of this fact, along with some extensions and applications. Specifically, we will discuss hitting times for the Ornstein-Uhlenbeck process, and a noisy Gaussian analogue of the "double bubble" problem.
I will first discuss doubling and desingularization constructions for minimal surfaces and applications on closed minimal surfaces in the round spheres, free boundary minimal surfaces in the unit ball, and self-shrinkers for the Mean Curvature flow. In the final part of the talk I will discuss my collaboration with Simon Brendle on constructions for Einstein metrics on four-manifolds and related geometric objects.
In the first part, I will talk about random subdivisions obtained from projections of polytopes. These are related to random polytopes and zeros of random tropical polynomials. In the second part, I will discuss results and open problems in neural coding, with emphasis on decoding grid cells.
We consider a drift-diffusion process on a smooth potential landscape with small noise. We give a new proof of the Eyring-Kramers formula which asymptotically characterizes the spectral gap of the generator of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald, Otto, Villani, and Westdickenberg and of the mean-difference estimate introduced by Chafai and Malrieu. The new proof exploits the idea that the process has two natural time-scales: a fast time-scale resulting from the fast convergence to a metastable state, and a slow time-scale resulting from exponentially long waiting times of jumps between metastable states. A nice feature of the argument is that it can be used to deduce an asymptotic formula for the log-Sobolev constant, which was previously unknown.
An admissible representation of a reductive p-adic group has its character as a distribution on the group, invariant under conjugation. The asymptotic behavior of the character is given by so-called Harish-Chandra–Howe local character expansion, which expressed the character near the identity in terms of a finite linear combination (of Fourier transforms of nilpotent orbital integrals). In this talk, we show examples about how the coefficients in this expansion arise as the numbers of rational points on varieties over the residue field, which will be certain covers of hyperelliptic curves in our example. We also talk about how the endoscopy transfer identity appears as geometric identities regarding the first cohomology of these curves.
The log-Sobolev inequality (LSI) is a very useful tool for analyzing high-dimensional situations. For example, the LSI can be used for deriving hydrodynamic limits, for estimating the error in stochastic homogenization, for deducing upper bounds on the mixing times of Markov chains, and even in the proof of the Poincaré conjecture by Perelman. For most applications, it is crucial that the constant in the LSI is uniform in the size of the underlying system. In this talk, we discuss when to expect a uniform LSI in the setting of unbounded spin systems. We will also explain a connection to the KLS conjecture.
Consider the plane with the discrete topology. A function belongs to its Fourier algebra if and only if that function is equal to the convolution of two square-summable functions. Call such a product weak on a set if its restriction to that set is square-summable. We show that weakness on the interior of a strictly convex subset of the plane implies weakness on the boundary of that subset. Here the words "interior" and "boundary" refer to the usual notions in the usual topology on the plane. We also show that weakness on the boundary follows from weakness on the interior of the complement of a strictly convex set in the plane. That is essentially due to V.A. Yudin, via a dual method. Our methods here are direct and visual.
The Hales--Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated result of Shelah from 1988 gives a significantly improved bound for this theorem. A key tool used in his proof, now known as the cube lemma, has become famous in its own right. Hoping to further improve Shelah's result, more than twenty years ago, Graham, Rothschild and Spencer asked whether there exists a polynoimal bound for this lemma. In this talk, we present the answer to their question and discuss numerous connections of the cube lemma with other problems in Ramsey theory.
Joint work with David Conlon (Oxford), Jacob Fox (MIT), and Benny Sudakov (ETH Zurich)
Note for Attendees
Refreshments will be served at 3:40pm in the Math Lounge area, MATH 125 before the colloquium.
We will review the state-of-the art techniques in the parallel direct solution of linear systems of equations and present several recent new research directions. This includes (i) fast methods for evaluating certain selected elements of a matrix function that can be used for solving the Kohn-Sham-equation without explicit diagonalization and (ii) stochastic optimization problems under uncertainty from power grid problems from electrical power grid systems. Several algorithmic and performance engineering advances are discussed to sove the underlying sparse linear algebra problems. The new developments include novel incomplete augmented multicore sparse factorizations, multicore- and GPU-based dense matrix implementations, and communication-avoiding Krylov solvers. We also improve the interprocess communication on Cray systems to solve e.g. 24-hour horizon power grid problems from electrical power grid systems of realistic size with up to 1.95 billion decision variables and 1.94 billion constraints. Full-scale results are reported on Cray XC30 and BG/Q, where we observe very good parallel efficiencies and solution times within a operationally defined time interval. To our knowledge, "real-time"-compatible performance on a broad range of architectures for this class of problems has not been possible prior to present work.
An elliptic curve in characteristic p is either ordinary or supersingular, depending on whether or not it has points of order p. It is known that elliptic curves are typically ordinary, and also exactly how many are super-singular for each prime p. However, for higher genus curves little is known. In this talk, we will discuss several higher genus generalizations of supersingular elliptic curves, focussing on the hyperelliptic case. In particular we discuss recent heuristics, computational results, and theorems on the proportion of hyperelliptic curves that are non-ordinary.
In this talk, we consider symmetries and conservation laws of FPDEs equation with Riemann-Liouville derivatives. First, we briefly review Lie group methods and the construction of conservation laws for PDEs. Within the framework of Lie group theory, we extend Lie group analysis to solve problems involving FPDEs. Finally, we give some examples to illustrate applications of the methods. Some open questions will be discussed.
Algebraic K-theory is a subtle and remarkable invariant of rings (as well as more general objects). In this talk, I will describe recent advances that demonstrate that it is the natural stable homotopy theory of higher categories, and I will explain how this description provides new approaches both to structures on algebraic K-theory and to an important web of conjectures of Waldhausen, Hopkins, and Rognes.
Note for Attendees
Refreshments will be served at 2:40pm in the Math Lounge area, MATH 125 before the colloquium.
I start with DWAVE, a successful BC company building and selling quantum computers. I go through the math, the chip and the controversies. I explain how one can program DWAVE machines and solve NP hard problems. I then talk about measurement based quantum computation (MBQC, also a BC product), lots of math, no controversies but no machines. I explain how it relates to Einstein–Podolsky–Rosen paradox and topos theory.
Note for Attendees
Tea and cookies beforehand in the IAM lounge, LSK 306
Polarized K3 surfaces of genus g can be thought of as families of canonical curves. As such, their moduli space K_g has similar properties to M_g. For instance, both are unirational for low values of g, and both have discrete Picard group. In this talk, we will use the explicit unirationality of K_g to compute its Picard number in a few cases, which verifies the Noether-Lefschetz conjecture for genus up to 10.
Azumaya Algebras are a generalization of central simple algebras over fields, and have been studied since the 1950s. In this talk, I shall explain how topological obstruction theory for PGLn bundles can be used to answer questions about Azumaya Algebras over rings.
Note for Attendees
Refreshments will be served at 3:40pm in the Math Lounge area, MATH 125 before the colloquium.
The Closest Point Method is a set of mathematical principles and associated numerical techniques for solving partial differential equations (PDEs) posed on curved surfaces or other general domains. The method works by embedding the surface in a higher-dimensional space and solving the PDE in that space using simple finite difference and interpolation schemes.
This presentation outlines how a chance encounter with instability improved our understanding of the method and is leading to new formulations with proven convergence properties.
We will also briefly survey some applications in thin-film flows, reaction-diffusion equations, bulk-surface coupling, point clouds, and image processing.
Abstract: We will present probabilistic and analytic properties of a class of degenerate diffusion operators arising in population genetics, the so-called generalized Kimura diffusion operators. Such processes arise as models for the evolution of gene frequencies. We will start by highlighting the main questions of interest and the mathematical difficulties in addressing them. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients, and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman-Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients, and on the a priori regularity of the weak solutions.
We parametrize genus 2 curves with a maximal isotropic (Z/3)2 in their Jacobian, together with an explicit description of the associated isogeny. This allows us to perform (3, 3)-isogeny descent on various simple principally polarized abelian surfaces and exhibit non-trivial 3-part in their Tate-Shafarevich groups. This is joint work with Victor Flynn and Damiano Testa.
The fractional Laplacian operator plays the same paradigmatic role in the theory of nonlocal operators that the Laplacian plays in the theory of local operators. We will present regularity results for solutions to problems defined by the fractional Laplacian operator with gradient perturbations. Our main results are the regularity of solutions in Sobolev spaces to the linear equation in the supercritical regime, when the operator is not elliptic, and the optimal regularity of solutions to the stationary obstacle problem in the supercritical regime.
This is joint work with Charles Epstein and Arshak Petrosyan.
Note for Attendees
Refreshments will be served at 2:40pm in the Math Lounge area, MATH 125 before the colloquium.
I will talk about the notion of "infinite root stack" of a logarithmic scheme, introduced by myself and Angelo Vistoli as part of my PhD thesis. It is a "limit" version of the generalization to log schemes of the stack of roots of a divisor on a variety, and we show, among other things, that its "bare" geometry closely reflects the "log" geometry of the base log scheme. After giving some motivation, I will briefly define log schemes and describe this infinite root construction. I will then state the results we get about it, and their relevance to log geometry, also in view of (hopefully) upcoming applications.
Environmental applications, such as predicting climate impacts and feedbacks in critical watersheds, pose significant challenges for modeling and simulation. These applications are inherently multiscale, and uncertainty about models and model coupling is common. Thus, to manage this complexity, new interdisciplinary community-based approaches are needed. Here, we highlight advances in open-source scientific libraries, frameworks, and software development methodologies that have led to a growing number of open-source analysis and simulation tools. We present results from our open-source simulator, Amanzi/ATS (Arctic Terrestrial Simulator), for subsurface contaminant transport at a representative waste site and coupled surface/subsurface flow in the Arctic.
Many scientific computing applications can be formulated as large-scale optimization problems, including inverse problems, medical and seismic imaging, classification in machine learning, data assimilation in weather prediction, and sparse difference graphs. While first-order methods have proven widely successful in recent years, recent developments suggest that matrix-free second-order methods, such as interior-point methods, can be competitive.
This talk has three parts. We first develop a modeling framework for a wide range of problems, and show how conjugate representations can be exploited to design a uniform interior point approach for this class. We then show a range of applications, focusing on modeling and special problem structure. Finally, we preview some recent work, which suggests that the conjugate representations admit very efficient matrix free methods in important special cases, and present some recent results for large scale extensions.
Note for Attendees
Refreshments will be served at 3:40pm in the Math Lounge area, MATH 125 before the colloquium.
Modeling and simulation are playing an increasingly critical role in understanding and predicting climate impacts and feedbacks in terrestrial systems. Managing the complexity of these process-rich integrated hydrologic and biogeochemical models requires flexible software designs that enable exploration of model features and model coupling. In addition, flexibility in meshing and robust discretization techniques are required to capture topographic features, such as hill slopes and rivers, and subsurface stratigraphy.
In this talk we highlight a flexible and extensible approach to multiphysics frameworks for these applications that specifies interfaces for coupled processes and automates weak and strong coupling strategies to manage this complexity. Process management is accomplished through a dual view of the model system: a high-level view ideal for model configuration represents the system of equations as a tree, where individual equations are associated with leaf nodes, and coupling strategies with internal nodes; and a low-level dynamically generated dependency graph that connects a variable to its dependencies, streamlining and automating model evaluation, easing model development, and ensuring models are modular and flexible. We use this multiphysics framework, dubbed Arcos, to support both model and algorithm development for environmental applications in the open-source code Amanzi. For example, we have developed infrastructure for general unstructured polyhedral meshing with a flexible operator-based implementation of the Mimetic Finite Difference method, and used it to simulate coupled surface/subsurface flow. Here we use a diffusive wave approximation for surface flow, and a Richards equation for subsurface flow. Coupling is accomplished by ensuring continuity of both pressure and fluxes from the surface to the subsurface, and the system can be solved using either sequential or implicit coupling. We show results for several benchmark problems, as well as physically relevant, large-scale simulation of rainfall on arctic tundra based upon LIDAR data from Barrow, Alaska. This demonstration shows the parallel performance of the code and its feasibility for use in watershed scale, high resolution simulations.
Active fluids, the novel class of non-equilibrium materials made up of self-driven constituents, is attracting growing interest due to its impact on cell biology, condensed matter physics, and nanotechnology. Despite their differences in composition, active fluids orchestrate cooperative actions across various length and time scales, and accompany energy conversion from one form to another. In this talk, I focus on a recent study of nonlinear dynamics and pattern formation of microtubule/motor-protein assemblies using multiscale modeling and simulation. I explain how the local microtubule-motor and microtubule-microtubule interactions manifest themselves at macroscopic scales through hydrodynamic instability, as well as the connections between the coherent flow structures and topological defects observed in active nematics. I also briefly discuss the new physics and phenomena in other systems, and numerical and theoretical tool development for the quantitative study of general complex fluids.
The problem of optimal transportation, which involves finding the most cost-efficient mapping between two measures, arises in many different applications. However, the numerical solution of this problem remains extremely challenging. After surveying several current applications, we describe a numerical method for the widely-studied case when the cost is quadratic. The solution is obtained by solving the Monge-Ampere equation, a fully nonlinear elliptic partial differential equation (PDE), coupled to anon-standard implicit boundary condition. Expressing this problem in terms of weak (viscosity) solutions enables us to construct a monotone finite difference approximation that provably converges to the correct solution. A range of challenging computational examples demonstrate the effectiveness of this method.
Darmon outlined a program which is suited to potentially resolving one parameter families of generalized Fermat equations. He gave explicit descriptions of Frey representations and conductor calculations for Fermat equations of signature (p,p,r). Somewhat less explicit results are stated for signature (r,r,p), and even less for signature (q,r,p).
For the equation (r,r,p), there are at least three competing Frey curve constructions: superelliptic curves of hypergeometric type due to Darmon, hyperelliptic curves due to Kraus, and elliptic curves with models over totally real fields due to Freitas.
I will survey these Frey curve constructions and end by giving explicit Frey hyperelliptic curves for signatures (2,r,p) and (3,5,p).
Unconventional fossil energy resources are revolutionizing the US energy market. While the techniques developed over the last 50 years lead to viable and profitable extraction of, e.g., trapped gas and hydrocarbons from almost-impermeable rock formations via hydraulic fracturing, the abysmal extraction rates (typically 15%) suggest the fluid mechanics of these processes is not well understood. In this talk, I will describe three basic theoretical fluid mechanics problems inspired by unconventional fossil fuel extraction. The first problem is flow in a deformable microchannel. Fluid-structure interaction couples the shape of the conduit to the flow through it, drastically altering the flow rate--pressure drop relation. Using perturbation methods, we show that the flow rate is a quartic polynomial of pressure drop for shallow channels, in contrast to the linear relation for rigid conduits. The second problem involves two-phase (gas-liquid) displacement in a horizontal Hele-Shaw cell with an elastic membrane as the top boundary. This problem arises at the pore-scale in enhanced oil recovery for large injection pressures. Once again, fluid-structure interaction alters the problem, leading to stabilization of the Saffman--Taylor (viscous fingering) instability below a critical flow rate. Using lubrication theory, we derive the stability threshold and show that it agrees well with recent experiments. The third problem involves the spread of a viscous liquid in a vertical Hele-Shaw cell with a variable thickness in the flow-wise direction, as a model for the spread of a plume of supercritical carbon dioxide through the non-uniform passages created by hydraulic fracturing. We show that the propagation regimes in such a shaped conduit are set by the direction of propagation. While the rate of spread in the direction of increasing gap thickness (and, hence, permeability) can be obtained by standard scaling techniques, the reverse scenario requires the construction of a so-called second-kind self-similar solution, leading to nontrivial exponents in the rate of spread.
Note for Attendees
Refreshments will be served at 3:45pm in the PIMS Lounge.
We say that a matrix F is a configuration in a matrix A if there is a submatrix of A which is a row and column permutation of F. We consider the following extremal function. Let G be a finite set of (0,1)-matrices. Let forb(m,G) denote the maximum number of columns in an m-rowed (0,1)-matrix A that has no repeated columns and no configuration F in the set G. It was already shown by Balogh and Bollobas that for any given k that if G consists of the three matrices I_k (identity matrix of order k),I_k^c (0-1 complement of I_k) and T_k (upper triangular (0,1)-matrix) then forb(m,G) is a constant where the constant is at least 2^{ck} for some constant c and at most 2^{2^k}. These three are unavoidable configurations in much the same way that a clique of size k and an independent set of size k are unavoidable in a large complete graph, you always get one or the other. We obtain a new argument (using Ramsey Theory) that brings the bound on forb(m,G) to 2^{ck^2}. This is joint work with Lincoln Lu.
Problems involving complicated deforming time-dependent boundaries or interfaces (i.e. codim-1 surfaces) are ubiquitous in the modeling of physical systems. The resulting Partial Differential Equations (PDEs) often have irregular, and even discontinuous solutions along these surfaces. In turn, the solution of these PDEs couples back to flow and hence deform the surfaces in question. The numerical approximation of such systems isnotoriously challenging.
In a regular Cartesian grid setting, I propose to replace the original PDE by another PDE which better approximates, in the discrete setting, the exact solution to the original PDE. I will provide 3 examples of this approach.
First, in the active penalty method one does not enforce the PDE boundary conditions directly, but rather solves the PDE in a larger domain without boundary (e.g. a flat torus) and adds a carefully constructed source or penalty term that mimics boundary conditions. I will show how to systematically construct penalty terms which improve the convergence rates of the penalized PDE, thereby allowing for higher-order finite-difference or Fourier-spectral numerical schemes to be applied to problems withnon-conforming boundaries.
Second, in the correction function method I tackle the problem of solving PDEs with jump discontinuities across a codim-1 surface, i.e. an interface. This is achieved by formulating an auxiliary local PDE which solution smoothly extends across the interface while enforcing the jump conditions. I will show that this approach is general, and can achieve arbitrary order of convergence while incurring (asymptotically) no additional computational cost.
Third, I will present methods for evolving in time arbitrary geometric objects such as boundaries or interfaces, but also general open/closed
surfaces with possibly no regularity (e.g. fractals). This is achieved by evolving in time the flow map and composing it with the initial conditions. This method fit naturally within the gradient-augmented level set framework and enables use of a two-grid approach to achieve arbitrary order of convergence and optimal efficiency.
Throughout the talk I will illustrate these approaches with simulations of various physical systems including problems from fluid dynamics,
electromagnetism, solid mechanics for which these methods may need to be combined.
Note for Attendees
PIMS Afternoon Tea will be served at 2:45 p.m. in the PIMS Lounge (ESB 4133).
I will describe a general technique to extend various results about non-degenerate quadratic and hermitian forms to systems of (possibly degenerate) hermitian forms. The idea is based on a certain categorical equivalence. Results that can be extended in this manner include: Witt’s cancellation theorem, Springer’s theorem, the weak Hasse principle (in certain cases), and statements about about finiteness of the genus.
(Joint work with Eva Bayer-Fluckiger and Daniel Moldovan)
Particulate flows are ubiquitous in environmental, geophysical and engineering processes. The intricate dynamics of these two-phase flows is governed by the momentum transfer between the continuous fluid phase and the dispersed particulate phase. When significant temperature differences exist between the fluid and particles and/or chemical reactions take place at the fluid/particle interfaces, the phases also exchange heat and/or mass, respectively. While some multi-phase processes may be successfully modelled at the continuum scale through closure approximations, an increasing number of applications require resolution across scales, e.g. dense suspensions, fluidized beds. Within a multi-scale micro/meso/macro-framework, we develop robust numerical models at the micro and meso-scales, based on a Distributed Lagrange Multiplier/Fictitious Domain method and a two-way Euler/Lagrange method, respectively. Particles, assumed to be of finite size, potentially collide with each other and these collisions are modeled with a Discrete Element Method. We discuss the mathematical issues related to modeling this type of flows and present the main numerical and computational features of our simulation methods. We also illustrate what can be gained from massively parallel computations performed with our numerical code PeliGRIFF, in terms of physical insight into both fundamental questions and applications from the chemical engineering and process industry. Finally, we explain how knowledge gained at the micro scale can cascade upwards and contribute to the development of enhanced meso and macro-scale models.
Speaker Biography: Dr Anthony Wachs received BS & MS degrees from University Louis Pasteur, Strasbourg and his PhD from the Institut National Polytechnique of Grenoble in 2000. He joined Institut Français du Pétrole in 2001 (now IFP Energies Nouvelles), passed his HDR in 2010 and is currently both scientific advisor and project manager. He leads a team of researchers that develop both mathematical models and robust computational algorithms for the resolution of multi-phase flows (www.peligriff.com). His main research interests are non-Newtonian Flows, Multiphase Flows and High Performance Computing. He collaborates extensively with academic groups in Canada, Brazil, France and Germany.
Note for Attendees
Refreshments will be served in the PIMS Lounge (ESB 4133) at 3:15p.m. before this colloquium.
The primitive equations are the main system of nonlinear partial differential equations on which modern weather and climate prediction models are based on. The Lie symmetries of the primitive equations are computed and the structure of the maximal Lie invariance algebra, which is infinite dimensional, is investigated. It is found that the maximal Lie invariance algebra for the case of a constant Coriolis force can be mapped to the case of vanishing Coriolis force. The same mapping allows one to transform the constantly rotating primitive equations to the equations in a resting reference frame. This mapping is used to obtain exact solutions for the rotating case from exact solutions from the nonrotating equations.
We will introduce the Bartnik mass and survey the progress toward its understanding. We will then present a new upper bound which is sharp in certain cases. It is contained in a recent joint work with C. Mantoulidis and has relevance to a conjecture of G. Gibbons.
Adhesive Dynamics is a method to simulate the dynamics of cell adhesion to surfaces. Adhesion receptors are modeled as reactive mechanical entities with adhesive tips, and the formation and breakage of adhesion molecules with cognate ligands is simulated using random number sampling. Once the bonds form, the contact points they make with surfaces are tracked, and a force balance is used to calculate the motion of the cell. Specific rheological laws relate stress (or strain) to bond failure rates, and the parameters of these laws dictate the quantitative sensitivity of adhesion molecules to force, and ultimately affect the dynamics of cell adhesion as a whole. We summarize major findings of cell adhesion that have been enabled by Adhesive Dynamics - the development of state diagrams of adhesion, that link distinct states of adhesion to molecular identity, such as leukocyte rolling; specification of what is required for firm arrest of a leukocyte; a description of the shear threshold effect in which adhesion increases with shear rate; and understanding how two molecules can act in synergy to secure adhesion that cannot be secured by either molecule alone. Finally, we show how signal transduction networks can be integrated within adhesive dynamics to understand how cell activation can lead to changes in adhesion and arrest, and use predictions to understand how cells might behave when molecular components are altered or eliminated in knock-out mice, or in various diseases due to molecular defects.
Oscillations of the sessile drop are of fundamental interest in a number of industrial applications, such as ink-jet printing and drop atomization. We generalize the stability analysis for the free inviscid drop (Rayleigh, 1879), focusing on the wetting properties of the solid substrate and mobility of the three-phase contact-line. We report oscillation frequencies and modal structures for the `symmetry-broken’ Rayleigh drop that display spectral splitting/reordering and compare with experiments. To organize and explain the hierarchy of frequencies, we construct a corresponding `periodic table of mode shapes’ from the spectral data. In addition to the oscillatory spectrum, we report a new hydrodynamic instability that has fundamental implications for fluid transport.
Profile: Dr Joshua Bostwick received bachelors degrees in Physics and Civil Engineering from University of Wisconsin-Milwaukee in 2005, and his PhD from Cornell University in 2011. He worked as a postdoctoral researcher at North Carolina State University and is currently Golovin Assistant Professor in the Department of Engineering Science and Applied Mathematics at Northwestern University. His research interests span surface tension, hydrodynamic instability, wetting and spreading, elastocapillarity, dynamical systems, constrained variational principles, symmetry methods.
Note for Attendees
Refreshments will be served at 3:40pm in the Math Lounge area, MATH 125 before the colloquium.
The dynamic motion of large human crowds is an ubiquitous phenomena in everyday life. First empirical studies on crowd motion started in the late 1950ties and spread into different fields like transportation research, psychology or urban and regional planning. Recently there has been a strong interest in crowd motion within the mathematical community, which initiated a lot of research on mathematical models, their analysis and simulations.
In this talk we focus on a fast exit scenario and consider a group, which wants to leave a room as quickly as possible. We present different modeling approaches, starting on the microscopic level and working our way up to the appropriate continuum limits. In particular we focus on Hughes model for pedestrian flow and give an interpretation from the mean field game perspective. Finally we discuss different challenges in the analysis and numerical simulations and illustrate the behaviour of the presented models with numerical simulations.
Maps describe the way a surface can be obtained by gluing polygons together, and as such they provide a handy tool to prove that closed surfaces can be classified by a genus parameter (both in the orientable and non-orientable case). I will review on this and on the equivalence between maps as polygon gluings and maps as graphs endowed with a rotation system. If time allows I will also review on a nice formula due to Harer and Zagier for the number of ways one can obtain the orientable surface of genus g by gluing pairwise the sides of a 2n-polygon.
(This is the first of a series of 3 talks on maps. The next two will occur later in March.)
The sharpness of the percolation phase transition, which is a crucial and much cited element of the theory, was first proved independently by Menshikov in 1986 and by Aizenman and Barsky in 1987. On February 11, 2015, a remarkable new proof was posted on arXiv by Duminil-Copin and Tassion. I will describe their proof, which is completed in two pages, in its entirety.
Apollonius's Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three. Apollonian circle packings are produced by repeating the construction of mutually tangent circles ll all remaining spaces. A remarkable consequence of Descartes' Theorem is if the initial four tangent circles have integral curvatures, then all of the circles in an Apollonian circle packing will have integral curvatures. This process results a sequence of integers with fascinating arithmetic properties.
In this talk, we will investigate the arithmetic properties of Apollonian circle packings. We will describe the Apollonian group action on the set of Descartes quadruples. We will talk about modular restrictions and density conjectures and theorems. Finally, we will show a correspondence between the root quadruples and reduced binary quadratic forms and answer an open question about finding the root quadruple of a given Descartes quadruple.
There are a number of seminal two-dimensional lattice models that are integrable, but have only been partially solved, in the sense that only some properties are fully known (e.g. the two-dimensional Ising model, where the free-energy is known, but not the susceptibility). Alternatively, critical properties are known for some lattices but not others. For
example, the critical point of the self-avoiding walk model is known rigorously for the honeycomb lattice, but not for other lattices. Similarly for the q-state Potts model and both bond and site percolation. The critical manifold of the former is known only for some lattices, likewise the percolation threshold is known only for some lattices.
A range of numerical procedures exist, based on exact enumeration, or other numerical work, such as calculating the eigenvalues of transfer matrices, which, when combined with various structural invariants seem to give exact results in those cases that are known to be exact, but can be used to give increasingly precise estimates in those cases which are not exactly known. Reasons for this partial success are not well understood. In this talk I will describe four such procedures, and demonstrate their performance, and speculate on
their partial success.
It is shown in this study that the Navier-Stokes equations allow an infinite-dimensional Lie group of point transformations, i.e., a group transforming solutions amongst each other. The Lie algebra of this symmetry group here depends on four arbitrary functions of time. Some new deformed solutions of Navier-Stokes equations in two and three dimensions are obtained by applying some of the elements of the symmetry group of these equations to their basic solutions. In order to explore the properties of deformed solutions, the analytic solutions are analyzed. It is noted that the corresponding deformed solutions behave as the basic solutions in the limiting sense for large time.
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. We first review some recent theoretical and computational studies of the 3D Euler equations. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain flow geometry. We then present strong numerical evidence that the 3D Euler equations develop finite time singularities. To resolve the nearly singular solution, we develop specially designed adaptive (moving) meshes with a maximum effective resolution of order 10^12 in each direction. A careful local analysis also suggests that the solution develops a highly anisotropic self-similar profile which is not of Leray type. A 1D model is proposed to study the mechanism of the finite time singularity. Very recently we prove rigorously that the 1D model develops finite time singularity.
This is a joint work of Prof. Guo Luo.
Note for Attendees
Coffee, tea and cookies served at 2:30pm in the PIMS Lounge, ESB 4133.
In this talk a method for the parallel generation of adaptive meshes using stochastic domain decomposition is presented. The method rests on numerically evaluating the stochastic representation of the exact solution of a linear elliptic or linear parabolic mesh generator for generating the mesh at the interfaces of the sub-domains. Unlike traditional domain decomposition, this method hence does not require iteration on the sub-domains or optimization of the transmission conditions to generate adaptive meshes over the entire domain. We show the generation of adaptive meshes for prescribed mesh density functions and study the scaling properties of the algorithm. A few physical examples for the parallel generation of adaptive meshes for Burgers equation and the shallow-water equations are presented. This is joint work with Ronald Haynes and Emily Walsh.
Given two smooth maps of manifolds f:M \to L and g:N \to L, if they are not transverse, the fibered product M \times_L N may not exist, or may not have the correct cohomological properties. In the world of derived manifolds, such a fibered product always exists as a smooth object, regardless of transversality. In this talk we will describe recent progress of ours with D. Roytenberg on giving an accessible geometric model for derived manifolds using differential graded manifolds.
We show that for an immersed two-sided minimal surface in R^3, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in R^3 of index 2, as conjectured by Choe. Moreover, we show that the index of an immersed two-sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface. (This is joint work with Otis Chodosh)
Ramanujan graphs are k-regular graphs admitting optimal connectivity properties (namely, optimal expanders). Infinite families of such graphs were first constructed by Lubotzky, Phillips and Sarnak in 1988 by relating the spectrum of a graph with certain representations of GL_2(Q_p). These ideas were generalized to simplical complexes by Lubotzky, Samuels and Vishne in 2005.
We will present a further generalization, showing that there is a natural way to relate spectral properties of simplicial complexes with certain representations of groups acting on their universal covers. Several results of this connection will be discussed. In particular, we strengthen the spectral properties of the complexes constructed by L-S-V. (Roughly speaking, we show that the complexes constructed by L-S-V have "optimal spectrum in all dimensions".)
We solve an open question raised by Le Gall and Lin. We study the probability of visiting a distant point $a \in Z^4$ by critical branching random walk starting from the origin. We prove that this probability is bounded by $1/(|a|^2 log|a|)$ up to a constant.
We are interested in classifying all finite groups which can act on a finite CW-complex homotopy equivalent to a sphere, such that all isotropy subgroups are rank one groups, i.e., they do not include Z/pxZ/p for any prime p. The equivalent question for free actions (all isotropy subgroups are trivial) has been answered completely by the works of P.A. Smith and R. Swan. For actions with rank one isotropy, we give a list of group theoretical conditions which guarantee the existence of such actions. Some of these conditions are necessary conditions depending on assumptions on fixed point subspaces. This is a joint work with Ian Hambleton.
A Diophantine m-tuple is a set A of m positive integers such that ab+1 is a perfect square for every pair a,b of distinct elements of A. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ways that might be of independent interest. The Erdős-Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials. (joint work with Scott Sitar)
Canfor Policy Rm 1600, SFU Harbour Centre, Downtown Vancouver
Fri 6 Mar 2015, 4:00pm-5:00pm
Abstract
Data assimilation is the process of systematically including (often noisy) data into a forecast. It is now widely used in numerical weather prediction and its positive impact on the accuracy of weather forecasts is unquestionable. Indeed improvements in our ability to forecast the weather over the last decade are a reflection on the increasing volume of data available, improved computational methods and (significantly) much improved algorithms for incorporating this data into the forecast. However, many problems remain, including dealing with the sheer volume of the data and the inherent complexity of the weather and climate, understanding the effects of data and model error, and of reducing spurious correlations between the data and the forecast.
In this talk I will give a survey of various techniques that are used operationally to implement data assimilation procedures in weather (and climate) forecasting including the Ensemble Kalman Filter, and the 4D-Var method.
I will discuss their various advantages and disadvantages, the nature of the errors and ways to minimise these. In particular I will show that the use of adaptive numerical methods can significantly improve the performance
of the 4D-Var method. Hopefully I will show that used carefully Data Assimilation techniques can significantly improve our ability to forecast the weather of Planet Earth.
Joint work with Mike Cullen and Chiara Piccolo at the Met Office.
Note for Attendees
Note SFU downtown venue. Reception at 3:30 pm (light refreshments).
Numerical weather prediction models can only operate at finite resolution. However, processes below the model resolution have an impact on the processes resolved by the model and therefore cannot be omitted in the model. The proper formulation of subgrid-scale processes in terms of resolved grid scale quantities is referred to as parameterization. The aim of this talk is to discuss a method for constructing parameterization schemes that preserve invariance properties. The method is based on group classifi cation of diff erential equations. By assuming a general functional dependency of the unknown subgrid-scale in terms of the known grid-scale quantities in a system of averaged diff erential equations turns the original unclosed di fferential equations into a class of diff erential equations which is approachable using tools from the classical group classi cation. The result of this procedure yields various forms of local closure ansatzes for the unresolved subgrid scale terms leading the closed diff erential equations having symmetry properties that are related to the original unaveraged diff erential equations.
Donaldson-Thomas invariants are fundamental deformation invariants of Calabi-Yau threefolds. We describe a recent conjecture of Oberdieck and Pandharipande which predicts that the (three variable) generating function for the Donaldson-Thomas invariants of K3xE (the product of a K3 surface and an elliptic curve) is given by the reciprocal of the Igusa cusp form of weight 10. For each fixed K3 surface of genus g, the conjecture predicts that the corresponding (two variable) generating function is given by a particular meromorphic Jacobi form. We prove the conjecture for K3 surfaces of genus 0 and genus 1. Our computation uses a new technique which mixes motivic and toric methods.
Electromagnetic inverse problems are now commonly solved in geophysical imaging applications. Many imaging techniques involve estimating the parameters of a PDE model from noisy measurements. This can be formulated as an optimization problem with constraints given by the PDE. The computational bottleneck are PDE simulations that need to be carried out for each measurement and at each iteration of the optimization algorithm. Most modern applications involve a very large number of measurements whose inversion requires optimization algorithms that converge quickly, but also allow for parallel and distributed computing.
In this talk, I will present recent developments in JEMI - a Julia package for electromagnetic inversion. JEMI is designed in a modular way and is thus offers great modeling potential. A particular focus of my talk will be on using Julia to adapt electromagnetic inversion codes to parallel systems.
Scintillating Scotoma is a phenomenon in the visual cortex which may signal the onset of migraine, or may happen for no apparent reason. Initial steps to model this use a stochastic reaction diffusion system. A stochastic version of Turing patterns, called quasi-patterns is introduced. This idea is analogous to oscillations sustained by noise in a stochatic ODE setting.
In a quantifiable way most groups, rings, and Lie algebras are nilpotent. In fact even the extension of two abelian groups, or two trivial algebras, has enough variation to match the total quantity of all finite groups, resp. finite-dimensional algebras. However, our most developed theories concern groups, rings, and algebras that are simple, semisimple, or highly related to simplicity.
In this talk I will demonstrate a simple way to convert questions about nilpotence into questions about simple and semisimple groups and nonassociative rings. The process is recursive and captures new structure in a positive proportion of all products. In fact 4/5 of the 11 million groups of size at most 1000 are explained by this mechanism. I will close with a a surprising characterization of the base case of these recursive techniques: they are products without zero-divisors and thus have storied histories in discrete and differential geometry.
Fix a graph G and place some number (random or otherwise) of sleeping frogs at each site, as well as one awake frog at the root. Set things in motion by having awake frogs perform independent simple random walk, waking any "sleepers" they encounter. Say the model is recurrent if the root is a.s. visited by infinitely many frogs and otherwise transient. When G is the rooted d-ary tree with one-frog-per-site we prove a phase transition from recurrence to transience as d increases. Alternatively, for fixed d with Poi(m)-frogs-per-site we prove a phase transition from transience to recurrence as m increases. The proofs use two different recursions and two different versions of stochastic domination. Several open problems will be discussed. Joint with Christopher Hoffman and Tobias Johnson.
This is a joint work with Susumu Hirose. We consider pseudo-Anosov elements of the mapping class groups on orientable surfaces. We are interested in a numerical invariant of pseudo-Anosovs, called the dilatation. The logarithm of the dilatation of a pseudo-Anosov mapping class is called the entropy. If we fix a surface, then the set of dilatations of pseudo-Anosovs defined on the surface is closed and discrete. In particular we can talk about a minimum of any subset of dilatations defined on the surface in question.
Penner proved that the minimal entropy of pseudo-Anosovs defined on a closed surface of genus g behaves like 1/g. Later Hironaka proved that the minimal entropy of pseudo-Anosovs in the handlebody subgroup on a closed surface of genus g also behaves like 1/g. We prove that the the minimal entropy of the hyperelliptic handlebody sugbroup of genus g has the same asymptotic behavior. (Our examples of pseudo-Anosovs improve the upper bound of the minimal entropy of the handlebody sugbroup given by Hironaka.) To do this, we study the spherical Hilden subgroup of the mapping class group defined on a sphere with 2n punctures, and we construct a sequence of pseudo-Anosovs with small dilatations in the spherical Hilden subgroups.
An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a beautiful product formula, purely in terms of congruence considerations involving that polynomial, for the size of such an isogeny class; an equidistribution hypothesis too strong to be true apparently calculates this cardinality.
I will give a new, transparent explanation, worked out with Julia Gordon, for this phenomenon. It turns out that Gekeler's formula computes an adelic orbital integral which, thanks to work of Langlands and Kottwitz, visibly calculates the desired quantity.
Let (M,g) be a compact surface without boundary. In this
lecture, we present some joint work with S. Nonnenmacher (Saclay)
giving the construction of logarithmic scale quasimodes of the
Laplace-Beltrami operator which concentrate around a given closed
hyperbolic geodesic. This result is related to a strengthened version
of the Quantum Unique Ergodicity conjecture and generalizes a previous
result of S. Brooks for logarithmic scale quasimodes on compact
hyperbolic surfaces. Our proof is microlocal and utilizes a quantum
Birkhoff normal form due to Sjöstrand as well as a result concerning
propagation around hyperbolic fixed points due to Combescure-Robert.
We discuss a surprising connection between Mirror Symmetry and the classification of Fano manifolds. This is joint work with Akhtar, Corti, Galkin, Golyshev, Kasprzyk, and Prince.
Given a hyperbolic surface S, consider any closed geodesic gamma on S. gamma is naturally embedded as a knot in the unit tangent bundle of S, and the complement of gamma is almost always a hyperbolic three manifold and thus has an intrinsic volume. In this talk I will describe a way to obtain an upper bound for this volume, linear with respect to the length of gamma. The proof goes through careful analysis of volumes for geodesics on the modular surface. This is joint work with Maxime Bergeron and Lior Silberman.
In the planar case a map can be seen as a connected graph embedded on the sphere (or in the plane) up to continuous deformation. The enumeration of (rooted) planar maps has started in the 60's with the seminal work of Tutte who found surprisingly simple counting formulas for several families of planar maps. We will briefly review on Tutte's method and present in details the more recent bijective approach, focusing on the Cori-Vauquelin-Schaeffer bijection for planar quadrangulations. This bijection has become famous since it makes it possible to trace the distances (from a distinguished vertex) in the map, and as such it has proven a fundamental tool in the recent proof that random planar quadrangulations (rescaled by n^{1/4}) converge to the so-called Brownian map.
This the second talk of a series of 3 talks, the 3rd one will focus on distance properties in random planar maps
Compressed sensing concerns the recovery of signals and images from seemingly incomplete data sets. Introduced nearly a decade ago, it has since become an intensive area of research in applied mathematics, engineering and computer science. However, many practical problems in which compressed sensing is applied, e.g. medical imaging, are not adequately explained by existing theory. In this talk I will present a new framework for compressed sensing that bridges this gap. This framework is based on replacing some of the standard principles of compressed sensing with new local notions; specifically, sparsity in levels, local coherence in levels and multilevel random subsampling. When combined, they lead to near-optimal recovery guarantees that explain the effectiveness of compressed sensing in such applications. Moreover, this framework is not just useful in understanding existing compressed sensing approaches. In the final part of this talk I will demonstrate how leveraging local sparsity through appropriately-designed locally incoherent sensing matrices also leads to substantially improved compressed sensing algorithms in a range of other applications.
Room 203 of the Graduate Student Centre (6371 Crescent Rd.), UBC
Thu 19 Mar 2015, 12:30pm-2:30pm
Details
This thesis consists of three parts. The common theme is finite group actions on algebraic curves defined over an arbitrary field k.
In Part I we classify finite group actions on irreducible conic curves defined over k. Equivalently, we classify finite (constant) subgroups of SO(q) up to conjugacy, where q is a nondegenerate quadratic form of rank 3 defined over k. In the case where k is the field of complex numbers, these groups were classified by F. Klein at the end of the 19th century. In recent papers of A. Beauville and X. Faber, this classification is extended to the case where k is arbitrary, but q is split. We further extend their results by classifying finite subgroups of SO(q) for any base field k of characteristic not 2 and any nondegenerate ternary quadratic form q.
In Part II we address the Hyperelliptic Lifting Problem (or HLP): Given a faithful G-action on the projective line defined over k and a double cover H of a finite group G, determine the conditions for the existence of a hyperelliptic curve C/k endowed with a faithful H-action that lifts the prescribed G-action on the projective line. In this thesis, we find a complete solution to the HLP in characteristic 0 for every faithful group action on the projective line and every exact sequence as above.
In Part III we determine whether, given a finite group G and a base field k of characteristic 0, there exists a strongly incompressible G-curve defined over k. Recall that a G-curve is an algebraic curve endowed with the action of a finite group G. A faithful G-curve C is called strongly incompressible if every dominant G-equivariant rational map of C onto a faithful G-variety is birational. We prove that strongly incompressible G-curves exist if G cannot act faithfully on the projective line over k. On the other hand, if G does embed into PGL(2,k), we show that the existence of strongly incompressible G-curves depends on finer arithmetic properties of k.
The talk concerns limit behaviors of stationary measures of diffusion processes generated from white-noise perturbed systems of ordinary differential equations. By relaxing the notion of Lyapunov functions associated with the stationary Fokker-Planck equations, new existence and non-existence results of stationary measures will be presented. As noises vanish, concentration and limit behaviors of stationary measures will be described with particular attentions paying to the special role played by multiplicative noises. Connections to problems such as stochastic stability, stochastic bifurcations, and the ergodicity hypothesis will also be discussed.
http://www.pims.math.ca/scientific-event/150319-pcyy
One way to quantify the level of singularity of a singular measure is to compute its local dimensions. For many interesting classes of measures, including self-similar measures and Cantor-like measures that satisfy a suitable separation condition, it is well known that the set of attainable values of the local dimensions is a closed interval. In contrast, convolutions of continuous measures often have an isolated point in their set of local dimensions. More generally, the structure of the set of local
dimensions of even self-similar measures which do not satisfy the separation condition can be surprising.
In this second talk, we again consider symmetries and conservation laws of FPDEs equation with Riemann-Liouville derivatives. Within the framework of Lie group theory, we extend Lie group analysis to solve problems involving FPDEs. Finally, we give further examples to illustrate applications of the methods. Some open questions will be discussed.
The most prominent basis of the ring of symmetric functions is that of Schur functions. This basis captures a significant amount of the interplay between algebraic combinatorics and fields such as representation theory and algebraic geometry. Recently, a natural refinement of Schur functions, called quasisymmetric Schur functions, was introduced by Haglund, Luoto, Mason, and van Willigenburg. While various analogues of Schur function properties were established for quasisymmetric Schur functions, one key property - that of a representation-theoretic interpretation - was lacking.
In this talk, I will start by giving a combinatorial description using diagrams for quasisymmetric Schur functions and then proceed to describe how they arise in the setting of the representation theory of the 0-Hecke algebra using easy to understand operations on diagrams. This is joint work with Steph van Willigenburg. The talk is aimed at a general audience and no knowledge of any of the above terms is assumed.
Note for Attendees
Refreshments will be served at 2:40pm in the Math Lounge area, MATH 125 before the colloquium.
Room 203 of the Graduate Student Centre (6371 Crescent Rd), UBC
Fri 20 Mar 2015, 4:00pm-6:00pm
Details
Given a Cantor-type subset Ω of a smooth curve in d-dimensional Euclidean space, we construct random examples of Euclidean sets that contain unit line segments with directions from Ω and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. We also develop a notion of finite order lacunarity for direction sets in arbitrary d-dimensional space, and use it to extend our construction to direction sets Ω that are sublacunary according to this definition. This generalizes to higher dimensions a pair of planar results due to Bateman and Katz. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set Ω is unbounded on the Lebesgue spaces of finite exponent.
In a two-stage assessment, students first complete and turn in the questions individually and then, working in small groups, answer the same questions again. This technique was first introduced in the UBC Faculty of Science in 2009 and is now being used in at least 20 science courses. In this session, we will discuss the advantages and disadvantages of two-stage assessments and describe some past experiences with this method at UBC and in the math department in particular. We will also consider different formats and options for implementing two-stage reviews or exams in your courses. Pizza and pop are provided.
We will survey recent developments dealing with characterization of absolutely almost simple algebraic groups having the same
isomorphism/isogeny classes of maximal tori over the field of
definition. These questions arose in the analysis of weakly commensurable Zariski-dense subgroups. While there are definitive results over number fields (which we will briefly review), the theory over general fields is only emerging. We will formulate the existing conjectures, outline their potential applications, and report on recent progress. Joint work with A. Rapinchuk and I. Rapinchuk.
Understanding physical controls on the variability of near-surface winds is of interest from the perspective of climate (as winds influence surface fluxes), of environmental hazards (particularly extreme winds), and of renewable energy. There is a long history of empirically-based probabilistic models of wind variability, but until recently relatively little physical attention has been paid to this problem.
In this talk, I will discuss how we have been using approaches from nonlinear time series, dynamical systems, and stochastic differential equations in the development of physically-based probabilistic models of near-surface wind variability. The focus will be on winds over land, characterized by a marked day/night contrast in the shape of the wind speed probability density function (pdf). I will first present an analysis of long time series of wind, temperature, and turbulence data from a 213m tower in Cabauw, Netherlands. After this, I will discuss an idealized stochastic model of the boundary layer momentum budget that captures some of the basic features of variations in the wind speed pdf.
The well-known Hammersley-Clifford Theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbour interaction. Following Petersen and Schmidt we utilise the formalism of cocycles for the homoclinic relation and introduce “Markov cocycles”, reparametrisations of Markov specifications. We exploit this formalism to deduce the conclusion of the Hammersley-Clifford Theorem for a family of Markov random fields which are outside the theorem’s purview (including Markov random fields whose support is the ddimensional “3-coloured chessboard”). On the other extreme, we construct a family of shift-invariant Markov random fields which are not given by any finite range shiftinvariant interaction.
The techniques that we use for this problem are further expanded upon to obtain the following results: Given a “four-cycle free” finite undirected graph H without self-loops, consider the corresponding ‘vertex’ shift, Hom(Zd, H) denoted by XH. We prove that XH has the pivot property, meaning that for all distinct configurations x, y in XH which differ only at finitely many sites there is a sequence of configurations (x = x1), x2 , . . . , (xn = y) in XH for which the successive configurations (xi, xi+1) differ exactly at a single site. Further if H is connected then we prove that XH is entropy minimal, meaning that every shift space strictly contained in XH has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the ‘lifts’ of the configurations in XH to their universal cover and the introduction of ‘height functions’ in this context.
Further we generalise the Hammersley-Clifford theorem with an added condition that the underlying graph is bipartite. Taking inspiration from Brightwell and Winkler we introduce a notion of folding for configuration spaces called strong configfolding to prove that if all Markov random fields supported on X are Gibbs with some nearest neighbour interaction so are Markov random fields supported on the “strong config-folds” and “strong config-unfolds” of X.
Bernstein and Breiner found a characterization of the catenoid that the area of a minimal annulus in a slab is bigger than that of the maximally stable catenoid in the same slab. We give a simpler proof of their theorem and extend the theorem to some minimal surfaces with genus (joint work with Benoit Daniel). New characterizations of the helicoid recently proved by Eunjoo Lee will be also presented.
The existence of interesting multiplicative cohomology theories for orbifolds was first suggested by string theorists, and orbifold products have been intensely studied by mathematicians for the last fifteen years. My work with S. Scherotzke focuses on the virtual orbifold product introduced by Lupercio et al. (2007). We construct a categorification of the virtual orbifold product that leverages the geometry of derived loop stacks. By work of Ben-Zvi Francis Nadler, this reveals connections between virtual orbifold products and Drinfeld centers of monoidal categories, thus answering a question of Hinich.
Motivated by subtle questions in Donaldson-Thomas theory, we study the spectrum of the inertia operator on the Grothendieck module of algebraic stacks. We hope to give an idea of what this statement means. Along the way, we encounter some elementary, but apparently new, questions about finite groups and matrix groups. Prerequisites for this talk: a little linear algebra, and a little group theory.
Note for Attendees
Refreshments will be served at 2:30 p.m. in ESB 4133, the PIMS Lounge.
Room 203, Graduate Student Centre (6371 Crescent Rd), UBC
Mon 30 Mar 2015, 9:00am-11:00am
Details
In 2011, Michael Bennett, Florian Luca and Jamie Mulholland showed that the equation involving a twisted sum of cubes has no pairwise coprime nonzero integer solutions for primes excluded from the set S where S is the set of primes q for which there exists an elliptic curve of conductor 18q, 36q, or 72q with at least one nontrivial rational 2-torsion point. In this dissertation, I present a solution that extends the result to include a subset of the primes in S; those primes q in S for which all curves with conductor 18q, 36q, or 72q with nontrivial rational 2-torsion have discriminants not of the form an integer squared or -3 times an integer squared. Using a similar approach, I will classify certain integer solutions to the equation of a twisted sum of fifth powers which in part generalizes work done from Billerey and Dieulefait in 2009.
Some functional analytic properties related to optimisation, such as the Krein-Milman Property and the Radon-Nikodym Property, for some Banach Algebras related to the Fourier Algebra are investigated.
Plant growth typically occurs through the coordinated anisotropic expansion of plant cells. Growth is regulated by hormones and is driven by high intracellular pressures generated by osmosis. This machinery allows a plant primary root, for example, to penetrate soil in a direction guided by gravity, while seeking out nutrients and avoiding obstacles. I will describe the biomechanical aspects of a computational multiscale model for root gravitropism that incorporates descriptions of cell walls as fibre-reinforced viscoelastic polymer networks and adopts upscaling approaches to efficiently describe the growth of multicellular tissues.
Period integrals are geometrical objects which can be realized as special functions, or sections of certain bundles. Their origin goes back to Euler, Gauss and Legendre in the study of complex algebraic curves. In their modern version, period integrals naturally arise in Hodge theory, and more recently in mathematical physics, and the theory of hypergeometric functions. I will give an overview of a recent program to use differential equations and D-module theory to study period integrals. Connections to hypergeometric functions of Gel'fand-Kapranov-Zelevinsky (GKZ) will also be considered. We will see that the theory is intimately related to a particular infinite dimensional representation of a reductive Lie algebra, and the topology of certain affine varieties. I will describe how the theory could help calculate period integrals, and offers new insights into the GKZ theory, and mirror symmetry for toric and flag varieties. This talk is based on joint works with S. Bloch, B. Lian, V. Srinivas, S-T. Yau, and X. Zhu.
When numerically solving a PDE in three dimensions, it is often necessary to generate a mesh on which to discretize the solution. Often this can be expensive to do. However, by using ideas from optimal transport it is possible both to construct a mesh quickly and cheaply, and also to prove that it has the necessary regularity properties to allow an accurate approximation of the solution of the PDE. In this talk I will describe these methods, prove results about their regularity and then apply them to some problems in meteorology.
Room 203 of the Graduate Student Centre (6371 Crescent Road), UBC
Tue 31 Mar 2015, 12:30pm-2:30pm
Details
In this thesis we study topological (continuous map on a compact metric space) and measure theoretical (measure preserving map on a probability space) dynamical systems.
Dynamical systems range from chaotic (random) to predictable (high structure). Structure and randomness can be represented with different forms of equicontinuity and sensitivity to initial conditions (sensitivity).
Inspired by the classical dichotomy between sensitivity and equicontinuity we define weak forms of topological and measure theoretical equicontinuity and strong forms of sensitivity for dynamical systems, and we study their relationships with spectral properties and sequence entropy. We also prove results of how measure theoretically equicontinuous cellular automata (a particular class of topological systems with close connections to computer science) behave in the long term.
The work of this thesis answers questions from - B. Scarpellini. Stability properties of flows with pure point spectrum. Journal of the London Mathematical Society, 2(3):451–464, 1982. - F. Blanchard and P. Tisseur. Some properties of cellular automata with equicontinuity points. Annales de l’Institut Henri Poincare (B) Probability and Statistics, 36(5):569 – 582, 2000.
Understanding the least quadratic nonresidue (mod p) is a classical problem, with a history stretching back to Gauss. The approach which has led to the strongest results uses character sums, objects which are ubiquitous in analytic number theory. I will discuss character sums, their connection to the least nonresidue, and some recent work of myself and Jonathan Bober (University of Bristol) on a promising new approach to the problem.
The largest and most difficult numerical approximation error to estimate is discretization error. Residual-based discretization error estimation methods are a category of error estimators that use an estimate of the source of discretization error and information about the specific application to estimate the discretization error using only one grid level. The higher-order terms are truncated from the discretized equations and are the local source of discretization error. The accuracy of the resulting discretization error estimate depends solely on the accuracy of the estimated truncation error. Residual-based methods require only one grid level compared to the more commonly used Richardson extrapolation which requires at least two. Reducing the required number of grid levels reduces computational expense and, since only one grid level is required, can be applied to unstructured grids where multiple quality grid levels are difficult to produce. The two residual-based discretization error estimators of interest are defect correction and error transport equations. The focus of this work is the development, improvement, and evaluation of various truncation error estimation methods considering the accuracy of the truncation error estimate and the resulting discretization error estimates. The minimum requirements for accurate truncation error estimation is specified along with proper treatment for several boundary conditions. The single grid methods require that the continuous operator be modified at the boundary to be consistent with the implemented boundary conditions. The methods are evaluated using various Euler applications. Defect correction showed to be more accurate for areas of larger discretization error; however, the cost was substantial (although cheaper than the primal problem) compared to the cost of solving the ETEs which was essentially free due to the linearization.
Sage Math Cloud (SMC) is an online cloud computing software based on the programming language Sage and founded in 2013. In this seminar, I will discuss using this software package for teaching based on its new 'course' functionality. I will show how easy it is to get set up with SMC and how one can organize a course using this software. In particular, one can easily type in LaTeX, assign and collect assignments, and share lecture notes and demonstrations. I will also show how simple it is to integrate computations into one's lectures, enabling students to make mathematics come alive via computer software. This talk is of particular interest for faculty teaching upper level courses and looking for an intuitive to use learning management system. We will discuss pros and cons of this software, as well as possible integration techniques for other classes.
Understanding the behavior of various Lp norms of high-energy eigenstates of the Laplacian on a compact Riemannian manifold is an important and interesting problem lying in the intersection of diverse brunches of mathematics. We will be mainly focusing on the L∞ norms and also be discussing on a related problem which is known as Quantum Unique Ergodicity. It is expected that when the manifold has negative sectional curvature and has enough symmetry from underlying congruence subgroup of the isometry group, the L∞ norm can be sharpened from the so-called trivial bound. We give a partial result in the case of hyperbolic manifolds of dimension 4 and 5. This is a joint work with my supervisor Lior Silberman.
Furstenberg-Katznelson multi-dimension Szemeredi theorem says that any subsets of Z^d with positive density must contain affine copies of any finite set in Z^d . Green and Tao extended this theorem to P^d where P is the set of primes and d=1, using a pseudo-randomness property of primes. Questions of extending Green-Tao result to higher dimensions left open for almost 10 years, due to some correlations between its elements. This problem is resolved by three methods in 2013. In this talk, I will describe one of the approaches, by transferring hypergraph regularity and energy increment method to the weighted setting. Then we can use it to prove simplex removal lemma on this weighted hypergraph which would imply multidimensional Green-Tao Theorem.
The computation of the semiclassical Schrödinger equation presents a number of difficult challenges because of the presence of high oscillation and the need to respect unitarity. Typical strategy involves a spectral method in space and Strang's splitting in time, but it is of low accuracy and sensitive to high oscillation. In this talk we sketch an alternative strategy, based on high-order symmetric Zassenhaus splittings, combined with spectral collocation, which preserve unitarity and whose accuracy is immune to high oscillation. These splittings, whose analysis requires Lie-algebraic techniques, can be implemented with large time steps and allow for an exceedingly affordable computation of underlying exponentials. The talk will be illustrated by the computation of different quantum phenomena.
In this gentle introduction to machine learning I will give an overview of the most popular algorithms for supervised and unsupervised learning. You will see that machine learning can be surprisingly simple, yet powerful, while always being mathematically appealing. I will finish by walking us through two example applications using Python's scikit-learn library.
The classical problem of Diophantine equations is to solve polynomial equations over the rationals. More generally, we may consider solutions over an extension of the rationals. If the equations define an elliptic curve (or more generally, an Abelian variety), there is more structure. In particular, the set of rational points forms a group which is finitely generated. What happens if we consider the same problem over an infinite extension (or equivalently, over an infinite tower of extensions)? The problem becomes very subtle and is the subject of current research. We shall describe some of the recent results in this area.
Note for Attendees
Refreshments will be served at 2:40 p.m. in the Math Lounge area (MATH 125) before the colloquium.
The Betti geometric Langlands program is a new physics-inspired variant of the [de Rham] geometric Langlands program, in which the geometry of algebraic curves is replaced by the topology of surfaces. It has close relations to representation theory of quantum groups, Hecke algebras and p-adic groups. I will describe some features of and progress on this program, following work with David Nadler as well as Adrien Brochier, David Jordan and Anatoly Preygel.
Galois cohomological invariants for algebraic groups are an arithmetic analogue of characteristic classes in algebraic topology; Totaro and Guillot connected them to unramified étale cohomology and Rost’s higher Chow Groups.
In his PhD thesis, Roberto Pirisi generalizes the theory of Galois cohomological invariants to smooth Artin stacks of finite type over a field, and computes the cohomological invariants for stacks of hyperelliptic curves of even genus. Furthermore he defines Rost’s higher Chow groups for Artin stacks.
In this thesis we study two problems, one concerning fusion energy and another superconductivity.
Magnetized target fusion reactors are a modern idea to generate hydrogen fusion energy on earth. The design entails magnetically confining a plasma and crushing it in an imploding shell of molten metal. The design has many unresolved questions in its feasibility as a power source and its efficiency. We study the problem with two approaches. Firstly, we use a coordinate transformation and implement a novel flux-limited, split-step, finite volume scheme for nonlinear coupled conservation laws and do a parameter sensitivity analysis for the performance. Secondly, by a careful series of asymptotic arguments, we establish a leading order expression for the plasma compression. This expression is qualitatively consistent with numerical simulations, but it also gives new insights into the device operation. We then infer key design parameters for the success of magnetized target fusion.
The second problem involves computational modelling of superconductors. In type II superconductors where the coherence length ξ is small compared to the London penetration depth λ, the London equation predicts that magnetic fields decay exponentially in magnitude with the depth into the superconductor with length scale λ, provided the surface is flat. Various measurements of λ using low energy muon spin rotation on superconductors such as Yttrium-Barium-Copper-Oxide measure field profiles that differ from this prediction. There seems to be a dead layer, a distance δ over which the magnetic field magnitude does not decay. Speculation has been made that this may be due to surface roughness. Surface roughness has been studied for a simple sinusoidal model of surface roughness. We extend this work firstly by using Atomic Force Microscopy data of Yttrium-Barium-Copper-Oxide (a type II superconductor) crystals and predicting the field profiles the crystals could produce with the London model given their actual surface geometry; and secondly, we consider how roughness could affect experimental values for λ and δ. We find that dead layers are unlikely due to roughness alone, that the measurement of λ may be influenced by surface roughness, and that the field orientation may be perturbed, negligibly, within the superconductor.
By the celebrated Hasse principle of Kneser, Harder and Chernousov, calculating Galois cohomology H1(K,G) of a simply connected simple algebraic K-group G over a number field K reduces to calculating H^1(R,G) over the field of real numbers R. In some cases, in particular, for the split simply connected R-group G of type E_7, the first calculations of H^1(R,G) appeared only in 2013 and 2014, in preprints of Jeffrey Adams, Brian Conrad, and the speaker and Zachi Evenor. All of these calculations rely on the speaker's note of 1988.
In this talk, based on joint work with Dmitry Timashev (in progress), I will explain the method of Kac diagrams for calculating the Galois cohomology set H1(R,G)^for a simply connected simple algebraic R-group G. I will use groups of type E_7 as an example. No prior knowledge of Galois cohomology or of groups of type E_7 will be assumed.
Directed polymer models are finite-temperature versions of first- and last-passage percolation on the lattice. In 1+1 dimensions, the free-energy of the directed polymer is conjecturally in the Tracy-Widom universality class at all finite temperatures. However, this has only been proven for a small class of polymers - the so-called solvable models that include Seppalainen's gamma polymers and the O'Connell-Yor semi-discrete polymer - with special sets of shapes and edge-weight distributions. We present some new results towards the universality conjecture in the intermediate disorder scaling regime.
Abstract: We derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. The theory also incorporates the change of the cross-section available to the fluid motion during the dynamics, sometimes called collapsible tubes. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. Using these methods, we derive the fully three dimensional equations of motion. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross-section. We also derive and analyze several analytical, fully nonlinear solutions of traveling wave type in two dimensions. Finally, we present results of preliminary experiments showing instability and re-stabilization elucidating the roles of rotation and boundary conditions. This research has been supported by NSERC and the University of Alberta.
Finite cubical complexes are abstract models for parallel processing systems. The vertices of a complex K are the states of the system, and the execution paths are morphisms of the corresponding path category P(K).
The theory of path categories and path 2-categories for finite oriented cubical and simplicial complexes will be reviewed. There is an algorithm for computing the path category P(K) of a finite complex K which is based on its path 2-category. This 2-category algorithm will be displayed, and complexity reduction methods for the algorithm will be discussed.
The 2-category algorithm works well only for toy examples. The size of the path category P(K) of a complex K can be an exponential function of the size of K. The algorithm has so far resisted parallelization.
One wants combinatorial local to global methods for addressing examples that are effectively infinite.
This will be the introduction to a weekly sequence of seminars on modern developments in symmetry methods for PDE. Topics will include how to find the:conservation laws for any DE system--the extension of Noether's theorem to non-variational systems, local symmetries, higher-order symmetries, invertible and local mappings (including linearizations through symmetries and conservation law multipliers), nonlocally related PDE systems, nonlocal symmetries, nonlocal conservation laws, nonlocal mappings, and the nonclassical method to obtain solutions of PDEs. The main emphasis of these seminars will be on how to find systematically symmetries and conservation laws (local and nonlocal) of a given PDE system and how to use systematically symmetries and conservation laws for related applications.
The first seminar will present an overview of topics.
Analyzing known elementary relations between U(p) operators and Picard functoriality of the Jacobians of each tower of modular curves of p-power level, we get fairly exact control of the ordinary part of the limit Barsotti-Tate groups and the (p-adically completed) ind-limit Mordell-Weil groups with respect to the weight Iwasawa algebra. Computing Galois cohomology of these controlled Galois modules, we obtain good control of the (ordinary part of) limit Selmer groups and limit Tate-Shafarevich groups.
Wavelets provide a mathematical tool that emerged in the 1980s from a synthesis of ideas in mathematics, physics, computer science and engineering. They are now used in a wide range of mathematical applications, and provide a mathematical way to "zoom in" on details, without losing track of the large picture. The talk will describe some of the essential features of the approach, and illustrate with examples.
Note for Attendees
There will be the Grad Reception on Monday, May 25th at 11:30-1:00 (lunch and awards presentation).
Room 203 of the Graduate Student Centre, 6371 Crescent Rd., UBC
Wed 27 May 2015, 12:30pm-2:30pm
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ABSTRACT: We present an analysis of the Gierer-Meinhardt model with saturation (GMS) on various curve geometries in two-dimensions. We derive a boundary fitted coordinate framework which translates an asymptotic two-component differential equation into a single component reaction diffusion equation with singular interface conditions. We create a numerical method that generalizes the solution of such a system to arbitrary two-dimensional curves and show how it extends to other models with singularity properties that are related to the Laplace operator. This numerical method is based on integrating logarithmic singularities which we handle by the method of product integration where logarithmic singularities are handled analytically with numerically interpolated densities.
In parallel with the numerical method, we present some analytical solutions to the GMS model on circular and slightly perturbed circular curve geometries. We see that for the regular circle, saturation leads to a hysteresis effect for two dynamically stable branches of equilibrium radii. For the near circle, we show that there are two distinct perturbations to the velocity profile, one which introduces angular dependence, and one which introduces a vertical shift caused by quadratic Fourier mode interactions. We perform a linear stability analysis to the true circle solution and show that there are two classes of eigenvalues leading to breakup or zigzag instabilities. For the breakup instabilities we show that the saturation parameter can completely stabilize perturbations that we show are always unstable without saturation and for the zigzag instabilities we show that the eigenvalues are given by the near-circle curve normal velocity. The breakup analysis is based on the reduction of an implicit non-local eigenvalue problem (NLEP) to a root finding problem. We derive conditions for which this eigenvalue problem can be made explicit and use it to analyze a stripe and ring geometry. This formulation allows us to classify certain technical properties of NLEPs such as instability bands and a Hopf bifurcation condition analytically.
The results for breakup and zigzag instabilities are verified with numerical simulations of the full model in both stripe and ring geometries. This includes confirmation of dominant breakup modes and demonstrating the stabilizing effect of saturation.
In this first lecture on the construction of the conservation laws for a DE system, we present the Direct Method. This leads to the direct construction of the CLs for essentially any DE system in a systematic framework. The classical Noether's Theorem only works for variational systems and also requires the construction of the Lagrangian. The Direct Method involves working directly with a given DE system. The second lecture will show explicitly how the Direct Method generalizes Noether's Theorem and overcomes all of its limitations.
In the second lecture on the construction of the conservation laws for a DE system, we review the standard and extended formulations of the classical Noether's Theorem. The classical Noether's Theorem only works for variational systems and requires the construction of the Lagrangian. We show how the Direct Method generalizes Noether's Theorem and overcomes its serious limitations. We also show how a symmetry maps a CL to another CL. Moreover, it will be shown that for a given PDE system, a solution pair consisting of a solution of its linearized system and a solution of the adjoint of its linearized system, directly yields a CL by a simple formula.
Often a given PDE system has no local symmetry and/or no local conservation law. Moreover, its local symmetries may not be useful for a problem at hand.
Aim: to extend existing methods for finding local symmetries and local CLs and their uses to PDE systems nonlocally related and equivalent to a given PDE system.
Two systematic and "natural" ways are presented for accomplishing this. In particular, for any PDE system, (1) each local CL as well as (2) each Lie point symmetry systematically yields a nonlocally related system. The first lecture will present the conservation law based method.
ABSTRACT: In the first part of this thesis, we prove the sharpness of the exponent range in the L^2 Fourier restriction theorem due to Mockenhaupt and Mitsis (with endpoint estimate due to Bak and Seeger). The proof is based on a random Cantor-type construction of Salem sets due to Laba and Pramanik. The key new idea is to embed in the Salem set a small deterministic Cantor set that disrupts the restriction estimate for the natural measure on the Salem set but does not disrupt the measure’s Fourier decay.
In the second part of this thesis, we prove a lower bound on the Fourier dimension of certain sets on the real line arising from Diophantine approximation. This generalizes theorems of Kaufman and Bluhm. As a consequence, we obtain new explicit examples of Salem sets. We apply our result to metrical Diophantine approximation and compute the Hausdorff dimension of sets arising from Diophantine approximation in new cases. We also prove a higher-dimensional analog of our result.
Room 200, Graduate Student Centre, 6371 Crescent Rd., UBC
Mon 22 Jun 2015, 12:30pm-2:30pm
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ABSTRACT: Many problems in climate modelling are characterized by their chaotic nature and multiple time scales. Stochastic parametrization methods can often simplify the analysis of such problems by using appropriate stochastic processes to account for degrees of freedom that are impractical to model explicitly, such that the statistical features of the reduced stochastic model are consistent with more complicated models and/or observational data. However, applying appropriate stochastic parametrizations is generally a non-trivial task. This is especially true when the statistics of the approximated processes exhibit non-Gaussian features, like a non-zero skewness or infinite variance. Such features are common in problems with nonlinear dynamics, anomalous diffusion processes, and multiple time scales. Two common topics in stochastic parameterization are model parameter estimation and the derivation of reduced stochastic models.
In this dissertation, we study both of these topics in the context of stochastic differential equation models, which are the preferred formalism for continuous-time modelling problems. The motivation for this analysis is given by problems in atmospheric or climate modelling. We estimate parameters of a dynamical model of sea surface vector winds using a method based on the properties of differential operators associated with the probabilistic evolution of the wind model. The parameter fields we obtain allow us to reproduce statistics of the vector wind data and inform us regarding the limitations of both the estimation method and the model itself. We also derive reduced stochastic models for a class of dynamical models with multiple time scales that are driven by α-stable stochastic forcing. The results of this project are then applied to derive a similar approximation for processes that are driven by a fast linear process experiencing additive and multiplicative Gaussian white noise forcing. The results of these chapters complement previous results for systems driven by Gaussian white noise and suggest a possible dynamical mechanism for the appearance of α-stable stochastic forcing in some climatic time series.
Given a field K, a polynomial f(x) with coefficients in a field K is said to be indecomposable over K if it can not be represented as a functional composition of lower degree polynomials in K[x]. Any polynomial of degree greater than 1 can clearly be represented as a composition of indecomposable polynomials. Such a representation, called a complete decomposition, does not need to be unique. Ritt in the 1920's described the extent of the non-uniqueness for complex polynomials. In so doing, Ritt exhibited some invariants of complete decompositions of complex polynomials. Building on the methods developed by Ritt, Fried and others, and by shifting to the setting of maps between curves, we extend and generalize known results on invariants of complete decompositions. These results are obtained jointly with Michael Zieve. We further present some methods for showing indecomposability and discuss applications of such results to Diophantine equations. These methods are described in a joint survey paper with Robert F. Tichy.
Room 203, Graduate Student Centre, 6371 Crescent Rd., UBC
Thu 25 Jun 2015, 9:00am-11:00am
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Abstract: Critical phenomena and phase transitions are important subjects in statistical mechanics and probability theory. They are connected to the phenomenon of universality that makes the study of mathematically simple models physically relevant. Examples of such models include ferromagnetic spin systems such as the Ising, O(n) and n-component |\varphi |^4 models, but also the self-avoiding walk that has been observed to formally correspond to a "zero-component" spin model.
The subject of this thesis is the extension and application of a rigorous renormalisation group method developed by Brydges and Slade to study the critical behaviour of the continuous-time weakly self-avoiding walk and of the n-component |\varphi |^4 model on the 4-dimensional lattice \bbbmath Z ^4. Although a "zero-component" vector is mathematically undefined (at least naively), we are able to interpret the weakly selfavoiding walk in a mathematically rigorous manner as the n=0 case of the n-component |\varphi |^4 model, and provide a unified treatment of both models.
For the |\varphi |^4 model, we determine the asymptotic decay of the critical correlation functions including the logarithmic corrections to Gaussian scaling, for n\ge 1. This extends previously known results for n=1 to all n\ge 1, and also observes new phenomena for n>1, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the "watermelon" network consisting of p weakly mutually and self-avoiding walks, for all p\ge 1, including the logarithmic corrections. This extends a previously known result for p=1, for which there is no logarithmic correction, to a much more general setting.
Room 200, Graduate Student Centre, 6371 Crescent Rd., UBC
Fri 26 Jun 2015, 12:30pm-2:30pm
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ABSTRACT: In my dissertation, I mainly consider two problems. First, I study the SU(3) Toda system, which comes from the theoretic physics and differential geometry. For this system, by analyzing the bubbling solutions when the parameters tends to the critical parameters, I could compute the degree jump when the parameters cross the critical parameter. As a consequence, I can obtain a partial result on the degree counting formula for the system and some existence result provided the parameters are restricted in some range. In the second part of my dissertation, I study a nonlinear Neumann elliptic problem, which comes from biology. For this problem, I construct a nontrivial solution, i.e., non-constant solution, which disproves a conjecture during the past two decades.
Room 203, Graduate Student Centre, 6371 Crescent Rd., UBC
Mon 29 Jun 2015, 12:30pm-2:30pm
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ABSTRACT: Formal functional integrals are commonly used as theoretical tools and as sources of intuition for predicting phase transitions of many-particle systems in Condensed Matter Physics. In this thesis, we derive rigorous versions of these functional integrals for two types of quantum many-particle systems.
We start with a brief review of quantum statistical mechanics and the formalism of coherent states, which form the basis for our analysis. For a mixed gas of bosons and/or fermions interacting on a finite lattice with a general Hamiltonian that preserves the total number of particles in each species, we rigorously derive a functional integral representation for the partition function, using a large-field cutoff for the boson fields. We then expand the resulting “action” in powers of the fields and find a recursion relation for the coefficients. In the case of a 2-body interaction (such as the Coulomb interaction), we also find bounds on the coefficients, which give a domain of analyticity for the action. This domain is large enough for use of the action in the functional integral, provided that the large-field cutoffs are taken to grow not too quickly. Next, we study a system of electrons and phonons interacting in a finite lattice, using the Holstein Hamiltonian. We again rigorously derive a coherent-state functional integral representation for the partition function of this system and then prove that the action in the functional integral is an entire-analytic function of the fields. However, since the Holstein Hamiltonian does not preserve the total number of bosons, our approach for the previous system requires some modification. In particular, we repeatedly use Duhamel expansions in powers of the interaction, rather than sums over particle numbers.
We introduce a new class of algebraic varieties which is closed under products and embraces, e.g., projective homogeneous varieties, generically split varieties, and standard norm varieties (associated to symbols in Galois cohomology groups). We prove for this class the characterization of canonical p-dimension in terms of algebraic cycles previously known for projective homogeneous and generically split varieties. As an application, we extend to this class the incompressibility criterion of products previously known only for certain type of projective homogeneous varieties.
It will be shown that from the symmetries or conservation law multipliers of a nonlinear PDE system one can determine systematically whether it can be mapped invertibly to some linear PDE system. We show to find systematically such a mapping when one exists.
It will also be shown that from the symmetries of a linear PDE system with variable coefficients, one can determine sytematically whether it can be mapped invertibly to some linear PDE system with constant coefficients. We show how to find systematically such a mapping when one exists. This leads to the solution of the problem posed by Kolmogorov on when can a diffusion process be mapped into a Wiener process.
These results will be extended systematically to noninvertible mappings.
Room 203, Graduate Student Centre, 6371 Crescent Road, UBC
Wed 22 Jul 2015, 4:00pm-6:00pm
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ABSTRACT:
In this thesis, we study a natural noncommutative lift of the ubiquitous Schur functions, called noncommutative Schur functions. These functions were introduced by Bessenrodt, Luoto and van Willigenburg and resemble Schur functions in many regards. We prove some new results for noncommutative Schur functions that are analogues of classical results, and demonstrate that the resulting combinatorics in this setting is equally rich. First we prove a Murnaghan-Nakayama rule for noncommutative Schur functions. In other words, we give an explicit combinatorial formula for expanding the product of a noncommutative power sum symmetric function and a noncommutative Schur function in terms of noncommutative Schur functions. In direct analogy to the classical Murnaghan-Nakayama rule, the summands are computed using a noncommutative analogue of border strips, and have coefficients ±1 determined by the height of these border strips. The rule is proved by interpreting the noncommutative Pieri rules for noncommutative Schur functions in terms of box adding operators on compositions. We proceed to give a backward jeu de taquin slide analogue on semistandard reverse composition tableaux. These tableaux were first studied by Haglund, Luoto, Mason and van Willigenburg when defining quasisymmetric Schur functions. Our algorithm for performing backward jeu de taquin slides on semistandard reverse composition tableaux results in a natural operator on compositions that we call the jdt operator. This operator in turn gives rise to a new poset structure on compositions whose maximal chains we enumerate. As an application, we also give new noncommutative Pieri rules for noncommutative Schur functions that use the jdt operators.
Room 200, Graduate Student Centre, 6371 Crescent Rd., UBC
Fri 24 Jul 2015, 12:30pm-2:30pm
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ABSTRACT: Self-avoiding walks appear ubiquitously in the mathematical study of linear polymers as it naturally captures their volume exclusion property. However, self-avoiding walks are very difficult to analyse with few rigourous results available.
In 2008, Alvarez et al. determined numerical results for the forces induced by a self-avoiding walk in an interactive slit. These results resembled the exact results for a directed model in the same setting by Brak et al., suggesting the physical consistency of directed walks as polymer models. In the directed walk model, three phases were identified in the infinite slit limit as well as the regions of attractive and repulsive forces induced by the polymer on the walls.
Via the kernel method, we extend the model to include two directed walks as a way to find exact enumerative results for studying the behaviour of ring polymers near an interactive wall, or walls.
We first consider a ring polymer near an interactive surface via two friendly walks that begin and end together along a single wall. We find an exact solution and provide a full analysis of the phase diagram, which admits three phase transitions.
The model is extended to include a second wall so that two friendly walks are confined in an interactive slit. We find and analyse the exact solution of two friendly walks tethered to different walls where single interactions are permitted. That is, each walk interacts with the wall it is tethered to. This model exhibits repulsive force only in the parameter space. While these results differ from the single polymer models, they are consistent with Alvarez et al.
Finally, we consider the model with double interactions, where each walk interacts with both walls. We are unable to find exact solutions via the kernel method. Instead, we use transfer matrices to obtain numerical results to identify regions of attractive and repulsive forces. The results we obtain are qualitatively similar to those presented in Alvarez et al. Furthermore, we provide evidence that the zero force curve does not satisfy any simple polynomial equation.
Room 200, Graduate Student Centre, 6371 Crescent Rd., UBC
Tue 28 Jul 2015, 9:00am-11:00am
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This dissertation studies semidirect products of a torus by a finite group from the representation theory point of view. The finite group of greatest interest is the cyclic group of prime order. Such semidirect products occur in nature as isotropy groups of Lie groups acting on themselves by conjugation and as normalizers of maximal tori in reductive linear algebraic groups.
The main results of this dissertation are
- the calculation of the representation ring of such semidirect products as an algebra over the integers for certain special cases,
- the adaptation of an algorithm from invariant theory to find finite presentations of representation rings,
- the computation of the topological K-theory of the classifying space of certain semidirect products,
- the demonstration that the equivariant K-theory of the projective unitary group of degree 2 acting on itself by conjugation is not a free module over its representation ring.
I will introduce a class of Calabi-Yau manifolds associated to the polytope tilings. Their mirrors provide new insights in the toric mirror symmetry, and are also closely related to certain modular forms. This is a joint work with Siu-Cheong Lau.
The multiplicative Horn problem asks what constraints the eigenvalues of two n x n unitary matrices place on the eigenvalues of their product. The solution of this problem, due to Belkale, Kumar, Woodward, and others, expresses these constraints as a convex polyhedron in 3n dimensions and describes the facets of this polyhedron more or less explicitly. I will explain how the vertices of the polyhedron may instead be described in terms of fixed points of a torus action on a symplectic stratified space, constructed as a quotient of the so-called universal group-valued implosion.
It will be shown that from the symmetries or conservation law multipliers of a nonlinear PDE system one can determine systematically whether it can be mapped invertibly to some linear PDE system. We show to find systematically such a mapping when one exists.
It will also be shown that from the symmetries of a linear PDE system with variable coefficients, one can determine systematically whether it can be mapped invertibly to some linear PDE system with constant coefficients. We show how to find systematically such a mapping when one exists. This leads to the solution of the problem posed by Kolmogorov on when can a diffusion process be mapped into a Wiener process.
These results will be extended systematically to noninvertible mappings
It will be shown that from the symmetries of a linear PDE system with variable coefficients, one can determine systematically whether it can be mapped invertibly to some linear PDE system with constant coefficients. We show how to find systematically such a mapping when one exists. This leads to the solution of the problem posed by Kolmogorov on when can a diffusion process be mapped into a Wiener process.
It will also be shown that from the conservation law multipliers of a nonlinear PDE system one can determine systematically whether it can be mapped invertibly to some linear PDE system. We show to find systematically such a mapping when one exists.
These results will be extended systematically to noninvertible mappings.
Practical information for everyone teaching at UBC for the first time: a quick overview of UBC rules, policies, key websites one would have to use, and other resources. Pizza lunch will be provided at noon.
The first part of this lecture will be a short discussion on how to find non-invertible mappings of linear PDEs with variable coefficients to linear PDEs with constant coefficient, focusing on parabolic PDEs. The main part of the lecture will focus on how to find invariant solution of DEs. There will be a presentation of Lie's systematic classical method based on the invariance of PDEs under Lie groups of point transformations--here a symmetry maps every solution of a PDE to another solution of the PDE and one looks for solutions that are themselves invariant. It will be shown how to generalize Lie's classical method to the systematic nonclassical method based on mapping only a subset of solutions of a PDE to solutions of the same PDE whereas the remaining solutions map into solutions of other PDEs. The aim is to find solutions of such submanifolds of that are themselves invariant. Illustrative examples yielding such nonclassical solutions include the Boussinesq equation and the Kompaneets equation.
In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramare, Elkies and many others, we complete the proof of Jacobi's observation.
Planar maps are of interest to several communities: In Combinatorics Tutte and others have enumerated them. In Physics they are a model for 2 dimensional quantum gravity. The past 15 years have seen great progress in this area using probabilistic tools. I will survey the topic, some of the significant progress made so far and big problems we hope to solve.
Note for Attendees
Refreshments will be served in MATH 125 before the colloquium.
Please join us at the Institute of Applied Mathematics to welcome new members at the start of a new academic year. We will have food and drink and will be light on speeches.
The Poisson-Nernst-Planck (PNP) system is a well-known model of ion transport with many applications in biology, engineering and physics. Cross-diffusion terms may describe the exclusion of steric effects. In this lecture, I shall introduce cross diffusion terms from the Lennard-Jones potential and show the analytical results as follows:
1. Stability of 1D boundary layer solutions to original Poisson-Nernst-Planck (PNP) systems
2. Multiple solutions of PNP systems with steric effects
Let F(x,y,z) be a real trivariate polynomial of constant degree, and let A,B,C be three sets of real numbers, each of size n.
How many roots can F have on A x B x C?
This setup arises in many interesting problems in combinatorial geometry, including distinct distances between points on curves, distinct distances from three points, collinear triples of points on curves (the `orchard problem'), unit-area triangles, triple intersection points of families of circles, and more.
This question has been studied by Elekes and R\'onyai and then by Elekes and Szab\'o about 15 years ago. One of their striking results is that, for the special case where F(x,y,z) = z-f(x,y), either F vanishes at only a subquadratic number (o(n^2)) of points of A x B x C, or else f must have one of the special forms f(x,y) = h(p(x)+q(y)) or f(x,y) = h(p(x)q(y)), for univariate polynomials p, q, h.
In this talk I will survey recent progress on this problem, in which the analysis is greatly simplified, and the bounds become sharp: If F does not have a special form, the number of roots is at most O(n^{11/6}). Moreover, the results also hold over the complex field. This yields significantly improved bounds for many geometric problems, as listed above.
The proofs use techniques from algebra and algebraic geometry, which are somewhat related to the recent growing body of work on algebraic techniques for incidences and distance problems, inspired by Guth and Katz's seminal papers.
Joint work with Orit Raz, Jozsef Solymosi, and Frank de Zeeuw (and others).
We consider random planar maps weighted by the critical Fortuin--Kasteleyn percolation model with parameter q \in (0,4). The study of these surfaces is central to the theory of Liouville quantum gravity.
We obtain rigorously the value of criticalexponents associated with the length of cluster interfaces, which is shown to be
$$
\frac{4}{\pi} \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right).
$$
This is consistent with physics predictions; in particular, applying the KPZ formula we recover the dimension of SLE curves.
Joint work with Benoit Laslier and Gourab Ray (Cambridge).
We combine the latest advances in modularity lifting with a 3-5-7 modularity switching argument to deduce modularity of 'most' elliptic curves over totally real fields. In particular, we show that all elliptic curves over real quadratic fields are modular. This talk is based on joint work with Bao Le Hung (Harvard) and Nuno Freitas (Bonn).
The subject of this talk is the problem of counting faces of various dimensions in a convex n-polytope. When the polytope is simplicial, McMullen's conjecture, proved in 1979 by Billera, Lee and Stanley, gives a complete description of all possible face numbers. One encodes the number of faces of dimension 0,1,...,n in a vector g with n/2 components. The result then is that g comes from a simplicial polytope if and only if the components of g are the dimensions of graded pieces in an algebra generated in degree 1.
In this talk I will explain the conjectural extensions of this theorem to simplicial spheres, non-simplicial polytopes and barycentric subdivisions. In every problem that involves counting faces, it is conjectured that there exists an algebra generated in degree 1 whose graded pieces give the face numbers.
Note for Attendees
Refreshments will be served in MATH 125 before the colloquium.
An Abelian variety (of complex dimension g) is an algebraic geometer's version of a torus — it is a variety which is topologically equivalent to a (real) 2g-dimensional torus. Geometers consider the problem of counting the number of curves on an Abelian variety subject to some set of constraints. In dimensions g=1,2, and 3, these geometric numbers have a surprising connections to number theory and combinatorics. They occur as the coefficients of Fourier expansions of various modular forms and they can also be determined in terms of combinatorics of 2D and 3D partitions (a.k.a. box counting). We illustrate this using only elementary ideas from topology and combinatorics in the case of g=1. For g=2 and g=3, we describe recent theorems and conjectures which complete determine the enumerative geometry of Abelian surfaces and threefolds in terms of Jacobi forms and in the process we indicate how Jacobi forms arise from the combinatorics of box counting.
Conical Kahler metrics have become an interesting topic in Kahler geometry, and played an important role in the solution of Yau-Tian-Donaldson conjecture. In this talk, we make use of approximation method of Guenancia-Paun to extend Tian-Zhang's maximal existence result of Kahler-Ricci flow to conic case. Finally if possible, we can talk a little about C^{2,\alpha}-estimate for conical Kahler-Ricci flow based on Tian's master thesis.
The chromatic symmetric function X_H of a hypergraph H is the generating function for all colorings of H so that no edge is monochromatic. When H is an ordinary graph, it is known that X_H is positive in the fundamental quasisymmetric functions F_S, but this is not the case for general hypergraphs. We exhibit a class of hypergraphs H --- hypertrees with prime-sized edges --- for which X_H is F-positive, and give an explicit combinatorial interpretation for the F-coefficients of X_H. We also present a conjecture that certain chromatic symmetric functions of hypergraphs are Schur-positive.
Circle packings provide a conformally natural way to draw triangulations of the sphere. Similarly and more generally, dessins d'enfants provide natural drawings of plane graphs. I will explain how this works, with an emphasis on trees. Then I will discuss a few results and a few questions in the uniformly random case.
We introduce a natural chain complex associated to a collection of subspaces of a vector spaces, and discuss the associated homology. We will give some background on Bieri-Neumann-Strebel invariants of groups, and show how the BNS invariant of a group leads to a nice subspace arrangement, whose associated homology is (yet) another invariant of the group. This can give a useful way of distinguishing between finitely presented groups - we will give some examples involving right-angled Artin groups.
Previous studies of shear-thinning fluids in pipe flow have shown that, although the time-averaged velocity profile is – as expected – axisymmetric when the flow is laminar or fully turbulent, contrary to expectation it exhibits marked asymmetry in the laminar-turbulent transition regime. Despite this strange effect being observed in different experimental facilities no satisfactory explanation yet exists. The general consensus of these previous experiments is that the location of the peak velocity remains at a fixed point in space (at least during one experimental realisation).
Here we present new experimental data obtained using 2D 3C stereo particle image velocimetry which demonstrates that, in fact, the asymmetry does not stay fixed in space and that it may be related to a linear instability occurring prior to classical transition to turbulence (i.e. the appearance of puffs/slugs). The experiments are performed using aqueous solutions of a xanthan gum (0.15wt %), which exhibits shear-thinning of the shear viscosity by about three-orders of magnitude and is only very weakly elastic in the range of shear rates probed in our pipe. To quantitatively describe the degree of azimuthal flow asymmetry we define an “asymmetry factor”. Our results once again confirm significant departures from axisymmetry in transitional flows of shear-thinning fluids but indicate that the asymmetry may occur slightly before transition. Furthermore at higher flowrates within transition, it can be seen that the asymmetry is not fixed in space but that, although it preferentially arises at certain azimuthal locations, there are short durations when the flow switches back to a quasi-axisymmetric state. Associated with some of these events, the flow can also briefly probe an asymmetric state of a different orientation to the preferred state.
Pretend that your car is a unit line segment. How do you perform a three point turn using zero area on the road? It turns out that this seemingly impossible driving stunt is related to the fundamental theorem of calculus! We will explore this connection and see how these ideas have been useful in many problems in mathematics.
I will present the results of our recent work on epithelial morphogenesis, a highly conserved set of processes that transform two-dimensional sheets of cells into complex three-dimensional structures. Such transformations play key roles during embryogenesis and their understanding is important both from a purely scientific standpoint and for the design of man-made tissues and organs. Our laboratory is using the eggshell morphogenesis in the fruit fly Drosophila melanogaster as a model for studying epithelial morphogenesis in a relatively simple setting, with a constant number of cells. Based on the combination of time lapse imaging studies and three-dimensional image reconstructions we found that epithelial morphogenesis in this system can be viewed as a two-stage process, in which buckling of a group of cells out of a curved sheet is followed by ordered cell rearrangements, generating a tube-like structure. A small number of cells involved in this transformation (<100) and their reproducible dynamics enable its computational modeling. I will present models based on vertex description of cell sheets and discuss some applied mathematical problems that arise in the analysis of these models.
References:
Osterfield M, Du X, Schüpbach T, Wieschaus E, Shvartsman SY. Three-dimensional epithelial morphogenesis in the developing Drosophila egg. Dev Cell. 2013 Feb 25;24(4):400-10.
Fletcher AG, Osterfield M, Baker RE, Shvartsman SY. Vertex models of epithelial morphogenesis. Biophys J. 2014 Jun 3;106(11):2291-304.
Osterfield M, Schüpbach T, Wieschaus E, Shvartsman SY. Diversity of epithelial morphogenesis during eggshell formation in drosophilids. Development. 2015 Jun 1;142(11):1971-7.
It has been recently shown by Chen-Donaldson-Sun that the existence of a Kähler-Einstein metric on a Fano manifold is equivalent to the property of K-stability. In general, however, this does not lead to an effective criterion for deciding whether such a metric exists, since verifying the property of K-stability requires one to consider infinitely many special degenerations called test configurations. I will discuss recent joint work with H. Süß in which we show that for Fano manifolds with complexity-one torus actions, there are only finitely many test configurations one needs to consider. This leads to an effective method for verifying K-stability, and hence the existence of a Kähler-Einstein metric. As an application, we provide new examples of Kähler-Einstein Fano threefolds.
The leading asymptotic large-scale behavior of the spatially bipartite entanglement entropy (EE) of the free Fermi gas at temperature T=0 is by now well understood. Here, we present and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at T>0. The leading large-scale term of this thermal EE turns out to be twice the first finite-size correction to the infinite-volume thermal entropy (density). However, it is given by a rather complicated integral derived from semiclassical trace formulas and differs, at least at high temperature, from simpler expressions previously obtained by arguments based on a conformal field theory. In the zero-temperature limit, the leading large-scale term of the thermal EE considerably simplifies and displays a \ln(1/T)-singularity which one may identify with the known logarithmic correction at T=0 to the so-called area-law scaling. This is joint work with Hajo Leschke and Alexander Sobolev.
We consider a model of random knots akin to the one proposed by Dunfield et. al.; a random knot diagram is a random immersion of the circle into the sphere with randomly assigned crossing signs. By studying diagrams as annotated planar maps, we are able to show that any given ``tangle diagram'' substructure almost certainly occurs many times in a random knot diagram with sufficiently many crossings. Thus, in this model, it is exponentially unlikely for a diagram with n crossings to represent an unknot as n \rightarrow \infty. This asymptotic behavior is similar to that seen in other models of random knots such as random lattice walks and random polygons.
We consider a classical problem of estimating the smallest singular value of random rectangular and square matrices with independent identically distributed entries. The novelty of our results lies in very weak, or nonexisting, moment assumptions on the distribution of the entries. We prove that, given a sufficiently “tall” N \times n rectangular matrix A = (a_{ij} ) with i.i.d. entries satisfying the condition \sup_{\lambda \in \mathbb{R}} \mathbb{P} \left\{ \lvert a_{ij} − \lambda \rvert \le 1 \right\} \le 1/2, the smallest singular value s_n (A) satisfies s_n(A) \gtrsim \sqrt{N} with probability very close to one.
Our second theorem is an extension of the fundamental result of Bai and Yin from the early 1990’s. Let \{a_{ij} \}^\infty_{i,j=1} be an infinite double array of i.i.d. random variables with zero mean and unit variance, and let (N_m )_{m=1}^\infty be an integer sequence satisfying \lim_{m \to \infty} \frac{N_m}{m} = r\) for some \(r \in (1, \infty). Then, denoting by A_m the N_m \times m top-left corner of the array \{a_{ij}\}, we have
\[ \lim_{m \to \infty} \frac{s_m(A_m)}{\sqrt{N_m}} = \sqrt{r}-1 \hspace{3mm}\mbox{ almost surely}.\]
This result does not require boundedness of any moments of a_{ij}'s higher than the 2-nd and resolves a long standing question regarding the weakest moment assumptions on the distribution of the entries sufficient for the convergence to hold.
(This is a continuation of the talk of 23 Septemer) We introduce a natural chain complex associated to a collection of subspaces of a vector spaces, and discuss the associated homology. We will give some background on Bieri-Neumann-Strebel invariants of groups, and show how the BNS invariant of a group leads to a nice subspace arrangement, whose associated homology is (yet) another invariant of the group. This can give a useful way of distinguishing between finitely presented groups - we will give some examples involving right-angled Artin groups.
Start labelling the vertices of the square grid with 0's and 1's with the condition that any pair of neighbouring vertices cannot both be labelled 1. If one considers the 1's to be the centres of small squares (rotated 45 degrees) then one has a picture of square-particles that cannot overlap. This problem of
"hard-squares" appears in different areas of mathematics - for example it has appeared separately as a lattice gas in statistical mechanics, as independent sets in combinatorics and as the golden-mean shift in symbolic dynamics.
A core question in this model is to quantify the number of legal configurations - the entropy. In this talk I will discuss the what is known about the entropy and describe our recent work finding rigorous and precise bounds for hard-squares and related problems.
This is work together with Yao-ban Chan.
Note for Attendees
Refreshments will be served in the MATH 125 lounge area at 2:45 p.m. before the colloquium.
This is a report on joint work with Zsolt Patakfalvi. We prove a strengthening of Kollár's Ampleness Lemma and use it to prove that any proper coarse moduli space of stable log-varieties of general type is projective. We also confirm the Iitaka-Viehweg conjecture on the subadditivity of log-Kodaira dimension for fiber spaces whose general fiber is of log general type.
Since the late eighties, when Bourgain proved his celebrated pointwise ergodic theorems, discrete analogues in harmonic analysis have come into vogue. Indeed, much work has been devoted to studying maximal functions and (maximal truncations of) singular integrals. The Carleson operator -- strictly more singular than either above operator -- is a natural operator to try adapt to the discrete setting: Eli Stein has been successful in "transferring" the continuous result to the discrete setting. In particular, he has established the L^2(\Z) boundedness of the discrete Carleson operator
\[ \sup_{\lambda} | \sum_{m \neq 0} f(n-m) e^{2\pi i \lambda m}/m |.\]
The purpose of this talk will be to study a discrete analogue of the quadratically modulated Carleson operator:
\[ \sup_{\lambda} | \sum_{m \neq 0} f(n-m) e^{2\pi i \lambda m^2}/m | \]
where \lambda ranges over certain subsets of (0,1].
This will be the first talk of the sequence, and very little background is required -- though the argument will eventually combine elements from analytic number theory, probability theory, and harmonic analysis.
In fluids, the presence of a dominant vortex has an organizing effect on the surrounding flow. We exploit this effect to simplify the stability problem for relative equilibria of (N+1) interacting point vortices, where N vortices have infinitesimal circulation and one vortex is strong in relation. Within this framework, existence and stability of equilibria reduces to characterizing critical points of a function defined on a circle. In the case that all vortices have circulation of the same sign, this function is potential-like, with minima corresponding to stable configurations and maxima and saddle points corresponding to unstable solutions. Surprisingly, typical stable configurations exhibit asymmetry.
A follow-up talk on multidimensional convolutional codes will be given on Thursday, October 8.
Coding theory - theory of error correcting codes - is one of the most interesting and applied parts of mathematics.Coding theory methods are often elegant applications of very basic concepts and methods of (abstract) algebra. IIn this talk we shall start by giving a brief general overview of this area before introducing the main topic of the talk, namely, convolutionalcodes.
These codes are mathematically more involved than the standard block codes as they posses a very rich algebraic structure. In this context the data is considered as a sequence. Even though the data is split into blocks of a fixed rate as in block codes, the relative position of each block in the sequence is taken into account. The blocks are not encoded independently and previous nodes in the sequence have an effect over the next encoded node. Because of this, convolutionalcodes have memory. From a mathematical point of view they can be seen as F[x]-submodules of F[x]^n, where F is a finite field and F[x] is the ring of polynomials over F.
The aim of this talk is to introduce this powerful class of codes, their properties and their use in practice. We shall conclude by presenting some of the most fascinating open problems in the design of convolutionalcodes.
I will consider the following classical harmonic map flow from a general two-dimensional domain D to S^2:
u_t=\Delta u +|\nabla u|^2 u, u: D \to S^2
We develop a parabolic gluing method to construct finite time blow-up solutions of Type II in general domains. We show that type II blow-up solutions with blow-up rate
(T-t)/\log^2 (T-t)
is stable and generic in arbitrary domains (without any symmetry). I will also discuss the construction of multiple blow-ups, reverse bubbling, bubbling trees, bubbling at infinity. As a by-product we can perform new geometric surgeries. (Joint work with Manuel del Pino and Juan Davila.)
Consider a social network modelled as a graph, with people and friendships represented by vertices and edges, respectively. Suppose that a person knows a piece of information, and as time passes, talks to other people and spreads it. How long it takes until everyone knows the rumour? The answer, which we call the "spread time", certainly depends on the graph's structure and how the rumour spreads. In this talk we discuss two well known randomized rumour spreading protocols (known as push&pull protocols) and survey the known results on their spread times on various graphs.
Based on joint work with H. Acan, A. Collevecchio, and N. Wormald.
It is classical that the zero set and the set of record times of a linear Brownian motion have Hausdorff dimension almost surely. Can we find a larger random subset on which a Brownian motion is monotone? Perhaps surprisingly, the answer is negative. We outline the short proof, which is an application of Kaufman's dimension doubling theorem for planar Brownian motion. This is a joint work with Yuval Peres.
I will discuss classification results for subgroups of Out(Fn) (analogous to Ivanov's classification of subgroups of mapping class groups of surfaces), and more generally of automorphism groups of free products. In particular, I will present a version of the Tits alternative for the automorphism group of a free product. This is partly joint work with Vincent Guirardel.
An introductory talk on one dimensional convolutional codes will be given on Tuesday, October 6.
Multidimensional (nD) convolutional codes generalize one dimensional (1D) convolutional codes and correspond to multidimensional systems widely studied in the systems theory literature. These codes have a practical potential in applications as they are very suitable to encode data recorded in n dimensions, e.g., pictures, videos, storage media, wireless applications, etc. However, in comparison to 1D convolutional codes, little is known in the area of nD convolutional codes and much more needs to be done to make it attractive for practical applications. From a mathematical point of view, these codes can be viewed as F[x_1,...,x_n]-submodules of F[x_1,...,x_n]^n, where F is a finite field and F[x_1,...,x_n] is the ring of polynomials in several variables over F. In this talk we present a construction of an nD convolutional code based on superregular matrices with excellent error-correcting capabilities.
Aircraft are designed to fly but also need to operate efficiently and safely as vehicles on the ground. The tricycle configuration of commercial aircraft presents challenges for manoeuvres, such as high-speed turns off a runway. The talk will present results of a collaboration with Airbus into the stability of ground manoeuvres, whose central idea is to employ tools from bifurcation analysis to relevant industry-validated aircraft models. Compared to standard extensive numerical simulations, this approach has been demonstrated to have potential efficiency benefits during the design stage. In particular, it allows for detailed studies of the nature of instabilities that need to be avoided in practice. As an example, we show that the sudden loss of lateral stability of a mid-size passenger aircraft turning on the ground is due to a canard phenomenon that arises due to a non-obvious slow-fast splitting.
This is joint work with Etienne Coetzee (Airbus), James Rankin (INRIA France), Mathieu Desroches (INRIA France) and Mark Lowenberg (University of Bristol).
Note for Attendees
Note this is the first of two talks at Robson Square campus. The second talk will be followed by a reception at 5:00.
Predicting the behaviour of a structure when subjected to an earthquake is an important problem from Civil Engineering. Here, we consider a planar post-tensioned frame, which can be modelled as a two-degree-of-freedom system that is equivalent to the analytical model of a tied rocking block on an elastic foundation. The frame remains structurally sound as long as the tilt angle of the frame does not exceed a certain maximal angle. A standard approach to studying the dynamics would be to run simulations, where it is assumed that the earthquake is a pure sine wave with varying frequency and amplitude. Such a brute-force approach establishes a region in the frequency-amplitude plane for which the structural stability of the frame eventually fails. We propose a novel approach that calculates the failure region in a much more efficient way by determining the failure boundary directly. Our method is based on continuation of a suitable two-point boundary value problem. Our computations demonstrate that the failure boundary is only piecewise smooth and the results highlight further interesting details of how the dynamics is organised in the frequency-amplitude plane.
Note for Attendees
Note this is the second of two talks at Robson Square campus. This talk will be followed by a reception at 5:00.
The Liouville function λ(n) is the completely multiplicative arithmetic function defined by λ(p) = –1 for each prime p. Pólya investigated its summatory function L(x) = Σn≤x λ(n), and showed for instance that the Riemann hypothesis would follow if L(x) never changed sign for large x. While it has been known since the work of Haselgrove in 1958 that L(x) changes sign infinitely often, oscillations in L(x) and related functions remain of interest in analytic number theory. We review some connections between oscillations in this function and its relatives with the Riemann hypothesis and other problems in number theory, and describe some recent work on this topic. In particular, we describe a method involving substantial computation that establishes new bounds on the magnitude of the oscillations of L(x). This is joint work with Tim Trudgian.
Abstract: We construct some ODE solutions for the fractional Yamabe problem in conformal geometry. The fractional curvature, a generalization of the usual scalar curvature, is defined from the conformal fractional Laplacian, which is a non-local operator constructed on the conformal infinity of a conformally compact Einstein manifold.
These ODE solutions are a generalization of the usual Delaunay and, in particular, solve the fractional Yamabe problem
$$ (-\Delta)^\gamma u= c_{n, {\gamma}}u^{\frac{n+2\gamma}{n-2\gamma}}, u>0 \ \mbox{in} \ \r^n \backslash \{0\},$$
with an isolated singularity at the origin.
This is a fractional order ODE for which new tools need to be developed. The key of the proof is the computation of the fractional Laplacian in polar coordinates.
The Hopf map eta is nilpotent in the stable homotopy groups of spheres. This is not so for the motivic Hopf map, considered as an element of the motivic stable homotopy groups of spheres. This suggests that the eta-local part of motivic stable homotopy theory is an interesting object of study. We will describe this for the base fields C and R.
A finite ergodic Markov chain exhibits cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Here we consider non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity condition, we establish the cutoff phenomenon, determine its precise window, and prove that the cutoff profile approaches a remarkably simple, universal shape. This is a joint work with Justin Salez (Paris-Diderot).
Compressed sensing concerns the recovery of signals and images from seemingly incomplete data sets. Introduced nearly a decade ago, it has since become an intensive area of research in applied mathematics, engineering and computer science. However, many practical problems in which compressed sensing is applied, e.g. imaging, are not fully explained by existing theory. In this talk I will present a new framework for compressed sensing that seeks to bridge this gap. This framework is based on replacing some standard principles of compressed sensing with new local notions; specifically, sparsity in levels, local coherence in levels and multilevel random subsampling. I will demonstrate a series of near-optimal recovery guarantees based on these local concepts that explains the effectiveness of compressed sensing in such applications. Moreover, this framework is not just useful in understanding existing compressed sensing approaches. In the final part of the talk I will demonstrate how leveraging local sparsity through appropriately-designed locally incoherent sensing matrices leads to substantially improved compressed sensing techniques in a range of other applications.
Despite the appearance sometimes given in textbooks, not all differential equations are posed on straight lines and rectangles. This talk will introduce some easy-to-use techniques for computing numerical solutions to partial differential equations (PDEs) posed on curved surfaces and other general domains.
I will show some applications in thin-film flows, reaction-diffusion equations, bulk-surface coupling, point clouds, and image processing.
The talk will also outline how a close encounter with instability improved our understanding and numerical analysis of these methods.
After introducing the class of the classifying stack of a (finite) group, BG, in the Grothendieck ring of algebraic stacks, I will present certain cohomological invariants for a group - the Ekedahl invariants.
I am going to show that the class of BG is trivial if G is a finite subgroup of GL_3(k) or if G is a finite linear (or projective) reflection group. (k is a algebraically closed field of characteristic zero.) I will also show that the Ekedahl invariants of the discrete 5-Heisenberg group are trivial.
These results relate naturally to Noether's Problem and to its obstruction, the Bogomolov multiplier.
At the end of the talk, I will link these results to the study of the motivic classes of the quotient varieties V/G by showing that such classes and the classes of BG exhibit the same combinatorial structure. Therefore, despite the title and technical terminology I will aim at making the talk enjoyable also by the combinatorial community.
In the first talk in this lecture series, we introduced discrete analogues in harmonic analysis, and discussed Bourgain's celebrated polynomial ergodic theorem. In this second lecture in the series, we will apply Bourgain's approximation arguments in the study of the discrete quadratic Carleson operator,
\[ C_\Lambda f(n) := \sup_{\lambda \in \Lambda} | \sum_{m \neq 0} e(\lambda m^2)/m f(n-m) | \]
where $\Lambda \subset [0,1]$ is a set of modulation parameters, and $e(t) := e^{2\pi i t}$.
In particular, we will reduce matters to a multi-frequency maximal multiplier theorem, which we will study using three separate techniques:
Comparison to a multi-frequency multi-frequency maximal multiplier theorem involving truncations of the Hilbert transform;
A Fourier-analytic entropy/chaining argument; and
A TT* argument from the theory of oscillatory integrals.
The Shimura varieties X attached to orthogonal and unitary groups come equipped with a large collection of so-called special cycles. Examples include Heegner divisors on modular curves and Hirzebruch-Zagier cycles on Hilbert modular surfaces. We will review work of Borcherds and Bruinier using regularised theta lifts for the pair (SL_2,O(V)) to construct Green currents for special divisors. Then we will explain how to construct other interesting currents on X using the dual pair (Sp_4,O(V)). We will show that one obtains currents in the image of the regulator map of a certain motivic complex of X. Finally, we will describe how an argument using the Siegel-Weil formula allows to relate the values of these currents to the product of a special value of an L-function and a period on a certain subgroup of Sp_4.
In this talk we shall be interested in discrete harmonic functions in cones (in particular, in the quarter plane). The generating function of these harmonic functions satisfies a functional equation (closed to a well-known functional equation that appears in the context of enumeration of confined walks in combinatorics). We shall show the link between these harmonic functions and a one-parameter family of conformal mappings. One of the motivations to that study is to condition (in the sense of Doob) random walks never to leave cones.
In Balmer's framework of tensor triangular geometry, the prime thick tensor ideals in a tensor triangulated category C form a space which admits a continuous map to the Zariski spectrum Spec^h(End_u(1)) of homogeneous prime ideals in the graded endomorphism ring of the unit object. (Here the grading is induced by an element u of the Picard group of C.) If C is the stable motivic homotopy category and u is the punctured affine line, then this endomorphism ring is the Milnor-Witt K-theory ring of the base field. I will describe work by my student, Riley Thornton, which completely determines the homogeneous Zariski spectrum of Milnor-Witt K-theory in terms of the orderings on the base field. I will then comment on work in progress which uses the structure of this spectrum to study the thick subcategories of the stable motivic homotopy category.
I will give an overview of things we know about c2 invariant of a graph. This is an invariant investigated principally by Brown and Schnetz which comes from counting points on the hypersurface defined by the Kirchhoff polynomial of a graph. This invariant predicts many properties of the Feynman integral of the graph. It connects with deep things like modular forms and motives. Many computations involving it come down to playing around with polynomials defined from the graph and so its also combinatorial. The fun and power of it come from the interplay of all three of these things.
In the second talk in the series on the discrete quadratic Carleson operators,
\[ C_\Lambda f(n) := \sup_{\lambda \in \Lambda} | \sum_{m \neq 0} e^{2\pi i \lambda m^2}/m f(n-m), \Lambda \subset [0,1] \]
we reduced matters to a multi-frequency maximal multiplier theorem.
In this third talk we will prove that the operator norm of the multi-frequency maximal multiplier is slowly growing in the number of distinguished frequencies. We use three separate techniques to do so; the highlight of our approach is a novel entropy argument.
Cox rings of algebraic varieties were defined by Hu and Keel in relation to the minimal model program. The main question in the theory is to determine if the Cox ring of a variety is finitely generated. Such varieties are called Mori Dream Spaces. In this talk I will discuss examples of varieties that are not Mori Dream Spaces. These include toric surfaces blown up at a point and the moduli spaces of rational curves with n points. This is a joint work with Jose Gonzalez.
This talks will describe the concept of earthquake early warning and will include a description of the different methods and approaches used to detect earthquake shaking. An overview of the work being done in BC to establish and implement an EEW network for BC will be presented. This includes a description of the hardware and software that has been, and is being, developed. The future directions of the network will be presented and the opportunities for collaboration with various research groups at UBC will be discussed.
We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global W^{1,p}-estimates of Calder\'{o}n-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument.
The talk is based on the joint work with L. Hoang (Texas Tech) and T. Nguyen (U. of Akron).
We study a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c(G) denote the number of cops needed to capture the robber in a graph G, and let tw(G) denote the treewidth of G. We show that if G is planar then c(G) = Theta(tw(G)), and there is a polynomial-time constant-factor approximation algorithm for computing c(G). We also determine, up to constant factors, the value of c(G) of the random graph G(n,p) for all admissible values of p, and show that when the average degree goes to infinity, c(G) is typically asymptotic to the domination number.
I will describe the role played by an Apollonian circle packing in the scaling limit of the abelian sandpile on the square grid Z^2. The sandpile solves a certain integer optimization problem. Associated to each circle in the packing is a locally optimal solution to that problem. Each locally optimal solution can be described by an infinite periodic pattern of sand, and the patterns associated to any four mutually tangent circles obey an analogue of the Descartes Circle Theorem. Joint work with Wesley Pegden and Charles Smart.
A stable spherical fibration is classified by a map X → BGL₁(S) and Lewis showed that if this map is an infinite loop map or an n-fold loop map then the Thom spectrum is an E_∞- or E_n-ring spectrum, respectively. Ando, Blumberg, Hopkins, Gepner and Rezk introduced a new approach to Thom spectra using the language of ∞-categories. Using their approach, we will explain how to apply some simple (∞-)category theory to study multiplicative structures on Thom spectra, proving a generalization of Lewis's theorem and moreover characterizing the ring structure by a universal property. As an application I'll discuss a new (slightly simpler) proof of a remarkable theorem of Mahowald's realizing the Eilenberg-MacLane spectrum HF₂ as a Thom spectrum of a double loop map.
Let G be a connected reductive algebraic group defined over a number field. The harmonic analysis of the adelic points of G leads to a decomposition of the regular representation into automorphic representations. The irreducible subrepresentations in this decomposition form the so-called discrete spectrum. When G=GLn the discrete spectrum has a nice description. Langlands' principle of functoriality suggests that the discrete spectrum of other groups might be described in terms of the discrete spectrum of GLn. Arthur has recently provided such a description for symplectic and special orthogonal groups in terms of sets of representations called A(rthur)-packets. The structure of A-packets is not well understood, and relies in part on their local analogues. We will outline a method for computing A-packets for real groups.
A variety X over a number field k is said to satisfy Hasse principle of it has a rational point provided it has a rational point over completions of k at all its places. Manin defined an obstruction in terms of the Brauer group to detect the failure of Hasse principle for varieties over number fields which is referred to as the Brauer-Manin obstruction. This obstruction is the only obstruction to Hasse principle for torsors under connected linear algebraic groups over k. We shall explain a reciprocity obstruction to Hasse principle for varieties over function fields of p-adic curves. This obstruction can be used to produce examples of principal homogeneous spaces under tori which fail Hasse principle over function fields of p-adic curves.
Do you like spheres? If not, you are wrong, and probably won't like this talk. To the remaining reasonable people out there hold on to your socks. It is an elementary fact that the 0-sphere S^0=\{\pm 1\} is the group \mathbb{Z}_2 and the circle S^1 is also the group e^{i\theta}. Is that just by chance, can all groups be given a group structure? If not what makes the ones that can so special? We will analyse which spheres have this property (turns out only 3.5ish do) and how to find the group structure when they do.
The question now becomes, what can we say about the spheres that can't be groups? Can we exploit the symmetries of S^n and the sphere groups above to salvage some geometric structure? To answer these questions we will begin by give a very elementary introduction to fibre bundles (emphasis on the ``very''), and discuss how we can think of higher order spheres as a bunch of copies (ie. fibres) of sphere groups by constructing the Hopf fibration. Time permitting, we will give an application of this abstract jargon to help visualize the spin \frac{1}{2}-system in quantum mechanics and introduce the bloch sphere.
This is meant to be very elementary, if you understood the first paragraph, you will understand this talk. The goal will be to focus on intuition and avoid technical details like the plague.
The classical theorem of Hasse-Minkowski asserts that a quadratic form over a number field represents zero nontrivially provided it represents zero nontrivially over its completions at all its places. We discuss analogous local global principles over function fields of p-adic curves. Such local-global principles in the general setting for homogeneous spaces have implications to the understanding of the arithmetic of these fields.
Aggregation is an ubiquitous natural phenomenon that pervades both the animal world and many inanimate physical systems. In the animal kingdom, group formation is observed across all levels from bacterial colonies and insect swarms to complex predator-prey interactions in fish, birds and mammals. Aggregation is also present in physical systems at all scales from the smallest (Bose-Einstein Condensates, DNA buckyball molecules, fluid vortices) to the largest (galaxies). The emergence of group behaviour is often a consequence of individuals (or atoms) following very simple rules, without any external coordination.
In this talk I will describe some simple models of swarms that are motivated by well-known physical systems. While they may not capture the fine details of biological interactions, these models are simple enough that many of their properties can be studied analytically in great detail. This in turn can shed light about the role of swarming in biological systems. For example, is swarming behaviour helpful in avoiding a predator? Conversely, the study of biological models can also lead to new insights into related physical models.
Note for Attendees
This is the IAM distinguished alumni lecture for 15-16. Tea served beforehand in ESB 4133 (the PIMS lounge).
A mesh-independent and second-order accurate multigrid strategy to solve constrained parabolic optimal control problems is presented. The resulting algorithms appear to be robust with respect to change of values of the control parameters and have the ability to accommodate constraints on the control also in the limit case of bang-bang control. Central to the development of these multigrid schemes is the design of iterative smoothers which can be formulated as local semi smooth Newton methods.
Motivated by a question of Duval and Reiner about eigenvalues of combinatorial Laplacians, we develop a generalization of (ordered) matroid theory to wider classes of simplicial complexes. In addition to all independence complexes of matroids, each such class contains all pure shifted simplicial complexes, and it retains a little piece of matroidal spirit/structure. To achieve this, we relax the various cryptomorphic definitions of a matroid. In contrast to the matroid setting, these relaxations are independent of each other, i.e., they produce different extensions. Imposing various combinations of these new axioms allows us to provide analogues of many classical matroid structures and properties. Examples of such properties include the Tutte polynomial, lexicographic shellability of the complex, the existence of a meaningful nbc-complex and its shellability, the Billera-Jia-Reiner quasisymmetric function, and many others. We then discuss the h-vectors of complexes that satisfy our relaxed version of the exchange axiom, extend Stanley's pure O-sequence conjecture about the h-vector of a matroid, solve this conjecture for the special case of shifted complexes, and speculate a bit about the general case. Based on joint works with Jeremy Martin, Ernest Chong and Steven Klee.
When a child randomly paints a coloring book, adjacent regions receive distinct colors whereas distant regions remain independent. It took mathematicians until 2014 to replicate this effect, when Holroyd and Liggett discovered the first stationary k-dependent q-colorings. In this talk, I will discuss an extension of Holroyd and Liggett's construction which associates a canonical insertion procedure to every finite graph. The known colorings turn out to be diamonds in the rough: apart from multipartite analogues, they are the only k-dependent processes which arise from finite graphs in this manner. Time permitting, I will present extensions of these results to weighted graphs and shifts of finite type. Joint work with Alexander Holroyd.
Braids represent mapping classes of the punctured disk,and hence braids induce automorphisms of the fundamental group of the punctured disk, i.e, automorphisms of the free groups. It is known that the free groups are bi-orderable. We consider which braid preserves some bi-ordering of the free group. Once we know a given braid preserves some biordering of the free group, the fundamental group of the mapping torus by the braid monodromy is bi-orderable. By using a criterion by Perron-Rolfsen together with a technique on the disk twists, we give new examples of links in the 3-sphere whose fundamental groups of the link exteriors are bi-orderable, for example, the Whitehead link, the minimally twisted 4- and 5- chain links. We also give an infinite sequence of pseudo-Anosov braids which do not preserve any bi-orderings of the free groups. As a corollary, it follows that the fundamental group of the Whitehead sister link (i.e, (-2,3,8)-pretzel link) exterior is not bi-orderable.
This will be largely a survey of classical results: I will recall the analytic class number formula, and the Minkowski–Siegel mass formula for the "number" of quadratic forms in a genus, as well as Tamagawa's reformulation of these results as a volume computation. Then I will discuss a similar formula for the number of elliptic curves in an isogeny class, and we will see that it can again appear in two versions: one is due to Gekeler (2003) and comes from probabilistic and equidistribution considerations, and the other is due to Langlands and Kottwitz and is based on a volume computation. This talk is motivated by the joint project with Jeff Achter, Ali Altug and Luis Garcia, where we explore the connection between these two formulas and generalize Gekeler's result to counting principally polarized Abelian varieties. The talk will be completely non-technical and is not aimed at the experts (who would find most of it very familiar).
Given a scheme X and a normal crossings divisor D in X, the Olsson fan of X and D is an algebraic stack that encodes the combinatorics of the components of D and their intersections. I will describe Olsson fans and show how they are constructed. Then I will discuss the moduli space of stable maps from curves into an Olsson fan, and highlighting a number of applications to Gromov-Witten theory.
A set that has Fourier dimension equal to its Hausdorff dimension is called a Salem set. Kaufman showed in 1981 that the well-approximable numbers of order \tau are a Salem set with dimension \frac{2}{\tau + 1}. In this joint work with Kyle Hambrook, we attempt to show that the well-approximable numbers in the p-adics of order \tau form a Salem set of dimension \frac{2}{\tau}.
Our goal is to develop large scale computational models of the human biomechanical system. Such models have a wide range of applications, ranging from computer graphics to human health. Despite the long history of research in this area, current models have significant shortcomings. I will first outline some of these problems and their solutions, including the proper accounting of muscle mass and the role of the giant protein titin in the molecular mechanisms of force production in muscles.
In the second part of the talk I will discuss numerical methods for simulating large scale musculoskeletal systems. Biological soft tissues are usually simulated using a Lagrangian discretization, following the standard practice in solid mechanics. However, realistic systems have many muscles and tendons that are highly constrained by each other and by connective tissues. Dealing with these constraints pose significant challenges for the traditional approach. Instead, I will advocate the use of Eulerian (and Eulerian-on-Lagrangian) discretizations. I will demonstrate recent results using this approach in simulating the tendon networks of the hand, human skin, and multiple muscles in contact.
The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of straight shape. No such product formula exists for the number of standard Young tableaux of skew shapes. In 2014, Naruse announced a formula for skew shapes as a positive sum of products of hook-lengths proved using equivariant cohomology and excited diagrams of Ikeda-Naruse and Kreiman. We prove Naruse's formula combinatorially and we give two q-analogues of this formula involving semistandard Young tableaux and reverse plane partitions of skew shape. The first q-analogue is proved algebraically. We show that the restricted Hillman-Grassl correspondence is a bijection explaining these q-analogues. Joint work with Igor Pak and Greta Panova.
The reduction of the local Galois representation attached to an ordinary (slope 0) form at a prime p away from the level is well known to be reducible. In this talk we shall survey what is known about the reduction when the slope is small but positive.
We concentrate on the case of slope 1, which was treated essentially completely in recent joint work with S. Bhattacharya and S. Rozensztajn. We show that while the local reduction is essentially reducible, its exact shape depends on the congruence class of the weight mod p-1. Moreover, we show that in each such class there is a further congruence class of weights mod p where the local reduction is irreducible. We also distinguish between the so called peu and trčs ramifiée cases in the relevant non-semisimple reducible cases.
The proof uses the compatibility between the p-adic and mod p Local Langlands Correspondences with respect to the process of reduction. This reduces the computation to the automorphic side. We will try and explain most of the key ingredients used in the proof in a self-contained way.
I will first talk generally about quantum mechanics and the nonlinear Schrodinger equation. I will then turn my attention to the focusing nonlinear Schrodinger equation which exhibits solitary wave solutions; the stability of which can be understood by studying the appropriate linearization operator. I will say something about the interesting resonance eigenvalue that appears on the edge of the essential spectrum and how it bifurcates.
We explore a relationship between combinatorics and certain moduli spaces appearing in symplectic geometry. The combinatorics comes from the theory of cluster algebras, a kind of unified theory of canonical bases in representation theory and algebraic geometry. Some basic features of cluster algebras are that they are defined from purely combinatorial data (for example, a quiver) and they are coordinate rings of varieties covered by algebraic tori with transition functions of a special, universal form. Despite the originally representation-theoretic motivation for the subject, connections between cluster theory and symplectic geometry emerged later through the appearance of similar formulae in wall-crossing and mirror symmetry.
We will discuss recent work expanding on this connection, in particular providing a universal framework for interpreting cluster varieties as moduli spaces of objects in Fukaya categories of Weinstein 4-manifolds. In simple examples these moduli spaces reduce to well-known ones, such as character varieties of surfaces and positroid cells in the Grassmannian. An accompanying theme, which plays a key role both technically and in relating the symplectic perspective to more standard representation-theoretic ones, is the role of categories of microlocal sheaves as topological counterparts of Fukaya categories. This is joint work with Vivek Shende, David Treumann, and Eric Zaslow.
Computational biology, spatio-temporal analysis, natural language processing and a range of other fields rely on increasingly complex probabilistic models to make predictions and take action. In practice, these models often need to incorporate high-dimensional latent variables, complex combinatorial spaces and various heterogeneous data-structures. Moreover, it is important to not only perform optimization on these models, but also to assess the uncertainty in predictions and reconstructions of latent states.
Monte Carlo methods have been used in the past several decades to approach these hard and important problems. Advances in probabilistic programming open the door for more widespread use of Monte Carlo, but computational efficiency remains a formidable challenge.
In this talk, I will provide some background on the state-of-the-art, and describe the progress that my collaborators and myself have been making towards more practical Monte Carlo methods. In particular, I describe Divide-and-Conquer Sequential Monte Carlo (D&C SMC), a method for performing inference on a collection of auxiliary distributions organized into a tree. In contrast to standard SMC samplers, D&C SMC exploits multiple populations of weighted particles, while still being an exact approximate method. D&C SMC is applicable to a broad class of probabilistic graphical models, including models with loops.
For any discretely self-similar, incompressible initial data which is arbitrarily large in weak L^3, we construct a forward discretely self-similar solution to the 3D Navier-Stokes equations in the whole space. This also gives a third construction of self-similar solutions for any -1-homogeneous initial data in weak L^3, improving those in by Jia-Sverak and Korobkov-Tsai for H\"older continuous data. Our method is based on a new, explicit a priori bound for the Leray equations. This is a joint work with Tai-Peng Tsai.
The c_2 invariant is an arithmetic graph invariant introduced by Brown and Schnetz in order to better understand Feynman integrals. I will look at what can be said about the c_2 invariant of 4-regular circulant graphs with one vertex removed. The answer is not much, but it's still interesting. Yes, this is the same talk as I gave at SFU on October 6, so don't come expecting something new.
Alice and Bob compete in a game of skill, making moves alternately until one or other reaches a winning position, at which the game ends. Or, perhaps neither player can force a win, in which case optimal play continues forever, and we say that the game is drawn.
What is the outcome of a typical game? That is, what happens if the game itself is chosen randomly, but is known to both players, who play optimally?
I will provide some answers (any many questions) in several settings, including trees, directed and undirected lattices, and point processes. The competitive nature of game play frequently brings out some of the subtlest and most fundamental properties of probabilistic models. We will encounter continuous and discontinuous phase transitions, hard-core models, probabilistic cellular automata, bootstrap percolation, maximum matching, and stable marriage.
Based on joint works with Riddhipratim Basu, Maria Deijfen, Irene Marcovici, James Martin and Johan Wastlund.
The Mahler measure of a multivariable polynomial P is given by the integral of log |P| where each of the variables moves on the unit circle and with respect to the Haar measure. In 1998 Boyd made a systematic numerical study of the Mahler measure of many polynomial families and found interesting conjectural relationships to special values of L-functions of elliptic curves. Recently, many of Boyd's conjectures have been proved by Burnault, Mellit, Rogers, and Zudilin. I will discuss some of those results and present new ones (in collaboration with D. Samart and W. Zudilin.)
The Koebe-Andreev-Thurston Circle Packing Theorem lets us draw planar graphs in a canonical way, so that the geometry of the drawing reveals analytic properties of the graph. Circle packing has proven particularly effective in the study of random walks on planar graphs, where it allows us to estimate various quantities in terms of their counterparts for Brownian motion in the plane.
In this talk, I will introduce the theory of circle packing and discuss work with Asaf Nachmias in which we use circle packing to study uniform spanning forests of planar graphs, a probability model closely related to random walk. We prove that the free uniform spanning forest of any bounded degree, proper planar graph is connected almost surely, answering positively a question of Benjamini, Lyons, Peres and Schramm.
Our proof is quantitative, and also shows that uniform spanning forests exhibit some of the same behaviour universally for all bounded degree transient triangulations, provided that one measures distances and areas in the triangulation using the hyperbolic geometry of its circle packing rather than with the usual graph metric and counting measure.
The concept of basis of a vector space over a field generalizes to the concept of generators of a module over a ring. However, generators carry very little information about the structure of the module, in contrast to bases, which are very useful in the study of vector spaces. Hilbert introduced the approach to describe the structure of modules by free resolutions. Hilbert's Syzygy Theorem shows that minimal free resolutions over a polynomial ring are finite. By a result of Serre, it follows that most minimal free resolutions over quotient rings are infinite. We will discuss the structure of such resolutions. The concept of matrix factorization was introduced by Eisenbud 35 year ago, and it describes completely the asymptotic structure of minimal free resolutions over a hypersurface. Matrix factorizations have applications in many fields of mathematics: for the study of cluster algebras, Cohen-Macaulay modules, knot theory, moduli of curves, quiver and group representations, and singularity theory. Starting with Kapustin and Li, physicists discovered amazing connections with string theory. In a current joint work with Eisenbud, we introduce the concept of matrix factorization for complete intersection rings and show that it suffices to describe the asymptotic structure of minimal free resolutions over complete intersections.
The numerical solution of ordinary differential equations is by now a very old subject. It's a surprise, therefore, that there's anything new to say about it at a basic level. This talk uses two very simple examples, the leaky bucket and the Dahlquist test problem, to demonstrate that there is indeed something new to say. This talk will be accessible to undergraduates. Joint work with Julia E. Jankowski, Yalçin Kaya, and Robert H.C. Moir.
Many solution techniques for inverse problems involve solving an optimization problem using a numerical method. For example, the Tikhonov regularized solution is commonly defined as the minimizer of a penalized least squares function. Uncertainty quantification (UQ), on the other hand, often requires sampling from the Bayesian posterior density function arising from the assumed physical model, measurement error model, and prior probability density function. In this talk, we bring these two computational approaches (numerical methods and sampling) together and present posterior sampling – and specifically Markov Chain Monte Carlo (MCMC) – methods for UQ that utilize existing numerical algorithms for solving inverse problems. In all cases, care is taken to make sure that the MCMC methods presented provide theoretically correct samples from the posterior density function. Moreover, we present MCMC methods for both linear and nonlinear inverse problems.
There are many interesting questions concerning the Nonlinear Schroedinger Equation such as the existence and stability of solitary wave solutions as well as the long time behaviour of solutions. These problems are made more complicated by the presence of a resonance eigenvalue. Such occurrences are special cases which serve to worsen time decay estimates and complicate resolvent expansions. We will talk about some particular perturbation results whose treatment is non-standard since a relevant linear operator has a resonance.
Discrepancy theory has been an important research area in combinatorics and geometry for several decades. Recently there has been a lot of activity in discrepancy theory, in two directions. The first is efficient algorithmic proofs for classical discrepancy results that previously had only non-constructive proofs. The second is matrix generalizations of some classical discrepancy results, the canonical example of which is the solution of the Kadison-Singer conjecture by Marcus-Spielman-Srivastava (Polya Prize 2014).
In this talk I will give an overview of the field of discrepancy theory, then discuss a self-contained result of my own that uses the Lovasz Local Lemma to solve a discrepancy question that was overlooked for many years.
I will give a second talk on the Lovasz Local Lemma, focusing on algorithms, on Wednesday November 25th in the Probability Seminar.
The Lovasz Local Lemma (LLL) is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events. Finding such an outcome by an efficient algorithm has been an active research topic for decades. Breakthrough work of Moser and Tardos (2009) presented an efficient algorithm for a general setting primarily characterized by a product structure on the probability space.
In this work we present an efficient algorithm for a much more general setting. Our main assumption is that there exist certain functions, called resampling oracles, that can be invoked to address the undesired occurrence of the events. We show that, in all scenarios to which the original LLL applies, there exist resampling oracles; and for essentially all known applications of the LLL we have designed efficient resampling oracles.
Our analysis is based on an alternative view of the LLL using multivariate polynomials, due to Shearer (and Scott and Sokal). Probabilists have also studied this topic under the name of "1-dependent hard-core processes".
In this talk we will give a broad overview of the Linnik problems concerning the equidistribution of integral points on homogeneous varieties. One particular example concerns the Heegner points, which are roots in the complex upper-half plane of certain quadratic forms. We will discuss certain "sparse" equidistribution problems concerning these points and give an application of an analog of Linnik's famous theorem on the first prime in an arithmetic progression. This is joint work with Riad Masri and Matt Young.
This week, I will talk about the famous Banach-Tarski paradox to give an brief introduction to the theory of amenability. The goal will be to give an intuition of "what does it means to be amenable, aside from having a mean ?"
This theory is not only great because it provides a field dedicated to making bad puns with "mean", "amen" or even "ramen", it also has a lot of interesting characterizations.
We will give a little overview of Banach-Tarski paradox to motivate the topic (and quickly re-attribute most of the result to Hausdorff), give a few alternative definitions and end up talking about graph theory.
Prerequisites : Be able to put up with my very average drawing skills, knowing how to pronounce Danish names always helps. Also, the axiom of choice is not optional, please don't be anti-choice axiom.
New molecular techniques such as DNA sequencing provide conceptual insights into microbial community metabolism and biogeochemical cycling in natural and engineered ecosystems. However, attempts to mechanistically integrate molecular data with biogeochemistry are faced with the inhibitory complexity of individual cells and a large number of unknown physiological parameters. Recent work suggests that biochemical pathways are, at ecosystem scales, strongly shaped by thermodynamic and stoichiometric constraints. Pathway-centric mathematical theories rooted in fluxes of matter and energy could thus provide holistic insight into microbial ecosystems and global biogeochemical fluxes.
Oxygen minimum zones (OMZ) are oxygen-depleted regions in the ocean that are dominated by microbial metabolism, thus constituting ideal systems for developing theories of microbial ecology. I will present our current efforts to model the biogeochemistry of an intensely studied OMZ off the coast of Vancouver Island using reaction-advection-diffusion models. In contrast to conventional approaches, we focus on individual enzymes catalyzing metabolic pathways and assume that energy fluxes translate directly to gene expression and biosynthesis. We use DNA, mRNA and protein sequence data, as well as geochemical depth profiles and process rate measurements to calibrate and validate our models.
In this talk I will introduce the generalization of relative Donaldson-Thomas theory to 3-dimensional smooth Deligne-Mumford stacks. We adopt Jun Li’s construction of expanded pairs and degenerations and prove an orbifold DT degeneration formula. I’ll also talk about the application in the case of local gerby curves, and its relationship to the work of Okounkov-Pandharipande and Maulik-Oblomkov.
In this two-part talk, we discuss a Fourier-analytic approach to solve translation-invariant systems of polynomial equations, when the variables lie in a dense subset of the integers. In the first part, we explain how discrete estimates in restriction theory come into play in this problem. In the second part, we show how to obtain weak restriction estimates by Bourgain's discrete version of the Tomas-Stein argument.
Allen-Cahn equation arises in the mathematical study of phase transition. Despite it's seemingly simple appearance, It has displayed very rich structure of solutions and involved with deep mathematics. In this talk, I will discuss the existence, symmetry and classification of saddle solutions of Allen-Cahn equation on the plane. In particular, I will describe the variational characterization of these solutions as a mountain pass solutions.
Poisson boundary provides an integral representation of all bounded harmonic functions. We say that a Markov chain satisfies the Liouville property if all bounded harmonic functions are constant, that is the Poisson boundary is trivial.
The first part of the talk is a gentle introduction to Poisson boundary. Then I will state a new condition that is equivalent to the Liouville property and provide a proof of this equivalence. This talk is based on an ongoing work and will be self-contained.
A curve is called rational if it can be parametrized by rational functions. Counting the number of rational curves in the plane that contain a given number of points has been an old and interesting problem. The degree of a curve is the degree of a polynomial equation defining it. We all know that there is only one curve of degree one through any given pair of points in the plane because this is just a straight line. The answer to similar questions for higher degrees quickly becomes more difficult and interesting. To simplify the problem, one may map a curve in the complex two-space to the real two-space by applying a component-wise absolute value and logarithm. The resulting object is for good reasons called an amoeba - I will show some pictures. The amoeba retracts to its spine which is a much simpler convex-geometric object that is also called a tropical curve. Such can be counted essentially by hand. A proof that the resulting counts coincide with the original problem goes via a recent theory of logarithmic Gromov-Witten invariants.
Note for Attendees
Refreshments will be served at 2:45pm in the Math Lounge area, MATH 125 before the colloquium.
Variable projection (VP) gained popularity as a technique for solving nonlinear least squares problems (NNLS) min_{x,y} ||F(x)y - b||^2. The VP approach is an itegrated algorithm in x that `projects out' the variable y at each iteration. The NLLS problem class had a range of applications, and we show that the 'projection' approach generalizes to a very broad setting, retaining the original flavour, with applications to nuisance parameter estimation, kernel learning, and non-smooth optimization. (Editar entrada)
Motivated by the problem of computing low-rank matrix completion solutions, we present an extension of the Frank-Wolfe method that is designed to induce near-optimal solutions on low-dimensional faces of the feasible region. We also present computational guarantees for the method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We then present computational results for large-scale matrix completion problems that demonstrate significant speed-ups in computing low-rank near-optimal solutions.
Abstract: Classical fluid dynamics arguments suggest that in certain
limits, the evolution of thin vortex filaments in an ideal incompressible
fluid should roughly be governed by an equation called the binormal
curvature flow. However, these classical arguments rely on assumptions
that are so unrealistic that it would be hard even to extract from them a
precise conjecture that admits any realistic possibility of a proof. We
present a different approach to this question that yields a reasonable
formulation of a conjecture and strong supporting evidence, and that
clarifies the very substantial obstacles to a full proof. Parts of the
talk are based on joint work with Didier Smets and with Christian Seis
In the simple case of a Bernoulli shift on two symbols, zero and one, by permuting the symbols, it is obvious that any two equal entropy shifts are isomorphic. We show that the isomorphism can be realized by a factor that maps a binary sequence to another that is coordinatewise smaller than or equal to the original sequence.
Consider the set of Hecke eigenforms of a fixed weight k and very large level N, and pick a prime p coprime to N. How are the eigenvalues of the operator Tp distributed? We will relate this question to a more abstract question about the distribution of local components of cuspidal automorphic representations. For a (reductive) algebraic group G defined over a p-adic field L, there is a measure, called the 'Plancherel measure', which is expected to serve as the limiting distribution under many circumstances. We'll define this measure, give a precise formulation of the problem, and discuss progress in this area of research. No prior knowledge of reductive groups or automorphic forms will be assumed.
The ADE braid group acts faithfully on the derived category of coherent sheaves on the resolution of the associated Kleinian singularity. In other words, the ADE braid groups are "2-linear" groups. In a similar spirit, the free group is a 2-linear group. In this talk we'll describe a few proofs of these results and explain how spherical twists and triangulated categories are related to some open problems in group theory.
I consider two different approaches for breaking scale invariance and restoring compactness for borderline variational problems involving the Hardy-Schrodinger operator -\Delta -\frac{\gamma}{|x|^2} on a domain containing the singularity 0, either in its interior or on its boundary. One consists of adding a linear perturbation, another exploits the geometry of the domain. I discuss the role of various ``positive singular mass theorems" that help account for the critical dimensions below which these approaches fail. This is a joint project with Frederic Robert.
The counting of walks in orthants is now a classical domain in enumerative combinatorics. In this talk, I will focus my attention on the growth constant for the number of such walks and show how the general framework of random walks in cones provides - via classical probabilistic tools - a unified solution to the problem of determining this growth constant. Joint work with Kilian Raschel.
In these talks we review basic facts about finite group actions and how they can be extended using homotopical methods. In the second talk we will describe some joint work with J.Grodal on constructing exotic group actions for certain rank two finite groups.
In 2010, Masser and Zannier proved that there are at most finitely many complex λ not equaling 0 or 1, such that two points on the Legendre elliptic curve y2=x(x−1)(x−λ) with x-coordinates 2 and 3 are simultaneously torsion. Recently, Stoll proved that there is in fact no such λ, and it is his result that inspires our work. In this talk we will focus on the Weierstrass family of elliptic curves y2=x3+λ, and show that in many instances there will be no parameter λ such that the points (a,*) and (b,*) are simultaneously torsion. In contrast to the original approach of Masser and Zannier, we will place this problem in the setting of arithmetic dynamics, by studying whether a and b are simultaneously preperiodic for a Lattes map. The results in this talk are obtained by analyzing the 2-adic behaviour of the iterates of the Lattes map.
In the summer of 1966, when I was taking a reading course from Paul Halmos, he told me: "If you want to study a question about operators on infinite dimensional Hilbert spaces, first formulate it in the setting of finite dimensional spaces. Answer it there, and only then move on to the infinite dimensional setting." While this admonition may seem naive, I want to show how taking it seriously can reveal interesting connections between operator algebra and the theory of analytic functions of noncommutative variables. Very briefly: The journey begins with the work of Murray and von Neumann on rings of operators. Much of their inspiration came from finite group representation theory, and the algebras they constructed are viewed by many as infinite dimensional versions of semisimple algebras. When trying to fit non-semisimple algebras into operator algebra, Baruch Solel and I were inspired by algebraic theories developed in the late 40s and were led to think about tensor algebras and the theory of quivers (i.e., finite directed graphs). The algebras we constructed could profitably be studied as spaces of analytic functions on the algebras' representations. These are the free functions to which the title refers. The presentation will be largely historical and non-technical. It will require background only from first-year graduate courses in algebra and analysis.
The geometric Satake equivalence relates the category of perverse sheaves on the affine Grassmannian and the representation category of a semisimple group G. We will discuss a quantum K-theoretic version of this equivalence. In this setup the representation category of G is replaced with (a quantum version) of coherent sheaves on G/G. This is joint work Joel Kamnitzer.
We study a branching Brownian motion $Z$ in $\mathbb{R}^d$, among obstacles scattered according to a Poisson random measure with a radially decaying intensity. Obstacles are balls with constant radius and each one works as a trap for the whole motion when hit by a particle. Considering a general offspring distribution, we derive the decay rate of the annealed probability that none of the particles of $Z$ hits a trap, asymptotically in time $t$. This proves to be a rich problem motivating the proof of a more general result about the speed of branching Brownian motion conditioned on non-extinction. We provide an appropriate ``skeleton" decomposition for the underlying Galton-Watson process when supercritical and show that the ``doomed" particles do not contribute to the asymptotic decay rate.
This is joint work with M. Caglar and M. Oz (Istanbul); to appear in AIHP.
In these talks we review basic facts about finite group actions and how they can be extended using homotopical methods. In the second talk we will describe some joint work with J.Grodal on constructing exotic group actions for certain rank two finite groups.
We take a look at two extremely counterintuitive results in mathematics which are consequences of the existence of infinite sets, one of which explicitly invokes the axiom of choice, and the other more subtly. One is Goodstein's theorem, the other might be called the "Prisoners with hats" problem. This talk should be accessible to anyone with an understanding of sets, posets, and equivalence relations.
The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit.
This talk is focused on the commutative case, where the laws of random variables are also described in terms of certain affinely flat structures on the formal moduli space of a naturally defined family attached to the given algebraic probability space, which the relevant category is the homotopy category of L_\infty-algebras. Time permitting, I will explain a example of homotopy probability space which law corresponds to variations Hodge structures on a toric hypersurface.
The response of the muscle-tissue unit (MTU) to activation and applied forces is affected by the architectural details as well as the material properties of this nearly-incompressible tissue. We will describe the (highly nonlinear) elastic equations governing this response for a fully three-dimensional, quasi-static, fully nonlinear and anisotropic MTU. We describe a three-field formulation for this problem, and present a DG discretization strategy. The scheme was implemented using {\tt deal.ii}. We present computational results about the effects of localized activation as well as the effects of fatty tissue on muscle response. This is joint with Sebastian Dominguez, Hadi Rahemi, David Ryan and James Wakeling.
Note for Attendees
Sushi lunch will be provided. We look forward to see you there.
In the 1980’s, Aldous and Broder independently proved that the collection of first-entry edges of a random on a finite graph is distributed as a uniform spanning tree of the graph; using this fact to sample the uniform spanning tree of a finite graph is known as the Aldous-Broder algorithm. In this talk, I will review the theorem of Aldous and Broder and discuss an extension of the Aldous-Broder algorithm to infinite graphs, in which the random walk is replaced by Sznitman’s random interlacement process. Time permitting, I will also show how this extension can be used to prove a few things about the wired uniform spanning forest.
In this talk I will give a introduction in a very useful tool – Controlled Algebra.
Controlled Algebra is a way of building new additive categories with better properties out of given ones. One application is the Definition of negative K-theory. Another application is a description of K-theoretic assembly maps as maps induced by additive functors.
A recent result of Karpenko and Merkurjev states that the essential dimension of a p-group G over a field K containing a primitive p'th root of unity is equal to the minimal dimension of faithful representations of G over K. Motivated by this result, it is then interesting to compute the minimal dimension of complex faithful representations of a given finite p-group. In this talk we will show how Lie algebraic method, namely Howe-Kirillov's orbit method, can be applied to answer this question for some classes of p-groups.
Uncertainty quantification (UQ) has been a hot research topic recently. UQ has a variety of applications, including hydrology, fluid mechanics, data assimilation, and weather forecasting. Among a large number of approaches, the high order numerical methods have become one of the important tools; and the relevant computational techniques and their mathematical theory have attracted great attention in recent years. This talk begins with a brief introduction to recent developments of high order numerical methods including Galerkin projection methods and stochastic collocation methods. The emphasis will be sample-based stochastic collocation methods, including random sampling, deterministic sampling and structured random sampling. We will also discuss approximating multivariate functions in unbounded domains by using discrete least squares projection with random point evaluations. Particular attention is given to functions with random Gaussian or gamma parameters.
Note for Attendees
This is an IAM/PIMS distinguished speaker. Tea will be served before the talk in the IAM lounge (LSK 306).
For a field F and a quadratic form Q defined on an n-dimensional vector space V over F, let G_Q, called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u,w in V form an edge if and only if Q(v-w)=1. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson problem. In the present talk, we will prove that for a local field F of characteristic zero, the Borel chromatic number of G_Q is infinite if and only if Q represents zero non-trivially over F. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009.
I am going to discuss Alexeev's and Brion's moduli space parametrizing maps
from broken toric varieties into a fixed toric variety V. Following ideas
of Olsson, I will explain how one can obtain a modular description of the
main irreducible component of Alexeev's and Brion's space, using an
analogous moduli space K(V) parametrizing logarithmic maps from broken
toric varieties into V. The resulting space K(V) is in fact a toric stack
-- it is a stacky enrichment of an appropriate Chow quotient of V. I will
conclude by explaining why K(V) and the Chow quotient stack coincide, and
describe explicitly the combinatorial data that determine the latter.
Let M be a submanifold of Euclidean space, and let X be a subset of N points, randomly sampled. Belkin and Niyogi showed that one can recover the Laplacian on functions on M as N gets large, by integrating the heat kernel. More recently, Singer and Wu use Principle Component Analysis to construct connection matrices between approximate tangent spaces for nearby points in X. This allows them to construct a rough Laplacian on 1-forms. Together with Ache, we show that by iterating the Laplace operator of Belkin and Niyogi, a la Bakry and Emery, and appealing to the Bochner formula, we can reconstruct the Ricci curvature on the approximate tangent spaces. Combining our work with the work of Singer and Wu, we are able to approximate the Hodge Laplacian on 1-forms.
MacMahon's classical theorem on boxed plane partitions states that the generating function of the plane partitions fitting in an $a\times b\times c$ box is equal to
where $H_q(n):=[0]_q![1]_q!\dots[n-1]_q!$ and $[n]_q!:=\prod_{i=1}^{n}(1+q+q^2+\dots+q^{i-1})$. By viewing a boxed plane partition as a lozenge tiling of a semi-regular hexagon, MacMahon's theorem yields a natural $q$-enumeration of lozenge tilings of the hexagon. However, such $q$-enumerations do not appear often in the domain of enumeration of lozenge tilings. In this talk, we consider a new $q$-enumeration of lozenge tilings of a hexagon with three bowtie-shaped regions removed from three non-consecutive sides.
The unweighted version of the result generalizes a problem posed by James Propp on enumeration of lozenge tilings of a hexagon of side-lengths $2n,2n+3,2n,2n+3,2n,2n+3$ (in cyclic order) with the central unit triangles on the $(2n+3)$-sides removed.
The field of enumeration of tilings dates back to the early 1900s when MacMahon proved his classical theorem on plane partitions fitting in a given box. The enumeration of tilings has become a subfield of combinatorics with applications and connections to diverse areas of mathematics. In this talk we will consider a connection between the enumeration of tilings and the theory of electrical networks.
The theory of electrical networks was studied systematically by Colin de Verdiere and Curtis, Ingerman, Moores, and Morrow in the 1990s. Associated with an electrical network is a "response matrix" that measures the response of the network to potential applied at the nodes. Kenyon and Wilson showed how to test the well-connectivity of an electrical network with n nodes by checking the positivity of n(n-1)/2 minors of the response matrix. Their test was based on the fact that any "contiguous minor" of a matrix is the generating function of domino tilings of a weighted Aztec diamond. They conjectured that a larger family of minors, "semicontiguous minors", can also be expressed in terms of domino tilings of certain regions. We prove this conjecture by describing explicitly the ``tiling expression" of the semicontiguous minors.
Note for Attendees
Refreshments will be served in MATH 125 before the colloquium.
As explained last week, Controlled Algebra is a way of building new additive categories with better properties out of given ones. This time we will use it to describe K-theoretic assembly maps as maps induced by additive functors.
Note for Attendees
This week the seminar is moved to 4:15pm, to allow people to attend the Colloquium.
A classical problem in combinatorial geometry is to determine the maximum number of incidences between a set of m points and n curves in the plane. If the curves are lines, then Szemerédi and Trotter proved that there could be at most O(m^{2/3}n^{2/3} + m + n) incidences, and this bound is tight. For other classes of curves, very few tight bounds are known. Work in this area progressed in the 80s and 90s, culminating in an incidence bound by Pach and Sharir in 1998 that applies to a very general class of curves. Since then, there have only been improvements for a few specific types of curves. In this talk I will discuss some new developments that improve upon Pach and Sharir's bound for a broad class of curves. A key innovation is the use of higher-dimensional incidence geometry, coupled with a new way of cutting collections of curves into segments so that the corresponding set of segments is better behaved than the original collection of curves. This is joint work with Micha Sharir.
The moduli space of stable curves is a community where families coexist peacefully. Unfortunately, this peacefulness comes at the price of conformity; you have to be "stable", and there is a strict screening process for families wishing to live there. If a family has undesirable members then they are barred from entering :(
Luckily, there is a process that a family can undergo to make themselves more desirable and be allowed entry. We examine this process when a nice family contains a single problem child. It may seems rather extreme, but this involves blowing up the family(!!) and gluing a pig's tail onto the problem child! :O
Given a collection of points and a collection of lines, circles, or other simple geometric objects, an incidence occurs when a point is contained in one of the objects. Incidence geometry is a branch of extremal combinatorics that studies the maximum number of incidences that can occur amongst all possible arrangements of the objects in question. It turns out that problems from diverse areas of mathematics can be phrased as incidence geometry questions, and often this is an effective way of tackling these problems.
There is a general phenomena in incidence geometry, which is the principle that higher dimensional incidence geometry problems often have fewer incidences than lower dimensional ones unless the objects arrange themselves into a low dimensional structure. For example, any collection of points and lines in three dimensions has relatively few incidences unless many of the points and lines cluster into a plane. I will discuss this phenomena, as well as some of its implications in discrete math and harmonic analysis.
Note for Attendees
Refreshments will be served in MATH 125 before the colloquium.
While large numbers of researchers have investigated efficient distributed-memory schemes for dense and sparse-direct linear algebra, relatively little work has been performed on extensions into the important fields of conic optimization and lattice reduction. (Perhaps surprising) performance barriers for distributed sparse Second-order Cone Programs will be discussed, and a case will be made for defaulting to explicitly storing quasi-constant edge-degree plus low-rank decompositions of the sparse KKT systems and then solving said systems via applying the iteratively-refined inverse of an a priori regularized, Symmetric Quasi-Semidefinite factorization as a preconditioner for Flexible GMRES(k). Recent work towards high-performance variants of lattice reduction schemes (LLL and BKZ 2.0) will also be briefly discussed to help make the case for the importance of high-precision arithmetic. Some practical issues related to the implementation of these techniques within the open source library Elemental (https://github.com/elemental/Elemental) will also be discussed.
In this talk I will give some relations between spaces of homomorphisms when the target group G is a real linear algebraic group, through homotopy stable decompositions of simplicial spaces. To obtain a simplicial space Hom(L,G) out of spaces of homomorphisms we think of L, a (suitable) family of finitely generated groups, as a cosimplicial group.
Also, if G=U, the colimit of the unitary groups U(m), I will show when the geometric realization of Hom(L,U) has an "E-infinity-ring-space" structure.
Random walk on a group is an established but still-growing field. I discuss aspects of recent work, alone and joint with Persi Diaconis, on random walks on unipotent groups. Among our results, we have a new local limit theorem for random walk on the Heisenberg group, which applies to arbitrary centered measures of compact support and obtains an optimal rate. There is also a mixing time bound of degree times diameter squared for the mixing time of random walk on some Cayley graphs of cyclic groups.
In this talk, I will first introduce the notions of Gibbs measures and thermodynamic limit on graded graphs, as they were defined by the Russian school (Vershik, Kerov, Olshanski,...). Then, I will present some results related to the thermodynamic limit on the graph of Zigzag diagrams: the latter is a graded graph whose set of vertices of degree n consists of words of length n-1 in two letters, and such that the edge structure is given by a simple combinatorial relation between words of consecutive lengths. This graph is related to the Young graph, and I will explain this relation by mapping paths on the graph of Zigzag diagrams to paths on the Young graph. "
The Lov\'{a}sz Local Lemma is a powerful technique from probabilistic combinatorics for treating many rare events with localized dependence structure. I discuss the local lemma and its application in my negative solution to the following problem of Erd\H{o}s.
A distinct covering system of congruences is a collection of arithmetic progressions
a_i \bmod m_i, \qquad 1 < m_1 < m_2 < ... < m_k
whose union is the integers. Can m_1 be arbitrarily large?
Note for Attendees
Refreshments will be served in MATH 125 before the colloquium.
Joint work with Michael Temkin, we extend the old results with Karu, Matsuki and Wlodarczyk from varieties to qe schemes and use this to prove factorization for various other categories.
Particle-laden flows are ubiquitous in environmental, geophysical and engineering processes. The intricate dynamics of these two-phase flows is governed by the momentum, heat and mass transfer between the continuous fluid phase and the dispersed particulate phase. While some multi-phase processes may be successfully modelled at the continuum scale through closure approximations, an increasing number of applications require resolution across scales, e.g. dense suspensions, fluidized beds. Within a multi-scale micro/meso/macro-framework, we develop robust numerical models at the micro and meso-scales, that both account for hydrodynamic interactions and particle/particle collisions. We present the mathematical issues related to modelling this type of flows together with the main numerical and computational features of our own simulation methods. Serial computations are almost not an option anymore and highly scalable codes on the most recent supercomputer architectures have become mandatory. We illustrate what can be gained from massively parallel computations in terms of physical insight into both fundamental questions and applications (essentially from the chemical engineering and process industry). We shortly discuss the next steps in the development of advanced numerical methods for particle-laden flows. Finally, we explain how knowledge gained at the micro scale can cascade upwards and contribute to the development of enhanced meso and macro-scale models.
Public health program managers and policy makers are regularly faced with complex decisions that affect the health and well-being of the public. Mathematical models and operations research tools can be used to consider diverse interacting factors, such as the epidemiology and clinical aspects of a condition, delivery methods of health services, and constraints on budget and resources. Creating reliable, data-driven models that are tractable and useful for informing policy decisions requires close collaboration of stakeholders and mathematicians.
This presentation will describe a health systems modeling project to inform the delivery of HIV health services in Vancouver, which illustrates the promise, challenges and rewards of cross-sector collaborations and offers possible avenues for the expanded use of modeling in public policy.
Registration is free and a light lunch will be provided. For more details and to register for this event, please visit http://www.pims.math.ca/industrial-event/160224-pllkv .
Note for Attendees
About the PIMS Lunchbox Series
The PIMS Lunch Box Series create a beneficial link between researchers in industry and academics in the mathematical sciences. Researchers with industrial and commercial concerns are invited to present their current work, demonstrating how the latest advances in the mathematical sciences apply to current, real-world issues in industrial processes such as seismic and medical imaging, financial mathematics and complex fluids. PIMS has been holding lunchbox lectures in Calgary for seven years and is now expanding to Vancouver. To learn more about PIMS, please visit www.pims.math.ca<http://www.pims.math.ca> .
In Euclidean space, symmetric stable process with indexα ∈ (0,2)is obtained by a time change of Brownian motion using the subordinator of indexα/2. By a similar subordination, there exists stable-like random walks with indices greater than two on various fractals and fractal-like graphs. However, existing methods to obtain transition probability estimates fail in this setting.
Davies' method is a technique introduced by E. B. Davies to obtain heat kernel upper bounds for uniformly elliptic operators in Euclidean space. This method was extended to general symmetric Markov semigroups by Carlen, Kusuoka and Stroock. In this talk, I will introduce some of the ideas involved in extending Davies method to obtain heat kernel bounds for stable-like processes with indices greater than two.
This talk is based on joint works with Laurent Saloff-Coste.
We will introduce basic notions of quantum mechanics, mostly employed in condensed matter physics, such as a separable Hilbert space, Bloch's theorem and Fermi surfaces. Then we will describe the problem of stability of Fermi surfaces and relate it to the mathematical concepts of Fredholm operators and homotopy classes. Equipped with these concepts we show that our proposed scheme yields a classification of topologically stable Fermi surfaces by K-1(X), where X is the Brillouin zone and K-1 is a well known functor in K-theory. We will show an explicit example when X = S1, known as the spectral flow and its relation to quantum anomalies. This is work in progress joint with Alejandro Adem and Gordon W. Semenoff.
The Lebesgue-Nagell equation is a Diophantine equation that arises in a variety of contexts, ranging from the classification of finite simple groups to Catalan’s problem. In this talk, we will discuss a variety of approaches to this equation that share the common thread of failing to solve it. This is joint work with Aaron Levin.
Perfect graphs are a class of graphs that behave particularly well with respect to coloring. In the 1960's Claude Berge made two conjectures about this class of graphs, that motivated a great deal of research, and by now they have both been solved.
The following remained open however: design a combinatorial algorithm that produces an optimal coloring of a perfect graph. Recently, we were able to make progress on this question, and we will discuss it in this talk. Last year, in joint work with Lo, Maffray, Trotignon and Vuskovic we were able to construct such an algorithm under the additional assumption that the input graph is square-free (contains no induced four-cycle). More recently, together with Lagoutte, Seymour and Spirkl, we solved another case of the problem, when the clique number of the input graph is fixed (and not part of the input).
The behaviour of elastic liquids does not follow simply from our understanding of both elastic solids and viscous liquids. Four anomalous behaviours will be discussed :-- (i) long wakes at low Reynolds numbers, (ii) large vortices upstream of a constriction, (iii) long times for capillary forces to squeeze a filament, and (iv) different devices measuring wildly different values of `the' extensional viscosity for the international standard liquid M1. Many features can be explained and understood using the simplest constitutive equation, that of an Oldroyd-B fluid, which generates the important ideas of tension in streamlines and delays for the stress to respond. This model fluid has however an undesirable negative viscosity under certain conditions, which can be regularized by requiring a finite extensibility of the underlying microstructure, in the FENE modification. This modification enables the remain anomalous behaviours to be understood, with a high extensional viscosity to increase drag and an anisotropy to create the long upstream vortices.
Note for Attendees
This is an IAM/PIMS distinguished speaker. Tea will be served before the talk in the IAM lounge (LSK 306).
Let S be a K3 surface. Generating series of Gromov-Witten invariants of the product geometry SxP1 are conjectured to be quasi-Jacobi forms. We sketch a proof of this conjecture for classes of degree 1 or 2 over P1 using genus bounds on hyperelliptic curves in K3 surfaces by Ciliberto and Knutsen. This has applications to a GW/Hilb correspondence for K3 surfaces, and curve counting on SxE, where E is an elliptic curve.
Many inverse problems in science and engineering involve multi-experiment data and thus require a large number of forward simulations. Dimensionality reduction techniques aim at reducing the number of forward solves by (randomly) subsampling the data. In the special case of (non-linear) least-squares estimation, we can interpret this compression of the data as a (low-rank) approximation of the noise covariance matrix. We show that this leads to different design criteria for the subsampling process. Furthermore, the resulting low-rank structure can be exploited when designing matrix-free methods for estimating (properties of) the posterior covariance matrix. Finally, we discuss the possibility of estimating the noise covariance matrix itself.
The parabolic Anderson model is the heat equation perturbed by a multiplicative noise. In case of Gaussian noise with non-trivial constant initial datum, the n-th moment of the solution grows exponentially fast in long term over the whole spatial domain. If the initial datum is localized, the moment grows exponentially only inside a space-time cone. Outside of the cone, the moment decays exponentially in long term. We will discuss how to specify these cones. The talk is based on a joint work with Jingyu Huang and David Nualart (available on arXiv:1509.00897).
Anyone can have a theorem but only a select few mathematicians have a Calculus named after them! One such is Tom Goodwillie who constructed a theory analogous to differential calculus for functors from spaces to spaces. I will give an introduction to the Goodwillie Calculus and explain why the identity functor has such an interesting "Taylor series".
The recent work of Minhyong Kim provides us with a very general framework for addressing questions of Diophantine finiteness for hyperbolic curves in an effective way. We will discuss some first steps towards making his strategy explicit for certain classes of higher genus curves, resulting in an effective version of Faltings' finiteness theorem. Together with the recent work of Jennifer Balakrishnan and Netan Dogra on bielliptic curves, these form the first examples of higher genus curves for which Kim's programme may be carried out.
You are Batman and your lover is captured by Two-face! Wanting to break the monotony of his daily routine, instead leaving her fate to a coin, he smugly decides to play the following game with you. He privately writes down two distinct numbers and reveals one to you, and asks ''Is this the bigger of the two?'', assuming that guessing correctly was equally random as flipping a coin. Little does he know that Batman is a trained probabilist and is uniquely ready for such an situation. Not only do you have a plan, you can execute it in such a way that the odds of winning are strictly better than 50/50! You save your lover, Two-face is defeated, you are a champion (at least with some probability better than 50%).
In this talk we will analyze the above problem and two others in the same vein. We will see unexpected uses of probability to determine unlikely solutions to seemingly arbitrary, hopeless situations to beat the odds in a way that is better than a pure guess.
There is no prerequisites for this talk other than a short attention span, a can-do attitude, and a hunger for pizza. Knowing what a random variable is won't hurt either, but is not necessary.
Aside from games of chance and a handful of textbook topics (e.g. opinion polls) there is little overlap between the content of an introductory course in mathematical probability and our everyday perception of chance. In this mostly non-mathematical talk I will give some illustrations of the broader scope of probability.
Why do your friends have more friends than you do, on average? How can we judge someone’s ability to assess probabilities of future geopolitical events, where the true probabilities are unknown? Were there unusually many candidates for the 2012 and 2016 Republican Presidential Nominations whose fortunes rose and fell? Why, in a long line at airport security, do you move forward a few paces and then wait half a minute before moving forward again? In what everyday contexts do ordinary people perceive uncertainty/unpredictability in terms of chance?
Note for Attendees
This lecture series was made possible by an endowment from Dr. Hugh Morris, former Board Chair and longtime friend of the mathematical sciences.
In this talk I will survey some recent and ongoing work of myself and collaborators (David Ben-Zvi, David Nadler, Hendrik Orem), and others, concerning certain topological field theories associated to a complex reductive group G. The basic example of such a theory, assigns the cohomology of the character variety (i.e. moduli of representations of the fundamental group) to a topological surface. To a point, it assigns the categorical group algebra of D-modules on G. I will discuss various approaches to studying this theory, including work from my thesis on parabolic induction and restriction functors, work in progress with Ben-Zvi and Nadler on a monoidal quantization of the the group scheme of regular centralizers using translation functors on Whittaker modules, and a categorical highest weight theorem with Ben-Zvi, Nadler and Orem. Our work is partly motivated by the "Arithmetic Harmonic Analysis" developed by Hausel, Rodriguez-Villegas, and Lettalier, to study the cohomology of character and quiver varieties.
This workshop is focused on teaching people of all genders how to support women by taking small, simple, everyday actions. It is designed for voluntary participants who already agree that sexism exists, that it should stop, and that they personally would like to help end it. Undergraduates, graduate students, staff, postdocs, and faculty are all welcome. Refreshments will be served to participants.
We encourage people of all genders to attend the workshop. Men are especially encouraged to learn more about effective approaches to supporting women in mathematics. People of other genders are encouraged to share their knowledge and personal experience. Please register in advance, at: http://tinyurl.com/UBCMathASW
Math Ally Skills Workshop
March 8, 2016
5:30pm–7:30pm (sign-in begins at 5:15pm)
room MATH 204
Facilitated by Laura Barton and Greg Martin
Laura Barton is a third year Bachelor of Computer Science student and member of the Focus on Women in Computer Science committee. She works to be both an ally and a positive role model for minorities in computer science, and believes we all play a part in creating an inclusive, vibrant culture that will benefit everyone.
Greg Martin is a professor in the Mathematics Department at UBC who has been working at being an active ally for several years. He has participated in the Anti-Violence Ally and Positive Space programs at UBC, and continues to write regularly about gender in STEM fields, including an academic paper on the underrepresentation of women in mathematics.
The classical method of Gelfand-Zetlin constructs bases in representations of gl_n using iterative restriction to smaller gl_k. For any semisimple Lie algebra this can be generalized using eigenvectors for maximal commutative subalgebra of universal envelopping algebras. In this way, we obtain a family of bases for representations of our Lie algebra. This family is parametrized by the moduli space of marked genus 0 real curves. The fundamental group of this moduli space is called the cactus group and thus we obtain an an action of the cactus group on one of these bases. This action of the cactus group matches an action defined combinatorially using crystals.
Convex optimization problems in a variety of applications have favorable objectives but complicating constraints, and first-order methods, often needed for large problems, are not immediately applicable. We propose a level-set approach that exchanges the roles of the objective and constraint functions, and instead approximately solves a sequence of parametric problems. We describe the theoretical and practical properties of this approach for a range of problems, including low-rank semidefinite optimization, which arise in matrix-completion applications.
Joint work with A. Aravkin, J. Burke, D. Drusvyatskiy, S. Roy.
We develop a KAM theory for tori with hyperbolic directions for PDEs coming mainly from fluid dynamics. One of the features of these PDEs is that they are strongly ill-posed. However, our method allows to construct specific quasi-periodic solutions. The format of the KAM theorem is a posteriori in a sense I will make precise and this allows to use several perturbative expansions to compute approximate solutions.
Baxter permutations are permutations whose matrix representations correspond to (renormalized) configurations of points given by the intersections (in the unit square [0,1]^2) of two plotted functions y=f(x) and x=g(y). We will review on their rich combinatorial properties and bijective connections,and give some open questions regarding the limit behaviour of random Baxter permutations.
Let C be a triangulated category. The Grothendieck group K_0(C) is defined as the abelian group generated by symbols [X], where X is an object of C, moduli the relation [X] = [X'] + [X''] for every exact triangle X' -> X -> X'' in C.
A simple consequence of this relation is that the double suspension functor X -> X[2] induces the identity map from K_0(C) to itself. In this talk, I will explain how this observation can be seen as the shadow of a certain rotation-invariance phenomenon in algebraic K-theory, and describe the connection of this phenomenon with the theory of "topological Fukaya categories" introduced by Dyckerhoff and Kapranov.
I'll talk about the paradoxes of Simpson and of St. Petersburg; both of these arise when all the information in a subject is reduced to the first moment or centre of mass of the probability.
If time permits, I'd like to address several cases in real life events (if possible, recent ones) where such paradoxes are employed to give information about a subject.
Let L be a positive definite lattice. There are only finitely many positive definite lattices L' which are isomorphic to L modulo N for every N > 0: in fact, there is a formula for the number of such lattices, called the Siegel mass formula. In this talk, I'll review the Siegel mass formula and explain how it was reformulated by Weil as a statement about volumes of adelic groups. I'll then describe some joint work with Dennis Gaitsgory on computing these volumes over function fields using ideas from topology: in particular, a nonabelian version of Poincare duality.
This seminar provides an overview of currently available methods for state estimation of linear, constrained and nonlinear dynamic systems. The seminar begins with a brief overview of the Kalman filter, which is the optimal estimator for a linear dynamic system subject to independent, normally distributed disturbances. Next, alternatives for treating nonlinear and constrained dynamic systems are discussed. Two complementary methods are presented in some detail: moving horizon estimation, which is based on optimization, and particle filtering, which is based on sampling. The advantages and disadvantages of these two approaches are presented. Topics for new research are suggested that address combining the best features of moving horizon estimators and particle filters.
Hall algebras play a prominent role in the interactions between algebraic geometry and representation theory. Recently, "refined" versions of them, called K-theoretic Hall algebras, were introduced by Schiffmann and Vasserot. They have notable connections with the geometric Langlands correspondence, the theory of quantum groups and gauge theories.
In the first part of the talk, I will give an overview of the theory of Hall algebras. In the second part, I will describe some (new) examples of K-theoretic Hall algebras. These algebras are related to some stacks of a certain kind of sheaves on noncompact surfaces. (Work in progress with Olivier Schiffmann.)
The deformation and transport of elastic filaments in viscous fluids play central roles in many biological and technological processes. Compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance may cause a significant alteration in the long-time sedimentation orientation and filament geometry. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes which match excellently with full numerical simulations. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. Embedding the analytical results for a single filament into a mean-field theory, we show how flexibility affects a well established concentration instability in a sedimenting suspension. Related topics will make cameo appearances, from a zoology of dynamics of settling bodies near walls to a new traction boundary integral equation for viscous fluid-body interactions.
Note for Attendees
Note this is a joint event with the SCAIM seminar, and the usual sushi lunch will be provided.
In algebraic combinatorics a central area of study is Schur functions. These functions were introduced early in the last century with respect to representation theory, and since then have become important in other areas such as quantum physics and algebraic geometry.
These functions also form a basis for the algebra of symmetric functions, which in turn forms a subalgebra of the algebra of quasisymmetric functions that itself impacts areas from category theory to card shuffling. Despite this strong connection, the existence of a natural quasisymmetric refinement of Schur functions was considered unlikely for many years.
In this talk we will meet such a natural refinement of Schur functions, called quasisymmetric Schur functions. Furthermore, we will see how these quasisymmetric Schur functions refine many well-known Schur function properties, with combinatorics that strongly reflects the classical case including diagrams, walks in the plane, and pattern avoidance in permutations.
This talk will require no prior knowledge of any of the above terms.
A boundary Harnack principle (BHP) was recently proved for Feller processes in metric measure spaces by Bogdan, Kumagai and Kwasnicki. In this talk I will first show how their method can be modified to obtain a BHP at infinity – a result which roughly says that two non-negative function which are harmonic in an unbounded set decay at the same rate at infinity. With BHP at hand, one can identify the Martin boundary of an unbounded set at infinity with a single Martin boundary point and show that, in case infinity is accessible, this point is minimal. I will also present analogous result for a finite Martin boundary point. The local character of these results implies that minimal thinness of a set at a minimal Martin boundary point is also a local property of that set near the boundary point. (Joint work with P.Kim and R.Song)
The classical EHP sequence is a partial answer to the question of how far the unit map of the loop-suspension adjunction fails to be a weak equivalence. It can be used to move information from stable to unstable homotopy theory. I will explain why there is an EHP sequence in A1 algebraic topology, and some implications this has for the unstable A1 homotopy groups of spheres.
A real number is said to be normal to base b, b≥2 an integer, if in its base b expansion each finite pattern of digits occurs asymptotically with the expected frequency. A real number is called absolutely normal if it is normal to all integer bases b≥2. So far, it remains an open problem to exhibit a natural example of such a number. We present a number of algorithms that compute absolutely normal numbers in the sense that they successively output the digits to a given base of a real number that can be shown to be absolutely normal. We analyze these algorithms with respect to computational complexity and speed of convergence to normality. Furthermore, we adapt one of the algorithms to the more general setting of Pisot numbers as bases and conclude with some open problems.
We will discuss the Cauchy problem for the cubic fourth order nonlinear Schrodinger equation (4NLS) on the circle. We first prove non-existence of solutions to (4NLS) for initial data lying strictly in negative Sobolev spaces, by using the short time Fourier restriction norm method. Then, we focus on the well-posedness issue of the renoramilzed 4NLS (so called the Wick ordered W4NLS). In particular, by performing normal form reductions infinite many times, we prove well-posedness of (W4NLS) in negative Sobolev spaces. This talk is based on a joint work with Tadahiro Oh (University of Edinburgh).
The Criminal Justice System is responsible for upholding public safety through the enforcement of laws, the apprehension, prosecution, and judging of suspects, and the administration of community and custodial sentences. It is highly complex, with interactions between police, prosecutors, judges, the court, and correctional services. Effective and efficient administration of justice is important for maintaining public safety.
We present an overview of operational research modeling applied to the Criminal Justice System. Two case studies are considered: a systems dynamics model of the impact of the 2010 impaired driving legislation in British Columbia and a queue network model of the impact of the Truth in Sentencing Act of Canada.
Registration is free and a light lunch will be provided. For more details and to register for this event, please visit http://www.pims.math.ca/industrial-event/160224-pllkv .
Note for Attendees
About the PIMS Lunchbox Series:
The PIMS Lunch Box Series create a beneficial link between researchers in industry and academics in the mathematical sciences. Researchers with industrial and commercial concerns are invited to present their current work, demonstrating how the latest advances in the mathematical sciences apply to current, real-world issues in industrial processes such as seismic and medical imaging, financial mathematics and complex fluids. PIMS has been holding lunchbox lectures in Calgary for seven years and is now expanding to Vancouver. To learn more about PIMS, please visit www.pims.math.ca<http://www.pims.math.ca> .
I will present three new results: (i) the Cramer transform of the uniform measure on a convex body is a universal self-concordant barrier; (ii) projected Langevin Monte Carlo (i.e. discretized reflected Brownian motion with drift) allows to sample from a log-concave measure in polynomial time; and (iii) Thompson sampling combined with a multi-scale exploration solves the Bayesian convex bandit problem. The unifying theme in these results is the interplay between concepts from convex geometry, probability theory, learning and information theory.
Wiles' proof of Fermat's Last Theorem gave birth to the "modular method" to attack Diophantine equations. Since then many other equations were solved using generalizations of this method. However, the success of the generalizations relies on a final "contradiction step" which is invisible in the original proof.
In this talk we will recall the modular method and discuss a tool known as "the symplectic argument" that sometimes allows to complete this extra step. We will also discuss some applications to Fermat-type equations, in particular, x3 + y3 = zp.
I will discuss the notion of entropy as it arises in information and coding theory and its central role in Shannon's classical source coding theorem. I will then, to the extent I can coherently manage, show how it generalizes to a more general notion of entropy as it arises in dynamical systems theory.
Abstract: Thin sheets exhibit a daunting array of patterns. A key difficulty in their analysis is that while we have many examples, we have no classification of the possible "patterns." I have explored an alternative viewpoint in a series of recent projects with Peter Bella, Hoai-Minh Nguyen, and others. Our goal is to identify the *scaling law* of the minimum elastic energy (with respect to the sheet thickness, and the other parameters of the problem). Success requires proving upper bounds and lower bounds that scale the same way. The upper bounds are usually easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-independent. In many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. My talk will give an overview of this activity, and details of some examples.
An Azumaya algebra of degree n is an algebra locally isomorphic to an nxn matrix algebra, a concept that generalizes that of central simple algebras over fields. The Brauer group consists of equivalence classes of Azumaya algebras and the index of a class in the Brauer group is defined to be the greatest common divisor of the degrees of all Azumaya algebras in that class.
Suppose p and q are relatively prime positive integers. Whereas an Azumaya algebra of degree pq need not, in general, decompose as a tensor product of algebras of degrees p and q, we show that a Brauer class of index pq does decompose as a sum of Brauer classes of indices p and q. The argument requires only the representation theory of GLn, and therefore establishes the result in contexts where one does not have recourse to the theory of fields, for instance in the theory of Azumaya algebras on topological spaces.
Room 200, Graduate Student Centre, 6371 Crescent Rd., UBC
Tue 5 Apr 2016, 11:00am-1:00pm
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ABSTRACT: This work leads to a simple and geometric framework for defining generalized Donaldson-Thomas invariants. We view the inertia construction of algebraic stacks as an operator on the Grothendieck groups of various categories of algebraic stacks. We expect to show that this operator is diagonalizable and the Donaldson-Thomas theory of various moduli spaces can be defined in term of the eigenprojections of this operator.
We are interested in showing that the inertia operator is (locally finite and) diagonalizable over for instance the field of rational functions of the motivic class of the affine line. This is proved for the Grothendieck group of Deligne-Mumford stacks and the category of quasi-split Artin stacks.
Motivated by the quasi-splitness condition we then develop a theory of linear algebraic stacks and algebroids, and define a space of stack functions over a linear algebraic stack. We prove diagonalization of the semisimple inertia for the space of stack functions. A different family of operators is then defined that are closely related to the semisimple inertia. These operators are diagonalizable on the Grothendieck ring itself (i.e. without inverting elements) and provide ways of computing eigenprojections of elements with respect to the semisimple inertia.
We then define a graded structure on the Hall algebra on the space of stack functions in terms of the eigenspaces of the semisimple inertia. The commutative and non-commutative products of the Hall algebra respect the graded structure of the Hall algebra and they coincide on the associated graded algebra. This result provides a geometric way of defining a Lie subalgebra of virtually indecomposables. An $\epsilon$-element associated to an algebroid is defined and shown to be in the space of virtually indecomposables.
There has been significant recent work on the theory and application of randomized coordinate descent algorithms, beginning with the work of Nesterov, who showed that a random-coordinate selection rule achieves the same convergence rate as the Gauss-Southwell selection rule. This result suggests that we should never use the Gauss-Southwell rule, as it is typically much more expensive than random selection. However, the empirical behaviours of these algorithms contradict this theoretical result: in applications where the computational costs of the selection rules are comparable, the Gauss-Southwell selection rule tends to perform substantially better than random coordinate selection. We give a simple analysis of the Gauss-Southwell rule showing that---except in extreme
cases---it's convergence rate is faster than choosing random coordinates. Further, we (i) show that exact coordinate optimization improves the convergence rate for certain sparse problems, (ii) propose a Gauss-Southwell-Lipschitz rule that gives an even faster convergence rate given knowledge of the Lipschitz constants of the partial derivatives, and (iii) analyze proximal-gradient variants of the Gauss-Southwell rule.
Joint work with Mark Schmidt, Issam Laradji, Michael Friedlander and Hoyt Koepke.
The irreducible smooth representations of Arthur class are the local components of automorphic representations. They are conjectured to be parametrized by the Arthur parameters, which form a subset of the usual Langlands parameters. The set of irreducible representations associated with a single Arthur parameter is called an Arthur packet. Following Arthur's classification theory of automorphic representations of symplectic and orthogonal groups, the Arthur packets are now known in these cases. On the other hand, Moeglin independently constructed these packets in the p-adic case by using very different methods. In this talk, I would like to describe a combinatorial procedure to study the structure of the Arthur packets following the works of Moeglin. As an application, we show the size of Arthur packets in these cases can be given by counting integral (or half-integral) points in certain polytopes.
We will study the edge of the support of 1-dimensional super-brownian motion. The local behaviour of the density, Hausdorff dimension of the boundary, and rates of convergence of certain solutions to singular semilinear heat equations studied in the pde literature are all expressed in terms of a particular eigenvalue of a killed Ornstein-Uhlenbeck operator. Time permitting, we will discuss some possible connections with pathwise uniqueness questions for some related stochastic pde's. This is joint work with Carl Mueller and Leonid Mytnik.
The Boussinesq equations concerned here model geophysical flows such as atmospheric fronts and ocean circulations. In addition, they play an important role in the study of Rayleigh-Benard convection. Mathematically the 2D Boussinesq equations serve as a lower-dimensional model of the 3D hydrodynamics equations. In fact, the 2D Boussinesq equations retain some key features of the 3D Euler and the Navier-Stokes equations such as the vortex stretching mechanism. The global regularity problem on the 2D Boussinesq equations with partial or fractional dissipation has attracted considerable attention in the last few years. This talk presents recent developments in this direction. In particular, we detail the global regularity result on the 2D Boussinesq equations with vertical dissipation as well as some recent work for the 2D Boussinesq equations with general critical dissipation.
ABSTRACT: We study stability of 3-dimensional quadratic AS-regular algebras and their moduli.
A quadratic algebra defined by a triple (E, L, σ) is stable if there is no node or line component of E fixed by σ. We first prove stability of the twisted homogeneous coordinate ring B(E, L, σ) by Riemann-Roch and exploiting noncommutativity. Then we lift stability to that of the quadratic algebra A(E, L, σ), by analyzing the central element c3 where B=A/(c3).
We study a coarse moduli space for each type, A, B, E, H, S. S-equivalence of strictly semistable algebras is studied. We compute automorphisms of AS-regular algebras and of those that appear in the boundary of the moduli. We found complete DM-stacks for 2,3-truncated algebras.
Type B algebra as Zhang twist of type A is studied. Exceptional algebras appear in the exceptional divisor of a blowing-up at a degenerate algebra in the moduli of 3-truncations. 2-unstable algebras are studied.
I will discuss a recent joint work with A. Malchiodi (Pisa) and M. Micallef (Warwick) in which we show that not every harmonic map can be approximated by a sequence of alpha-harmonic maps.
For any non-negative integer n there exist n-cusped hyperbolic 3-manifolds of minimal possible volume. They are sometimes not unique. For example there are exactly two distinct minimal 1-cusped examples: the figure eight complement, and another which is not a knot complement. Similarly there are distinct 2-cusped examples. I will show how these examples differ in terms of properties of their fundamental groups. In particular, in the pairs of examples in the 1 or 2 cusped case, one has bi-orderable fundamental group while the other’s group is not orderable. This is a preliminary announcement of work in progress with Eiko Kin (Osaka).
Room 200, Graduate Student Centre, 6371 Crescent Rd., UBC
Thu 14 Apr 2016, 12:30pm-2:30pm
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ABSTRACT: In this thesis, we study methods in the proof of Green-Tao’s theorem on existence of arithmetic progressions in dense subsets of primes (or almost primes) and prove some of its generalizations. Firstly, we study the Goldston-Yildirim sieve on almost primes which is used to prove pseudorandom conditions for almost primes in the work of Green and Tao. We combine the Goldston-Yildirim sieve with circle method of Birch to obtain a lower bound on the number of almost prime solutions to high rank Diophantine systems. Secondly, we apply the transference principle used in the original proof of Green-Tao’s result (simplified by Gowers using Hahn-Banach’s Theorem) to obtain a lower bound on the number of affine copies of a corner configurations in dense subsets of prime lattices in higher dimensions. The main difficulty is to deal with unbounded dual functions. Finally, we prove a hypergraph regularity lemma and removal lemma on weighted hypergraphs and obtain a lower bound on the number affine copies of any configuration in prime lattices. The key technique is the energy increment on parametric family of weighted probability spaces.
Abstract:
The Prisoner’s Dilemma game, the working horse for studying social traps, has recently undergone a remarkable rejuvenation. New results allow to characterize partner strategies, competitive strategies, and ZD strategies. If a player uses a partner strategy, both players can fairly share the social optimum; but a co-player preferring an unfair solution will be penalized by obtaining a reduced payoff. A player using a competitive strategy never obtains less than the co-player. A player using a ZD strategy unilaterally enforces a linear relation between the two players payoffs. These properties hold for all possible strategies of the co-player and thus cover a vast range of behaviors. The new results will be embedded in an overview covering a wide field of well-established theoretical and experimental results.
Speaker: Karl Sigmund was Professor of Mathematics at the University of Vienna from 1974-2013, and is one of the pioneers of evolutionary game theory. He also worked on ergodic theory and dynamical systems, and biomathematics. More recently, he has increasingly turned to the history of science - with books, exhibitions, and films on the Vienna Circle.
Note for Attendees
Please register for this free event at http://events.pwias.ubc.ca/special-events
The Peter Wall Institute for Advanced Studies is located at 6331 Crescent Road, UBC campus, Vancouver.
This Public Talk is presented in collaboration with the UBC Department of Mathematics.
Earth Sciences Bldg. (ESB) 2012, 2207 Main Mall, UBC
Thu 21 Apr 2016, 4:00pm-5:00pm
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Abstract: Albert Einstein is usually associated with Zürich, Bern, Berlin or Princeton rather than Vienna. However, his Viennese contacts were many-sided and highly relevant for his work. Ernst Mach had a major influence on the general theory of relativity (the “Mach principle”), and Kurt Gödel discovered its most paradoxical consequence (the possibility of time travels into the past). In addition, Friedrich Adler, Hans Thirring, Erwin Schrödinger and the thinkers of the Vienna Circle were important companions of Einstein. In this lecture, Dr. Sigmund will not only trace a highly dramatic story replete with murder and flight, fakes and nervous breakdowns, but also highlight a central topic of general relativity: the interplay of gravitation and inertia, and their effects on rotating bodies.
Note for Attendees
Reception before the lecture will be held in ESB 4133 from 3:30--4:00 p.m.
Let k be a field. A classical result of Manin and Swinnerton-Dyer states that every del Pezzo surface of degree 5, defined over k, is a k-rational variety. It is a standard fact that such a surface is a k-form of the moduli space M_(0,5) of genus 0 curves equipped with 5 ordered points. In this talk, we shall, more generally, consider k-forms of the moduli spaces M_{g,n} of curves of genus g, with n marked points. These are, in most cases, moduli spaces for genus g curves, equipped with an embedding of a fixed degree n 'etale algebra. The main result is that k-forms of M_{0,n} are always k-rational if n is odd. On the contrary, if n is even, there exists a field k and a k-form of M_{0,n} which is not (retract) k-rational. Results in higher genus will also be discussed. This is joint work with Zinovy Reichstein.
We consider the critical behaviour of long-range O(n) models
for n greater than or equal to 0. For n=1,2,3,... these
are phi^4 spin models. For n=0 it is the weakly self-avoiding walk.
We prove existence of critical exponents for the susceptibility
and the specific heat, below the upper critical dimension.
This is a rigorous version of the epsilon expansion in physics.
The proof is based on a rigorous renormalisation group method
developed in previous work with Bauerschmidt and Brydges.
Many complex systems involving a large number of independent variables have come to be very well understood. One such example is Bernoulli percolation on a planar lattice. However, how to adapt the techniques to closely related models, such as continuum percolation in $\mathbb{R}^2$, may be far from obvious. We will describe some techniques of this kind that recently has been developed for Poisson Boolean percolation. We will focus on a certain two-stage construction that allows for a reduction to the discrete setting, where a larger arsenal of techniques is available for the study of phenomena such as sharp thresholds and noise sensitivity. Joint work with Vincent Tassion and Augusto Teixeira.
We will present some recent results about bond percolation on the hypercube {0,1}^m in the "slightly" subcritical phase, that is, just below the critical percolation scaling window. We estimate the size, diameter and mixing time of the largest components. A difficulty that arises only in the subcritical phase is that the cluster of the largest size does not attain the largest possible diameter. Therefore, we are able to analyze rather accurately the cluster of largest diameter, but not the cluster of largest size, leaving some interesting open problems.
Abstract: Let (M,g_0) be a compact n-dimensional Riemannian manifold with a finite number of singular points, where at each singular point the metric is asymptotic to a cone over a compact (n-1)-dimensional manifold with curvature operator greater or equal to one. We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like C/t. The initial metric is attained in Gromov-Hausdorff distance and smoothly away from the singular points. To construct this solution, we desingularize the initial metric by glueing in expanding solitons with positive curvature operator, each asymptotic to the cone at the singular point, at a small scale s. Localizing a recent stability result of Deruelle-Lamm for such expanding solutions, we show that there exists a solution from the desingularized initial metric for a uniform time T>0, independent of the glueing scale s. The solution is then obtained by letting s->0. We also show that the so obtained limiting solution has the corresponding expanding soliton as a forward tangent flow at each initial singular point. This is joint work with P. Gianniotis.
Abstract: During World War II Hedy Lamarr, a striking Hollywood actress, together with George Antheil, a radical composer, invented and patented a secret signaling system for the remote control of torpedoes. The ideas in this patent have since developed into one of the ingredients in modern digital wireless communications. The unlikely biography of these two characters, along with some of the more modern developments in wireless communications will be described.
Bio: Mark is a UBC alum and former math department faculty member. He is most well known for discovering Intersection Cohomology Theory with Robert MacPherson. Intersection Cohomology has had a profound impact on several areas of math, particularly Representation Theory, Algebraic Topology and Algebraic Geometry.
About the Niven Lectures: Ivan Niven was a famous number theorist and expositor; his textbooks won numerous awards, have been translated into many languages and are widely used to this day. Niven was born in Vancouver in 1915, earned his Bachelor’s and Master’s degrees at UBC in 1934 and 1936 and his Ph.D. at the University of Chicago in 1938. He was a faculty member at the University of Oregon from 1947 until his retirement in 1982. The annual Niven Lecture Series, held at UBC since 2005, is funded in part through a generous bequest from Ivan and Betty Niven to the UBC Mathematics Department.
The cohomology of certain p-adic Lie groups plays a central role in chromatic homotopy theory. In this talk I will discuss these groups and discuss some of their elementary group theoretical structure. Then I will explain some recent group cohomology calculations relevant for stable homotopy theory. In particular I will highlight the result of some highly non-trivial calculation resulting in a surprising isomorphism which cries out for a conceptual explanation.
We consider mixed-type jump processes on metric measure spaces and prove the stability of two-sided heat kernel estimates, heat kernel upper bounds, and parabolic Harnack inequalities. We establish their stable equivalent characterizations in terms of the jump kernels, modifications of cut-off Sobolev inequalities, and the Poincar\'e inequalities. In particular, we prove the stability of heat kernel estimates for \alpha -stable-like processes even with \alpha\ge 2, which has been one of the major open problems in this area. We will also explain applications to stochastic processes on fractals.
This is a joint work with Z.Q. Chen (Seattle) and J. Wang (Fuzhou).
Scalable parallel optimization methods are gaining importance for a wide range of machine learning applications, for example, as implemented in the machine learning library of the Apache Spark distributed data processing environment. I will discuss our work on accelerating parallel algorithms for two common applications in this area: matrix factorization for recommendation systems, and line search methods for problems such as logistic regression.
For the recommendation application, we accelerate the standard Alternating Least Squares (ALS) optimisation algorithm using a nonlinear conjugate gradient (NCG) wrapper around the ALS iterations. In parallel numerical experiments on a 16 node cluster with 256 computing cores, we demonstrate that the combined ALS-NCG method requires many fewer iterations and less time (with acceleration factors of 4 and more) than standalone ALS to reach movie rankings with high accuracy on the MovieLens 20M dataset and synthetic datasets with up to nearly 1 billion ratings (http://arxiv.org/abs/1508.03110).
The second part of the talk discusses a new type of parallel line search for large-scale unconstrained minimization of smooth loss functions such as logistic regression. We present a new line search technique that computes more accurate minima by evaluating a Taylor polynomial approximation to the loss function, which also reduces the parallel communication costs, resulting in overall efficiency gains of a factor of 2 or more in parallel compared to existing approaches (http://arxiv.org/abs/1510.08345).
This is joint work with Mike Hynes.
Room 203, Graduate Student Centre, 6371 Crescent Rd., UBC
Fri 22 Jul 2016, 12:30pm-2:30pm
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Abstract: Over the last few decades, there has been a growing interest in a measure-theoretical property of Gibbs distributions known as strong spatial mixing (SSM). SSM has connections with decay of correlations, uniqueness of equilibrium states, approximation algorithms for counting problems, and has been particularly useful for proving special representation formulas and the existence of efficient approximation algorithms for (topological) pressure. We look into conditions for the existence of Gibbs distributions satisfying SSM, with special emphasis in hard constrained models, and apply this for pressure representation and approximation techniques in Z^d lattice models.
Given a locally finite countable graph G and a finite graph H, we consider Hom(G,H) the set of graph homomorphisms from G to H, and we study Gibbs measures supported on Hom(G,H). We develop some sufficient and other necessary conditions on Hom(G,H) for the existence of Gibbs specifications satisfying SSM (with exponential decay). In particular, we introduce a new combinatorial condition on the support of Gibbs distributions called topological strong spatial mixing (TSSM). We establish many useful properties of TSSM for studying SSM on systems with hard constraints, and we prove that TSSM combined with SSM is sufficient for having an efficient approximation algorithm for pressure. We also show that TSSM is, in fact, necessary for SSM to hold at high decay rate.
Later, we prove a new pressure representation theorem for nearest-neighbour Gibbs interactions on Z^d shift spaces, and apply this to obtain efficient approximation algorithms for pressure in the Z^2 (ferromagnetic) Potts, (multi-type) Widom-Rowlinson, and hard-core lattice gas models. For Potts, the results apply to every inverse temperature except the critical. For Widom-Rowlinson and hard-core lattice gas, they apply to certain subsets of both the subcritical and supercritical regions. The main novelty of this work is in the latter, where SSM cannot hold.
Room 203, Graduate Student Centre, 6371 Crescent Road, UBC
Mon 25 Jul 2016, 12:30pm-2:30pm
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Abstract: The goal of this thesis is to understand the topology of representation varieties. To be more precise, let G be a complex reductive linear algebraic group and let K be a maximal compact subgroup of G. Given a nilpotent group Γ generated by r elements, we consider the representation spaces Hom(Γ,G) and Hom(Γ,K) with the natural topology induced from an embedding into Gr and Kr respectively. Our main result shows that there is a strong deformation retraction of Hom(Γ,G) onto Hom(Γ,K). We also obtain a strong deformation retraction of the geometric invariant theory quotient Hom(Γ,G)//G onto the ordinary quotient Hom(Γ,K)/K. Using these deformations, we then describe the topology of these spaces.
Aperiodic substitutive dynamical systems are recognizable, i.e., given a substitution \zeta and an element x of the two-sided shift (X_\zeta, T) generated by \zeta , there is a unique y \in \Omega_\zeta (up to a power of the shift map T) such that x = T^k \zeta(y) for some k. This was shown by Mossé (1996) for primitive substitutions and by Bezuglyi, Kwiatkowski and Medynets (2009) in the general case. We consider shifts generated by a sequence of substitutions, so-called S-adic shifts. This includes shifts generated by morphic words, and we show that certain morphic words are not recognizable. More precisely, there exist substitutions \zeta and \theta such that non-trivial relations \theta(x) = T^k \theta(y) with x, y \in X_\zeta hold. On the other hand, we prove recognizability for a large class of S-adic shifts using S-adic Rauzy fractals.
This is joint work with Valérie Berthé, Jörg Thuswaldner and Reem Yassawi.
The connection between textiles and mathematics is intimate but not often explored, possibly because textiles and fiber arts have traditionally been the domain of women while mathematics was viewed as a male endeavour. How times have changed! Today, textiles and mathematics, like art and science, are recognized for their interwoven, complimentary attributes. In this presentation, mathematics professor Gerda de Vries will examine the connection between textiles and mathematics, in the context of both traditional and contemporary quilts. In a sense, every quilt is a mathematical object, by virtue of the fact that it has shape and dimension. But some quilts are more mathematical than others, and in very different ways. She will show how mathematical concepts such as symmetry, fractals, and algorithmic design show up in the world of quilting through serendipitous and intentional design.
This lecture is for a general audience. A background in mathematics is not needed, nor the ability to sew!
For any dimension\alpha < n,we define a type of random Cantor set in\mathbb{R}^nand prove near optimal decoupling and restriction bounds. This is joint-work with Izabella Laba.
Descriptive convergence estimates or bounds for Krylov subspace iterative methods for nonsymmetric matrix systems are keenly desired but remain elusive. In the case of symmetric (self-adjoint) matrices, bounds based on eigenvalues can be usefully descriptive of observed convergence; an important consequence is that there are rigorous criteria for what constitutes a good preconditioner for symmetric matrices. For nonsymmetric matrices preconditioning must generally be
heuristically motivated.
Such comments apply quite generally, however there is one class of nonsymmetric matrices for which we have recently been able to rigorously prove descriptive convergence bounds, namely real Toeplitz (constant diagonal) matrices. Our results apply regardless of non-normality or any 'degree' of nonsymmetry.
Gil Strang proposed the use of circulant matrices (and the FFT) for preconditioning symmetric Toeplitz matrix systems in 1986 and there is now a well-developed theory which guarantees rapid convergence of the conjugate gradient method for such preconditioned positive definite symmetric systems.
In this talk we describe our recent approach which provides a preconditioned MINRES method with the same guarantees for real nonsymmetric Toeplitz systems regardless of the non-normality, and demonstrate the application of the approach for time-dependent PDE problems.
This is joint work with Jennifer Pestana (Strathclyde University) and Elle McDonald (Oxford University).
The work deals with the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the Laplace operator in a fractional power. The proof of existence of solutions relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used.
The Brownian motion on the Unitary group \mathrm{U}(N) has a large-N (spectral) limit: for each fixed time, the histogram of eigenvalues converges almost surely to a deterministic law with a (mostly) smooth density on the circle. This was proved by Philippe Biane in the late 1990s. One can think of this as a companion to Wigner's semicircle law for Hermitian Gaussian random matrices: the latter is really about the Brownian motion on the Unitary Lie algebra, and so it is compelling that some of the same behavior carries over to the Lie group.
In this lecture, I will talk about two finer properties of the large-N limit of Unitary Brownian motion.
In joint work with Guillaume Cébron, following related work of Thierry Lévy and Mylčne Maďda, we showed that the bulk fluctuations (a.k.a. linear statistics) of the eigenvalues are Gaussian, with an explicit covariance that generalizes the Haar unitary case studied by Evans and Diaconis.
In joint work with Benoît Collins and Antoine Dahlqvist, we showed that the largest (angle) eigenvalue of the Brownian motion has an explicit almost sure limit.
In both cases, we proved multi-dimensional versions of the theorems, which I will also describe if time permits.
We initiate the study of optimal transportation of exact differential k–forms and introduce various distances as minimal actions. Our study involves dual maximization problems with constraints on the codifferential of k–forms. When k < n, only some directional derivatives of a vector field are controlled. This is in contrast with prior studies of optimal transportation of volume forms (k = n), where the full gradient of a scalar function is controlled. Furthermore, our study involves paths of bounded variations on the set of k–currents. This talk is based a joint work with B. Dacorogna and O. Kneuss.
Note for Attendees
Refreshments will be served in ESB 4133 (the PIMS Lounge).
Given a smooth surface, the generating series of Euler characteristics of its Hilbert schemes of points can be given in closed form by (a specialisation of) Goettsche's formula. I will discuss a generalisation of this formula to surfaces with rational double points. A certain representation of the affine Lie algebra corresponding to the surface singularity (via the McKay correspondence), and its crystal basis theory, play an important role in our approach. Joint work with András Némethi and Balázs Szendrői.
In this talk, we will discuss the Trudinger-Moser and Caffarelli-Kohn-Nirenberg inequalities in the settings where the classical Schwarz rearrangement cannot be used. We will then talk about some approaches to study the maximizers for these problems. This is joint work with Guozhen Lu.
The study of convolution powers of a finitely supported probability distribution \phi on the d-dimensional square lattice is central to random walk theory. For instance, the nth convolution power \phi^{(n)} is the distribution
of the nth step of the associated random walk and is described by the classical local limit theorem. When such distributions take on complex values, their convolution powers exhibit surprising and disparate behaviors not seen in the probabilistic setting. In this talk, I will discuss new results concerning the asymptotic behavior of convolution powers of complex-valued functions on \mathbb{Z}^d, specifically generalized local limit theorems and sup-norm estimates. This joint work with Laurent Saloff-Coste extends previous results by I. J. Shoenberg, T. N. E. Greville, P. Diaconis and L. Saloff-Coste.
The classifying functor from categories to topological spaces provides a way of constructing spaces with certain properties or structure from categories with similar properties of structure. An important example of this is the construction of infinite loop spaces from symmetric monoidal categories. The particular kinds of extra structure can typically be encoded by monads on the category of small categories. In order to provide more flexibility in the kinds of morphisms allowed, one can work with the associated 2-monad in the 2-category of categories, functors, and natural transformations. In this talk I will give the categorical setup required, and I will give examples of interest to homotopy theorists. I will also outline how this method of working can give general statements about strictifications and comparisons of homotopy theories. This is partially based on work with two different sets of collaborators: Nick Gurski, Niles Johnson, and Marc Stephan; Bert Guillou, Peter May, and Mona Merling.
Let g>2 be a positive integer, and let M_g be the moduli space of smooth curves of genus g over \mathbb{C}. The classical Franchetta conjecture asserts that the Picard group of the generic curve C_{\mu} over Mg is freely generated by its cotangent bundle. It was proved by Arbarello and Cornalba in 1980, Then Mestrano ('87) and Kouvidakis ('91) deducted the Strong Franchetta conjecture, which asserts that the rational points of the relative picard scheme Pic_{C_{\mu} / \mu} are precisely the multiples of the cotangent bundle.
We will show that a suitably modified version of the Franchetta conjecture holds for a different moduli problem over \mathbb{C}, that of principally polarised abelian varieties (p.p.a.v.) of genus g\geq 3 with n-level structure. The abelian Franchetta conjecture states that the generic p.p.a.v. of genus g with n-level structure X_{g,n} has Picard group isomorphic to \mathbb{Z} \oplus (\mathbb{Z}/n\mathbb{Z})^{2g}, where the free part is generated by the bundle inducing the polarization, and the torsion part comes from the level structure.
In the abelian case, the ``weak" statement immediately implies the corresponding ``strong" statement regarding the rational points of the relative Picard scheme. Using duality, we will use this to compute the Picard group of the universal abelian variety \mathscr{X}_{g,n} over the moduli stack \mathscr{C}_{g,n} of genus g p.p.a.v. with n-level structure.
For any polynomial P(y), and any Calder\'{o}n-Zygmund kernel, K, the operator below maps L^1 to weak L^1.
\sup _{\epsilon >0} |\int_{|y| > \epsilon} f (x-y) e ^{2 \pi i P (y) } K(y) dy |.
The bound is only a function of the degree of the polynomial P, the dimension, and on the kernel K. The same bound, without maximal truncations, is a special case of a result due to Chanillo and Christ (1987).
For many modern scientific and machine learning applications, it is often desired to use statistical learning methods that are scalable, highly predictive, highly interpretable, and robust. However, generally speaking, current methods do not simultaneously enjoy all of these properties. We introduce the Union of Intersections (UoI) framework for designing such “all-in-one” methods. In particular, within the context of Lasso, we introduce UoI Lasso and describe its superior theoretical properties under less restrictive conditions. We further demonstrate the performance of UoI Lasso on a variety of benchmark biomedical data: extraction of meaningful functional networks from human electrophysiology recordings and dramatically more parsimonious prediction of behavioral and physiological phenotypes from genetic data.
Note for Attendees
Tea served before the talk in ESB 4133 (PIMS lounge)
Room 200, Graduate Student Centre, 6371 Crescent Road, UBC
Tue 20 Sep 2016, 12:30pm-2:30pm
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Abstract:
Microbial metabolic activity drives biogeochemical cycling in virtually every ecosystem. Yet, microbial ecology and its role in ecosystem biochemistry remain poorly understood, partly because the enormous diversity found in microbial communities hinders their modeling. Despite this diversity, the bulk of global biogeochemical fluxes is driven by a few metabolic pathways encoded by a small set of genes, which through time have spread across microbial clades that can replace each other within metabolic niches. Hence, the question arises whether the dynamics of these pathways can be modeled regardless of the hosting organisms, for example based on environmental conditions. Such a pathway-centric paradigm would greatly simplify the modeling of microbial processes at ecosystem scales.
Here I investigate the applicability of a pathway-centric paradigm for microbial ecology. By examining microbial communities in replicate "miniature" aquatic environments, I show that similar ecosystems can exhibit similar metabolic functional community structure, despite highly variable taxonomic composition within individual functional groups. Further, using data from a recent ocean survey I show that environmental conditions strongly explain the distribution of microbial metabolic functional groups across the world's oceans, but only poorly explain the taxonomic composition within individual functional groups. Using statistical tools and mathematical models I conclude that biotic interactions, such as competition and predation, likely underlie much of the taxonomic variation within functional groups observed in the aforementioned studies. The above findings strongly support a pathway-centric paradigm, in which the distribution and activity of microbial metabolic pathways is strongly determined by energetic and stoichiometric constraints, whereas additional mechanisms shape the taxonomic composition within metabolic guilds.
These findings motivated me to explore concrete pathway-centric mathematical models for specific ecosystems. Notably, I constructed a biogeochemical model for Saanich Inlet, a seasonally anoxic fjord with biogeochemistry analogous to oxygen minimum zones. The model describes the dynamics of individual microbial metabolic pathways involved in carbon, nitrogen and sulfur cycling, and largely explains geochemical depth profiles as well as DNA, mRNA and protein sequence data. This work yields insight into ocean biogeochemistry and demonstrates the potential of pathway-centric models for microbial ecology.
Many data analysis applications require the solution of optimization problems involving a sum of large number of functions. We consider the problem of minimizing a sum of n functions over a convex constraint set. Algorithms that carefully sub-sample to reduce n can improve the computational efficiency, while maintaining the original convergence properties. For second order methods, we first consider a general class of problems and give quantitative convergence results for variants of Newtons methods where the Hessian or the gradient is uniformly sub-sampled. We then show that, given certain assumptions, we can extend our analysis and apply non-uniform sampling which results in modified algorithms exhibiting more robustness and better dependence on problem specific quantities, such as the condition number.
We consider the initial-value problem for the 3d Navier-Stokes equation when the initial vorticity is supported on a circle. Such initial datum is in certain function spaces where perturbation theory works for small data, but not for large data, even for short times, and there are good reasons to believe that this is not just a technicality. We prove global existence and uniqueness for large data in the class of axi-symmetric solutions. The main tools are Nash-type estimates and certain monotone quantities. Uniqueness in the class of solutions which are not necessarily axi-symmetric remains a difficult open problem, which we plan to discuss briefly. Joint work with Thierry Gallay.
We discuss random walks on groups acting on hyperbolic spaces (e.g. the Poincaré disk), and their limiting behaviour on the boundary. The limiting distribution of the random walk (the harmonic measure) is of particular interest for description of bounded harmonic functions on the group (the Poisson boundary). We consider the Hausdorff dimension of harmonic measure on the boundary and give a formula in terms of the entropy and the drift under a general moment condition. Related recent results are also discussed during the talk.
Using some ideas of Atiyah and Segal and a pushforward map defined using deformation groupoids we explain how to endow to the twisted geometric K-homology groups of a discrete group with an external product. Using the Baum-Connes assembly maps one can transfer this product to the twisted K-theory groups of the reduced group C*-algebra. This is a joint work with Noé Barcenas and Paulo Carrillo.
We explain some of the recent results in concerning PDEs describing fluid flows, as well as some of the difficulties. Model equations will also be discussed.
I will report on joint work with D. Carchedi, S. Scherotzke and N. Sibilla, about a comparison between two objects obtained from a fs log scheme over the complex numbers: the "infinite root stack" and the "Kato-Nakayama space". I will also hint at more recent work that explains how parabolic sheaves (with real or rational weights) interact with the picture.
I will be as little technical as possible and focus on examples rather than on the general theory.
One of the distinguishing features of many living systems is their ability to move, to self-propel, to be active. Through their motion, either voluntarily or involuntarily, living systems are able self-assemble: birds flock, fish school, bacteria swarm, etc. But such behavior is not limited to living systems. Recent advances in colloid chemistry have led to the development of synthetic, nonliving particles that are able to undergo autonomous motion by converting chemical energy into mechanical motion and work – chemical swimming. This swimming or intrinsic activity imparts new behaviors to active matter that distinguish it from equilibrium condensed matter systems. For example, active matter generates its own internal pressure (or stress), which can drive it far from equilibrium and free it from conventional thermodynamic constraints, and by so doing active matter can control and direct its own behavior and that of its surroundings. In this talk I will discuss our recent work on swimmers and on the origin of a new source for stress that is responsible for self-assembly and pattern formation in active matter.
Let $m$ be a radial multiplier supported in a compact subset away from the origin. For dimensions $d\ge 2$, it is conjectured that the multiplier operator $T_m$ is bounded on $L^p(R^d)$ if and only if the kernel $K=\hat{m}$ is in $L^p(R^d)$, for the range $1<p<2d/(d+1)$. Note that there are no a priori assumptions on the regularity of the multiplier. This conjecture belongs near the top of the tree of a number of important related conjectures in harmonic analysis, including the Local Smoothing, Bochner-Riesz, Restriction, and Kakeya conjectures. We discuss new progress on this conjecture in dimensions $d=3$ and $d=4$. Our method of proof will rely on a geometric argument involving sizes of multiple intersections of three-dimensional annuli.
In this talk we will present recent local and global well-posedness results on the one dimensional periodic fractional Schrodinger equation. We will also talk about construction of Gibbs' measures on certain Sobolev spaces and how we can prove almost sure global well-posedness using this construction.
Schur functions form an important basis for the space of symmetric functions and show up in areas from representation theory to quantum mechanics. Given an appropriate diagram of boxes, we construct its corresponding Schur function by counting the numbers of tableaux: fillings of these boxes with positive integers that satisfy some simple conditions. We then form the Schur-positivity partially ordered set by comparing these numbers of tableaux. In this talk, we present some new results of how order relations in this partially ordered set can be derived from properties of the diagrams. We then present some progress toward long-standing conjectures.
The success of probability theory in the analysis of linear, or even certain semi-linear, parabolic and elliptic pde’s is well documented. In spite of various attempts to find a stochastic foothold for the analysis of Navier-Stokes equations and related quasilinear equations, the problem remains a substantial challenge. That said, the quest can lead to new stochastic structures and problems that relate to modern probability in fundamental ways. In this talk I will try to indicate this with a few explicit examples largely stemming from the Lejan-Sznitman multiplicative cascade/branching random walk framework for Navier-Stokes equations.
This talk is primarily based on recent joint work with Radu Dascaliuc, Nicholas Michalowski, and Enrique Thomann with partial support from the National Science Foundation.
Mixing Voevodsky's filtration in motivic homotopy and Dugger's in C_2-equivariant homotopy theory leads to an interesting filtration on the C_2-equivariant motivic homotopy category. In this talk, I'll introduce these slice filtrations and talk about some joint work with P. A. Ostvaer, where we compute the resulting zero slice of the equivariant motivic sphere spectrum.
The Langlands program, initiated in the 1960s, is a set of conjectures predicting a unification of number theory and the representation theory of groups. More precisely, the Langlands correspondence provides a way to interpret results in number theory in terms of group theory, and vice versa.
In this talk we sketch a few aspects of the local Langlands correspondence using elementary examples. We then comment on some questions raised by the emerging "mod p" Langlands program.
In this talk, I will present a CR-version of Lichnerowicz--Obata type theorem in a closed pseudo-Hermitian CR manifolds. It includes the lower bound estimates for the first positive eigenvalue for the both sub-Laplacian and Kohn Laplacian. I will also provide Obata type theorem associated to the sub-Laplacian and Kohn Laplacian on a closed pseudo-Hermitian manifold. As an application, we give some rigidity theorem when lower bound of eigenvalue is achieved. This is based on a joint work with X. Wang and a joint work with Duong N. Son and Wang. I will also talk about some ongoing work in this topic.
Given two matrices A,B we say that A is a Berge hypergraph of B if there is a submatrix of B, say matrix D, and a row and column permutation of A, say matrix C, so that C<=D. Define Av(m,F) to be the set of all m-rowed (0,1)-matrices with no repeated columns and no Berge hypergraph F. Define Bh(m,F) to be the maximum, over all matrices A in Av(m,F), of the number of columns of A. We are interested in determining the asymptotic growth of Bh(m,F) for specific F. We show some techniques we can use to this end and mention the general results determined for F with 5 or fewer rows. We also show that if F is the vertex-edge incidence matrix of a tree then bh(m,F) has a linear bound. When F is the vertex-edge (s+t)x(st) incidence matrix of the bipartite graph K_{s,t} we show that finding Bh(m,F) relates to determining ex(m,K_r,K_{s,t} ), the maximum number of complete subgraphs K_r in a m-vertex graph avoiding K_{s,t} as a subgraph. Recent papers by Alon and others have solved some cases.
A deterministic model may sometimes seem to be a good description of the dynamics of an observed system but may have a long-term stable constant limit, whereas observations of the system itself show a noisy pattern. An example is semi-arid vegetation patterns. Adding noise to the model may well reveal the pattern. In this talk I show some photos, talk about some math, and show some simulations. This is not a magic show.
Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n – 3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). See http://www.math.cmu.edu/~ctsourak/ran.html for an example.
We prove that the diameter of a RAN is asymptotic to c log(n) in probability, where c ≈ 1.668 is the solution of an explicit equation. The proof adapts a technique of Broutin and Devroye for estimating the height of random trees.
We also prove that there exists a fixed s<1, such that eventually every self-avoiding walk in this graph has length less than n^s, which verifies a conjecture of Cooper and Frieze. Using a similar technique, we show that if r < d are fixed constants, then every r-ary subtree of a random d-ary recursive tree on n vertices has less than n^b vertices, for some b=b(d,r)<1.
Based on joint work with A. Collevecchio, E. Ebrahimzadeh, L. Farczadi, P. Gao, C. Sato, N. Wormald, and J. Zung.
We will begin by briefly discussing the non-linear Schroedinger (NLS) equation and the corresponding classical Strichartz estimates. We will then introduce a so-called Wiener randomization of initial data and indicate how it leads to an improvement of the classical Strichartz estimates. As a toy application, we will show how, in contrast with the deterministic case, the energy-critical cubic NLS in four dimensions is almost surely well-posed with respect to randomized initial data below the energy space. This is a joint work with Tadahiro Oh (University of Edinburgh) and Oana Pocovnicu (Heriot-Watt University).
In this talk, we will discuss a new proof of the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber for semi-simple perverse sheaves of geometric origin on complex algebraic varieties. This proof follows from rather formal considerations of higher algebra, stable motivic homotopy theory and Grothendieck's six functors, avoiding both the positive-characteristic methods of the original proof, and the delicate analysis of degenerations of mixed Hodge structures involved in M. Saito's proof.
The Cahn-Hilliard equation models the motion of interfaces between several phases. The underlying energy functional includes a potential for which different types were proposed in the literature. We consider smooth and nonsmooth potentials with a focus on the latter. In the nonsmooth case, we apply a function space-based algorithm, which combines a Moreau-Yosida regularization technique with a semismooth Newton method. We apply classical finite element methods to discretize the problems in space. At the heart of our method lies the solution of large and sparse fully discrete systems of linear equations. Block preconditioners using effective Schur complement approximations are presented. For the smooth systems, we derive optimal preconditioners, which are proven to be robust with respect to crucial model parameters. Further, we prove that the use of the same preconditioners give poor approximations for the nonsmooth formulations. The preconditioners we present for the nonsmooth problems incorporate the regularization terms. Extensive numerical experiments show an outstanding behavior of our developed preconditioners. Our strategy applies to different Cahn-Hilliard problems including phase separation and coarsening processes, image inpainting, and two-phase flows.
The Teichm\"uller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. It was introduced by M. Rupflin and P. Topping in 2012. The objective of the flow is to find branched minimal immersions on a given surface. We will give some background on the flow and then describe some recent work. In particular we show that if the flow exists for all times then in a certain sense the maps (sub-)converge to a collection of branched minimal immersions with no loss of energy (even when allowing for degeneration of the metric at infinity). We also construct an example of a smooth flow where the image of the limit maps is disconnected. This is joint work with M. Rupflin and P. Topping.
The Volumes of voronoi cells in random trees have a surprisingly simple law. I will present a proof of this and related conjectures about the Brownian sphere. Joint with Louigi Addario-Berry, Christina Goldschmidt, Guillaume Chapuy, and Eric Fusy.
Carlsson conjectured that if a finite complex admits a free action by an elementary abelian p-group of rank n, then the sum of its mod-p Betti numbers is at least 2^n. For the prime p=2, he reduced the conjecture to an algebraic problem which he solved for low n. In this talk, we will retrace Carlsson's journey through homological and commutative algebra. The following week, I will report on joint work in progress with Jeremiah Heller with the goal of extending Carlsson's methods to all primes.
The end of UBC's current license with Blackboard Learn (badged locally as Connect) is approaching, and we need to assess future options for the application that sits at the core of our Learning Technology ecosystem.
We understand that the technology provided by Connect has not been satisfactory for faculty in the Mathematics Department. We'd like to hear about your needs, in order that these might inform the selection of the new LMS tool.
We will start by introducing Langlands parameters and their enhanced versions. Then we will explain how to mirror the expected classification of irreducible smooth representations of p-adic groups into a classification of corresponding enhanced Langlands parameters. We will illustrate our construction with simple examples. It is joint work with Ahmed Moussaoui and Maarten Solleveld.
The Murnaghan-Nakayama rule expresses the product of a
Schur function with a Newton power sum in the basis of Schur
functions. As the power sums generate the algebra of
symmetric functions, the Murnaghan-Nakayama rule is as
fundamental as the Pieri rule. Interesting, the resulting
formulas from the Murnaghan-Nakayama rule are significantly
more compact than those from the Pieri formula. In
geometry, a Murnaghan-Nakayama formula computes the
intersection of Schubert cycles with tautological classes
coming from the Chern character.
In this talk, I will discuss some background, and then some
recent work with Andrew Morrison establishing
Murnaghan-Nakayama rules for Schubert polynomials and for
the quantum cohomology of the Grassmannian. The results I
discuss are contained in the preprint arXiv:1507.06569.
I will survey the equidistribution problem for eigenfunctions of the Laplace operator, especially from the point of view of eigenfunction bounds. For the case of locally symmetric spaces I will discuss positive results for exact eigenfunctions (especially joint work with Anantharaman) and negative results for approximate eigenfunctions (joint work with Eswarathasan).
Note for Attendees
Refreshments will be served in MATH 125 at 2:45pm before the colloquium.
Motivated by algebraic geometry, one studies non-commutative analogs of resolutions of singularities. In short, a non-commutative resolution (=NCR) of a commutative ring R is an endomorphism ring of a certain R-module of finite global dimension. However, it is not clear how to construct non-commutative resolutions in general and which structure they have. The most prominent example of NCRs comes from the classical McKay correspondence that relates the geometry of so-called Kleinian surface singularities with the representation theory of finite subgroups of SL(2,\mathbb{C}).
In this talk we will first review this fascinating result, exhibiting the connection to the ubiquitious Coxeter-Dynkin diagrams. Moreover, we will comment on an algebraic version of the correspondence, due to Maurice Auslander.
This leads to joint work in progress with Ragnar Buchweitz and Colin Ingalls about a version of the McKay correspondence when G in GL(n,\mathbb{C}) is a finite group generated by reflections: The group G acts linearly on the polynomial ring S in n variables over \mathbb{C}. When G is generated by reflections, then the discriminant D of the group action of G on S is a hypersurface with a singular locus of codimension 1. We give a natural construction of a NCR of the coordinate ring of D as a quotient of the skew group ring A=S*G. We will explain this construction, which gives a new view on Knörrer's periodicity theorem for matrix factorizations and allows to extend Auslander's theorem to reflection groups.
"Which numbers (and in particular, which primes) are sums of two rational cubes" is a classical and still-not-entirely-solved Diophantine problem. I'll talk about how it turns into a problem about rational points on elliptic curves, and how it can then be attacked using the modern machinery of the arithmetic of elliptic curves. In particular, proving that certain primes can be written as a sum of two cubes can be accomplished by constructing a Heegner-type point and proving it's nonzero. This is a subtle question and has been carried out in different ways by Elkies and by Dasgupta-Voight. My work (in process) gives a new method to carry this out, based on a new construction I've given of a certain type of anticyclotomic p-adic L-function.
Following the new format for the number theory seminar, this talk will consist of two 45-minute parts. The first 45 minutes will be expository and is intended to be accessible for graduate students in number theory. There will be a short break (when people are welcome to leave), and the second 45 minutes will be at a higher level.
We investigate the existence of solutions to a nonlinear elliptic problem involving the critical Sobolev exponent for a Polyharmomic operator on a Riemannian manifold M. We first show that the best constant of the Sobolev embedding on a manifold can be chosen as close as one wants to the Euclidean one, and as a consequence derive the existence of minimizers when the energy functional goes below a quantified threshold. Next, higher energy solutions are obtained by Coron's topological method, provided that the minimizing solution does not exist and the manifold satisfies a certain topological assumption. To perform the topological argument, we obtain a decomposition of Palais-Smale sequences as a sum of bubbles and adapt Lions's concentration-compactness lemma.
Supercharacter theory, a generalization of character theory, was developed originally by Carlos Andre and then picked up and studied more extensively by Persi Diaconis and I. M. Isaacs. This new development in finite group representation theory has spurred much exciting work, leading to a range of results in algebraic combinatorics and number theory. I will begin at the (very) beginning, so no prior experience in supercharacter theory is required.
Individual cells regulate their morphology in response to environmental cues, selective pressures and signalling. The precise mechanism(s) through which cells control their shape is not well understood. Studies have shown that cell morphology has important implications for nutrient uptake, motility, proliferation, etc. For example, a change in bacterial cell diameter of 0.2 μm can change the energy required for chemotaxis by a factor of 10^5. Automatic classification and counting can facilitate a systematic investigation of cell morphology. Furthermore, it is a useful tool for diagnosis of diseases like leukemia that are characterized by cell shape deformation.
In this talk, I will describe techniques for image segmentation and feature extraction that we use to build a descriptor of cell shape. This descriptor is used to classify cells using unsupervised learning (PCA, hierarchical clustering) methods. I will briefly discuss the advantages and limitations of supervised learning (deep neural networks, convolutional neural networks) methods.
I will talk about an intrinsic approach to uniformization of a graph's geometry based on extremal discrete metrics. This method allows one to generalize known results about planar graphs that rely heavily on the theory of circle packings, and to obtain new information even for well-studied models of planar graphs like the uniform infinite triangulation (UIPT) and quadrangulation (UIPQ).
It yields a short proof of Benjamini and Schramm's result that a distributional limit of bounded-degree planar graphs is almost surely recurrent. The same argument resolves a 2001 conjecture of those authors since it works also for H-minor-free graphs (and, in fact, a substantial generalization known as region intersection graphs).
Gurel-Gurevich and Nachmias recently solved a central open problem by showing that UIPT and UIPQ are almost surely recurrent. By combining extremal discrete metrics with methods from spectral geometry, I will present a new proof of this fact that also gives explicit control on divergence of the Green function. If g(x, T) denotes the number of returns to the root at time T, we show that almost surely (over the choice of the random rooted graph), g(x,T) grows asymptotically at least as fast as log log T. This is a consequence of a general result that holds for any distributional limit of H-minor-free graphs and provides lower bounds on return probabilities based on the tails of the degree distribution of the root.
Finally, I will discuss extensions to general graphs and, in particular, give a characterization of the almost sure spectral dimension of a distributional limit in terms of the "conformal growth rate" of any sequence that approaches the limit graph. This has consequences for graphs that can be sphere-packed in R^d for d > 2.
Carlsson conjectured that if a finite complex admits a free action by an elementary abelian p-group of rank n, then the sum of its mod-p Betti numbers is at least 2^n. For the prime p=2, he reduced the conjecture to an algebraic problem which he solved for low n. In this talk, I will report on joint work in progress with Jeremiah Heller with the goal of extending Carlsson's methods to all primes. The crucial ingredient is a new notion of Koszul p-complexes.
Particle-laden flows are ubiquitous in environmental, geophysical and engineering processes. The intricate dynamics of these two-phase flows is governed by momentum transfer between the continuous fluid phase and the dispersed particulate phase. In this talk, I will suggest a multi-scale modelling framework for particle-laden flows and address mathematical and computational issues related to the numerical simulation of this type of flow. Based on various simulation results, I will present what can already be achieved with existing numerical models as well as the next stages in the development of faster and more accurate solution methods as e.g. fast Navier-Stokes solvers on cartesian grids, Adaptive Mesh Refinement or highly scalable numerical tools.
Note for Attendees
Refreshments will be served in MATH 125 at 2:45pm before the colloquium.
In 2007 G.A.Karagulyan proved that the maximal directional Hilbert transform over a set U of directions in the plane cannot be extended to a (2,q)-strongly bounded operator for any q in the range 1 ≤ q < +∞. In this talk, I will describe my proof of the generalization of this result and show that, for any 1 < p ≤ 2, any 1 ≤ q < +∞ and any dimension n ≥ 2, the maximal directional Hilbert transform over an infinite set of directions in IR^n, cannot be extended to a (p,q)-strongly bounded operator.
I will review the notion of a t-structure and discuss some recent uses of t-structures on categories of coherent sheaves in (geometric) representation theory. After reviewing some traditional methods to obtain t-structures I will present a new construction that uses categorical Lie actions. As an application one recovers the category of "exotic sheaves", used in a recent proof of Lusztig's conjectures on a canonical bases for the Grothendieck group of Springer fibers by Bezrukavnikov and Mirković. The new construction is purely geometric, instead of using deep results from modular representation theory. This is joint work with Sabin Cautis.
Solving Maxwell's equations for earth science applications requires the discretization of large domains with sufficiently small mesh to capture local conductivity variation. Multiscale methods are discretization techniques that allow to use a coarse mesh and still obtain accuracy that is obtained through finer meshes. However, when considering the multiscale solution of vector equations, basic operator properties are not conserved. In this talk we will show how to extend multiscale methods for vector quantities and demonstrate their use for Maxwell's equations.
Abstract: We will discuss two types of curvature flows designed to prove isoperimetric type inequalities. The first one is a mean curvature type flow, it was introduced in a previous joint work with Junfang Li in space forms. In a recent joint paper with Junfang Li and Mu-Tao Wang, we consider a similar normalized hypersurface flow in the more general ambient setting of warped product spaces with general base. Under a natural necessary condition, the flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. Under another condition with is related to the notion of “photon sphere” in general relativity, we establish the regularity and convergence of the flow, thereby solve the isoperimetric problem in warped product spaces. In a similar spirit, I will discuss a inverse mean curvature type flow in hyperbolic space to deal with Alexandrov-Fenchel type isoperimetric inequalities.
Lattice theory is the study of an intriguing class of partially ordered sets known as lattices. Within this subject, there are many hard open problems. One such problem is to determine which countable lattices are sublattices of free lattices. In this talk, we will describe past progress on this problem and propose possible methods of attack using finite and finite width lattices. This research was done while I was a masters student at the University of Calgary.
We consider models of gradient type, which is the density of a collection of real-valued random variables \phi :=\{\phi_x: x \in \Lambda\} given by Z^{-1}\exp({-\sum\nolimits_{j \sim k}V(\phi_j-\phi_k)}). We focus our study on the case that V(\nabla\phi) = [1+(\nabla\phi)^2]^\alpha with 0 < \alpha < 1/2, which is a non-convex potential. We introduce an auxiliary field t_{jk} for each edge and represent the model as the marginal of a model with log-cancave density. Based on this method, we prove that finite moments of the fields \left<[v \cdot \phi]^p \right> are bounded uniformly in the volume. This leads to the existence of infinite volume measures. Also, every ergodic infinite volume Gibbs measure with mean zero for the potential V above scales to a Gaussian free field.
For this survey talk I will bring a bestiary of algebraic structures that are often less well known than they deserve to be. As I will explain, these are all related to symmetries and topology, and they have interesting symmetries of their own. This observation leads to new homology computations, and those will be the results that I will present along the way. Part of this work is joint with N. Wahl.
In 1969, Sato and Hörmander introduced the notion of wave front set of a distribution in the real context. This concept gives a better understanding of operations on distributions such as product or pullback and it plays an important role in the theory of partial differential equations. In 1981, Howe introduced a notion of wave front set for some Lie group representations and in 1985, Heifetz gave an analogous version in the p-adic context. In this talk, in the t-adic context in characteristic zero, using Cluckers-Loeser motivic integration we will present analogous constructions of test functions, distributions and wave front sets. In particular, we will explain how definability can be used as a substitute for topological compactness of the sphere in the real and p-adic contexts to obtain finiteness. This a joint work with R. Cluckers, and F. Loeser.
The "nonpositive immersion" property is a condition on a 2-complex X that generalizes being a surface. When X has this property, itsfundamental group appears to have has some very nice properties which I will discuss. I will spend the remainder of the talk outlining a proof that the nonpositive immersion property holds for a 2-complex obtained by attaching a single 2-cell to a graph. This was proven recently with Joseph Helfer and also independently by Lars Louder and Henry Wilton.
Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that have recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.
Note for Attendees
Refreshments (A light reception) are served in ESB 4133 from 2:30pm-3:00pm before the colloquium.
The minimal multiple of the diagonal to admit a decomposition in the sense of Bloch and Srinivas is called the torsion order of a smooth projective variety. It is bounded above by the greatest common divisor of the degrees of all unirational parameterizations, and is a stable birational invariant. Recently, a degeneration method initiated by Voisin, and developed by Colliot-Thélčne and Pirutka, has led to a breakthrough in establishing lower bounds for the torsion order, hence obstructions to stable rationality. The power of this method lies in its mix of inputs from algebraic cycles, Hodge theory, algebraic K-theory, birational geometry, and singularity theory. I will survey the state of the art of this theory, which includes recent work of Chatzistamatiou and Levine, as well as provide some new examples.
A Besicovich set is a compact subset of R^d that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that every Besicovich set in R^d must have dimension d. I will discuss some new trilinear Kakeya-type bounds in R^4, and how these bounds can be used to obtain improved bounds on the dimension of certain sets in R^4 that resemble Kakeya sets. This is joint work with Larry Guth.
We give a new definition of the derived Maurer-Cartan locus MC^*(L), as a functor from differential graded Lie algebras to cosimplicial schemes, whose definition is sufficiently straightforward that it generalizes well to other settings such as analytic geometry. If L is differential graded Lie algebra, let L_+ be the truncation of L in positive degrees i>0. We prove that the differential graded algebra of functions on the cosimplicial scheme MC^*(L) is quasi-isomorphic to the Chevalley-Eilenberg complex of L_+, which is the usual definition of the derived Maurer-Cartan locus in characteristic zero.
Mixed-integer convex optimization problems are convex problems with the additional (non-convex) constraints that some variables may take only integer values. Despite the past decades' advances in algorithms and technology for both mixed-integer *linear* and *continuous, convex* optimization, mixed-integer convex optimization problems have remained relatively more challenging and less widely used in practice. In this talk, we describe our recent algorithmic work on mixed-integer convex optimization which has yielded advances over the state of the art, including the globally optimal solution of open benchmark problems. Based on our developments, we have released Pajarito, an open-source solver written in Julia and accessible from popular optimization modeling frameworks. Pajarito is immediately useful for solving challenging mixed combinatorial continuous problems arising from engineering and statistical applications.
There is a well-known technique due to Schoen-Yau from the late 70s which uses (stable) minimal hypersurfaces to find topological implications of a (closed) manifold's ability to support positive scalar curvature metrics. In this talk, we describe a version of this technique for manifolds with boundary and discuss how it can be used to study bordisms of positive scalar curvature metrics.
In the last few years there has been significant advancement in understanding the possible range of behaviors of the volume growth and of the entropy and rate of escape of random walks on groups. Bartholdi and Erschler constructed the first family of intermediate growth groups whose volume growth function follows any prescribed nice enough function in the exponent range [\alpha_0,1] for some explicit \alpha_0 \approx 0.7674. We discuss a variant of a construction of Kassabov and Pak which provides an alternative proof of the Bartholdi-Erschler result. Different behaviors of entropy of random walks on these two families of groups allow us to deduce a result concerning possible joint behavior of intermediate volume growth and entropy of random walks within a certain range of parameters. Joint with Gidi Amir.
In this talk I shall discuss the results which were obtained by several people: M.Avdeeva, F.Cellarosi, Dong Li and myself. Moebius function is one of the most important functions in number theory also connected with Riemann hypothesis. Its simplest part leads to the well-known in probability theory Dickman-de Bruijn distribution. The so-called Mirsky formulas allow to construct the dynamical system with pure point spectrum corresponding to the square of the Moebius function. In the last part of the talk the famous Hall theorem about the variance of the number of square-free numbers will be discussed.
Note for Attendees
Refreshments are served in ESB 4133 from 2:30pm-3:00pm before the colloquium.
We will not assume any prior knowledge of stacks for this talk. Toric stacks, like toric varieties, form a concrete class of examples which are particularly amenable to computation. We give an introduction to the subject and explain how we have used toric stacks to obtain an unexpected result in cycle theory. We end the talk by discussing some conjectures recently introduced by myself and Dan Edidin.
The space of smoothly embedded n-spheres in Rn+1 is the quotient space Mn:=Emb(Sn,Rn+1)/Diff(Sn). In 1959 Smale proved that M1 is contractible and conjectured that M2 is contractible as well, a fact that was proved by Hatcher in 1983.
While it is known that not all Mn are contractible, for n\get 3 no single homotopy group of Mn is known. Even knowing whether the Mn are path connected or not would be extremely interesting. For instance, if M3 is not path connected, the 4-d smooth Poincare conjecture can not hold true.
In this talk, I will first explain how mean curvature flow can assist in studying the topology of geometric relatives of Mn. I will first illustrate how the theory of 1-d mean curvature flow (aka curve shortening flow) yields a very simple proof of Smale's theorem about the contractibility of M1. I will then describe a recent joint work with Reto Buzzno and Robert Haslhofer, utilizing mean curvature flow with surgery to prove that the space of 2-convex embedded spheres is path connected.
We study the following game on a finite graph G = (V, E). At the start, each edge is assigned an integer n_e \ge 0, n = \sum_{e \in E} n_e. In round t, 1 \le t \le n, a uniformly random vertex v \in V is chosen and one of the edges f incident with v is selected by the player. The value assigned to f is then decreased by 1. The player wins, if the configuration (0, \dots, 0) is reached; in other words, the edge values never go negative. Our main result is that there is a phase transition: as n \to \infty , the probability that the player wins approaches a constant c_G > 0 when (n_e/n : e \in E) converges to a point in the interior of a certain convex set \mathcal{R}_G, and goes to 0 exponentially when (n_e/n : e \in E) is bounded away from \mathcal{R}_G. We also obtain upper bounds in the near-critical region, that is when (n_e/n : e \in E) lies close to \partial\mathcal{R}_G. We supply quantitative error bounds in our arguments.
Given a fiber sequence of n-fold loop spaces X-->Y-->Z, and morphism of n-fold loop spaces Y-->Pic(R) for R an E_{n+1}-ring spectrum, we describe a method of producing a new morphism of (n-1)-fold loop spaces Z-->Pic(MX), where MX is the Thom spectrum associated to the composition X-->Y-->Pic(R). This new morphism has associated Thom spectrum MY, but constructed directly as an MX-module. In particular this induces a relative Thom isomorphism for MY over MX: MY⊗_{MX} MY = MY⊗Z. We will see a rough description of this construction as well as many examples allowing us to find equivalent forms of relative smash products of spectra like MString, MSpin, HZ/2, X(n) and many others. We also describe a way to use this construction to identify certain obstructions to giving a complex orientation on an associative ring spectrum.
Associated to a fibration E --> B with homotopy finite fiber is a stable wrong way map LB --> LE of free loop spaces coming from the transfer map in THH. This transfer is defined under the same hypotheses as the Becker-Gottlieb transfer, but on different objects. I will use duality in bicategories to explain why the THH transfer contains the Becker-Gottlieb transfer as a direct summand. The corresponding result for the A-theory transfer may then be deduced as a corollary. When the fibration is a smooth fiber bundle, the same methods give a three step description of the THH transfer in terms of explicit geometry over the free loop space. (Joint work with C. Malkiewich)
Let K/\mathbb{Q} be a number field with ring of integers \mathbb{Z}_K. If K has class number one, the set of irreducible elements of \mathbb{Z}_K coincides with the set of prime elements; in general, this need not be the case. One is led to wonder: Do statements about primes in \mathbb{Z} have analogues for irreducibles in \mathbb{Z}_K, for a general choice of K? This talk concerns two instances where the answer is yes. We will discuss the maximal order of the number of irreducible divisors of an element of \mathbb{Z}_K, and we will provide an asymptotic formula for the number of irreducible elements of norm up to x belonging to a given arithmetic progression.
It has long been known that 3D time-dependent Navier-Stokes equations for incompressible fluids admit the classical conservation laws (CL) of mass, momentum, angular momentum and centre-of-mass theorem. For inviscid flows, i.e. Eulers equation, this is extended by the conservation of helicity and energy. Employing the “direct method” (DM) by Anco, Bluman (1997) it has been shown that this set of conservation laws is complete for primitive variables. The DM is a substantial generalization of Noethers theorem and does not rely on a variational principle, and, further, is directly applicable to any type of differential equation, even dissipative ones. With this an additional infinite set of CL for Navier-Stokes equations in vorticity formulation are derived. Various examples are shown.
Interesting enough, even more CLs exist for Euler and Navier-Stokes equations in spatially reduced coordinate systems such as for plane, axisymmetric and helically symmetric flows. E.g. an infinite set of CLs for the generalization of helicity has been derived, and, surprisingly, even new CL for plane flows haven been identified.
In the theory of automorphic representations the study of L-functions plays a key role. A common method to study the analytic behavior of such functions (and, in fact, proving that they are meromorphic functions) is the Rankin-Selberg method. In this method an integral representation, with good analytic properties, is attached to the L-function. Many examples of Rankin-Selberg integrals were studied along the years. However most examples rely on the uniqueness of certain models of the representation (most popular in use is the Whittaker model but many other, such as the Peterson bilinear form and Bessel model, are used). In a pioneering paper ("A new-way to get Euler products", Krelle, 1988) I. Piatetski-Shapiro and S. Rallis suggested a remarkable mechanism that makes it possible to use integrals containing a "non-unique model" by a slight strengthening of the unramified computations.
In the first part of my talk we will have a crash-course on cuspidal automorphic representations and the new-way mechanism via the classical example of Hecke's integral representation for L-functions of cuspidal representations of GL_2.
In the second part of my talk I will present a joint work with N. Gurevich in which we proved that a family of Rankin-Selberg integrals representing the standard twisted L-function of a cuspidal representation of the exceptional group of type G_2. In its unfolded form (a term which will be explained in the talk), the integrals contain a non-unique model and we apply the new-way mechanism. The unramified computation gives rise to two interesting objects: the generating function of the L-function and its approximations. If time permits, I will discuss the possible poles of this L-function and some applications to the theory of cuspidal representations of G_2.
The development of the new discontinuous Galerkin (DG) framework BoSSS (bounded support spectral solver) starting in 2007. Solvers for incompressible as well as compressible single and multi-phase flows were implemented.
The code features a modern object-oriented design and is of course MPI-parallel. Within the development cycle, we use unit-testing to ensure software quality: this covers a wide range of tests, form very simple ones that test e.g. accuracy of implemented quadrature rules or the derivatives of a scalar field to complex MPI-parallel Navier-Stokes simulations and convergence tests. All these checks and tests are automatically executed by a dedicated server, whenever a developer commits changes to the GIT-repository.
BoSSS supports arbitrary partitioning of the grid, on an arbitrary number of compute-nodes at start-up time, which is an important perquisite for the adaptive meshing which is addressed by this proposal. Quite recently, support for mixed meshes (combining e.g. triangles and quads) and hanging nodes has been added.
The most outstanding feature of the code is the support for multiphase-flows and immersed boundary methods. For both of these applications, the interface is described by a level-set-method. The novelty is that we can demonstrate arbitrarily high spectral convergence in the presence of curved interfaces. This becomes possible due to a novel numerical integration technique for implicitly defined surfaces, the so-called hierarchical moment fitting (HMF) technique.
We will discuss Liouville quantum gravity as a scaling limit theory for random planar maps. In particular, we will focus on a recent approach called peanosphere or mating of trees and provide several applications of this framework.
I will describe lower bounds (i.e. constructions) for several hypergraph Ramsey problems. These constructions settle old conjectures of Erdos-Hajnal on classical Ramsey numbers as well as more recent questions due to Conlon-Fox-Lee-Sudakov and others on generalized Ramsey numbers and the Erdos-Rogers problem. Most of this is joint work with Andrew Suk.
I will discuss a method of coding of geodesics on quotients of the hyperbolic plane by Fuchsian groups using boundary maps and “reduction theory”. These maps are piecewise fractional-linear given by generators of the Fuchsian group, and the orbit of a point under the boundary map defines its boundary expansion. For compact surfaces they are generalizations of the Bowen-Series map, and for the modular surface are related to a family of (a,b)-continued fractions. For the natural extensions of the boundary maps Zagier’s Reduction Theory Conjecture (RTC) holds: for the appropriate open sets of parameters they have attractors with finite rectangular structure to which (almost) every point is mapped after finitely many iterations. I will also explain how the RTC is used for representing the geodesic flow as a special flow over a cross-section of “reduced” geodesics parametrized by the attractor. This was proved for the modular group and generalizes for Fuchsian groups that satisfy the RTC. The talk is based on joint works with Ilie Ugarcovici.
Note for Attendees
Refreshments will be served at 2:45pm in MATH 126.
Our talk investigates the subtle relationship between the size of the level sets of the (bilinear) Kakeya function and the corresponding geometric distribution of the points within these level sets. Under suitable conditions, our goal is to characterize the situation in which the size of these level sets is maximal and thus to provide qualitative and quantitative information about the extremizers associated with the (bilinear) Kakeya function.
Our analysis will involve additive combinatorics (e.g. Plünnecke sum-set estimate) and incidence geometry (e.g. Szemeredi-Trotter) techniques and relates with a class of problems including Bourgain's sum-product theorem and Katz-Tao ring conjecture.
1. Let A be a 2x2 matrix over any universal field with any characteristic and An be any power of A. Our main proposition describes each component of An, the n‐th power, in terms of eigenvalues of A and values of well known cyclotomic rational function F(n,X,Y) at those eigenvalues. When A={0,1;1,1}, the case of Fibonacci, X is the golden number 1.61803... and Y=‐0.61803..., and F(n) is the well known sequence.
2. We apply algebraic results in Part 1 to the case of real quadratic fields and study the action of the group of units on the ring of integers of the quadratic fields. We will introduce a family of infinitely many real quadratic fields parametrised by odd integers q. In the classical case q=1.
Biography:
Professor Takashi Ono is a world‐renowned and highly accomplished mathematician, specializing in number theory and algebraic groups. He has made major contributions to the field of number theory, and his work is considered a cornerstone of arithmetic algebraic geometry. Having received his PhD from Nagoya University in 1958 under the supervision of Shokichi Iyanaga, Takashi Ono moved to the Institute of Advanced Study in Princeton, New Jersey, where he held a fellowship from 1959‐1961 at the invitation of J. Robert Oppenheimer. He then spent three years working as a mathematics professor at the University of British Columbia, followed by five years at the University of Pennsylvania. He then moved to Johns Hopkins University in 1969 and remained there until his retirement in 2011. Professor Ono has received a number of honours, including an invitation to speak at the International Congress of Mathematicians in Moscow 1966 and election as a Fellow of the American Mathematical Society in 2012. His mathematical contributions are complemented by the talent and hard work of his family: his eldest son is the musician Momoro Ono, his youngest son is the mathematician Ken Ono and his middle son is Santa Ono, President and Vice‐Chancellor of the University of British Columbia.
By physical considerations Huang, Katz and Klemm conjecture that the generating series of Donaldson-Thomas invariants of an elliptic Calabi-Yau threefold is a Jacobi form. In this talk I will explain a mathematical approach to proving part of their conjecture. The method uses an autoequivalence of the derived category, and wallcrossing techniques developed by Toda. This leads to strong structure results for curve counting invariants. As a leading example we will discuss the elliptic fibration over P2 in degree 1.
We study the computational hardness of problems whose inputs are obtained by applying random noise to worst-case instances. For an appropriate notion of noise we show that a number of classical NP-hard problems on graphs remain essentially as hard on the noisy instances as they are in the worst-case.
Focusing on the Graph Coloring problem, we establish the following result: Given any graph G, let H be a random subgraph of G obtained by deleting the edges of G independently with probability 0.5. We show that if \chi(G) is large, then \chi(H) is also large with high probability. This means that the chromatic number of any graph is ``robust'' to random edge deletions.
We extend several results in the potential theory of random walk to critical branching random walk. In the supercritical dimensions (d\geq 5), we introduce branching capacity for any finite subset of \Z^d and establish its connections with the hitting probability by branching random walk and branching recurrence. In the critical dimension (d=4), we also establish the asymptotics of the hitting probability and some related results.
An ordered group (G,<) is a group G together with a strict total ordering < of its elements which is invariant under left- and right-multiplication. If such an ordering exists for a group, the group is said to be orderable. It is easy to see that if G and H are orderable, then so is their direct product. In 1949, A. A. Vinogradov proved that if G and H are orderable groups, then the free product G*H is also orderable. I’ll show that such an ordering can be constructed in a functorial manner, in the category of ordered groups and order-preserving homomorphisms, using an algebraic trick due to G. Bergman. This was motivated by a certain question in the theory of the braid groups B_n and the Artin representation of B_n in the automorphism group Aut(F_n) of a free group.
The Calabi flow is a fourth order nonlinear parabolic flow, introduced by Calabi in 1980s, and it aims to find Kahler metrics with constant scalar curvature (or more generally extremal Kahler metrics). We prove that the Calabi flow can have a unique smooth short time solution with continuous initial metric. As a byproduct, we prove some elementary but new Schauder type estimates for biharmonic heat equation on compact manifolds. This is a joint work with Yu Zeng (University of Rochester). Our result partially answers a problem proposed by Xiuxiong Chen.
Higher-order methods that use multiway and multilinear correlations are necessary to identify important structures in complex data from biology,
neuroscience, ecology, systems engineering, and sociology. We will study our recent generalization of spectral clustering to higher-order structures in depth. This will include a generalization of the Cheeger inequality (a concise statement about the approximation quality) to higher-order structures in networks including network motifs. This is easy to implement and seamlessly enables spectral clustering-style methods for directed, signed, and many other types of complex networks. If there is time, we will see software demonstrations in the Julia language for reproducibility. I will also briefly highlight recent methods we have developed that use new types of stochastic processes and random walks to study these data involving tensor eigenvectors. These topics motivate a number of exciting open questions at the intersection of numerical linear algebra, optimization, and data analysis.
The Rogers-Ramanujan identities and Monstrous moonshine are among the deepest results which occur at the interface of number theory and representation theory. In this lecture the speaker will discuss these identities, and describe recent work with Duncan, Griffin on Warnaar on their recent generalizations. This will include a framework of Rogers-Ramanujan identities and singular moduli, and the theory of umbral Moonshine.
Note for Attendees
Refreshments (A light reception) will be served in ESB 4133 from 2:30pm-3:00pm before the colloquium.
Motivated by problems arising in the applied sciences, I will tell the story of what might reasonably be called a "forgotten function". It was discovered in the late 1800s, but has hardly ever been used in the physical sciences even though, as I will show, its applications in science and engineering turn out to be many and varied.
In particular, I will survey a new theoretical approach to solving problems in what mathematicians call "multiply connected" domains. These are ubiquitous in applications; whenever two or more objects or entities (airfoils, bacteria, vortices, inhomogeneities in an elastic medium, black holes!...) interact in some ambient medium the analysis may call for the methods I will discuss.
Some illustrative example problems from applications, especially in fluid dynamics, will be described and their solutions explicitly constructed. I will also describe freely available numerical codes that we have developed for the computation of this "forgotten function" in order to promote its use.
We hope to demonstrate that the new methods are sufficiently general that they provide broad scope for tackling a variety of problems.
I will give an introduction to gauged Gromov-Witten theory. The theory naturally leads to studying compactifications of the moduli space of G bundles on nodal curves, which I'll discuss briefly. Then I'll focus on a version of gauged Gromov-Witten theory developed by Woodward and Gonzalez and I'll present a theorem which is joint work with Woodward and Gonzalez on the properness of the moduli of scaled gauged maps satisfying a stability condition introduced by Mundet and Schmitt.
Fermat's Last Theorem states that the equation x^n + y^n = z^n for n > 2 has no integer solutions such that xyz \neq 0. It's proof was completed in 1995 by the groundbreaking work of Andrew Wiles on the modularity of semistable elliptic curves over Q. From its proof a new revolutionary method to attack Diophantine equations was born. This method, now known as the modular method, builds on the work of Frey, Serre, Ribet, Mazur and makes use of the Galois representations attached to modular forms and elliptic curves.
In the first part of this talk, guided by the proof of FLT, we will introduce the tools and sketch the basic strategy behind the modular method. In the second part, we will discuss the main obstacles that arise when we try to apply the method to other type of equations or over number fields.
For the past two years I have been using a "recitation model" to teach the MATH 100/MATH 101 sequence at Vantage College. Instead of three lectures a week, students have one large lecture (run by me) and two small problem-based recitations (run by graduate and undergraduate instructors). I will be testing this model in a mainstream section of MATH 101 next term.
I will discuss the strengths and weaknesses of the model, and whether or not it should be used for our large first-year Math classes.
We deal with a class of the semilinear elliptic equations of the Hénon-type in hyperbolic space. The problem involves a logarithm weight in the Poincaré ball model, bringing singularities on the boundary. Considering radial functions, a compact Sobolev embedding result is proved, which extends a former Ni result made for a unit ball in R^N. Combining this compactness embedding with the Mountain Pass Theorem, an existence result is established.
In this talk we will define and explore an infinite family of vector bundles, known as vector bundles of conformal blocks, on the moduli space M0,n of marked curves. These bundles arise from data associated to a simple Lie algebra. We will show a correspondence (in certain cases) of the rank of these bundles with coefficients in the cohomology of the Grassmannian. This correspondence allows us to use a formula for computing "quantum Kostka" numbers and explicitly characterize families of bundles of rank one by enumerating Young tableaux. We will show these results and illuminate the methods involved.
The uniform spanning forests (USFs) of an infinite graph G are defined to be infinite volume limits of uniformly chosen spanning trees of finite subgraphs of G. These limits can be taken with respect to two extremal boundary conditions, yielding the free uniform spanning forest (FUSF) and wired uniform spanning forest (WUSF). While the wired uniform spanning forest has been quite well understood since the seminal paper of Benjamini, Lyons, Peres and Schramm (’01), the FUSF is less understood, and some very basic questions about it remain open. In this talk I will introduce a new tool in the study of USFs, called update tolerance, and describe how update tolerance can be used to prove, among other things, that the FUSF has either one or infinitely many connected components on any infinite Cayley graph, and that components of either the FUSF and WUSF are indistinguishable from each other by invariantly defined properties on any infinite Cayley graph. Another crucial component of these proofs is the Mass-Transport Principle, which I will also give an introduction to.
Abstract:
This thesis examines the derivation and implementation of the discrete adjoint method for several time-stepping methods. Our results are important for gradient-based numerical optimization in the context of large-scale parameter estimation problems that are constrained by nonlinear time-dependent PDEs. To this end, we discuss finding the gradient and the action of the Hessian of the data misfit function with respect to three sets of parameters: model parameters, source parameters and the initial condition. We also discuss the closely related topic of computing the action of the sensitivity matrix on a vector, which is required when performing a sensitivity analysis. The gradient and Hessian of the data misfit function with respect to these parameters requires the derivatives of the misfit with respect to the simulated data, and we give the procedures for computing these derivatives for several data misfit functions that are of use in seismic imaging and elsewhere.
The methods we consider can be divided into two categories, linear multistep (LM) methods and Runge-Kutta (RK) methods, and several variants of these are discussed. Regular LM and RK methods can be used for ODE systems arising from the semi-discretization of general nonlinear time-dependent PDEs, whereas implicit-explicit and staggered variants can be applied when the PDE has a more specialized form. Exponential time-differencing RK methods are also discussed. Our motivation is the application of the discrete adjoint method to high-order time-stepping methods, but the approach taken here does not exclude lower-order methods. Within each class, each time-stepping method has an associated adjoint method and we give details on its implementation.
All of the algorithms have been implemented in MATLAB using an object-oriented design and are written with extensibility in mind. It is illustrated numerically that the adjoint methods have the same order of accuracy as their corresponding forward methods, and for linear PDEs we give a simple proof that this must always be the case. The applicability of some of the methods developed here to pattern formation problems is demonstrated using the Swift-Hohenberg model.
I will discuss joint work with Pierre-Andre Chiappori and Robert McCann on the Monge-Kantorovich problem of transporting a probability measure on \mathbb{R}^n to another on the real line. We introduce a nestededness criterion relating the cost to the marginals, under which it is possible to solve this problem uniquely (and essentially explicitly), by constructing an optimal map one level set at a time. I plan to discuss examples for which the nestedness condition holds, as well as some for which it fails; some of these examples arise from a matching problem in economics which originally motivated our work. If time permits, I will also briefly discuss how level set dynamics can be used to develop a local regularity theory in the nested case
My research concerns models of gradient type with sub-quadratic actions and their scaling limits. The model of gradient type is the density of a collection of real-valued random variables ϕ’s given by Z^{-1}e^({-ΣV(ϕ_j-ϕ_k)}). We focus our study on the case that V(t) = [1+t^2]^a with 0 < a < 1/2, which is a non-convex potential.
The first result concerns the thermodynamic limits of the model of gradient type. We introduce an auxiliary field t for each edge and represent the model as the marginal of a model with log-concave density. Based on this method, we prove that finite moments of the fields are bounded uniformly in the volume for the finite volume measure. This bound leads to the existence of infinite volume measures.
The second result is the random walk representation and the scaling limit of the translation-invariant, ergodic gradient infinite volume Gibbs measure. We represent every infinite volume Gibbs measure as a mixture over Gaussian gradient measures with a random coupling constant ω for each edge. With such representation, we give estimations on the decay of the two point correlation function. Then by the quenched functional central limit theorem in random conductance model, we prove that every ergodic, infinite volume Gibbs measure with mean zero for the potential V above scales to a Gaussian free field.
Consider a real Gaussian stationary process, either on Z or on R. That is, a stochastic process, invariant under translations, whose finite marginals are centred multi-variate Gaussians. The persistence of such a process on [0,T] is the probability that it remains positive throughout this interval.
The relation between the decay of the persistence as T tends to infinity and the covariance function of the process has been investigated since the 1950s with motivations stemming from probability, engineering and mathematical physics. Nonetheless, until recently, good estimates were known only for particular cases, or when the covariance kernel of the process is either non-negative or summable.
In the talk we discuss a new spectral point of view on persistence which greatly simplifies its analysis. This is then used to obtain better bounds in a very general setting.
The Ising model, and its generalisation, the Potts model, are two classical graph-colouring models for magnetism and antiferromagnetism. Albeit their simple formulation, these models were instrumental in explaining many real-world magnetic phenomena and have found various applications in physics, biology and computer science.
While our understanding of these models as modeling magnets has been constantly improving since the early twentieth century, little progress was made in treatment of Potts antiferromagnets.
In the talk, after a historical introduction to magnets, antiferromagnets and their modeling, we will describe how application of modern combinatorial and probabilistic methods resulted in recent breakthroughs in the rigorous analysis of Potts antiferromagnets.
We discuss L^p estimates for some multilinear singular integral forms and their applications to sharp quantitative norm convergence of ergodic averages with respect to two commuting transformations, quantitative cancellation estimates for the simplex Hilbert transform, and a question on side lengths of corners in dense subsets of the Euclidean space.
In this talk, I will discuss work giving in many cases a complete description of the group of rational sections of the relative Jacobian of a linear system of curves on a surface. By specializing to the case of spectral curves, we are able to determine very explicitly the group of sections of the Hitchin fibration. We will also discuss work in progress to extend this work to principal G-Higgs bundles for more general groups G.
The aim of this talk is to give an overview of some recent results in two interconnected areas:
*a) **Random graphs, random trees, and complex networks: *The last decade of the 20th century saw significant growth in the availability of empirical data on networks, and their relevance in our daily lives. This stimulated activity in a multitude of fields to formulate and study models of network formation and dynamic processes on networks to understand real-world systems.
One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on n vertices and degree exponent \tau>3, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}. In other words, the degree exponent determines the universality class the random graph belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjecture.
More generally, recent research has provided strong evidence to believe that several objects, including
(i) components under critical percolation,
(ii) the vacant set left by a random walk, and
(iii) the minimal spanning tree,
constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the Gromov-Hausdorff-Prokhorov sense, and these limiting objects are universal under some general assumptions. We report on recent progress in proving these conjectures.
*b) Stochastic geometry:* In contrast, less precise results are known in the case of spatial systems. We discuss a recent result concerning the length of spatial minimal spanning trees that answers a question raised by Kesten and Lee in the 90's, the proof of which relies on a variation of Stein's method and a quantification of the classical Burton-Keane argument in percolation theory.
Based on joint work with Louigi Addario-Berry, Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.
RIDC (revisionist integral deferred correction) methods are a class of time integrators well-suited to parallel computing. RIDC methods can achieve high-order accuracy in wall-clock time comparable to forward Euler. The methods use a predictor and multiple corrector steps. Each corrector is lagged by one time step; the predictor and each of the correctors can then be computed in parallel. This presentation introduces RIDC methods and demonstrates their effectiveness on some test problems.
Motivated by the presence of empirical data on a wide array of real-world networks, there has been an explosion in the number of random graph models proposed to explain various phenomenon observed in real-world systems including power law degree distribution and small world phenomenon. A major general theme in this area, since the time of Erdos and Renyi, has been understanding the properties of components in the critical regime. In the past five years, significant progress has been made in establishing scaling limits of critical random graphs and various constructs on random graphs when they are viewed as metric measure spaces, and understanding the general universality principles underlying such scaling limits. Further, striking connections between these questions and some of the central models of stochastic coalescence, and random interlacements and vacant set left by random walks have emerged. In this talk, we survey some recent results in this area and discuss general methodology for proving such results.
Everyone understands the statement: "The probability of rain tomorrow is .90, or 90%. For a probabilist, many questions have answers of this type. The problem is to find that number.
I will talk about a random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels. We are particularly interested in the high-dimensional case when d is large. Upon observing a graph, we want to tell if it was generated from this geometric model, or from the Erdos-Renyi model. We show that there exists a computationally efficient procedure to do this which is almost optimal (in an information-theoretic sense). The key insight is based on a new statistic which we call "signed triangles". To prove optimality we compute the total variation distance between Wishart matrices and the Gaussian Orthogonal Ensemble. This is joint work with Sebastien Bubeck, Jian Ding, Ronen Eldan, and Jacob Richey.
From networks to genomics, large amounts of data are increasingly available and play critical roles in helping us understand complex systems. Statistical inference is crucial in discovering the underlying structures present in these systems, whether this concerns the time evolution of a network, an underlying geometric structure, or reconstructing a DNA sequence from partial and noisy information. In this talk I will discuss several fundamental detection and estimation problems in these areas.
I will present an overview of recent developments in source detection and estimation in randomly growing graphs. For example, can one detect the influence of the initial seed graph? How good are root-finding algorithms? I will also discuss inference in random geometric graphs: can one detect and estimate an underlying high-dimensional geometric structure? Finally, I will discuss statistical error correction algorithms for DNA sequencing that are motivated by DNA storage, which aims to use synthetic DNA as a high-density, durable, and easy-to-manipulate storage medium of digital data
Fossil record indicates that the emergence of arches in human ancestral feet coincided with a transition from an arboreal to a terrestrial lifestyle. Propulsive forces exerted during walking and running load the foot under bending, which is distinct from those experienced during arboreal locomotion. I will present mathematical models with varying levels of detail to illustrate a simple function of the transverse arch. Just as we curve a dollar bill in the transverse direction to stiffen it while inserting it in a vending machine, the transverse arch of the human foot stiffens it for bending deformations. A fundamental interplay of geometry and mechanics underlies this stiffening -- curvature couples the soft out-of-plane bending mode to the stiff in-plane stretching deformation. In addition to presenting a functional interpretation of the transverse arch of the foot, this study also identifies a classification of flat feet based on the skeletal geometry and mechanics.
Note for Attendees
Reception before the talk in ESB 4133 (PIMS lounge). This is the annual IAM alumni lecture.
Tautological relations are certain equations in the Chow ring of the moduli space of curves. I will discuss a family of such relations, first conjectured by A. Pixton, that arises by studying moduli spaces of ramified covers of the projective line. These relations can be used to recover a number of well-known facts about the moduli space of curves, as well as to generate very special equations known as topological recursion relations. This is joint work with various subsets of S. Grushevsky, F. Janda, X. Wang, and D. Zakharov.
First suggested by Witten in the early 1990's, the Landau-Ginzburg/Calabi-Yau correspondence studies a relationship between spaces of maps from curves to the quintic 3-fold (the Calabi-Yau side) and spaces of curves along with 5th roots of their canonical bundle (the Landau-Ginzburg side). The correspondence was put on a firm mathematical footing in 2008 when Chiodo and Ruan proved a precise statement for the case of genus-zero curves, along with an explicit conjecture for the higher-genus correspondence. In this talk, I will begin by describing the motivation and the mathematical formulation of the LG/CY correspondence, and I will report on recent work with Shuai Guo that verifies the higher-genus correspondence in the case of genus-one curves.
Many problems in scientific computing require the approximation of smooth, high-dimensional functions from limited amounts of data. For instance, a typical problem in uncertainty quantification involves identifying the parameter dependence of the output of a computational model. Complex physical systems involve models with multiple parameters, resulting in multivariate functions of many variables. Although the amount of data may be large, the curse of dimensionality
essentially prohibits collecting or processing sufficient data to approximate the unknown function using classical techniques.
In this talk, I will give an overview of the approximation of smooth, high-dimensional functions using sparse polynomial expansions. I will focus on the application of techniques from compressed sensing to this problem, and discuss the extent to which such approaches overcome the curse of dimensionality. If time, I will also discuss several extensions, including dealing with corrupted and/or unstructured data, the effect of model error and incorporating additional information such as gradient data. I will also highlight several challenges and open problems.
This is joint work with Casie Bao, Simone Brugiapaglia and Yi Sui (SFU).
As we use more and more group work in our classes, should we consider introducing it into our assessments? One model that has been used are so-called two-stage assessments, where students first complete and turn in the questions individually and then, working in small groups, answer the same questions again. This technique was first introduced in the UBC Faculty of Science in 2009 and is now being used in at least 20 science courses.
In this session, we will discuss a study of feasibility and effectiveness of two-stage quizzes as introduced into two mathematics courses at UBC with a total of 834 students. We examine both short and long term retention resulting from introducing group assessments, we analyze results of collaborative learning based on question type and group composition. Finally, we present student and instructor feedback as well as discuss future directions of implementation and research.
The boundary Harnack principle (BHP) is a fundamental tool to understand the behaviour of positive harmonic functions near the boundary of a domain. For instance, the BHP implies a concrete description of the Martin boundary of a domain in geometric terms. Other applications of BHP include Carleson estimate, Fatou's theorem, and heat kernel estimates for diffusions killed upon exiting a domain. In this talk, I will discuss a recent extension of BHP that provides new examples of diffusions satisfying BHP even in R^n.
Harnack inequalities have proved to be a powerful tool in PDE (regularity estimates), geometry (geometric flows) and probability (heat kernel estimates). In the early 1990s Grigor'yan and Saloff-Coste gave a characterisation of the parabolic Harnack inequality (PHI). This characterisation implies that PHI is stable under perturbations (quasi-isometries). In this talk, I will provide an introduction to Harnack inequalities and discuss the stability of elliptic Harnack inequality.
This is joint work with Martin Barlow.
Note for Attendees
Tea and cookies will be served before this special colloquium.
We discuss two recent scaling limit results for discrete dynamics converging to stochastic PDEs. The first is the asymmetric simple exclusion process in contact with sources and sinks at boundaries, called Open ASEP. We prove that under weakly asymmetric scaling the height function converges to the KPZ equation with Neumann boundary conditions. The second is the Glauber dynamics of the Blume-Capel model (a generalization of Ising model), in two dimensions with Kac potential. We prove that the averaged spin field converges to the stochastic quantization equations. The main purpose of this talk is to discuss the general issues one needs to address when passing from discrete to continuum, the common challenge in the proofs of such scaling limit theorems, and how we overcome these difficulties in the two specific models. (Based on joint works with Ivan Corwin and Hendrik Weber)
Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.
Stem cell fate is regulated by cues from the cellular microenvironment, including biophysical and biochemical cues presented by the extracellular matrix. Matrix regulation of cell fate has broad implications from disease to regeneration. In this talk, I will discuss our work aimed at determining how biophysical and biochemical cues from the matrix act to drive the fate and function of mesenchymal stem cells. In particular, I will discuss how the pathological differentiation of heart valve progenitor cells is modulated by the extracellular matrix and how we are using microdevices to dissect how matrix mechanical and biochemical cues act in concert to regulate mesenchymal stem cell differentiation and tissue regeneration.
Note for Attendees
Reception preceding the talk in ESB 4133 (the PIMS lounge). This is part of the IAM/PIMS distinguished colloquium series.
The study of Grothendieck rings of varieties in the context of real algebraic geometry has begun since the apparition of motivic integration. Several such rings are of interest, depending notably on the class of functions we are interested in. We will discuss recent progress in the cases of real algebraic varieties and of arc-symmetric sets.
Chiral differential operators (CDOs) are a vertex algebra analog of the associative algebra of differential operators. Originally introduced by mathematicians, Witten explained how CDOs arise as the perturbative part of the curved beta-gamma system with target X. I will describe recent work with Gorbounov and Williams in which we construct the BV quantization of this theory and use a combination of factorization algebras and formal geometry to recover CDOs. At the end, I hope to discuss how the techniques we developed apply to a broad class of nonlinear sigma models, including source manifolds of higher dimension.
MATH 1102 Topology/Algebra/Math Physics seminar [We will start at 4:10pm]
Tue 24 Jan 2017, 4:10pm-5:10pm
Details
We construct two homotopy functors from the category of homotopy commutative algebras to the category of pro-unipotent group and the category of completed commutative Hopf algebras such that
(1) the group is isomorphic to the pro unipotent fundamental group of smooth connected and based manifold (M,*) and
(2) the Hopf algebra is that of homotopy functionals defined by Chen’s iterated path integrals over closed and based loops on M
if the homotopy commutative algebra is quasi-isomorphic to the algebra of differential forms on M.
These constructions can be quantized in appropriate sense after interpreting them as doing “classical field theory” such that the pro unipotent fundamental group is the structure in the space of all "classical expectations" and the completed Hopf algebra is the algebra of “classical observables”. The quantization involves a generalized deformation quantization of homotopy commutative algebra into topologically free homotopy associative algebra along the direction of compatible homotopy Lie algebra.
Binding of molecules, ions or proteins to small target sites is a generic step of cell activation. This process relies on rare stochastic events where a particle located in a large bulk has to find small and often hidden targets. I will present in this talk a hybrid discrete-continuum model that takes into account both a stochastic regime governed by rare events and a continuous regime in the bulk, in the context of vesicular release at chemical synapses.
In a first part, I computed the mean time for a Brownian particle to arrive at a narrow opening defined as the small cylinder joining two tangent spheres. This models the binding of calcium ions on the SNARE complex, a process that triggers vesicular release. Using this result, I developed a model to study how vesicles and calcium channels organization shape such process.
In a second part, I will present a model for the pre-synaptic terminal built using the results described above. This model was formulated in an initial stage as a reaction-diffusion problem in a confined microdomain, where Brownian particles have to bind to small target sites. I coarse-grained this model into a system of mass action equations coupled to a set of Markov equations, which allows to obtain analytical results and to realize fast stochastic simulations.
Factorization algebras are local-to-global objects that play a role in quantum field theory akin to the role of sheaves in geometry: they conveniently organize complicated information. In the talk I will introduce this notion, give some concrete examples, and then explain how factorization algebras mediate between QFT and higher algebra. An important example will be Chern-Simons theory; ongoing work with Costello and Francis recovers quantum groups with formal parameter by combining Koszul duality with Feynman diagrams.
Note for Attendees
Tea and cookies will be served in the PIMS Lounge ESB 4133 at 2:45 p.m. before this special colloquium.
Topology is full of ineffective arguments constructing objects and equivalences by algebra.
One of the great early achievements of algebraic topology was the work of Thom, followed by Milnor and Wall, on cobordism theory, which describes when a compact smooth (oriented) manifold is the boundary of some compact manifold with boundary. This method is typical of the problems that arise in the use of algebraic methods and is an early example of one of the dominant philosophies of geometric topology. The question we study is to what extent the complexity of a manifold can be used to bound, when it exists, the minimum necessary complexity of something that it bounds.
The goal of this talk is to explain generally some of the issues of making topology less ineffective.
We shall show that there are polynomial size nullcobordisms in a suitable sense. This is joint work with Greg Chambers, Dominic Dotterer and Fedor Manin.
We discuss recent work showing that in type A_n the category of equivariant perverse coherent sheaves on the affine Grassmannian categorifies the cluster algebra associated to the BPS quiver of pure N=2 gauge theory. Physically, this can be understood as a statement about line operators in this theory, following ideas of Gaiotto-Moore-Neitzke, Costello, and Kapustin-Saulina -- in short, coherent IC sheaves are the precise algebro-geometric counterparts of Wilson-'t Hooft line operators. The proof relies on techniques developed by Kang-Kashiwara-Kim-Oh in the setting of KLR algebras. A key moral is that the appearance of cluster structures is in large part forced by the compatibility between chiral and tensor structures on the category in question (i.e. by formal features of holomorphic-topological field theory). This is joint work with Sabin Cautis.
Note for Attendees
Refreshments will be served at 3:15pm in the MATH 125 Lounge Area.
The discovery of so-called canonical (or crystal) bases of quantum groups by Lusztig and Kashiwara was one of the major milestones of Lie theory during the past three decades. The larger theory of canonical bases that grew out of their work (and others') has continually revealed surprising connections to other areas of mathematics such as algebraic and symplectic geometry. In this talk we sample some of these connections, for example to the geometry of character varieties of surfaces, and explain how ideas from theoretical physics provide an organizing framework for many mathematically disparate results. A central role is played by the flexible combinatorial language of cluster algebras, which provides a useful lingua franca for the subject.
Note for Attendees
Refreshments will be served at 2:45 p.m. before this special colloquium.
Adiabatic theory studies dynamical systems featuring separation of motion into slow and fast components. When the system is driven these components start mixing. The basic premise of adiabatic theory is that this mixing is small for slow driving. Furthermore, the resulting driven motion is irreversible in the slow component but reversible in the fast component. In this talk, I first illustrate this phenomena on a spinning top and state the classical adiabatic theorem of quantum mechanics. Then, I will show extensions of the theory to abstract Banach spaces and stochastic equations, culminating with an adiabatic theorem for extended systems where the separation holds uniformly in the volume of the system. An important corollary of this theorem is a proof of the Kubo formula for gapped interacting systems.
In 1997 Caporaso, Harris and Mazur, motivated by uniformity results in Diophantine Geometry, proposed a conjecture about fibered powers of families of varieties of general type. In particular they conjecture that, if X -> B is a family whose general fiber is a variety of general type, then for large n, the n-th fiber power of X over B dominates a variety of general type. The conjecture has been proved by Abramovich and used to deduce interesting results on the distribution of rational points on projective varieties. I will discuss recent work, joint with Kenny Ascher (Brown), generalizing this to pairs (X,D) of a projective scheme and a divisor, and the new challenges that arise when one tries to obtain analogous results for the distribution of integral points in quasi projective varieties.
Statistics of measurement outcomes in quantum mechanics is described by a map that associates a matrix to each possible measurement outcome. The probability of this outcome is then given by a trace of this matrix. Probability of a sequence of measurement outcomes is computed in the same way from a product of associated matrices. In this talk I will describe two results related to this setting. A theorem giving optimal conditions for uniqueness of the associated invariant measure on the projective sphere, and a theorem describing large deviation theory in the case when the matrices commute. The latter problem received lots of recent attention following experiments of S.~Haroche.
BC Safety Authority’s accident prevention strategy follows a risk based platform, called “bow tie” for ongoing management and prevention of major incidents. The bow tie method draws on the direct experience of safety management team and the data collected from various safety oversight programs in order to identify hazards and to properly incorporate critical controls into management systems. In addition, it raises awareness and improves understanding and risk knowledge of the potential major incidents and the reliance on critical controls that prevent those accidents from occurring. Opportunities exist for greater connectivity among various databases to maximize the risk knowledge, especially the pre incident data based on inspections and post incident data.
Note: this event is the inaugural Vancouver Data Science Colloquium, co-sponsored by CANSSI, the Data Sciences Institute (DSI), the Institute for Applied Mathematics (IAM) and PIMS.
We present mod-two cohomology of both symmetric and alternating groups as (almost) Hopf rings.
As is being seen in subjects such as representation stability, in these settings one gains substantial insight by considering all cases together, binding them with some structure. For these group cohomologies, that structure is a transfer or induction product, akin to taking external tensor product of S_n and S_m representations and inducing up to S_{n+m}. This product along with cup product and a standard coproduct together define a ring object in the category of coalgebras in the setting of symmetric groups, and are close enough to such an object for practical purposes in the setting of symmetric groups.
Settings such as general linear groups over finite fields now beg for investigation, as do a number of questions internal to topology (e.g. Margolis homology, working towards Morava K-theory) and at the interface with algebra (e.g. how do analogous structures interface with current understanding of modular representation theory of symmetric groups).
We will aim for the talk to be accessible to algebraists. While topology will be mentioned throughout, it will be presented in a supporting role, familiar to topologists but treatable as a black box when necessary to algebraists.
ESB 2012 (Note the unusual time: 4-5 pm on Wednesday, Feb 1. )
Wed 1 Feb 2017, 4:00pm-5:00pm
Abstract
This talk addresses how the calculus-of-variations is applied to non-linear elasticity. In physically realistic classical models, energy-minimizing deformations may not be smooth enough to satisfy the variational Euler-Lagrange equations. However, with a second-gradient model we guarantee sufficient regularity to rigorously prove energy-minimizers satisfy such an equation and maintain incompressibility and/or global injectivity.
The constraints of incompressibility and self-contact introduce subtle challenges of infinite-dimensional non-linear analysis. I will discuss the techniques and assumptions that we use to prove the existence of a distributional pressure for the incompressibility constraint and a measure-valued surface traction for the self-contact constraint. This work was part of my dissertation research done under the supervision of Professor Timothy J. Healey at Cornell University.
I will present an adiabatic theorem for the driven dynamics of ground state projections of a smooth family of many-body gapped quantum systems. The diabatic error is uniformly bounded in the volume of the interacting system. As an corollary, Kubo’s formula of linear response theory can be obtained in the thermodynamic limit.
While thermal phases of matter (liquid water vs solid ice) are mathematically well understood, the theoretical understanding of ground state phases (at zero temperature) is incomplete. The discoveries of last year’s physics Nobel laureates provide a framework valid for noninteracting systems with topological properties, and I shall discuss current efforts to extend some of it to phases of strongly interacting matter.
Note for Attendees
Tea and cookies will be served at 2:45 pm in the PIMS Lounge (ESB 4133) before this colloquium.
The Hodge theory of surfaces provides a link between enumerative geometry and modular forms, via the cohomological theta correspondence. I will present an approach to studying the Gromov-Witten invariants of Weierstrass fibrations over P^2, proving part of a conjectural formula coming from topological string theory.
We introduce SPMR, a new family of methods for iteratively solving saddle-point systems using a minimum or quasi-minimum residual approach. No symmetry assumptions are made. The basic mechanism underlying the method is a novel simultaneous bidiagonalization procedure that yields a simplified saddle-point matrix on a projected Krylov-like subspace, and allows for a monotonic short-recurrence iterative scheme. We develop a few variants, demonstrate the advantages of our approach, derive optimality conditions, and discuss connections to existing methods. Numerical experiments illustrate the merits of this new family of methods.
This is joint work with Ron Estrin.
Schur functions form an important basis for the space of symmetric functions and show up in areas from representation theory to quantum mechanics. We continue to investigate the problem of which ribbon Schur functions are most Schur-positive; that is, when the difference of two ribbon Schur functions is a nonnegative linear combination of Schur functions. In particular, we describe techniques leading to a powerful necessary condition and time permitting, lay out how to continue these methods in resolving long-standing conjectures.
Bootstrap percolation with threshold r on a graph G evolves as follows: initially some of its vertices are infected, and then any vertex with at least r infected neighbors becomes infected. On the Erdos–Renyi graph G(n,p) we identify the sharp threshold for p, above which there is with high probability a set of size r whose infection results in the infection of the entire graph.
Joint work with Omer Angel.
Part I: Representations of reductive groups over local fields
In this talk, we describe the representations of certain reductive groups over local fields and the representations of Weil groups. Then we review the class field theory for local fields and the Langlands correspondence for these reductive groups. The talk will be a brief overview of the representation theory of reductive groups over local fields, largely based on examples of low-rank groups.
Part II: Endoscopic classification of essentially tame supercuspidal representations for quasi-split classical groups
Continue from the last talk, we specify our reductive group \mathbf{G} to be a quasi-split classical group (special orthogonal, symplectic, unitary) over a p-adic field of odd residual characteristic. We describe the endoscopic classification, proved by Arthur and Mok, of certain supercuspidal representations and their L-packets of \mathbf{G}, under some regularity and tameness conditions. These representations can be described by inducing types constructed by Bushnell-Kutzko, Stevens, or Yu, and they correspond to Langlands parameters related to characters of elliptic maximal tori of \mathbf{G}.
(This work is partly joint with Corinne Blondel.)
In Boltzmann equation, the interplay among free transport, collision, and boundary yields rich phenomena in regularity of solutions. In this talk, we will first introduce the logarithmic singularity both on macroscopic and microscopic variables due to the boundary. Then, we will discuss the regularity of stationary solutions in a convex domain. Finally, we will provide the analysis that realizes our observation.
Coffee and cookie will be provided before the seminar at the PIMS lounge.
Prof. I-Kun Chen is currently a Senior Lecturer at the Department of Applied Analysis and Complex Dynamical Systems, Kyoto University, http://www.acs.i.kyoto-u.ac.jp/en.html . He is visiting UBC between Feb 8-22, 2017.
The stretched Sierpinski gasket, SSG for short, is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. It has also been called ``deformed Sierpinski gasket'' or ``Hanoi attractor''. As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing of the Brownian motions on the intervals. In fact, there have been several works in this direction. There still remains, however, ``reminiscence'' of the Sierpinski gasket in the geometric structure of SSG and the same should therefore be expected for diffusions. This paper shows that this is the case. In this work, we identify all the completely symmetric resistance forms on SSG. A completely symmetric resistance form is a resistance form whose restriction to every contractive copy of SSG in itself is invariant under all geometrical symmetries of the copy, which constitute the symmetry group of the triangle. We prove that completely symmetric resistance forms on SSG can be sums of the Dirichlet integrals on the intervals with some particular weights, or a linear combination of a resistance form of the former kind and the standard resistance form on the Sierpinski gasket.
In recent years, important families of symmetric group representations have come to be better understood through the perspective of representation stability, a viewpoint introduced and developed by Thomas Church, Jordan Ellenberg, and Benson Farb, among others. A fundamental example of representation stability is the S_n-module structure for the $i$-th cohomology of the configuration space of n distinct, labeled points in the plane, or more generally in a connected, orientable manifold, as $i$ is held fixed and n grows. For the plane, this translates to Whitney homology of the partition lattice via an S_n-equivariant version of the Goresky-MacPherson formula. This talk will survey the combinatorial literature regarding the partition lattice and discuss what new things this can tell us about representation stability for configuration spaces. In particular, we deduce new, sharp stability bounds and representation theoretic structure through a combination of symmetric function technology and poset topology. This is a joint work with Vic Reiner.
I will discuss some aspects of the tight connection between the study of certain dynamical systems and number theory (most notably to questions in Diophantine approximation and the geometry of numbers).
I will describe how deterministic and stochastic dynamic optimal mass transports are to Mean Field Games what the classical calculus of variations offers to classical mechanics.
As its name is meant to suggest, the subject of p-adic Hodge theory was historically concerned with the relationship between different cohomology theories attached to p-adic algebraic varieties. Within p-adic Hodge theory, the concept of a perfectoid space (discussed in my PIMS lectures) arose quite naturally and has led to improvements in the subject which were in some sense "expected".
However, it also had several "unexpected" applications rather far afield. We'll survey three of these: Deligne's weight-monodromy conjecture (Scholze); Galois representations associated to torsion cohomology of arithmetic groups (Scholze, Caraiani-Scholze); and the direct summand conjecture of commutative algebra (Andre, Bhatt).
Joint work with Kazuhiro Bessho. Classical population genetic theory generally assumes either a fully haploid or fully diploid life cycle. However, many
organisms exhibit more complex life cycles, with both free-living haploid and diploid stages. Here we ask what the probability of fixation is for selected alleles in organisms with selection acting on both haploid and diploid phases of a sexual life cycle, based on both a Moran model and a Wright–Fisher model. Applying a branching process approximation, we obtain an accurate fixation probability assuming that the population is large and the net effect of the mutation is beneficial. We also find the diffusion approximation for the fixation probability, which is accurate even in small populations and for deleterious alleles, as long as selection is weak. These fixation probabilities from branching process and diffusion approximations are similar when selection is weak for beneficial mutations that are not fully recessive. In many cases, particularly when one phase predominates, the fixation probability differs substantially for haploid-diploid organisms compared to either fully haploid or diploid species.
The diversity of mathematics and roles of mathematicians in industry has expanded greatly over the last decade. The first part of this talk is a survey of work that I have done as an applied mathematician working in industry. Areas include modeling groundwater flow for soil remediation, image editing, content-based image search, and machine learning for searching large satellite images.
The second part of this talk will focus on a particular use of mathematics in indexing high dimensional floating-point vectors using a search engine intended for text. The result is a database search method that can combine "fuzzy" data, such as image textures or audio recordings, with text-based data such as document titles or descriptive keywords in an infrastructure easy for a software engineer to build and maintain.
The ultimate purpose of this talk is to highlight areas of mathematics of particular use to my work in industry, and hence of particular interest to future industrial practitioners.
Note for Attendees
This is a talk in the new BC Data Colloquium.
Convene beforehand for PIMS tea in the PIMS lounge at 14:45
Our work on modulo p smooth representations of reductive p-adic groups (classification of irreducible admissible representations, functoriality properties,Satake homomorphisms, with Abe, Henniart, Herzig, Ollivier), raises questions for which a kind of answer is required if we want to venture further.
This talk is part of the PIMS focus semester on the mod p Langlands program.
There are many fundamental problems concerning sequences that arise in many areas of mathematics and computation. Typical problems include finding or avoiding patterns;
testing or validating various `random-like’ behavior; analyzing or comparing different statistics, etc. In this talk, we will examine various notions of regularity or irregularity for sequences and mention numerous open problems.
Note for Attendees
Refreshments will be served in ESB 4133 from 2:45pm-3:00pm.
Classically, there are two objects that are particularly interesting to algebraic geometers: hyperelliptic curves and quadrics. The connection between these two seemingly unrelated objects was first revealed by M. Reid, which roughly says that there’s a correspondence between hyperelliptic curves and pencil of quadrics. I’ll give a brief review of Reid’s work and then describe a higher degree generalization of the correspondence.
A Toda system is a nonlinear second order elliptic system with exponential nonlinearity. It is very commonly observed in physics and has many ties with algebraic geometry. From analytic viewpoints it is challenging since the solutions do not have symmetry, maximum principles cannot be applied and the structures of global solutions are incredibly complicated. In this joint work with Chang-shou Lin, Juncheng Wei and Wen Yang, we use a unified approach to discuss all rank two singular Toda systems. First for local systems we prove that all weak limits of mass concentration belong to a very small finite set. Then for systems defined on compact Riemann surface we establish some new estimates. Our approach is a combination of delicate blowup analysis and fundamental tools from algebraic geometry.
The study of infinite geodesics in first passage percolation have received a lot of attention since the work of Hoffman [`05] and [08], Damron and Hanson [`14] and more recently Ahlberg and Hoffman [`16]. In this talk we discuss further advances in this topic presenting a family of measures for which it is possible to completely characterize the set of infinite geodesics. This is joint work with Christopher Hoffman.
A classical approach to understanding spaces of homomorphisms is to describe its connected components. We mainly focus in the spaces Hom(N_{n,q},SU(2)), where N_{n,q} denotes the free q-nilpotent group on n-generators. We show that the connected components arising from non-commuting q-nilpotent n-tuples in SU(2) are homeomorphic to RP^3 and we give the exact number of these. We prove it by showing a seemingly unknown result about SU(2) that states: all non-abelian nilpotent subgroups are conjugated to the quaternion group Q_8 or to the generalized quaternion groups Q_{2^q}, of order 2^q. Some applications of this result are the stable homotopy type of Hom(N_{n,q},SU(2)); a homotopy description of the classifying spaces B(q,SU(2)) of transitionally q-nilpotent principal SU(2)-bundles, and its derived versions for SO(3) and U(2). If time permits I'll also show some cohomology calculations for the spaces B(r,Q_{2^q}) for low values of r.
This is joint work with Omar Antolín Camarena.
Recently, Venkatesh introduced a global version of the derived Hecke algebra to explain extra endomorphisms on the cohomology of arithmetic manifolds: the crucial local construction is a derived version of the spherical Hecke algebra of a reductive p-adic group.
Working with p-torsion coefficients, we will describe a Satake homomorphism for the derived spherical Hecke algebra of a p-adic group. This allows us to understand its structure well enough to attack some global questions, which are work in progress.
(This talk is part of the PIMS focus semester on the mod p Langlands program).
SFU Harbour Centre Campus in Downtown Vancouver, Room 1530
Fri 10 Mar 2017, 12:00pm-1:00pm
Abstract
Mathematics is integral to every aspect of game development including character and level creation, movement, player input, NPC behaviour, physics simulations, and real-time rendering. Fortunately for game designers, most of this computation is conveniently supplied by software developers and/or handled by existing game engines. However, when designing a game, lots of systems and mechanics are dependent on numbers such as weapons ranges, jump heights, experience points, damage, rewards, currency, etc., many of which can have complex inter-relationships. Although much of the math may be basic, a good understanding of the underlying equations as well as the fields of logic, probability, and statistics can be incredibly beneficial to a designer, especially when it comes to game design and balancing. This lecture will give an overview of how even the most basic knowledge of these fields can benefit a game designer.
The talk will first present some classical results on the automorphisms of complex projective curves (or alternatively, of compact Riemann surfaces). We will then discuss the automorphism groups of projective algebraic varieties of higher dimensions; in particular, their "connected part" (which can be arbitrary) and their "discrete part" (of which little is known).
Note for Attendees
Refreshments will be served in ESB 4133 from 2:45pm-3:00pm.
Let G be a connected algebraic k-group acting on a normal k-variety, where k is a field. We will show that X is covered by open G-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding into the projectivization of a G-linearized vector bundle on an abelian variety A, where A is a quotient of G. This generalizes a classical result of Sumihiro for actions of smooth connected algebraic groups.
Many chemical systems exhibit a regular pattern of precipitate bands known as Liesegang rings in tribute to the chemist Raphael E. Liesegang who demonstrated them using a reaction of silver nitrate and potassium dichromate. A variety of theories have been studied to try and understand how these patterns develop and one of the most seminal papers on the topic was a mathematical model developed by Keller and Rubinow using a supersaturation framework. This model predicted the formation of primary and secondary bands using heuristic arguments and assumptions about the underlying equations. In this talk we present the Keller-Rubinow model from a formal asymptotic perspective and discuss the dynamics and formation of the primary precipitation band. Furthermore we show that secondary bands are infeasible under the model as posed thus necessitating the development of a new model. We will provide preliminary insights into such a new model and its resolution towards secondary bands.
This is part II of the February 28 talk. Original abstract: I will describe how deterministic and stochastic dynamic optimal mass transports are to Mean Field Games what the classical calculus of variations offers to classical mechanics.
The correlation length of order p for the |\varphi|^4 spin model (a continuous-spin version of the O(n) model) is a normalization of the p-th moment of its two-point function. We will outline the proof (based on a renormalisation group method of Bauerschmidt, Brydges, and Slade) that, in the upper-critical dimension 4, this quantity undergoes mean-field scaling with a logarithmic correction as the critical point for this model is approached from above (for sufficiently weak coupling). Via a supersymmetric integral representation, this result also extends to the weakly self-avoiding walk with a contact attraction, for which the correlation length of order p is closely related to the mean p-th displacement of the walk. This is joint work with Roland Bauerschmidt, Gordon Slade, and Alexandre Tomberg.
Given a variety X defined over Z, there are two problems which a priori seem to not have a lot to do with each other:
1. Describe the singularities of the complex variety X(C)
2. Fix a prime p and describe how the cardinality of X(Z/p^rZ) depends on r.
A surprising result from the 80s concerning the second problem is that the Poincaré series of X - a formal power series having the above cardinalities as coefficients - is a rational function.
In this talk, I will explain this in more detail and I will present a new notion of stratifications which contributes to both problems: On the one hand, such a stratification specializes to a stratification of X(C) (which has stronger regularity properties than classical Whitney stratifications); on the other hand, using those stratifications, one obtains a (new) geometric proof of the rationality of Poincare series.
In this talk on joint work with REBECCA BELLOVIN we discuss the “local ε-isomorphism” conjecture of Fukaya and Kato for (crystalline) families of G_{Q_p}-representations. This can be regarded as a local analogue of the global Iwasawa main conjecture for families, extending earlier work of Kato for rank one modules, of Benois and Berger for crystalline representations with respect to the cyclotomic extension as well as of Loeffler, Venjakob and Zerbes for crystalline representations with respect to abelian p-adic Lie extensions of Q_p. Nakamura has shown Kato’s conjecture for (ϕ,\Gamma)-modules over the Robba ring, which means in particular only after inverting p, for rank one and trianguline families. The main ingredient of (the integrality part of) the proof consists of the construction of families of Wach modules generalizing work of Wach and Berger and following Kisin’s approach via a corresponding moduli space.
(This talk is part of the PIMS focus semester on the mod p Langlands program).
What conditions must be placed on a set E \subset \mathbb{R}^n in order to guarantee that E contains certain finite arrangements of points? We will discuss some conditions on a set E that guarantee the existence of point configurations, and constructions of large sets E that avoid them. This work was completed jointly with Malabika Pramanik.
Note for Attendees
Refreshments will be served at 2:45 p.m. in MATH 125 before this colloquium.
Many useful and interesting optimization problems can be cast in a convex composite form min_x h(c(x)), where h is a non-smooth but convex function and c is a smooth nonlinear or linear mapping. The non-smoothness of the outer function prevents traditional methods such as the Gauss-Newton method from converging quickly, which is problematic for large scale problems. In this talk, we will explore level set methods, aka the SPGL1 'trick', for solving this class of problem
when we can easily project on to the level sets of h(z). The resulting subproblems will be smooth and have simple constraints, which are amenable to smooth optimization methods such as LBFGS. We also use the variable projection technique, which gives us an alternate interpretation as computing the minimal distance between the level set of h and the image of our nonlinear mapping c. We will demonstrate the effectiveness of this technique on a number of convex and non-convex problems, including cosparsity-based compressed sensing for seismic data interpolation, audio signal declipping, robust tensor PCA/completion, and more.
We prove two related concentration inequalities concerning the points of elliptic curves contained within subsets of F_p^2. In particular we investigate the probability of a large difference between the portions of quadratic residues and non-residues in the image of f(S), where S is a subset of F_p and f is a cubic, or higher degree polynomial.
On March 22nd, we'll be discussing the article entitled "Turning Routine Exercises Into Activities that Teach Inquiry: A Practical Guide". This is a great article for anyone interested in finding out more about inquiry based learning and how it can be easily implemented. This could be especially relevant for anyone teaching students who are still relatively new to proofs and mathematical thinking.
Anyone interested is encouraged to attend. Snacks are provided.
This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse. Among the mathematicians with vignettes in this talk are Euler, Weierstrass, Fejer, Banach, Kolmogorov, Weyl, Cotlar, Fourier, Paley, Hausdorff and Schur.
Note for Attendees
A light reception will be served in ESB 4133, PIMS Lounge, from 2:30pm - 3:00pm
After defining the spectral theory of orthogonal polynomials on the unit circle (OPUC) and real line (OPRL), I'll describe Verblunsky's version of Szego's theorem as a sum rule for OPUC and the Killip--Simon sum rule for OPRL and their spectral consequences. Next I'll explain the original proof of Killip--Simon using representation theorems for meromorphic Herglotz functions. Finally I'll focus on recent work of Gambo, Nagel and Rouault who obtain the sum rules using large deviations for random matrices.
Note for Attendees
A light reception will be hosted at PIMS, ESB 4133 from 2:30pm- 3:00pm.
Let E be an elliptic curve over the rationals. Let K be an imaginary quadratic field and H_K the corresponding Hilbert class field. We discuss recent results on the arithmetic of E over H_K as K varies (joint with H. Hida and Y. Tian).
In this talk, I will discuss the formation of singularity in Lagrangian mean curvature flow and study the space of such singularity. The goal is to show a compactness result on the space of such singularity.
This is a joint work with Professor Jingyi Chen.
Note for Attendees
Refreshments will be served at 2:45 p.m. in MATH 125 before this colloquium.
Several patterns emerge in collections of Betti tables associated to the powers of a fixed ideal. For example, Wheildon and others demonstrated that the shapes of the nonzero entires of these tables eventually stabilize when the fixed ideal has generators of the same degree. In this talk, I will discuss patterns in the graded Betti numbers of these and other graded systems of ideals. In particular, I will describe ways in which the Betti tables may stabilize, and how different types of stabilization are reflected in the corresponding Boij-Soederberg decompositions.
A nonlinear optimization method is proposed for the solution of inverse scattering problems in the frequency domain, when the scattered field is governed by the Helmholtz equation. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type iteration. Instead of a grid-based discrete representation, the unknown wave speed is projected to a particular finite-dimensional basis of
eigenfunctions, which is iteratively adapted during the optimization. Truncating the adaptive eigenspace (AE) basis at a (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. Both analytical and numerical evidence underpins the accuracy of the AE representation. Numerical experiments demonstrate the efficiency and robustness to missing or noisy data of the resulting adaptive eigenspace inversion (AEI) method.
On Wed March 29th we'll continue the discussion from the previous week on simple ways to introduce inquiry based learning into a class. The article is "Turning Routine Exercises Into Activities that Teach Inquiry A Practical Guide" as before. Please bring a "routine" exercise or lecture example, as well as a more "inquiry based" version of the same example/exercise. The goal will be to discuss the challenges of turning routine exercises into inquiry based exercises.
Some questions for discussion:
Which kind of topics, examples or exercises lend themselves well to inquiry based learning?
What challenges did you encounter when trying to create your inquiry based exercise?
What challenges do you anticipate when implementing your inquiry based exercise in lecture or homework?
LetA be an isotropic, sub-gaussianm by nmatrix. We prove that the process Z_x = ||A x||_2 – m^(.5) ||x||_2has sub-gaussian increments. Using this, we show that for any bounded setT in R^n, the deviation of∥Ax∥2around its mean is uniformly bounded by the Gaussian complexity ofT. In other words, we give a simple sufficient condition for a random sub-Gaussian matrix to be well conditioned when restricted to a subset of R^n. We also prove a local version of this theorem, which allows for unbounded sets. These theorems have various applications, such as a general theory of compressed sensing. We discuss some applications and point to open (probabilistic) questions that remain.
A group is left-orderable if it has a strict total ordering that is invariant under multiplication from the left. For countable groups, this is equivalent to acting on the real line by order-preserving homeomorphisms. A group being circularly orderable has a slightly trickier algebraic definition than left-orderability, but in the countable case boils down, as expected, to the existence of a orientation-preserving action by homeomorphisms on the circle.
The set of all left-orderings of a group forms a topological space, and similarly, so does the set of all circular orderings. I will provide an introduction to these spaces, and discuss recent progress towards understanding the structure of groups whose spaces of circular orderings are “degenerate”, in the sense that they consist simply of a finite set of points with the discrete topology. This is joint work with Cristobal Rivas and Kathryn Mann.
Let F be a finite extension field of {\mathbb Q}_p. Let {\mathcal H} be the pro-p-Iwahori-Hecke algebra for {\rm GL}_n(F), with coefficients in the residue field k of {\mathcal O}_F (or a finite extension of it). We are going to discuss an exact functor D from the category of supersingular {\mathcal H}-modules to the category of (\psi,\Gamma)-modules over k((X)). The latter category generalizes in a straightforward way the one defined and studied by Colmez in the case F={\mathbb Q}_p; in particular, it admits an exact functor to the category of (\varphi,\Gamma)-modules, and hence to that of Galois representations. Our main result today is that the functor D is almost fully faithful, i.e. it is fully faithful when restricted to the category of supersingular {\mathcal H}-modules satisfying a very mild additional assumption.
(This talk is part of the PIMS focus semester on the mod p Langlands program).
Fluorescence microscopy allows experimental biologists to obtain quantitative data about different cellular processes. This data is then an important part of mathematical models to further understand the biological system. In this talk I will focus on two fluorescence techniques and their application to cell adhesion and immune activation. Cell adhesion to the extracellular matrix is fundamental for shape and stability of multicellular organisms. Experimentally, these adhesions can be observed using fluorescence recovery after photobleaching (FRAP). We built an ODE model to analyze changes in collected FRAP data under different mutations, and by fitting the model, identified necessary conditions for stable adhesions. In the second part, immune cell activation is believed to be triggered by clustering of membrane receptors. Experimentally this system requires lower and more precise fluorescence labelling, obtained using stochastic optical reconstruction microscopy (STORM). STORM uses photoswitchable fluorophores to achieve resolutions at or below 20nm, with the down side of possibly observing a given fluorophore multiple times in the process. We are developing a mathematical model to estimate the number of fluorophores present in the experiment. We apply a Markov chain model to describe the temporal dynamics, and a Gaussian mixture model for the spatial information. This approach will enhance a microscopy technique that is already widely used in biological applications, and will allow more precise analysis of receptor cluster formation and its effects on immune cell signaling.
Recently Bridgeland has introduced the notion of a BPS structure, which is meant to encode the output of unrefined Donaldson-Thomas theory. He studied an associated Riemann-Hilbert problem and found a relation with Gromov-Witten invariants in the case of the conifold. In this talk I will try to give an overview of this work, ending with some potential new directions to explore.
In this talk, we examine two methods for frequency extrapolation. Frequency extrapolation is the problem of utilizing data processing techniques to obtain the entire spectrum of an objective signal while only a middle band is sampled. This problem is well-posed for signals with special structures, such as those with a few non-zeros. The study is motivated by seismic inversion. Due to physical constraints, data obtained from a seismic survey is severely limiting in both the low and high frequency extent for the purposes of inversion. In particular, the missing low frequencies are known to be extremely essential, because when missing, most inversion algorithms are very likely to get stuck in local minima. In the numerical section, I will demonstrate the efficacy of the proposed methods in manufacturing low frequencies with real seismic examples and visualized inversion results.
We will discuss about the regularity of the Gauss curvature flow: the optimal C^{1,\frac{1}{n-1}} regularity of degenerate solutions with flat sides and the interior C^{\infty} regularity of strictly convex solutions.
L^p inequalities for certain Littlewood-Paley functionals arising from Lévy processes will be discussed. These are motivated by applications to the L^p boundedness of Fourier multiples which give L^p regularity of solutions to some non-local operators, including the fractional Laplacian. Non-local operates have been extensively studied in recent years by researchers in analysis, probability and PDE. The relevant Fourier multiples have been studied using the deep sharp martingale transform inequalities of Burkholder. The proofs here, although not sharp, are completely elementary and use nothing more than Itô’s formula and arguments similar to those for the classical case of the Laplacian.
Penalized regression methods permit data analysts to fit models with more adjustable parameters than data points by imposing strong prior assumptions on the relationships between variables. As a result classical tools of uncertainty assessment no longer work. I have been involved in one approach to uncertainty assessment for problems of this sort. I will try to summarize the issues and contrast the current approaches.
I will discuss finite group actions on integral or rational homology 3-spheres. The main examples for this talk are the Brieskorn integral homology 3-spheres M(p,q,r) arising from isolated singularities, which bound smooth 4-manifolds with definite intersection forms. In addition, there are special infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. We ask whether the free periodic actions on Brieskorn spheres extend to smooth actions with isolated fixed points on one of these associated 4-manifolds.
This lecture is an invitation to real algebraic geometry, along with computational aspects, ranging from bitangents and K3 surfaces to eigenvectors and ranks of tensors. We present an experimental study - with many pictures - of smooth curves of degree six in the real plane. The number 64 refers to rigid isotopy types in the Rokhlin-Nikulin classification.
Note for Attendees
Refreshments will be served at 2:45 p.m. in PIMS lounge before this colloquium
We present an analog of the total variation image reconstruction approach by Rudin, Osher, Fatemi (1992) for images defined on smooth surfaces, together with a proper analytical framework. The problem is defined in terms of quantities intrinsic to the surface and it is therefore independent of the parametrization. It is shown that the Fenchel predual of the total variation problem is a quadratic optimization problem for the vector-valued predual variable with pointwise constraints on the surface. The predual problem is solved using a function space interior point method, and discretized by conforming Raviart-Thomas finite elements on a triangulation of the surface. As in the flat case, the predual variable serves as an edge detector. Numerical examples including denoising and un-erasing problems with both gray-scale and color images on complex 3D geometries are presented.
I will present a joint work with Martin Li. Minimal surfaces with freeboundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with freeboundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with freeboundary in any compact smooth Euclidean domain. The minimal surfaces with freeboundary were constructed using the min-max method. I will explain the basic ideas behind the min-max theory as well as our new contributions.
The fact that |xy| = |x| |y| for real, complex, quaternion and octonion numbers x and y leads to the open question of finding all possible maps f: R^r X R^s ---> R^n satisfying |f(x,y)| = |x| |y| for each x in R^r and y in R^s. For example, that no such f can exist when r=s=n=16 was a celebrated result in classical algebra.
When the components of f(x,y) are required to be bilinear forms in the components of x and y with integer coefficients, one can crudely encode f by a colored matrix M of r rows and s columns, using n colors but avoiding certain "forbidden configurations". In 1981, S. Yuzvinsky conjectured that the chromatic number for this kind of coloring of M should be given by a certain function of r and s already familiar to topologists. In this talk I shall prove his conjecture for a majority of values of r and s, including the case of square matrices r=s. I shall explain how each step in my combinatorial proof was indeed suggested by topological considerations.
This talk is dedicated to the memory of professor Erhard Luft (1933-2017).
A lattice L is the set of vectors arising from integer linear combinations of given basis vectors in R^n. Given some vector x, the Closest Vector Problem (CVP) is to find a vector v in L of minimum l_2-norm distance to x. CVP is a fundamental problem for lattices with many applications, and it is in general NP Hard.
A zonotopal lattice is given as the set of integer points {v | Mv = 0} when M is a totally unimodular matrix. We show how to adapt the Cancel and Tighten algorithm of Karzanov and McCormick to solve CVP for zonotopal lattices in O(n^3) time via the Seymour decomposition of totally unimodular matrices. The algorithm uses the decomposition to reduce the problem to a series of subproblems that are piecewise linear convex circulation and co-circulation network flow problems.
by
Britta Peis, Robert Scheidweiler (RWTH Aachen)
S. Thomas McCormick (UBC Sauder)
Frank Vallentin (Cologne)
The independence polynomial has been widely studied in algebraic graph theory, in statistical physics, and in algorithms for counting and sampling problems. Seminal results of Weitz (2006) and Sly (2010) have shown that in bounded-degree graphs the independence polynomial can be efficiently approximated if the argument is real, positive and below a certain threshold, whereas above that threshold the polynomial is hard to approximate. Furthermore, this threshold exactly corresponds to a phase transition in physics, which demarcates the region within which the Gibbs measure has correlation decay.
We consider the problem of computing the independence polynomial with a negative (or even complex) argument, whose magnitude is less than the smallest magnitude of any root of the polynomial. We show that there is a fully-polynomial-time approximation scheme (FPTAS) for such an argument. This result actually holds much more generally for the multivariate independence polynomial, in which each vertex has its own activity. Our proof uses a novel multivariate form of the correlation decay technique.
This FPTAS can be used to give a constructive algorithm for the Lovasz Local Lemma in probabilistic combinatorics.
One important construction in the theory of metric spaces is the tight span. The tight span of a metric space can be thought of as a generalization of the idea of a convex hull in linear spaces and is the basis for much work in the study and visualization of finite metric spaces. Motivated by problems in phylogenetics, we have developed a generalization of the concept of metric spaces, which we call diversities. In a diversity, every subset of points in the space corresponds to a number, not just pairs, and there is a more general version of the triangle inequality. Besides encompassing a number of interesting examples as special cases, diversities have a natural tight span construction with corresponding theory. I will give an introduction to tight span theory for metric spaces and then show how it extends to our theory of diversities. I will conclude by demonstrating the relation between diversities and Steiner tree packing in graphs. This is joint work with David Bryant (University of Otago, New Zealand).
The Goodwillie derivatives of the identity functor on pointed spaces form an operad in spectra that is very closely related to the Lie operad. I will describe the mod 2 homology operations for algebras over this operad. This talk will not assume prior knowledge of either operads or Goodwillie calculus. Sadly, due to time constraints, it also won't explain any Goodwillie calculus! (Instead I will start from a combinatorial description of the derivatives of the identity functor.)
Let K be a field and let P be a polynomial. What can we say about the field generated by the roots of P and of all its iterates? I will discuss some questions motivated by this general problem when K is a p-adic field. Along the way, we'll see Coleman power series, p-adic dynamical systems and a little bit of p-adic Hodge theory.
(This talk is part of the PIMS focus semester on the mod p Langlands program).
We prove that any sequence of conformally branched compact Lagrangian self-shrinkers in four dimensional Euclidean space with uniform area upper bound and fixed genus has a convergent subsequence, if the conformal structures do not degenerate. When the genus is one, we can drop the assumption on non-degeneracy the conformal structures. We also show that there is no branched immersion of Lagrangian self-shrinking sphere. When the area bound is small, we show that any such Lagrangian self-shrinking torus is embedded with uniform curvature estimates. For a general area bound, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Lojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori.
Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori, along which the Lagrangian condition is preserved, area is decreasing, and the compact type I singularities with a fixed area upper bound can be perturbed away in finitely many steps. This is a Lagrangian version of the construction for embedded surfaces by Colding-Minicozzi.
In the noncompact situation, we derive a parabolic Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on l-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in Euclidean spaces with uniform bounded second fundamental form. This generalizes a result of Wang for compact immersions. We also prove an Omori-Yau maximum principle for properly immersed self-shrinkers.
We present Possession Sketches, a new machine learning method for organizing and exploring a database of basketball player-tracks. Our method organizes basketball possessions by offensive structure. We first develop a model for populating a dictionary of short, repeated, and spatially registered actions. Each action corresponds to an interpretable type of player movement. We examine statistical patterns in these actions, and show how they can be used to describe individual player behavior. Leveraging this vocabulary of actions, we develop a hierarchical model that describes interactions between players. Our approach draws on the topic-modeling literature, extending Latent Dirichlet Allocation (LDA) through a novel representation of player movement data which uses techniques common in animation and video game design. We show that our model is able to group together possessions with similar offensive structure, allowing for efficient search and exploration of the entire database of player-tracking data. We show that our model finds repeated offensive structure in teams (e.g. strategy), providing a much more sophisticated, yet interpretable lens into basketball player-tracking data.
Note for Attendees
This is a talk in the BC Data Colloquium. Reception beforehand in ESB 4133 (the PIMS lounge).
It has been said that mathematics is the poetry of science. Some of the fundamental values of poetry parallel those of mathematics, and mathematical research is, in many ways, an art. Professor Cédric Villani will discuss the interface between mathematics and art, showing how both these disciplines seek to illuminate hidden beauty in the world.
Note for Attendees
This lecture with Cédric Villani is sold out. There will be a standby line available on the night of the lecture.
Stability of classical mechanical systems is very old problem, still very much under scrutiny nowadays. Some techniques can be applied transversally in different problems.
***
Cédric Villani is the Director of the Institut Henri Poincaré, France’s prime and oldest international institute for research in mathematical sciences. He has received many mathematical awards, including the Fields Medal in 2010, often considered the most prestigious in mathematics. Prof. Villani is a specialist of mathematical analysis applied to problems of statistical physics, geometry and probability. His books on gas theory and optimal transport theory have become classics.
Note for Attendees
Join us for coffee and cookies prior to the talk from 2:45pm - 3:00pm : ESB 4133
A saturated fusion system associated to a finite group G encodes the p-structure of the group as the Sylow p-subgroup enriched with additional conjugation. The fusion system contains just the right amount of algebraic information to for instance reconstruct the p-completion of BG, but not BG itself. Abstract saturated fusion systems F without ambient groups exist, and these have (p-completed) classifying spaces BF as well.
In a joint project with Tomer Schlank and Nat Stapleton, we combine the theory of abstract fusion systems with the work by Hopkins-Kuhn-Ravenel and Stapleton on transchromatic character maps, and we generalize several results from finite groups to fusion systems.
A main ingredient of this project is studying the free loop spaces L(BG) and L(BF) for groups and fusion systems, and constructing transfer maps from L(BG) to L(BH) when H is a subgroup of G.
Simplicial commutative rings are one of the first steps into "derived" rings that one can take. Many constructions for general E_infty-ring spectra or even Z-algebras are simpler in the world ofsimplicial commutative rings; however, from a purelyhomotopy-theoretic or categorical picture they are slightlymysterious. I will explain ongoing work with Bhargav Bhatt on anextended theory of "generalized rings" which extends this category toallow nonconnective objects. Many "equational" constructions whichcannot work with E_infty-rings extend well to generalized rings.
I will talk about the problem of studying the reduction modulo p of crystalline representations of dimension 2 of the Galois group of Q_p. In particular, I will be interested in the following situation: fix Hodge-Tate weights and a residual representation, and consider the locus parametrizing crystalline representations with the given weights and reduction modulo p. What can be said about this locus in general?
(This talk is part of the PIMS focus semester on the mod p Langlands program).
Abstract: It has been more than hundred years since the death of Henri Poincaré, the world’s greatest mathematician (as we like to say in France, and abroad as well). From the onset, Poincaré’s work and writings, and himself as a universalist and continue to be one of the main symbols of creativity of the human mind and spirit. His errors, however prove that even the greatest, make mistakes.
We consider the set of isomorphism classes of elliptic curves over a finite field k from a probabilistic point of view. Let t_E denote the trace of the Frobenius endomorphism, A a finite abelian group, 1_A the characteristic function of the event that there is a subgroup of E(k) isomorphic to A, and R a non-negative integer. In joint work with N. Kaplan we give explicit formulas for the expectation of t_E^R 1_A in terms of elementary number theory functions and traces of Hecke operators on spaces of classical modular forms. The formulas are necessarily complicated but quite usable in practice as one knows a lot about these spaces of modular forms.
The emphasis of this thesis is on critical parabolic problems, in particular, the harmonic map heat from the plane to S2, and nonlinear focusing heat equations with an algebraic nonlinearity.
The focus of this work has been on long-time dynamics, stability and singularity formation, and the investigation of the role of special, soliton-like, solutions to the asymptotic behaviour of solutions.
Harmonic Map Heat Flow: We consider m-corotational solutions to the harmonic map heat flow from R2 to S2. We first work in a class of maps with trivial topology and energy of the initial data below two times the energy of the stationary harmonic map solutions. We give a new proof of global existence and decay. The proof is based on the "concentration-compactness plus rigidity" approach of Kenig and Merle and relies on the dissipation of the energy and a profile decomposition. We also treat m-corotational maps (m greater than 4) with non-trivial topology and energy of the initial data less than three times the energy of the stationary harmonic map solutions. Through a new stability argument we rule out finite-time blow-up and show that the global solution asymptotically converges to a harmonic map.
Nonlinear Heat Equation: We also study solutions of the focusing energy-critical nonlinear heat equation. We show that solutions emanating from initial data with energy and kinetic energy below those of the stationary solutions are global and decay to zero. To prove that global solutions dissipate to zero we rely on a refined small data theory, L2-dissipation and an approximation argument. We then follow the "concentration-compactness plus rigidity" roadmap of Kenig and Merle (and in particular the approach taken by Kenig and Koch for Navier-Stokes) to exclude finite-time blow-up.
The talk will be divided into 2 parts. First, we will theoretically justify the applicability of compressed sensing (CS) in real-life applications. To do so, I will introduce CS theorems compatible with physical acquisition constraints. These new results do not only encompass structure in the acquisition but also structured sparsity of the signal of interest. Then, we will present a new way to generate subsampling schemes that can be implemented on real sensors and that give good reconstruction results. This work relies on measure projection and will be illustrated in the case of MRI.
This is joint work with Frederic Naud (Avignon). After reviewing general results about resonances on (asymptotically) hyperbolic manifolds, we discuss some recent results on the distribution of resonances for infinite index congruence subgroups of SL(2,Z), as well as some conjectures.
The Gauss curvature of a convex body can be seen as a measure on the unit sphere (with some properties). For such a measure \mu , Alexandrov problem consists in proving the existence of a convex body whose curvature measure is \mu . In the Euclidean space, this problem is equivalent to an optimal transport problem on the sphere.
In this talk I will consider Alexandrov problem for convex bodies of the hyperbolic space. After defining the curvature measure, I will explain how to relate this problem to a non linear Kantorovich problem on the sphere and how to solve it.
We will present different ways to compute the p-adi etale cohomology of layers of the Drinfeld tower, and give applications to the p-adic local Langlands correspondence. (Lecture 1 of 4)
On a complex variety, you can integrate a differential form over a cycle to get a period. For instance, an elliptic curve has two periods, whose quotient gives an element of the upper half plane. There is a family of concepts (Hodge decomposition, variation of Hodge structures, Shimura varieties) arising from the study of periods on families of complex varieties. What if the complex variety is replaced with a rigid-analytic variety over a p-adic field? We will review work of Tate, Fontaine, Kedlaya-Liu, Scholze and others that falls under the domain of p-adic Hodge theory. One goal will be to understand the surprising Hodge-Tate period map, defined by Scholze, attached to the modular curve at infinite level. (Lecture 1 of 4)
We will present different ways to compute the p-adi etale cohomology of layers of the Drinfeld tower, and give applications to the p-adic local Langlands correspondence. (Lecture 2 of 4)
On a complex variety, you can integrate a differential form over a cycle to get a period. For instance, an elliptic curve has two periods, whose quotient gives an element of the upper half plane. There is a family of concepts (Hodge decomposition, variation of Hodge structures, Shimura varieties) arising from the study of periods on families of complex varieties. What if the complex variety is replaced with a rigid-analytic variety over a p-adic field? We will review work of Tate, Fontaine, Kedlaya-Liu, Scholze and others that falls under the domain of p-adic Hodge theory. One goal will be to understand the surprising Hodge-Tate period map, defined by Scholze, attached to the modular curve at infinite level. (Lecture 2 of 4)
We will present different ways to compute the p-adi etale cohomology of layers of the Drinfeld tower, and give applications to the p-adic local Langlands correspondence. (Lecture 3 of 4)
On a complex variety, you can integrate a differential form over a cycle to get a period. For instance, an elliptic curve has two periods, whose quotient gives an element of the upper half plane. There is a family of concepts (Hodge decomposition, variation of Hodge structures, Shimura varieties) arising from the study of periods on families of complex varieties. What if the complex variety is replaced with a rigid-analytic variety over a p-adic field? We will review work of Tate, Fontaine, Kedlaya-Liu, Scholze and others that falls under the domain of p-adic Hodge theory. One goal will be to understand the surprising Hodge-Tate period map, defined by Scholze, attached to the modular curve at infinite level. (Lecture 3 of 4)
We will present different ways to compute the p-adi etale cohomology of layers of the Drinfeld tower, and give applications to the p-adic local Langlands correspondence. (Lecture 4 of 4)
The classical theory of (phi, Gamma)-modules relates continuous p-adic representations of the Galois group of a p-adic field with modules over a certain mildly noncommutative ring. That ring admits a description in terms of a group algebra over Z_p which is crucial for Colmez's p-adic local Langlands correspondence for GL_2(Q_p). We describe a method for applying a key property of perfectoid spaces, the analytic analogue of Drinfeld's lemma, to the construction of "multivariate (phi, Gamma)-modules" corresponding to p-adic Galois representations in more exotic ways. Based on joint work with Annie Carter and Gergely Zabradi.
On a complex variety, you can integrate a differential form over a cycle to get a period. For instance, an elliptic curve has two periods, whose quotient gives an element of the upper half plane. There is a family of concepts (Hodge decomposition, variation of Hodge structures, Shimura varieties) arising from the study of periods on families of complex varieties. What if the complex variety is replaced with a rigid-analytic variety over a p-adic field? We will review work of Tate, Fontaine, Kedlaya-Liu, Scholze and others that falls under the domain of p-adic Hodge theory. One goal will be to understand the surprising Hodge-Tate period map, defined by Scholze, attached to the modular curve at infinite level. (Lecture 4 of 4)
Suppose that rho is an irreducible automorphic n-dimensional global p-adic Galois representation that is upper-triangular locally at p. In previous work with Breuil we constructed a unitary representation of GL_n(Q_p) on a p-adic Banach space (depending only on rho locally at p) that is an extension of finitely many principal series, and we conjectured that this representation occurs globally in a space of p-adic automorphic forms cut out by rho. In work in progress we prove many new cases of this conjecture, assuming that rho is moreover crystalline with distinct Hodge-Tate weights.
The Laplacian matrices of graphs arise in many fields, including Machine
Learning, Computer Vision, Optimization, Computational Science, and of
course Network Analysis. We will explain what these matrices are and why
they appear in so many applications.
We then survey recent ideas that allow us to solve systems of linear
equations in Laplacian matrices in nearly linear time, emphasizing the
utility of graph sparsification---the approximation of a graph by a sparser
one---and a recent algorithm of Kyng and Sachdeva that uses random sampling
to accelerate Gaussian Elimination.
Networks describe how things are connected, and are ubiquitous in science and society. Networks can be very concrete, like road networks connecting cities or networks of wires connecting computers. They can represent more abstract connections such as friendship on Facebook. Networks are widely used to model connections between things that have no real connections. For example, Biologists try to understand how cells work by studying networks connecting proteins that interact with each other, and Economists try to understand markets by studying networks connecting institutions that trade with each other.
Questions we ask about a network include "which components of the network are the most important?", "how well do things like information, cars, or disease spread though the network?", and "does the network have a governing structure?".
I will explain how mathematicians address these questions by modeling networks as physical objects, imagining that the connections are springs, electrical resistors, or pipes that carry fluid, and analyzing the resulting systems.
About the Niven Lectures: Ivan Niven was a famous number theorist and expositor; his textbooks have won numerous awards and have been translated into many languages. They are widely used to this day. Niven was born in Vancouver in 1915, earned his Bachelor's and Master's degrees at UBC in 1934 and 1936 and his Ph.D. at the University of Chicago in 1938. He was a faculty member at the University of Oregon since 1947 until his retirement in 1982. The annual Niven Lecture, held at UBC since 2005, is funded in part through a generous bequest from Ivan and Betty Niven to the UBC Mathematics Department.
Elastoplastic deformations play a tremendous role in industrial forming. Many of these processes happen at non-isothermal conditions. Therefore, the optimization of such problems is of interest not only mathematically but also for applications.
In this talk we will present the analysis of the existence of a global solution of an optimal control problem governed by a thermovisco(elasto)plastic model. We will point out the difficulties arising from the nonlinear coupling of the heat equation with the mechanical part of the model. Finally, we will discuss first numerical results.
The talk is based on joint work with Roland Herzog and Christian Meyer.
ABSTRACT
The central concern of this thesis is the study of critical behaviour in models of statistical physics in the upper-critical dimension. We study a generalized n-component lattice |φ|4 model and a model of weakly self-avoiding walk with nearest-neighbour contact self-attraction on the Euclidean lattice Zd. By utilizing a supersymmetric integral representation involving boson and fermion fields, the two models are studied in a unified manner.
Our main result, which is contingent on a small coupling hypothesis, identifies the precise leading-order asymptotics of the two-point function, susceptibility, and finite-order correlation length of both models in d = 4. In particular, we show that the critical two-point function satisfies mean-field scaling whereas the near-critical susceptibility and finite-order correlation length exhibit logarithmic corrections to mean-field behaviour. The proof employs a renormalisation group method of Bauerschmidt, Brydges, and Slade based on a finite-range covariance decomposition and requires two extensions to this method.
The first extension, which is required for the computation of the finite-order correlation length (even for the ordinary weakly self-avoiding walk and |φ|4 model), is an improvement of the norms used to control the evolution of the renormalisation group. This allows us to obtain improved error estimates in the massive regime of the renormalisation group flow.
The second extension involves the identification of critical parameters for models initialized with a non-zero error coordinate coupled to a marginal/relevant coordinate. This allows us, for example, to realize the two-point function and susceptibility for the walk with self-attraction as a small perturbation of the corresponding quantities without self-attraction, whose asymptotic behaviour was determined by Bauerschmidt, Brydges, and Slade. This establishes a form of universality
We present a unified approach to quadratic asymptote of solutions to a class of fully nonlinear elliptic equations over exterior domains, including Monge-Ampere equations (previously known), special Lagrangian equations, quadratic Hessian equations, and inverse harmonic Hessian equations. This is joint work with Dongsheng Li and Zhisu Li.
With super-resolution microscopy techniques such as Direct Stochastic Optical Reconstruction Microscopy (dSTORM), it is possible to image fluorescently labeled proteins on a cell membrane with high precision. Often, the extent to which such proteins cluster is biologically meaningful; for example, in B-cells, clustering of the B-cell receptor (BCR) is associated with increased intracellular signaling and B-cell activation, and spontaneous BCR clustering can cause chronic active BCR signaling that results in an aggressive B-cell malignancy. Computational methods are therefore needed to make quantifiable comparisons between the observed clustering in different data sets, such as for different cell types or different experimental conditions.
Inspired by the success of graph-based clustering algorithms such as PhenoGraph in other research areas, we developed StormGraph, a graph-based clustering algorithm for analyzing Single Molecule Localization Microscopy (SMLM) data such as would be obtained by dSTORM.
This talk will present StormGraph, which distinguishes clusters from random background and assigns individual localizations to specific clusters,
allowing for a detailed analysis of statistics such as cluster area. The utility of StormGraph will be illustrated on dSTORM data of BCRs imaged on malignant B-cells.
The study of spaces of homomorphisms from a discrete group to a compact Lie group has led to the definition of a new cohomology theory, called commutative K-theory. This theory, which was first introduced by Adem and Gomez, is a refinement of classical topological K-theory. It is defined using vector bundles which can be represented by commuting cocycles. I will begin the talk by discussing some general properties of the "classifying space for commutativity in a Lie group". Specialising to the unitary groups, I will show that the classifying space for commutative complex K-theory is precisely the E-Infinity ring space underlying the ku-group ring of BU(1). If time permits, I will mention some results about the real variant of commutative K-theory.
Abstract:
We prove several theorems concerning random walks, harmonic functions, percolation, uniform spanning forests, and circle packing. We study these models primarily on planar graphs and on unimodular random rooted graphs, although some of our results hold for more general classes of graphs. Broadly speaking, we are interested in the interplay between the geometry of a graph and the behaviour of probabilistic processes on that graph. Material taken from a total of nine papers is included. We have also included an extended introduction explaining the background to these papers.
University of British Columbia, Earth Science Bldg. Room 1013
Fri 16 Jun 2017, 4:00pm-5:00pm
Abstract
We will survey some of the mathematical objects arising naturally in probability theory, as well as some of their surprising properties. In particular, we will see how one of these objects was involved in the confirmation of the existence of atoms over 100 years ago and how new properties of related objects are still being discovered today.
Last time we have seen that the spectrum for commutative K-theory is equivalent to the ku-group ring of BU(1). In this second talk I will show how this induces a splitting of the space BcomU as a product of all the stages in the Whitehead tower for BU. I will then explain the relationship between the homotopy groups of BcomU and certain rational characteristic classes introduced by Adem and Gómez. Finally, I would like to explain an alternative but equivalent picture using Segal's Dold-Thom functor for K-homology theory.
A brief introduction of translation-invariant kernel based methods, aka radial basis function methods, for function approximation and PDEs will be given. These methods do not require meshes, but in return, we have to deal with highly ill-conditioned linear systems. In this talk, we will introduce an adaptive algorithm that selects appropriate column subspaces that ensure linear independency.
Room 202, Anthropology and Sociology Building, UBC
Mon 10 Jul 2017, 9:00am-11:00am
Details
Oral Defense Abstract: In this thesis, we investigate the existence of ground state solutions associated to the fractional Hardy-Schrödinger operator on Euclidean space and its bounded domains. In the process, we extend several results known about the classical Laplacian to the non-local operators described by its fractional powers. Our analysis show that the most important parameter in the problems we consider is the intensity of the corresponding Hardy potential. The maximal threshold for such an intensity is the best constant in the fractional Hardy inequality, which is computable in terms of the dimension and the fractional exponent of the Laplacian. However, the analysis of corresponding non-linear equations in borderline Sobolev-critical regimes give rise to another threshold for the allowable intensity. Solutions exist for all positive linear perturbations of the equation, if the intensity is below this new threshold. However, once the intensity is beyond it, we had to introduce a notion of Hardy-Schrödinger Mass associated to the domain under study and the linear perturbation. We then show that ground state solutions exist when such a mass is positive. We then study the effect of non-linear perturbations, where we show that the existence of ground state solutions for large intensities, is determined by a subtle combination of the mass (i.e, the geometry of the domain) and the size of the nonlinearity of the perturbations.
ABSTRACT: This thesis investigates the geometry of random spaces.
Geodesics in Random Surfaces
The Brownian map, developed by Le Gall and Miermont, is a random metric space arising as the scaling limit of random planar maps. Its construction involves Aldous’ continuum random tree, the canonical random real tree, and Brownian motion, an almost surely continuous but nowhere differentiable path. As a result, the Brownian map is a non-differentiable surface with a fractal geometry that is much closer to that of a real tree than a smooth surface.
A key feature, observed by Le Gall, is the confluence of geodesics phenomenon, which states that any two geodesics to a typical point coalesce before reaching the point. We show that, in fact, geodesics to anywhere near a typical point pass through a common confluence point. This leads to information about special points that had remained largely mysterious.
Our main result is the almost everywhere continuity and uniform stability of the cut locus of the Brownian map. We also classify geodesic networks that are dense and find the Hausdorff dimension of the set of pairs that are joined by each type of network.
Susceptibility of Random Graphs
Given a graph G=(V,E) and an initial set I of active vertices in V, the r-neighbour bootstrap percolation process, attributed to Chalupa, Leath and Reich, is a cellular automaton that evolves by activating vertices with at least r active neighbours. If all vertices in V are activated eventually, we say that I is contagious. A graph with a small contagious set is called susceptible.
Bootstrap percolation has been analyzed on deterministic graphs, such as grids, lattices and trees. More recent work studies the model on random graphs, such as the fundamental Erdős–Rényi graph G(n,p).
We study thresholds for the susceptibility of G(n,p), refining approximations by Feige, Krivelevich and Reichman. Along the way, we develop large deviation estimates, which complement central limit theorems of Janson, Łuczak, Turova and Vallier. We also study graph bootstrap percolation, a variation due to Bollobás. Our main result identifies the sharp threshold for K4-percolation, solving a problem of Balogh, Bollobás and Morris.
We discuss a family of Cantor sets on which the equilibrium measure coincides with the corresponding Hausdorff measure. Several applications are presented. Some asymptotics for orthogonal polynomials (Widom factors) are given for non Parreau-Widom sets. Our main concern is the concept of the Szeg˝o class for singular measures.
Room 202, Anthropology and Sociology Building, UBC
Thu 20 Jul 2017, 12:30pm-2:30pm
Details
Oral Defense Abstract: Let G be a real linear algebraic group and L a finitely generated cosimplicial group. We prove that the space of homomorphisms Hom(Ln,G) has a homotopy stable decomposition for each n ≥ 1. When G is a compact Lie group, we show that
the decomposition is G-equivariant with respect to the induced action of conjugation by elements of G. In particular, under these hypotheses on G, we obtain stable decompositions for Hom(Fn /Γ,G) and Rep(Fn /Γq,G) respectively, where Fn /Γq are the finitely generated free nilpotent groups of nilpotency class q − 1. The spaces Hom(Ln,G) assemble into a simplicial space Hom(L,G). When G = U we show that its geometric realization B(L,U) has a non-unital E∞ -ring space structure whenever Hom(L0,U(m)) is path connected for all m ≥ 1. We describe the connected components of Hom(Fn /Γq,SU(2)) arising from non-commuting q-nilpotent n-tuples. We prove this by showing that all these n-tuples are conjugated to n-tuples consisting of elements in the the generalized quaternion groups Q2^q ⊂ SU(2), of order 2^q . Using this result, we exhibit the homotopy type of ΣHom(Fn/Γq,SU(2)) and a homotopy description of the classifying spaces B(q,SU(2)) of transitionally (q-1)-nilpotent principal SU(2)-bundles. The above computations are also done for SO(3) and U(2). We also include cohomology calculations for the spaces B(r,Q2^q ) for low values of r. Finally we compute the integral cohomology ring of BcomG_1 for the Lie groups G=SO(3), SU(2) and U(2)..
Hamilton-Jacobi-Bellman (HJB) equations are nonlinear partial differential equation that arise as optimality conditions of stochastic control problems. HJB equations often possess a variety of difficulties, e.g., discontinuous coefficients, vanishing viscosity, unknown boundary conditions, etc. One particular issue is the handling of non-existing second-order derivatives. In this presentation we focus on the discretization of HJB equations with a fully implicit timestepping scheme based on a semi-Lagrangian approach. We discuss the algorithmic idea in the context of a finite difference approximation and present numerical examples.
Auditorium Annex (AUDX) Room 142, 1924 West Mall, UBC
Thu 10 Aug 2017, 10:00am-12:00pm
Details
Abstract:
This thesis concerns critical branching random walks. We focus on the supercritical (d>=5) and critical (d=4) dimensions.
In this thesis, we extend the potential theory for random walk to critical branching random walk. In the supercritical dimensions, we introduce branching capacity for every finite subset of Z^d and construct its connections with critical branching random walk through the following three perspectives.
i) The visiting probability of a finite set by a critical branching random walk starting far away;
ii) Branching recurrence and branching transience;
iii) Local limit of branching random walk in torus conditioned on the total size.
In the critical dimension, we also construct some parallel results. On the one hand, we give the asymptotics of visiting a finite set and the convergence of the conditional hitting point. On the other hand, we establish the asymptotics of the range of a branching random walk conditioned on the total size.
Also in this thesis, we analyze a small game which we call the Majority-Markov game and give an optimal strategy.
ABSTRACT:
The present thesis is split in two parts. The first deals with the focusing Nonlinear Schrödinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.
The second considers the perturbed energy critical focusing Nonlinear Schrödinger Equation in three dimensions. We construct solitary wave solutions for focusing subcritical perturbations as well as defocusing supercritical perturbations. The construction relies on the resolvent expansion, which, is singular due to the presence of a resonance. Specializing to pure power focusing subcritical perturbations we demonstrate, via variational arguments, the existence of a ground state soliton, which, is then shown to be the previously constructed solution. Finally, we achieve a dynamical theorem which characterizes the fate of solutions whose initial data are below the action of the ground state. Such solutions will either scatter or blow-up in finite time depending on their initial data.
ABSTRACT: Inverse problems are an important class of problems found in many areas of science and engineering. In these problems, one aims to estimate unknown parameters of a physical system through indirect multi-experiment measurements. Inverse problems arise in a number of fields including seismology, medical imaging, and astronomy, among others.
An important aspect of inverse problems is the quality of the acquired data itself. Real-world data acquisition restrictions, such as time and budget constraints, often result in measured data with missing entries. Many inversion algorithms assume that the input data is fully sampled and relatively noise free and produce poor results when these assumptions are violated. Given the multidimensional nature of real-world data, we propose a new low-rank optimization method on the smooth manifold of Hierarchical Tucker tensors. Tensors that exhibit this low-rank structure can be recovered from solving this non-convex program in an efficient manner. We successfully interpolate realistically sized seismic data volumes using this approach.
If our low-rank tensor is corrupted with non-Gaussian noise, the resulting optimization program can be formulated as a convex-composite problem. This class of problems involves minimizing a non-smooth but convex objective composed with a nonlinear smooth mapping. In this thesis, we develop a level set method for solving composite-convex problems and prove that the resulting subproblems converge linearly. We demonstrate that this method is competitive when applied to examples in noisy tensor completion, analysis-based compressed sensing, audio declipping, total-variation deblurring and denoising, and one-bit compressed sensing.
With respect to solving the inverse problem itself, we introduce a new software design framework that manages the cognitive complexity of the various components involved. Our framework is modular by design, which enables us to easily integrate and replace components such as linear solvers, finite difference stencils, preconditioners, and parallelization schemes. As a result, a researcher using this framework can formulate her algorithms with respect to high-level components such as objective functions and hessian operators. We showcase the ease with which one can prototype such algorithms in a 2D test problem and, with little code modification, apply the same method to large-scale 3D problems.
In many applications one wishes to minimize an objective function that is not convex and is not differentiable at its minimizers. We discuss two algorithms for minimization of nonsmooth, nonconvex functions. Gradient Sampling is a simple method that, although computationally intensive, has a nice convergence theory. The method is robust and the convergence theory has recently been extended to constrained problems. BFGS is a well known method, developed for smooth problems, but which is remarkably effective for nonsmooth problems too. Although our theoretical results in the nonsmooth case are quite limited, we have made some remarkable empirical observations and have had broad success in applications. Limited Memory BFGS is a popular extension for large problems, and it is also applicable to the nonsmooth case, although our experience with it is more mixed. Throughout the talk we illustrate the ideas through examples, some very easy and some very challenging. Our work is with Jim Burke (U. Washington) and Adrian Lewis (Cornell).
For those in Vancouver on the Labour Day weekend, we have a really spectacular public event, on Friday evening (1st September). Lord Rees, from Cambridge (UK) will give a public lecture on "Mars and the Multiverse"; the abstract is below. Rees, one of the 20th century's most important astrophysicists, has also written on many other topics. He is well known around the world as a public speaker, particularly on our place in the universe as we now understand it, how to reconcile the findings of astronomy with religion, and the future of the human race. The talk he will give at UBC will allow extensive audience participation and dialogue with him, with plenty of opportunity for questions and discussion with him. This promises to be a really exceptional event. The talk will be suitable for the general public, including high school students.
Abstract:
In the year 2017, unmanned spacecraft have visited all the planets of our Solar System, and even some of their moons, and extensively explored Mars.
Many thousands of planets have been found orbiting other stars -- some of these planets even resembling our Earth. Looking further afield, observers have probed galaxies and the massive back holes at their centres, and checked models of their evolution by detecting them all the way back to their formation. Indeed we can trace pre-galactic history back to a nanosecond after the 'Big Bang'. But the key features of our universe -- its expansion rate, geometry and content -- were established far earlier even than this, at a time that precision measurements are now trying to probe.
All these advances pose key questions: What does the long-range cosmic future hold? Should we be surprised at the emergence of life? Is physical reality even vaster than what we can see? Are there many 'big bangs' and many universes? In this illustrated talk I will address these questions.
Martin Rees is one of the pioneers of modern astrophysics, and played a key role in our understanding of black holes. A Professor at Cambridge University, he has also been successively Astronomer Royal (UK), and President of the Royal Society (London). He is well-known to the general public for his popular books and for his public lectures - the one he will give at UBC in the evening of Sept 1st is for the general public, from 16 years upwards.
Given a compact Kähler manifold X and a biholomorphic self-map g of X, the topological entropy of g plays an important role in the study of dynamical system (X, g). In this talk, I first talk about a generalization of a surface result, that is, a parabolic automorphism of a compact Kähler surface preserves an elliptic fibration, to hyperkähler manifolds. We give a criterion for the existence of equivariant fibrations on ‘certain’ hyperkähler manifolds from a dynamical viewpoint. Next, I will generalize a finiteness result for the null-entropy subset of a commutative automorphism group due to Dinh–Sibony (2004), to arbitrary virtually solvable groups G of maximal dynamical rank. This is based on joint work with T.-C. Dinh, J. Keum, and D.-Q. Zhang.
We explore how Fourier-Motzkin elimination, a standard tool in finite dimensional linear programming, can be used to understand the duality theory of more general optimization problems, including semi-infinite linear, convex and conic programming.
This is joint work with Amitabh Basu (Johns Hopkins) and Kipp Martin (University of Chicago).
I will discuss some recent results and conjectures relating knot invariants (such as HOMFLY-PT polynomial and Khovanov-Rozansky homology) to algebraic geometry of Hilbert schemes of points on the plane. All notions will be introduced in the talk, no preliminary knowledge is assumed. This is a joint work with Andrei Negut and Jacob Rasmussen.
Visual computing is a wide area that includes computer graphics and image processing, where the ``eyeball-norm'' rules.
I will briefly discuss two case studies involving numerical methods and analysis applied to this area. The first involves motion simulation and calibration of soft objects such as cloth, plants and skin. The governing elastodynamics PDE system, discretized in space already at the variational level using co-rotated FEM, leads to a large, expensive to assemble, dynamical system in time, where the damped motion may mask highly oscillatory stiffness. Geometric integration ideas are making their way into visual computing research these days in search for more quantitative computations.
The other case study involves some image processing problems where there is a premium for local approaches that do not necessarily use underlying PDEs. I will demonstrate and discuss.
The examples used are from several published or submitted papers.
Course description: The Julia programming language is designed to be a high level language for numerical computing, that is as fast as C or Fortran, despite employing a high level syntax. Since its first release in 2012 it has been continually improved and build a fast growing community around it.
The aim of this course is to give an overview of the key concepts of the Julia programming language as well as explain the advantages over other languages designed for numerical computing, as e.g. Matlab or R. Furthermore it demonstrates how readily available packages developed with Julia can be used to solve common problems occurring in numerical analysis, such as - linear systems of equations - non linear systems of equations - ordinary differential equations - linear programs. The course will also cover the basic tasks frequently encountered by numerical analysts: benchmarking, plotting and debugging. If time permits we will also explore possibilities for using Julia in deep learning applications.
Note for Attendees
Future Dates/Times of this ongoing minicourse: Wednesday, Sep 20, 3:00-5:30pm, Thursday, Sep 21 3:00-5:30pm.
We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can be retrieved. This is a joint work with Pierre Germain and Tej-Eddine Ghoul.
We characterize the GFF as the only random distribution which is conformally invariant and satisfies a domain Markov property. Joint work with Nathanael Berestycki and Ellen Powell.
Course description: The Julia programming language is designed to be a high level language for numerical computing, that is as fast as C or Fortran, despite employing a high level syntax. Since its first release in 2012 it has been continually improved and build a fast growing community around it.
The aim of this course is to give an overview of the key concepts of the Julia programming language as well as explain the advantages over other languages designed for numerical computing, as e.g. Matlab or R. Furthermore it demonstrates how readily available packages developed with Julia can be used to solve common problems occurring in numerical analysis, such as - linear systems of equations - non linear systems of equations - ordinary differential equations - linear programs. The course will also cover the basic tasks frequently encountered by numerical analysts: benchmarking, plotting and debugging. If time permits we will also explore possibilities for using Julia in deep learning applications.
Hopf invariants, a basic construction in homotopy theory, are closely related to Lusternik–Schnirelmann category which, in turn, can be defined as the sectional category of a certain evaluation map. In this talk I'll introduce the notion of Hopf invariants for general fibrations and exhibit a connection between the Hopf invariants for a product fibration and those for the factors. Applications will be drawn to the motion planning problem.
Course description: The Julia programming language is designed to be a high level language for numerical computing, that is as fast as C or Fortran, despite employing a high level syntax. Since its first release in 2012 it has been continually improved and build a fast growing community around it.
The aim of this course is to give an overview of the key concepts of the Julia programming language as well as explain the advantages over other languages designed for numerical computing, as e.g. Matlab or R. Furthermore it demonstrates how readily available packages developed with Julia can be used to solve common problems occurring in numerical analysis, such as - linear systems of equations - non linear systems of equations - ordinary differential equations - linear programs. The course will also cover the basic tasks frequently encountered by numerical analysts: benchmarking, plotting and debugging. If time permits we will also explore possibilities for using Julia in deep learning applications.
Probability measures are key objects in many scientific and engineering areas that deal with randomness, distributions, data sets, etc. When coupled with optimization, many interesting questions naturally arise. In this talk, I will explain a few of such questions from the point of view of optimal transport theory, which gives a natural and robust framework for studying probability measures. These involve among others, matching probability measures in an optimal way following certain rules (e.g martingale), as well as finding geometric averages between probability measures.
Note for Attendees
Light refreshments will be served in ESB 4133, the PIMS Lounge before this colloquium.
The notion of a power structure is closely related to that of a lambda ring. It is a powerful way to encode operations on certain generating functions. Gusein-Zade, Luengo, and Melle-Hernandez have defined a power structure over the Grothendieck ring of varieties. I will discuss an analog of this on a version of the Grothendieck ring of pretriangulated categories, whose elements represent enhancements of derived categories of coherent sheaves on varieties.
We establish a poset structure on combinatorial bases of polynomials, defined by positive expansions. These bases include the well-studied Schubert polynomials, Demazure characters and Demazure atoms, as well as the recently-introduced slide and quasi-key bases. The product of a Schur polynomial and an element of a basis in the poset expands positively in that basis; in particular we give the first Littlewood-Richardson rule for the product of a Schur polynomial and a quasi-key polynomial, extending the rule of Haglund, Luoto, Mason and van Willigenburg for quasi-Schur polynomials. We also establish bijections connecting combinatorial models for these polynomials, including semi-skyline fillings and quasi-key tableaux.
The field of pattern-avoiding permutations was introduced by Knuth in the 1960s as a way of characterising certain data structures.
Since then, it has grown into an important area in its own right. There are a number of classical problems, among which is the number of 1324-avoiding
permutations. We will give some history, and then give details of a new algorithm we have developed for the generating function for this problem.
As a result we can count these up to length 50.
A new method of analysis we have developed, which can in some circumstances be an alternative to Monte Carlo analysis, reveals some interesting features.
In particular, we conjecture that the generating function is not D-finite, and has asymptotics that include a stretched-exponential term.
(Joint work with Andrew Conway and Paul Zinn-Justin).
The late, great Mark Kac often said that his seminars assumed zero knowledge but infinite wisdom.
This seminar only assumes zero knowledge and finite wisdom.
As an instructor, you have probably observed that a student's performance on the first midterm is strongly correlated with their final grade in the course, and that the lowest performing students on the midterm are often “at risk” of failing the course. In this lunch series I'll discuss two alternative ways of identifying at risk students, in the context of first term calculus. The first involves using data from the pre-calculus diagnostic test and the math attitude survey (MAPS). The second method involves mining data from the first webwork assignment. I'll discuss the variables identified as most important for predicting grades, and how the predictive model I have developed will be used as part of an intervention this semester in Math 180. Lastly, I'll discus some research I have done on “outlier” students whose final grade is far above what would be expected based on their first midterm score alone.
The topic of the talk is motivated by certain questions on analytic side of (non-commutative) Iwasawa theory, namely the integrality properties of the p-adic L-function. Though we are motivated by these properties on the analytic side of Iwasawa theory, our work deals with the algebraic objects, called Iwasawa modules. We will discuss certain (non-primitive) Iwasawa modules that have a free resolution of length one over appropriate Iwasawa algebras. Iftimepermits, we will consider an application of this result involving an elliptic curve E over Q with a cyclic isogeny of degree p^2. This is joint work with Alexandra Nichifor.
Vantage College is an academic program integrating content and language instruction for first-year international students at UBC. Specially designed MATH 100 and MATH 101 courses have been offered to Vantage Science students since the program started in 2014.
In this presentation, I will give an overview of Vantage College, describe some innovations in first-year Mathematics instruction that have been developed in the program, and make some recommendations about how the Mathematics Department might work with Vantage College in the coming years.
Note for Attendees
Light refreshments will be served in ESB 4133, the PIMS Lounge before this colloquium.
L-theory is often dubbed as "the K-theory" of quadratic forms. It has been used in a crucial way in surgery theory, to determine if two manifolds are cobordant. I will explain how it is easily defined in the derived setting by considering "derived" quadratic forms, and how I have used derived algebraic geometry to prove a rigidity result for L-theory. This will give an application of derived methods to a non-derived problem.
In this talk we will explore deep neural networks from a dynamical systems point of view. We will show that the learning problem can be cast as a path planning problem with PDE constraint. This opens the door to conventional Computational techniques that can speed up the learning process and avoid some of the local minima.
Let B be the unit ball of a norm in the d-dimensional space and assume that V is a finite subset of B, and the sum of the vectors in V is the zero vector. A theorem of Steinitz from 1914 says that there is an ordering v_1,...,v_n of the vectors in V such that every partial sum along this ordering has norm at most 2d. In the lecture several versions and various extensions of this theorem will be explained.
We present a new condition for the existence of long-range order in discrete spin systems, which emphasizes the role of entropy and high dimension. The condition applies to all symmetric nearest-neighbor discrete spin systems with an internal symmetry of `dominant phases'. Specific applications include a proof of Kotecký's conjecture (1985) on anti-ferromagnetic Potts models, a strengthening of results of Lebowitz-Gallavotti (1971) and Runnels-Lebowitz (1975) on Widom-Rowlinson models and of Burton-Steif (1994) on shifts of finite type, and a significant extension of results of Engbers-Galvin (2012) on random graph homomorphisms on the hypercube. No background in statistical physics will be assumed and all terms will be explained thoroughly. Joint work with Ron Peled.
I will explain some new techniques for computing the homology of braid groups with coefficients in a certain class of exponential representations that arise in a natural way from braided monoidal categories. Surprisingly (at least to me), these techniques are related to fundamental objects — Nichols algebras — in the theory of quantum groups and the classification theory of Hopf algebras. These techniques can be used to establish part of a function field analogue of Malle’s conjecture on the distribution of Galois groups. I will not discuss this application much in the topology seminar, but will focus on it in the colloquium. This is joint work with Jordan Ellenberg and TriThang Tran.
In this talk I will present results that come from a joint work with Clemens Fuchs and Christina Karolus from University of Salzburg (Austria). We considered a sequence of polynomials (Gn(x))∞n=0 in C[x] satisfying a linear recurrence relation of order d ≥ 2:
Gn+d(x) = Ad−1(x)Gn+d−1(x) + · · · + A0(x)Gn(x), n ∈ N,
determined by A0 , A1 , . . . , Ad−1 , G0 , G1 , . . . , Gd−1 ∈ C[x], and we asked about the properties of g(x), h(x) ∈ C[x] such that Gn (x) = g(h(x)), deg g ≥ 2, deg h ≥ 2. Our work was inspired by Zannier’s results about lacunary polynomials. The possible ways of writing a polynomial as a composition of lower degree polynomials were studied by many authors. There are applications to several areas of mathematics. In my talk I will address about some Diophantine applications
The inverse Galois problem asks whether every finite group occurs as the Galois group of an extension of the rational numbers. In 2002, Malle made this existential question more numerical, by conjecturing an asymptotic formula on the growth of the number of fields with a given Galois group, as a function of discriminant. One may reformulate this question in a function field context, replacing the rational numbers with the field of rational functions in positive characteristic. In joint work with Jordan Ellenberg and TriThang Tran, we show that Malle’s conjectured formula does give an upper bound on that distribution. Our methods are very topological, relying on new tools for computing the homology of Hurwitz moduli spaces of branched covers. We will elide these technicalities in this talk — they will be the focus of the topology seminar earlier this week — and focus on the number theoretic results and how to reformulate them in topological terms.
Note for Attendees
Light refreshments will be served at 2:45pm in ESB 4133, the PIMS Lounge before this colloquium.
Richard Stanley introduced the chromatic symmetric function X_G of a simple graph G, which is the sum of all possible proper colorings with colors {1,2,3,...} coded as monomials in commuting variables. These formal power series are symmetric functions and generalize the chromatic polynomial. Soojin Cho and Stephanie van Willigenburg found that, given a sequence of connected graphs G_1,G_2,... where G_i has i vertices, { X_{G_i} } forms a basis for the algebra of symmetric functions. This provides a multitude of new bases since they also discovered that only the sequence of complete graphs provides a basis that is equivalent to a classical basis, namely the elementary symmetric functions. This talk will discuss new results on chromatic symmetric functions using these new and old bases, and additionally we will also resolve Stanley's e-Positivity of Claw-Contractible-Free Graphs. This is joint work with Angele Hamel and Stephanie van Willigenburg.
The subject of critical phenomena in statistical mechanics is a rich source of interesting and difficult mathematical problems that touch on combinatorics, probability, and mathematical physics. Self-avoiding walks and lattice spin systems provide fundamental examples. This talk will address recent progress in computing critical exponents for these models, using a rigorous version of Wilson's renormalisation group method.
In the late 1930s, Paul Erdős attended a seminar at Cornell University given by Mark Kac, who suspected that divisibility by primes satisfies a certain "statistical independence" condition. If this were true, the central limit theorem could be used to show that the number of distinct prime factors of n, as n varies over the natural numbers, is normally distributed, with mean loglog n and standard deviation (loglog n)^(1/2). Erdős used sieve methods to confirm Kac's intuition, and the resulting Erdős-Kac theorem is a foundational result in the field of probabilistic number theory. Many different proofs of and variations on the Erdős-Kac theorem have been given in the intervening decades. This talk will highlight some of these results and the techniques used to obtain them, including recent work of the speaker and Greg Martin (UBC).
Given a set S of positive measure on the unit circle, a set of integers K is an interpolation set (IS) for S if for any data {c(k)} in l^2(K) there exists a function f in L^2(S) such that its Fourier
coefficients satisfy f^(k)=c(k) for all k in K. In the talk I will discuss the relationship between the concept of IS and the existence of arbitrarily
long arithmetic progressions with specified lengths and step sizes in K. Multidimensional analogue and recent developments will also be considered.
Based on joint work with A. Olevskii.
The subject of the talk is two-dimensional cyclic quotients, i.e. two-dimensional toric singularities. We introduce the classical work of Koll'ar/Shephard-Barron relating the components of their deformations and the so-called P-resolutions, we give several combinatorial descriptions of both gadgets, and we will focus on two special components among them - the Artin component allowing a simultaneous resolution and the qG-deformations preserving the Q-Gorenstein property. That is, it becomes important that several (or all) reflexive powers of the dualizing sheaf fit into the deformation as well. We will study this property in dependence on the exponent r. While the answers are already known for deformations over reduced base spaces (char = 0), we will now focus on the infinitesimal theory. (joint work with János Kollár)
Statistical and probabilistic methods are promising approaches to solving inverse problems - the process of recovering unknown parameters from indirect measurements. Of these, the Bayesian methods provide a principled approach to incorporating our existing beliefs about the parameters (the prior model) and randomness in the data. These approaches are at the forefront of extensive current investigation. Overwhelmingly, Gaussian prior models are used in Bayesian inverse problems since they provide mathematically simple and computationally efficient formulations of important inverse problems. Unfortunately, these priors fail to capture a range of important properties including sparsity and natural constraints such as positivity, and so we are motivated to study non-Gaussian priors. In this talk we introduce the theory of well-posed Bayesian inverse problems with non-Gaussian priors in infinite dimensions. We show that the well-posedness of a Bayesian inverse problem relies on a balance between the growth rate of the forward map and the tail decay of the prior. Next, we turn our attention to a concrete application of non-Gaussian priors in recovery of sparse or compressible parameters. We construct new classes of prior measures based on the Gamma distribution and develop a Markov Chain Monte Carlo algorithm for exploring the posterior measures that arise from our compressible priors in infinite dimensions.
We study chemotaxis equations coupled to the Navier-Stokes equations, which is a mathematical model describing the dynamics of oxygen, swimming bacteria (Bacillus subtilis) living in viscous incompressible fluids. It is, in general, not known if regular solutions with sufficiently smooth initial data exist globally in time or develop a singularity in a finite time. We discuss existence of regular solutions and asymptotics as well as temporal decays of solutions, under a certain type of conditions of parameters (chemotatic sensitivity and consumption rate) or initial data, as time tends to infinity.
We consider an i.i.d. balanced environment omega(x,e)=omega(x,-e), genuinely d dimensional on the lattice and show that there exist a positive constant C and a random radius R(omega) with streched exponential tail such that every non negative omega harmonic function u on the ball B_{2r} of radius 2r>R(omega), we have max_{B_r} u <= C min_{B_r} u. Our proof relies on a quantitative quenched invariance principle for the corresponding random walk in balanced random environment and a careful analysis of the directed percolation cluster. This result extends Martins Barlow's Harnack's inequality for i.i.d. bond percolation to the directed case. This is joint work with N. Berger, M. Cohen, and X. Guo.
The local zeta function of a variety X over a finite field F_q is defined to be Z(X,t) = \exp\sum_{n > 0}\frac{|X(F_{q^n})|}{n}. This invariant depends only on the point counts of X over extensions of F_q. We discuss how Z(X,t) can be considered as a group homomorphism of K-groups and show how to lift it to a map between K-theory spectra.
The efficacy of machine learning algorithms hinges on their ability to generalize from data (i.e., when does small training error guarantee small test error). However, the theory of generalizability remains largely unexplained, particularly for monstrous networks that achieve ground-breaking performance on complex tasks like image classification, language translation, or the game Go. The goal of this talk is to showcase how a simple yet elegant phenomenon in probability - concentration of measure - informs the formalization of "learning from data". By way of Hoeffding's inequality, we show how PAC learning uses concentration of measure to address the problem of generalizability in machine learning.
This talk will be accessible to all graduate students and post-docs who've heard the phrase "Markov inequality" at least once. I chose the subject matter because it contains elegant math that yields deep results in an equally elegant formalization of a high-level salient concept. This talk is related but tangential to the subject matter of the probability reading group this term (i.e., I'd be excited to discuss how these results might generalize to more exciting cases).
The dimer model on a finite bipartite graph is a uniformly chosen perfect matching, i.e., a set of edges which cover every vertex exactly once. It is a classical model of mathematical physics, going back to work of Kasteleyn and Temeperley/Fisher in the 1960s, with connections to many topics including determinantal processes, random matrix theory, algebraic combinatorics, discrete complex analysis, etc.
A central object for the dimer model is a notion of height function introduced by Thurston, which turns the dimer model into a random discrete surface. I will discuss a series of recent results with Benoit Laslier (Paris) and Gourab Ray (Victoria) where we establish the convergence of the height function to a scaling limit in a variety of situations. This includes simply connected domains of the plane with arbitrary linear boundary conditions for the height, in which case the limit is the Gaussian free field, and Temperleyan graphs drawn on Riemann surfaces. In all these cases the scaling limit is universal (i.e., independent of the details of the graph) and conformally invariant.
A key new idea in our approach is to exploit "imaginary geometry" couplings between the Gaussian free field and Schramm's celebrated SLE curves.
Note for Attendees
Light refreshments will be served at 2:45pm in ESB 4133, the PIMS Lounge before this colloquium.
Crouzeix's conjecture is among the most intriguing developments in matrix theory in recent years. Made in 2004 by Michel Crouzeix, it postulates that, for any polynomial p and any matrix A, ||p(A)|| <= 2 max(|p(z)|: z in W(A)), where the norm is the 2-norm and W(A) is the field of values (numerical range) of A, that is the set of points attained by v*Av for some vector v of unit length. Crouzeix proved in 2007 that the inequality above holds if 2 is replaced by 11.08, and very recently this was greatly improved by Palencia, replacing 2 by 1+sqrt(2). Furthermore, it is known that the conjecture holds in a number of special cases, including n=2. We use nonsmooth optimization to investigate the conjecture numerically by attempting to minimize the “Crouzeix ratio”, defined as the quotient with numerator the right-hand side and denominator the left-hand side of the conjectured inequality. We present numerical results that lead to some theorems and further conjectures, including variational analysis of the Crouzeix ratio at conjectured global minimizers. All the computations strongly support the truth of Crouzeix’s conjecture. This is joint work with Anne Greenbaum and Adrian Lewis.
Note for Attendees
Refreshments before talk in room ESB 4133 (the PIMS Lounge).
There is by now an extensive theory of rational Chow rings of stacks of smooth curves. The integral version of these Chow rings is not as well understood. I will survey what is known, including some recent developments.
The Corradi-Hajnal Theorem gives a minimum-degree condition for the existence of a given number of vertex-disjoint cycles in a simple graph. We discuss a number of variations on the Corradi-Hajnal Theorem, changing both the nature of the necessary condition (for example, minimum degree sum instead of minimum degree) and the kind of subgraph whose existence is desired. We also briefly discuss the connections between these types of theorems and equitable graph colourings.
The mod pSteenrod algebra is the (Hopf) algebra of stable operations on mod pcohomology, and in part measures the subtle behavior of p-local homotopy theory(as opposed rational homotopy theory, which is much simpler).A classical theorem of Dold-Thom tells us that the infinitesymmetric power of the n-dimensional sphere is the Eilenberg-Maclanespace K(Z, n),and one can use an appropriate modification of thisconstruction to compute the dual Steenrod algebra. The infinitesymmetric power of the sphere spectrum has a filtration whosek-thcofiber miraculously turns out to be the Steinbergsummand(from modular representation theory of GL_k(F_p)) of theclassifying space of (Z/p)^k. This opens the door for slickcomputations - for example, the Milnorindecomposables canbe picked out as explicit cells.
In this talk, I will introduce the concepts and resultschronologically. I will also include hands-on homotopy theory computations as time permits.
Since 2010, the math department at UBC has been gradually adopting the WeBWork online homework system in most first and second year courses. Instructors typically give students several days to complete their WeBWork assignments, and allow students several attempts at each problem. Despite this, many students struggle to complete their online assignments. In this talk, I'll discuss how the timing of answer submissions recorded in WeBWork log files can reveal the behaviour patterns of students who struggle with WeBWork. In particular, students who don't complete the WeBWork start the assignments closer to the deadline, have shorter login sessions and don't persist for as long once they encounter a problem they can't solve. I'll discuss what these observations reveal about the mindset of struggling students, and how assignments could be restructured to help increase their completion rate.
Semigroups are useful in solving a large class of problems known as evolution equations. These kind of equations often appear in many disciplines including physics, chemistry, biology and engineering. I will be discussing an introduction to theory of C0-semigroups on Banach spaces. We then discuss uniformly continuous semigroups and prove some important results and characterizations related to them. Following which, we present examples of how new semigroups can be constructed out of a given semigroup. We associate a generator to a C0-semigroup and illustrate their properties by means of some theorems. Subsequently we prove Hille-Yosida theorem, which provides a necessary and sufficient condition for an unbounded operator on a Banach space to be a generator of a contraction C0-semigroup. This was proved in 1948 by mathematicians E. Hille and K. Yosida. We also prove Feller-Miyadera-Phillips theorem which generalizes Hille-Yosida theorem. This was proved around 1952. If time permits I'll do some other interesting stuff related to theory of C_0-semigroups.
The Banana manifold (or bananafold for short), is a compact Calabi-Yau threefold X which fibers over P^1 with Abelian surface fibers. It has 12 singular fibers which are non-normal toric surfaces whose torus invariant curves are a banana configuration: three P^1’s joined at two points, each of which locally look like the coordinate axes in C^3. We show that the Donaldson-Thomas partition function of X (for curve classes in the fibers) has an explicit product formula which, after a change of variables is the same as the generating function for the equivariant elliptic genera of Hilb(C^2), the Hilbert scheme of points in the plane.
Transitional phenomena are ubiquitous in fluid dynamics and other nonlinear systems; they occur whenever there are multiple states in which a system can reside. Frequently, we are able to investigate when and how a system transitions from one state to another by performing a linear stability analysis and obtaining critical thresholds for various parameters beyond which our original state becomes "unstable". However, there are numerous examples for which such an approach does not work. Perhaps the most widely studied scenarios in fluid mechanics for which a linear stability analysis fails to predict transition are the canonical homogeneous shear flows of plane Couette flow and pipe flow. Both of these flows have a laminar (quiescent) solution to the Navier-Stokes equations which is linearly stable at all flow rates, and yet sustained turbulent dynamics are observed in plane Couette flow and in pipe flow for sufficiently rapid flow. Such systems are "two-state" systems for which both the laminar flow and turbulence co-exist as (locally) stable solutions to the Navier-Stokes equations. Recent developments in "generalised nonlinear stability theory" (Pringle & Kerswell, 2010) allow us to find minimal perturbation amplitudes, in a nonlinear sense, to transition between two (linearly) stable flow states. However, the full interpretation of the results of nonlinear stability theory is possible only when interpreting fluid flows in the language of dynamical systems. Drawing from the recent focus of interpreting turbulence in terms of coherent structures rather than statistics, Eaves & Caulfield (2015) interpreted the minimal thresholds for transition to turbulence in statically stable density-stratified plane Couette flow with a focus on coherent structures, and demonstrated that the effect of stratification has an unexpectedly significant impact on the transition scenario. In this talk, I will outline the methodology behind nonlinear stability theory, explain what turbulence is from a dynamical systems point of view, and outline how these two ideas were utilised in my thesis work on stratified shear flow. I will conclude with a brief overview of ongoing work and other extensions to this rapidly developing field of research.
In this talk I will present some results on stochastic heat equations driven by a Gaussian noises. I will focus on Feynman-Kac representation of the solution and the moments of the solution. Both lower and upper bounds for the L^p moments of the solution are obtained which is relevant to the so-called intermittency. The Driving Gaussian noises include fractional Brownian fields of Hurst parameters greater or smaller than 1/2.
In episode 2 of the series, I will turn my attention to the setting of G-equivariant stable homotopy theory, where G is an abelianp-group. Analogous to the classical case, we can use symmetric powers of the equivariant sphere to filter H\underline{\F}_p, and the cofibers are Steinbergsummands of equivariant classifying spaces. We then study howthe cells of these spaces split after smashing with H\underline{\F}_pin the case G=C_p. When p=2, the result is a decomposition of H\underline{\F}_2 \sm H\underline{\F}_2 whose generators correspond torepresentation spheres, while at odd primes, we see something more unusual.
We introduce a class of polytopes which we call endoskeletal. The structure of an endoskeletal polytope is determined by its internal skeleton and its volume is given by a strikingly simple formula involving a single determinant. A rudimentary knowledge of undergraduate mathematics is necessary and sufficient for understanding this talk.
Note for Attendees
Light refreshments will be served at 2:45pm in ESB 4133, the PIMS Lounge before this colloquium.
Resolvent degree is an invariant of a branched cover which quantifies how "hard" is it to specify a point in the cover given a point under it in the base. Historically, this was applied to the branched cover P^n/S_{n-1} -> P^n/S_n, from the moduli of degree n polynomials with a specified root to the moduli of degree n polynomials. Classical enumerative problems and congruence subgroups provide two additional sources of branched covers to which this invariant applies. In ongoing joint work with Benson Farb, we develop the theory of resolvent degree as an extension of Buhler and Reichstein's theory of essential dimension. We apply this theory to systematize an array of classical results and to relate the complexity of seemingly different problems such as finding roots of polynomials, lines on cubic surfaces, and level structures on intermediate Jacobians.
(This talk is held at SFU, and is being viewed at UBC via videoconference.) Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine
Conjecture. Loosely speaking, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by the Fourier expansions of a distinguished set of modular functions. This conjecture arose from astonishing coincidences noticed by finite group theorists and arithmetic geometers in the 1970s. Recently, mathematical physicists have revisited moonshine, and they discovered evidence of undiscovered moonshine which some believe will have applications to string theory and 3d quantum gravity. The speaker and his collaborators have been developing the mathematical facets of this theory, and have proved the conjectures which have been formulated. These results include a proof of the Umbral Moonshine Conjecture, and Moonshine for the first sporadic finite simple group which does not occur as a subgroup or subquotient of the Monster. The most recent Moonshine (announced here) yields unexpected applications to the arithmetic elliptic curves thanks to theorems related to the Birch and Swinnerton-Dyer Conjecture and the Main Conjectures of Iwasawa theory for modular forms. This is joint work with John Duncan, Michael Griffin and Michael Mertens.
Optimal stopping problems can be viewed as a problem to calculate the space and time dependent value function, which solves a nonlinear, possible non-smooth and degenerate, parabolic PDE known as an Hamilton-Jacobi-Bellman (HJB) equation. These equations are well understood using the theory of viscosity solutions, and the optimal stopping policy can be retrieved when there is some regularity and non-degeneracy of solution.
The HJB equation is commonly derived from a dynamic programming principle (DPP). After adding a probabilistic constraint, the optimal policies no longer satisfy this DPP. Instead, we can reach the HJB equation by a method related to optimal transportation, and recover a DPP for a Lagrangian-relaxation of the problem. The resulting HJB equation remains coupled through the constraint with the optimal policy (and another parabolic PDE). Solving the HJB and recovery of the optimal stopping policy is aided by considering the ``piecewise-monotonic’' structure of the stopping set.
Schur functions form the most interesting and important basis for the algebra of symmetric functions. They have connections to representation theory and algebraic geometry, and satisfy a multitude of beautiful combinatorial identities. We investigate an algebraic relationship between ribbon Schur functions, a generalization of Schur functions. More specifically, we consider when the difference between two ribbon Schur functions is a single Schur function. We will see that this near-equality phenomenon occurs for fourteen infinite families and we will present conditions under which these are the only possibilities.
The term ‘anomalous diffusion’ is used in the physics literature to refer to Markovprocesses with the property that E|X_t-X_0|^2 grows eitherfaster or slower than linearly. In this talk I will give a surveyof results in this area, including random walks and diffusionon the Sierpinski gasket and other exact fractals, and randomexamples such as critical percolation clusters and the uniformspanning tree.
Basic questions in analytic number theory concern the density of one set in another (e.g. square-free integers in all integers). Motivated by Weil's number field/function field dictionary, we introduce a topological analogue measuring the “homological density” of one space in another. In arithmetic, Euler products can be used to show that many seemingly different densities coincide in the limit. By combining methods from manifold topology and algebraic combinatorics, we discover analogous coincidences for limiting homological densities arising from spaces of 0-cycles (e.g. configuration spaces of points) on smooth manifolds and complex varieties. We do not yet understand why these topological coincidences occur. This is joint work with Benson Farb and Melanie Wood.
In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann's Xi-function. This hyperbolicity has only been proved for degrees d=1, 2, 3. We prove the hyperbolicity of 100% of the Jensen polynomials of every degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.
Note for Attendees
Join us for light refreshments in ESB 4133 from 3:00-3:30pm.
Schur functions form the most interesting and important basis for the algebra of symmetric functions. They have connections to representation theory and algebraic geometry, and satisfy a multitude of beautiful combinatorial identities. We investigate an algebraic relationship between ribbon Schur functions, a generalization of Schur functions. More specifically, we consider when the difference between two ribbon Schur functions is a single Schur function. We will see that this near-equality phenomenon occurs for fourteen infinite families and we will present conditions under which these are the only possibilities.
Eikonal equation is a fundamental nonlinear PDE that find vast applications. One particular example is to compute geodesic distance on a curved surface through solving an eikonal equation defined on the surface (surface eikonal equations). However, there are only very few literatures on solving surface eikonal equations numerically, due to the complication from the surface geometry. In this talk, we present a simple and efficient numerical algorithm to solve surface eikonal
equations on general implicit surfaces.
where Q is a non-negative function and n>2. Geometrically, if u is a solution to \eqref{1} then Q is the Q-curvature of the conformal metric g_u = e^{2u}|dx|^2 (|dx|^2 is the Euclidean metric on \mathbb{R}^n), and \kappa is the total Q-curvature of g_u.
Under certain assumptions on Q around origin and at infinity, we prove the existence of solution to \eqref{1} for every \kappa > 0.
PreTeXt is a new markup language for describing structured scholary documents, such as research articles and textbooks. It was designed originally to meet the demands of communicating mathematics, but has now been used to author books on computer science, physics, music theory, and composition (writing). It is the basis for the mobile edition of UBC's CLP calculus text. A key feature of PreTeXt is high-fidelity conversions to print, PDF, online (HTML), Jupyter notebooks, and soon EPUB.
After an introduction, I will demonstrate some of the more interesting extra capabilities of the online versions, including embedded live Sage code and WeBWorK automated homework problems. Recent advances in producing Jupyter notebooks will also be demonstrated.
Regulators of the actin cytoskeleton such Rho GTPases can modulate forces developed in cells by promoting actomyosin contraction. At the same time, through mechanosensing, tension is known to affect the activity of Rho GTPases.
What happens when these effects act in concert? Using a minimal model (1 GTPase coupled to a Kelvin-Voigt element), we show that two-way feedback between signaling (“RhoA”) and mechanical tension (stretching) leads to a spectrum of cell behaviors, including contracted or relaxed cells, and cells that oscillate between these extremes. When such “model cells” are connected to one another in a row or in a 2D sheet (“epithelium”), we observe waves of contraction/relaxation and GTPase activity sweeping through the tissue. The minimal model lends itself to full bifurcation analysis, and suggests a mechanism that explains behavior observed in the context of development and collective cell behavior.
I will begin with a gentle introduction to "conformal welding" from the probabilistic viewpoint, which is at the heart of Scott Sheffields "quantum zipper" as well as Malliavin's and his coauthors work on Brownian measures on the group of circle homeomorphisms. Then I will describe a conformal welding problem involving to the CRT, discuss the existence of its solution (joint work with Peter Lin), and describe how it arises as the limit of certain dessin d'enfants.
In the 1970s, work of Adams, Baird, Bousfield, and Ravenel gave a description of the orders of the KU[1/2]-local stable homotopy groups of spheres as the denominators of special values of the Riemann zeta-function. Meanwhile, Lichtenbaum conjectured a formula, ultimately proven 30 years later as a consequence of the Iwasawa main theorem and the norm residue theorem, relating the orders of the algebraic K-groups of totally real number rings to special values of their Dedekind zeta-functions. In this talk I will describe two general approaches, an analytic approach and an algebraic approach, to a general kind of number theory that arises in any tensor triangulated category: this is a general framework for the above results and gneralizations of them, and which aims to describe the orders of Bousfield-localized stable homotopy groups of finite spectra in terms of special values of L-functions. Then I'll show off some new results in this framework, in particular, a "universal" description of the KU-local homotopy groups of the Moore spectrum S/p in terms of L-values, and as a consequence, a proof of a certain (infinite) family of cases of Leopoldt's conjecture, by counting orders of homotopy groups.
We study a generalization of the quadratic sieve to a geometric setting. We apply this to counting points of bounded height on an l-cyclic cover over the rational function field and we consider a question of Serre. In addition to the geometric quadratic sieve, we use Fourier analysis over function fields, deep results of Deligne and Katz about cancellation of mixed character sums over finite fields, and a bound on the number of points of bounded height due to Browning and Vishe.
This is joint work with A. Bucur, A. C. Cojocaru, and L. B. Pierce.
Random models lead to a precise and comprehensive theory of compressed sensing and matrix completion. The number of random linear measurements needed to recover a sparse signal, or a low-rank matrix, or, more generally, a structured signal, are now well understood. This is appealing in practice since it helps to determine the pros and cons of different models and gives a benchmark for success. Nevertheless, a practitioner with a fixed data set will wonder: Can they apply theory based on randomness? Is there any hope to get the same guarantees? We discuss these questions in compressed sensing and matrix completion, which, surprisingly, seem to have divergent answers.
Yaniv Plan is the 2016 winner of the PIMS UBC Math Sciences Prize.
Note for Attendees
Light refreshments will be served at 2:45pm in ESB 4133, the PIMS Lounge before this colloquium.
This talk is concerned with the question of the minimal number of equations necessary to define a given projective variety scheme-theoretically. Every hypersurface is cut out by a single polynomial scheme-theoretically (also set-theoretically and ideal theoretically). Therefore the question is more interesting if a variety has a higher codimension. In this talk, we focus on the case when the codimension is two. If a variety in projective n-space has codimension two, then the minimal number of polynomials necessary to cut out the variety scheme-theoretically is between 2 and n+1. However the varieties cut out by fewer than n+1, but more than 2 polynomials seem very rare. The main goal of this talk is to discuss conditions for a non-singular surface in projective four-space to be cut out by three polynomials.
Complex individual-based models abound in epidemiology and ecology. Fitting these models to data is a challenging problem: methodologies can be inaccessible to all but specialists, there may be challenges in adequately describing uncertainty in model fitting, and the complex models may take a long time to run, requiring parameter selection procedures. Approximate Bayesian Computation has been proposed as a likelihood-free method in resolving these issues, however requires careful selection of summary statistics and annealing scheme. I compare this procedure directly to standard methodologies where the likelihood exists, Markov-chain Monte Carlo and maximum likelihood. This is then applied to a complex individual-based simulation for lymphatic filariasis, a human parasitic disease, which affects over 120 million individuals internationally. Finally, I will discuss a new approach to individual-based model fitting by constructing a synthetic likelihood using mixture density networks.
Showing asymptotic stability in one dimensional nonlinear Klein-Gordon equations is a notoriously difficult problem. In this talk I will describe an approach based on virial estimates which allows to prove it in case when only odd perturbations are allowed. In particular I will discuss asymptotic stability of the kink in the \phi^4 model.
We study the moduli space of smooth complete intersections of two quadrics by relating it to the geometry of the singular members of the corresponding pencil. We give a new description for this parameter space by using the fact that two quadrics can be simultaneously diagonalized. Using this description we can compute the Picard group, which always happens to be cyclic. For example, we show that the Picard group of the moduli stack of smooth degree 4 Del Pezzo surfaces is Z/4Z.
Pattern formation is quite recurrent in the natural world such as in the stripes or dots in some animals' coat. The morphogen hypothesis introduced by Turing in 1952 has been used and studied extensively to explain such patterns on several different domain shapes. In this talk we use the spherical cap domain. This is motivated by the shape of the the tip of a conifer embryo, where branching patterns emerge as the tip flattens. Previous results have been achieved to characterize the different patterns obtained on the cap for different radius and curvature values for a constant domain in time. Here we work with a non-autonomous domain with slowly decreasing curvature. We start with previously obtained results from center manifold reduction and finite elements methods. After that we continue by broadly introducing the closest point method for solving PDEs, explain how we use the method on a flattening spherical cap and end with some very preliminary results.
It has long been known that many interesting questions in probability have a formulation in the language of spin systems. However, it has been only rather recently that the methods developed for spin systems were applied to finally obtain answers to some of these questions. In this talk, I will discuss three such questions, about the weakly self-avoiding walk, the vertex reinforced jump process, and random band matrices. I will then show the audience some technical lemmas that are at the heart of the analysis of spin systems.
The algebraic K-theory, due to Quillen, of a field is related to a theory defined by Milnor called Milnor K-theory and denoted K^M. In the 1980s, Andrei Suslin constructed a map K_n(F) -> K^M_n(F), and conjectured that the image was the subgroup (n-1)! K^M_n(F). He also proved the conjecture for n<=3. For n=5, we reinterpret the construction as a construction in the A1 homotopy groups of spheres and BGL, and by calculating these groups, show that the conjecture is true in this case as well. This represents part of a joint project with Aravind Asok, Jean Fasel and Kirsten Wickelgren.
Classical toric varieties come in two flavours: Normal toric varieties are given by rational fans in R^n. A (not necessarily normal) affine toric variety is given by finite subset A of Z^n. When A is homogeneous, it is projective. Applications of mathematics have long studied the positive real part of a toric variety as the main object, where the points A may be arbitrary points in R^n. For example, in 1963 Birch showed that such an irrational toric variety is homeomorphic to the convex hull of the set A.
Recent work showing that all Hausdorff limits of translates of irrational toric varieties are toric degenerations suggested the need for a theory of irrational toric varieties associated to arbitrary fans in R^n. These are R^n_>-equivariant cell complexes dual to the fan. Among the pleasing parallels with the classical theory is that the space of Hausdorff limits of the irrational projective toric variety of a finite set A in R^n is homeomorphic to the secondary polytope of A.
There are some proofs I'm never going to get around to understanding. Shinichi Mochizuki's 2012 proof of the ABC conjecture amounts to 500 pages. The best mathematicians still struggle to understand Mochizuki's techniques today. The entire classification of finite simple groups amounts to 100000-200000 pages!
Mathematical logic tells us that theorems exist whose shortest proofs are arbitrarily unreadable. However, mathematicians have recently discovered that any `computationally feasible' theorem has a `random proof' which is only 3 letters long! In this talk I present the basic ideas behind these `probabilistically checkable proofs', with applications to computational complexity, cryptography, and the foundational limitations of machine learning.
The talk requires no background; only a vague sense of mathematical curiosity is required to enjoy the discussion the most important results in theoretical computing science in the past 20 years.
Let X be an algebraic variety over a field k. Which representations of pi_1(X) arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over X? We study this question by analyzing the action of the Galois group of k on the fundamental group of X, and prove several fundamental structural results about this action.
As a sample application of our techniques, we show that if X is a normal variety over a field of characteristic zero, and p is a prime, then there exists an integer N=N(X,p) such that any non-trivial p-adic representation of the fundamental group of X, which arises from geometry, is non-trivial mod p^N.
This thesis consists of two parts. In the first, we address the issue that — unlike their deterministic counterparts — stochastic differential equations have never been formulated as stationary states of some energy functionals on spaces of stochastic processes. We show how the self-dual variational calculus can remedy the situation by providing a natural variational approach for the resolution of a number of non-linear stochastic partial differential equations driven by monotone operators and additive or non-additive noise. Such operators can be gradients of convex energy or in divergence form. These equations are used to model population dynamics in biology, evolution of a fluid velocity and the turbulence in physics and also in modelling of stock prices and the risky assets in finance.
In the second part of the thesis, we use methods from optimal transport to address functional inequalities on the n-dimensional sphere. In particular, we prove Energy-Entropy duality formulas that yield and improve the celebrated Moser-Onofri inequalities on 2-dimensional sphere.
I am working with colleagues on several projects modelling Lithium Ion batteries. Experimental results are fit to simple models of performance and "State of Health", a vaguely defined measure of the change of battery characteristics with use and age. This project is a collaboration with a local company, JTT electronics. Opening the hood of a battery reveals an interesting mix of multi-scale and multi-phase transport. Recent progress on categorizing models in the literature based on an asymptotic parameter, and computational approaches to the resulting structure, will be shown. All projects are "work in progress". Collaborators include Arman Bonakdarpour, Bhushan Gopaluni, Matt Hennessey, David Kong, Iain Moyles, and Tim Myers.
The Min system is an important regulator network involved in E-coli cell division. However, although the effects and chemicals involved in this network are known, there still remain a variety of hypotheses about the exact reactions and mechanisms involved. Recent experimental work by Vecchiarelli et al. has demonstrated a large variety of reaction-diffusion induced patterns on a 2d membrane. In this talk I discuss some data processing on Vecchiarelli's data, and initial forays into the ``nucleation question'' (that is, how the particular patterns observed appear from an initially ``Homogeneous'' membrane).
Bootstrap percolation is a fundamental cellular automaton model for nucleation. Despite its simplicity, the model holds many surprises. I'll focus on how growth from sparse random seeds is affected by sparse random impurities in the medium. The answer will involve using recent oriented surface technology to construct a stegosaurus.
Based on joint work with Janko Gravner and David Sivakoff.
As of Tuesday, Nov 28th: unfortunately this colloquium has been cancelled. ESB 2012
Wed 29 Nov 2017, 5:00pm-6:00pm
Abstract
A recent wave of contributions in machine learning center on the concept of generative models for extremely complex data such as natural images. These approaches provide principled ways to use deep network architectures, large datasets and automatic differentiation to come up with algorithms that are able to synthesize realistic images. We will present in this talk how optimal transport is gradually establishing itself as a valuable tool to carry out this estimation procedure.
Note for Attendees
This Colloquium is a series of talks at UBC by Marco Cuturi. He will be giving two other lectures on Thursday, Nov 30 and Friday Dec 1, 2017.
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.
In this informal lunch talk, I will introduce several standard mathematical tools for exploring the effectiveness of multiple choice exam questions: covariance, binning, and item response theory. We will examine three tests analyzed using these methods and discuss what the results imply about how multiple choice questions should be written and how tests should be constructed. The tools above enable us to find questions that have a varying levels of discrimination between high scoring students and low scoring students, as well as rate such questions difficulty. Using this information, we will discuss what features of a question contribute to or detract from its effectiveness in measuring student performance. Finally, we will discuss the construction of the multiple choice portion of an exam and what kinds of distributions of questions serve to most effectively measure students knowledge.
Optimal transport theory provides practitioners from statistics, imaging, graphics or machine learning with a very powerful toolbox to compare probability measures. These tools translate however in their original form into computational schemes that can become intractable or suffer from instability (such as non-differentiability or estimation bias). We will present in these two lectures how a few insights from optimization theory and in particular a careful regularization can result in tools that are considerably easier to implement, run faster because they can take advantage of parallel hardware and behave better from a statistical perspective. We will highlight applications from diverse areas, from graphics and brain imaging to text analysis and parametric estimation.
Note for Attendees
This is the first of a two part lecture. The second part will be given on Dec, 1 2017, in ESB 2012 at the same time. Details for Part II are available here.
Systems of points with Coulomb, logarithmic (or more generally inverse powers of the distance) interactions arise in various settings: an instance is the classical Coulomb gas which in some cases happens to be a random matrix ensemble, another is vortices in the Ginzburg-Landau model of superconductivity, where one observes in certain regimes the emergence of densely packed point vortices forming perfect triangular lattice patterns named Abrikosov lattices, a third is the study of Fekete points which arise in approximation theory. After reviewing the motivations, we will take a point of view based on the detailed expansion of the interaction energy to describe the microscopic behavior of the systems and its statistical mechanics. In particular a Central Limit Theorem for fluctuations and a Large Deviations Principle for the microscopic point processes are given. This allows to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions.
The main results are joint with Thomas Leblé and also based on previous works with Etienne Sandier, Nicolas Rougerie and Mircea Petrache.
Note for Attendees
Light refreshments will be served at 2:45pm in ESB 4133, the PIMS Lounge before this colloquium.
Optimal transport theory provides practitioners from statistics, imaging, graphics or machine learning with a very powerful toolbox to compare probability measures. These tools translate however in their original form into computational schemes that can become intractable or suffer from instability (such as non-differentiability or estimation bias). We will present in these two lectures how a few insights from optimization theory and in particular a careful regularization can result in tools that are considerably easier to implement, run faster because they can take advantage of parallel hardware and behave better from a statistical perspective. We will highlight applications from diverse areas, from graphics and brain imaging to text analysis and parametric estimation.
Note for Attendees
This is the second of a two part lecture. The first part will be given on Nov 30 2017, in ESB 2012 at the same time. Details for Part I are available here.
Speaker: Ping Xu, Penn State
Time: 15:00-16:00 in MATH 126
Title: Formal exponential maps and L-infinity structures
Abstract: Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The infinite jets of these classical exponential maps are related to the Poincare-Birkhoff-Witt isomorphism and the complete symbols of differential operators. We will investigate the question how to extend these maps to Z-graded manifolds by introducing formal exponential maps in a purely algebraic way. As an application, we will prove an analogue of a theorem of Kapranov in the dg manifold context: there is an L-infinity structure in connection with the Atiyah class of a dg manifold.
Speaker: Hsuan-Yi Liao, Penn State
Time: 16:15-17:15 in MATH 126
Title: A formality theorem for differential graded manifolds
Abstract: A dg (differential graded) manifold consists of a Z-graded manifold and a homological vector field. A wide range of geometric objects can be described in terms of dg manifolds, including Lie algebras, Lie algebroids, L-infintiy algebras and derived critical loci. We establish a formality theorem for finite-dimensional smooth dg manifolds: there is a "good” L-infinity quasi-isomorphism from the twisted dgla of polyvector fields to the twisted dgla of polydifferential operators. As an application, we prove the Kontsevich-Shoikhet conjecture: a Kontsevich-Duflo type theorem holds for all finite-dimensional smooth dg manifolds, which extends the classical Duflo theorem in Lie theory.
We develop a mimetic multiscale method to simulate quasi-static Maxwell's equations in frequency domain. This is especially useful for extensive geophysical models that include small-scale features. Applying the concept of multiscale methods, we avoid setting up a large and costly system of equations on the fine mesh where the material parameters are discretized on. Instead, we build and solve a system on a much coarser mesh. For doing that, it is inevitable to interpolate
between fine and coarse meshes. The construction of this coarse-to-fine interpolation is done by solving local optimization problems on each coarse cell incorporating the fine-mesh features. With our method being mimetic, the properties of the continuous differential operators are preserved in their discrete counterparts and thus, the resulting simulations do not contain spurious modes.
Gender equality is a major issue within science communities. While efforts to be proactive and bring awareness to gender inequality have been made in recent years, still only 1/8th of academic scientists are women. A recent paper published in PLOS Computational Biology (Hashe et al. 2017) highlights the under-representation of women in biology, computational biology, and computer science. Here, I will present their findings and lead a general discussion on women in science and ways we may help to close the gender gap.
Let M be a smooth manifold, let Diff(M) denote the topological group of sel-diffeomorphisms, and let BDiff(M) denote the "classifying space”. For any paracompact space X, there is a one-one correspondence between the set of homotopy classes [X, BDiff(M)] and the set of isomorphism classes of fibre bundles over X with fibre M. The classifying space BDiff(M) is referred to as the "moduli space of manifolds of type M". The study of the homotopy type of these spaces occupies a central place in smooth topology.
In this talk I will discuss some contemporary approaches to studying the homotopy/homology of BDiff(M), for varying M. In particular I will discuss the results of Madsen and Weiss on the stable moduli spaces of Riemann surfaces and the results of Galatius and Randal-Williams on the stable moduli spaces of manifolds of dimension 2n. I will then present recent work of mine pertaining to the moduli spaces of odd dimensional manifolds, and manifolds with boundary, and discuss connections to cobordism categories and surgery theory.
Note for Attendees
Refreshments will be served in MATX 1100 at 2:30p.m. before this Mathematics Colloquium.
In this talk I will show how to use parametrized Morse theory to construct a map from the infinite loopspace of certain Thom spectrum, MTSpin(d), into the space of positive scalar curvature metrics on a closed spin manifold of dimension d > 4. My main novel construction is a cobordism category consisting of cobordisms equipped with a choice of Morse function, whose critical points occupy a prescribed range of degrees. My first result identifies the homotopy type of the classifying space of this topological category with the infinite loopspace of another Thom spectrum related that is related to MTSpin(d), and built out of the space of Morse jets on Euclidean space. The result can viewed as an analogue of the well known theorem of Galatius, Madsen, Tillmann, and Weiss, for manifolds equipped with the extra geometric structure of a choice of admissible Morse function.
In the second part of the talk I will show how to use this cobordism category to probe the homotopy type of the space of positive scalar curvature metrics, R^{+}(M), on a closed, spin manifold M when dim(M) > 4. This uses a parametrized version of the Gromov-Lawson construction developed by Walsh and Chernysh. Our main result detects many non-trivial homotopy groups in the space of positive scalar curvature metrics R^{+}(M). It in particular gives an alternative proof and extension of a recent breakthrough theorem of Botvinnik, Ebert, and Randal-Williams.
Abstract: In this dissertation, we combine the work of A. Aizenbud and D. Gourevitch on Schwartz functions on Nash manifolds, and the work of F. du Cloux on Schwartz inductions, to develop a toolbox of Schwartz analysis on algebraic groups. We then use these tools to study the intertwining operators between parabolic inductions, and the behaviour of intertwining distributions on certain open subsets. Finally we use our results to give new proof of results of F. Bruhat, on irreducibilities of degenerate principal series and minimal principal series.
The HIV prevalence in the MSM population of Vancouver is slowly increasing,
but the amount of Syphilis cases is increasing very rapidly. Since Syphilis
facilitates the infection of HIV, there is a certain relation between the
infected populations. In this model, we explore the
co-operation of these two STD's and see what the influence of testing rate
is on the prevalence.
Heegaard Floer homology provides a suite of invariants for studying three-manifolds, introduced by Ozsváth and Szabó. This theory has, more recently, been expanded to treat manifolds with boundary through bordered Floer homology, providing the tools required to answer natural questions that arise when decomposing a three-manifold along a surface. This talk aims to provide a brief overview of Heegaard Floer theory, give a sense for some of the questions driving its study, and point to some recent progress on answering these.
Note for Attendees
Refreshments will be served in MATX 1100 at 2:30 p.m. before this Mathematics Colloquium.
The Heegaard Floer homology of a manifold with torus boundary can be expressed as a collection of immersed curves (possibly decorated with local systems). This provides a geometric structure theorem, interpreting the algebraic invariants that arise in bordered Floer homology. From this point of view, the Heegaard Floer homology of a closed manifold obtained by gluing manifolds (with boundary) along a torus may be recovered as the Lagrangian intersection Floer homology of the associated curves. In practice, this reduces gluing problems to simple minimal intersection counts. I'll set up this machinery and describe some of the applications that follow. This is joint work with Jonathan Hanselman and Jake Rasmussen.
This talk will illustrate some patterns in the homology of the configuration space F_k(M), the space of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena -- relationships between unstable homology classes in different degrees -- established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius--Kupers--Randal-Williams.
The Torelli subgroups of mapping class groups are fundamental objects in low-dimensional topology, through some basic questions about their structure remain open. In this talk I will describe these groups, and how to use tools from representation theory to establish patterns their homology. This project is joint with Jeremy Miller and Peter Patzt. These "representation stability" results are an application of advances in a general algebraic framework for studying sequences of group representations.
Equivariant formality, a notion in equivariant topology introduced by Goresky-Kottwitz-Macpherson, is a desirable property of spaces with group actions. Examples of equivariantly formal spaces include compact symplectic manifolds equipped with Hamiltonian compact Lie group actions and projective varieties equipped with linear algebraic torus actions. Less is known about compact homogeneous spaces G/K equipped with the isotropy action of K, which is not necessarily of maximal rank. In this talk we will review previous attempts of characterizing equivariant formality of G/K, and present our recent results on this problem using an analogue of equivariant formality in K-theory. Part of the work presented in this talk is joint with Jeffrey Carlson.
Optimal maps in R^n to disconnected targets necessarily contain discontinuities (i.e.~tears). But how smooth are these tears? When the target components are suitably separated by hyperplanes, non-smooth versions of the implicit function theorem can be developed which show the tears are hypersurfaces given as differences of convex functions --- DC for short. If in addition the targets are convex the tears are actually C^{1,\alpha}. Similarly, under suitable affine independence assumptions, singularities of multiplicity k lie on DC rectifiable submanifolds of dimension n+1-k. These are stable with respect to W_\infty perturbations of the target measure. Moreover, there is at most one singularity of multiplicity n. This represents joint work with Jun Kitagawa.
Hardy and Littlewood initiated the study of the 2k-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. In this talk we consider the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula for the sixth moment. This builds on earlier work of Ivic and of Conrey-Gonek.
We will discuss some results concerning l^p boundedness of maximal functions related to diophantine problems which are susceptible to the circle method.
The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.
Floer homology has been a central tool to study global aspects of symplectic topology, which is based on pseudoholomorphic curve techniques proposed by Gromov. In this talk, we introduce a so-called wrapped Floer homology. Roughly speaking, this is a certain homology generated by intersection points of two Lagrangians and its differential is given by counting solutions to perturbed Cauchy-Riemann equation. We investigate an entropy-type invariant, called the slow volume growth, of certain symplectomorphisms and give a uniform lower bound of the growth using wrapped Floer homology. We apply our results to examples from real symplectic manifolds, including A_k-singularities and complements of a complex hypersurface. This is joint work with Myeonggi Kwon and Junyoung Lee.
One of the ways we understand Teichmuller space, endowed with the Teichmuller metric, is by studying Teichmuller discs. They exist in great abundance: There is a Teichmuller disc through any point and in any direction. Typically, their projection to moduli space is dense, and yet infinitely often their projection is a closed subvariety of moduli space called a Teichmuller curve. Recently, in joint work with Eskin, McMullen, and Mukamel, we discovered the first non-trivial examples of higher dimensional analogues of Teichmuller discs, namely totally geodesic submanifolds.
In this talk, we will explain that such higher dimensional totally geodesic submanifolds are much more rigid and rare than Teichmuller discs: Each must cover a closed subvariety of moduli space, and only finitely many such subvarieties exist in each moduli space. This result is an application of joint work with Eskin and Filip on the Kontsevich-Zorich cocycle. One of the goals of the talk will be to explain what this cocycle is and why it lies at the heart of Teichmuller dynamics.
Let E be an arbitrary subset of R^n. Given real valued functions f defined on E and g defined on R^n, the classical obstacle problem asks for a minimizer of the Dirichlet energy subject to the following two constraints: (1) F = f on E and (2) F lies above g on R^n. In this talk, we will discuss how to use extension theory to construct (almost) solutions directly. We will also explain several recent results that will help lay the foundation for building a complete theory revolving around the belief that any variational problems that can be solved using PDE theory can also be dealt with using extension theory.
In two dimensions, many self-interacting processes are described by the Schramm-Loewner Evolution SLE(kappa), a family of random fractal path joining two boundary points of an underlying domain D.These continuous paths arise as the scaling limits of various discrete self-interacting paths, such as loop-erased random walk.
A self-interacting process has the locality property if it does not "feel" the boundary of its domain D until it hits the boundary. Among the two-dimensional processes known as Schramm Loewner Evolution SLE(kappa), it is known that only one, SLE(6), satisfies the locality property.In this talk, I will describe the key properties that identify SLE(6) - the Domain Markov Property, conformal invariance, and the (classical) Locality Property - and introduce a "non-local" form of locality also satisfied by SLE(6), describing the behaviour of the process when it first encloses a target set.
Ectodermal derivatives such as teeth, hair, feathers or scales share similar morphological features and spatial patterning mechanisms. From the mathematical point of view, pioneering works of Alan Turing showed that spatial-temporal self-organization structures can emerge from reaction-diffusion systems. However, recent biological and mathematical studies give evidence that there is a substantial difference in pattern generation between static and growing domains. The latter may contain a key to understanding the problem of sequential patterning in developmental biology.
In this talk we present a macroscopic model of gene expression dynamics in the growing field where molars appear sequentially. Our model mimics the expression of the Edar gene during the formation of signaling centers, from where future teeth originate. We rely on a reaction-diffusion system of an activator-inhibitor type on a dynamically evolving tissue. The key element is not only the tissue growth but also its non-constant properties, which affect the reaction kinetics, depending on the presence of the activator. The purpose of the model is twofold. On one hand it describes a sequential formation of individual spots through Turing instability mechanism. On the other hand, it produces the activator up-regulation waves starting at distal field thanks to reaction functions containing bistable solutions. We present numerical studies of two dynamics on growing domain: under wild conditions and under a mutation regulating the inhibitor concentrations. For a fixed and fully matured domain, we analyze the effect of chemotaxis on the wavelength of Turing patterns and, as a consequence, on the merging of signaling centers that is observed in some biological conditions.
We consider an instance of the phase-retrieval problem, where one wishes to recover a signal (viewed as a vector) from the noisy magnitudes of its inner products with locally supported vectors. Such measurements arise, for example, in ptychography, which is an imaging technique used in lense-less X-ray microscopes and in optical microscopes with increased fields of view.
Starting with the setup where the signal is one-dimensional, we present theoretical and numerical results on an approach that has two important properties. First, it allows deterministic (rather than random) measurement constructions, which we give examples of. Second, it uses a robust, fast recovery algorithm that consists of solving a system of linear equations in a lifted space, followed by simply calculating an eigenvector. We also present extensions, including to the two-dimensional setting.
You may have heard that if you rotate an electron through a full turn, then it points in the opposite direction from how it started. You may have tried to make sense of that by twisting your arms around in knots or turning in a circle until you fell over. And you may have overheard probabilists in this department talking about partition functions and magnetism, and wondered how it was all related.
Allow me to connect the dots with what I've learned while getting up to speed on the physics of spin systems. In the first part, we'll take a physics-style look at how the topology of rotation groups makes magnets work. In the second part, we'll use a classical identity in algebraic graph theory (Kirchoff's matrix-tree theorem) to work out some thermodynamic properties of ridiculously simplified materials. The only background required is undergraduate linear algebra. If you've diagonalized something in the last fifteen years, you're good.
We discuss two related problems that arise in the acquisition and processing of high-dimensional data. First, we consider distance-preserving fast binary embeddings. Here we propose fast methods to replace points from a set \mathcal{X} \subset \R^N with points in a lower-dimensional cube \{\pm 1\}^m, which we endow with an appropriate function to approximate Euclidean distances in the original space.
Second, we consider a problem in the quantization (i.e., digitization) of compressed sensing measurements. Here, we deal with measurements arising from the so-called bounded orthonormal systems and partial circulant ensembles, which arise naturally in compressed sensing applications. In both these problems we show state-of-the art error bounds, and to our knowledge, some of our results are the first of their kind.
In this talk I will explain how to endow the moduli space of objects in certain wrapped Fukaya categories with a cluster structure. This will be achieved by counts of holomorphic disks between Lagrangians which are described by trivalent graphs. In particular, we will be recovering the Fock-Goncharov coordinates on the moduli space of flat connections on a surface and provide a symplectic interpretation for the non-Abelianization procedure via spectral networks.
Insects are first evolved to fly, and to fly is not to fall. How does an insect fly, why does it fly so well, and how can we infer its ‘thoughts’ from its flight dynamics? We have been seeking mechanistic explanations of the complex movement of insect flight. Starting from the Navier-Stokes equations governing the unsteady aerodynamics of flapping flight, we worked to build a theoretical framework for computing flight. This has led to new interpretations and predictions of the functions of an insect’s internal machinery that orchestrate its flight. I will discuss our recent computational and experimental studies of the balancing act of dragonflies and fruit flies: how a dragonfly recovers from falling upside-down and how a fly balances in air. In each case, the physics of flight informs us about the neural feedback circuitries underlying their fast reflexes.
Note for Attendees
There will be a reception before the talk in ESB 4133 (the PIMS lounge). This is a talk in the IAM/PIMS distinguished series.
In this talk we will study complex affine varieties via symplectic topology. First, I will explain how to describe their complex structures, up to deformation, using Legendrian knots. Second, we will focus on the study of these Legendrian knots and provide techniques to distinguish them or show they are isotopic. Then, we will apply them to obtain new results about complex affine manifolds. In particular, we will recover the mirror symmetry functor from the perspective of Legendrian knot theory.
Note for Attendees
Refreshments will be served before this colloquium in MATH 125, the Math Lounge area, at 3:45 p.m.
Consider a contraction pi: X -> Y from a smooth Calabi-Yau 3-fold to a singular one. (This is half of an "extremal transition;" the other half would be a smoothing of Y.) In many examples there is an intermediate object called an "exoflop" -- a category of matrix factorizations, derived-equivalent to X, where the critical locus of the superpotential looks like Y with a P^1 sticking out of it, and objects of D(X) that will be killed by pi_* correspond to objects supported at the far end of the P^1. I will discuss one or two interesting examples. This is joint work with Paul Aspinwall.
What if we could imitate spider silk glands to produce biodegradable materials with properties similar to rubber or plastic? In our interdisciplinary team of fluid dynamicists, chemical engineers and material scientists, my role as mathematician is to try and answer this question from the numerical perspective. In this context, I am working on a problem of multiphase flow that includes advection, diffusion, chemical reaction, osmosis and viscoplastic behavior.
When it comes to the numerical solution of such a model that is based on a real-life problem, I am a strong advocate of so-called mimetic methods, i.e. discretisation schemes which preserve the physical properties of the system also at the discrete level. Following this philosophy,
* the transition between viscous flow and plastic creep is treated in a genuinely nonsmooth fashion and not simply smoothed out,
* the discretisation of the Navier-Stokes equations is pressure-robust,
* conservation of mass and momentum are respected,
* maximum principles are preserved,
* numerical diffusion is limited to an absolute minimum.
Additionally, the algorithm should clearly be stable, efficient and accurate for both steady and unsteady flow problems.
In this talk, I will show how we can couple fast algorithms from convex optimisation, a finite-element discretisation and algebraic flux correction to attain these objectives. Some videos of various flow configurations are included as well!
We use Lusternik-Schnirelmann Theory to study the topology of the space of closed embedded minimal hypersurfaces on a manifold of dimension between 3 and 7 and positive Ricci curvature. Combined with the works of Marques-Neves we can also obtain some information on the geometry of the minimal hypersurfaces they found.
As the source of new genetic variation, mutations constitute a fundamental process in evolution. While most mutations lower fitness, rare beneficial mutations are essential for adaptation to changing environments. Thus, understanding the effects of mutations and estimating their rate is of strong interest in evolutionary biology. The necessity to treat rare mutational events stochastically has also stimulated a rich mathematical literature. Typically, mutations are modelled simply as an instantaneous change of type, occurring at a fixed rate. However, the underlying biology is more complex. I will present two recent projects delving deeper into mutational mechanisms and their consequences. Firstly, mutations can exhibit a multi-generational delay in phenotypic expression. Secondly, individuals within a population can vary in their propensity to mutate. Through analytical and simulation methods, we investigated the impact of these biological complexities on (a) population fitness and capacity to evolve, and (b) our ability to accurately infer mutation rates from data. I will conclude by discussing some future directions to incorporate these insights into more realistic models and to quantify the distribution of mutation rate empirically.
Note for Attendees
Math 126 is behind a locked glass door. Latecomers without access should knock loudly!
A plane set is called a t-Furstenberg set for some t in (0,1), if it has an at least t-dimensional intersection with some line in each direction (here and in the sequel dimension refers to Hausdorff dimension). Classical results are that every t-Furstenberg set has dimension at least 2t, and at least t + 1/2.
We generalize these estimates for families of affine subspaces. As a result, we prove that the union of any s-dimensional family of k-dimensional affine subspaces is at least k + s/(k+1) -dimensional, and is exactly k + s -dimensional if s is at most 1.
Based on joint work with Tamas Keleti and Andras Mathe.
Biological populations facing severe environmental change must adapt in order to avoid extinction. This so-called “evolutionary rescue” scenario is relevant to many applied problems, including pathogen evolution of drug resistance during the treatment of infectious diseases. Understanding what drives the rescue process gives rise to interesting mathematical modelling challenges arising from two key features: demographic and evolutionary processes occur on the same timescale, and stochasticity is inherent in the emergence of rare well-adapted mutants. In this talk, I will present recent work on population dynamics in changing environments, merging biological realism with tractable stochastic models. Firstly, I will describe a model of drug resistance evolution in chronic viral infections, which serves as a case study for a novel mathematical approach yielding analytical approximations for the probability of rescue. Secondly, I will present a combined theoretical and experimental investigation of the classical problem of establishment (non-extinction) of new lineages, using antibiotic-resistant bacteria as a model system. Finally, I will discuss some future directions in modelling antibiotic treatment to predict optimal dosing strategies, and in developing a general theoretical framework for evolutionary rescue that unites approaches to distinct applied problems.
Note for Attendees
Refreshments will be served at 2:45 p.m. in ESB 4133, the PIMS Lounge.
Ordinary differential equation models are used extensively within mathematics as descriptions of processes in the real world. However, they are rarely employed by statisticians and there is a paucity of methods for combining differential equation models with data. This talk provides a survey of recently developed statistical methods for estimating parameters from data, conducting model criticism and improvement for differential equation models in the light of data, and designing experiments that yield optimal estimates of parameters. It ends with some perspectives on the current state of the field and open problems.
Note for Attendees
Reception before the talk in ESB 4133 (the PIMS lounge). This is in the IAM/PIMS distinguished speaker series.
Nakajima quiver varieties lie at the interface of geometry and representation theory. I will discuss a particular example, hyperpolygon space, which arises from star-shaped quivers. The simplest of these varieties is a noncompact complex surface admitting the structure of an "instanton", and therefore fits nicely into the Kronheimer-Nakajima classification of ALE hyperkaehler 4-manifolds, which is a geometric realization of the McKay correspondence for finite subgroups of SU(2). For more general hyperpolygon spaces, we speculate on how this classification might be extended by studying the asymptotic geometry of the variety. In moduli-theoretic terms, this involves driving the stability parameter for the quotient to an irregular value. This is joint work in progress with Harmut Weiss, building on previous work with Jonathan Fisher.
Quasisymmetric maps are fruitful generalizations of conformal maps. Quasisymmetric uniformization problem seeks for extensions of uniformization theorem beyond the classical context of Riemann surfaces.
The goal of this talk is to show that quasisymmetric uniformization problem is closely related to random walks and diffusions. I will explain how the existence of quasisymmetric maps is equivalent to heat kernel estimates for the simple random walk on a family of planar graphs. The same methods also apply to diffusions on a class of fractals homeomorphic to the 2-sphere.
These ideas will be illustrated using snowballs and their graph approximations. Snowballs are high dimensional analogues of Koch snowflake.
Many types of large cells have multiple nuclei. In long muscle cells, nuclei are distributed almost uniformly along their length, which is crucial for cell function. However, the underlying positioning mechanisms remain unclear. We examine computationally the hypothesis that a force balance generated by microtubules positions the nuclei. Rather than assuming what the forces are, we allow for various types of forces between pairs of nuclei and between the nuclei and the cell boundary. Mathematically, this means that we start with a great number of potential models. We then use a reverse engineering approach by screening the models and requiring their predictions to fit imaging data on nuclei positions from hundreds of muscle cells of Drosophila larva. Computational screens result in a small number of feasible models, the most adequate of which suggests that the nuclei repel each other and the cell boundary with forces that decrease with distance.
This suggests that microtubules growing from nuclear envelopes push on neighboring nuclei and the cell boundary. We support this hypothesis with stochastic microscopic simulations. Using statistical and analytical tools such as correlation and bifurcation analysis, we make several nontrivial predictions: An increased nuclear density near the cell poles, zigzag patterns in wider cells, and correlations between the cell width and elongated nuclear shapes, all of which we confirm by image analysis of the experimental data.
This is joint work with Mary Baylies, Alex Mogilner and Stefanie Windner.
Note for Attendees
Refreshments: PIMS tea will be served at 2:45 in ESB 4133.
ESB 4133 (PIMS Lounge) (please note the unusual date)
Thu 25 Jan 2018, 11:00am-12:00pm
Abstract
We derive a universal criterion for the preservation of the lake at rest solution in general mesh-based and meshless numerical schemes for the shallow-water equations with bottom topography. The main idea is a careful mimetic design for the spatial derivative operators in the momentum flux equation that is paired with a compatible averaging rule for the water column height arising in the bottom topography source term. The resulting numerical schemes for the shallow-water equations are called well-balanced.
Based on a well-balanced RBF-FD discretization of the shallow-water equations, we develop a meshless tsunami propagation and inundation model. The moving wet-dry interface between the incoming wave and the shoreline is handled using RBF generated extrapolation, yielding a truly meshless tsunami model. Several numerical results are presented that demonstrate excellent agreement of the resulting model with standard one- and two-dimensional benchmark tests.
This is joint work with Rüdiger Brecht, Scott MacLachlan and Jörn Behrens.
Although the zeta function \zeta(s) is often named after Riemann, it was Euler who discovered many of its remarkable properties. After making his name on the evaluation of \zeta(2), Euler was able to obtain similar formulas at all positive even integers, and defined putative values at negative integers, where the series does not converge. Euler showed these values at negative integers were all rational numbers. A comparison with the values at positive integers led him to guess the functional equation relating \zeta(s) to \zeta(1-s) (which was proved about one hundred years later by Riemann). I will begin by exposing some of this work, then show how the values at negative integers can be used to compute the dimension of certain spaces of automorphic forms. In a special case the dimension turns out to be 1, and this leads to a construction of local systems with exceptional Galois groups on the projective line (minus two points) over a finite field.
PDE models can be a powerful tool for understanding emerging structures and patterns, such as aggregates and traveling waves formed by large populations of cells. As a specific example, I will discuss myxobacteria, which, due to their co-operative nature, lie on the boundary between uni- and multicellular organisms. I will present a novel age-structured, continuous macroscopic model. The derivation is based on simple interaction rules and set within the SOH (Self-Organized Hydrodynamics) framework. The strength of this combined approach is that microscopic information can be incorporated into the particle model in a straight-forward manner, whilst the continuous model can be analyzed using mathematical tools, such as stability and asymptotic analysis.
It has been suggested that myxobacteria are not able to react to signals immediately after they have reversed their direction. Our analysis reveals that this insensitivity period is not necessary for wave formation, but is essential for wave synchronization. A more mathematical focus will be the existence and stability of such traveling waves moving in two opposing waves frames. Fascinatingly, while the wave profiles do not change, the wave composition does, and the fractions of reversible and non- reversible bacteria form waves traveling in the opposite direction. I will discuss the explicit construction of such waves and show simulation results.
This is joint work with Pierre Degond and Hui Yu.
Note for Attendees
Refreshments will be served in ESB 4133 (PIMS Lounge) at 2:45 p.m. before this colloquium.
The essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. I will explain how the representation type of a finitely-generated algebra (finite, tame, wild) is determined by the essential dimension of the functors of its n-dimensional representations and I will introduce new numerical invariants for algebras. I will then illustrate the theorem and explicitely determine the invariants in the case of quiver algebras.
We show that the Hausdorff dimension of the boundary of d-dimensional super-Brownian motion is 0, if d=1; 4-2^{3/2}, if d=2; and (9-17^{1/2}})/2, if d=3. This is a joint work with Leonid Mytnik.
Given a group G the space of n-tuples of pairwise commuting elements can be assembled into a subspace Bcom(G) of the classifying space BG. This space classifies certain types of principal bundles, but unlike BG it is not always aspherical for discrete groups.
The universal cover can be studied using posets, and when G is an extraspecial p-group, I will show that it has the homotopy type of a wedge of spheres. Extraspecial groups appear as the basic observables in quantum computation, and I will also talk about some recent applications of Bcom(G).
I will present a new quantitative genetic model of adaptation to a changing environment. The mathematical analysis will use small variance asymptotics introduced by Diekmann et al in 2005 to derive information on the equilibrium. The framework can handle sexual and asexual reproduction. Heuristics can be made to guess the lineages of the population inside the equilibrium, as shown by numerical simulations.
The longstanding view in chemistry and biology is that high-affinity, tight-binding interactions are optimal for many essential functions, such as receptor-ligand interactions. Yet, an increasing number of biological systems are emerging that challenge this view, finding instead that low-affinity, rapidly unbinding dynamics can be essential for optimal function. These mechanisms have been poorly understood in the past due to the inability to directly observe such fleeting interactions and the lack of a theoretical framework to mechanistically understand how they work. In fact, it is only by tracking the motion of effector nanoprobes that afford detection of multiple such interactions simultaneously, coupled with inferences by stochastic modeling, Bayesian statistics, and bioimaging tools, that we recently begin to obtain definitive evidence substantiating the consequences of these interactions. A common theme has begun to emerge: rapidly diffusing third-party molecular anchors with weak, short-lived affinities play a major role for self organization of micron-scale living systems. My talk will demonstrate how these ideas can answer a longstanding question: how mucosal barriers selectively impede transport of pathogens and toxic particles, while allowing diffusion of nutrients.
Note for Attendees
A light reception will be served at the PIMS Lounge, 4133 from 3:00pm.
Molecular crowding has recognized consequences for biological function. However, there are also circumstances in which un-crowding is important that is, when molecules must evacuate from a region before a given process can occur. One example is offered by the T cell, where large surface molecules must evacuate from a region to allow for the T cell to interact with its target, thereby facilitating immune function. Evacuation is fundamentally stochastic and spatial, since diffusion is a major driver. Studies of molecular evacuation present mathematical and computational challenges. For example, in some scenarios, it is a rare event, making straightforward simulation unfeasible. To obtain a complete picture of diffusional evacuation, we use a combination of perturbation theory and numerical simulation. I will also show evidence of persistent un-crowding in living fungal cells. Based on our understanding of diffusional evacuation, we know that diffusion alone cannot explain these observations. I will discuss our current efforts to quantify and resolve how fungal cells control un-crowding.
Modular functions play an important role in many aspects of number theory. The theory of complex multiplication, one of the grand achievements of the subject in the 19th century, asserts that the values of modular functions at quadratic imaginary arguments generate (essentially all) abelian extensions of imaginary quadratic fields. Kronecker's famous ``Jugendtraum", which later came to be known as Hilbert’s twelfth problem concerns the generalization of this theory to other base fields. I will describe an ongoing work in collaboration with Jan Vonk which identifies a class of functions that seem to play the role of modular functions for real quadratic fields. A key difference with the classical setting is that they are meromorphic functions of a p-adic variable (defined in the framework of “rigid analysis” introduced by Tate) rather than of a complex variable. An important role in this theory of ``rigid modular cocycles" is played by the p-modular group {\bf SL}_2({\rm bf Z}[1/p]) whose cohomology was studied by Serre and Adem.
Biography
Born in 1965 in Paris, France, Darmon moved to Canada in 1968. He received a bachelor's degree from McGill University in 1987 and a PhD in mathematics from Harvard University in 1991, under the supervision of Benedict Gross. He then held a postdoctoral position at Princeton University, under the mentorship of Andrew Wiles. It was around this time that Wiles gained worldwide fame for his proof of Fermat's Last Theorem.
In 1994, Darmon joined the faculty of McGill University, where he is currently a James McGill Professor in the Department of Mathematics and Statistics. His other honors include the André Aisenstadt Prize (1997), the Coxeter-James Prize of the Canadian Mathematical Society (1998), the Ribenboim Prize of the Canadian Number Theory Association (2002), and the John L. Synge Award of the Royal Society of Canada (2008). He was elected a Fellow of the Royal Society of Canada in 2003 and received the 2017 AMS Cole Prize in Number Theory for his contributions to the arithmetic of elliptic curves and modular forms.
Note for Attendees
Refreshments will be served from 3:00pm at the PIMS Lounge: ESB 4133.
A famous theorem by Gabriel asserts that two Noetherian schemes X, Y are isomorphic if and only if the categories Coh(X), Coh(Y) are isomorphic. This theorem has been extended in many directions, including algebraic spaces and stacks (if we consider the monoid structure given by tensor product). One more idea to extend the theorem is the following: let X be a scheme of finite type over a field k, and consider the subcategory of Coh(X) given by sheaves supported in dimension at most d-1. We can form the quotient of Coh(X) by this subcategory, which we will call C_d(X). This category should contain enough information to describe the geometry of X "up to subsets of dimension d-1". In a joint work in progress with John Calabrese, we show that this is indeed true, i.e. to any isomorphism f: C_d(Y) ---> C_d(X) we can associate an isomorphism f': U---> V, where U and V are open subset respectively of X and Y whose complement have dimension at most d-1. Additionally, this isomorphism is unique up to subsets of dimension at most d-1. As a corollary of this result, we show that the automorphisms of C_d(X) are in bijection with the set {"automorphisms of X up to subsets of dimension d-1"} x {"line bundles on X up to subsets dimension d-1"}.
In this talk, I will first introduce the Monge-Ampere eigenvalue problem on general bounded convex domains and related analysis including the Brunn-Minkowski inequality for the eigenvalue. Then I will discuss the recent resolution, in joint work with Ovidiu Savin, of global smoothness of the eigenfunctions of the Monge-Ampere operator on smooth, bounded and uniformly convex domains in all dimensions. A key ingredient in our analysis is boundary Schauder estimates for certain degenerate Monge-Ampere equations.
Consider the Markov process in the space of binary trees in which, at each step, you delete a random leaf and then grow a new leaf in a random location on the tree. In 2000, Aldous conjectured that it should have a continuum analogue, which would be a continuum random tree-valued diffusion. We will discuss a family of projectively consistent Markov chains that are projections of this tree, and discuss how these representations can be passed to the continuum.This is joint work with Soumik Pal, Douglas Rizzolo, and Matthias Winkel.
In the past five years biotechnological innovations have enabled the measurement of transcriptome-wide gene expression in single-cells. However, the destructive nature of the measurement process precludes genuine time-series analysis of e.g. differentiating cells. This has led to the pseudo time estimation (or cell ordering) problem: given static gene expression measurements alone, can we (approximately) infer the developmental progression (or "pseudotime") of each cell? In this talk I will introduce the problem from the typical perspective of manifold learning before re-casting it as a (Bayesian) latent variable problem. I will discuss approaches including nonlinear factor analysis and Gaussian Process Latent Variable Models, before introducing a new class of covariate-adjusted latent variable models that can infer such pseudotimes in the presence of heterogeneous environmental and genetic backgrounds.
Much is known about cloud formation and their behaviour at large scales (kilometers). Considerably less, in atmospheric science, addresses the fluid mechanics dictating smaller scale motions that determine the shapes of cloud edges. Only recently (2015) has the mechanism for the formation of a holepunch cloud been understood; a curious phenomena whereby a growing circular hole in a shallow cloud layer opens up due to a disturbance typically initiated via aircraft. We present a two-dimensional thermodynamic model for the edge motion of a convectively stable cloud under thermodynamic conditions that are near saturation. The proposed model couples stratified fluid mechanics through the Boussinesq equations linked to the theory of moist thermodynamics. This leads to a free boundary model for an interface separating clear and cloudy air. The presence of the two phases of moist air (clear/cloudy) induces derivative discontinuities across the boundary. We are adapting the immersed interface method (IIM), a finite difference approach, to compute the Poisson inversion for pressure to second order accuracy. We then demonstrate the application of this IIM approach to motions of the top of a fog layer. It is confirmed that a propagating wave on the clear/fog interface obeys a gravity wave dispersion relation.
An ordered graph is simple graph with a linearly ordered vertex set. The Turán type extremal theory can be extended to ordered graph by forbidding the appearance of an ordered graph as a subgraph: we ask for the maximal number of edges in an ordered graph having no subgraph order-siomorphic to a given pattern.
Some of the classical results of Turán type extremal graph theory carry over to this setting, while others lead to hard questions. In this talk I survey old and recent results in the area.
Here is my favorite conjecture: If the forbidden pattern is a cycle-free ordered graph which is bipartite with one partite class preceding the other, then the corresponding extremal function (the maximal number of edges of an n vertex ordered graph without this as a subgraph) is o(n^c) for any c>1. This has been proven for a large class of forbidden patterns (joint work with Dániel Korándi, István Thomon and Craig Weidert), but it is open in general and in particular it is also open for a particular ordered path on 8 vertices.
We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres. This is joint work with I. Hambleton.
In both health and disease, cells interact with one another through cellular adhesions. Normal development, wound healing, and metastasis all depend on these interactions. These phenomena are commonly studied using continuum models (partial differential equations). However, a mathematical description of cell adhesion in such tissue models had remained a challenge until 2006, when Armstrong et. al. proposed the use of an integro-partial differential equation (iPDE) model. The initial success of the model was the replication of the cell-sorting experiments of Steinberg. Since then this approach has proven popular in applications to embryogenesis, wound healing, and cancer cell invasions. In this talk, I present a first derivation of the non-local (iPDE) model from an individual description of cell movement. The key to the derivation is the extension of the biological concept of a cell's polarization vector to the mathematical world. This derivation allows me to elucidate in detail how cell level properties such as cell-size of density of adhesion molecules affect tissue level phenomena. I will also present a study of the steady-states of the non-local cell adhesion model on an interval with periodic boundary conditions. The importance of steady-states is that these are the patterns observed in nature and tissues (e.g. cell-sorting experiments). I combine global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the mathematical properties of the non-local term to obtain a global bifurcation result for the first branch of non-trivial solutions. I will extend the non-local cell adhesion model to a bounded domain with no-flux boundary conditions.
The dynamics of the T immune cell membrane and the motion of its surface-bound receptors can be analyzed using a sophisticated microscopy technique called Total Internal Reflection Fluorescence Microscopy (TIRF), where receptors can be tagged with light-emitting particles that are illuminated by a laser. Methods in probability and numerical analysis, such as the Finite-State Hidden Markov Model and the Metropolis-Hastings algorithm, were applied to the trajectories of the receptors from the microscopy images using single-particle tracking to estimate parameters such the diffusivity and Markov state transition probabilities of the receptors. This, however, is very computationally expensive, taking days on a supercomputer for the data analysis to complete. There is also another issue involving the Finite-State Hidden Markov Model: Before applying the model, the user must first choose and fix the number of states to model in the system. This is a significant limitation as it disables the model from adjusting to new data and it increases the possibility of over/under-fitting data and cherry-picking data. In this presentation, we will explore TIRF, the Metropolis-Hastings Algorithm, and an approach to reduce computation time: an Embarrassingly Parallel Monte Carlo Markov Chain (MCMC) heuristic. We will also discuss the potential of using the newly developed Infinite Hidden Markov Model, which aims to overcome the limitation of fixing a finite number of states by allowing an arbitrary number of states to dynamically model data, chosen from an infinite-sized state space.
The Szemerédi-Trotter theorem gives a sharp upper bound on the maximum number of incidences between any finite sets of points and lines living in the real plane; this has also been extended to the complex plane. We can also ask for such an incidence bound over arbitrary fields. I will talk about two results in this direction in work joint with Frank de Zeeuw. The study of incidence bounds over the reals has found many applications in additive combinatorics; in arbitrary fields this utility remains true, and I shall present some of these applications.
Leveraging physical contacts to interact with our surroundings is an essential skill to achieve any physical task, but contact-rich, dynamically changing environments often create significant challenges to autonomous robotic locomotion and manipulation. Unexpected slippage or loss of contact can cause a balance controller to fail during locomotion, incidental contacts with unseen obstacles can disrupt a manipulator during a pick-and-place task, and large impulse induced by contacts can result in irreparable damage to the robot hardware. While there exists computationally tractable contact models to aid the development of robust contro policies, the discontinuities inherent in the contact phenomenon introduce non-differentiability in the equations of motion, rendering traditional approaches to optimal control ineffective. In this talk, I will show that, with intelligent contact control and planning algorithms, the challenge of handling contact can become a solution. The first part of the talk focuses on a model-based approach to controlling a deformable robot for locomotion. The control algorithm leverages both static and dynamic contact friction by solving an optimization with non-differentiable linear complementarity constraints efficiently. The second part of the talk focuses on a model-free reinforcement learning approach to minimizing the damage of humanoid falls. We formulate the control problem as a Markov Decision Process that solves for a contact sequence with the ground such that the maximal impulse incurred during the fall is minimized. Lastly, I will mention some work we have done in the area of data-driven haptic perception for robot-assisted dressing tasks.
Note for Attendees
Reception before the talk in ESB 4133 (the PIMS lounge). This talk is in the IAM/PIMS distinguished speaker series.
Fujita famously conjectured that for a d-dimensional smooth projective variety X with ample divisor H, mH+K_X is basepoint free whenever m\geq d+1. I will discuss recent joint work with Klaus Altmann in which we show this conjecture is true whenever X admits an effective action by a torus of dimension d-1.
One of the central problems in computational mathematics is to fit a suitable model to observed data. Mathematically, this can be posed as a nonlinear least-squares problem. Standard methods for solving such problems are based on the Gauss-Newton and Newton approximation, solved either within a trust-region or with an additional regularization term (e.g., the Levenberg-Marquardt method).
I will describe a method that combines Gauss-Newton and Newton approximations, where appropriate, to produce a hybrid method that exhibits better convergence properties. I will then describe a newly proposed algorithm, the tensor-Newton method, that minimizes a tensor model locally. Since this shares the sum-of- squares nature of the problem being solved, it makes better use of second derivative information that has been computed than the traditional Newton approximation.
Part of the motivation of this work is improving the fitting capabilities of the widely used data analysis and visualization package Mantid. As well as standard test examples, I present results on real-world examples from ISIS, a pulsed neutron and muon source located at the Rutherford Appleton Laboratory. The algorithms described in this talk are available as part of the open source nonlinear least-squares solver RALFit (https://github.com/ralna/RALFit).
A sorting network is a shortest path from the identity to the reverse permutation in the Cayley graph of S_n generated by adjacent transpositions. An n-element uniform random sorting network displays many striking global properties as n approaches infinity. For example, scaled trajectories of the elements 1, 2, ... n converge to sine curves and the 1/2-way permutation matrix measure converges to the projected surface area measure of the 2-sphere.
In this talk, I will discuss how the local structure of random sorting networks can be used to find a global limit, proving these statements and more.
Using p-adic local Langlands correspondence for GL2(Qp), we prove that the support of patched modules constructed by Caraiani, Emerton, Gee, Geraghty, Paskunas, and Shin meet every irreducible component of the potentially semistable deformation ring. This gives a new proof of the Breuil-Mézard conjecture for 2-dimensional representations of the absolute Galois group of Qp when p > 2, which is new in the case p = 3 and \bar{r} a twist of an extension of the trivial character by the mod p cyclotomic character. As a consequence, a local restriction in the proof of Fontaine-Mazur conjecture by Kisin is removed.
Most of the contemporary mathematics we use is built on set theory. To avoid any possible complications, the axioms describing such a system have to be as precise as possible. As a pedantic person who tries to pay attention to logical details (since I find it difficult to trust even my own judgement), I will discuss about various topics and situations which I feel many of us should handle more delicately. Then I will use this opportunity as a segue to talk about axioms, set theory, functions, and some logic.
We shall survey over the general issue of
`specializations which preserve a property',
for a parametrized family of algebraic varieties.
We shall limit ourselves to a few examples.
We shall start by recalling typical contexts like
Bertini and Hilbert Irreducibility theorems,
illustrating some new result.
Then we shall jump to much more recent instances,
related to algebraic families of abelian varieties.
Note for Attendees
Refreshments will be served in ESB 4133 from 2:45 p.m.-3:00 p.m.
Let E/Q be an elliptic curve. In Iwasawa Theory, we study the behaviours of E over a tower of number fields. For example, it is known that the Mordell Weil ranks of E over all p-power cyclotomic extensions of Q are bounded when p does not divide the conductor of E. Surprisingly, the techniques required to show this are very different depending on the number of points on the finite curve when we consider E reduced modulo p. The easier case is when E has "ordinary" reduction at p and the more difficult case is when E has "supersingular" reduction at p. I will review the Iwasawa-theoretic tools used to study the behaviours of E over cyclotomic fields in these two cases. I will also discuss some recent developments on the Iwasawa theory of elliptic curves over quadratic extensions of Q.
This is talk 1 of 3 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
In this talk, we will address the following question: given an algebraic group G acting on a variety X, when does thequotientX/G exist? We will provide an answer to this question in the case that G is reductive by giving necessary and sufficient conditions for thequotientto exist. We will discuss various applications to equivariant geometry, moduli problems and Bridgeland stability.
A spectral method for the solution of integral and differential equations is generally understood to be an expansion of the solution in a Fourier series. Chebyshev polynomials are also often the preferred basis set for many problems. In kinetic theory, the Sonine polynomials have been used for decades for the solution of the Boltzmann equation and the calculation of transport coefficients. This talk will focus on the use of nonclassical polynomials orthonormal with respect to an appropriate weight function chosen dependent on the problem considered. The associated quadrature rules are also used in the pseudospectral solution of several different problems in kinetic theory and quantum mechanics. The recurrence coefficients in the three term recurrence relation for the nonclassical polynomials define the Jacobi matrix, J, and are determined numerically with the Gautschi-Stieltjes procedure. The quadrature points are the eigenvalues of J and the weights are the first components of the i th eigenfunction. This methodology is applied to the solution of the Fokker-Planck equation(1), the Schroedinger equation (2), the evaluation of integrals in quantum chemistry (3) and for nuclear reaction rate coefficients (4).
(1) Pseudospectral solution of the Fokker-Planck equation: the eigenvalue spectrum and the approach to equilibirum. J. Stat. Phys. 164, 1379-1393 (2016).
(2) Pseudospectral method of solution of the Schroedinger equation with nonclassical polynomials; the Morse and Poschl-Teller (SUSY) potentials. J. Comput. Theor. Chem. 1084, 51-58 (2016).
(3) A novel Rys quadrature algorithm for use in the calculation of electron repulsion integrals. J. Comput. Theor. Chem. 1074, 178-184 (2015).
(4) An efficient nonclassical quadrature for the calculation of nonresonant nuclear fusion reaction rate coefficients from cross section data. Comp. Phys. Comm. 205, 61-69 (2016).
In this talk, we will look at the question of existence of blowing-up solutions for smooth perturbations of energy-critical elliptic nonlinear Schrödinger equations on a closed manifold. From a result of Olivier Druet, we know that in dimensions different from 3 and 6, a necessary condition for the existence of blowing-up solutions with bounded energy is that the linear part of the limit equation agrees with the conformal Laplacian at least at one blow-up point. I will present new existence results in situations where the limit equation is different from the Yamabe equation away from the blow-up point. I will also discuss the special role played by the dimension 6. This is a joint work with Frederic Robert.
In this talk I will survey results related to incidences between points and algebraic curves in dimensions 2,3 and 4, and will concentrate on the problem of incidences between points and restricted families of curves in R^3, e.g, "unit circles" is a restricted family inside the family of circles in R^3. I will show a new upper bound for incidences between points and unit circles in R^3, and state more general results and other applications as well.
Let E/Q be an elliptic curve. In Iwasawa Theory, we study the behaviours of E over a tower of number fields. For example, it is known that the Mordell Weil ranks of E over all p-power cyclotomic extensions of Q are bounded when p does not divide the conductor of E. Surprisingly, the techniques required to show this are very different depending on the number of points on the finite curve when we consider E reduced modulo p. The easier case is when E has "ordinary" reduction at p and the more difficult case is when E has "supersingular" reduction at p. I will review the Iwasawa-theoretic tools used to study the behaviours of E over cyclotomic fields in these two cases. I will also discuss some recent developments on the Iwasawa theory of elliptic curves over quadratic extensions of Q.
This is talk 2 of 3 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
Super-Brownian motion is a measure-valued Markov process which arises as the scaling limit of several discrete models, including branching random walk. In dimension one, it has a continuous density. In this talk I will discuss the construction of a boundary local time for the density, which is a random measure supported on the boundary of its zero set. I will then show how a close analysis of the right endpoint of the density's support is used to prove that the local time is positive almost surely (when the process itself is non-zero). An application energy method using the local time then gives an almost sure characterization of the Hausdorff dimension of the boundary of the zero set, completing a result which was previously only known to hold with positive probability.
Locust swarms pose a major threat to agriculture, notably in North Africa and the Middle East. In the early stages of aggregation, locusts form hopper bands. These are coordinated groups that march in columnar structures that are often kilometers long and may contain millions of individuals. We propose a model for the formation of locust hopper bands that incorporates intermittent motion, alignment with neighbors, and social attraction, all behaviors that have been validated in experiments. Using a particle-in-cell computational method, we simulate swarms of up to a million individuals, which is several orders of magnitude larger than what has previously appeared in the locust modeling literature. We observe hopper bands in this model forming as a fingering instability. Our model also allows homogenization to yield a system of partial integro-differential evolution equations. We identify a bifurcation from a uniform marching state to columnar structures, suggestive of the formation of hopper bands.
Let E/Q be an elliptic curve. In Iwasawa Theory, we study the behaviours of E over a tower of number fields. For example, it is known that the Mordell Weil ranks of E over all p-power cyclotomic extensions of Q are bounded when p does not divide the conductor of E. Surprisingly, the techniques required to show this are very different depending on the number of points on the finite curve when we consider E reduced modulo p. The easier case is when E has "ordinary" reduction at p and the more difficult case is when E has "supersingular" reduction at p. I will review the Iwasawa-theoretic tools used to study the behaviours of E over cyclotomic fields in these two cases. I will also discuss some recent developments on the Iwasawa theory of elliptic curves over quadratic extensions of Q.
This is talk 3 of 3 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
Enumerative geometry studies the enumeration of geometric structures, however there are also strong links to many other areas of mathematics. A quintessential example of this is the study of Hurwitz numbers which dates back to the 19th century. Using Hurwitz numbers I will explicitly describe links between geometry, combinatorics, representation theory and physics. I will then discuss recent progress in the subject using modern techniques.
Note for Attendees
Refreshments will be served in ESB 4133 from 2:45 p.m.-3:00 p.m.
Let X be the variety of Borel subgroups of a split semisimple algebraic group G over a field, twisted by a generic G-torsor. Conjecturally, the canonical epimorphism of the Chow ring CH(X) onto the associated graded ring GK(X) of the topological filtration on the Grothendieck ring K(X) is an isomorphism. We prove new cases G=Spin(11) and G=Spin(12) of this conjecture. On an equivalent note, we compute the Chow ring CH(Y) of the highest orthogonal grassmannian Y for the generic 11- and 12-dimensional quadratic forms of trivial discriminant and Clifford invariant. In particular, we describe the torsion subgroup of the Chow group CH(Y) and determine its order which is equal to 16 777 216. On the other hand, we show that the Chow group of 0-cycles on Y is torsion-free.
In open pit mining, one must dig a pit, that is, excavate upper layers of ground to reach valuable minerals. The walls of the pit must satisfy some geomechanical constraints (maximum slope constraints) so as not to collapse. The _ultimate pit limits _problem is to determine an optimal pit, the total volume to be extracted so as to maximize total net profits. We set up the problem in a continuous space framework (as opposed to discretized space, such as with block models), and we show, under weak assumptions, the existence of an optimum pit. For this, we formulate an infinite-dimensional, optimal transportation problem of the Kantorovich type, where the cost function is lower semi-continuous and is allowed to take the value +infinity. We show that this transportation problem is a strong dual to the optimum pit problem, and also yields optimality (complementary slackness) conditions. This approach has the potential of leading to novel algorithmic approaches, yet to be explored, to the ultimate pit limits and related mine planning problems.
This is joint work with Ivar Ekeland (CEREMADE, Université Paris-Dauphine).
Analytic torsion is a classical invariant for compact Riemannian manifolds. The Cheeger-Mueller Theorem relates it to its combinatorial equivalent, the Reidemeister torsion. This can be exploited to study the torsion homology of certain arithmetic lattices as in recent work of Bergeron and Venkatesh.
In my talk I want to explain the definition of analytic torsion for congruence quotients of X=SL(n,R)/SO(n) (which are non-compact). Further, I want to discuss the behavior of the analytic torsion in the limit N->infinity for the spaces G(N)\X with G(N) the principal congruence subgroup of level N in SL(n, R). (Joint work with W. Mueller.)
We consider a Monge-Kantorovich optimal transport problem on the unit simplex with a cost function given by the log of the Euclidean inner product. We show that the transport is the large deviation limit of multiplication by the Dirichlet (or, gamma) process and suitable normalization. This is a multiplicative counterpart to the Wasserstein-2 transport that is carried by adding Brownian motion to an initial mass distribution (called the Schrödinger problem by Léonard). The potential function and the Lagrangian of this transport appear to be closely related to the Wasserstein diffusion (Brownian motion on the Wasserstein space) put forward by Sturm and other coauthors, although it is unclear what the exact nature of this relationship is.
We present a mathematical model of the core mechanism responsible for the regulation of polarized growth dynamics by the small GTPase Cdc42. The model is based on the competition of growth zones of Cdc42 localized at the cell tips for a common substrate (inactive Cdc42) that diffuses in the cytosol. We consider several potential ways of implementing negative feedback between Cd42 and its GEF in this model that would be consistent with the observed oscillations of Cdc42 in fission yeast. We analyze the bifurcations in this model as the cell length increases, and total amount of Cdc42 and GEF increase. Symmetric antiphase oscillations at two tips emerge via saddle-homoclinic bifurcations or Hopf bifurcations. We find that a stable oscillation and a stable steady state can coexist, which is consistent with the experimental finding that only 50% of bipolar cells oscillate. Our model suggests that negative feedback is more likely to be acting through inhibition of GEF association rather than upregulation of GEF dissociation.
p-Adic modular forms have first been defined by J.-P. Serre as q-expansions and have later been interpreted geometrically by N. Katz as sections of certain modular line bundles over the ordinary locus of the relevant modular curves. Katz also defined overconvergent modular forms of integer weights as overconvergent sections of the modular line bundles of that weight. Many years later H. Hida and respectively R. Coleman defined ordinary, respectively finite slope overconvergent modular forms of arbitrary, p-adic weight as q-expansions and using these Coleman and Mazur constructed at the end of the 90's the famous eigencurve. Recently, together with Andreatta, Pilloni and Stevens we have been able to geometrically redefine the overconvergent modular forms of Hida and Coleman and so we were able to generalize these constructions to Hilbert and Siegel modular forms.
This is talk 1 of 2 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
Abstract: We discuss four different problems. The first, a joint work with Malabika Pramanik, concerns large subsets of \mathbb{R}^n that do not contain various types of configurations. We show that a collection of v points satisfying a continuously differentiable v-variate equation in \mathbb{R} can be avoided by a set of Hausdorff dimension \frac{1}{v-1} and Minkowski dimension 1. The second problem concerns large subsets of vector spaces over non-archimedean local fields that do not contain configurations. Results analogous to the real-variable cases are obtained in this setting. The third problem is the construction of measure-zero Besicovitch-type sets in K^n for non-archimedean local fields K. This construction is based on a Euclidean construction of Wisewell and an earlier construction of Sawyer. The fourth problem, a joint work with Kyle Hambrook, is the construction of an explicit Salem set in \mathbb{Q}_p. This set is based on a Euclidean construction of Kaufman.
p-Adic modular forms have first been defined by J.-P. Serre as q-expansions and have later been interpreted geometrically by N. Katz as sections of certain modular line bundles over the ordinary locus of the relevant modular curves. Katz also defined overconvergent modular forms of integer weights as overconvergent sections of the modular line bundles of that weight. Many years later H. Hida and respectively R. Coleman defined ordinary, respectively finite slope overconvergent modular forms of arbitrary, p-adic weight as q-expansions and using these Coleman and Mazur constructed at the end of the 90's the famous eigencurve. Recently, together with Andreatta, Pilloni and Stevens we have been able to geometrically redefine the overconvergent modular forms of Hida and Coleman and so we were able to generalize these constructions to Hilbert and Siegel modular forms.
This is talk 2 of 2 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
Individually and collectively, cells are organized systems with many interacting parts. Mathematical models allow us to infer behaviour at one level of organization from information at another level. In this talk, I will share two biological questions that are answered through the development of new mathematical approaches and novel models.
(1) Molecular motors are responsible for transporting material along molecular tracks (microtubules) in cells. Typically, transport is described by a system of reaction-advection-diffusion partial differential equations (PDEs). To understand how the behaviour of many molecular motors, various model parameters, and nonlinear interactions affect the overall transport process at the cellular level, I develop an asymptotic quasi-steady-state approach, reducing the full PDE system to a single nonlinear PDE. I find that the approximating PDE is a conservation law for the total density of motors within the cell, with effective diffusion and velocity that depend nonlinearly on the motor densities and model parameters.
(2) Protein regulators (GTPases) modulate cell shape and forces exerted by cells. Meanwhile, cells sense forces such as tension. The implications of this two-way feedback on cell behaviour is of interest to biologists. I explore this question by developing a simple mathematical model for GTPase signalling and cell mechanics. The model explains a spectrum of behaviours, including relaxed or contracted cells and cells that oscillate between these extremes. Through bifurcation analysis, I find that changes in single cell behaviour can be explained by the strength of feedback from tension to signalling. When such model cells are connected to one another in a row or in a 2D sheet, waves of contraction/relaxation propagate through the tissue. Model predictions are qualitatively consistent with developmental-biology observations.
This is joint work with Dhananjay Bhaskar, Leah Edelstein-Keshet, Tim Small, and Michael Ward.
Note for Attendees
Refreshments will be served in ESB 4133 from 2:45 p.m.-3:00 p.m.
Environmental governance can be viewed as the process by which a group of individuals builds a set of feedbacks into their social and economic systems to maintain some set of stable structures that promote wellbeing. These feedbacks often take the form of institutions, the rules and norms that structure repeated human interactions. An institutional statement such as "if the fishery biomass, forest cover, groundwater level, etc. is below (above) a certain value, then extraction must (may) be adjusted downward (upward)" can be mathematically formalized as a feedback policy in a dynamical system. As such, dynamical systems theory provides a powerful set of tools to study institutions, governance, and environmental policy. In this talk, I will discuss several dynamic models of social-ecological systems that, when combined with experimental and comparative case-study techniques, can be used to explore the very rich space of environmental governance structures observed in practice, and how they may be used to address the challenge of good environmental governance.
Note for Attendees
Reception beforehand in ESB 4133 (the PIMS lounge). Marty is the distinguished IAM Alumni Lecturer this year.
The Hikita conjecture relates the cohomology ring of a symplectic resolution to the coordinate ring of another such resolution. I will explain this conjecture, and present a new version of the conjecture involving the quantum cohomology ring. There will be an emphasis on explicit examples.
Kitaev’s quantum double models provide a rich class of examples of two-dimensional lattice models with topological order in the ground states and a spectrum described by anyonic elementary excitations. The infinite volume ground states of the abelian quantum double models come in a number of equivalence classes called superselection sectors. We prove that the superselection structure remains unchanged under uniformly small perturbations of the quantum double Hamiltonians. (joint work with Matthew Cha and Pieter Naaijkens)
Zarankiewicz’s problem asks for the largest possible number of edges in a graph with $n$ vertices that does not contain K_{s,t} for some fixed integers $s, t$. Recently, Fox, Pach, Sheffer, Sulk and Zahl considered this problem for semi-algebraic graphs, the ones whose vertices are points in Euclidean spaces and edges are defined by some semi-algebraic relations. They found an upper bound that only depends on the dimensions of those Euclidean spaces; this result is a vast generalization of the well-known Szemer\'edi-Trotter theorem and has many geometric applications. In this talk, we will explain this result and how to extend it to hypergraphs. Our proof uses a packing result in VC-dimension theory and the polynomial partitioning method. As an application, we find an upper bound for the number of unit d × d minors in a d × n matrix with no repeated columns.
Abstract:
In this dissertation we study the maximal directional Hilbert transform operator associated with a set U of directions in the n-dimensional Euclidean space. This operator shall be denoted by H U. We discuss in detail the proof of the (p,p)-weak unboundedness of H U in all dimensions n ≥ 2 and all Lebesgue exponents 1 < p < +∞ if U contains infinitely many directions in IR^n.
This unboundedness result for H U is an immediate consequence of a lower estimate for the (p,p) norm of the operatorH U that we prove if the cardinality of U (denoted by #U) is finite. In this case, we prove that the aforementioned operator norm is bounded from below by the square root of log(#U) up to a positive constant depending only on p and n, for any exponent p in the range 1 < p < +∞ and any n ≥ 2.
These results were first proved by G. A. Karagulyan in the case n = p = 2. The structure of our argument follows Karagulyan’s, but includes the results that are necessary for the extension of the lower estimate to the case 1 < p < +∞ and to all dimensions n ≥ 2.
Finally, a review of the scientific literature on H U and related topics is also included.
Abstract:
In this thesis we investigate generalizations of a theorem by Masser and Zannier concerning torsion specializations of sections in a fibered product of two elliptic surfaces.
We consider the Weierstrass family of elliptic curves 𝐸𝐸𝑡𝑡∶𝑦𝑦2=𝑥𝑥3+𝑡𝑡 and points 𝑃𝑃𝑡𝑡(𝑎𝑎)=(𝑎𝑎,√𝑎𝑎3+𝑡𝑡)in 𝐸𝐸𝑡𝑡parametrized by non-zero 𝑡𝑡.
Given 𝛼𝛼,𝛽𝛽algebraic over 𝑄𝑄2 with rational ratio, we provide an explicit description for the set of parameters 𝑡𝑡=𝜆𝜆 such that 𝑃𝑃𝜆𝜆(𝛼𝛼) and 𝑃𝑃𝜆𝜆(𝛽𝛽) are simultaneously torsion for 𝐸𝐸𝜆𝜆. In particular, we prove that the aforementioned set is empty unless 𝛼𝛼/𝛽𝛽∈{−2,−1/2}. Furthermore, we show that this set is empty even when 𝛼𝛼/𝛽𝛽∉𝑄𝑄 provided that 𝛼𝛼 and 𝛽𝛽 have distinct 2-adic absolute values and the ramification index of 𝛼𝛼/𝛽𝛽 over 𝑄𝑄2is coprime with 6.
Our methods are dynamical. Using our techniques, we derive a recent result of Stoll concerning the Legendre family of elliptic curves 𝐸𝐸𝑡𝑡:𝑦𝑦2=𝑥𝑥(𝑥𝑥−1)(𝑥𝑥−𝑡𝑡), which itself strengthened earlier work of Masser and Zannier by establishing, as a special case, that there is no complex parameter 𝑡𝑡=𝜆𝜆∉{0,1} such that the points with x-coordinates 𝑎𝑎 and 𝑏𝑏 are both torsion in 𝐸𝐸𝜆𝜆, provided 𝑎𝑎,𝑏𝑏 have distinct reduction modulo 2.
We also consider an extension of Masser and Zannier's theorem in the spirit of Bogomolov's conjecture.
Let 𝐸𝐸→𝐵𝐵 be an elliptic surface defined over a number field 𝐾𝐾, where 𝐵𝐵 is a smooth projective curve, and let 𝑃𝑃:𝐵𝐵→𝐸𝐸 be a section defined over 𝐾𝐾 with non-zero canonical height. We use Silverman's results concerning the variation of the Neron-Tate height in elliptic surfaces, together with complex-dynamical arguments to show that the function 𝑡𝑡→ℎ𝐸𝐸𝑡𝑡(𝑃𝑃𝑡𝑡) satisfies the hypothesis of Thuillier and Yuan's equidistribution theorems. Thus, we obtain the equidistribution of points 𝑡𝑡∈𝐵𝐵 where 𝑃𝑃𝑡𝑡 is torsion. Finally, combined with Masser and Zannier's theorems, we prove the Bogomolov-type extension of their theorem. More precisely, we show that there is a positive lower bound on the height ℎ𝐴𝐴𝑡𝑡(𝑃𝑃𝑡𝑡), after excluding finitely many points 𝑡𝑡∈𝐵𝐵, for any `non- special' section 𝑃𝑃 of a family of abelian varieties 𝐴𝐴→𝐵𝐵 that split as a product of elliptic curves.
The Lambertian distribution, also known as Knudsen's Law, is a model for random reflections of light or gas particles from rough surfaces. I will present a mathematical "justification" of the Lambertian distribution. Then I will discuss a deterministic model inspired by stirring coffee. The analysis of the model will be partly deterministic, and partly based on the Lambertian distribution.
Joint work with O. Angel, M. Duarte, C.-E. Gauthier, J. San Martin, and S. Sheffield.
Computations in RO(G)-graded Bredon cohomology can be challenging and are not well understood, even for G=C_2, the cyclic group of order two. In this talk I will present a structure theorem for RO(C_2)-graded cohomology with constant Z/2 coefficients that substantially simplifies computations. The structure theorem says the cohomology of any finite C_2-CW complex decomposes as a direct sum of two basic pieces: shifted copies of the cohomology of a point and shifted copies of the cohomologies of spheres with the antipodal action. I will give some examples and sketch the proof, which depends on a Toda bracket calculation.
Consider a smooth projective variety over a number field. The image of the associated (complex) Abel--Jacobi map inside the (transcendental) intermediate Jacobian is an abelian variety. We show that this abelian variety admits a distinguished model over the number field. Among other applications, this tool allows us to answer a recent question of Mazur; recover an old result of Deligne; and give new constructions of period maps over arithmetic bases.
Clustering appears in many guises, playing important roles in diverse areas of cell biology. One such guise is the spatial clustering of proteins on the membrane of a cell. The ability of cell membrane proteins to cluster in response to stimuli is important to the normal function of many cells, but spontaneous, uncontrolled clustering can lead to cancer. Biologists are therefore keen to analyse protein clustering to better understand how cells function and gain insight into related diseases. This quest is assisted by super-resolution microscopy techniques that enable single molecules to be imaged down to nanoscale precision. In this talk, I will outline StormGraph, a graph-based clustering algorithm that I have developed for the analysis of protein clustering in super-resolution microscopy data. Using simulated data, I have found StormGraph to recover ground-truth clusters more accurately than current leading algorithms, and I have demonstrated its use on super-resolution microscopy data from normal and cancerous B-cells, our antibody-producing immune cells.
I will also provide a brief overview of how I intend to use clustering in multi-dimensional proteomic space to potentially improve personalized cancer therapies in the future. Tumours are heterogeneous populations of cells, and the activity of various signalling proteins can differ between cells within the same tumour. This intratumour heterogeneity is a key driver of resistance to cancer therapies, and should therefore be considered if trying to develop effective personalized therapies. I am working to develop suitable experiments and computational analysis to analyse this heterogeneity in B-cell tumours.
Note for Attendees
Refreshments will be served in ESB 4133 from 2:45 p.m.-3:00 p.m.
The nilpotent cone has very special geometry which encodes interesting representation theoretic information. It is expected that many of its special properties have analogues for general “symplectic singularities.” This talk will discuss one such analogy for a class of symplectic singularities called hypertoric varieties. The main result, joint with T. Braden, can be described as a duality between nearby and vanishing cycle sheaves on Gale dual hypertoric varieties.
The process of optimally fitting a differential-equation model to data is usually approached in an iterative manner by solving the equations numerically with some choice of parameters and using some algorithm (e.g. gradient descent) to improve the choice of parameters with successive steps of the iteration. We propose a new method that steps back from an exact numerical method and instead allows the numerical solution to emerge as part of the optimization. We introduce the objective function (1-s) || x - data ||^2 + s || Dx - f(x;p) ||^2 where x is the model values, Dx=f(x;p) is the differential equation in discrete form (i.e. Dx is the discretization of the differential operator), where we must optimize for model values x and parameters p. We use s to implement niches in a genetic optimization algorithm and extract the best fit in the limit as s approaches 1. This method bypasses the need for implicit solution methods and, interestingly, admits conservative quantities, which allow us to gauge the accuracy of our optimization. I will discuss the theory, benefits, and examples of the method.
In classical ramification theory, we consider extensions of complete discrete valuation rings with perfect residue fields. We would like to study arbitrary valuation rings with possibly imperfect residue fields and possibly non-discrete valuations of rank ≥ 1, since many interesting complications arise for such rings. In particular, defect may occur (i.e. we can have a non-trivial extension, such that there is no extension of the residue field or the value group).
We present some new results for Artin-Schreier extensions of arbitrary valuation fields in positive characteristic p. These results relate the “higher ramification ideal” of the extension with the ideal generated by the inverses of Artin-Schreier generators via the norm map. We also introduce a generalization and further refinement of Kato’s refined Swan conductor in this case. Similar results are true in mixed characteristic (0, p).
This is talk 1 of 2 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
I will discuss some recent joint work with Lu Wang in which we prove that a shrinking gradient Ricci soliton which agrees to infinite order at spatial infinity with a generalized cylinder along some end must be isometric to the cylinder on that end. When the shrinker is complete, it must be globally isometric to the cylinder or else to a Z_2-quotient. This work belongs to a larger program aimed at obtaining a structural classification of complete noncompact shrinking solitons.
In classical ramification theory, we consider extensions of complete discrete valuation rings with perfect residue fields. We would like to study arbitrary valuation rings with possibly imperfect residue fields and possibly non-discrete valuations of rank ≥ 1, since many interesting complications arise for such rings. In particular, defect may occur (i.e. we can have a non-trivial extension, such that there is no extension of the residue field or the value group).
We present some new results for Artin-Schreier extensions of arbitrary valuation fields in positive characteristic p. These results relate the “higher ramification ideal” of the extension with the ideal generated by the inverses of Artin-Schreier generators via the norm map. We also introduce a generalization and further refinement of Kato’s refined Swan conductor in this case. Similar results are true in mixed characteristic (0, p).
This is talk 2 of 2 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
First-passage percolation is a classical random growth model which comes from statistical physics. We will discuss recent results about the relationship between the limiting shape in first passage percolation and the structure of the infinite geodesics. This includes a solution to the midpoint problem of Benjamini, Kalai and Schramm. This is joint work with Daniel Ahlberg.
Understanding incidence, spread, prevalence and control of an infectious disease requires a multidisciplinary approach that encompasses many fields of inquiry in Natural and Social Sciences. Several biological, environmental and economic/social/demographic factors govern the disease spread in a population. The overall pattern of a disease incidence is an outcome of the interaction of all these processes acting at different scales - from genetic epidemiology to public health - making it a complex multi-scale and interdisciplinary study.
Mathematical modelling of the disease process has been one of the oldest areas of study in Mathematical Biology. It has contributed significantly to the understanding of basic infection process, predicting future incidence to aid in taking immediate control measures, drug discovery, and health policy development. It uses application of concepts from different areas in mathematics, statistics and computational algorithms for data analysis and visualization. Each theoretical approach incorporates information from the biological, environmental, and social sciences, and offers understanding at different scales.
In this talk I will outline studies at three different scales to highlight the type of data required, variety of methods of analysis, and kinds of inferences/information that the analysis offers. I will show that comparative whole genome analysis of HIV-1, the pathogen responsible for AIDS, offers some insights into the differential evolution of HIV-1 genes; Understanding HIV-1 Reverse Transcriptase (RT) wild-type and mutant protein structures using graph theory allows us to uncover the drug resistance mechanisms in RT-drug mutants. Finally, at the population level modelling of disease spread, I will discuss our studies of Malaria using mathematical, statistical, and graphical approaches suitable for a diversity of fine and coarse-grained data from India.
Note for Attendees
Refreshments are served in the PIMS lounge (ESB 4th floor) at 2:45PM
Prof. Sinha is an International Visiting Research Scholar at the Peter Wall Institute for Advanced Studies (PWIAS); her hosts in the Department of Mathematics are Daniel Coombs and Leah Keshet. This public talk is shared between PWIAS, the UBC Mathematics Department, and the Pacific Institute for Mathematical Sciences, where she is also affiliated during her visit to UBC.
We show that Lubin--Tate spectra at the prime 2 are Real oriented and Real Landweber exact. The proof is by application of the Goerss--Hopkins--Miller theorem to algebras with involution. For each height n, we compute the entire homotopy fixed point spectral sequence for E_n with its C_2-action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these C_2-fixed points.
We consider the family of D_4-quartic fields ordered by the Artinconductors of the corresponding 2-dimensional irreducible Galoisrepresentations. In this talk, I will describe ways to compute the numberof such D_4 fields with bounded conductor. Traditionally, there have beentwo approaches to counting quartic fields, using arithmetic invariant theory in combination of geometry-of-number techniques, and applying Kummertheory together with L-function methods. Both of these strategies fallshort in the case of D_4 fields since counting quartic fields containing aquadratic subfield of large discriminant is difficult. However, whenordering by conductor, these techniques can be utilized due to additionalalgebraic structure that the Galois closures of such quartic fields have,arising from the outer automorphism of D_4. This result is joint work withAli Altug, Arul Shankar, and Kevin Wilson.
The symmetry group of a knot in the three-sphere is the mapping class group of the knot’s exterior. Elements of order two with fixed point set meeting the boundary of the knot exterior are called strong inversions, and a pair (K,h) is called a strongly invertible knot when h is a strong inversion in the symmetry group of K. Studying the equivalence of strongly invertible knots amounts to studying conjugacy classes of strong inversions. I will discuss how to construct invariants of strongly invertible knots using Khovanov homology. Some of this is joint work with Mike Snape and, time permitting, I will also discuss work in progress with Andrew Lobb.
We shall place in a general context the following result recently (*) obtained jointly with Yuri Bilu (Bordeaux), Sanoli Gun (Chennai) and Florian Luca (Johannesburg).
Theorem. Let τ(·) be the classical Ramanujan τ-function and let k be a positive integer such that τ(n) ≠ 0 for 1 ≤ n ≤ k/2. (This is known to be true for k < 1023 , and, conjecturally, for all k.) Further, let σ be a permutation of the set {1, ..., k}. We show that there exist infinitely many positive integers m such that |τ(m + σ(1))| < |τ(m + σ(2))| < ... < |τ(m + σ(k))|.
The proof uses sieve method, Sato-Tate conjecture, recurrence relations for the values of τ at prime power values.
(*) Hopefully to appear in 2018
In this talk, I will present universality results for the edge transport properties of interacting, 2d topological insulators. I will mostly focus on the case of quantum Hall systems, displaying single mode edge currents. After reviewing recent results for the bulk transport properties, I will present a theorem establishing the universality of the edge conductance and the emergence of spin-charge separation for the edge modes. Combined with well-known results for noninteracting systems, our theorem implies the validity of the bulk-edge correspondence for a class of weakly interacting 2d lattice models, including for instance the interacting Haldane model. The proof is based on rigorous renormalization group methods, and on the combination of chiral Ward identities for the effective 1d QFT describing the infrared scaling limit of the edge currents, together with lattice Ward identities for the original lattice model. Joint work with G. Antinucci (UZH/Tuebingen) and with V. Mastropietro (Milan).
The asymptotics of the second-largest eigenvalue in random regular graphs (also referred to as the Alon conjecture) have been computed by Joel Friedman in his celebrated 2004 paper. Recently, a new proof of this result has been given by Charles Bordenave, using the non-backtracking operator and the Ihara-Bass formula. In the same spirit, we have been able to translate Bordenave's ideas to bipartite biregular graphs in order to calculate the asymptotical value of the second-largest pair of eigenvalues, and obtained a similar spectral gap result. Applications include community detection in equitable graphs or frames, matrix completion, and the construction of channels for efficient and tractable error-correcting codes (Tanner codes). This work is joint with Gerandy Brito and Kameron Harris.
Diffuse Large B-Cell Lymphoma (DLBCL), a non-Hodgkin lymphoma, is the most common blood cancer and comprises more than two subtypes. The Activated B-Cell like (ABC) subtype has inferior survival rates, and is typically characterized by constitutive signalling that resembles B-cell activation following antigen engagement. However, there is significant heterogeneity observed clinically within the ABC subtype of DLBCL, with various mutations able to give rise to this oncogenic signalling. When present within an individual patient's tumour, this kind of heterogeneity can lead to drug resistance due to evolutionary selection for cells with mutations that confer drug resistance. Optimized personalized therapies should therefore account for any underlying intratumour heterogeneity to prevent or delay the onset of drug resistance. In this work-in-progress talk, I will present our work towards developing a pipeline using mass cytometry -- a technique that enables the measurement of over 30 proteins simultaneously in single cells -- and computational analysis to study proteomic heterogeneity, especially at the level of intracellular signalling, in DLBCL samples. Since the 'ground-truth' cellular populations (clusters in proteomic or mutational feature space) that make up a heterogeneous tumour are not known for real tumours, we have devised novel mass cytometry experiments to simulate a heterogeneous DLBCL sample using cell lines as 'ground truth' populations. This novel data will facilitate the improvement of existing, and development of new, computational algorithms for analysing heterogeneity and signalling in tumours.
Contextuality is a property of quantum mechanics that sets it apart from classical physics. Recently, it has been established as a necessary ingredient that any quantum computation must have in order to provide a speedup over conventional classical computation [1], [2]. It has thus become a resource.
In my talk, I will first review the notion of quantum contextuality, and then explain how it is a useful commodity in quantum computation---for the models of quantum computation with magic states and measurement-based quantum computation (MBQC).
I’ll end with a cohomological picture underlying MBQC and contextuality [3], which is a focus of current research in my group.
[1] M. Howard et al., Nature (London) 510, 351 (2014).
[2] R. Raussendorf, Phys. Rev. A 88, 022322 (2013).
[3] C. Okay et al., Quantum Information and Computation 17, 1135-1166 (2017).
Abstract: The hyperkahler quotient construction (introduced by Hitchin et al in the 1980s) allows us to construct new hyperkahler spaces from suitable group actions on hyperkahler manifolds. This construction is an analogue of symplectic reduction (introduced by Marsden and Weinstein in the 1970s), and both are closely related to the quotient construction for complex reductive group actions in algebraic geometry provided by Mumford's geometric invariant theory (GIT). Hyperkahler implosion is in turn an analogue of symplectic implosion (introduced in a 2002 paper of Guillemin, Jeffrey and Sjamaar) which is related to a generalised version of GIT providing quotients for non-reductive group actions in algebraic geometry.
Moduli spaces arise naturally in classification problems in geometry. The study of the moduli spaces of nonsingular complex projective curves (or equivalently of compact Riemann surfaces) goes back to Riemann himself in the nineteenth century. The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT), developed in the 1960s. Here a projective curve is stable if it has only nodes as singularities and its automorphism group is finite. The aim of this talk is to describe these moduli spaces and outline their GIT construction, and then explain how recent methods from non-reductive GIT can help us to classify the singularities of unstable curves in such a way that we can construct moduli spaces of unstable curves (of fixed singularity type).
Note for Attendees
Light refreshments will be served at 2:45pm in ESB 4133, the PIMS Lounge before this colloquium.
The existence of critical points for the Moser-Trudinger inequality for large energies has been open for a long time. We will first show how a collaboration with G. Mancini allows to recast the Moser-Trudinger inequality and the existence of its extremals (originally due to L. Carleson and A. Chang) under a new light, based on sharp energy estimate. Building upon a recent subtle work of O. Druet and P-D. Thizy, in a work in progress with O. Druet, A. Malchiodi and P-D. Thizy, we use these estimates to compute the Leray-Schauder degree of the Moser-Trudinger equation (via a suitable use of the Poincaré-Hopf theorem), hence proving that for any bounded non-simply connected domain the Moser-Trudinger inequality admits critical points of arbitrarily high energy.
Frequently, the interests of a group do not align with those of its members. An individual could, for instance, do well by considering the collective needs in its actions but many times, it can gain even more benefits within the group by pursuing personal interests to the detriment of the entire community. This social dilemma is at the heart of public good interactions, and of particular importance when the production of a public resource is essential for the survival of a population. This scenario occurs, for example, when microbes secrete substances which grant microbial communities resistance to antibiotic drugs.
The arising dynamics for the public good producing cooperative and the freeriding non-cooperative subpopulations have previously been analyzed by Professor Hauert and Professor Doebeli and extended by Wakano et al. into a spatial setting, in which the diffusing microbes form clusters and showcase rich patterns. As many microbes sense chemical gradients - and with that the public good - directional movement can lead to the aggregation of cooperative clusters and the exploitation through the defective subpopulation alike. In this talk, I will incorporate chemotactic migration in the aforementioned models and discuss how this extension affects the composition of the subpopulation, and whether cooperation can be maintained.
This talk also showcases some parts of my research that are still in progress, and I'm happy to hear your feedback.
Braverman and Kazhdan have conjectured the existence of summation formulae that are essentially equivalent to the analytic continuation and functional equation of Langlands L-functions in great generality. Motivated by their conjectures and related conjectures of L. Lafforgue, Ngo, and Sakellaridis, Baiying Liu and I have proven a summation formula analogous to the Poisson summation formula for the subscheme cut out of three quadratic spaces (V_i,Q_i) of even dimension by the equation
Q_1(v_1)=Q_2(v_2)=Q_3(V_3).
I will sketch the proof of this formula in the first portion of the talk. In the second portion, time permitting, I will discuss how these summation formulae lead to functional equations for period integrals for automorphic representations of
GL_{n_1} \times GL_{n_2} \times GL_{n_3}
where the n_i are arbitrary, and speculate on the relationship between these period integrals and Langlands L functions.
Abstract:
Cell migration is a complex interplay of biochemical and biophysical mechanisms. I investigate the link between individual and collective cell behaviour using mathematical and computational modelling. Specifically, I study: (1) cell-cell interactions in a discrete framework with a spatial sensing range, (2) migration of a cluster of cells during zebrafish (Danio rerio) development, and (3) collective migration of cancer cells and their interactions with the extracellular-matrix (ECM).
My 1D model (1), is approximated by a continuum equation and investigated using asymptotic approximations, steady-state analysis, and linear stability analysis. Analysis and computations characterize regimes corresponding to cell clustering, and provide a link between micro and macro-scale parameters. Results suggest that drift (i.e. due to chemotaxis), can disrupt the formation of cellular aggregates.
In (2), I investigate spontaneous polarization of a cell-cluster (the posterior lateral line primordium, PLLP) in zebrafish development. I use a cell-based computational framework (HyDiCell3D) coupled with a differential equation model to track the segregation and migration of the PLLP. My model includes mutual inhibition between the diffusible growth factors Wnt and FGF. I find that a non-uniform degradation of an extracellular chemokine (CXCL12a) and chemotaxis is essential for long range cohesive migration. Results compare favourably with data from the Chitnis lab (NIH).
I continue using HyDiCell3D in (3) to elucidate mechanisms that facilitate cancer invasion. I focus on: wound healing in a cell-sheet (2D epithelium), and cell-clusters (3D spheroids) embedded in ECM with internal signalling mediated by podocalyxin, a trans-membrane molecule. Experimental data from the Roskelley lab (UBC) motivates the model derivation. I use the models to investigate the role of cell-cell and cell-ECM adhesion in collective migration as well as the emergence of a distinct phenotype (leader-cells) that guides the migration. ECM induced disruption in the localization of podocalyxin on the cell membrane is captured in the model along with morphological changes of spheroids. The model predicts that cell polarity and cell division axis influence the invasive potential. Lastly, I develop quantitative methods for image analysis and automated tracking of cells in a densely packed environment to compare modelling results and biological data.
I will discuss some upper bounds for the length of a shortest periodic geodesic, and the smallest area of a closed minimal surface on closed Riemannian manifolds of dimension 4 with Ricci curvature between -1 and 1. These are the first bounds that use information about the Ricci curvature rather than sectional curvature of the manifold. (Joint with Nan Wu).
I will also give examples of Riemannian metrics on the 3-disk demonstrating that the maximal area of 2-spheres arising during an ``optimal" homotopy contracting its boundary cannot be majorized in terms of the volume and diameter of the 3-disc and the area of its boundary. This contrasts with earlier 2-dimensional results of Y. Liokomovich, A. Nabutovsky and R. Rotman and answers a question of P. Papasoglu. On the other hand I will show that such an upper bound exists if, instead of the volume, one is allowed to use the first homological filing function of the 3-disc. (Joint with Parker Glynn-Adey).
As biotechnologies for data collection become more efficient and mathematical modeling becomes more ubiquitous in the life sciences, analyzing both high-dimensional experimental measurements and high-dimensional spaces for model parameters is of the utmost importance. We present a perspective inspired by differential geometry that allows for the exploration of complex datasets such as these. In the case of single-cell leukemia data we present a novel statistic for testing differential biomarker correlations across patients and within specific cell phenotypes. A key innovation here is that the statistic is agnostic to the clustering of single cells and can be used in a wide variety of situations. Finally, we consider a case in which the data of interest are parameter sets for a nonlinear model of signal transduction and present an approach for clustering the model dynamics. We motivate how the aforementioned perspective can be used to avoid global bifurcation analysis and consider how parameter sets with distinct dynamic clusters contrast.
Morphogenesis, the shaping of organisms, organs and tissues is driven by chemical signals and physical forces. It is still poorly understood how cells are able to collectively form intricate patterns, like for instance vascular networks. In particular, we were concerned with how interactions between the cell and the extracellular matrix (a protein network surrounding tissues that supports cells and guides cell migration) regulates morphogenesis. My PhD has mainly focused on how physical forces may drive morphogenesis. Lab experiments have shown that the mechanical properties of the matrix, such as its stiffness, regulate morphogenesis. In this presentation I will focus on my work on mechanical cell-matrix interactions. We developed a multiscale model that describe cells and the matrix and their interactions through physical forces. In this model, cells are represented by the Cellular Potts Model. The deformations in the ECM are calculated using a Finite Element Method. We model a mechanical feedback between cells and the ECM, where 1) cells pull on the ECM, 2) strains are generated in the ECM, and 3) cells preferentially extend protrusions oriented with strain. Similar to lab experiments, simulations show that cells are able to generate vascular like patterns on matrices of intermediate stiffness. Lab experiments where the matrix is uniaxially stretched, show that cells orient parallel to stretch. Model results on cells on a stretched matrices with and without traction forces indicate that cell traction forces amplify cell orientation parallel to stretch. Furthermore, they allow cells to organize into strings in the direction of stretch. I will also show an extension of this model. Stiffness sensing is mediated by transmembrane integrin molecules, which behave as ‘catch bonds’ whose strength increases under tension. Focal adhesions, which are large assemblies of these integrins, grow larger on stiffer substrates. We included such dynamics in our multiscale model. This second model explains how cell shape depends on matrix stiffness and how cells are able to durotact (move up a stiffness gradient). This model gives a more molecular understanding of how cells respond to matrix stiffness.
In growing plant cells, parallel ordering of microtubules (MTs) influences the direction of cell expansion. Models of MT growth in the plane and on polyhedral surfaces have shown that growing-MT encounters lead to the formation of ordered arrays. The polyhedral surfaces models assume that when a MT crosses an edge, it emerges on the adjacent face at the same angle with the edge as the incident angle (i.e. following geodesics). This assumption ignores the MT mechanics - an elastic rod constrained to a rigid surface ought to deflect away from a geodesic when such a deflection decreases its energy. Here, we show this principle for a growing elastic rod on a cylindrical surface with one end clamped. We write down an energy functional that accounts for the bending energy of the rod and derive the associated Euler-Lagrange equation getting a two-variable boundary value problem. Minima and their stability can be found analytically in some cases. The system has a locus of saddle-nodes with a pitchfork in the symmetric case. In general, growing rods deflect away from high curvature directions and toward the flat axial direction, as expected. A rod growing circumferentially continues to grow circumferentially until a critical length (the pitchfork) after which it buckles up or down the cylindrical wall. Our results indicate that, for consistency with observations, the growing tip of MTs ought to be no longer than the radius of curvature of the cell.
Predicting the impacts of environmental change on species requires a mechanistic understanding of biological processes such as foraging, migration, and reproduction. However, the continuous behavioural data needed to assess how these processes change through time is often impossible to gather, particularly for Arctic and marine species. Thus, ecologists increasingly rely on animal telemetry to monitor activity patterns. In this talk, I will demonstrate how emerging statistical methods and movement data can be used to model the behaviour of a range of species (e.g. polar bear, rhinoceros auklet), and discuss how the information provided by movement models can help us answer fundamental ecological questions and solve conservation problems.
We first study the general theory of Kähler-Ricci flow on non-compact complex manifolds. By using a parabolic Schwarz lemma and a local scalar curvature estimate, we prove a general existence theorem for Kähler metrics lying in the 𝐶𝐶∞-closure of complete bounded curvature Kähler metrics that are uniformly equivalent to a fixed background metric. In particular we do not assume any curvature bounds. Next, we compare the maximal existence time of two complete bounded curvature solutions by using the equivalence of the initial metrics and using this, we also estimate the maximal existence time of a complete bounded curvature solution in terms of the curvature bound of a background metric. We also prove a uniqueness theorem for Kähler-Ricci flow which slightly improves the result of Chen-Zhu in the Kähler case.
We apply the above results to study the Kähler-Ricci flow on some specific non-compact complex manifolds. We first study the Kähler-Ricci flow on ℂ𝑛𝑛. By applying our general existence theorem and existence time estimate, we show that any complete non-negatively curved U(n)-invariant Kähler metric admits a long-time U(n)-invariant solution to the Kähler-Ricci flow, and the solution converges to the standard Euclidean metric after rescaling.
Then we study the Kähler-Ricci flow on a quasi-projective manifold 𝑀𝑀∖𝐷𝐷. By modifying the approximation theorem of Blocki-Kolodziej and applying a general existence theorem of Lott-Zhang, we construct a Kähler-Ricci flow solution starting from certain smooth Kähler metrics. In particular, if the metric is the restriction of a smooth Kähler metric in the ambient space 𝑀𝑀, then the solution instantaneously becomes complete and has cusp singularity at 𝐷𝐷. We also produce a solution starting from some complete metrics that may not have bounded curvature, and the solution is likewise complete with cusp singularity for positive time. On the other hand, if the initial data has bounded curvature and is asymptotic to the standard cusp model at 𝐷𝐷 in a certain sense, we find the maximal existence time of the corresponding complete bounded curvature solution to the Kähler-Ricci flow.
In this work, we focus on three problems. First, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in 3-dimensional Euclidean space tends to infinity as its genus or the number of boundary components tends to infinity.
Second, we consider the relationship between the kth normalized eigenvalue of the Dirichlet-to-Neumann map (the kth Steklov eigenvalue) and the geometry of rotationally symmetric Mobius bands. More specifically, we look at the problem of finding a metric that maximizes the kth Steklov eigenvalue among all rotationally symmetric metrics on the Mobius band. We show that such a metric can always be found and that it is realized by the induced metric on a free boundary minimal Mobius band in the 4-dimension Euclidean ball.
Third, we consider the existence problem for harmonic maps into CAT(1) spaces. Given a continuous finite energy map ϕ from a compact Riemann surface Σ into a compact locally CAT(1) space X, we use the technique of harmonic replacement to prove that either there exists a harmonic map u : Σ → X homotopic to ϕ or there exists a conformal harmonic map v from the two-sphere into X. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.
Human African Trypanosomiasis (HAT) is a vector-borne disease endemic to rural areas of Sub-Saharan Africa. Over the last 150 years the disease has been a serious challenge to the people of Africa, with multiple outbreaks resulting the deaths of hundreds of thousands of people. Recent work by local programs and NGOs has had a large impact on controlling HAT through interventions like screening and vector control, and has brought the total number of recorded cases of HAT to its lowest point ever. However, this low number of cases brings with it a peculiar set of challenges in the pursuit of elimination of HAT. In this talk we will discuss how mathematical modeling and new data analysis methods can be brought to bear on some of these challenges. In particular, we will be using traditional models to assess the future impact of various interventions, to demonstrate the need for finer case data, and we will be using an equation-free data analysis method (Dynamic Mode Decomposition) to identify hotspots of disease activity and areas of low treatment coverage in the Democratic Republic of the Congo.
We start with a general introduction to pro-objects in topological and geometric settings, with descriptions of well-known examples, including Postnikov towers, absolute Galois groups, and etale homotopy types.
This is followed by a brief discussion of homotopy theories of pro-objects.
Homotopy theoretic techniques are applied to give the first correct comparison of finite descent and Galois cohomological descent for simplicial presheaves on the etale site of a field. Examples arise in the study of algebraic K-theory with finite coefficients over a field.
Abstract: In October 2017, I found myself testifying for hours in a Federal court. I had not been arrested. Rather I was attempting to quantify gerrymandering using analysis which grew from asking if a surprising 2012 election was in fact surprising. It hinged on probing the geopolitical structure of North Carolina using a Markov Chain Monte Carlo algorithm. I will start at the beginning and describe the mathematical ideas involved in our analysis. And then explain some of the conclusions we have reached. The talk will be accessible to undergraduates. In fact, this project began as a sequence of undergraduate research projects and undergraduates continue to be involved to this day.
About the Niven Lecture: Ivan Niven was a famous number theorist and expositor; his textbooks have won numerous awards and have been translated into many languages. They are widely used to this day. Niven was born in Vancouver in 1915, earned his Bachelor's and Master's degrees at UBC in 1934 and 1936 and his Ph.D. at the University of Chicago in 1938. He was a faculty member at the University of Oregon since 1947 until his retirement in 1982. The annual Niven Lecture, held at UBC since 2005, is funded in part through a generous bequest from Ivan and Betty Niven to the UBC Mathematics Department.
Note for Attendees
The Grad Reception on Monday, May 28th will happen from 11:15 a.m. - 12:45 p.m. (lunch and awards presentation).
Dr. Denise Daley PhD, is an Associate Professor in the Department of Medicine at the University of British Columbia. Dr. Daley is trained as a statistical geneticist with PhDs in Epidemiology and Biostatics and she currently holds a Canada Research Chair in Genetic Epidemiology of Complex Diseases. Dr. Daley is a Principal Investigator at the Centre for Heart and Lung Innovation at St. Paul’s Hospital in Vancouver, where she currently studies complex diseases such as asthma, food allergies, cancer, heart disease, and healthy aging. In particular, she has focused on why some children get asthma/allergic disease and others do not. Dr. Daley is currently investigating genes that may predispose children to developing asthma, and how a combination of genetic variations can interact with gender and the environment to produce the condition.
This presentation will focus on the concepts, principles and results of genetic association studies both candidate gene and genome-wide association studies, and the statistical models used to identify associations. A brief discussion of how this information can be used to identify individuals at risk for disease and the implications for clinical risk management.
Abstract:
Individually and collectively, cells are organized systems with many interacting parts. Mathematical models allow us to infer behaviour at one level of organization from information at another level. In this thesis, I explore two biological questions that are answered through the development of new mathematical approaches and novel models.
(1) Molecular motors are responsible for transporting material along molecular tracks (microtubules) in cells. Typically, transport is described by a system of reaction-advection-diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the PDE system. To understand how nonlinear reactions affect the overall transport process at the cellular level, I extend the QSS approach to certain nonlinear reaction models, reducing the full PDE system to a single nonlinear PDE. I find that the approximating PDE is a conservation law for the total density of motors within the cell, with effective diffusion and velocity that depend nonlinearly on the motor densities and model parameters. Cell-scale predictions about the organization and distribution of motors can be drawn from these effective parameters.
(2) Rho GTPases are a family of protein regulators that modulate cell shape and forces exerted by cells. Meanwhile, cells sense forces such as tension. The implications of this two-way feedback on cell behaviour is of interest to biologists. I explore this question by developing a simple mathematical model for GTPase signalling and cell mechanics. The model explains a spectrum of behaviours, including relaxed or contracted cells and cells that oscillate between these extremes. Through bifurcation analysis, I find that changes in single cell behaviour can be explained by the strength of feedback from tension to signalling. When such model cells are connected to one another in a row or in a 2D sheet, waves of contraction/relaxation propagate through the tissue. Model predictions are qualitatively consistent with developmental-biology observations such as the volume fluctuations in a cellular monolayer. The model suggests a mechanism for the organization of tissue-scale behaviours from signalling and mechanics, which could be extended to specific experimental systems.
Environmental challenges, such as pollution and anti-biotics, can cause populations to decline towards extinction. But declining populations can also be rescued from extinction by sufficiently fast adaptive evolution. In this talk I’ll describe some simple mathematical models we’ve created and analyzed to predict when and how evolutionary rescue will occur. In particular I’ll talk about how we’ve used branching process theory and a geometrical representation of trait space to predict how many mutations evolutionary rescue is likely to take and what the characteristics of these mutations will be given population survival.
We propose a mathematical model based on non-equilibrium thermodynamics to describe the mechanical behavior of an active polymer gel created by the inclusion of molecular motors in its solvent. When activated, these motors attach to the chains of the polymer network and shorten them creating a global contraction of the gel, which mimics the active behavior of a cytoskeleton. The power generated by these motors is obtained by ATP hydrolysis reaction, which transduces chemical energy into mechanical work. The model is based on the Flory and Rehner theory for polymer network swelling and considers species diffusion to describe the transient passive behavior of the gel. The active behavior is modeled defining a volumetric density of mechanical power generated by the motors, through ATP hydrolysis, which increases the strain energy of the polymer network. The latter is depicted by an increment of the crosslink density in the polymer network, reducing the entropy of the polymer network. The model is finally adapted to the problem of uniaxial contraction of a slab of gel and compared with experimental results, showing good agreement.
Using variational and gluing techniques, we construct solutions to local and non-local semilinear elliptic equations. The new fractional elliptic gluing method can be used to obtain a counterexample for the fractional De Giorgi conjecture for ˝<s<1. Singular solutions to the fractional Yamabe problem are constructed. We also establish the necessary condition for the existence of solutions for the linearly perturbed nonlinear Schrödinger equation in the hyperbolic space.
Changes in cell-substrate adhesion are believed to signal the onset of cancer metastasis, but such changes must be quantified against background levels of intrinsic heterogeneity between cells. Variations in cell-substrate adhesion strengths can be probed through biophysical measurements of cell detachment from substrates upon the application of an external force. I will describe theoretical and experimental investigations of the detachment of cells adhered to substrates when these cells are subjected to fluid shear. I will present a theoretical framework within which we calculate the fraction of detached cells as a function of shear stress for fast ramps as well as for the decay in the fraction of detached cells at fixed shear stress as a function of time. Using HEK and 3T3 fibroblast cells as experimental model systems, characteristic force scales for cell adhesion as well as characteristic detachment times are extracted. Variations in adhesion across cell types are especially prominent when cell detachment is probed by applying a time-varying shear stress. These methods can be applied to characterizing changes in cell adhesion in a variety of contexts, including metastasis.
What can mathematics tell us about the treatment of cancer? In this talk I will present some of work that I have done in the modeling of tumor growth and treatment over the last fifteen years.
Cancer is a myriad of individual diseases, with the common feature that an individual's own cells have become malignant. Thus, the treatment of cancer poses great challenges, since an attack must be mounted against cells that are nearly identical to normal cells. Mathematical models that describe tumor growth in tissue, the immune response, and the administration of different therapies can suggest treatment strategies that optimize treatment efficacy and minimize negative side-effects.
However, the inherent complexity of the immune system and the spatial heterogeneity of human tissue gives rise to mathematical models that pose unique challenges for the mathematician. In this talk I will give a few examples of how doctors, immunologists, and mathematicians can work together to understand the development of the disease and to design effective treatments.
This talk is intended for a general audience: no knowledge of biology or advanced mathematics will be assumed.
Biography: A California native, Professor Radunskaya received her Ph.D. in Mathematics from Stanford University. She has been a faculty member in the Math Department Pomona College since 1994. In her research, she specializes in ergodic theory, dynamical systems, and applications to various "real-world" problems. Some current research projects involve mathematical models of cancer immunotherapy, developing strategies for targeted drug delivery to the brain, and studying stochastic perturbations of dynamical systems. Prior to her academic career, Professor Radunskaya worked extensively as a cellist and composer. . Her music, described as "techno-clectic", combines traditional forms with improvisation, acoustic sounds with electronic, computer-generated, and found sounds, and abstract structures with narrative visual and sonic elements.
Contrary to popular belief, Professor Radunskaya thinks that anyone can succeed in mathematics, and she has committed herself to increasing the participation of women and underrepresented groups in the mathematical sciences. She is currently the President of the Association for Women in Mathematics, and co-directs the EDGE (Enhancing Diversity in Graduate Education) program, which won a "Mathematics Program that Makes a Difference" award from the American Mathematics Society in 2007, and a Presidential Award for Excellence in Science, Mathematics and Engineering Mentoring (PAESMEM) in 2015.
Professor Radunskaya was recently elected as a Fellow of the American Math Society, and she is the recipient of several awards, including a WIG teaching award in 2012, and the 2017 AAAS Mentor award.
We prove that the right equivalence class of a super potential in complete free algebra is determined by its Jacobi algebra and the canonical class in its 0-th Hochschild homology represented by the super potential, assuming the Jacobi algebra is finite dimensional. This is a noncommutative version of the famous Mather-Yau theorem in isolated hyper surface singularities. As a consequence, we prove a rigidity theorem for Ginzburg dg-algebra. I will discuss some applications of these results in three dimensional birational geometry. This is a joint work with Guisong Zhou (1803.06128).
The Dyson game is an explicitly solvable N player dynamic game that admits Dyson Brownian motion as a Nash equilibrium. The game is motivated by the real world phenomenon found in the spacing of buses, parked cars and perched birds, which exhibit random matrix statistics (i.e. Dyson Brownian motion). We find the optimal repulsion parameter (universality class) of the equilibrium depends on the information available to the players, furthering the understanding of an open problem in random matrix theory proposed by Deift. The limiting mean field game has a local cost term, which depends on the optimal universality class due to the nontrivial asymptotic behavior of the players. We solve the mean field game master equation and the associated Hamilton-Jacobi equation on Wasserstein space exactly, and we discuss how generalizing our results will require answering novel questions on the analysis of these equations on infinite dimensional spaces.
Any cobordism spectrum M associated to a classifying space BG is naturally equipped with a so-called Thom diagonal M->M ^ BG+ and Thom isomorphism M ^ M -> M ^ BG+. In the homotopy category, these maps give M the structure of a cotorsor for the coalgebra BG+. I'll describe how this structure can be lifted to the derived setting by proving a more general theorem about derived or "stacky" quotients of En-rings in ∞-categories. This theorem states that, given an action by an n-fold loop space G on an En-ring R, the stacky or derived quotient R//G is naturally a BG+-cotorsor (the fact that this specializes to a result for cobordism spectra follows from the work of Ando, Blumberg, Gepner, Hopkins and Rezk on Thom spectra and orientations). This result can be reinterpreted in the setting of noncommutative spectral algebraic geometry as saying that Spec(R//G) is a principal Spec(BG+)-bundle (or torsor) over Spec(R) in the category of affine En-varieties. If there is time, I'll speculate wildly about applications to geometry and noncommutative algebra.
Note for Attendees
The time of this seminar has changed. It now starts at 2:45, and precedes PIMS tea (which is now at a later time).
The classification of Fano varieties is unknown beyond dimension 3; however, many Fano fourfolds are expected to be GIT theoretic subvarieties of either toric varieties or quiver flag varieties. Quiver flag varieties are a generalization of type A flag varieties and are GIT quotients of vector spaces. In this talk, I will discuss my recent work on quiver flag varieties, including a proof of the Abelian/non-Abelian correspondence for quiver flag varieties, and its application in the large scale computer search for Fano fourfolds that I have carried out in joint work with T. Coates and A. Kasprzyk. We find 139 new Fano fourfolds. I will also discuss the importance of these subvarieties as a testing ground for the conjectures of Coates, Corti, Galkin, Golyshev, Kasprzyk and Tveiten on mirror symmetry for Fano varieties.
The quintessential example of a Darboux integrable differential equation is the Liouville equation
u_{xy} = -2 {\rm e}^u,
whose general solution is given by
u= \log \frac{ f'(x) g'(y)}{(f(x)-g(y))^2}.
Daboux integrability is classically related to the existence of intermediate integrals (or Riemann invariants) which in turn allow an explicit closed form formula to be derived for these equations.
Motivated by work of E. Vessiot, I will describe a differential geometric construction which provides a fundamental description of Darboux integrable systems in terms of superposition of differential systems and the quotient theory of differential systems by Lie groups.
The general theory will be discussed (I won't assume familiarity with differential systems), and demonstrated with examples. If time permits some interesting properties of these systems will be shown based on the existence of the quotient representation of Darboux integrable systems.
We will describe a geometric interpretation of Khovanov homology as Lagrangian Floer homology of two immersed curves in the 4-punctured 2-dimensional sphere. The main ingredient is a construction which translates Khovanov (or Bar-Natan) invariant of a 4-ended tangle to an immersed curve. It is inspired by a result of [Hedden, Herald, Hogancamp, Kirk], which embeds 4-ended reduced Khovanov arc algebra (or, equivalently, Bar-Natan dotted cobordism algebra) into the Fukaya category of the 4-punctured sphere. The main tool we will use is a category of peculiar modules, introduced by Zibrowius, which is a model for the Fukaya category of 2-sphere with 4 discs removed. This is joint work with Liam Watson and Claudius Zibrowius.
The goal of this talk is to create a correspondence between the representation theory of algebraic groups and the topology of Lie groups. The idea is to study the Hodge theory of the classifying stack of a reductive group over a field of characteristic p, the case of characteristic 0 having been studied by Behrend, Bott, Simpson and Teleman. The approach yields new calculations in representation theory, motivated by topology.
A fundamental problem of algebraic geometry is to determine which algebraic varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. We discuss the history of the problem. Some dramatic progress in the past 5 years uses a new tool in this context: the Chow group of algebraic cycles.
Note for Attendees
Refreshments will be served in ESB 4133 from 3:45 p.m.-4:00 p.m.
The Batyrev--Manin conjecture gives a prediction for the asymptotic growth rate of rational points on varieties over number fields when we order the points by height. The Malle conjecture predicts the asymptotic growth rate for number fields of degree d when they are ordered by discriminant. The two conjectures have the same form and it is natural to ask if they are, in fact, one and the same. We develop a theory of point counts on stacks and give a conjecture for their growth rate which specializes to the two aforementioned conjectures. This is joint work with Jordan Ellenberg and David Zureick-Brown.
We describe a penalty function for constrained nonlinear programs, originally proposed by Fletcher (1970). This penalty function is smooth and exact, so that minimizers of the original problem are minimizers of the penalty function for a sufficiently large (but finite) penalty parameter. The main computational kernel required to evaluate this penalty function and its derivatives is solving augmented least-squares like systems. The penalty function can then be efficiently evaluated for problems where good preconditions exist, such as for PDE-constrained optimization problems. We discuss extensions to regularized problems, problems with inequality constraints, and the use of inexact evaluations. We provide some preliminary numerical results on some standard optimization test problems and PDE-constrained problems.
This is joint work with Michael Friedlander, Dominique Orban and Michael Saunders.
Church-Ellenberg-Farb introduced the theory of FI-modules to explain the phenomenon of representation stability of the cohomology of configuration spaces. I will explain the basics of how this story goes, and then explain how to extend their analysis to two generalized types of configuration spaces. Furthermore, I will explain how the Grothendieck-Lefschetz formula connects these topological stability phenomena to stabilization of statistics for polynomials and rational maps over finite fields.
In my talk I will explain (partially conjectural) relation between
1) Homology of Hilbert scheme of points on singular curves
2) Knot homology of the links of curve singularities
3) Space functions on the moduli space of maps from the formal disc to the curve singularities.
I will center my talk around discussion of the case of cuspidal curve x^m=y^n and its singularity. In this case it is now known that 1) 2) and 3) are essentially equal. Talk is based on the joint projects with Gorsky, Rozansky, Rasmussen, Shende and Yun.
Moving boundary problems are found in many contexts including coastlines and the seabed on the large scale to atherosclerosis and blood clotting on the micro scale. The coupling between the fluid flow and evolving solid bodies is an important consideration when modelling these problems. We have run numerical simulations for the melting front of ice for both interior and exterior flows; classical examples of the Stefan problem. We also modelled erosion and colloidal deposition problems which involved other constitutive equations. The terminal shape of an eroding cylinder in cross flow was a rounded triangular profile pointed upstream, which agreed well with experimental results. All of these problems require careful attention to deformation of the mesh at the interface, and we have developed a node shuffle algorithm for preserving mesh quality at these boundaries.
The Heegaard Floer theory of a 4-ended tangle takes the form of an immersed curve (with possibly non-trivial local system) on the boundary of the tangle minus the tangle ends. The Heegaard Floer homology of a link can be computed as the Lagrangian intersection theory of the Heegaard Floer homologies of two 4-ended tangles obtained by splitting the link along an embedded 2-sphere.
I will outline the construction of the tangle invariant, with particular focus on the action of the mapping class group of the 4-punctured sphere. I will then discuss the current state of symmetry properties for this invariant in the light of the mutation conjecture.
Mathematical models for population dynamics have a long history in biomathematics. They are tools to explore the effects of birth and death, species interaction, landscape quality and spatial movement on the persistence, spread and spatial distribution of a species. One particular question is how spatial variation in landscape attributes affects the dynamics of populations, for example in the context of species invasions. A relatively recent approach to this question divides a landscape into "patches" and incorporates small-scale individual movement information to predict large-scale population dynamics. In this talk, I will review several aspects of this growing body of literature. I will include empirical evidence, model derivation, basic model outcomes, analytical challenges and some future ideas. The talk is aimed at a general mathbio audience.
Although integers are ubiquitous in quantum physics, they have different mathematical origins. In this colloquium, I will give a glimpse of how integers arise as either topological invariants or as analytic indices, as is the case in the so-called quantum Hall effect. I will explain the difficulties arising in extending well-known arguments when one relaxes the approximation that the particles effectively do not interact with each other in matter. Recent advances have made such realistic generalizations possible.
Note for Attendees
Refreshments will be served in ESB 4133 from 3:45 p.m.-4:00 p.m.
I will present nonlinear dynamics for distributed decision-making that derive from principles of symmetry and bifurcation. Inspired by studies of animal groups, including house-hunting honeybees and schooling fish, the nonlinear dynamics describe a group of interacting agents that can manage flexibility as well as stability in response to a changing environment.
Naomi Ehrich Leonard is Edwin S. Wilsey Professor of Mechanical and Aerospace Engineering and associated faculty in Applied and Computational Mathematics at Princeton University. She is a MacArthur Fellow, and Fellow of the American Academy of Arts and Sciences, SIAM, IEEE, IFAC, and ASME. She received her BSE in Mechanical Engineering from Princeton University and her PhD in Electrical Engineering from the University of Maryland. Her research is in control and dynamics with application to multi-agent systems, mobile sensor networks, collective animal behavior, and human decision dynamics.
Let N be the conormal variety of a Schubert variety X. In this talk, we discuss the geometry of N in two cases, when X is cominuscule, and when X is a divisor. In particular, we present a resolution of singularities and a system of defining equations for N, and also describe certain cases when N is normal, Cohen-Macaulay, and Frobenius split. Time permitting, we will also illustrate the close relationship between N and orbital varieties, and discuss the consequences of the above constructions for orbital varieties.
A lattice-indexed family of stochastic processes has quasi-cycle oscillations if its otherwise-damped oscillations are sustained by noise. Such a family performs the reaction part of a stochastic reaction-diffusion system when we insert a local Mexican Hat-type, difference of Gaussians, coupling on a one-dimensional and on a two-dimensional lattice. In one dimension we find that the phases of the quasi-cycles synchronize (establish a relatively constant relationship, or phase locking) rapidly at coupling strengths lower than those required to produce spatial patterns of their amplitudes. The patterns of phase locking persist and evolve but do not induce patterns in the amplitudes. In two dimensions the amplitude patterns form more quickly, but there remain parameter regimes in which phase patterns form without being accompanied by clear amplitude patterns. At higher coupling strengths we find patterns both of phase synchronization and of amplitude (resembling Turing patterns) corresponding to the patterns of phase synchronization. Specific properties of these patterns are controlled by the parameters of the reaction and of the Mexican Hat coupling.
Self-similar solutions and translating solitons are not only special solutions of mean curvature flow (MCF) but a key role in the study of singularities of MCF. They have received a lot of attention. We introduce some examples of self-similar solutions and translating solitons for the mean curvature flow (MCF) and give rigidity results of some of them. We also investigate self-similar solutions and translating solitons to the inverse mean curvature flow (IMCF) in Euclidean space.
This is the first of a two part expository talk on the Balog-Szemerédi-Gowers theorem. This theorem, originally due to Balog and Szemerédi and later strengthened by Gowers, is one of the most important tools in additive combinatorics, with applications ranging from additive number theory to combinatorial geometry and harmonic analysis. In part one we will motivate the result by highlighting its importance and usefulness. In part two we will present a proof of the result. The proof that we present is a variant of Gowers’ graph theoretic proof, due to Sudakov, Szemerédi and Vu.
Given two homotopy equivalent manifolds with different dimensions, it is natural to ask if the smaller one embeds in the larger one. We will discuss this problem in the case of four-manifolds homotopy equivalent to surfaces.
We introduce a Markov chain model to study evolution of a continuous trait based on population genetics. It corresponds to individual-based model which includes frequency dependent selection caused by m-player game interactions and stochastic fluctuations due to random genetic drift and mutation. We prove that under a proper scaling limit as the population size increases the system converges to the solution of replicator-mutator equations. Our result establishes an affirmative mathematical base to the adaptive dynamics formulation employed in the theory of the mathematical biology.
Note for Attendees
Refreshments (PIMS Tea) are served at 2:45PM in the PIMS lounge.
I will discuss results and open problems about harmonic functions for finitely generated groups, with an emphasis on polynomial vs. super-polynomial growth. By "polynomial growth" I am implying to at least two different notions:
1. The volume growth of balls in the group (with respect to the word metric).
2. The growth of the sup norm of a harmonic function on a ball.
The investigation is partly motivated by Kleiner’s proof for Gromov’s theorem on groups of polynomial growth, by Ozawa's more recent proof of Gromov's theorem and by some related conjectures and hypothetical future applications to problems in geometric group theory.
Most of the results I will present in this talk will be at least a few years old and based on joint work with Ariel Yadin and Perl, Tointon and Yadin.
The affine Grassmannian, though a somewhat esoteric looking object at first sight, is a fundamental algebro-geometric construction lying at the heart of a series of ideas connecting number theory (and the Langlands program) to geometric representation theory, low dimensional topology and mathematical physics.
Historically it is popular to study the category of constructible perverse sheaves on the affine Grassmannian. This leads to the *constructible* Satake category and the celebrated (geometric) Satake equivalence.
More recently it has become apparent that it makes sense to also study the category of perverse *coherent* sheaves (the coherent Satake category). Motivated by certain ideas in mathematical physics this category is conjecturally governed by a cluster algebra structure.
We will illustrate the geometry of the affine Grassmannian in an elementary way, discuss what we mean by a cluster algebra structure and then describe a solution to this conjecture in the case of general linear groups.
Note for Attendees
Refreshments will be served in ESB 4133 from 3:45 p.m.-4:00 p.m.
The Allen--Cahn equation is a semi-linear PDE that produces minimal surfaces via a certain singular limit. We will describe recent work proving index, multiplicity, and curvature estimates in the context of an Allen--Cahn min-max construction in a 3-manifold. Our results imply, for example, that in a 3-manifold with a generic metric, for every positive integer p, there is an embedded two-sided minimal surface of Morse index p. This is joint with Otis Chodosh.
This is the second of a two part expository talk on the Balog-Szemerédi-Gowers theorem. This theorem, originally due to Balog and Szemerédi and later strengthened by Gowers, is one of the most important tools in additive combinatorics, with applications ranging from additive number theory to combinatorial geometry and harmonic analysis. In part one we will motivate the result by highlighting its importance and usefulness. In part two we will present a proof of the result. The proof that we present is a variant of Gowers’ graph theoretic proof, due to Sudakov, Szemerédi and Vu.
For the local time L_t^x of super-Brownian motion X starting from \delta_0, we study its asymptotic behavior as x\to 0. In d=3, we find a normalization \psi(x)=((2\pi^2)^{-1} \log (1/|x|))^{1/2} such that (L_t^x-(2\pi|x|)^{-1})/\psi(x) converges in distribution to standard normal as x\to 0. In d=2, we show that L_t^x-\pi^{-1} \log (1/|x|) converges a.s. as x\to 0. We also consider general initial conditions and get some renormalization results. The behavior of the local time allows us to derive a second order term in the asymptotic behavior of a related semilinear elliptic equation.
Azumaya algebras over a commutative ring R are generalizations of central simple algebras over a field k, and both are "twisted matrix algebras". In this, they bear the same relationship to a noncommutative ring of matrices Mat_n(k) that a vector bundles (or projective modules) bear to vector spaces. That is, they are bundles of algebras. In this talk, I will show that thinking about Azumaya algebras from the algebraic-topological point of view, as bundles of algebras, is fruitful, both in producing examples of algebras with interesting properties, and in proving certain results about such algebras that are difficult to prove by direct, algebraic methods.
Note for Attendees
Refreshments will be served in ESB 4133 from 3:45 p.m.-4:00 p.m.
The derived category of a hypersurface is equivalent to the category of matrix factorizations of a certain function on the total space of a line bundle over the ambient space. The hypersurface is smooth if and only if the critical locus of the function is compact. I will present a construction through which a resolution of singularities of the hypersurface corresponds to a compactification of the critical locus of the function, which can be very interesting in examples. This is joint work with Paul Aspinwall and Ed Segal.
This is joint work with Jeffrey Dawson, Linyuan Lu and Attila Sali.
We define simple matrices as those whose entries are chosen from {0,1,2} and for which no columns are repeated. We consider the extremal problem of how many columns can an m-rowed simple matrix A have, subject to the condition that A avoids certain submatrices.
We consider some forbidden submatrices (configurations) that seem to force A to behave like a (0,1)-matrix. Define T_k(a,b,c) to be the kxk matrix with b's on the diagonal, a's below the diagonal and c's above the diagonal. These generalize the identity and triangular matrices. There are such 8 matrices to forbid which seem to force A to behave like a (0,1)-matrix. Some preliminary results are given.
The main goal of this talk is to introduce transitionally commutative principal G-bundles and to show that they can be classified homotopically using a space B_comG that is called the classifying space for commutativity in G. In the second half of the talk I provide some examples of bundles that are trivial as principal G-bundles but not as transitionally commutative bundles.
A traditional population-genetics approach studies geneaologies in a population of a fixed size, which forms the basis of several spectral theories of finite samples. In contrast, a population of tumor cells typically experiences an exponential growth phase in its initial progression, which is far from constant population size. In this work, I develop two different numerical procedures, one of which is based on forward-in-time and the other is based on backward-in-time treatment, to derive allele frequency spectrum in such exponentially growing cancer cell populations. We find significance bias toward singletons both analytically and numerically, which reflects the fact that most observed mutations have recent origins in a growing population.
This work was done in collaboration with Prof. Hideki Innan.
In this talk I will introduce Dyson Brownian motion. Roughly put, this is a stochastic process given by the eigenvalues of an (N x N) Hermitian matrix whose entries are Brownian motions. I will discuss its connection to random matrix theory and give several other descriptions of the process: these include, on one hand, a system of stochastic differential equations with repulsive drift, and on the other, the distribution of N one-dimensional Brownian motions that are conditioned not to collide.
The (1d) KPZ universality class contains random growth models, directed random polymers, stochastic Hamilton-Jacobi equations (e.g. the eponymous Kardar-Parisi-Zhang equation). It is characterized by unusual scale of fluctuations, some of which appeared earlier in random matrix theory, and which depend on the initial data. The explanation is that on large scales everything approaches a special scaling invariant Markov process, the KPZ fixed point. It is obtained by solving one model in the class, TASEP, and passing to the limit. Both TASEP and the KPZ fixed point turn out to have a novel structure: "stochastic integrable systems" (Joint work with Konstantin Matetski and Daniel Remenik).
Note for Attendees
Refreshments will be served in ESB 4133 from 3:45 p.m.-4:00 p.m.
I will outline the construction of a natural bivariant theory extending algebraic bordism, which will yield an extension of algebraic cobordism to singular varieties. I will also discuss the connections of this theory to algebraic K-theory and to intersection theory.
ESB 4133 (PIMS Lounge) Please note it starts 30 mins. earlier than the usual time.
Tue 23 Oct 2018, 12:00pm-1:00pm
Abstract
The quadratic assignment problem, QAP, has many applications ranging from the planning of building locations of a university, to the positioning of modules on a computer chip (VLSI design), to the design of keyboards. This problem is arguably one of the hardest of the NP-hard problems, as problems with dimension 30 are still considered hard to solve to optimality.
The QAP in the trace formulation is modelled as the minimization of a quadratic function over the permutation matrices. The set of permutation matrices can be represented by quadratic constraints. Relaxations of these constraints are used in branch and bound solution methods. These relaxations include the eigenvalue and projected eigenvalue relaxations, as well as various semidefinite programming, SDP, and doubly nonnegative, DNN, relaxations. These latter relaxations are particularly strong and often solve the QAP to optimality. However, they can be extremely expensive to solve.
We show that the combination of an alternating directions method of multipliers, ADMM, in combination with facial reduction works extremely well in solving the very difficult DNN relaxation.
We consider the Cauchy problem of 3D incompressible Navier-Stokes equations for uniformly locally square integrable initial data. The existence of a time-global weak solution has been known, when the square integral of the initial datum on a ball vanishes as the ball goes to infinity. For non-decaying data, however, the only known global solutions are either for perturbations of constants or when the velocity gradients are in Lp with finite p. In this talk, I will outline how to construct global weak solutions for general non-decaying initial data whose local oscillations slowly decay.
This is a joint work with Tai-Peng Tsai.
This is the first part of a two part exposition on the triangle-free process. The triangle-free process begins on an empty graph and adds edges at random, provided no triangle is created with the existing edges. One of the original motivations for this process came from Ramsey Theory. Spencer conjectured that the maximum size of an independent set in a graph resulting from the process should be relatively small, and so the triangle-free process would provide constructions for lower bounds on the Ramsey number R(3,t). Recently, Bohman and Keevash obtained new estimates on independence number of such graphs, which gives a lower bound on R(3,t) within a factor of 4+o(1) of the best know upper bound. In this first part I will discuss random graph processes, with an emphasis on the triangle-free process and the odd-cycle-free process.
While vaccines are available and are effective in protecting against colonisation and disease with Streptococcus pneumoniae, their effectiveness is limited by strain (serotype) replacement following widespread vaccination. Understanding the post-vaccination balance of serotypes would present the opportunity to achieve a final population composed of the most benign (non-invasive) strains. However, the complex ecology of the pneumococcus makes it difficult to predict the post-vaccination balance of strains. Recently, Corander et al proposed that there is widespread apparent negative frequency-dependent selection (NFDS) in the pneumococcus (Corander et al 2017 Nat. Ecol. Evol.).
Here, we use this principle to develop a deterministic model of pneumococcal strain dynamics, and use the model to make predictions about the ecological response of the pneumococcal population to new candidate vaccine strategies. We find that we can identify formulations that out-perform existing formulations in the model. Furthermore, it is possible to obtain a final model population that scores as well as the currently used formulation, using a vaccine strategy with fewer serotypes -- these formulations would be much less costly to produce than current vaccines. We suggest that this approach could provide a template for principled vaccine design based on global surveillance data and genomics.
This is joint work with N. Croucher.
How can individuals cooperate to satisfy local constraints without a central authority? Examples might include autonomous drones navigating in a swarm, or university departments scheduling their seminars. Individuals can make random choices and communicate with each other, but all must follow the same procedure.
How small can we make the "coding radius" - the distance to which an individual must communicate? In the setting of the integer line Z, there is a surprising universal answer that applies to every non-trivial constraint problem. In d-dimensional Euclidean space, answers are available for the key case of proper colouring; it turns out that there is a huge difference between 3 and 4 colors. Finally, I'll mention how changing the question slightly has led to the discovery of an amazing mathematical object that seemingly has no right to exist.
Communication-avoiding and pipelined variants of Krylov solvers are critical for the scalability of linear system solvers on future exascale architectures. We present low synchronization variants of iterated classical (CGS) and modified Gram-Schmidt (MGS) algorithms that require one and two global reduction communication steps. Derivations of low synchronization algorithms are based on observations by Ruhe. Our main contribution is to introduce a backward normalization lag into the compact WY form of MGS resulting in a {\cal O}(\eps)\kappa(A) stable GMRES algorithm that requires only one global synchronization per iteration. The reduction operations are overlapped with computations and pipelined to increase speed. Further improvements in performance are achieved by accelerating GMRES BLAS-2 operations on GPUs. Extensions to re-cycled Krylov iterations are explored.
Co-authors/collaborators: Kasia Swirydowicz (NREL), Julien Langou (CU Denver), Shreyas Ananthan (NREL), Ulrike Yang (LLNL).
This is the second part of a two part series on the triangle-free process. The triangle-free process begins on an empty graph and adds edges at random, provided no triangle is created with the existing edges. One of the original motivations for this process came from Ramsey Theory. Spencer conjectured that the maximum size of an independent set in a graph resulting from the process should be relatively small, and so the triangle-free process would provide constructions for lower bounds on the Ramsey number R(3,t). I will present Erdos, Suen, and Winkler's proof that the triangle-free process gives a lower bound on R(3,t) within a logarithmic factor of the best possible. We will see that the triangle-free process results in fewer edges than the odd-cycle free process.
Chikungunya is a viral disease transmitted to humans by infected mosquitoes: Aedes aegypti and Aedes albopictus. It causes fever and severe joint pain. Other symptoms include muscle pain, headache, nausea, fatigue and rash. Joint pain is often debilitating and can vary in duration. Some of the main mathematical methods to simulate Chikungunya spread are set as ordinary differential equations over compartmental models, SEIR for host and sei for vectors. We propose a spatio-temporal description of chikungunya spread using a cellular automata over unstructured triangular meshes.
This talk presents joint work with Ed Perkins which can be considered a followup to the paper "Super-Brownian motion and the spatial Lambda-Fleming-Viot process" by Chetwynd-Diggle and Etheridge, Electron. J. Probab. (2018). In that paper the authors prove that certain rescaled SLFV processes converge to super-Brownian motion. We obtain this convergence under a natural scaling not covered in the Chetwynd-Diggle Etheridge paper.
Complete chaos is impossible! Is the idea behind Ramsey Theory anyways. The Ramsey number R(s,t) is the least number n such that any graph on n vertices with edges coloured red and blue has a red s-clique or a blue t-clique. These numbers are famously hard to compute, or obtain comparable upper and lower bounds for. Many of the best bounds on Ramsey numbers use randomness. Indeed, one of the first applications of randomness in graph theory was by Erdos in 1947 to obtain a lower bound on the Ramsey numbers R(s,s). 70 years later we've been able to improve his bound by only a factor of 2! In my talk I will introduce some of the main problems in Ramsey Theory and outline a few of ways randomness has been used to obtain estimates on Ramsey numbers.
Milnor fibers are invariants that arise in the study of hypersurface singularities. A major open conjecture predicts that for hyperplane arrangements, the Betti numbers of the Milnor fiber depend only on the combinatorics of the arrangement. I will discuss how tropical geometry can be used to study related invariants, the virtual Hodge numbers of a hyperplane arrangement's Milnor fiber. This talk is based on joint work with Max Kutler.
The number of inversions of a permutation is an important statistic that arises in many contexts, including as the minimum number of simple transpositions needed to express the permutation and, equivalently, as the rank function for weak Bruhat order on the symmetric group. In this talk, I’ll describe an analogous statistic on the reduced expressions for a given permutation that turns the Coxeter graph for a permutation into a ranked poset with unique maximal element. This statistic simplifies greatly when shifting our paradigm from reduced expressions to balanced tableaux, and I’ll use this simplification to give an elementary proof computing the diameter of the Coxeter graph for the long permutation.
This talk is elementary and assumes no background other than passing familiarity with the symmetric group.
The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll use branched surfaces to build taut foliations for manifolds obtained by surgery on positive 3-braid closures. No background in Heegaard-Floer or foliation theories will be assumed.
GTPases are a family of signalling proteins that controls cell shape by regulating the actin cytoskeleton. Understanding the dynamics of GTPase activity is essential toward deciphering the mechanisms behind cell motility and filopodia formation. In this talk, I will present several modelling approaches with ODEs and PDEs, numerical simulations and analytical results on their bifurcation behaviors.
In this talk, I will present an L^\infty-estimate for non-local parabolic SPDEs with measurable jumping kernels, using an improved version of stochastic De Giorgi iteration. Then a priori Holder estimate will be given for solutions of these SPDEs.
We present a thorough rheological study of the BMP-EVP model. The constitutive model consists of an Upper-convected Maxwell equation for the stress coupled with a kinetic equation for the fluidity (this equation accounts for the breakdown and reformation of the fluid structure). Through a modification, we can now control the shear-thinning power law in the model. We test the model in simple rheometric flows: simple shear, uniaxial elongation and large amplitude oscillatory test. The results compare well with published data for Carbopol gel ("ideal" viscoplastic material).
Schubert calculus began in the 1880s when Hermann Schubert began asking enumerative questions in geometry, such as how many lines in space are incident to four given lines. Efforts to build a rigorous foundation for these questions led to the development of cohomology rings and modern intersection theory. In the 1980s, Lascoux and Schutzenberger defined an explicit basis for polynomials, called Schubert polynomials, whose structure constants precisely compute these intersection numbers. For the special case of the grassmannian sub variety of the complete flag manifold, these polynomials are Schur polynomials and the classical Littlewood—Richardson rule gives their structure constants by enumerating Yamanouchi tableaux, which are certain ways of putting numbers into boxes. In this talk, I’ll survey combinatorial models for Schubert polynomials that generalize these tableaux models for Schur polynomials and lead to new cases for computing structure constants by enumerating new ways of putting numbers in boxes.
Note for Attendees
Refreshments will be served in ESB 4133 from 3:45 p.m.-4:00 p.m.
In this work we examine notions of alignment with respect to a generalized form of Holder's inequality. We show that tightness of this generalized inequality often serves as an optimality condition for many important convex optimization problems, such as atomic regularization in machine learning, gauge duality, and linear programming. In particular, when the primal and dual variables are aligned, the support of the primal solution can be recovered from the dual solution, which is a property often exploited in two-stage methods for sparse optimization. We show that many popular convex optimization methods (e.g. proximal gradient and conditional gradient) can be interpreted as "aligning methods", which allows for a more geometric view of method and problem properties.
Given a smooth projective variety X and a smooth divisor D \subset X, one can study the enumerative geometry of counting curves in X with tangency conditions along D. There are two theories associated to it: relative Gromov-Witten invariants of (X,D) and orbifold Gromov-Witten invariants of the r-th root stack X_{D,r}. For sufficiently large r, Abramovich-Cadman-Wise proved that genus zero relative invariants are equal to the genus zero orbifold invariants of root stacks (with a counterexample in genus 1). We show that higher genus orbifold Gromov-Witten invariants of X_{D,r} are polynomials in r and the constant terms are exactly higher genus relative Gromov-Witten invariants of (X,D). If time permits, I will also talk about further results in genus zero which allows us to study structures of genus zero relative Gromov-Witten theory, e.g. Givental formalism for genus zero relative invariants. This is based on joint work with Hisan-Hua Tseng, Honglu Fan and Longting Wu.
Algebraic K-theory is an invariant defined on categories that records how object in the category are related by exact sequences --- it is a homotopical version of the classical Euler characteristic. However, there are many categories of interest that do not have exact sequences, but instead have cutting and pasting operations. For example, the category of varieties or the category of polytopes. I'll describe how to define a higher algebraic K-theory for categories like this, and show that it's not so different from the case of more algebraic categories. Even better, theorems like Quillen's Devissage and Localization can be proved internal to these structures. Time permitting, I'll describe how the cutting and pasting of polytopes is intimately related to the weight filtration on the algebraic K-theory of fields.
Sub-Gaussian heat kernel estimates are typical of fractal graphs. We show that sub-Gaussian estimates on graphs follow from a Poincaré inequality, capacity upper bound, and a slow volume growth condition. An important feature of this work is that we do not assume elliptic Harnack inequality, cutoff Sobolev inequality, or exit time bounds.
Question: imagine you are located somewhere inside a maďze maze that extends all over the plane. Could you find an exit towards infinity?
To be (just) a (little) bit more accurate, consider the following model. Suppose maďze grows in the Euclidean plane (denoted R2 from now onwards) forming walls parallel to the axes of R2. The walls have length 1 and they grow between every two neighbouring points of the standard whole numbers lattice (henceforth denoted Z2). The walls are set up in such a way that every point in Z2 is adjacent to a wall (there are no points left alone) and so that, for whatever the closed square K of R2 may be, there always exists a path from whatever point inside A = K - the walls of maďze to the exterior of K and such that you never backtrack (nor cross through a wall). From these conditions, it should be intuitively clear that from whatever point in R2 (not in the walls) there exist paths that never backtrack and that go on forever. How many such paths exist? We shall see that in almost every case, the answer is surprising. To respond this we shall introduce the language of graphs and the uniform spanning tree. The talk will aim at the undergraduate level and exceeding formalities will be avoided.
The Grothendieck ring of varieties is the target of a rich invariant associated to any algebraic variety which witnesses the interplay between geometric, topological and arithmetic properties of the variety. The motivic Hilbert zeta function is the generating series for classes in this ring associated to a certain compactification of the unordered configuration space, the Hilbert scheme of points, of a variety. In this talk I will discuss the behavior of the motivic Hilbert zeta function of a reduced curve with arbitrary singularities. For planar singularities, there is a large body of work detailing beautiful connections with enumerative geometry, representation theory and topology. I will discuss some conjectural extensions of this picture to non-planar curves.
Quantum information processing holds great promise, yet large-scale, general purpose, universal quantum computers capable of solving hard problems are not yet available, despite 20+ years of immense worldwide effort and large investments. However, special-purpose quantum information processors, such as the quantum simulators originally envisioned by Feynman, now appear to be within reach. Another type of currently operational special-purpose quantum information processor is a "quantum annealer," designed to speed up the solution to classical optimization problems. “Quantum supremacy” has meanwhile been identified as an intermediate target, allowing the current generation of quantum computers to demonstrate superiority against classical computers. After a brief introduction to “what is quantum computing,” this talk will review these developments and their broader impacts, with an eye toward the long-term prospects of quantum computers.
Bio. Daniel Lidar is the Viterbi Professor of Engineering at USC, and a professor of Electrical Engineering, Chemistry, and Physics. He holds a Ph.D. in physics from the Hebrew University of Jerusalem. He did his postdoctoral work at UC Berkeley. Prior to joining USC in 2005 he was a faculty member at the University of Toronto. His main research interest is quantum information processing, where he works on quantum control, quantum error correction, the theory of open quantum systems, quantum algorithms, and theoretical as well as experimental adiabatic quantum computation. He is the Director of the USC Center for Quantum Information Science and Technology, and is the co-Director (Scientific Director) of the USC-Lockheed Martin Center for Quantum Computing. Lidar is a recipient of a Sloan Research Fellowship, a Guggenheim Fellowship and is a Fellow of the AAAS, APS, and IEEE.
The IAM Lecture Series on Quantum Computing is generously sponsored by 1QBit, DWave, and PIMS.
Note for Attendees
Please join us for refreshments immediately preceding Daniel's talk, 3:30-4p in LSK 306.
For my work supporting Bifrost, a programming environment built for visual effects artists, I am seeking a good “default” preconditioner for solving linear systems or accelerating nonlinear solvers in problems arising in physics simulation and geometric problems. Narrowing the focus, I’m looking particularly at large, sparse, symmetric-positive definite matrices with either an explicit or implied sum-of-squares structure, most notably finite element stiffness matrices for elliptic problems. On the other hand, I do not want to make further assumptions such as the signs of off-diagonal nonzeros, whether the vector of all ones plays a special role, etc. which may break down in the face of situations such as biharmonic-like shell problems and matrix rescalings to avoid unnecessary ill-conditioning. In this talk I’ll discuss my current experiments with preconditioners inspired by Shklarski and Toledo’s "fretsaw extension”, itself based on ideas from support graph preconditioning. Rather than viewing this as a way to approximate the graph of the original matrix with a tree plus a few extra edges (which implies an efficient exact Cholesky factorization), I introduce two new ideas:
1) sparsification viewed as partial assembly of the stiffness matrix, guided by exactly what an efficient Cholesky factorization needs, and
2) adding constraints to the semi-assembled system to keep it equivalent to the original problem, rather than just an approximation.
While the constrained semi-assembled system cannot be efficiently factored in its entirety, it opens up more avenues for how to approach preconditioning, which hopefully will lead to the desired solver characteristics.
A Smith inequality refers to a rank inequality between the homology of a space with a G action and the homology of its fixed set. In the case of G = Z/2, I will discuss an analog of this statement for the Knot Floer homology of periodic knots, including a conjectural filtered refinement. These inequalities appear to give new restrictions on the Alexander polynomials of periodic alternating and periodic L-space knots.
Microbial communities play major roles in human health and disease and dominate global biochemical cycles. Many properties of these communities emerge from interactions between its member species and cannot be understood based on the properties of its members living in isolation. The ubiquity of these emergent properties has led some researchers to conclude that microbial communities evolve by multi-level selection (MLS) where selection acts at both the levels of individuals and communities. However, this view is controversial and whether MLS plays a role in natural microbial communities has been heatedly debated in the recent literature. The main controversy in the debate is whether community level properties can be inherited. Whether this is the case will depend on the quantitative details of how communities are assembled. For example, the community composition of host-associated communities that are vertically-transmitted between host generations will remain relatively constant through time, allowing for high degrees of heritability of community level traits. In contrast, the composition of communities assembled from the environment can fluctuate strongly in time, leading to low levels of heritability. In general, we need a quantitative method to predict the degree of heritability based on the dynamics of community assembly. Here we present such a method which is based on a recently published MLS framework that explicitly models both individual and community level dynamics. We are attempting to extract generalized rules that can predict the degree of heritability of community level traits for naturally occurring microbial communities. With this framework, we hope to change the focus of the ongoing debate from the question of whether microbial communities can evolve by MLS to the more productive question of which communities can evolve by MLS.
Note for Attendees
Following the seminar there will be PIMS tea at 3:45PM
Take the n by n box in the square lattice and fill it with integers where the integers on adjacent vertices only differ by +1 or -1. How large can a typical value be?
We prove that the variance at a point blows up like log n which in particular implies that such a function is delocalized. This is a consequence of a Russo Seymour Welsh theory that we build for such functions. I will also discuss a connection with the six-vertex model (also called the square ice model).
Joint work with Hugo Duminil Copin, Matan Harel, Benoit Laslier and Aran Raoufi.
Recently Xiuxiong Chen and Jingrui Cheng have made a breakthrough on the existence of constant scalar curvature metrics on compact Kahler metrics, in view of Calabi-Donaldson program and Yau-Tian-Donaldson conjecture. The essential new input is a highly nontrivial a priori estimates for scalar curvature type equation, which is a fully nonlinear fourth order elliptic PDE. We will discuss the exciting developments regarding the existence of constant scalar curvature metrics and their extensions.
We introduce a construction call the {\em saturated blowup} of an Artin stack with good moduli space. The saturated blowup is a birational map of stacks which induces a proper birational map on good moduli spaces. Given an Artin stack {\mathcal X} with good moduli space X, there is a canonical sequence of saturated blowups which produces a stack whose rigidification is a DM stack. When the stack is smooth, all of the stacks in the sequence of saturated blowups are also smooth. This construction generalizes earlier work of Kirwan and Reichstein in geometric invariant theory and the talk is based on joint work with David Rydh.
There is growing concern about fairness in algorithmic decision making: Is it treating different groups fairly? How can we make it fairer? And what do we even mean by fair? In this talk I will discuss some of our work on this topic, focusing on the setting of online decision making. For instance, a classic result states that given a collection of predictors, one can adaptively combine them to perform nearly as well as the best in hindsight (achieve “no regret”) without any stochastic assumptions. Can one extend this guarantee so that if the predictors are themselves fair, the overall combination is as well? I will discuss this and other issues.
If you wish to meet with Avrim on Monday, please email Anna Eberhard with a range of times that you're available, and we'll try to add you to his schedule.
The IAM Distinguished Colloquium Series is sponsored in part by PIMS.
Note for Attendees
Please join us for refreshments immediately preceding Avrim's talk, 3:30-4:00pm in LSK 306.
To a 3-manifold with torus boundary, we can associate an element of the Fukaya category of the punctured torus—that is, a collection of immersed curves in the torus, decorated with local systems—such that when two such manifolds are glued the Heegaard Floer homology of resulting 3-manifold is recovered from Floer homology of the corresponding curves. These curves are a reformulation of the bordered Heegaard Floer defined by Lipshitz, Ozsvath, and Thurston, but their geometric nature makes them more user friendly. We will discuss some properties of these curves and some applications, including invariance of Heegaard Floer homology under genus one mutation and a rank inequality for Heegaard Floer homology of toroidal manifolds. This is joint work with J. Rasmussen and L. Watson.
The Min system in E. coli is one of the simplest known biological systems that demonstrates diverse complex dynamic behavior or transduces local interactions into a global signal. Various mathematical models of the Min system show behaviors that are qualitatively similar to dynamic behaviors of the Min system that have been observed in experiments, but no model has been quantitatively compared to time-course data. In this talk, I will discuss extracting time-course data for model fitting from experimental measurements of the Min system and fitting established and novel biochemistry-based models to the time-course data using a homotopy-minimization method for parameter estimation in differential equations. Comparing models to time-course data allows me to make precise distinctions between biochemical assumptions in the various models. I will discuss how my modeling and fitting supports a novel model, which suggests that a regular, ordered, stability-switching mechanism underlies the emergent, dynamic behavior of the Min system.
In '99, David Aldous conjectured that a certain natural "random walk" on the space of binary combinatorial trees should have a continuum analogue, which would be a diffusion on the Gromov-Hausdorff space of continuum trees. This talk discusses ongoing work by F-Pal-Rizzolo-Winkel that has recently verified this conjecture with a path-wise construction of the diffusion. This construction combines our work on dynamics of certain projections of the combinatorial tree-valued random walk with our previous construction of interval-partition-valued diffusions.
In early 90's, there are two mathematical theories motivated by physics, Verlinde theory counting the number of generalized theta function on a curve and quantum cohomology counting the number of holomorphic maps. Early explicit computation shows that the level l GL_n Verlinde algebra is isomorphic to quantum cohomology ring of Grassmanian G(n, n+l). In 1993, Witten gave a conceptual explanation of this isomorphism, by proposing an equivalence between the quantum field theories which govern the level-l GL Verlinde algebra and the quantum cohomology of the Grassmannian. His physical derivation of the equivalence naturally leads to a mathematical problem: these two objects are conceptually isomorphic (without referring to detailed computations). In the talk, I will explain a K-theoretic version of Witten's conjecture motivated by our observation that Verlinde invariants is a K-theoretic invariant and hence should be compared to the quantum K-theory of Grassmanian. The main content of the talk is a reformation of the conjecture as the consequence of wall-crossing problem of GLSM and the resolution of the conjecture in rank 2. This is a joint work with Ming Zhang.
Mirror symmetry describes a surprising duality between symplectic geometry and complex geometry, coming from a duality between certain models of string theory. Of particular interest in these theories is the question: "what are the observable particles?". These particles are described by certain nonlinear partial differential equations; the deformed Hermitian-Yang-Mills equation, and the special Lagrangian equation. These equations do not always have solutions, corresponding to the fact that not all particles are stable-- some particles decay into small constituent parts. It has long been expected that these decays are predicted by purely algebraic structures. I will describe how such an algebraic structure appears in the study of the deformed Hermitian-Yang-Mills equation through the study of certain convex functions on infinite dimensional manifolds. Time permitting, I will also explain how these results, coupled with mirror symmetry, provide insight on the special Lagrangian equation. This is joint work with S.-T. Yau.
Note for Attendees
Refreshments will be served in ESB 4133 from 3:45 p.m.-4:00 p.m.
Cell migration is a fundamentally important phenomenon underlying wound healing, tissue development, immune response and cancer metastasis. Understanding basic physics of the cell migration presented a great challenge until, in the last three decades, a combination of biological, biophysical and mathematical approaches shed light on basic mechanisms of the cell migration. I will describe models, based on nonlinear partial differential equations and free boundary problems, which predicted that individual cells do not linger in a symmetric stationary state for too long, but rather spontaneously break symmetry and initiate motility. The cells can either crawl straight, or turn, depending on mechanical parameters. I will show how experimental data supported the models, and I will also review current computational challenges.
Bio: Professor Alex Mogilner received M.Eng. degree in Engineering Physics in 1985 from the Ural Polytechnic Institute. He received PhD degree in Physics from the USSR Academy of Sciences in 1990. He did research in Mathematical Physics until 1992, when he started studying Mathematical Biology at the University of British Columbia. After receiving PhD degree (adviser Leah Edelstein-Keshet) in Applied Mathematics from UBC in 1995, Alex worked at UC Berkeley with George Oster as a postdoctoral researcher, and in 1996 he came to the Math Department at the University of California at Davis as an Assistant Professor. He became an Associate professor in 1999, and in 2002 he became a Professor at the Department of Mathematics and Department of Neurobiology, Physiology and Behavior at UC Davis. Since 2014, Dr. Mogilner is a Professor of Mathematics and Biology at Courant Institute and Department of Biology at the New York University. Alex's areas of expertise include Mathematical Biology, Cell Biology and Biophysics; he does research on mathematical and computational modeling of cell motility, cell division and galvanotaxis. Alex published about 130 papers in high impact journals including Nature, Science, PNAS. He developed models of keratocyte motility, polymerization ratchet, and search-and-capture mechanism of spindle assembly. His research is/was supported by NIH and NSF grants, Army Office of Research and by United States-Israel Binational Science Foundation. Alex served on editorial boards of many journals including Cell, Biophysical Journal, Current Biology, Journal of Cell Biology, Bulletin of Mathematical Biology, Molecular Biology of the Cell. He gave plenary talks and organized many international conferences on mathematical biology and cell biophysics, and taught at many summer schools. Dr. Mogilner was a panel chair at NIH.
Note for Attendees
Please join us for refreshments immediately preceding Alex's talk, 3:30-4:00p.m. in LSK 306.
In 1987, Aizenman, Kesten, and Newman proved that percolation on Z^d always has at most one infinite cluster a.s. While their proof has mostly been eclipsed by the more general and arguably more elegant proof of Burton and Keane, the Aizenman-Kesten-Newman proof is more quantitative and yields interesting bounds on certain two-arm probabilities. In this talk, I will discuss a new variation on these bounds that holds universally over a large class of graphs, and is stronger even in the case of Z^d. I will then show how this new bound can be applied to derive new results on percolation in various "infinite-dimensional" settings.
In joint work with Hans-Joachim Baues, we study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of Baues on the strictification of secondary cohomology operations. As an application, we show that the secondary integral Steenrod algebra is strictifiable.
Using bordered Floer theory, we construct an invariant for 3-orbifolds with singular set a knot that generalizes the hat flavor of Heegaard Floer homology. We show that for a large class of 3-orbifolds the orbifold invariant behaves like HF-hat in that the orbifold invariant, together with a relative Z_2-grading, categorifies the order of H_1^orb. When the 3-orbifold arises as Dehn surgery on an integer-framed knot in S^3, we use the {-1,0,1}-valued knot invariant epsilon to determine the relationship between the orbifold invariant and HF-hat of the 3-manifold underlying the 3-orbifold.
The theory of Mean Field Games was invented roughly a decade ago simultaneously by Lasry-Lions on the one hand and Caines-Huang-Malhamé on the other hand. The aim of both groups was to study Nash equilibria of differential games with infinitely many players. A fundamental object — introduced by Lions in his lectures — that fully characterizes these equilibria is the so-called master equation. This is an infinite dimensional nonlocal Hamilton-Jacobi equation set on the space of Borel probability measures endowed with a distance arising in the Monge-Kantorovich optimal transport problem. A central question in the theory is the global well-posedness of this equation in various settings. After an introduction, in this talk, we will focus on master equations in absence of noise in the dynamics of the agents. Because of lack of smoothing effect (in the absence of diffusion), previously only a short time existence result of classical solutions (due to Gangbo-Swiech) was available. The highly nonlocal nature of the equation prevents us from developing a theory of viscosity solutions in this setting. In the second half of the talk — as part of an ongoing joint work with W. Gangbo — we present a possible approach to construct global classical solutions when the data satisfies a suitable convexity/monotonicity condition.
Abstract:
A mathematical model of a dynamical process, often in the form of a system of differential equations, serves to elucidate underlying dynamical structure and behavior of the process that may otherwise remain opaque. However, model parameters are often unknown and may need to be estimated from data for a model to be informative. Model parameters in differential equations are estimated using numerical solutions in numerical-integration-based methods or using solution approximations in non-numerical integration methods. Numerical-integration-based methods can demand extensive computation, especially in large, stiff systems that require implicit methods for stability. Non-numerical integration methods are computationally more efficient, but do not provide an impartial measure of how well a model fits data, a measure required for the testability of a model. In this dissertation, I propose a new method that steps back from a numerical-integration-based method, and instead allows an optimal data-fitting numerical solution to emerge as part of an optimization process. This method bypasses the need for implicit solution methods, which can be computationally intensive, seems to be more robust than numerical-integration-based methods, and, interestingly, admits conservation principles and integral representations, which allow me to gauge the accuracy of my optimization.
The Min system is one of the simplest known biological systems that demonstrates diverse complex dynamic behavior or transduces local interactions into a global signal. Various mathematical models of the Min system show behaviors that are qualitatively similar to dynamic behaviors of the Min system that have been observed in experiments, but no model has been quantitatively compared to time-course data. In this dissertation, I extract time-course data for model fitting from experimental measurements of the Min system and fit established and novel biochemistry-based models to the time-course data using my parameter estimation method for differential equations. Comparing models to time-course data allows me to make precise distinctions between biochemical assumptions in the various models. My modeling and fitting supports a novel model, which suggests that a regular, ordered, stability-switching mechanism underlies the emergent, dynamic behavior of the Min system.
In the last few decades, the theory of optimal transportation has blossomed into a powerful tool for exploring applications both within and outside mathematics. Its impact is felt in such far flung areas as geometry, analysis, dynamics, partial differential equations, economics, machine learning, weather prediction, and computer vision. The basic problem is to transport one probability density onto other, while minimizing a given cost c(x,y) per unit transported. In the vast majority of applications, the probability densities live on spaces with the same (finite) dimension. After briefly surveying a few highlights from this theory, we focus our attention on what can be said when the densities instead live on spaces with two different (yet finite) dimensions. Although the answer can still be characterized as the solution to a fully nonlinear differential equation, it now becomes badly nonlocal in general. Remarkably however, one can identify conditions under which the equation becomes local, elliptic, and amenable to further analysis.
Obtaining accurate samples from the hard-core model (and related models) is an important problem at the intersection of probability theory, statistical mechanics, and theoretical computer science. I will introduce this problem and describe an algorithm that is efficient at low temperatures on lattices — that is, efficient in precisely the regime in which Glauber dynamics take an exponential time to mix. The algorithm is based on classical tools from statistical mechanics, the cluster expansions and Pirogov-Sinai theory, and Barvinok’s approach to polynomial approximation.
Based on joint work with Will Perkins and Guus Regts.
Error correcting codes protect information by encoding it into a larger system with some redundancy, allowing the original logical information to be recovered even if part of the encoded system becomes lost or corrupted. In this talk I will introduce the theory of quantum error correcting codes (QECC), which depart from the classical case in surprising and beautiful ways in order to surmount some uniquely quantum hurdles: quantum information cannot be copied, and general measurements will disturb quantum states. The success story of QECC culminates in the proof that they enable fault-tolerant quantum computation, overcoming the inherent fragility of quantum states and operations. But apart from this central application in quantum computation, there is also a link between QECC and condensed matter physics, which comes from the correspondence between locally checkable codes and local interactions in quantum spin models. This correspondence is relatively well-studied in the case of topologically ordered systems with a finite spectral gap, but in this talk I will also describe recent work on the frontier of this subject which finds QECC in the low energy space of gapless quantum spin chains, as well as QECC with nearly optimal parameters in the ground space of high dimensional gapless local Hamiltonians.
Bio. Elizabeth Crosson is an Assistant Professor of Physics at the University of New Mexico's Center for Quantum Information and Control. Prior to joining UNM in 2018, Crosson earned her PhD in physics at the University of Washington in 2015, and then spent three years as a postdoc at the Caltech Institute for Quantum Information and Matter. Her research involves several topics from the theory of quantum computation, including quantum algorithms, Hamiltonian complexity theory, rigorous classical simulations of quantum spin systems, and quantum error correction.
The IAM Lecture Series on Quantum Computing is generously sponsored by 1QBit, DWave, and PIMS.
Note for Attendees
Refreshments will be served in LSK 306 immediately following the talk, 4:00-5:00 p.m.
Exceptional groups (over arbitrary rings) are related to octonion algebras, triality and exceptional Jordan algebras. I will talk about recent results of an approach to these objects using certain torsors (principal homogeneous spaces) under smaller exceptional groups, and explain how an explicit understanding of these torsors provides insight into the objects and their interrelations.
Dynamical sampling is a new area in sampling theory that deals with processing a time series of evolving signals {A^nx, n=0,1,2,...} and aims at recovering the signal x from its coarsely sampled evolving states. In the talk, I will present why and how the temporal dynamics can compensate for insufficient spatial information. In particular, I will show how the dynamical sampling problem is linked to compressed sensing. Another more challenging problem arises when the operator A is also unknown and we want to recover both x and A. This problem exhibits features that occur similar to many fundamental problems in engineering such as deconvolution, and super-resolution and the matrix completion problem. We propose reconstruction algorithms with provable guarantees by employing ideas from the classical Prony method, matrix pencil method, and the ESPRIT method. Finally, I will conclude with some open questions and future research directions.
This is a special joint Mathematics of Information and Applications/SCAIM seminar.
Walks in the plane taking unit-length steps north and east from (0,0) to (n,n) never dropping below x=y, and parking cars subject to preferences, are two intriguing ingredients in a formula conjectured in 2005, now famously known as the shuffle conjecture.
Here we describe the combinatorial tools needed to state the conjecture. We also give key parts and people in its history, including its eventual algebraic solution by Carlsson and Mellit, which was published in the Journal of the American Mathematical Society in 2018. Finally, we conclude with some remaining open problems.
(This is a practice run for my JMM Invited Address, so is intended for a general audience and feedback is welcome.)
Microbial communities routinely have several alternative stable states observed for the same environmental parameters. A possibility of sudden and irreversible transitions between these states (regime shifts) complicates external manipulation of these systems. Can we predict which specific perturbations may induce such regime shifts and which would have only a transient effect? Here I will describe several new conceptual models that exhibit these emergent phenomena. Two of our models [2,3] were inspired by a decades-old economics work: the stable marriage or stable allocation problem, developed by Gale and Shapley in the 1960s and awarded the Nobel prize in economics in 2012. Using only the ranked tables of nutrient preferences and competitive abilities of microbes, we can determine all stable states and specific perturbations driving the system from one state to another.
It is an open problem whether there exists an exotic smooth structure on S^1 × S^3. Motivated by this problem, Akbulut asked if there is a simply connected homology cobordism from a homology sphere with non-trivial Rochlin invariant to itself. The main progress toward this question was made by Taubes who used his celebrated technique of gauge theory on manifolds with periodic ends to give a negative answer to this question for various homology spheres including the Poincare homology sphere. In this talk, I will present a different technique which can be used to generalize Taubes' result in various directions. In particular, we address Akbulut's question for a family of Brieskorn homology spheres, which is not covered by Taubes' theorem.
Systems of interacting agents arise in many areas of science, such as particle systems in physics, opinion dynamics in social science and flocking and swarming models in biology. Inferring the laws of interaction of agents from observational data is a fundamental challenge in a wide variety of disciplines. We propose a non-parametric statistical learning approach to estimate the governing laws of distance-based interactions, with no reference or assumption about their analytical form, from data consisting trajectories of interacting agents. We present efficient regression algorithms to learn the interaction kernel and develop a learning theory addressing consistency and optimal learning rate of the estimators. Especially, we show that despite the high-dimensionality of the systems, optimal learning rates can still be achieved, equal to that of a one-dimensional regression problem. We demonstrate the efficiency of the algorithms on various examples. Finally, I will conclude with some open questions and future research directions.
Note for Attendees
Pre-colloquium refreshments will be served at 3:45 p.m. in MATH 125 (Lounge).
Graphical semi-supervised learning is the problem of labelling the verticess of a graph given the labels of a a few vertices along with geometric information about the graph. Such problems have attracted a lot of attention in machine learning for classification of large datasets. In this talk we discuss consistency and perturbation properties of the probit approach to semi-supervised learning-- an approach that relaxes semi-supervised learning to a convex optimization problem. We show that the probit solution is unique and the predicted labels are consistent with the true labels of the vertices under some conditions. Furthermore, we study the probit approach in the large data limit where the number of vertices tends to infinity. In this limit, the probit approach converges to a convex optimization problem for functions. We then present analogous consistency and perturbation results for this limiting optimization problem.
Given a principally polarized abelian variety A over a number field (or a function field), one can naturally extract two real numbers that capture the ``complexity'' of A: one is the Faltings height and the other is the N\'eron-Tate height (of a symmetric effective divisor defining the polarization). I will discuss a precise relationship between these two numbers, relating them to some subtle invariants arising from tropical geometry (more precisely, from Berkovich analytic spaces). (Joint work with Robin de Jong.)
Inverse problems (the problem of inferring an unknown parameter from indirect and noisy measurements) are ubiquitous in science and engineering. The Bayesian approach to inverse problems provides a probabilistic framework in which prior knowledge about the unknown parameter is combined with indirect measurements to give an improved estimate of the unknown. Furthermore, the Bayesian approach allows for rigorous estimation of uncertainties associated with the estimated value of the unknown. In this talk we give a brief introduction to Bayesian inverse problems with a focus on modelling of prior knowledge with non-Guassian probability measures. We will discuss theoretical aspects of Bayesian inverse problems such as their well-posedness as well as applications and algorithms for their solution.
Note for Attendees
Pre-colloquium refreshments will be served at 2:30 p.m. in ESB 4133 (PIMS Lounge).
Fractal uncertainty principle states that no function can be localized close to a fractal set simultaneously in position and momentum. The strongest version so far has been obtained in one dimension by Bourgain and the speaker with recent higher dimensional advances by Han and Schlag. I will present the following application of the fractal uncertainty principle (joint work with Jin): on hyperbolic surfaces, the mass of an eigenfunction on an open set is bounded from below independently of the energy. As shown by Jin, these results lead to observability and control for the Schrödinger equation.
Church-Ellenberg-Farb introduced the theory of FI-modules to explain the phenomenon of representation stability of the cohomology of configuration spaces. I will explain the basics of how this story goes, and then explain how to extend their analysis to two generalized types of configuration spaces. Furthermore, I will explain how the Grothendieck-Lefschetz formula connects these topological stability phenomena to stabilization of statistics for polynomials and rational maps over finite fields.
We present a collection of results, based on a novel operator maximal inequality approach, providing precise relations between the time it takes a Markov chain to converge to equilibrium and the time required for it to exit from small sets. These refine results of Aldous and Lovasz & Winkler. Among the applications are:
(1) A general characterization of an abrupt convergence to equilibrium phenomenon known as cutoff. Specializing this to Ramanujan graphs and trees.
(2) Proving that the return probability decay is not geometrically robust (resolving a problem of Aldous, Diaconis - Saloff-Coste and Kozma).
(3) Random walk in evolving environment.
Note for Attendees
Pre-colloquium refreshments will be served in MATH 125 at 3:45 p.m.
In this talk we introduce an unusual eigenvalue problem that arises in fluid-structure interaction problems: the Jones eigenmode problem, described first by D.S. Jones in the eighties. The Jones eigenvalue problem was stated in the context of fluid-structure interaction problems where a bounded elastic body is immersed in an unbounded inviscid compressible fluid. In this model, there may exist frequencies such that the elastic obstacle sustains time-harmonic displacements whose normal components as well as tractions are identically zero on the boundary, and the fluid-structure problem fails to possess unique solutions.
The starting point of our work is the mathematical eigenvalue problem. In this unusual problem the existence of eigenvalues intimately depends on the shape of the boundary; indeed, it has been proved that almost all domains with infinitely smooth boundary do not possess such modes. The situation for Lipschitz domains has not been deeply studied. In this paper we describe these eigenmodes for a range of planar domains. Analytic expressions are obtained for simple domains, and we confirm the existence of these modes for a range of other shapes numerically using a finite element strategy.
The exclusion process is one of the most basic and best studied processes in the literature on interacting particle systems, with connections to card shuffling and statistical mechanics. It has been one of the major examples driving the study of mixing-times. In the exclusion process on an n-vertex graph we have k black particles and n-k white particles, one per site. Each edge rings at rate 1. When an edge rings, the particles occupying its end-points switch positions. Oliveira conjectured that the order of the mixing time of the process is at most that of the mixing-time of k independent particles. Together with Richard Pymar we verify this up to a constant factor for d-regular (or bounded degree) graphs in various cases:
(1) the degree d is at least logarithmic in n, or (2) the spectral-gap of a single walk is small (at most log number of vertices to the power 4) or (3) when the number of particles k is roughly n^a for some constant 0<a<1.
In these cases our analysis yields a probabilistic proof of Aldous' famous spectral-gap conjecture (resolved by Caputo et al.). We also prove a general bound which (when k > n^c) is within a \log \log n factor from Oliveira's conjecture. As applications we get new mixing bounds:
(a) O(\log n \log \log n) for expanders, (b) order \log (dk) for the hypercube {0,1}^d and (c) order (diameter)^2 \log k for vertex-transitive graphs of moderate growth and for the giant component of supercritical percolation on a torus.
Abstract: The partner model is a stochastic SIS model of infection spread over a dynamic network of monogamous partnerships. In previous work, Edwards, Foxall and van den Driessche identify a threshold in parameter space for spread of the infection and show the time to extinction of the infection is of order log(N) below the threshold, where N is population size, and grows exponentially in N above the threshold. Later, Foxall shows the time to extinction at threshold is of order sqrt(N). Here we go further and derive a single-variable diffusion limit for the number of infectious individuals rescaled by sqrt(N) in both population and time, and show convergence in distribution of the rescaled extinction time. Since the model has effectively four variables and two relevant time scales, the proof features a succession of probability estimates to control trajectories, as well as an averaging result to contend with the fast partnership dynamics.
(This is joint work with Rick Durrett and Anirban Basak.)
Nonparametric density estimation is a challenging statistical problem -- in general the maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints allows a path forward. In this talk I will discuss non-parametric density estimation under total positivity (i.e. log-supermodularity). Though they possess very special structure, totally positive random variables are quite common in real world data and exhibit appealing mathematical properties. Given i.i.d. samples from a totally positive distribution, we prove that the MLE exists with probability one if there are at least 3 samples. We characterize the domain of the MLE, and give algorithms to compute it. If the observations are 2-dimensional or binary, we show that the logarithm of the MLE is a piecewise linear function and can be computed via a certain convex program. Finally, I will discuss statistical guarantees for the convergence of the MLE, and will conclude with a variety of further research directions.
In this talk we explore a variety of individual-based stochastic models inspired from ecology, epidemiology, game theory and the social sciences. Our goal in each case is to understand the (global) population-level behaviour in terms of the (local) interaction rules. We pay special attention to large-population limit processes, both deterministic and stochastic, and to phase transitions that occur as model parameters are varied.
Note for Attendees
Pre-colloquium refreshments will be served in MATH 125 at 3:45 p.m.
Tensor decomposition has many applications. However, it is often a hard problem. In this talk I will discuss a family of tensors, called orthogonally decomposable tensors, which retain some of the properties of matrices that general tensors don't. A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of n orthonormal vectors. Such tensors are interesting because their decomposition can be found efficiently. We study their spectral properties and give a formula for all of their eigenvectors. We also give equations defining all real symmetric orthogonally decomposable tensors. Analogously, we study nonsymmetric orthogonally decomposable tensors, describing their singular vector tuples and giving polynomial equations that define them. In an attempt to extend the definition to a larger set of tensors, we define tight-frame decomposable tensors and study their properties. Finally, I will conclude with some open questions and future research directions.
Note for Attendees
Refreshments will be served at 2:30 p.m. in ESB 4133 (PIMS Lounge).
Vinogradov's Mean Value Theorem was proven separately by Wooley's efficient congruencing method and Bourgain-Demeter-Guth's decoupling method. While similarities between the methods have been observed no precise dictionary has been written. We give a proof of l^2 decoupling for the parabola inspired by efficient congruencing in two dimensions. We will mention where tools like ball inflation and l^2 L^2 decoupling come into play. Making this proof quantitative also allows us to match a bound obtained by Bourgain for the discrete Fourier restriction problem in two dimensions without resorting to using the divisor bound.
Consider the problem of recovering a rank 1 tensor of order k that has been subject to Gaussian noise. It is information theoretically possible to recover the tensor with a finite number of samples via maximum likelihood estimation, however, it is expected that one needs a polynomially diverging number of samples to efficiently recover it. What is the cause if this large statistical-to-algorithmic gap? To understand this interesting question of high dimensional statistics, we connect it to an intimately related question: constrained optimization of a typical homogenous polynomials on the sphere in high dimensions. We show that the estimation threshold is related to a geometric analogue of the BBP transition for matrices for the landscape of these polynomials. We then study the threshold for efficient recovery for a simple class of algorithms, Langevin dynamics and gradient descent. We view this problem in terms of a broader class of polynomial optimization problems and propose a mechanism or success/failure of recovery in terms of the strength of the signal on the high entropy region of the initialization.
Note for Attendees
Pre-colloquium refreshments will be served at 3:45 p.m. in MATH 125 lounge.
This is a joint work with B. Calmes, V. Petrov, N. Semenov and K. Zainoulline. In the talk I will discuss a connection between direct sum decompositions of the Chow motive with Z-coefficients of a homogeneous space of a group G, and representations of affine nil Hecke algebras defined in terms of root system of G. This connnection can be used in two directions: prove indecomposability of certain motives as well as get some structural results about Hecke algebras.
The generalized null space decomposition (GNSD), better known as the staircase decomposition, is a unitary reduction of a general matrix A of order n to a block upper triangular form that reveals the structure of the Jordan blocks of A corresponding to a zero eigenvalue. The reduction was introduced by Kublanovskaya in 1968. It was extended first by Ruhe and then by Golub and Wilkinson, who based the reduction on the singular value decomposition. If A has large Jordan blocks, the complexity of these algorithms can approach the order of n^4. We outline an alternative algorithm, based on repeated updates of a QR decomposition of A, that is guaranteed to be of order n^3. Numerical experiments confirm the stability of this algorithm, which turns out to produce essentially the same form as that of Golub and Wilkinson. The effect of errors in A on the ability to recover the original structure is investigated empirically. Several applications are discussed, including the computation of the Drazin inverse.
Joint work with G.W. (Pete) Stewart and Nicola Guglielmi.
Abstract: Indirect reciprocity explores how humans act when their reputation is at stake. The corresponding models describe how reputations change over time, depending on the actions people take, and on the social norms they employ. The first part of the talk gives a brief overview of the field. It discusses how the reputation dynamics within a population can be modelled mathematically, and which social norms have been proposed to stabilize cooperation in the long run. The second part shows that these previously proposed norms fail once the models take into account that information is typically private and noisy. Finally, we prove in the last part that cooperation can be sustained in noisy environments if individuals have access to stochastic norms.
I will survey recent progress on understanding dynamical thresholds in two problems: spherical spin glasses and spiked tensor models. I will begin by reviewing an elementary approach to studying spherical spin glass dynamics based on differential inequalities for one-time observables. Using this approach, one can obtain an approximate phase diagram for the evolution of the energy H and its gradient under Langevin dynamics. I will then turn to how these ideas can be used to understand the algorithmic thresholds in spiked tensor models discussed in Monday's talk. In particular, I will discuss the key ideas behind the proof of these thresholds by combining global regularity estimates for the landscape with point-wise estimates for the initialization. This is joint work with Ben Arous-Gheissari.
This is a joint work with G.Garkusha. In the talk I will discuss the constructionof fibrant replacements for spectra consisting of Thom spaces (suspension spectra of varieties and algebraic cobordism MGL being the motivating examples) that uses the theory of framed correspondences. As a consequence we get a description of the infinite loop space of MGL in terms of Hilbert schemes.
When two identical nonlinear oscillators are coupled through a 1-D bulk diffusion field, new patterns of synchronization occur that would be absent in the uncoupled system. Furthermore, if the two oscillators are quiescent, the effect of the coupling can be to turn the oscillations on. Mathematically, the models consist of systems of nonlinear ODEs coupled with linear diffusive PDEs. Through a detailed bifurcation analysis of three different examples, we reveal some of the underlying mechanisms behind phenomena as diverse as the diffusion sensing of reacting agents, the synchronization of chaotic oscillations and the formation of membrane-bound patterns at the cell-scale level.
Iterated games are the baseline model to explain how cooperation can evolve in repeated interactions. The basic idea is that individuals are more likely to cooperate if they can expect their beneficiaries to remember and to return their cooperative acts in future. However, six years ago, William Press and Freeman Dyson have shown that certain repeated games also allow individuals to employ extortionate strategies. By using an extortionate strategy, players can guarantee that they systematically outperform their opponent, irrespective of the opponent’s reaction. In this talk, I will first present a simple proof for the existence of these extortionate strategies. This proof applies to both, finitely and infinitely repeated games with arbitrarily many players. I will then discuss under which circumstances such strategies can emerge in an evolutionary process, and how successful they are against actual human players. In the end, I will discuss a few interesting generalizations and open problems.
Note for Attendees
Pre-colloquium refreshments will be served in MATH 125 at 3:45 p.m.
Abstract:
For a reliable simulation of systems subject to noise, it is necessary to characterize the noise properly and develop efficient algorithms. In the first part of this talk, I will present an extension of Karhunen-Loeve (K-L) expansion to model and simulate multiple correlated random processes. The method finds the appropriate expansion for each correlated random process by releasing the bi-orthogonal condition of the K-L expansion. I will address the convergence and computational efficiency, in addition to some explicit formulae and analytical results. In the remaining talk, I will discuss numerical methods to effectively compute the propagation of uncertainty in parameterized stochastic differential equations. Joint response-excitation PDF equation generalizes the existing PDF equations and enables us to compute the PDF of the solution to system subject to non-Gaussian colored noise. An adaptive discontinuous Galerkin method combined with probabilistic collocation method is developed to resolve both local and discontinuous dynamics, while low-rank tensor method is employed in case of high-dimensionality. For anisotropic parameterized stochastic PDEs, we develop a reduced basis method using ANOVA decomposition to automatically identify the important dimensions and appropriate resolution in each dimension. The effectiveness of the methods is demonstrated in high-dimensional stochastic PDEs.
Recent advances in single-cell gene sequencing data and high-dimensional data analysis techniques are bringing in new opportunities in modeling biological systems. In this talk, I discuss different approaches to develop mathematical models from single-cell data. For high-dimensional single-cell gene sequencing data, dimension reduction techniques are applied to find the trajectories of cell states in the reduced differentiation space, then modeled as directed and random movement on the abstracted graph with PDEs. Normal hematopoiesis differentiation and abnormal processes of acute myeloid leukemia (AML) progression are simulated, and the model can predict the emergence of cells in novel intermediate states of differentiation consistent with immunophenotypic characterizations of AML. In addition, we develop representations of multi-correlated stochastic processes for correlated time series cell data, by releasing the bi-orthogonal condition of Karhunen-Loeve expansion. Convergence and computational efficiency of the methods are addressed. Finally, for fluorescence in situ hybridization data that provides spatial-temporal patterns of cells, we develop tumor growth model incorporating dynamics of drug resistance. It is demonstrated that assuming continuous cell state may result in different dynamics of anti-cancer drug resistance when compared with the predictions of classical discrete models, and its implications in designing therapies are studied.
Note for Attendees
Pre-colloquium refreshments will be served in ESB 4133 at 2:30 p.m.
Let G=G(N,p) be an Erd\H{o}s--R\'enyi graph on N vertices (where each pair is connected by an edge independently with probability p). We view N as going to infinity, with p possibly going to zero with N. What is the probability that G contains twice as many triangles as we would expect? I will discuss recent progress on this ``infamous upper tail" problem, and more generally on tail estimates for counts of any fixed subgraph. These problems serve as a test bed for the emerging theory of \emph{nonlinear large deviations}, and also connect with issues in extending the theory of \emph{graph limits} to handle sparse graphs. In particular, I will discuss our approach to the upper tail problems via new versions of the classic regularity and counting lemmas from extremal combinatorics, specially tailored to the study of random graphs in the large deviations regime. This talk is based on joint work with Amir Dembo.
Note for Attendees
Pre-colloquium refreshments will be served in MATH 125 at 3:45 p.m.
Let G/B be a flag variety over C, where G is a simple algebraic group with a simply laced Dynkin diagram, and B is a Borel subgroup. The Bruhat decomposition of G defines subvarieties of G/B called Schubert subvarieties. The codimension 1 Schubert subvarieties are called Schubert divisors. The Chow ring of G/B is generated as an abelian group by the classes of all Schubert varieties, and is "almost" generated as a ring by the classes of Schubert divisors. More precisely, an integer multiple of each element of G/B can be written as a polynomial in Schubert divisors with integer coefficients. In particular, each product of Schubert divisors is a linear combination of Schubert varieties with integer coefficients.
In the first part of my talk I am going to speak about the coefficients of these linear combinations. In particular, I am going to explain how to check if a coefficient of such a linear combination is nonzero and if such a coefficient equals 1. In the second part of my talk, I will say something about an application of my result, namely, how it makes it possible estimate so-called canonical dimension of flag varieties and groups over non-algebraically-closed fields.
Compressed sensing theory explains why Lasso programs recover structured high-dimensional signals with minimax order-optimal error. Yet, the optimal choice of the program’s governing parameter is often unknown in practice. It is still unclear how variation of the governing parameter impacts recovery error in compressed sensing, which is otherwise provably stable and robust. We establish a novel notion of instability in Lasso programs when the measurement matrix is identity. This is the proximal denoising setup. We prove asymptotic cusp-like behaviour of the risk as a function of the parameter choice, and illustrate the theory with numerical simulations. For example, a 0.1% underestimate of a Lasso parameter can increase the error significantly; and a 50% underestimate can cause the error to increase by a factor of 109. We hope that revealing parameter instability regimes of Lasso programs helps to inform a practitioner’s choice. Finally, we discuss how these results extend to their more general Lasso counterparts.
Statistics of the characteristic polynomial for large Haar unitary matrices U at points on the unit circle have received considerable attention due to similarities with the statistics of the Riemann zeta function far up the critical axis. While the best-known instances of this universality phenomenon concern statistics of \emph{zeros} for these functions (eigenvalues of U), there is strong evidence that the analogy also applies to extreme values.
Towards the more modest goal of understanding this universality phenomenon within the class of distributions on the unitary group, in this talk we consider the characteristic polynomial \chi_N(z) for an N\times N Haar permutation matrix. Our main result is a law of large numbers for (the logarithm of) the maximum modulus of \chi_N(z) over the unit circle. The main idea is to uncover a multi-scale structure in the distribution of the random field \chi_N(z), and to adapt a well-known second moment argument for the maximum of a branching random walk. Unlike the analogous problem for the Haar unitary, the distribution of \chi_N(z) is sensitive to Diophantine properties of the argument of z. To deal with this we borrow tools from the Hardy--Littlewood circle method in analytic number theory. Based on joint work with Ofer Zeitouni.
VC-dimension is a combinatorial measure for the complexity of a set system. In binary classification, the VC-dimension of the learning model characterizes the amount of training data needed to learn the model. We prove lower and upper bounds for the VC-dimension of neural networks with piecewise linear activation functions, that are within a logarithmic factor of one another. Based on joint work with Peter Bartlett, Nick Harvey, and Christopher Liaw.
Distribution learning lies at the intersection of statistics, theoretical computer science and machine learning. We give an overview of this area, with the problem of learning mixtures of high-dimensional Gaussians as a running example. In particular, we prove nearly tight sample complexity bounds for this problem in the density estimation model. Based on joint work with Hassan Ashtiani, Shai Ben-David, Luc Devroye, Nick Harvey, Christopher Liaw, Yani Plan, and Tommy Reddad.
Note for Attendees
Pre-colloquium refreshments will be served at 3:45 p.m. in MATH 125 lounge.
R.Sharifi formulated remarkable conjectures which relate the arithmetic of cyclotomic fields to Eisenstein quotient of the homology groups of modular curves. In this talk, I will give a brief introduction to Sharifi's conjectures and explain the statuses of these conjectures.
Abstract: Translation of mRNA into protein is one of the most essential biological processes underlying life. As this process is mediated by the traffic of molecular machines (known as ribosomes), it has been modeled by the Totally Asymmetric Simple Exclusion Process (TASEP), a classical stochastic process for non-equilibrium particle systems. Recently, progress in sequencing technology has given access to quantitative information about the local dynamics of translation. However, the interpretation of this data is still elusive, and requires not only to investigate the theoretical properties of the TASEP, but also the development of new methods and tools based on this model. In such a perspective, I will present analytical, statistical and computational approaches that I developed for the TASEP to 1) study the extent of traffic jams by deriving the density of isolated particles, 2) obtain the phase diagram that describes different regimes of the flux and 3) infer the particle hoping rate function from their observed density. Finally, I will show how applying these methods and tools to experimental data has led us to investigate other stochastic phenomena and mathematical questions occurring at various scales, as the diffusion of molecules in a confined ribosomal sub-compartment, optimal control of ribosomal populations to optimize translation in single cell, or evolutionary processes that have shaped the ribosome geometry or the usage of synonymous codons.
Points in a dynamical system are just like us - they're creatures of habit. That is, they like returning to their neighborhoods not just once, but infinitely often. Roughly, there are two types of systems we are interested in - those with a measure-preserving transformation and those with a continuous transformation - and on both sides we observe the same phenomenon. In this talk I will discuss recurrence in dynamical systems - specifically, the Poincaré Recurrence theorem for measure-preserving dynamical systems and the notion of Non-wandering sets for continuous transformations. If time permits, we can discuss applications/implications of what this means in Dynamical System Theory and Ergodic Theory.
In this talk, I will present two recent studies that respectively revisit the theories of first passage time, and interacting particle process. First, I will analyze a generic planar diffusion process, inspired by neuronal dynamics. The specificity of this process is that the density of escape time from a characteristic boundary is non-Poissonian, and presents an oscillatory decay, which is quantified by using asymptotic methods to derive the spectrum of the Fokker-Planck operator. Secondly, I will present a generalization of the totally asymmetric simple exclusion process (TASEP), which is a widely used model for transport phenomena in non-equilibrium particle systems. To model protein synthesis, the TASEP needs to be generalized, such that particles occupy an extended number of sites and move at arbitrary site-dependent rates. While classical ansatz and mean-field approximations cannot be applied to analyze this generalized model, we developed another method based on the hydrodynamic limit of the process. Deriving and solving this limit yield analytical formulas for the particle flux and density, and a complete description of phase transitions. Surprisingly, the phase diagram and key quantities of the process depend on few parameters, which has important consequences on the control and regulation of traffic in such particle systems.
Note for Attendees
Pre-colloquium refreshments will be served at 2:30 p.m. in ESB 4133 lounge.
The Grothendieck ring of algebraic stacks was introduced by Ekedahl in 2009. It may be viewed as a localization of the more common Grothendieck ring of varieties. If G is a finite group, then the class {BG} of its classifying stack BG is equal to 1 in many cases, but there are examples for which {BG}\neq 1. When G is connected, {BG} has been computed in many cases in a long series of papers, and it always turned out that {BG}*{G}=1. We exhibit the first example of a connected group G for which {BG}*{G}\neq 1. As a consequence, we produce an infinite family of non-constant finite étale group schemes A such that {BA}\neq 1.
Density functional theory has been a popular tool in solid state physics and quantum chemistry for electronic structure calculation. However, current functionals used in density functional theory face difficulties when dealing with strongly correlated systems. In this talk, we examine the regime where the electrons are strictly correlated. This gives rise to a multimarginal optimal transport problem, a direct extension of the optimal transport problem that has applications in economics and operations research as well. In particular we introduce methods from convex optimization to provide a lower bound to the cost of the multimarginal transport problem with a practical running time. We further propose projection schemes based on tensor decomposition to obtain upper bounds to the energy. Numerical experiments demonstrate a gap of order 10^{-3} to 10^{-2} between the upper and lower bounds.
Note for Attendees
Pre-colloquium refreshments will be served in the MATH 125 lounge at 3:45 p.m.
Deep neural-network provides an alternative method for compressing high-dimensional functions arising from partial differential equations (PDE). In this talk, we focus on using artificial neural-networks for solving PDEs in two ways: (1) Using neural-networks to represent mappings between PDE coefficients and solutions. (2) Constructing a solution space with neural-networks when solving for a PDE, and obtaining the neural-network parameterized solution via optimization. We apply the methods developed in scattering problems, stochastic homogenization, and when studying transition between states in stochastic systems.
There are numerous genetic, molecular and physical factors that all need to be carefully orchestrated to create the face. When one or more of these processes goes awry during embryonic development, birth defects such as cleft lip result. In the past, our work has focused on several molecular signals and how they regulate facial growth. Here we investigate in an unbiased manner, the extrinsic and intrinsic factors contributing to shaping of the face. This presentation will describe new methods to track embryonic mesenchymal cells by marking cell nuclei in a defined culture system. These tracking data were analyzed over time and space to determine the order and disorder in the tissue. K-means clustering revealed a surprising degree of coordination in regions of the tissue. The normalization of the data throughout the whole centre of the face showed that there was a high degree of right-left symmetry. These reflected data support strong genetic control over cell movements. Furthermore, the block of intrinsic cytoskeletal remodeling with a drug, completely disrupted these conserved patterns of cell movements. Taken together the collaboration between biologists and mathematicians has shed new light on the fundamental mechanisms driving facial morphogenesis.
In the 1970s, Serre formulated his remarkable conjecture that every two-dimensional mod-p Galois representation of the absolute Galois group of ℚ, which is odd and irreducible, should come from a modular form. He later refined his conjecture, giving a precise recipe for the weight and level of the modular form. Both the "weak form" and "strong form" of Serre's conjecture are now theorems, due to the work of many mathematicians (Khare-Wintenberger, Kisin, Edixhoven, Ribet, and others). In this talk, we will discuss how to generalize Serre's weight recipe when the Galois representation is replaced by a homomorphism from an absolute Galois group to the Langlands dual of a rank 2 unitary group. This is joint work with Stefano Morra.
This talk focuses on estimating temporal couplings of stochastic processes with optimal transport (OT), motivated by applications in developmental biology and cellular reprogramming. We present experimental evidence for the fact that the temporal couplings of a developmental stochastic process are well-approximated by (entropic) optimal transport, over short time-scales. We collect 315,000 single cell RNA-seq expression profiles at 40 time points over 18 days of stem cell reprogramming, and we demonstrate that OT can accurately interpolate the distribution of cells at held-out time points. Our analysis leads to biological discoveries about cellular reprogramming, including new ways to enhance reprogramming efficiency. Our approach provides a general framework for investigating cellular differentiation, and poses some interesting questions in theoretical statistics.
Note for Attendees
Pre-colloquium refreshments will be served at 3:45 pm in MATH 125.
If you have no idea what mathematical biology is, or even what applied math looks like, then this seminar is for you. In this talk, we will discuss a multitude of problems in mathematical biology. I will present topics from a range of biological contexts, including cell biology, ecology, physiology and evolution, and a range of mathematical methods, including game theory, cellular potts models, and differential equations. Since there is such a range of problems to cover and interests to satisfy, there will be an opportunity for the audience to choose which topics we will discuss in depth.
Calculus of variations and partial differential equations have been seeing great progress over the last decade, and some of the most important driving forces in these developments are optimal transport and free boundary theories. The recent Fields medalist, Alessio Figalli, has made groundbreaking results in these directions, and it is worth to bring a few leading researchers in these areas, to discuss the state of the art and the future directions, especially to inspire young researchers in the PIMS community.
Main Speakers
Alessio Figalli, ETH Zurich: Regularity of interfaces in phase transitions via obstacle problems
Francesco Maggi, University of Texas at Austin: Soap films, soap bubbles, and almost critical points in geometric variational problems
Robert McCann, University of Toronto: Displacement convexity of Boltzmann's entropy characterizes positive energy in general relativity
Ovidiu Savin, Columbia University: Sharp $W^{2,p}$ regularity results in the optimal transport problem between convex domains.
The so-called Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase change, for example ice melting to water. An important goal is to describe the structure of the interface separating the two phases. In its stationary version, the Stefan problem can be reduced to the classical obstacle problem, which consists in finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. The aim of this talk is to give a general overview of the classical theory of the obstacle problem, and then discuss recent developments on the structure of interfaces, both in the static and the parabolic settings.
now MATH 100 (please note this change of room within the MATH building)
Sat 9 Feb 2019, 9:00am-5:30pm
Abstract
Calculus of variations and partial differential equations have been seeing great progress over the last decade, and some of the most important driving forces in these developments are optimal transport and free boundary theories. The recent Fields medalist, Alessio Figalli, has made groundbreaking results in these directions, and it is worth to bring a few leading researchers in these areas, to discuss the state of the art and the future directions, especially to inspire young researchers in the PIMS community.
Main Speakers
Alessio Figalli, ETH Zurich: Regularity of interfaces in phase transitions via obstacle problems
Francesco Maggi, University of Texas at Austin: Soap films, soap bubbles, and almost critical points in geometric variational problems
Robert McCann, University of Toronto: Displacement convexity of Boltzmann's entropy characterizes positive energy in general relativity
Ovidiu Savin, Columbia University: Sharp $W^{2,p}$ regularity results in the optimal transport problem between convex domains.
In 2013, Esnault and Srinivas proved that as in the de Rham cohomology over the field of complex numbers, the algebraic entropy of an automorphism of a smooth projective surface over a finite field $\mathbb{F}_q$ is taken on the span of the Néron–Severi group inside of $\ell$-adic cohomology. Later, motivated by this and Weil's Riemann Hypothesis, Truong asked whether the spectral radius $\chi_{2k}(f)$ of the pullback $f^*: H^{2k}(X, \mathbb{Q}_\ell) \to H^{2k}(X, \mathbb{Q}_\ell)$ is the same as the spectral radius $\lambda_k(f)$ of the pullback $f^*: N^k(X)_\mathbb{R} \to N^k(X)_\mathbb{R}$, where $f: X \to X$ is a surjective self-morphism of a smooth projective variety $X$ of dimension $n$ defined over an algebraically closed field $\mathbb{k}$ and $N^k(X)$ denotes the finitely generated abelian group of algebraic $(n-k)$-cycles modulo the numerical equivalence. He has shown that $\displaystyle \max_{0\le i\le 2n} \chi_{i}(f) = \max_{0\le k\le n} \lambda_{k}(f)$. We give an affirmative answer to his question in the case of abelian varieties and $k=1$.
The conjugate gradient algorithm (CG) is a widely used iterative method for solving large symmetric positive definite linear systems Ax=b. Beginning in the 1980's (and, even earlier, in the thesis of Paige dating back to 1971), a great deal of research has been aimed at explaining the behavior of the Lanczos and conjugate gradient algorithms in finite precision arithmetic. The algorithms do not behave the way exact arithmetic theory predicts, yet they are widely used and often deliver impressive results. Explanations were derived, under the assumption that the finite precision computation satisfied certain properties that could be shown to hold for particular (now standard) implementations. Names associated with this early work include Paige, Druskin and Knizhnerman, Greenbaum and Strakos. With the advent of parallel computing, different variants of the conjugate gradient algorithm were proposed to take better advantage of parallelism; see, e.g, Chronopoulas and Gear, On the efficient implementation of preconditioned s-step conjugate gradient methods on multiprocessors with memory hierarchy, Parallel Comput. 11 (1989), pp. 37-53, and Ghysels and Vanroose, Hiding global synchronization latency in the preconditioned conjugate gradient algorithm, Parallel Comput. 40 (2014), pp. 224-238.All of these are mathematically equivalent to the original Hestenes and Stiefel algorithm, but none have been shown to satisfy the hypotheses used in the analysis of the 1980's. Does this mean that they perform poorly? Or does it simply mean that the assumptions of the early analysis were sufficient but not necessary conditions for good behavior? Lower precision arithmetic is now becoming more popular. What will be the effect of this on CG codes? I will discuss these issues and what insight might be gained from the analysis in order to devise stable and efficient finite precision implementations.
We introduce the notion of a spoke algebra, which encodes the data of a C_p-equivariant cohomology equipped with a coherent system of norms. This is a generalization of the notion of an ``\mathbb{E}_{\sigma}-algebra" for the group C_2 and the sign representation \sigma. We explain several naturally occurring examples (such as certain mapping spaces) and then describe several applications. The first is a method for delooping a suitably structured space by an irreducible, 2-dimensional C_p-representation in two steps. The second is the construction of certain interesting C_p-spectra related to the odd primary Kervaire invariant problem using a version of Koszul duality for spoke algebras. This is joint work with Jeremy Hahn.
A purity conjecture due to Grothendieck and Auslander-Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension >1. The combination of several works of Gabber settles the conjecture except for some cases that concern p-torsion Brauer classes in mixed characteristic (0, p). We will discuss an approach to the mixed characteristic case via the tilting equivalence for perfectoid rings, as well as a more general purity conjecture for the Brauer group of some singular schemes.
The deformation space of a rational map is a submanifold of Teichmueller space parameterizing maps that are combinatorially and dynamically equivalent. S. Koch and I recently showed that this space need not be connected, and can have infinitely many components. In this talk we describe some tools from algebraic geometry, and the topology of coverings that reveal the obstructions to connectedness, and give some evidence that connectedness may be achieved by passing to augmented Teichmueller space.
In this talk, we discuss the following question first popularized in the 1980s by the then startling new computer renderings of the Mandelbrot set. Given a rational map from the Riemann sphere to itself, how does the behavior of the map near the forward orbits of the critical points determine the rational map itself? This circle of ideas, developed by Douady-Hubbard, Thurston, Milnor and others draws from classical ideas in complex analysis, topology and algebraic geometry. We will give a brief survey, and state some recent results and open questions.
Note for Attendees
Refreshments will be served at 2:30 p.m. in ESB 4133 (Lounge).
The singular value decomposition (SVD) is one of the core computations of today's scientific applications and data analysis tools. The main goal is to compute a compact representation of a high dimensional operator, a matrix, or a set of data that best resembles the original in its most important features. Thus, SVD is widely used in scientific computing and machine learning, including low rank factorizations, graph learning, unsupervised learning, compression and analysis of images and text.
The popularity of the SVD has resulted in an increased diversity of methods and implementations that exploit specific features of the input data (e.g., dense/sparse matrix, data distributed among the computing devices, data from queries or batch access, spectral decay) and certain constraints on the computed solutions (e.g., few/many number of singular values and singular vectors computed, targeted part of the spectrum, accuracy). The use of the proper method and the customization of the settings can significantly reduce the cost.
In this talk, we'll overview the most relevant methods in terms of computing cost and accuracy (direct methods, iterative methods, online methods), including the most recent advances in randomized and online SVD solvers. We present what parameters have the biggest impact on the computational cost and the quality of the solution, and some intuition for their tuning. Finally, we discuss the current state of the software on widely used platforms (MATLAB, Python's numpy/scipy and R) as well as high-performance solvers with support for multicore, GPU, and distributed memory.
I will discuss two recent results obtained in collaboration with I. Parissis (U Basque Country). The first is a sharp L^2 estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth.
The second result is a sharp L^4 estimate for the Fourier multiplier associated to a polygon of N sides in R^2, and a sharp form of the two parameter Meyer's lemma. These results improve on the usual ones obtained via weighted norm inequalities and rely on a novel Carleson measure estimate for directional square functions of time-frequency nature.
I will talk about a new approach to computing the motivic weight of the stack of G-bundles on a curve. The idea is to associate a motivic weight to certain ind-schemes, such as the affine Grassmannian and the scheme of maps X -> G, where X is an affine curve, using Bittner's calculus of 6 operations. I hope that this will eventually lead to a proof of a conjectural formula for the motivic weight of the stack of bundles in terms of special values of Kapranov's zeta function.
In the presentation we will discuss our research program concerning the study of extreme vortex events in viscous incompressible flows. These vortex states arise as the flows saturating certain fundamental mathematical estimates, such as the bounds on the maximum enstrophy growth in 3D (Lu & Doering, 2008). They are therefore intimately related to the question of singularity formation in the 3D Navier-Stokes system, known as the hydrodynamic blow-up problem. We demonstrate how new insights concerning such questions can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable discrete gradient flows. More specifically, such an optimization formulation allows one to identify "extreme" initial data which, subject to certain constraints, leads to the most singular flow evolution. In offering a systematic approach to finding flow solutions which may saturate known estimates, the proposed paradigm provides a bridge between mathematical analysis and scientific computation. In particular, it makes it possible to determine whether or not certain mathematical estimates are "sharp", in the sense that they can be realized by actual vector fields, or if these estimates may still be improved. In the presentation we will review a number of results concerning 2D and 3D flows characterized by the maximum possible growth of different Sobolev norms of the solutions. Even when extreme initial data is used, high-resolution computations for the 3D Navier-Stokes system reveals no tendency for singularity formation in finite time.
[Joint work with Diego Ayala, Dongfang Yun and Di Kang]
The Fukaya category of a Weinstein manifolds W algebraically package all Lagrangian submanifolds of W into an A-infinity category. In this talk I will motivate why studying Lagragian submanifolds in symplectic manifolds is relevant and then I will give an overview of the definition of wrapped Fukaya categories. I will explain a theorem stating that for Weinstein manifolds a particular finite collection of Lagrangians generates this category. Finally we will see some element of the proof and applications of this generation criterion. This is joint work with G. Dimitroglou-Rizell, P. Ghiggini and R. Golovko.
Insulin is an essential hormone that regulates nutrient homeostasis. Insufficient insulin results in diabetes, one of the most prevalent and costly diseases. Although the primary actions of insulin are to induce glucose uptake and metabolism in distant tissues, including muscle, fat and liver, the insulin secreting pancreatic beta-cells contain a high number of insulin receptors and known to respond to the hormone. On a minute-to-minute time-scale, insulin has been reported to have negative feedback effects on its own secretion, and we have data suggesting that the actions of insulin may be context-dependent, potentially depending on the ambient glucose levels (which are primarily controlled by glucose). Insulin has also been reported to have positive effects on its own synthesis and on the survival of the beta-cells over a timescale of months. Within beta-cells, insulin production is inherently stressful and exerts a negative effect on beta-cell proliferation that is most pronounced at a young age. We have also recently found that single beta-cells can exist in ‘bursting’ states of elevated insulin production that account for a significant proportion of the previous described heterogeneity in this cell type. Thus, using a variety of experimental approaches, we seek to understand context-dependent insulin feedback signalling on single beta-cells and their collective populations and we are interested in collaborating to build quantitative and testable models that could be used to explain the pathogenesis of diabetes. We also interested in expanding models to include other tissues and other soluble factors that are also relevant in nutrient homeostasis and diabetes.
Optimal transport (OT) problems, initiated by G. Monge 200 years ago and refined by L. Kantorovich in the 1940’s, provide --among other things-- a quantitative way for measuring correlations between probability distributions. Martingale optimal transports (MOT) and their Skorokhod embeddings in Brownian motion lead to optimal transport by stopping time (OTST) problems. These are important variations on OT, with applications to financial mathematics and probability theory. In OTST, one specifies a stochastic state process and a cost, and minimizes the expected cost over stopping times with a given state distribution.
In this talk, I will focus on the case where the state process is d-dimensional Brownian motion and the cost is given by the Euclidean distance. I will discuss new results involving dual variational principles, their attainment, as well as characterizations of the optimal stopping times as a hitting time of barriers given by solutions of corresponding obstacle problems. I will also discuss how these results generalize for other processes and costs and relate them to other aspects of probability theory. This talk is based on joint work with Nassif Ghoussoub, Young-Heon Kim and Tongseok Lim.
Conjectures on the statistics of elliptic curves are generally formulated with the assumption that the curves in question are ordered by their conductors. However, when proving results on the statistics of elliptic curves, the curves are usually ordered by (naive) height. There are two main reasons for doing so: first, it is difficult to rule out the possibility that there are many elliptic curves with large height but small discriminant; second, it is difficult to rule out the possibility that there are many elliptic curves with large discriminant but small conductor. In this talk, we will focus on the second question. We assume the j-invariant is bounded and count the number of elliptic curves having bounded conductor and whose discriminant is bounded by their conductor to the power 7/4. We also count elliptic curves where the quotient of discriminant by conductor is squarefree and prove that the average size of their 2-Selmer groups is 3.
This is joint work with Ananth Shankar and Arul Shankar.
Probability theory is a game of heads or tails. This Friday we will verify how far we can get just by flipping coins - and taking limits.
We will discuss models for two physical phenomena. The first one is well-known: the movement of a pollen grain. Starting with coin tossings, we can describe the trajectory of pollen at a microscopic scale. If we look with macroscopic lenses, the outcome is Brownian motion. The second model to discuss is percolation. A percolation process appears when water moves through porous rock. Our starting point will be to more coin flipping. This time, we will find a mathematical jewel in the intersection of probability, complex analysis, and mathematical physics: the Schramm-Loewner evolution (SLE).
C. Voisin proved that no two distinct points on a very general surface of degree $\ge 7$ in ${\mathbb P}^3$ are rationally equivalent. She conjectured that the same holds for a very general sextic surface. We settled this conjecture by improving her method which makes use of the global jet spaces. This is a joint work with James D. Lewis and Mao Sheng.
I will discuss joint work with Lior Silberman in which we describe the topology of spaces of representations of Baumslag-Solitar groups into complex reductive algebraic groups.
My research program is motivated by fascination with bird flight. My laboratory group uses a multi-disciplinary approach that includes biomechanics, physiology, and neuroscience to examine flight ability. Our current research is organized around two topics: 1) how birds morph their wings and what benefits this provides; and 2) how optic flow signals are encoded in the avian brain and used to guide their flight. As we gain understanding of flight mechanisms, we further endeavor to apply comparative approaches that provide deeper insight into avian ecology and evolution.
We study the local limits of uniform triangulations chosen uniformly over those with fixed size and genus in the regime where the genus is proportional to the size. We show that they converge to the Planar Stochastic Hyperbolic Triangulations introduced by Curien. This generalizes the convergence of uniform planar triangulations to the UIPT of Angel and Schramm, and proves a conjecture of Benjamini and Curien. As a consequence, we obtain new asymptotics on the enumeration of high genus triangulations.
Joint work with Baptiste Louf.
Have you ever wondered how the immune system works? How does our body recognize that something “alien” got inside us? What does it mean to have or to develop an autoimmune disease? How come an HIV vaccine doesn’t exist yet? ... do you wonder why a math grad student is giving this talk? Well, I will give a brief introduction to immunology and to the history of mathematics and immunology interactions in this seminar. I am planning to present two immunological problems and their solution through mathematical models. At the end of the seminar, you will know how to identify diffusion vs chemotaxis from cell tracking.
Also, I will give a short description of the meaning of March 8th, International Women’s Day.
Wi-Fi, the most commonly used radio technology, is based on the ever evolving IEEE 802.11 standard. One of the ongoing endeavours of the IEEE 802.11 organization is the standardization of so-called wake-up radios. In this talk I will describe a problem that arose during the standardization of wake-up radio technology. The problem is related to the spectral characteristics of the wake-up signals, and I will explain how it can be solved by means of randomization techniques. No previous knowledge of wireless communications is assumed, and all the necessary background will be explained. Although the talk is essentially about applied Fourier analysis, it touches upon softer subjects that are highly relevant in today's economy, namely innovation, inventiveness and standardization.
We will discuss the decoupling theorems proved by Bourgain and Demeter in 2014. Then we will talk about the refined decoupling inequality which was applied in the recent progress on Falconer distance problem (joint with Guth, Iosevich and Ou).
The neural crest offers a unique model system to study cell migration mechanisms during vertebrate embryogenesis since migrating cells are accessible to in vivo time-lapse imaging and manipulation. We recently discovered by single cell profiling that lead cells of neural crest migratory streams express a distinct set of genes. By combining agent-based modeling and experiment, we are testing the function of novel genes for their role in cell invasion. I will discuss the function of two of these genes in the context of our current cell-induced gradient model of neural crest cell migration.
STORM is a super-resolution technique that uses photoswitchable fluorophores to achieve resolutions at or below 20nm. A downside of STORM is the possibility of recording several blinks from one fluorophore, affecting the estimation of the number of molecules detected in the image. I constructed a mathematical model to identify unique fluorophores in STORM images by independently using the localization and the time series of the observations. The temporal sequence is described with a Markov chain approach and their spatial distribution with a Gaussian mixture model. To estimate the parameter values, I implemented a maximum likelihood procedure which requires a mixed optimization. I have tested my protocol in simulated data and I will use it to improve STORM images of B-cell surface receptors. B-cell receptors distribution on the membrane has been related to B-cell activation. This model will enhance a microscopy technique that is already widely used in biological applications and will allow to deeper analyze immune cells signaling.
The Grassmannian manifold of k-planes in \mathbb{R}^n has a group action if \mathbb{R}^n is taken to be a real representation of the group. When the group is C_2, the Schubert cell construction of the Grassmannian generalizes to an equivariant representation-cell structure. However, this generalization is not unique; an identification of representation with \mathbb{R}^n must be chosen.
I am interested in computing the RO(C_2)-graded Bredon cohomology of these spaces. Although a theorem of Kronholm dictates that this must be free, determining the degrees of the generators is nontrivial. The ambiguity introduced by the choice mentioned above turns out to be an asset for this task. Using a computation by Dan Dugger of the cohomology of an infinite equivariant Grassmannian, and some theorems about equivariant flag manifolds, I will present a way to succeed in finding the cohomologies of several infinite families of finite-dimensional equivariant Grassmannians.
After reviewing old work with Teitelbaum, in which we constructed the character variety X of the additive group o_L in a finite extension L/Q_p and established the Fourier isomorphism for the distribution algebra of o_L, I will briefly report on more recent work with Berger and Xie, in which we establish the theory of (\varphi_L,\Gamma_L)-modules over X and relate it to Galois representations. If time permits I will discuss an ongoing project with Venjakob. Our goal is to use this theory over X for Iwasawa theory.
If E is a compact set of Hausdorff dimension s > 1 on the plane, then its distance set was conjectured (by Falconer 1985) to have positive Lebesgue measure. Joint with Guth, Iosevich and Ou, we show that dim E > 5/4 is a sufficient condition. This improves upon Wolff's result for dim E > 4/3.
Note for Attendees
Refreshments will be served at 2:30 p.m. in ESB 4133 (Lounge).
In this talk I will discuss joint work with J. Achter and C. Vial showing that the image of the Abel--Jacobi map on algebraically trivial cycles descends to the field of definition for smooth projective varieties defined over subfields of the complex numbers. The main focus will be on applications to topics such as: descending cohomology geometrically, a conjecture of Orlov regarding the derived category and Hodge theory, and motivated admissible normal functions.
One of the goals of quantum information theory is to understand quantum systems from the standpoint of computational complexity. How difficult is it to compute fundamental properties of a quantum system or simulate a particular system over time? Physicists have been using computers for decades to understand various aspects of quantum systems, but these methods are typically heuristic and achieve success on only limited classes of systems. This talk will give an overview of recent developments in the effort to understand these problems from a formal complexity-theoretic point of view. In particular, one of the most basic properties of a system is its lowest energy state or ground state. I will survey results on the complexity of ground states and the computational resources required to compute them. I will also discuss heuristics to find ground states on more near-term quantum computers.
Note for Attendees
Please join the reception immediately preceding the talk (same venue), 3:30-4:00. Send email to Anna Eberhard <anna.eberhard@ubc.ca> if you would like to meet with Professor Irani on that Monday.
The standard iterative methods for solving linear and nonlinear systems of equations are all synchronous, meaning that in the parallel execution of these methods where some processors may complete an iteration before other processors (for example, due to load imbalance), the fastest processors must wait for the slowest processors before continuing to the next iteration. This talk will discuss parallel iterative methods that operate asynchronously, meaning that the processors never wait for each other, but instead proceed using whatever iterate values are already available from other processors. Processor idle time is thus eliminated, but questions arise about the convergence of these methods. Asynchronous iterative methods will be introduced using simple fixed-point iterative methods for linear systems, before discussing asynchronous versions of rapidly converging methods, in particular, optimized Schwarz and multigrid methods.
Cutting plane techniques are key to solving large scale optimization problems with mixed-integer variables. Modern approaches to cutting plane theory shows that the concept of sublinearity is a unifying way to organize these ideas. This leads to a rich interplay of ideas between convex analysis and geometry, geometry of numbers and functional analysis. We will survey this modern viewpoint of cutting plane theory. No background in mixed-integer optimization or convex analysis will be assumed.
Note for Attendees
Please join the reception immediately preceding the talk (same venue), 3:30-4:00.
Send email to Anna Eberhard <anna.eberhard@ubc.ca> if you would like to meet with Professor Basu during his visit to UBC, Tuesday to Thursday.
We investigate the scaling limit of the range (the set of visited vertices) for a general class of critical lattice models, starting from a single initial particle at the origin. Conditions are given on the random sets and an associated ``ancestral relation" under which, conditional on long-term survival, the rescaled ranges converge weakly to the range of super-Brownian motion as random sets. These hypotheses also give precise asymptotics for the limiting behaviour of the probability of exiting a large ball. Applications include voter models, contact processes, oriented percolation and lattice trees. This is joint work with Mark Holmes and also features work of Akira Sakai and Gord Slade.
Endoscopy is an important case of the Langlands functoriality conjecture, which describes a transfer of automorphic representations on reductive groups that has broad implications for number theory. On the other hand, p-adic automorphic forms are refinements of classical automorphic forms, and their relation to functoriality remains unclear. Eigenvarieties are rigid analytic objects that parametrise p-adic automorphic forms. In this talk, I will introduce these ideas and discuss a generalisation to GSp(4) of a recent result of Judith Ludwig relating the endoscopy of SL(2) to the geometry of the associated eigenvariety. This is joint work with Baskar Balasubramanyam.
It is well-known that a closed convex set C containing the origin in its interior can be represented as the 1-sublevel set of its gauge function. If the set C is compact, then the gauge is the unique sublinear function whose 1-sublevel coincides with C. However, if C is not compact, there can be multiple different sublinear functions whose 1-sublevels coincide with C. We call any such function a sublinear representation of C. It is not hard to see that the gauge of C is the largest sublinear representation of C, with respect to pointwise dominance. We show that there is a unique smallest sublinear representation f^ of C, i.e., f <= f for any other sublinear representation f of C. The gauge, which is the largest sublinear representation of C, is well-known to be equal to the support function of the polar of C. We associate the notion of a “prepolar” with other sublinear representations and show that the geometric analog of the smallest sublinear representation is the concept of the smallest “prepolar”, with respect to set inclusion. This smallest “prepolar” has an explicit description, just like the classical polar.
Let A be a finite-dimensional algebra. A fundamental theorem of Drozd shows that the complexity of the representation theory of A belongs to exactly one of three rather distinct classes, called finite, tame or wild representation type. I will explain how the notion of essential dimension determines the representation type of A. I will go further and define new numerical invariants of A that refine the representation type of A. I will then determine these invariants explicitly in the special case of quiver algebras.
Note for Attendees
Refreshments will be served at 2:30 p.m. in ESB 4133 (Lounge).
O. Forster proved that over a ring R of Krull dimension d a finite module M of rank at most n can be generated by n+d elements. Generalizing this in great measure U. First and Z. Reichstein showed that any finite R-algebra A can be generated by n+d elements if each A\otimes_R k(\mathfrak{p}), for \mathfrak{p}\in \mathrm{MaxSpec}(R), is generated by n elements. It is natural to ask if the upper bounds can be improved. For modules over rings R. Swan produced examples to match the upper bound. Recently B. Williams obtained weaker lower bounds in the context of Azumaya algebras. In this paper we investigate this question for étale algebras. We show that the upper bound is indeed sharp. Our main result is a construction of universal varieties for degree-2 étale algebras equipped with a set of r generators and explicit examples realizing the upper bound of First & Reichstein. This is joint work with Ben Williams.
The conjugate gradient algorithm (CG) is a widely used iterative method for solving large symmetric positive definite linear systems Ax=b. Beginning in the 1980's (and, even earlier, in the thesis of Paige dating back to 1971), a great deal of research has been aimed at explaining the behavior of the Lanczos and conjugate gradient algorithms in finite precision arithmetic. The algorithms do not behave the way exact arithmetic theory predicts, yet they are widely used and often deliver impressive results. Explanations were derived, under the assumption that the finite precision computation satisfied certain properties that could be shown to hold for particular (now standard) implementations. Names associated with this early work include Paige, Druskin and Knizhnerman, Greenbaum and Strakos.
With the advent of parallel computing, different variants of the conjugate gradient algorithm were proposed to take better advantage of parallelism; see, e.g, Chronopoulas and Gear, On the efficient implementation of preconditioned s-step conjugate gradient methods on multiprocessors with memory hierarchy, Parallel Comput. 11 (1989), pp. 37-53, and Ghysels and Vanroose, Hiding global synchronization latency in the preconditioned conjugate gradient algorithm, Parallel Comput. 40 (2014), pp. 224-238.
All of these are mathematically equivalent to the original Hestenes and Stiefel algorithm, but none have been shown to satisfy the hypotheses used in the analysis of the 1980's. Does this mean that they perform poorly? Or does it simply mean that the assumptions of the early analysis were sufficient but not necessary conditions for good behavior? Lower precision arithmetic is now becoming more popular. What will be the effect of this on CG codes? I will discuss these issues and what insight might be gained from the analysis in order to devise stable and efficient finite precision implementations.
Epstein-Barr virus (EBV) is a ubiquitous infection worldwide and is associated with the development of several kinds of cancers. Rates of EBV replication and disease are higher in individuals who are co-infected with HIV-1; however, the causes of this remain unknown. Here, we developed a mathematical model to describe the dynamics of EBV infection within the tonsils and analyzed oral EBV shedding data in a cohort of adults from Uganda to predict the role of HIV-1 in determining infection severity.
We shall give an overview of what Khovanov cohomology and its relatives have told us about knot concordance over the past 15 years. No knowledge required. Includes joint work with Lukas Lewark and Peter Feller.
The phi^4 model is a popular toy model in the classical statistical mechanics of magnets, and in quantum field theory. It has a phase transition, and we discuss the behavior of thermodynamic quantities as its critical point is approached. One of the major successes of the renormalization group technique has been to explain why this model features logarithmic corrections to the scaling predictions for the blow up of these quantities. We present some old and some new results that have been obtained by this technique, and describe the key ideas of the method.
In this talk, I will give an introduction to the Picard curves and how we can compute important invariants. In addition, I will describe a new algorithm of computing Picard curves with given set of bad reduction solving S-unit equations. This is joint work with Irene Bouw, Jeroen Sijsling and Stefan Wewers.
Knots are ubiquitous in our life, and their peculiarities certainly caught the eyes of mathematicians, who have been keen to classify them over the last two centuries. Hence it’d be my fortune to present knot theory, catering to the general audience. We’ll first take a glimpse of the history through polynomial means of classifying knots, and hopefully touch some of the entertaining aspects via applications to arts and sciences. Then proceeding to the second part of the talk with slight algebraic topology involved, we’ll have a taste of classifying the knot by the fundamental group of its complement, namely the knot group.
Motivated by cotyledon patterns in conifer embryos, we study pattern emergence in solutions of systems reaction-diffusion partial differential equations on evolving curved surfaces. We start by reviewing results on a static domain. Then, we go over the reduction to a non-autonomous normal form in the case of a flattening spherical cap. Finally we show numerical results, using the closest point method, for a variety of different surfaces.
Note for Attendees
Refreshments will be served at 2:30 p.m. in ESB 4133 (Lounge).
This talk discusses a new and unified approach to three main themes in Harmonic Analysis: (I) The Linear Hilbert Transform and Maximal Operator along variable curves; (II) Carleson Type operators in the presence of curvature; (III) The bilinear Hilbert transform and maximal operator along variable curves.
Our talk will focus on the following two aspects (A): a presentation of the historical evolution of our three themes with a special accent on their powerful inter-connections, and (B): a description of our most recent results, and if time permits, an exposition of the main ideas and methods.
I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a result on concentration of vorticity at the boundary for symmetric flows and the justification of Prandtl approximation for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains, quantifying the effect of curvature on the pressure correction.
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then we discuss our main result which addresses the existence and behavior of convex non-compact inverse mean curvature flow.
The key ingredient is a priori interior in time estimate on inverse mean curvature written in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.
Single particle tracking is a powerful tool to study the mobility of molecules in the cell membrane. The most common approaches in analyzing these kinds of data are mean squared displacement and analyses with one or more hidden Markov states. However, in most experiments, positional measurements contain systematic and random errors, and to achieve proper fits, we must take these errors into account. In this work, we develop a hidden Markov model with multiple diffusive states. Our goal is to estimate the diffusion coefficients and transition probabilities between different states incorporating uncertainty due to measurement error in a rational way. Moreover, we also develop a Bayesian nonparametrics framework to estimate the number of states in the hidden Markov model, and then using information from the data we find the optimal Markov Model that describes that data. We test our methods using simulated data and present results using particle tracks obtained from surface receptor molecules on B cells.
The first order language on graphs consists of sentences, or graph properties, that are expressible using the relations of vertex equality (x = y) and vertex adjacency (x \sim y). In their 1988 paper, Shelah and Spencer showed that when 0 < \alpha < 1 is an irrational number, the Erdös-Rényi random graph G(n, n^{-\alpha}) satisfies the zero-one law for the first order language on graphs, i.e. for any first order sentence A, the limit \lim_{n \rightarrow \infty} \mathbb{P}\left[G(n, n^{-\alpha}) \text{ satisfies } A\right] exists and equals either 0 or 1.
It has been a long-standing question in mathematical logic as to whether the existential fragment of a given logic is less expressive compared to the full logic. In particular, one may consider whether, for certain edge probability functions, the random graph G(n, p(n)) satisfies the zero-one law for the existential fragment but not for the entire logic. In his 2012 paper, Zhukovskii showed that the minimum 0 < \alpha < 1 such that G(n, n^{-\alpha}) satisfies the zero-one law for all first order sentences of quantifier depth at most k, is \frac{1}{k-2}. At first glance, it seems that this bound can be moved significantly for existential first order sentences of quantifier depth at most k, but this is not true, since we show that the minimum positive \alpha such that the zero-one law fails for the existential fragment is \frac{1}{k - 2 - O(k^{-2})}.
Joint with Maksim Zhukovskii, Department of Discrete Mathematics, Moscow Institute of Physics and Technology.
I will describe how to use the recent theory of prismatic cohomology of Bhatt--Morrow--Scholze to say something about the de Rham cohomology of classifying spaces of algebraic groups such as PGL_p in characteristic p. This leads to interesting new phenomena, including new obstruction classes to period=index for p-torsion classes in characteristic p.
I will present a recent proof of the Multiplicity One Conjecture in Min-max theory. This conjecture was raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one. As direct corollaries, it implies the generalized Yau's conjecture for such manifolds with positive Ricci curvature, which says that there exist a sequence of minimal hypersurfaces with area tending to infinity, and the Weighted Morse Index Bound Conjecture by Marques and Neves.
In many large-scale inverse problems and optimization problems, the objective function (to be minimized) is composed of many terms, each term or group of terms requiring an expensive simulation/computation. In the cases discussed here, we need to solve a large linear system. As a result, in a realistic setting each optimization step may require hundreds or thousands of large linear system solves, which creates an overwhelming computational burden.
In an inverse problem, we typically try to infer parameters describing (the interior of) a medium from measurements on the surface. The optimization minimizes the misfit, the difference between actual measurements and predicted measurements computed using the underlying physical model and a given set of parameters. In topology optimization, we try to find the distribution of a limited amount of material(s) to define a structure that has, for example, maximum stiffness under a large number of loading conditions.
We discuss several techniques that will make these optimizations much cheaper. Model reduction can drastically reduce the size of the systems to be solved, but typically does not reduce the number of solves required. In addition, computing the reduced model itself introduces a substantial cost, though typically much less then for the overall optimization. Randomization uses stochastic techniques to estimate function values, gradients, and possibly (approximate) Hessians with a greatly reduced number of linear solves but of the original size. Combinations of these methods promise to be very efficient.
Identifying and visualizing meaningful clusters within high-dimensional (HD) data is an important but challenging problem with applications to molecular biology and biomedicine. For example, mass cytometry (CyTOF) is a high-throughput single-cell technology that can quantify the abundance of >30 proteins simultaneously in single cells. CyTOF is commonly used to to investigate phenotypic heterogeneity in tumours, but this demands identification of biologically meaningful clusters from CyTOF data.
The "curse of dimensionality" poses computational challenges when analyzing HD data. Another major difficulty common to most clustering algorithms is choosing optimal values for input parameters such as the number of clusters, K. I will present a new measure of similarity between clusters in HD data using local dimensionality reduction followed by triangulation of alpha shapes. Using this new approach, which I call ASTRICS, HD data can initially be over-clustered using an existing clustering algorithm with a very conservative choice of K. ASTRICS then generates a K x K similarity matrix, which can also be interpreted as a weighted graph, for the K clusters. In turn, this can be used as input for any similarity- or graph-based clustering algorithm, and force-directed layout of the graph can be used to visualize the initial K clusters in two or three dimensions.
The introduction of ASTRICS as a fully automated, intermediate step in clustering or visualization of HD data alleviates some of the difficulties of parameter selection. Some community detection (i.e. clustering) algorithms for graphs do not require any input parameters to be specified by the user. Otherwise, the visualization afforded by ASTRICS can be used to guide parameter selection for the final clustering step. In this talk, I will demonstrate application of ASTRICS to clustering and visualizing CyTOF data. I will also illustrate the broader utility of ASTRICS beyond biology by applying it to the popular MNIST digital image dataset.
This is a preliminary report on an ongoing project with Jack Morava on the braid groups and categorification.
The braid groups can be seen as the fundamental group of the space of monic square-free complex polynomials.
I will discuss the role of Sylvester's discriminant to understand the strange fact that the braid groups abelianize
to the integers, while the subgroup of pure braids has a much larger abelianization.
We give a nearly efficient *deterministic* algorithm for approximately counting the number of 0/1-coordinate-points in high-dimensional polytopes. The two main technical tools are: a new multidimensional Berry--Eseen theorem under limited independence; and, a new multidimensional Littlewood--Offord theorem for polytopes.
Joint work with Rocco Servedio (Columbia) and Li-Yang Tan (Stanford)
One of the main themes in Iwasawa theory is that various L-functions that occur in number theory (e.g., Dirichlet L-functions, the Hasse-Weil L-function for an elliptic curve, and L-functions attached to modular forms) have p-adic analogues. These so-called p-adic L-functions are analytic functions of a p-adic variable. In most case, these functions have only finitely many zeros. The most interesting thing of all is that these zeros have a purely algebraic interpretation (proven in many cases, conjectured in others). This talk will be an introduction to the above circle of ideas centered around joint work with Nike Vatsal concerning p-adic analogues of Artin L-functions.
As climate warms, different phenotypes, such as different flowering time in plants or breeding date in birds, are favored by natural selection. To persist, species must therefore change their geographical distribution to track the climate to which they were adapted to, and/or their phenotypic distribution to adapt to new climates. Quantitative genetics models describing joint changes in phenotypic and geographical distributions have been developed in the nineties to better understand the challenges faced by species under climate change. We have built on this work by examining how the life cycle of species affects and jointly evolves with these dynamics: I will present a few examples of work in progress where collaboration between mathematicians and evolutionary biologists have led to new insights on how age-structure, mode of reproduction, mutation and dispersal affect the response of species to climate change.
Let H^(u) be the Hilbert transform along the parabola (t; ut^2).
For a set U of real numbers consider the maximal function
H^U f = sup {H^(u) f : u in U}.
There are sharp results for the dependence of the L^p operator norm on U when 2 < p < 1, and close to sharp results when 1 < p <= 2.
The results are proved for families of Hilbert transforms along more general non-flat homogeneous curves.
Joint work with Shaoming Guo, Joris Roos and Po-Lam Yung.
Computational modelling is now firmly established as part of modern engineering design. This is especially true in fields such as aviation and turbomachinery, where the strive for increased aeronautical efficiency means that engineers are increasingly in need of high-fidelity simulations to understand the influence of even small design changes on overall aerodynamic performance. A step towards this is to consider moving away from present industry-standard techniques such as RANS, which is implemented at the heart of most commerical software, towards transient large eddy simulations (LES), which offer increased fidelity with an associated increase in computational cost. To enable LES simulations in reasonable timescales therefore requires an increase in solver speed that, to date, commercial software has not been able to provide, despite dramatic increases in the calculation power of modern computational hardware.
In this talk, I will discuss how high-order methods, having been originally developed in academia some 30 years ago, can be used to overcome these issues and deliver high-fidelity LES simulations at reasonable computational cost for high-end engineering problems. I will broadly outline the formulation and capabilities of high-order methods and the spectral/hp element method in particular, as well as highlighting some of the many challenges that need to be overcome in this setting, such as stabilisation and mesh generation. One of the key issues in the use of high-order methods is their complexity in terms of their implementation, both in CFD and more generally for problems outside of fluids. I will therefore discuss their implementation in Nektar++ (www.nektar.info), an open-source framework for the spectral/hp element method, and show how this has been leveraged to perform simulations of complex, industrial geometries.
Bio: Dr. David Moxey is a senior lecturer in Engineering at the University of Exeter, UK, having previously been appointed as lecturer in 2017. Previously, he worked as a research fellow in the Department of Aeronautics at Imperial College London, and holds a Ph.D. and M.Math. in mathematics from the University of Warwick. His research interests lie broadly at the intersection of computational engineering, applied mathematics and high performance computing, with a particular focus on computational fluid dynamics and high-order numerical methods. From a fluids perspective, he is interested in better understanding the transition to turbulence, and how this impacts on real-world problems in the aeronautics industry, by developing efficient numerical tools for high-fidelity CFD simulations. To achieve this, he is a leader of the Nektar++ project: an efficient, robust and massively parallel framework for the high-order spectral element method. Together with modern computing technology, his research aims to show how the spectral element method can overcome current bottlenecks in present-day numerical methods.
Abstract: Guidobaldo del Monte (1545-1647), a patron and friend of Galileo Galilei, believed he had witnessed the creation of something out of nothing when he established mathematically that zero equals one. He thereby thought that he had proved the existence of God! Although I daren't be so bold with my claims, I am willing to prove instead that one equals two. And, moreover, just to convince you that I am right I will do so multiple times over, drawing upon a wide spectrum of mathematical techniques: algebra and arithmetic, probability and mechanics, pure thought and physical action! Will you be able to find fault with any of my "proofs?"
I will discuss the following question: Suppose f and g are real bounded measurable functions with the property that i[f (P ), g(Q)] is a non-negative operator. Here P = −id/dx and Q is multiplication by x in L2(R). What can be said about f and g? This is joint work with Tom Kriete.
When a group acts faithfully by orientation-preserving homeomorphisms on S^1, one can sometimes use the action on S^1 to prove the existence of a faithful order-preserving action by homeomorphisms on the real line. This can be reworded in algebraic terms by using circular orderings and left-orderings of groups: a circularly orderable group may secretly admit a left-ordering, though the existence of such an ordering may not be apparent.
In this talk I will review some classical results that use cohomology and circular orderings of a group to detect the existence of left-orderings. I'll also present one new technique for determining when a circularly orderable group admits a left-ordering. As a bonus, when a group is circularly orderable but NOT left-orderable, our new approach determines a subset of the natural numbers that precisely encodes the obstruction to the existence of a left-ordering. This is joint work with Ty Ghaswala and Jason Bell.
A primary goal of evolution biology is to understand the mechanisms that generate and shape the vast diversity of life. From stable polymorphisms at susceptibility loci to the maintenance of sexual reproduction, pathogens are thought to play an import role in the maintenance of genetic diversity of their hosts. Indeed as evidenced by the death toll of the Plague in 14th century Europe to the decimation of African undulates by Rinderpest, infectious pathogens can exert strong selective pressures on their hosts. Using methods from Epidemiology we modelled coevolution between hosts and their infectious pathogens. We explore how coevolution, epidemiology, and stochasticity shape the genetic diversity of hosts.
There is a rich interplay between the degree of regularity of a group action on the circle and the allowable algebraic structure of the group. I will outline some highlights of this theory, culminating in a construction due to Kim and myself of groups of every possible critical regularity $\alpha \in [1,\infty)$.
In the neighborhood of a singular point, a real analytic curve in the plane consists of a finite number of branches. Each of these branches intersects a small circle around the singular point in two points. The main purpose of this talk is to give an complete description of those analytic chord diagrams. On our way, we shall meet some interesting concepts from computer science, graph theory and operads.
Note for Attendees
Reception at 2:30pm | Lecture at 3:00pm - 4:30 pm.
Event Registration: There is no fee for attending these talks, however, for logistical purposes, please register for any or one of these events through the site,
https://www.eventbrite.ca/e/etienne-ghys-vancouver-tour-tickets-58608339120
A common problem of atomistic materials modelling is to determine properties of crystalline defects, such as structure, energetics, or mobility, from which meso-scopic material properties or coarse-grained models (e.g., Kinetic Monte-Carlo, Discrete Dislocation Dynamics, Griffith-type fracture laws) can be derived. This leads to a range of atomistic (discrete) models for crystals and defects. In this lecture I will give an overview over different popular multiscale schemes for approximating these models, and in particular how a numerical analysis structure can be imposed on them.
Note for Attendees
Refreshments will be served at 3:15 p.m. in MATH 125 Lounge.
Accurate molecular simulation requires computationally expensive quantum chemistry models that makes simulating complex material phenomena or large molecules intractable. The past decade has seen a revival of interatomic potentials (IPs), which are computationally cheap but traditionally inaccurate surrogate models, re-casting their construction as a “machine learning” problem.
In the first part of the talk I will explain how this problem can be formalised as an unusual infinite-dimensional approximation problem, with many structures that can be exploited to make it tractable. In particular, our initial results indicate that the curse of dimensionality can be almost completely overcome.
In the second part of the talk I will introduce a practical regression scheme which (1) realises such an approximation and (2) at the same time aims to resolves a long-standing challenge to construct high-dimensional approximations for IPs that "extrapolate well" outside of a limited training set.
Abstract: After briefly reviewing a number of experiments on dewetting and evaporating thin films/drops of simple and complex liquids, I introduce the concept of a gradient dynamics description of the evolution of interface-dominated films and drops on solid substrates. First, the case of films/drops of simple non-volatile liquid is discussed, and illustrated with results on droplet patterns and sliding droplets. As a further example, the diffusion equation is formulated as a gradient dynamics. The obtained elements are combined into a thermodynamically consistent gradient dynamics formulation for films of mixtures and surfactant suspensions [1,2].
Next, such models are employed to investigate the out-of-equilibrium process of the deposition of line patterns at receding contact lines for evaporatively dewetting solutions/suspensions [3] and in Langmuir Blodgett transfer [4]. Finally, I discuss how to combine the presented thin-film dynamics with bioactive elements to obtain models for the osmotic spreading of biofilms growing on moist agars [5]. I conclude with a summary and outlook.
[1] U. Thiele, D. Todorova, H. Lopez, Phys. Rev. Lett. 111, 117801 (2013)
[2] Thiele, U.; Archer, A.Pismen, L., Phys. Rev. Fluids 1, 083903 (2016).
[3] U. Thiele, Adv. Colloid Interface Sci. 206, 399-413 (2014). [4] M.H. Köpf and U. Thiele, Nonlinearity 27, 2711-2734 (2014).
[5] Trinschek, S.; John, K.; Lecuyer, S., Thiele, U., Phys. Rev. Lett. 119, 078003 (2017).
[6] Trinschek, S.; John, K. Thiele, U., Soft Matter 14, 4464-4476 (2018)
Bio: Uwe Thiele is a Professor of Theoretical Physics in Münster, Germany. After his PhD (Dresden) he worked as research associate in Madrid, Berkeley, Dresden and Augsburg, before becoming faculty first at Loughborough University and later in Münster. There he is a director of the Institute of Theoretical Physics and also speaker of the Center of Nonlinear Science (CeNoS) of the University of Münster.
Maryam Mirzakhani, born in Tehran on May 12 1977, was the first and currently only woman to be awarded the Fields Medal, the most prestigious award in mathematics. In honour of her achievements and to recognize other ‘Women of the Mathematical Sciences,” PIMS is proud to host the inaugural Maryam Mirzakhani Day at UBC.
This event will consist of a reception, four short talks on 'Women of the Mathematical Sciences' by faculty members from UBC, and a panel Q&A session where the audience can interact with the four speakers and ask questions on a number of topics related to women in STEM.
Program
Refreshments
Part 1: Tribute to Maryam Mirzakhani
Part 2: Panel Session: four short talks on inspiring 'Women of the Mathematical Sciences'
Part 3: Open discussion and Q&A
Panel Speakers
Anne Condon – Computer Science
Agnes d’Entremont – Mechanical Engineering
Leah Edelstein-Keshet – Mathematics
Sujatha Ramdorai – Mathematics
Note for Attendees
The event is free but we request that attendees register so we can ensure we have sufficient refreshments and seating.Please registerhere.
Cellular organization is regulated by a complex network of interactions between cytoskeletal regulators and the cytoskeleton itself. Here I will utilize mathematical modeling discuss how these complex interactions can give rise to a range of morphological cellular behaviors. I’ll discuss 1) how Rho GTPase dynamics give rise to a diverse array of cellular morphologies observed experimentally. 2) How feedback interactions between the cytoskeleton and the proteins responsible for its remodeling influence the dynamics of actin remodeling. 3) How interactions between the extra cellular matrix and Rho GTPase signaling modulate cellular migration. And 4) how feedbacks between GTPase signaling and membrane tension provide a mechanical means for cells to adapt to high concentration signaling environments. Together these investigations paint a picture where Rho GTPases form a signaling hub of cytoskeletal regulation and feedbacks between them and cellular remodeling lead to a diverse array of cellular dynamics.
Tissue growth requires cells of various types to organise themselves into the appropriate patterns and structures to produce viable, functional tissue. Similar processes occur in tissue repair (e.g. wound healing) or in biofilms (communities of bacteria or yeast cells). Understanding how this organisation is coordinated is therefore an important basic problem in biology. I will present a brief overview of recent work on pattern formation in vitro (in biofilms, and in interacting cell populations) using both continuum and individual-cell based models. A particular focus of our work has been on using image processing methods and spatial statistics (such as pair-correlation functions) to quantify both experimental data and model outputs, to facilitate comparisons between the two.
In an article published in 2009, David Benson described, for a finite group G, the mod p homology of the space \Omega(BG^{\wedge}_p) -- the loop space of the p-completion of BG -- in purely algebraic terms. In joint work with Carlos Broto and Ran Levi, we have tried to better understand Benson's result by generalizing it. Among other things, we showed that when \mathcal{C} is a small category, |\mathcal{C}| its geometric realization, R a commutative ring, and |\mathcal{C}|_R^+ a plus construction of |\mathcal{C}| with respect to homology with coefficients in R, then H_*(\Omega(|\mathcal{C}|_R^+);R) is the homology of any chain complex of projective R\mathcal{C}-modules that satisfy certain conditions. Benson's theorem is then the special case where \mathcal{C} is the category associated to a finite group G and R=\mathbb{F}_p, so that p-completion appears as a special case of the plus construction.
First, we review recent experiments on and biophysical modelling approaches for the early stages of osmotically spreading biofilms at an agar-air interface (e.g. [1,2]). Doing so, we highlight important experimental features and successes/limitations of the various models. In particular, it is pointed out that modelling has paid little attention to the physico-chemical interactions of the film and the agar (adhesion, wettability, etc) [3]. We propose to incorporate these surface forces in the form of a wetting potential that accounts for finite contact angles at the three-phase contact line where biofilm, agar and gas phase meet.
Second, we establish the basic modelling principles of thin-film hydrodynamics for the dynamics of free surface films of mixtures and suspensions where all aspects of capillarity and wettability may, in principle, depend on the local film composition. We argue that in a passive (non-bioactive) limit one has to be able to write all such models in the form of a gradient dynamics. The passive model is then extended by bioactive terms like bacterial proliferation and matrix or biosurfactant production to reach a set of simplified models for the growth dynamics of biofilms [4].
Finally, we employ such models to investigate two phenomena: (i) It is shown that surface forces determine whether a biofilm can expand laterally over a substrate. In particular, we discuss modelling results and experimental evidence related to a transition between continuous and arrested spreading for Bacillus subtilis biofilms [5]. In the case of arrested spreading, the lateral expansion of the biofilm is confined, albeit the colony is biologically active. However, a small reduction in the surface tension of the biofilm is sufficient to induce spreading. (ii) As second phenomenon we discuss the relation of fingering instabilities of an advancing biofilm edge and the production of biosurfactant within the biofilm. As a result we distinguish four dynamical (morphological) modes of biofilm growth [6]. We conclude with an outlook.
[1] Fauvart, M. et al., Surface tension gradient control of bacterial swarming in colonies of Pseudomonas aeruginosa, Soft Matter, 2012, 8, 70-76.
[2] Seminara, A. et al., Osmotic spreading of Bacillus subtilis biofilms driven by an extracellular matrix, Proc. Natl. Acad. Sci. U. S. A., 2012, 109, 1116-1121.
[3] Tuson, H., Weibel, D. Bacteria-surface interactions, Soft Matter, 2013, 9, 4368-4380.
[4] Trinschek, S.; John, K.; Thiele, U., From a thin film model for passive suspensions towards the description of osmotic biofilm spreading, AIMS Materials Science, 2016, 3, 1138-1159.
[5] Trinschek, S.; John, K.; Lecuyer, S.; Thiele, U., Continuous vs. arrested spreading of biofilms at solid-gas interfaces - the role of surface forces, Phys. Rev. Lett., 2017, 119, 078003.
[6] Trinschek, S.; John, K.; Thiele, U.; Modelling of surfactant-driven front instabilities in spreading bacterial colonies Soft Matter, 2018, 14, 4464-4476.
Abstract: The speaker will give an introduction to predator prey math modeling and then bridge it with recent research on animal territory.
About the Niven Lectures: Ivan Niven was a famous number theorist and expositor; his textbooks won numerous awards, have been translated into many languages and are widely used to this day. Niven was born in Vancouver in 1915, earned his Bachelor's and Master's degrees at UBC in 1934 and 1936 and his Ph.D. at the University of Chicago in 1938. He was a faculty member at the University of Oregon from 1947 until his retirement in 1982. The annual Niven Lecture Series, held at UBC since 2005, is funded in part through a generous bequest from Ivan and Betty Niven to the UBC Mathematics Department.
Note for Attendees
The Mathematics Grad Reception will be held in the MATH 125/126 Lounge prior to the Niven Lecture.
ESB 4127 This seminar start time has been revised to 3:15pm.
Wed 29 May 2019, 3:15pm-4:15pm
Abstract
Cellular movement plays important roles in many (patho)physiological processes, such as immune cell response, growth of neuronal axons and cancer. The regulation of this movement depends on the interaction of several key proteins implicated in the development of cellular polarity (consisting of a front and a back) and the formation of protein complexes called adhesions. Adhesions anchor the cell to its substrate, allowing it to migrate. In CHO cells, three classes of adhesion can be identified based on size and dynamic properties: nascent adhesions, focal complexes and focal adhesions. When cells extends forward at the front, nascent adhesions assemble and anchor the leading edge to the substrate, while focal adhesions at the back disassemble, allowing detachment, retraction and forward movement. The dynamics of these processes are controlled by a number of regulatory factors, occurring on both cell-wide and adhesion-level scales. The coordination of these regulatory factors is complex, but insights into their dynamics can be gained from the use of mathematical/biophysical modeling techniques which integrate many of these components together. In this talk, I will present our recently developed molecularly-explicit and mechanosensitive models of cell polarity and adhesion dynamics to explore how local regulation of key adhesion proteins (including paxillin, rho family of GTPases and integrin) produce cell-wide polarization and nascent adhesion assembly/disassembly. The dynamics associated with various parameter regimes will be presented and insights into the mechanisms regulating adhesion dynamics will be provided.
Carlsson conjectured that if a finite CW complex admits a free action by an elementary abelian p-group G of rank n, then the sum of its mod-p Betti numbers is at least 2^n. In 2017, Iyengar and Walker constructed equivariant chain complexes that are counterexamples to an algebraic version of this conjecture. Their work raised the question if these chain complexes can be realized topologically by free G-spaces to produce counterexamples to Carlsson’s conjecture. In this talk, I will explain multiplicative properties of the spectral sequence obtained by filtering the mod-p cochains of a space with a free p-group action by powers of the augmentation ideal and deduce that the counterexamples can not be realized topologically. This is joint work with Henrik Rüping.
The classifying space of the nth projective unitary group, BPUn, is a fundamental mathematical object. However, unlike its close cousins BUn, BSUn, etc., surprisingly little has been known to its cohomology. I will briefly introduce the roles of BPUn in the study of the topological period-index problem and anomalies in theoretical physics, and present recent progresses on the computation of its cohomology.
In the context of algebraic geometry, BPGLn, the classifying stack of the projective linear group PGLn, is an analog of BPUn, while the theory of Chow rings plays a similar role as singular cohomology. I will point out where the above computation pass to the Chow ring of BPGLn.
Quivers with potential appear naturally in the study of the deformation theory of objects in 3D Calabi-Yau categories, for example the deformation of vector bundles on 3D Calabi-Yau manifolds. They provide a deep link between geometry of Calabi-Yau manifolds to some aspects of representation theory, for example cluster algebras, quantum enveloping algebras, etc. In this talk, I will survey some recent progress in non commutative differential calculus of quivers with potentials, and show how this leads to new results in birational geometry and Donaldson-Thomas theory.
There is by now an extensive theory of rational Chow rings of moduli spaces of smooth curves. The integral version of these Chow rings is not as well understood. I will survey what is known. In the last part of the talk I will discuss the Chow ring of the stack of stable curves of genus 2, which has been recently calculated by Eric Larson. I will present a different approach to the calculation, which offers an interesting point of view on stack of stable curves of genus 2. This part is joint work with Andrea Di Lorenzo.
We know that large random combinatorial structures (e.g., graphs, tilings, Ising configurations) with given parameters tend to look alike. But what do they look like?
In the case of permutations, we can in some cases answer this question with the help of limit structures called “permutons,” together with a variational principle. We’ll present some examples, showing very nice apparent behavior and uncovering some intriguing contrasts with the case of graphs and their limit structures (graphons).
This is joint work with Rick Kenyon, Dan Kral and Charles Radin.
Consider the Monge-Kantorovich problem of transporting densities $\rho_0$ to $\rho_1$ on $\rr^d$ with a strictly convex cost function. A popular relaxation of the problem is the one-parameter family called the entropic cost problem. The entropic cost J_h, h>0, is significantly faster to compute and $h J_h$ is known to converge to the optimal transport cost as $h$ goes to zero. We will give an overview of various ideas in this field, including discrete approximations, gamma convergence and particle systems. Finally we will discuss Gaussian approximations to Schrodinger bridges as $h$ approaches zero. As a consequence we obtain ``gradient flows’’ of entropy even in cases where the cost function is not a metric.
Note for Attendees
* This is a joint event between probability and DG-MP-PDE.
Abstract: Computer (in silico) experiments in this area have given researchers invaluable insights and in some cases re-directed experimental research and theory. With mathematics and in silico experiments we will explore the experimentally observed photoreceptor death and rescue in retinal degeneration and provide a framework for future physiological investigations and potential ways to circumvent blindness.
Speaker Biography:
Dr. Erika T. Camacho grew up in East Los Angeles and was taught by Jaime Escalante at Garfield High School. She received her B.A. in Mathematics and Economics from Wellesley College. After earning her Ph.D. in applied mathematics at Cornell University, she spent a year as a researcher at Los Alamos National Laboratory. She then held a tenure-track faculty position at Loyola Marymount University before joining the faculty at ASU in 2007. She was the recipient of the 2014 Presidential Award for Excellence in Science, Mathematics and Engineering Mentoring (PAESMEM) and was only the 4th female mathematician to have been bestowed this honor. She is also the recipient of the 2019 AAAS mentor award. She was a 2013-2014 MLK Visiting Assistant Professor of Mathematics at MIT. She cofounded the Applied Mathematical Sciences Summer Institute (AMSSI) and co-directed other summer programs dedicated to the recruitment of undergraduate women, underrepresented minorities, and those that might not otherwise have the opportunity.
I will present a blow up construction - jointly with Kihyun Kim - on the self-dual Chern-Simons-Schr\"odinger equation (CSS), also known as a gauged nonlinear Schr\"odinger equation (NLS). CSS is $L^{2}$-critical, admits solitons, and has the psuedoconformal symmetry. These features are similar to the $L^{2}$-critical NLS. In this work, we consider pseudoconformal blow-up solutions under $m$-equivariance, $m\geq1$. Our result is threefold. Firstly, we construct a pseudoconformal blow-up solution $u$ with given asymptotic profile $z^{\ast}$:
as $t\to0^{-}$, where $Q(r)e^{im\theta}$ is a static solution. Secondly, we show that such blow-up solutions are unique in a suitable class. Lastly, yet most importantly, we exhibit an instability mechanism of $u$. We construct a continuous family of solutions $u^{(\eta)}$, $0\leq\eta\ll1$, such that $u^{(0)}=u$ and for $\eta>0$, $u^{(\eta)}$ is a global scattering solution. Moreover, we exhibit a rotational instability as $\eta\to0^{+}$: $u^{(\eta)}$ takes an abrupt spatial rotation by the angle
\[
\Big(\frac{m+1}{m}\Big)\pi
\]
on the time interval $|t|\lesssim\eta$.
We are inspired by works in the $L^{2}$-critical NLS. In the seminal work of Bourgain and Wang (1997), they constructed such pseudoconformal blow-up solutions. Merle, Rapha\"el, and Szeftel (2013) showed an instability of Bourgain-Wang solutions. Although CSS shares many features with NLS, there are essential differences and obstacles over NLS. Firstly, the soliton profile to CSS shows a slow polynomial decay $r^{-(m+2)}$. This causes many technical issues for small $m$. Secondly, due to the nonlocal nonlinearities, there are strong long-range interactions even between functions in far different scales. This leads to a nontrivial correction of our blow-up ansatz. Lastly, the instability mechanism of CSS is completely different from that of NLS. Here, the phase rotation is the main source of the instability. On the other hand, the self-dual structure of CSS is our sponsor to overcome these obstacles. We exploited the self-duality in many places such as the linearization, spectral properties, and construction of modified profiles.
In the talks, I will present background of the problem, main theorems, and outline of the proof with emphasis on heuristics of main features, such as the long- range interaction between blow up profile and asymptotic profile $z$, and rotational instability mechanism.
The three medians of any triangle intersect in its barycenter and dissect the triangle into six smaller "children" triangles. Let D_n be a Markov chain of triangles with D_n chosen uniformly among the children of D_{n-1}. P. Diaconis and L. Miclo (2011) show that almost surely the "flatness" of D_n, namely its maximal edge length divided by its minimal height length, converges to infinity exponentially fast. In the limit the vertices of D_n become collinear. D. Mannion (1988, 1990) and S. Volkov (2013) show the same results for different Markov chains of triangles D_n, where the vertices of D_n are either independently and uniformly chosen in the interior of D_{n-1} (Mannion) or independently and uniformly on each edge of D_{n-1} (Volkov). We formulate the above results as special cases of a more general theorem about certain random walks on the group of complex 3 x 3 matrices.
The classical random walk isomorphism theorems relate the square of a Gaussian free field to the local time of a corresponding random walk. These relations and their supersymmetric versions have been used in both directions: in the study of (non-Gaussian) spin systems and field theories using random walk techniques and in the study of self-interacting walks in terms of fields. I will present non-Gaussian analogues that relate hyperbolic sigma models to linearly reinforced random walks. As an application, we show that the vertex-reinforced jump process is recurrent in two dimensions.
This is joint work with Andrew Swan and Tyler Helmuth.
These are algebraic surfaces with the Betti numbers of the complex projective plane, and are called Q-homology projective planes. We describe recent progress in the study of such surfaces including smooth examples, the fake projective planes. We also discuss open questions on Montgomery-Yang problem.
Let K be a field, let t : K ŕ K be an automorphism of order 1 or 2. Let F denote the subfield of t-invariant elements in K. Then either K=F or K/F is a quadratic Galois extension. Given a central simple K-algebra A, a t-involution of A is an anti-automorphism s: A ŕ A satisfying s2 = id_A and which restricting to t on the center K. The involution s is said to be of the first kind if K=F and of the second kind if K/F is quadratic Galois. A classical theorem of Albert gives a necessary and sufficient for A to have a t-involution.
Suppose now that R is a commutative ring, t: R ŕ R is an automorphism of order 1 or 2 and S is the fixed subring of t. Over R, the role of central simple algebras is played by an Azumaya R-algebra. In this context, Albert's theorem fails, but Saltman showed that the condition given by Albert determines when an Azumaya algebra A is Brauer equivalent to another Azumaya algebra admitting a t-involution, provided S=R (first kind) or R/S is quadratic etale (second kind). This was extended to Azumaya algebras over schemes by Knus, Parimala and Srinivias.
I will discuss recent work with Ben Williams in which we treat the case where R is neither S nor a quadratic etale extension of S. (Our results also apply in the even more general context of locally ringed spaces.) In this case, the t-involutions can be regarded as being "of the third kind". This setting features new phenomena and raises interesting open questions.
Relevant definitions will be recalled during the talk.
Let G be a transitive nonamenable graph, and consider supercritical Bernoulli bond percolation on G. We prove that the probability that the origin lies in a finite cluster of size n decays exponentially in n. We deduce that:
Every infinite cluster has anchored expansion almost surely. This answers positively a question of Benjamini, Lyons, and Schramm (1997).
Various observables including the percolation probability and the truncated susceptibility are analytic functions of p throughout the entire supercritical phase.
Starting with one red ball and one black ball in an urn, repeat the following - choose a ball from the urn at random, observe the colour, put it back in the urn and ADD another ball of the same colour. This simple model is called Polya’s urn and is an example of a random process with ``reinforcement’’ (e.g. if red is selected first then after this first iteration we have 2 red balls and 1 black ball in the urn, so red is more likely to be selected again in the second iteration).
Beginning with Polya’s urn, this talk will take a tour through some of the weird and wonderful behaviour that has been observed or conjectured for various random processes with reinforcement.
In this talk we introduce a mathematical model to describe temporal processes like embryonic development and cellular reprogramming. We consider stochastic processes in gene expression space to represent developing populations of cells, and we use optimal transport to recover the temporal couplings of the process. We apply these ideas to study 315,000 single-cell RNA-sequencing profiles collected at 40 time points over 18 days of reprogramming fibroblasts into induced pluripotent stem cells. To validate the optimal transport model, we demonstrate that it can accurately predict developmental states at held-out time points. We construct a high-resolution map of reprogramming that rediscovers known features; uncovers new alternative cell fates including neural- and placental-like cells; predicts the origin and fate of any cell class; and implicates regulatory models in particular trajectories. Of these findings, we highlight the transcription factor Obox6 and the paracrine signaling factor GDF9, which we experimentally show enhance reprogramming efficiency. Our approach provides a general framework for investigating cellular differentiation, and poses some interesting theoretical questions.
In this talk, we will show some recent results on the 3D Navier-Stokes equations, which were obtained by our group during these years. For incompressible problem, we will show the large data existence of the solution for the generalized N-S equations, partial large data problems of the N-S equations, and so on. Furthermore, we will show a result for compressible N-S equations.
Solvers for the Boolean satisfiability problem have been increasingly used to solve hard problems from many fields and now routinely solve problems with millions of variables. Combinatorial problems are a natural target, as SAT solvers contain excellent combinatorial search algorithms. Despite this, SAT solvers can fail on small problems, for example when properties of the problem cannot be concisely expressed in Boolean logic. We describe a new combinatorial search method that allows properties to be specified using a computer algebra system (CAS), thereby combining the expressiveness of a CAS with the search power of SAT solvers.
In this talk we describe how our SAT+CAS system MathCheck has verified, partially verified, or found new counterexamples to conjectures from design theory, graph theory, and number theory. In particular, we have classified Williamson matrices up to order 70, quaternary Golay sequence pairs up to length 28, best matrices up to order 7, verified the Ruskey–Savage and Norine conjectures up to larger bounds than had previously been verified, found the smallest counterexample of the Williamson conjecture, and found three new counterexamples to a conjecture on good matrices. Currently we are using the system to verify the nonexistence of projective planes of order 10.
The translation of proteins is a key part of the central dogma of biology that underlies life. Thus, unraveling the dynamics of translation and understanding how this process is regulated across scales, systems and species, is fundamental. In light of recent experimental data, I will present theoretical and computational tools, which we developed for the past few years, to study translation at the molecular, mesoscopic (single rNA), and macroscopic levels. By combining methods and models from computational geometry, stochastic analysis and systems biology, with sequencing, Cryo-EM and other experimental data, our results highlight the fundamental and complex role played by the ribosomes in modulating the dynamics of translation and its efficiency, which also leads to many challenging and still open questions.
The optimal transportation problem, which originated in the work of Gaspard Monge in 1781, provides a fundamental and quantitave way to measure the distance between probability distributions. It has led to many successful applications in PDEs, Geometry, Statistics and Probability Theory. Recently, and motivated by problems in Financial Mathematics, variations on this problem were introduced by requiring the transport plans to abide by certain "fairness rules," such as following martingale paths. One then specifies a stochastic state process and a costing procedure, and minimize the expected cost over stopping times with a given state distribution. Recent work has uncovered deep connections between this type of constrained optimal transportation problems, the celebrated Skorokhod embeddings of probability distributions in Brownian motion, and Hamilton-Jacobi variational inequalities.
Speaker Biography:
Nassif Ghoussoub obtained his Ph.D. from Université Pierre et Marie Curie in 1975. Shortly thereafter he joined the Mathematics Department at UBC, where he is currently Distinguished University Professor. Professor Ghoussoub’s contributions have been recognized by the Coxeter-James, Jeffery-Williams and David Borwein Awards of the Canadian Mathematical Society, honorary doctorates from Université Paris-Dauphine and the University of Victoria, and Fellowship in the American Mathematical Society and the Royal Society of Canada. He is an Officer of the Order of Canada.
About the Prize:
The CRM-Fields-PIMS prize is the premier Canadian award for research achievements in the mathematical sciences. It is awarded jointly by the three Canadian mathematics institutes. The winner receives a monetary award and an invitation to present a lecture at each institute within one year after the award is announced. A list of previous winners can be found on the PIMS website.
Note for Attendees
Light refreshments will be served in the PIMS lounge from 2:30p.m.
To study the difference between motivic invariants of the moduli spaces of coherent sheaves on a smooth surface and that on the blown-up surface, Nakajima-Yoshioka constructed a sequence of flip-like diagrams connecting these moduli spaces. In this talk, I will explain birational geometric property of Nakajima-Yoshioka's wall crossing diagram. It turned out that it realizes a minimal model program.
Coding theory has been associated mainly with maintaining reliability in communication and storage systems. Yet work published already in the 1950s by Von Neumann, Moore, and Shannon also considered the use of error handling techniques to maintain reliability of computation devices. Still, until very recently, the prevailing paradigm of designing computing devices appeared to put all of the effort of the reliability guarantee on the hardware design. Distributed computing, as well as the introduction of new nanoscale accelerator devices, are two examples of recent developments that have revived the interest in coding for reliable computation.
In this talk, we focus on accelerators that perform vector–matrix multiplication. We describe code designs that guarantee computation reliability under two paradigms: exact computation, which is needed when accelerating ordinary ALU computations, and approximate computation, which suits learning applications. In the latter, we view the computation to be over the real field, and introduce new tools for analyzing and synthesizing coding schemes for this model. In particular, we show connections between the code design problem and certain extremal problems of convex polygons
The TFDW (Thomas-Fermi-Dirac-Weizacker) model has been used to describe certain electron configurations of molecules. On the other hand, the Ohta-Kawasaki model arises in the context of diblock copolymer melts. In this talk, we discuss results concerning compactness of minimizing sequences of the TFDW model and a variant of the Ohta-Kawasaki model.
In 1993, Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank 1 boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a special case of the conjecture, and show that a weaker decomposition into boolean trees always exists. This is joint work with Joseph Doolittle.
A coupled PDE-ODE model used to describe communication between dynamically active signaling compartments (biological cells) is analyzed using strong localized perturbation theory. In the limit D >> 1, the coupled model is reduced into a nonlinear system of ODEs, which is then used to study global coupling and synchronous oscillations among the cells. In addition, this reduced system is used to study quorum sensing and phase synchronization among the cells.
Take n objects and put them in a container. What is the configuration space of all the ways they can fit in the container without intersecting? How does the topology of that configuration space change depending on the size of the objects and the size of the container? We will look at configurations of segments in a disk, of squares in a rectangle, and of disks in an infinite strip. In the latter two cases, the configuration space is homotopy equivalent to a polyhedral cell complex that can be studied combinatorially.
Note for Attendees
The Topology seminar has been moved to ESB 4127 (the Math Bio seminar is now in ESB 4133)
Consider a system of N particles moving according to Brownian motions and branching at rate one. Each time a particle branches, the particle in the system furthest from the origin is killed. It turns out that we can use results about a related partial differential equation known as a free boundary problem to control the long term behavior of this particle system for large N. This is joint work with Julien Berestycki, Eric Brunet and James Nolen.
The Wasserstein space, which is the space of probability measures endowed with the so-called (quadratic) Wasserstein distance coming from optimal transport, can formally be seen as a Riemannian manifold of infinite dimension. We propose, through a variational approach, a definition of harmonic mappings defined over a domain of R^n and valued in the Wasserstein space. As the latter has nonnegative curvature, we cannot rely on the theory of Koorevaar, Schoen and Jost about harmonic mappings valued in metric spaces and we use arguments based on optimal transport instead. We manage to recover a fairly satisfying theory which captures some key features of harmonicity.
Suppose two populations with different genetic types live close to each other and can interbreed, but hybrid offspring have a lower evolutionary fitness. The interface between such populations is known as a hybrid zone. We can model this situation using a stochastic process. I will discuss a result on the motion of the interface, which is related to a well-known PDE result connecting the Allen-Cahn equation and mean curvature flow.
If we take a simplified model in only one spatial dimension, we can trace the ancestral lineages of individuals backwards in time and (in some cases) determine the asymptotic behavior of the genealogy (or family tree) of a set of individuals. Several interesting questions about the genealogies remain open.
Partly based on joint work with Alison Etheridge and Nic Freeman.
Note for Attendees
Refreshments will be served at 2:30 p.m. in ESB 4133 (Lounge).
For a large class of GIT quotients X=W//G, Ciocan-Fontanine—Kim—Maulik and many others have developed the theory of epsilon-stable quasimaps. The conjectured wall-crossing formula of cohomological epsilon-stable quasimap invariants for all targets in all genera has been recently proved by Yang Zhou.
In this talk, we will introduce permutation-equivariant K-theoretic epsilon-stable quasimap invariants with level structure and prove their wall-crossing formulae for all targets in all genera. In particular, it will recover the genus-0 K-theoretic toric mirror theorem by Givental-Tonita and Givental, and the genus-0 mirror theorem for quantum K-theory with level structure by Ruan-Zhang. It is based on joint work in progress with Yang Zhou.
The possibility that a discrete process can be fruitfully approximated by a continuous one, with the latter involving a differential system, is fascinating. Important theoretical insights, as well as significant computational efficiency gains may lie in store. A great success story in this regard are the Navier-Stokes equations, which model many phenomena in fluid flow rather well. Recent years saw many attempts to formulate more such continuous limits, and thus harvest theoretical and practical advantages, in diverse areas including mathematical biology, image processing, game theory, computational optimization, and machine learning.
Caution must be applied as well, however. In fact, it is often the case that the given discrete process is richer in possibilities than its continuous differential system limit, and that a further study of the discrete process is practically rewarding. Furthermore, there are situations where the continuous limit process may provide important qualitative, but not quantitative, information about the actual discrete process. In this talk will present and discuss several case studies of such continuous limits and demonstrate success as well as cause for caution. Consequences will follow.
In the talk we present a new method for calculating decay rates of eigenfunctions for eigenvalues below the threshold of the essential spectrum. The method is also applicable for eigenvalues at the threshold provided that the eigenfunction exists. In fact, our a-priori bounds are the crucial first step for the proof that bound states at the thresholds exist. We consider the example of a helium atom and show that the decay rate of an eigenfunction at the threshold of the essential spectrum behaves as exp(− C|x|∞ ), where |x|∞ = maxj (|x|j ) is the maximum of electron coordinates. Moreover we show a similar result for the Froehlich Bipolaron.
The concept of Hopf algebras originated from the theory of algebraic groups and algebraic topology in the mid 20th century. Hopf structures have numerous applications in many other mathematical branches, and now it is a familiar concept in representation theory as the class functions and superclass functions of some tower of groups have Hopf structures. In these Hopf structures, representation theoretic functors give the product and coproduct. In this talk, we give a brief introduction to normal lattice supercharacter theories, and then we construct a Hopf structure by using these supercharacter theories.
The understanding of the growth of degrees of iterates of a rational self-map of a projective variety is a fundamental problem in holomorphic dynamics. I shall review some basic results of the theory and discuss some recent directions of research.
I will talk about a new geometric inequality: the Sphere Covering Inequality. The inequality states that the total area of two distinct surfaces with Gaussian curvature less than 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least 4π. In other words, the areas of these surfaces must cover the whole unit sphere after a proper rearrangement. We apply the Sphere Covering Inequality to solve several open problems about uniqueness and symmetry of solutions of mean field type equations. In particular we apply this inequality to prove an old conjecture of A. Chang and P. Yang about the best constant of a Moser-Trudinger type inequality.
We first establish a new Minkowski formula for free boundary hypersurfaces in the unit ball and use it to completely solve the stabilty problem of CMC free boundary hypersurfaces. Then using the Minkowski formula we introduce a inverse curvature type flow for free boundary hypersurfaces and prove the convergence. As an application, we establish Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball. The talk is based on the joint work with Chao Xia and also with Julian Scheuer.
A pair of skew shapes are said to be (skew) equivalent if they admit the same number of semistandard tableaux of any weight; i.e. if their associated skew Schur functions are equal. A complete characterization is known for ribbon shapes (those with no 2×2 square) but the general case is more complicated. I will give a brief survey of existing results in the area and discuss some recent work of my own.
The growth regulator auxin plays a central role in development across plants. Auxin spatial patterning is critical in the phyllotactic arrangement of leaves along a stem, the shapes of the leaves themselves, and venation within leaves. These patterns depend on polar auxin transport (PAT) at the cellular level, particularly the preferential allocation of PIN efflux proteins to certain areas of the plasma membrane. Two general mechanisms have been studied: an up-the-gradient (UTG) allocation dependent on neighbouring-cell auxin concentrations, and a with-the-flux (WTF) allocation dependent on the flow of auxin across walls. We developed a combined UTG+WTF model for leaf venation. The model simulates intracellular and membrane kinetics and intercellular transport, and is solved for a 2D leaf of several hundred cells. We find that vein initiation in the leaf margin and cell polarization towards new veins is UTG-driven, while WTF is critical for vein extension. UTG is important for joining veins to form a network structure. The model produces the experimentally observed succession of effects when PAT is increasingly inhibited by NPA treatment. Venation patterns are highly correlated with leaf shape; this model enables the investigation of how PAT dynamics contribute to the diversity of leaf shapes across plants.
The classifying space for commutativity, denoted by B_\text{com} G, of a Lie group G is assembled from commuting tuples in G as a subspace of the usual classifying space BG. The resulting space classifies principal G-bundles whose transition functions generate an abelian subgroup of G whenever they are simultaneously defined. The relationship between the homotopy type of G and the space B_\text{com} G is much more interesting, and non-trivial compared to the case of BG. In this talk, I will present a work, joint with Ben Williams, where we study the mod-\ell homotopy type of B_\text{com} G at a prime \ell. The techniques involve a homotopy colimit decomposition over a topological category generalizing the construction of Adem-Gomez and application of results on mapping spaces between classifying spaces of compact Lie groups due to Dwyer-Wilkerson. We show that for a connected compact Lie group the mod-\ell homotopy type of B_\text{com}G depends on the mod-\ell homotopy type of BG.
The (α,β)-superprocess is a spatial branching model associated to an α-stable spatial motion and a (1+β)-stable branching mechanism. Technically, it is a measure-valued Markov process, but this talk concerns the absolutely continuous parameter regime, in which the random measure has a density. After introducing this process and some classical results, I will discuss some newly proven path properties of the density. These include (i) strict positivity of the density at a fixed time (for certain values of α and β) and (ii) a classification of the measures which the density “charges” almost surely, and of the measures which the density fails to charge with positive probability, when conditioned on survival. The duality between the superprocess and a fractional PDE is central to our method, and I will discuss how the probabilistic statements above correspond to new results about solutions to the PDE.
Don’t worry, I’m NOT announcing a proof of Riemann Hypothesis. Rather, in this talk, I will like to give a brief overview of Riemann Hypothesis, and explain why it is such an important conjecture in number theory.
Roughly speaking, Riemann Hypothesis is a result regarding the (non-trivial) zeroes of the Riemann zeta-functions. You might then wonder: what does that have anything to do with number theory, and in particular prime numbers? Well then, come join this tour as we journey from Euler’s proof of infinitude of primes, to Riemann penning his only paper in number theory, and beyond! Time permitting, I’ll briefly talk about recent work that was done towards *avoiding* Riemann Hypothesis in certain number-theoretic results, with terms and conditions.
If you’ve have ever wondered what all the fuss Riemann Hypothesis is about, this talk is for you!
Don’t worry, I’m NOT announcing a proof of Riemann Hypothesis. Rather, in this talk, I will like to give a brief overview of Riemann Hypothesis, and explain why it is such an important conjecture in number theory.
Roughly speaking, Riemann Hypothesis is a result regarding the (non-trivial) zeroes of the Riemann zeta-functions. You might then wonder: what does that have anything to do with number theory, and in particular prime numbers? Well then, come join this tour as we journey from Euler’s proof of infinitude of primes, to Riemann penning his only paper in number theory, and beyond! Time permitting, I’ll briefly talk about recent work that was done towards *avoiding* Riemann Hypothesis in certain number-theoretic results, with terms and conditions.
If you’ve have ever wondered what all the fuss Riemann Hypothesis is about, this talk is for you!
Particle-laden flows are ubiquitous in many environmental flows and man-made processes. Their intricate dynamics is governed by momentum transfer between fluid and particles and is strongly multi-scale. I will discuss mathematical, computational and modelling issues related to the accurate, fast and realistic computing of particle-laden flows. Specifically, I will elaborate on the three following problems: (i) a massively scalable and fast Navier-Stokes solver on Cartesian grid, (ii) an extension of a distributed Lagrange multiplier/fictitious domain method to octree adaptive grid and (iii) an attempt to design realistic drag law models using a stochastic approach and machine-learning in the context of a multi-scale analysis of particle-laden flows.
Note for Attendees
Light refreshments will be served in the PIMS lounge at 2:30 p.m.
We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant’s Nodal Domain Theorem. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n ≥ 3. We show that 0 is generically not an eigenvalue of the conformal Laplacian. If time permits, we shall discuss related results for manifolds with boundary, and for weighted graphs. This is joint work with Y. Canzani, R. Gover, R. Ponge, A. Hassannezhad, M. Levitin, M. Karpukhin, G. Cox and Y. Sire.
Coulomb branches have recently been given a good mathematical footing thanks to work of Braverman-Finkelberg-Nakajima. We will discuss their categorical structure. For concreteness we focus on the massless case which leads us to the category of coherent sheaves on the affine Grassmannian (the so called coherent Satake category).
This category is conjecturally governed by a cluster algebra structure. We will describe a solution to this conjecture in the case of general linear groups and discuss extensions of this result to more general Coulomb branches of 4D N=2 gauge theories. This is joint work with Harold Williams.
Generative Adversarial Networks have been shown to be powerful in generating content. To this end, they have been studied intensively in the last few years. Nonetheless, training these networks requires solving a saddle point problem that is difficult to solve and slowly converging. Motivated from techniques in the registration of point clouds and by the fluid flow formulation of mass transport, we investigate a new formulation that is based on strict minimization, without the need for the maximization. The formulation views the problem as a matching problem rather than an adversarial one and thus allows us to quickly converge and obtain meaningful metrics in the optimization path.
Given a direction set in Euclidean space, we consider a maximal function for the directional Hilbert transforms associated with the direction set. For each finite p>1, it is known that this maximal function is bounded on L^p if and only if the direction set is finite. This raises the following quantitative problems;
1) What is a sharp uniform upper bound on the L^p-operator norm of the maximal function that depends only on the cardinality of direction sets?
2) Under which geometric assumptions on direction sets can this uniform bound be improved?
We will study these problems for the p=2 case using polynomial partitioning tools from discrete geometry and an almost-orthogonality principle for the maximal function. This is a joint work with Malabika Pramanik.
In this talk, I will introduce some recent developments of free boundary minimal hypersurfaces, including compactness, generic finiteness and index estimates. As an application, we also give the existence of infinitely many minimal hypersurfaces with non-empty free boundary under some weak assumptions. Part of the work is joint with Q.Guang, M.Li and X.Zhou
One of the best studied experimental models for selforganization in embryonic development is the formation of skeletal elements in tetrapod limbs, in particular in the chicken and the mouse. Here cells aggregate to form chondrogenic condensations, which later turn into cartilage, then bone. This behavior is also seen in vitro in so-called micromass experiments. Several models for the underlying pattern forming mechanism have been proposed, notably Turing-type reaction-diffusion mechanisms and positional information mechanisms based on spatiotemporal gradients of signaling molecules. However the exact regulatory mechanisms for this process are far from understood. We present a model based on the experimentally established dynamics of a multiscale regulatory network consisting of two glycan-binding proteins expressed early in chick limb development: CG (chicken galectin)-1A, CG-8 and their counterreceptors. The model consists of a system of partial differential equations containing a nonlocal term to represent cell-cell adhesion, adapted from the work of Armstrong, Painter and Sherratt. Due to the high dimensionality of the problem, published results have only been in one spatial dimension. We present new results in two spatial dimensions, which allow for exploration of the topology of two dimensional patterns that can be generated. The full model recapitulates qualitatively and quantitatively the experimental results of network perturbation and leads to new predictions.
This talk is based on joint work with S. A. Newman (NY Medical College), R. Bhat (Indian Institute of Science, Bangalore) and J. Zhang (West. Wash Univ.) .
For a symmetric diffusion, we show that the two-sided sub-Gaussian heat kernel estimate implies singularity of energy measures with respect to the reference measure. This gives an affirmative answer to a conjecture of Barlow (2003).
Harmonic maps have found various applications in Teichmuller theory. While Teichmuller space describes conformal (or hyperbolic) structures on a surface, harmonic maps provide a parametrization on Teichmuller space (via their Hopf differentials) as well as a distance function (the Teichmuller metric). Beginning with a 2-dimensional simplicial complex, we first describe several spaces of metrics. We then show the existence of harmonic maps between such spaces, in the hopes of adding the analytic tools of harmonic maps to the Teichmuller spaces of simplicial complexes. This is joint work with Victoria Gras Andreu.
We explore classes of graphs on which a large number of pursuers are required to capture an evader. We give a lower bound for the cop number of graphs of high girth that improves a result of P. Frankl. We also consider lower bounds for the cop number of various algebraically constructed graph classes. In particular, we present a class of directed graphs with cop number (1-o(1)) \sqrt{n}, which is greater than any lower bound currently known for any directed graph class.
The Hedtrich lab is developing human-based organ models with a current focus on skin and lung. They are specifically interested in the modeling of inflammatory and genetic diseases in vitro and use the organ models to study (patho)physiological mechanism. In this talk, Dr. Hedtrich will give an overview of the different approaches in her lab has with emphasize on their work done in atopic diseases and the atopic march.
Conway mutation is an operation on links that is notoriously difficult to detect: it preserves many link invariants such as the signature, the Alexander polynomial or, more generally, the HOMFLY polynomial. Baldwin and Levine conjectured that δ-graded link Floer homology also belongs in this list—despite the fact that *bigraded* link Floer homology can distinguish some mutant knots such as the famous Kinoshita-Terasaka and Conway knots.
In [arXiv:1909.04267], I proved Baldwin and Levine's conjecture by studying symmetry properties of peculiar modules, an immersed curve invariant of 4-ended tangles. In this talk, I will sketch this proof.
We describe detailed two-sided inequalities for the three-player ruin problem and related estimates for killed random walks in inner-uniform finite subset of grids and other graphs. The key is the simultaneous use of the appropriate Doob transform, doubling and Poincaré inequalities. This talk is based on joint work with Persi Diaconis and Kelsey Houston-Edwards.
The "doubling property" refers to the property (of a metric measure space) that max{Vol(B(x,2r))/Vol(B(x,r)): r>0} is bounded. We consider the following question: do we have good control of the doubling property for left-invariant geometries on a given compact Lie group? For instance, on the group SU(2) (which, as a manifold, is the 3-sphere) what can we say of the doubling constant of a left-invariant geometry? We will discuss the conjecture that, for any compact Lie group G, there is a constant D(G) such that max{Vol_g(2r)/Vol_g(r): r>0} is bounded by D(G) uniformly over all left-invariant metric g. This is true in the case of SU(2). This talk is based on joint work with Nate Eldredge and Maria Gordina.
Note for Attendees
Refreshments will be served at 2:30 p.m. in ESB 4133 (Lounge).
The Hilbert scheme parameterizing n points on a K3 surface X is a holomorphic symplectic manifold with rich properties. In the 90s it was discovered that the generating function for the Euler characteristics of the Hilbert schemes is related to both modular forms and the enumerative geometry of rational curves on X. We show how this beautiful story generalizes to K3 surfaces with a symplectic action of a group G. Namely, the Euler characteristics of the "G-fixed Hilbert schemes” parametrizing G-invariant collections of points on X are related to modular forms of level |G| and the enumerative geometry of rational curves on the stack quotient [X/G] . These ideas lead to some beautiful new product formulas for theta functions associated to root lattices. The picture also generalizes to refinements of the Euler characteristic such as chi_y genus and elliptic genus leading to connections with Jacobi forms and Siegel modular forms.
The binomial coefficients are some of the most widely-used tools in enumerative combinatorics. Yet, like most tools we use regularly, we often get trapped into thinking of them only in certain ways. In this talk we'll connect some common uses of the binomial coefficients with some uncommon ones, taking a look at topics such as Leibniz's generalized product rule, a finite difference formula for binomial sums, and determinants of matrices containing binomial coefficients.
Fluid dynamics governs many phenomena on the Earth’s surface and interior, from the emplacement of fluid magma, to the viscous deformation of mountain ranges on the longest timescales. Understanding these processes presents a challenge to traditional modelling techniques. However, simplifying models of the leading-order features of the flow can give insight into the dominant physical balances at play. In this talk I will address two geophysical processes: the formation of fold-thrust belts and the dynamics of shallow magmatic intrusions. Although geophysically distinct, these two problems both involve the interplay between viscous flow and elastic deformation so inform the modelling of one another.
Fold-thrust belts are formed at convergent plate margins, where accretion of weak sediments to the front of the overriding plate creates continually flexural subsidence of the underthrusting plate. The analysis shows that the evolution of fold-thrust belts can be dominated by gravitational spreading or vertical thickening and can be readily applied to mountain ranges seen in the field.
The propagation of shallow magmatic intrusions is governed by the interplay between elastically deforming sedimentary layers, the viscous flow of magma beneath, and the requirement to fracture rock layers at the front. This process can be described by a model for elastic-plated gravity currents, which gives rises to two dynamical regimes of spreading; viscosity dominant spreading and fracture toughness (or adhesion) dominant spreading. Experiments using clear, PDMS sheets confirm these regimes and demonstrate the formation of a vapour tip between the fluid front and the fracture front.
Bio: Dr. Ball did her undergrad in Mathematics at the University of Cambridge. She then stayed on at Cambridge doing a PhD in Earth Sciences working with Prof. Jerome Neufeld. During her PhD, she did a summer project at the Geophysical Fluid Dynamics summer school at WHOI working with Neil Balmforth and Ian Hewitt. She has now started as a postdoc with Neil working on viscoplastic fluids.
Although convolution on Euclidean space and the Heisenberg group satisfy the same L^p bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.
The nonlinear Schrödinger equation with derivative cubic nonlinearity (dNLS) is a model quasilinear dispersive equation. It admits a family of solitons, which are orbitally stable in the energy space. After a review of the many interesting properties of dNLS, we will present a result of orbital stability of multi-solitons configurations in the energy space, and some ingredients of the proof
Inspired by a problem in joint source-channel coding, we introduce a new notion of similarity between graphs, termed graph information ratio. We discuss various properties of this measure, including in particular metric structure and partial ordering of graphs, an information ratio power inequality, relations to graph homomorphism, algebraic identities and inequalities, and more.
A diffusion-type operator biologically significant in neuroscience is a difference of Gaussian functions used as a spatial-convolution kernel (Mexican-Hat operator). We are interested in the dynamics inherent in a neural structure such as visual cortex modeled by stochastic neural field equations, a class of stochastic differential-integral equations using the Mexican-Hat kernel. We find that spatially smoothed noise, in a field of Ornstein-Uhlenbeck processes, without direct spatial coupling, causes pattern formation. Our analysis of the interaction between coupling and noise-sharing yields optimal parameter combinations for the formation of spatial pattern.
Joint work with PH Baxendale and LM Ward, recently on line, Phys.Rev.E
Which 3-manifolds smoothly embed in the 4-sphere? This seemingly simple question turns out to be rather subtle. Using Donaldson's theorem, we derive strong restrictions to embedding a Seifert fibered space over an orientable base surface, which in particular gives a complete classification when e is greater than k/2, where k is the number of exceptional fibers and e is the normalized central weight. Our results point towards a couple of interesting conjectures which I'll discuss. This is joint work with Duncan McCoy.
What is the minimal possible diameter of a hyperbolic surface (i.e. with constant curvature equal to -1) of genus g? We will prove that it is asymptotic to log(g). While the lower bound follows from a simple volume growth argument, the upper bound is obtained by considering a model of random hyperbolic surfaces, which we analyse by adapting techniques from the study of random graphs. Based on joint work with Nicolas Curien and Bram Petri.
Avigilon Corporation Atrium, 555 Robson Street, Vancouver (Robson at Seymour)
Thu 24 Oct 2019, 6:00pm-7:00pm
Details
/Abstract: When you load a webpage, watch a video, send an email, or do any other task on the internet, packets of information travel along a path from you to your destination. But where, physically, is this path? We answer this question by using traceroutes, the speed of light, a directed graph, and a linear program.
Please note:The event is free & refreshments are included. For more details and to register, clickhere.
More about bcdata
Thebcdata initiativeis a collaborative effort to build a vibrant community intertwining people from government, industry, universities and not-for-profits toward three goals:
Share knowledge and identify opportunities emerging from the data explosion
Develop training experiences and career pathways for young mathematical scientists (mathematicians, statisticians, computer scientists) and support organizations in recruiting talent
Forge long-term research and development partnerships to improve society and advance industry in British Columbia
The bcdata community meets regularly for the bcdata colloquium, a networking event with presentations on various data science topics. The initiative also hosts problem solving workshops in the summer, where government, industry, and not-for-profit partners are invited to contribute challenge problems and collaborate with interdisciplinary teams of students and university researchers. The 2017 and 2018,workshopshave catalyzed long-term research and development partnerships within the bcdata community.
Mutation is a process that makes a non-trivial local change to a knot; the distinction between the resulting pair of knots is difficult to detect. I will present a vaguely historical account of why mutant pairs are difficult to distinguish. This will focus on symmetries of certain relative invariants associated with Khovanov homology: I will explain why Khovanov homology cannot detect mutation, but that in certain instances, the presence of an appropriate involution gives way to a new refinement of Khovanov homology that is able to separate mutant pairs.
The Coulomb branches of 3d N=4 gauge theories were recently given a mathematical definition by Braverman, Finkelberg, and Nakajima. These very interesting algebraic varieties were already discussed in Sabin Cautis's talk a few weeks ago, but since they may be unfamiliar I will overview their definition and properties, and discuss some interesting examples. Finally, I will discuss my joint work with Nakajima where we give a generalization of the definition of Coulomb branches, which allows us to realize affine Grassmannian slices of all finite types.
VENUE CHANGE: now CEME 1203 (instead of CEME 2202)
Mon 28 Oct 2019, 4:00pm-5:00pm
Details
Abstract: The BMP model [1] is an attractive Thixo-Elast-Visco-Plastic (TEVP) model that is popular and widely used both because of its intuitive use of the fluidity as a structural parameter and its ability to fit rheological data of a wide variety of materials. When the zero-shear fluidity (ϕ0) is set to zero the fluid exhibits a yield stress and the inelastic version of this model dates back to the 1970s [2]. Elastic linear instabilities of plane Poiseuille flows of the BMP model have recently been extensively studied in [3]. Here we focus on flow stability using this constitutive model as a relaxation of an ideal yield stress fluid (i.e. setting ϕ0=0 throughout). In particular we are interested in how the model compares with that of a simple yield stress fluid from the perspective of flow stability.
Borrowing the notation of [3], we first show that the limiting case of zero timescale ratio and zero thixoelastic number (inelastic and non-thixotropic) does indeed recover the key dynamic characteristics of a simple yield stress fluid: (i) supporting a static solution in the presence of applied forcing for yield stress above a well-defined critical value; (ii) global stability and finite time decay to zero for yield stresses above the critical value; see [4]. Both these features are not shared by many TVP models currently studied. Having determined the underlying dynamics of the yield stress system, we then relax inelastic and non-thixotropic assumptions to explore how static stability is affected. These results are illustrated with a simple 1D example: a pressure driven plane Poiseuille flow for which the forcing is suddenly stopped.
[1] F. Bautista , J.M. de Santos , J.E. Puig , O. Manero, “Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model.” Journal of Non-Newtonian Fluid Mechanics 80 (1999) 93–113.
[2] A.G. Fredrickson, “A model for the thixotropy of suspensions.” AIChE Journal 16 (1970) 436–441.
[3] H.A. Castillo, H.J. Wilson, “Elastic instabilities in pressure-driven channel flow of thixotropic-viscoelasto-plastic fluids.” Journal of Non-Newtonian Fluid Mechanics 261 (2018) 10–24.
[4] I. Karimfazli, I.A. Frigaard, “Flow, onset and stability: Qualitative analysis of yield stress fluid flow in enclosures.” Journal of Non-Newtonian Fluid Mechanics 238 (2016) 224–232.
Different from most deep learning settings, we focus on the situation where we have only one or a few large-scale examples. We also assume that only partial labels are available. Examples include video segmentation and segmentation of 3D geophysical imaging results where the ground-truth labels are available in a small number of boreholes.
First, we discuss how we can use partial/projected loss-functions and regularization of the neural-network output to train on partial examples and mitigate low prediction quality away from labeled pixels/voxels. Second, we present a fully reversible network that enables training on large-scale data. We induce and exploit the reversibility of networks based on certain partial-differential-equations. As a result, no storage is required for all the network states to compute a gradient, and the memory requirements for training are independent of network depth. A fully reversible network can train directly from video-to-video without resorting to slice-by-slice based methods and therefore simplifies previous approaches.
Throughout the talk, we highlight similarities and differences with well-known computational techniques from PDE-constrained optimization.
Joint work with Eldad Haber, Justin Granek, and Keegan Lensink.
In this talk, we consider families of finite sets that we call shellable and that have been characterized by Chang and Hirst and Hughes as being the families of sets that admit unique solutions to Hall's matching problem. We prove that shellable families can be characterized by using a generalized notion of hook-lengths; hook-lengths originate from the hook-length formula which is used to determine the number of standard Young tableaux on partition shapes. Then, we introduce a natural generalization of standard skew tableaux and Edelman and Greene's balanced tableau, then prove existence results about such a generalization using our characterization of shellable families. We also calculate the average number of such tableaux using a hook-length formula.
In Section 11.5 of their book, Freedman and Quinn showed that an aspherical 4-manifold M with polycyclic fundamental group pi is determined by pi and the boundary of M. We revisit this result, with a shift in emphasis. We show that if Mis aspherical and pi is elementary amenable then pi is 1, ZZ, a Baumslag-Solitar group BS(1,m) or (torsion free) polycyclic of Hirsch length 3 or 4. We characterise the possible boundaries, except for the cases with pi=BS(1,m) for some m.
A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology and show how the random switching can `rescue' species from extinction.
The problem of determining conditions under which a rational map can exist between a pair of twisted flag varieties plays an important general role in the study of linear algebraic groups and their torsors over arbitrary fields. A non-trivial special case arising in the algebraic theory of quadratic forms amounts to studying the extent to which an anisotropic quadratic form can become isotropic over the function field of a quadric. To this end, let p and q be a pair of anisotropic non-degenerate quadratic forms over a field, and let k be the dimension of the anisotropic part of q over the function field of the quadric p=0. We then make the general conjecture that the dimension of q must lie within k of an integer multiple of 2^{s+1}, where s is such that 2^s < \mathrm{dim}(p) \leq 2^{s+1}. This generalizes a well-known "separation theorem" of D. Hoffmann, bridging the gap between it and some other classical results on Witt kernels of function fields of quadrics. Note that the statement holds trivially if k \geq 2^s - 1. In this talk, I will discuss recent work that confirms the claim in the case where k \leq 2^{s-1} + 2^{s-2}, and more generally when \mathrm{dim}(p) > 2k - 2^{s-1}.
Using an example to illustrate the process, I will explain how an Arthur parameter \psi for a p-adic group G determines a category P_\psi of equivariant perverse sheaves on a moduli space X_\psi of Langlands parameters for G and then how the microlocal perspective on P_\psi reveals the local Arthur packet \Pi_\psi attached to \psi . This talk will not assume you already know how to compute Arthur packets for p-adic groups but rather will show how to compute these things directly using geometric tools -- that's really one of the main points of this perspective. Joint with Andrew Fiori, Ahmed Moussaoui, James Mracek and Bin Xu.
I will discuss how to see symmetries of an alternating knot from any reduced alternating diagram. As a consequence we will see that the quotient of an alternating periodic knot is alternating, and that any alternating knot which is p and q periodic for odd primes p and q is also p*q periodic (c.f. the trefoil which is 2 and 3 but not 6 periodic). The main tool for this is the study of flypes, but we will also mention an approach using spanning surfaces.
Reptiles have the ability to replace their teeth continuously throughout life. Replacement occurs in highly patterned waves passing from the back to the front of the mouth, in alternating tooth positions. Although this pattern has been recognized for over 100 years, the formation and maintenance of this pattern is not well understood. In this presentation, I will present work being undertaken in the Richman lab at UBC to attempt and understand the mechanisms of continuous tooth replacement in a reptile model organism, the leopard gecko. We performed tooth removal surgeries on adult geckos and tracked tooth replacement for over one year post-surgery. The pattern of tooth replacement was analyzed, and preliminary results suggest that signaling between adjacent teeth might be the primary control over the alternating replacement pattern.
Chase-escape is a competitive growth process in which red particles spread to adjacent uncoloured sites while blue particles overtake and kill adjacent red particles. We can think of this model as prey escaping from pursuing predators. If the red particles spread fast enough, both particle types occupy infinitely many sites with positive probability. Otherwise, both almost surely occupy only finitely many sites. In this talk, we introduce the modification that red particles die at some rate. When the underlying graph is a d-ary tree, chase-escape with death exhibits a new phase in which blue almost surely occupies finitely many sites, while red reaches infinity with positive probability. Moreover, the critical behaviour, which we precisely characterize, is different with the presence of death. Many of our arguments make use of novel connections to analytic combinatorics.
Joint work with Erin Beckman, Keisha Cook, Nicole Eikmeier and Matthew Junge
Join the second meeting of the She&They&Allies. This time we will have a potluck snack on November 7th, 3:30 pm (MATX 1101) and we propose the following agenda. Agenda for the meeting: - Summary of survey results - Group mission and guidelines (draft to be edited) - Structure the group into four committees: social, education, outreach, treasury.
We want everyone to feel welcome and we also want our time to be valued. We encourage you to give us any suggestions and to come prepared to be in charge of/join one of the committees!
Our budget is limited but mingling is a.s. better with food, so please consider bringing a snack to share with everyone! (not required for entry tho!)
Every number field K has a maximal unramified extension K^un, with Galois group Gal(K^un/K) (whose abelianization is the class group of K). As K varies, we ask about the distribution of the groups Gal(K^un/K). We give a conjecture about this distribution, which we also conjecture in the function field analog. We give some results about Gal(K^un/K) that motivate us to build certain random groups whose distributions appear in our conjectures. We give theorems in the function field case (as the size of the finite field goes to infinity) that support these new conjectures. In particular, our distributions abelianize to the Cohen-Lenstra-Martinet distributions for class groups, and so our function field theorems give support to (suitably modified) versions of the Cohen-Lenstra-Martinet heuristics.
This talk is on joint work with Yuan Liu and David Zureick-Brown.
A random nxm matrix gives a random linear transformation from \Z^m to \Z^n (between vectors with integral coordinates). Asking for the probability that such a map is injective is a question of the non-vanishing of determinants. In this talk, we discuss the probability that such a map is surjective, which is a more subtle integral question. We show that when m=n+u, for u at least 1, as n goes to infinity, the surjectivity probability is a non-zero product of inverse values of the Riemann zeta function. This probability is universal, i.e. we prove that it does not depend on the distribution from which you choose independent entries of the matrix, and this probability also arises in the Cohen-Lenstra heuristics predicting the distribution of class groups of real quadratic fields. This talk is on joint work with Hoi Nguyen.
Note for Attendees
Refreshments will be served at 2:30 pm in the PIMS Lounge.
The classical Erdos-Ko-Rado theorem in extremal combinatorics states the following: Given a family F of k-subsets of an n-set that is "pairwise intersecting" (meaning A \cap B \neq \emptyset for all A,B in F) it follows that F can be no larger than \binom{n-1}{k-1}. A commonly-studied extension is whether the same bound can be applied to F if any d elements of F obey some given intersectional structure, for d greater than 2. We will discuss the history of these problems as well as new results on some long-open conjectures.
A major challenge in mathematical analysis of infectious diseases is that the epidemic process is usually only partially observed. Although we might be able to identify when an individual became symptomatic, rarely can we observe when infection began or from whom it was transmitted. This means that the likelihood of the observed data is computationally intractable for any more than a handful of infected cases. Although data-augmented Markov Chain Monte Carlo methods are generally considered the gold standard for analysis of partial epidemic data since they employ a tractable augmented likelihood, they also often struggle for large population sizes. I will describe a new approach which seeks to instead approximate the likelihood by exploiting the underlying structure of the epidemic model, without the need for augmentation. We regard this approach as a useful and adaptable addition to the toolkit for analysing infectious disease data, and I will provide examples of applications to real outbreaks and a variety of disease transmission models.
We will attempt, in a colloquium-like fashion, to introduce the basic concepts of quantum mechanics and functional analysis necessary to derive the connection between twisted equivariant K-theory of a d-1 dimensional torus and quasi-adiabatic stability of Fermi surfaces of non-interacting fermions on a crystalline half-solid with a gapped bulk condition. This is part of the speaker's PhD thesis and is an extension of joint work with A.Adem, O.A. Camarena and G.W. Semenoff.
Abstract for the first talk: We consider a Galois extension L/K of number fields. It has been known since the work of Dirichlet at the end of the Nineteenth Century that in some cases there are relations among the class numbers of K, of L and of the intermediate subextensions. Dirichlet's results have been put in a broader and conceptually clearer contest by Brauer's work on special values of Artin L functions at s=0. In this talk I will report on a joint work with L. Caputo focusing on the case of extensions whose Galois group is dihedral. I will briefly survey the history of the problem and I will eventually discuss a purely algebraic proof of the class number relation, relying only upon class field theory and free of any analytical tool.
Abstract for the second talk: In the second talk, I plan to start by going into more details of the proof of the main theorem discussed in the first talk. I will then move on towards an "Iwasawa-theoretic" result, which is part of a second joint work with L. Caputo, discussing the behaviour of class numbers in pro-dihedral towers of number fields that we call "fake Z_p-extensions of dihedral type". The main result combines tools from classical abelian Iwasawa theory with the class-number relation discussed in the first talk.
This will be a talk on a few questions in the philosophy of mathematics, with a good, healthy dose of perspective from the history of mathematics. I'm not an expert in either of these fields, but that hasn't stopped me before.
Here is a question: do the things that mathematicians claim to study---say, prime numbers, or Sobolev spaces, or the Borel hierarchy---actually exist? If they do exist, do they exist in the same sense that, say, toothbrushes and the planet Jupiter exist? If mathematical objects do not exist in this sense, then in what sense do they exist? How does this sense explain how we know things about mathematical objects and apply this knowledge to "real-world" questions?
If mathematical objects do not exist, then in what sense is the statement, "There exists a group of order two" true? If it isn't true, must it be false? If it false, then why do we treat it differently from false statements like, "There exist two non-isomorphic groups of order two"?
I will present a few contemporary takes on these questions, and discuss how the nature of the question has changed over time, particularly in ancient Greece and in the late 19th and early 20th centuries. No background in philosophy is presumed, and there will be lots of good, hearty mathematical examples. My main goal in giving this talk is to make anyone with existing strong views on this issue, including me, feel less secure in those views.
How to approximate intractable integrals? This is an old problem which is still a pain point in many disciplines (including mine, Bayesian inference, but also statistical mechanics, computational chemistry, combinatorics, etc).
The vast majority of current work on this problem (HMC, SGLD, variational) is based on mimicking the field of optimization, in particular gradient based methods, and as a consequence focusses on Riemann integrals. This severely limits the applicability of these methods, making them inadequate to the wide range of problems requiring the full expressivity of Lebesgue integrals, for example integrals over phylogenetic tree spaces or other mixed combinatorial-continuous problems arising in networks models, record linkage and feature allocation.
I will describe novel perspectives on the problem of approximating Lebesgue integrals, coming from the nascent field of non-reversible Monte Carlo methods. In particular, I will present an adaptive, non-reversible Parallel Tempering (PT) allowing MCMC exploration of challenging problems such as single cell phylogenetic trees.
By analyzing the behaviour of PT algorithms using a novel asymptotic regime, a sharp divide emerges in the behaviour and performance of reversible versus non-reversible PT schemes: the performance of the former eventually collapses as the number of parallel cores used increases whereas non-reversible benefits from arbitrarily many available parallel cores. These theoretical results are exploited to develop an adaptive scheme approximating the optimal annealing schedule.
My group is also interested in making these advanced non-reversible Monte Carlo methods easily available to data scientists. To do so, we have designed a Bayesian modelling language to perform inference over arbitrary data types using non-reversible, highly parallel algorithms.
Note for Attendees
Refreshments will be served at 2:30 p.m. in ESB 4133 (PIMS Lounge)
We study the birational self-maps of the projective plane that induce bijections on the k-rational points for a given field k. These form a subgroup BCr_2(k) inside the Cremona group. The elements of BCr_2(k) are called Biregular Cremona transformations. We show that BCr_2(k) is not finitely-generated under a mild hypothesis on the field k. When k is a finite field, we study the possible permutations induced on the k-rational points of the plane. This is joint work with Kuan-Wen Lai, Masahiro Nakahara and Susanna Zimmermann.
Abstract: We introduce a novel triple layer core-annular method. The method has broad application in lubricated transport of heavy viscous oils to reduce the frictional pressure gradient and ensure the continued flow. In this method, we purposefully positioned a shaped unyielded visco-plastic fluid (skin layer) at the interfaces to eliminate interfacial instabilities. Specifically, the skin layer is shaped which allows for lubrication force to develop as the core rises under the action of transverse buoyancy forces due to density differences between layers. In this talk, we address the feasibility of the method. Also, we show how to sculpt the interface in a very controlled way for a desirable interface. Finally, we give an overview of 3D triple-layer computations and the buoyant motion of the core.
Bio: Parisa is a graduate research assistant and teaching fellow in the Department of Mechanical Engineering at UBC. She works on “visco-plastically lubricated multi-layer flows” under supervision of Prof. Ian Frigaard at UBC and Prof. Sarah Hormozi at Ohio University.
Deep learning (DL) is transforming whole industries as complicated decision-making processes are being automated by neural networks trained on real-world data. Yet as these tools are increasingly being applied to critical problems in medicine, science, and engineering, many questions about their stability, reliability, and approximation capabilities remain. Such questions include: how many data points are sufficient to train a neural network on simple approximation tasks, and how robust are these trained architectures to noise in the data? In this work we seek to quantify the capabilities of deep neural networks (DNNs), both theoretically and numerically. Recently published results show that these architectures allow for the same convergence rates as best-in-class schemes, e.g., h,p-adaptive finite element and spectral approximations. Our own analysis confirms that DNNs afford the same sample complexity estimates as compressed sensing (CS) on sparse polynomial approximation problems. In exploring the approximation capabilities of DNNs, we also present numerical experiments on a series of simple tests in high-dimensional function approximation, with comparisons to results achieved with CS on the same problems. Our numerical experiments show that standard methods of training and initialization often yield DNNs which fail to achieve the rates of convergence suggested by theory. We conclude with a discussion of the conditioning of the DL problem.
We gratefully acknowledge generous financial support for the SCAIM seminar by PIMS and the IAM.
The chromatic polynomial was generalized to the chromatic symmetric function by Stanley in his seminal 1995 paper. This function is currently experiencing a flourishing renaissance, in particular the study of the positivity of chromatic symmetric functions when expanded into the basis of elementary symmetric functions, that is, e-positivity.
In this talk we approach the question of e-positivity from various angles. Most pertinently we resolve the 1995 statement of Stanley that no known graph exists that is not contractible to the claw, and whose chromatic symmetric function is not e-positive.
This is joint work with Soojin Cho, Samantha Dahlberg, Angele Foley and Adrian She, and no prior knowledge is assumed.
The green sea turtle Chelonia midas travels for thousands of miles from the coast of Brazil to a small island in the Atlantic Ocean, Ascension Island. There the turtles lay their eggs into the warm sand on the beach. It is a classic scientific challenge to understand the navigational skills of the turtles and several orienteering mechanisms are discussed, such as geomagnetic information, chemotaxis, Atlantic flow patterns etc.
In this talk I will present a mathematical model for the homing of sea turtles and discuss how it can be used to identify the navigational mechanisms of sea turtles.
(joint work with K.J. Painter).
Note for Attendees
This is Part I of a special "Double Feature" Math Biology Seminar on November 20, 2019.
‘Koszul duality’ is a phenomenon which algebraists are fond of, and has previously been studied in the context of '(bordered) Heegaard Floer homology' by Lipshitz, Ozsváth and Thurston. In this talk, I shall discuss an occurrence of Koszul duality which links older constructions in Heegaard Floer homology with the bordered Heegaard Floer homology of three-manifolds with torus boundary. I shan’t assume any existing knowledge of Koszul duality or any form of Heegaard Floer homology.
Abstract: Many phytoplankton species are motile, propelling themselves through water with velocities ranging from 10 to 1000 microns per second. Daily, phytoplankton needs to migrate vertically from and towards the ocean surface for light and to find nutrients such as dissolved oxygen. To travel through the water column, they need to fight against gravity (by swimming) and fluid turbulence which can make their journey longer.
The spatial distribution of unicellular organisms in the Ocean is observed to be heterogeneous. This communication demonstrates that heterogeneity can be generated ex novo at the smallest scales of turbulent flows via an active coupling between motility and hydrodynamic shear, and stands in direct contrast with the aforementioned mechanism that considers phytoplankton cells to act as passive tracers. It is often observed that cells migrate across the water column as chains. The purpose of our study is to elaborate on this observation as a potential benefit to swim as a chain in turbulence.
We carried out numerical simulations of the coupled system of homogeneous isotropic turbulence and gyrotactic cell trajectories through Lagrangian tracking. Realistic flows are obtained by randomly forcing large-scale fluid motions and solving Navier-Stokes equations through direct numerical simulations for the resultant turbulent motion. This flow is seeded with hundreds of thousands of cells and statistical analysis is carried out to find out the physical mechanisms. References
•Chain formation can enhance the vertical migration of phytoplankton through turbulence. S. Lovecchio, E. Climent, R. Stocker and W. M. Durham (2019) accepted in Sciences advance
•Turbulent fluid acceleration generates clusters of gyrotactic microorganisms. F. De Lillo, M. Cencini, W.M. Durham, Barry, R. Stocker, E. Climent and G. Boffetta (2014). Physical Review Letters – 112, 044502.
•Turbulence drives microscale patches of motile phytoplankton. W. M. Durham, E. Climent, M. Barry, F. De Lillo G. Boffetta, M. Cencini and R. Stocker. Nature Communications (2013) – 4:2148.
•Gyrotaxis in a steady vortical flow. W.M. Durham, E. Climent and R. Stocker. (2011). Physical Review Letters – 106, 238102.
Bio:
Eric Climent is a full professor at University of Toulouse, France, and the current head of the Fluid Mechanics Institute in Toulouse that counts about 200 researchers in the field of Fluid Mechanics. Eric graduated his Master degree in 1993 and PhD degree in 1996 from University of Toulouse. From 1998 to 2007, he was an Associate professor at University of Strasbourg and University of Toulouse. From 1999 to 2003, he spent about 1 year and a half at Brown University as a visiting professor. His main area of research is the multi-scale modelling and numerical simulation of dispersed phase flows (bubbles, droplets and rigid particles) with potentially additional effects as chemical reactions, magnetic forces, interface properties and biological interactions.
Different types of cells, i.e. from different tissues, typically look quite different from each other. Even when cultured on two-dimensional surfaces like glass slides or tissue culture polystyrene under identical conditions, cells adopt different shapes. These shapes are in general functions of the cytoskeletal properties of those cells, itself a subset of what we can call the “state” of the cell. Experimental evidence over several decades has indicated that for some cell types, imposed changes in shape lead to changes in cellular differentiation and other properties. Conversely there is increasing evidence that some changes in cell state can lead to stereotypical changes in cell shape. We have developed a large number of morphological parameters to quantify cell shape and cytoskeletal morphology. Using these parameters to quantify morphologies of different cell lines, as well as cells in different experimental conditions, we show that quantifiers of cell shape and cytoskeletal texture can be used to discriminate between different cell states. A neural network is able to correctly classify different cell states with high accuracy. Using projections of the data to lower-dimensional shape space, we find that we can distinguish between similar and dissimilar changes in shape. We use this method to identify similarities in shape changes between breast cancer and osteosarcoma cell lines accompanying the acquisition of invasive characteristics. Our data indicates that cellular morphology is a powerful and sensitive window into the physiological state of the cell, and underline the need to develop mechanistic models that relate cell state to cell shape.
Note for Attendees
This is Part II of a special "Double Feature" Math-Biology Seminar on Nov 20, 2019. It will follow directly after the PIMS Tea.
Maximal tori in reductive groups form the foundation for many constructions in $p$-adic representation theory. Many of these constructions place constraints on the tori involved, requiring that they split over unramified or tamely ramified extensions of the ground field. When the residue characteristic is small, wild tori occur even for groups of small rank. Such tori complicate standard tools used to construct representations, such as Bruhat-Tits buildings, Néron models and the Moy-Prasad filtration. In an effort to aid in the study of representations in small characteristic, I will present an early version of an online database of p-adic tori.
Swimming bacteria with helical flagella are self-propelled micro-swimmers in nature, and the swimming strategies of such bacteria vary depending on the number and the position of flagella on the cell body. In this talk, I will introduce two microorganisms, multi-flagellated E. coli and single-flagellated Vibrio A. The Kirchhoff rod theory is used to model the elastic helical flagella and the rod-shaped cell body is represented by a hollow ellipsoid that can translate and rotate as a neutrally buoyant rigid body interacting with a surrounding fluid. The hydrodynamic interaction between the fluid and the bacteria is described by the regularized version of Stokes flow. I will focus on how bacteria can swim and reorient swimming course for survival and how Mathematics can help to understand the swimming mechanism of such bacteria.
Stein's spherical maximal theorem has a supplement, due to Duoandikoetxea and Rubio de Francia, which observes that stronger results for L^p(R^n) boundedness hold for maximal functions with supremums restricted to 'sparse' sets -- the primary example of a 'sparse' set being the collection of dyadic numbers {2^j:j=1,...}. Magyar, Stein, and Wainger have also given a discrete analogue of Stein's spherical maximal theorem, which gives rise to the natural question of whether or not one can show similar results for 'sparse' maximal functions in the discrete setting. Working in the discrete setting, however, gives rise to extra complications. Indeed, an example due to Zienkiewicz shows that a discrete analogue of the work of Duoandikoetxea and Rubio de Francia is impossible. Still, though, certain improvements are expected to hold. This talk will focus on some results in this direction, and includes joint work with Kevin Hughes.
I will introduce the universal precobordism theory, which generalizes algebraic cobordism of Levine-Morel to arbitrary quasi-projective schemes over a Noetherian base ring A. In the main part of the talk I will outline the proof of projective bundle formula for this new cohomology theory. The usual proof techniques based on resolution of singularities and weak factorization break down in this generality, so we have to use an alternative approach based on carefully studying the structure of precobordism rings of varieties with line bundles, which were inspired by a paper of Lee-Pandharipande. The talk is based on joint work with Shoji Yokura.
In this talk I will introduce the quantum phase-estimation problem as well as the iterative algorithm by Kitaev that is commonly used to solve it. I show how Kitaev's algorithm can be modified to obtain an asymptotically minimal sampling complexity. Two novel algorithms for obtaining an initial phase estimate, used to start the iterative estimation process, are given along with their theoretical sample complexity.
We gratefully acknowledge generous financial support for the SCAIM seminar by PIMS and the IAM.
A group is said to be torsion-free if it has no elements of finite order. An example is the group, under composition, of self-homeomorphisms (continuous maps with continuous inverses) of the interval I = [0, 1] fixed on the boundary {0, 1}. In fact this group has the stronger property of being left-orderable, meaning that the elements of the group can be ordered in a way that is invariant under left-multiplication.. If one restricts to piecewise-linear (PL) homeomorphisms, there exists a two-sided (bi-)ordering, an even stronger property of groups.
I will discuss joint work with Danny Calegari concerning groups of homeomorphisms of the cube [0, 1]^n fixed on the boundary. In the PL category, this group is left-orderable, but not bi-orderable, for all n>1. Also I will report on recent work of James Hyde showing that left-orderability fails for n>1 in the topological category.
We study the existence and regularity of harmonic and biharmonic almost complex structures.
Harmonic almost complex structures were introduced by C. Wood in 1990s. Since then there are considerate interest.
For harmonic almost complex structure, we prove that the seminal regularity results in the theory of harmonic maps hold similarly in the setting of harmonic almost complex structures. We follow the classical work of Schoen-Uhlenbeck, and recent advance of Cheeger-Naber. A new ingredient is the construction of a comparison almost complex structure used in the regularity, which are different from the classical work of Schoen-Uhlenbeck.
We introduce the notion of biharmonic almost complex structure and study the existence and regularity, in particular in dimension four. Weakly biharmonic almost complex structures are smooth in dimension four. We also prove existence results in a fixed homotopy class and our method relies on a new extension theorem for Sobolev maps (and almost complex structures).
Abstract:
Many aquatic environments are characterized by regions where water density varies over depth, often due to temperature or salinity gradients. These ‘pycnoclines’ are associated with intense biological activity and can affect carbon fluxes by slowing the descent of particles. We explore the effects of stratification on the fundamental hydrodynamics of settling particles, rising drops, and small organisms. Our results show that the presence of vertical density gradients in the water column can substantially affect the settling dynamics of a particle, interaction between a pair of particles, and settling rates and microstructure of a suspension of particles. We further demonstrate an unexpected effect of buoyancy, potentially affecting a broad range of processes at pycnoclines in oceans and lakes. In particular, stratification has a major effect on the flow field, energy expenditure, and nutrient uptake of small organisms.
Bio:
Prof. Ardekani is currently an associate professor at the Purdue University. Prior to joining Purdue in 2014, she was O’Hara Assistant Professor at the University of Notre Dame and a Shapiro Postdoctoral Fellow at MIT. In summer 2016, she was a visiting professor at the Institut de Mécanique des Fluides de Toulouse. She graduated from University of California Irvine with her Ph.D. in 2009 and from Sharif University of Technology with her B.S., in 2003. Prof. Ardekani was honored with the Presidential Early Career Award for Scientists and Engineers (PECASE) from President Obama in 2016, an NSF CAREER award in 2012, and the Purdue College of Engineering Award of Excellence in Early Career Research in 2019. Prof. Ardekani also received the Society of Women Engineers Fellowship and the Amelia Earhart Award in 2007 and 2008, and the Schlumberger Foundation Faculty for the Future Award in 2009 and 2010. Prof. Ardekani is the Associate Editor of ASME Applied Mechanics Review, Editorial Board Member for Scientific Reports, and Editorial Advisory Board Member of the International Journal of Multiphase Flows. Prof. Ardekani has organized sixteen conferences/symposia/sessions at SES, SOR, ParCFD, AIChE, IUTAM, Frontiers in Applied and Computational Mathematics, and Aspen Center for Physics. She has served as a co-chair of the Acrivos Award Committee in 2018.
A subset of the integers P is called predictive if for all zero-entropy processes X_i; i in Z, X_0 can be determined by X_i; i in P. The classical formula for entropy shows that the set of natural numbers forms a predictive set. In joint work with Benjamin Weiss, we will explore some necessary and some sufficient conditions for a set to be predictive. These sets are related to Riesz sets (as defined by Y. Meyer) which arise in the study of singular measures. This and several questions will be discussed during the talk.
In this talk, I will define Donaldson-Thomas type invariants for non-commutative projective Calabi-Yau-3 schemes whose associated graded algebras are finite over their centers. As an example, I will discuss the local structure of Hilbert schemes of points on the quantum Fermat quintic threefold, and compute some of its invariants.
Lattice cohomology is a combinatorial invariant of isolated singularities of complex surfaces introduced by Ozsvath Stipsicz and Szabo, and independently by Nemethi. The construction, inspired by Heegaard Floer homology, was recently employed by Nemethi to prove that a complex surface singularity is rational if and only if the fundamental group of its link is left orderable. I will give an introduction to the topic and discuss a few conjectures/new results. Some of these speculations are ongoing work in collaboration with Irving Dai.
Physicists have long been interested in answering homotopical questions about (appropriately topologized) spaces of quantum systems --- for example, the connected components of such spaces classify phases of matter (in the solid-liquid-gas sense). Recent evidence suggests that such spaces may also be of interest to pure mathematicians, because in many cases they have the same homotopy types as objects of fundamental interest in topology. I will describe two examples of this phenomenon. First, an example from condensed matter: the classification of topological phases of matter leads to rich category theory, and, conjecturally, to a relationship between cobordism groups and a higher-categorical version of Galois theory. Second, an example from high energy physics: the space of minimally supersymmetric 2D quantum field theories provides, conjecturally, an analytic model for universal elliptic cohomology.
The elliptic genus is a powerful deformation invariant of 2D SQFTs: if it is nonzero, then it protects the SQFT from admitting a deformation to one with spontaneous supersymmetry breaking. I will describe a "secondary" invariant, defined in terms of mock modularity, that goes beyond the elliptic genus, protecting SQFTs with vanishing elliptic genus. The existence of this invariant supports the hypothesis that the space of minimally supersymmetric 2D SQFTs provides a geometric model for universal elliptic cohomology.
She&They&Allies will have our December social event this Wednesday! Since it's finals season and many of us are busy marking or writing exams, this event will be a chance to take a break from the mess of finals and relax for a spell. We will put on a movie, drink hot chocolate and/or hot cider, enjoy some snacks and socialize. All are welcome.
Max and the team will present an overview of teaching and researching with Maple, as well as a summary of what’s new in Maple2019, followed by a discussion/tutorial where attendees can ask questions and we can walk through examples.
We will discuss complex geometry in the non-Kahler setting. The focus will be on a certain class of manifolds proposed in the physics literature as heterotic string compactifications. These are non-Kahler Calabi-Yau threefolds which were first introduced by C. Hull and A. Strominger. We propose to study this geometry by using the Anomaly flow, which is a nonlinear flow of non-Kahler metrics. This talk will contain joint works with T. Fei, D.H. Phong, and X.-W. Zhang.
We will discuss the Anomaly flow and its reduction in certain special cases. Though the Anomaly flow is a fully nonlinear system, under certain ansatze it leads to new nonlinear scalar partial differential equations. We will present estimates and techniques for studying the behavior of this flow. This talk will contain joint works with T. Fei, D.H. Phong, and X.-W. Zhang.
Over the past three decades, symplectic geometers have constructed powerful curve-counting invariants of symplectic manifolds. The chief example is the Fukaya category, which revealed a deep connection with algebraic geometry via Kontsevich's Homological Mirror Symmetry conjecture. In this talk, I will describe my program to relate the Fukaya categories of different symplectic manifolds. The key objects are "witch balls" (coupled systems of PDEs whose domain is the Riemann sphere decorated with circles and points), as well as the configuration spaces of these domains, which are posets called "2-associahedra". I will describe applications to symplectic geometry, algebraic geometry, and topology.
Abstract: A Lagrangian correspondence between symplectic manifolds induces a functor between their respective Fukaya categories. I will begin by introducing this construction, along with a family of abstract polytopes called 2-associahedra (introduced in math/1709.00119), which control the coherences among this collection of functors. Next, I will describe new joint work with Alexei Oblomkov (math/1910.02037), in which we construct a compactification of the moduli space of configurations of pointed vertical lines in C^2 modulo affine transformations (x,y) -> (ax+b,ay+c). These spaces are proper complex varieties with toric lci singularities, which are equipped with forgetful maps to \overline{M}_{0,r}. Our work yields a smooth structure on the 2-associahedra, thus completing one of the last remaining steps toward a complete functoriality structure for the Fukaya category.
One of the interests of low-dimensional topologists is understanding which smooth 4-manifolds can bound a given 3-manifold, or, as a special case, understanding the set of 3-manifolds up to so-called homology cobordism (to be defined in the talk). This question turns out to have applications to the study of triangulations of high-dimensional manifolds, and is a natural proving ground for Floer-theoretic techniques of studying 3-manifolds. In this talk, we will give some structure theorems about the homology cobordism group, and show that there are three-manifolds that are very far from having any of the seven non-hyperbolic Thurston geometries. This talk includes joint work with I. Dai, K. Hendricks, J. Hom, L. Truong, and I. Zemke.
Note for Attendees
There will be coffee/cookies etc. in MATH 126 beforehand 2:15pm-3:00pm.
For periodic links, we show that the Khovanov space of Lipshitz-Sarkar admits a natural cyclic group action, and identify its fixed point set. As an application, we prove that the Khovanov homology (with coefficients in the field of p elements) of a p-periodic link has rank greater than or equal to that of the annular Khovanov homology of the quotient link. This talk includes joint work with Aaron Mazel-Gee, Mike Willis, and Melissa Zhang.
Different mathematical models (direct and indirect transmission models) were used to understand and analyze different infectious diseases dynamics and possible prevention and/or elimination strategies.
As a first step, an age of infection model with heterogeneous mixing and indirect transmission was considered. The simplest form of SIRP epidemic model was introduced and served as a basis for other models. Most mathematical results in this part were based on the basic reproduction number and the final size relation.
The epidemic model was further extended to incorporate the effect of diffusion and how the coupled PDE-ODE system could be reduced to an ODE system with a diffusion term. A novel approach to modeling air-transmitted diseases using an ODE system was proposed, and showed how the reduced ODE system approximates the coupled PDE-ODE system.
In addition, a deterministic compartmental model of the co-interaction of HIV and infectious syphilis transmission (direct transmission) among gay, bisexual and other men who have sex with men (gbMSM) was developed and used to examine the impact of syphilis infection on the HIV epidemic, and vice versa. Analytical expressions for the reproduction number and necessary conditions under which disease-free and endemic equilibria are asymptotically stable were established. Numerical simulations were performed and used to support the analytical results.
Finally, the co-interaction model was modified to assess the impact of combining different HIV and syphilis interventions on HIV incidence, HIV prevalence, syphilis incidence and all-cause mortality among gbMSM in British Columbia from 2019 to 2028. Plausible strategies for the elimination of both diseases were evaluated. According to our model predictions and based on the World Health Organization (WHO) threshold for disease elimination, we suggested the most effective strategies to eradicate the HIV and syphilis epidemics over a 10-year intervention period.
The results of the research suggest diverse ways in which infectious diseases could be modeled, and possible ways to improve the health of individuals and reduce the overall disease burden, ultimately resulting in improved epidemic control.
In 1963, Lieb introduced an effective theory to approximate the ground state energy of a system of Bosons interacting with each other via a repulsive pair potential, in the thermodynamic limit. Lieb showed that in one dimension, this effective theory predicts a ground state energy that differs at most by 20% from its exact value, for any density. The main idea is that instead of considering marginals of the square of the wave function, as in Hartree theory, we consider marginals of the wave function itself, which is positive in the ground state. The effective theory Lieb obtained is a non-linear integro-differential equation, whose non-linearity is an auto-convolution. In this talk, I will discuss some recent work about this effective equation. In particular, we proved the existence of a solution. We also proved that the ground state energy obtained from this simplified equation agrees exactly with that of the full N-body system at asymptotically low and at high densities. In fact, preliminary numerical work has shown that, for some potentials, the ground state energy can be computed in this way with an error of at most 5% over the entire range of densities. This is joint work with E. Carlen and E.H. Lieb.
A liquid crystal is a phase of matter in which order and disorder coexist: for some degrees of freedom, there is order, whereas for others, disorder. Such materials were discovered in the late XIXth century, but it took over a century to understand, from microscopic models, how such phases form. In 1979, O. Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of such a liquid crystal phase. In this setting, this amounts to showing that dimers spontaneously align, but do not fully crystallize: there is no translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. In this talk, I will discuss a recent proof of this conjecture, that is, a proof of the emergence of a liquid crystal phase in the Heilmann-Lieb model. This is joint work with E.H. Lieb.
We study the relative Gromov-Witten theory on T*P^1 \times P^1 and show that certain equivariant limits give us the relative invariants on P^1\times \P^1. By formulating the quantum multiplications on Hilb(T*P^1) computed by Davesh Maulik and Alexei Oblomkov as vertex operators and computing the product expansion, we demonstrate how to get the insertion and tangency operators computed by Yaim Cooper and Rahul Pandharipande in the equivariant limits.
Results on a two way convection-diffusion problem; an asymptotic error analysis on a piece-wise uniform grid; and a framework for understanding quadrature errors in Finite Element Methods are presented. These results were never published, although I learned something new working on each project. I will discuss the reasons (which are of mixed quality) why I never finished these projects. The discussion will hopefully be useful to the audience as they decide which projects to see to completion and which to put on the back burner (possibly forever).
This talk is devoted to variational problems on the set of probability measures which involve optimal transport between unequal dimensional spaces. In particular, we study the minimization of a functional consisting of the sum of a term reflecting the cost of (unequal dimensional) optimal transport between one fixed and one free marginal, and another functional of the free marginal. A motivating application is the study of Cournot-Nash equilibria, when the continuous space of agents has more heterogeneity (that is, is higher dimensional) than the space of strategies. For a variety of different forms of the term described above, we show that a nestedness condition, which is known to yield much improved tractability of the optimal transport problem, holds for any minimizer. This represents joint work with Luca Nenna, and builds on earlier joint work with Pierre-Andre Chiappori and Robert McCann.
The cohomology ring of the Grassmannian, the set of k-dimensional subspaces in n-dimensional space, can be described by Schur functions, a symmetric function that are characters of the special linear Lie group. To study the K-theory ring, the corresponding objects we use are (symmetric) Grothendieck polynomials. Demazure characters can be considered as partial Schur functions and are characters of representations of the subgroup of upper triangular matrices. The K-theoretic analog of Demazure characters are known as Lascoux polynomials, but they currently have no representation theoretic or geometric interpretation. In joint work with Valentin Buciumas and Katherine Weber, we give the first known combinatorial interpretation for Lascoux polynomials by describing a colored version of the 5-vertex model of Motegi and Sakai. In this talk, we will discuss Lascoux polynomials, the colored 5-vertex model, and the corresponding combinatorial interpretation from our result. No knowledge of the material will be assumed.
In low-dimensional geometry and topology, there is a classical construction that takes a holomorphic quadratic differential on a surface and produces a PGL(2)-local system. This construction provides a local homeomorphism from the moduli space of quadratic differentials to the moduli space of local systems. In this talk, I will propose a categorical generalization of this construction. In this generalization, the space of quadratic differentials is replaced by a complex manifold parametrizing Bridgeland stability conditions on a certain 3-Calabi-Yau triangulated category, while the space of local systems is replaced by a cluster variety. I will describe a local homeomorphism from the space of stability conditions to the cluster variety in a large class of examples and explain how it preserves the structures of these two spaces.
Recent advances in technology have led to a monumental increase in large-scale data across many platforms. One mathematical model that has gained a lot of recent attention is the use of sparsity. Sparsity captures the idea that high dimensional signals often contain a very small amount of intrinsic information. Using this notion, one may design efficient low-dimensional representations of large-scale data as well as robust reconstruction methods for those representations. Binary, or one-bit, representations of data for example, arise naturally in many applications, and are appealing in both hardware implementations and algorithm design. In this talk, we provide a brief background to sparsity and 1-bit measurements, and present new results on the problem of data classification with low computation and resource costs. We illustrate the utility of the proposed approach on recently acquired data about Lyme disease.
Bio:Deanna Needell earned her PhD from UC Davis before working as a postdoctoral fellow at Stanford University. She is currently a full professor of mathematics at UCLA. She has earned many awards including the IEEE Best Young Author award, the Hottest paper in Applied and Computational Harmonic Analysis award, the Alfred P. Sloan fellowship, an NSF CAREER and NSF BIGDATA award, and the IMA prize in Applied Mathematics. She was a research professor fellow at MSRI last Fall and is now a (semi-) long term visitor at Simons this Fall. She also serves as associate editor for IEEE Signal Processing Letters, Linear Algebra and its Applications, the SIAM Journal on Imaging Sciences, and Transactions in Mathematics and its Applications as well as on the organizing committee for SIAM sessions and the Association for Women in Mathematics.
Note for Attendees
Reception to follow, at 4:00, in the IAM Lounge (LSK 306).
We study Caccioppoli’'s inequalities of the non-stationary Stokes equations and Navier-Stokes equations. Our analysis is local near boundary and we prove that, in contrast to the interior case, the Caccioppoli’'s inequalities of the Stokes equations and the Navier-Stokes equations, in general, fail near boundary. This is the joint work with Dr. Chang, Tong-Keun.
In Euclidean space, a periodic configuration is a union of finitely many translations of a lattice. In particular, energy-minimizing periodic configurations are those which possess minimum energy. Finding energy-minimizing periodic configurations is an interesting problem, not only because of its theoretical significance in physics, but also its connection with the famous sphere packing problem. On the other hand, the search of energy-minimizing periodic configurations is notoriously difficult and very few theoretical results are known. Nevertheless, an insightful idea due to Cohn, Kumar, Reiher and Schurmann, enables us to study the energy-minimizing periodic configurations from a combinatorial viewpoint. Roughly speaking, among pairs of energy-minimizing periodic configurations, they revealed a remarkable symmetry named formal duality. Furthermore, they translated the formal duality into a purely combinatorial context, where the corresponding configuration was called a formally dual pair, which is a pair of subsets in a finite abelian group satisfying a subtle difference-set-like property. In this talk, we will give an overview and present some recent results of formally dual pairs, which involve constructions, nonexistence results and characterizations.
This is joint work with Alexander Pott and Robert Schuller.
We study the κ-color cyclic particle system on the one-dimensional integer lattice, first introduced by Bramson and Griffeath. In their original article they show that almost surely, every site changes its color infinitely often if κ ∈ {3, 4} and only finitely many times if κ ≥ 5. In addition, they conjecture that for κ ∈ {3, 4} the system clusters, that is, for any pair of sites x, y, with probability tending to 1 as t → ∞, x and y have the same color at time t. Here we prove that conjecture.
Mumford's GIT quotient is one way to construct moduli spaces that parametrize classes of algebro-geometric objects. It turns out there is an interesting structure on the set of unstable points discarded in the GIT quotients. In this talk I would aim to describe:
1. the stratification of the unstable points and its variation caused by different choices of linearizations;
2. a wall and chamber decomposition analogous to Variation of Geometric Invariant Theory Quotient;
3. examples and results in the case of projective toric varieties.
Error bounds for the conjugate gradient method using minimal polynomials are well known. What are less well known are comparable bounds for indefinite systems using minimal polynomials over two intervals. The key result goes back to Akhieser, with significant enhancements by Fischer.
We gratefully acknowledge generous financial support for the SCAIM seminar by PIMS and the IAM.
We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrödinger equation with nonlinear exponent close to the mass critical case and for any space dimension. These profiles bifurcate from the ground state solitary wave. In this talk, we present the argument which relies on the matched asymptotics method and we derive an exponentially smallness condition on the Sobolev critical exponent as conjectured by Sulem and Sulem in 1997.
This is a joint work with Yvan Martel and Pierre Raphaël.
Breast cancer remains the second leading cause of cancer-related death with metastasis accounting around 90% of the total deaths. Specialized subcellular structures termed invadopodia play a critical role in metastasis, aiding tumor cell dissemination to distant sites. Invadopodia have documented roles in aiding tumor cells movement into (intravasation) and out of (extravasation) the blood vessels. While movement through the hematogenous system is well characterised, we are limited in our understanding of dissemination through the lymphatics. In this study we explore the role of invadopodia in aiding tumor cells invasion through the lymphatics. To impair invadopodia formation, knockout (KO) of key invadopodial regulatory protein Tks5 was performed in human breast cancer cells MDA-MB-231. Invadopodia formation in Tks5-KO cells was found to be completely abolished. We assessed cell invasion across a lymphatic monolayer and found a significant reduction in lymphatic invasion for Tks5-KO cells. Next, using transendothelial electrical resistance (TEER) we measured lymphatic tight junction integrity and found that control cells were able to reduce lymphatic tight junctions but this was significantly impaired in Tks5-KO. Overall, the inability of Tks5-KO cells to form invadopodia compromised their ability to invade through the lymphatics suggesting that invadopodia aid tumor cells invasion through the lymphatic system. Current studies are expanding on this work to better understand the role of invadopodia in lymphatic dissemination through the use of live cell imaging and bioluminescent imaging of progression and lymphatic invasion in mice.
In this talk, I will discuss a surprising connection between finite groups of Lie type and string topology of classifying spaces of compact connected Lie groups recently discovered by Jesper Grodal and myself: the cohomology of a finite group of Lie type is a module over the cohomology of the free loop space of the classifying space of the corresponding compact Lie group when the latter cohomology groups are equipped with a string topological multiplication. This module structure provides in particular a new perspective towards the Tezuka conjecture asserting that under certain conditions, the cohomologies of the two objects are isomorphic.
The notion of conformal walk dimension serves as abridgebetweenelliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that the finiteness of the conformal walk dimension characterizes the elliptic Harnack inequality.
Roughly speaking, the conformal walk dimension is the infimum of all possible values of the walk dimension that can be attained by a time-change of the process and by a quasisymmetric change of the metric. Two natural questions arise (a) What are the possible values of the conformal walk dimension? (b) When is the infimum attained? In this talk, I will explain the answer to (a), and mention partial progress towards (b).
This talk is based on joint work with Naotaka Kajino.
I will talk about mathematical modelling reports from around the world for the emerging novel coronavirus epidemic. I will explain their mathematical methods, findings and potential significance. At the moment I am not working on my own model, but this is a rapidly evolving situation so by Friday, who knows? This will be an informal talk suitable for a general mathematical audience.
In this talk, I will present the tame automorphisms group preserving an affine quadric threefold. The main focus of my talk is the understanding of the degree sequences induced by the elements of this group. Precisely, I will explain how one can apply some ideas from geometric group theory in combination with valuative techniques to show that the values of the dynamical degrees of these tame automorphisms admit a spectral gap. Finally, I will apply these techniques to study random walks on this particular group.
This is the first part of a two part exposition on the joints problem. The solution of the joints problem (up to a constant factor) by Guth and Katz was one of the early successes of the polynomial method and has since been streamlined and generalized in various directions, culminating in Yu and Zhao’s recent result which establishes the correct bound up to a multiplicative 1+o(1) factor. In this first part, we will present the state-of-the-art on the joints problem up to the result of Yu and Zhao.
To a finite group G and an integer q > 0 one can associate the coset-poset of nilpotent subgroups of G of nilpotency class less than q. The nerve of this poset is a simplicial complex denoted E(q,G). It is closely related to a filtration of the classifying space BG studied by Adem, Cohen and Torres-Giese. In this talk I will address a question of C. Okay which asks for the relationship between simple connectivity of E(q,G) and nilpotency of G. This is ongoing work with Bernardo Villarreal.
We investigate heat kernel estimates for regular Dirichlet forms without killing terms on metric measure spaces. For a local Dirichlet from, the heat kernel admits a sub-Gaussian or a Gaussian estimate, whilst for a non-local Dirichlet from, the heat kernel admits a stable-like estimate. We will give equivalent conditions to the heat kernel estimate for both local and non-local Dirichlet forms. This talk is based on several joint papers respectively with Alexander Grigor'yan, Eryan Hu, and Ka-Sing Lau.
We give an example of an algebraic torus T such that the torsion subgroup of the Chow group CH^2(BT) is non-trivial. This answers a question of Blinstein and Merkurjev.
The Riesz-Sobolev inequality and Brascamp-Lieb inequality both involve multilinear integrals (without any oscillatory factor). There are many results about the implicit constants in these inequalities, from finiteness to optimality. In particular, I will describe recent work studying maximizing configurations for a generalization of the Riesz-Sobolev inequality. I will also describe a generalized weak Brascamp-Lieb inequality which leads to a generalized multilinear Kakeya inequality.
Over the past few decades, a lot of attention has been drawn to large-scale streaming data analysis, where researchers are faced with huge amount of high-dimensional data acquired in a stream fashion. In this case, conventional algorithms that compute the result from scratch whenever a new data comes are highly inefficient. To handle this problem, we propose a new incremental regularized least squares algorithm that is applied to supervised dimensionality reduction of large-scale streaming data with focus on linear discriminant analysis. Experimental results on real-world data sets demonstrate the effectiveness and efficiency of our algorithms.
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We gratefully acknowledge generous financial support for the SCAIM seminar by PIMS and the IAM.
In this second part of our exposition on the joints problem we showcase Yu and Zhao’s `Joints tightened’. This exciting application of the polynomial method has resulted in a rare occurrence in incidence geometry; the exact constant has been determined, i.e. the upper and lower bounds match up to a multiplicative 1+o(1) factor. Their (not long) proof will be discussed in detail, and the key new ideas will be highlighted.
In vector-borne epidemic models there is often a substantial difference between the vector (usually insects) and host (usually humans) time scales. This makes it possible to view the model as a singular perturbation problem and use the quasi-steady state (equilibrium for the vector population in terms of the host population) to decompose the model into two problems of lower dimension and obtain a final size estimate for the epidemic.
Degenerate random environments (DRE) are a class of models of random media that arise naturally when considering random walks in non-elliptic random environments.
In this talk we will motivate and introduce this class of models, discuss percolation-type properties of some of them, and present some simple implications for random walks in non-elliptic random environments. (Based on joint work with Tom Salisbury).
Superprocesses are measure-valued Markov processes describing the evolution of populations undergoing random spatial motion and independent branching (reproduction). In this talk I will discuss the dual relationship of superprocesses with certain nonlinear PDE and explore how properties of solutions to the dual PDE can be used to infer path properties of the superprocess, and vice-versa. In particular, I will describe some new hitting properties of the (\alpha,\beta)-superprocess, and how these properties correspond to a new (partial) characterization of the admissible initial traces for a fractional nonlinear PDE.
Note for Attendees
Refreshments will be served in ESB 4148, the PIMS Lounge from 2:30-3:00p.m.
Regularity has long been known to fail for minimal submanifolds of Euclidean space with large dimension, or large codimension, the latter in part because of no maximum principle for systems of PDEs. If, however, a minimal submanifold of Euclidean 2n-space has the gradient form (x,Du(x)) for a (Lagrangian) potential u(x), then u(x) satisfies a single elliptic PDE with a maximum principle, called the special Lagrangian equation (SLE), so there is hope for a regularity theory of continuous (viscosity) solutions. At the same time, the equation itself says the phase function, or the trace of arctan D^2u(x), is constant, so the PDE fails the uniform ellipticity condition exemplified by the ordinary trace of D^2u(x), which precludes standard methods in regularity theory. As a compromise, for large phases, solutions are smooth in the interior of a domain, and for smaller phases, counterexamples exist. One last puzzle remained: in 2009, [Chen-Warren-Yuan] showed a regularity estimate assuming instead smoothness and convexity. The convexity condition is necessary given semiconvex counterexamples, but the smoothness assumption could not be weakened to merely convex viscosity solutions. In our preprint [Chen-Shankar-Yuan], we establish interior regularity for convex viscosity solutions of the SLE. Our approach combines elementary convex duality, geometric measure theory, and maximum principle ideas
We consider the thin one-phase free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in the half-space plus the area of the positivity set of that function restricted to the boundary. I will provide a rather complete picture of the (partial) regularity of the free boundary, providing content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight related to an anomalous diffusion on the boundary. The approach does not follow the standard one introduced in the seminal work of Alt and Caffarelli. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE. This opens several directions of research that I will try to describe.
In a famous paper from 2003, Fock and Goncharov conjectured that the algebra of regular functions on a cluster variety has a canonical basis parametrized by the tropicalization of a dual cluster variety. In this talk, I will show how to construct this canonical basis in an interesting class of examples. Using ideas from the representation theory of quivers, I will construct graded vector spaces which categorify the elements of the canonical basis. These graded vector spaces are closely related to spaces of BPS states in supersymmetric quantum field theories.
We introduce a class of UV-regularized two-body interactions for fermions in $\R^d$ and prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a step towards this result, we also prove a propagation bound of Lieb-Robinson type for one-particle Schr\“odinger operators. We apply the propagation bound to prove the existence of a strongly continuous infinite-volume dynamics on the CAR algebra.
In many bacteria, the segregation of their DNA is actively transported by a two protein system. One of the proteins acts as a substrate and binds to DNA in an ATP bound form, while the other stimulates its phosphatase activity, causing it to unbind after conversion to an ADP bound form. The result is a burnt-bridges style locomotion where the activity of the proteins generates a spatial gradient of the substrate that can drive motion. When this machinery is segregating low-copy plasmids, experiments show that the plasmids oscillate along the cell length, eventually placing themselves regularly along the cell. However it is unclear whether these oscillations persist as plasmids continue to replicate, or if system moves to a stable fixed point? Here I will present a deterministic model for the spatial dynamics of plasmids under the control of this two protein system. We find that over the course of the cell cycle, through a competition between spatial confinement and fluctuations in the amount of free substrate protein, the system can transition from a stable point to oscillations, then back to a stable point again. The prediction is that the system measure's cell length via oscillations but eventually gets pushed into a fixed point that faithfully partitions the genetic information.
Consider two translation-invariant continuous-time processes X=(X_t) and Y=(Y_t). The two processes are isomorphic if there exists an invertible (bimeasurable) map from X to Y which commutes with translations. The map is finitary if in order to determine a portion of Y one only needs to see a large portion of X. When does such a finitary map exist? We investigate this question, showing, for example, that Brownian motion reflected on an interval is finitarily isomorphic to a Poisson point process (thereby answering a question of Kosloff and Soo).
I will describe why the trivial knot S2-->S4 has non-unique spanning discs up to isotopy. This comes from a chain of deductions that include a description of the low-dimensional homotopy-groups of embeddings of S1 in S1xSn(for n>2), a group structure on the isotopy-classes of reducing discs of S1xDn, and the action of the diffeomorphism group Diff(S1xSn) on the embedding space Emb(S1, S1xSn). Roughly speaking, these results say there is no direct translation from dimension 3 to 4, for the Hatcher-Ivanov theorems on spaces of incompressible surfaces. Or said another way, isotopy in dimension 4 is more closely analogous to isotopy in high dimensions.
In this talk we will first discuss Crowd motions in emergency evacuation setting. Then we will discuss its relevance to the transport of incompressible fluids. We formulate these motions as ``gradient flows", i.e. an evolution that moves to dissipate their free energy as fast as possible. The gradient flow structure allows variational methods to study time-dependent problems. Our goal is to establish global-time existence of solutions past potential topological or geometrical singularities. We will survey relevant results in the literature, open problems, and then report a recent result.
Note for Attendees
Refreshments will be served in ESB 4148 from 2:30-3:00 p.m.
I will discuss the notion of hybrid spaces introduced by Berkovich and further developed by Boucksom and Jonsson in order to understand various problems concerning degenerations of complex manifolds. Applications to complex dynamical systems will be presented.
We discuss some recent advances in developing meshfree methods based on radial basis function generated finite differences (RBF-FD) for numerically solving partial differential equations (PDEs) on surfaces. The primary advantages of these methods are 1) they only require a set of nodes on the surface of interest and the corresponding normal vectors; 2) they can give high orders of accuracy; and 3) they algorithmically accessible. Commonly perceived disadvantages are that these methods require too many tuning parameters and that they are not well suited for advection-dominated problems. A goal of this talk will be to demonstrate how to overcome these issues with the use of polyharmonic spline kernels augmented with polynomials and semi-Lagrangian advection methods.
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We gratefully acknowledge generous financial support for the SCAIM seminar by PIMS and the IAM.
We will discuss Gradient flow structures in parabolic type PDEs and interface motions. Such structure introduces physical interpretations to time-dependent PDEs in terms of energy dissipation, and allows a weak notion of global-time solutions to nonlinear PDE problems. In broader terms, we will discuss minimizing movements, which approximates dissipative PDE systems with a discrete-time optimization problem. Lecture 1 will be a general introduction. Lecture 2 and 3 will discuss specific examples of minimizing movements: the mean curvature flow and nonlinear diffusions with Darcy's law.
The many different cell types and states result in large part from the different sets of genes they express. Gene expression level is encoded in the DNA sequence of the genome, and is interpreted by sequence-specific proteins called "transcription factors" (TFs). While characteristics of how TFs work are known, we lack a quantitative understanding of their function. Here, I describe a strategy using random DNA for building such a quantitative understanding, using yeast as a model system. I will provide a basic overview of how TFs recognize DNA, and why random DNA provides ideal gene regulatory "big data" for learning the relationship between DNA sequence and expression level. Using this strategy, we train highly complex models that learned a great deal about the biochemistry of transcriptional regulation, and gain insight into the activities of regulatory mutations.
We consider the random Cayley graph of a finite group G formed by picking k random generators uniformly at random:
(1) We prove universality of cutoff (for the random walk) and a concentration of measure phenomenon in the Abelian setup (namely, that all but elements lie at distance [R-o(R),R+o(R)] from the origin, where R is the minimal ball in Z^k of size at least |G|), provided k-d(G) >> 1 where d(G) is the size of the smallest generating set of G. As conjectured by Aldous and Diaconis, the cutoff time is typically independent of the algebraic structure (it is given by the time at which the entropy of a random walk on Z^k is log |G|).
(2) We prove analogous results for the Heisenberg groups of d x d uni-upper triangular matrices with entries defined mod p, for p prime.
(3) Lastly, we resolve a conjecture of D. Wilson that if G is a group of size at most 2^d then for all k the mixing time of random walk on a Cayley graph of G with k random generators is as rapid as that of Z_2^d and likewise.
In the first half of my talk, I will give an overview of the Iwasawa main conjecture for motives. In particular, I will discuss some tools developped in recent years that have been used to prove partial results towards this conjecture. I will also discuss some partial results towards the Birch and Swinnerton-Dyer conjecture obtained using Iwasawa Theory.
In the second half of my talk, I will discuss recent progress on constructing Euler systems for Rankin-Selberg products of modular forms. I will discuss how different treatments are required depending on the reduction types of the modular forms.
We will discuss Gradient flow structures in parabolic type PDEs and interface motions. Such structure introduces physical interpretations to time-dependent PDEs in terms of energy dissipation, and allows a weak notion of global-time solutions to nonlinear PDE problems. In broader terms, we will discuss minimizing movements, which approximates dissipative PDE systems with a discrete-time optimization problem. Lecture 1 will be a general introduction. Lecture 2 and 3 will discuss specific examples of minimizing movements: the mean curvature flow and nonlinear diffusions with Darcy's law.
Single-particle tracking is a powerful tool to study the mobility of molecules in the cell membrane. The most common approaches in analyzing single-particle tracks are mean square displacement analysis, and Markov models. In this talk, I will present a constrained hidden Markov model developed for a specific type of experimental data. Moreover, in most experiments, positional measurements contain systematic and random errors, and to optimize fitting, we must take these errors into account. We considered a hidden Markov model with two diffusive states and developed a method to estimate the diffusion coefficients and transition probabilities between the two states while incorporating uncertainty due to measurement error in a rational way. Here, I will introduce this model and show the results on simulated data. I will also talk about a nonparametric Bayesian framework that we developed which incorporates the Markov model parameter estimation and also estimates the number of states in the hidden Markov model.
Note for Attendees
Refreshments will be served in ESB 4148, the PIMS Lounge from 2:30-3:00 p.m.
Let k be a field, A be a finite-dimensional k-algebra (not necessarily commutative, associative or unital), and R be a commutative ring containing k. An R-algebra B is called an R-form of A if there exists a faithfully flat ring extension S/R such that B and A become isomorphic after tensoring with S. In this talk, based on joint work with Uriya First, I will be interested in the following question: if A can be generated by n elements as a k-algebra, how many elements are required to generate B as an R-algebra? For example, if A is an n-dimensional k-algebra with trivial (zero) multiplication, then an R-form of A is the same thing as a projective R-module. Otto Forster (1964) showed that every projective R-module B can be generated by n + d elements, where d is the Krull dimension of R. Richard Swan subsequently showed that this number is optimal. I will discuss generalizations of Forster's result to other types of algebras, in particular to Azumaya algebras.
Most modern problems in science and engineering are described on irregular geometries or free boundaries that are notoriously difficult to handle numerically. In addition, the differences in length scale and the limitation of computational resources necessitate the use of adaptive grids for their numerical approximations. I will discuss a numerical strategies based on the level-set method, sharp treatment of boundary conditions and Quad/Oc-tree cartesian grids on massively parallel architecture. I will also consider some applications from materials, fluid dynamics and biology.
Talk 1: Nilima Nigam, SFU
Title: An overview of finite element methods ŕ la Dominik Schoetzau
Finite element methods occupy a central and ubiquitous place in much of modern engineering and scientific computing, to the point where they are now treated as 'black boxes'. I'll give a quick tour of the (somewhat unclear) history of these methods. The years following the late '90s saw intense activity in the study of the Discontinuous Galerkin (DG) methods and the hp methods. I'll briefly describe some of these important ideas, using the lasting and influential contributions of Dominik Schoetzau as guideposts. This talk is dedicated to his 50th birthday.
Talk 2: Chen Greif, UBC
Title: Preconditioners for the time-harmonic Maxwell equations
Back in 2007, Dominik and I published an article that introduced a new preconditioning technique for iteratively solving linear systems arising from finite element discretization of the mixed formulation of the time-harmonic Maxwell equations. The preconditioners are motivated by spectral equivalence properties of the discrete operators. In the paper we provided a complete spectral analysis, and showed that the eigenvalues of the preconditioned saddle point matrix are strongly clustered. The analytical observations were accompanied by numerical results demonstrating the scalability of the proposed approach. This was one of five articles Dominik and I jointly published, and it remains until this day one of my favourite pieces of work. I will describe the main results of this work and other joint projects, and will talk about the joy that my collaboration with Dominik has given me.
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We gratefully acknowledge generous financial support for the SCAIM seminar by PIMS and the IAM.
We will discuss Gradient flow structures in parabolic type PDEs and interface motions. Such structure introduces physical interpretations to time-dependent PDEs in terms of energy dissipation, and allows a weak notion of global-time solutions to nonlinear PDE problems. In broader terms, we will discuss minimizing movements, which approximates dissipative PDE systems with a discrete-time optimization problem. Lecture 1 will be a general introduction. Lecture 2 and 3 will discuss specific examples of minimizing movements: the mean curvature flow and nonlinear diffusions with Darcy's law.
In the first part of the talk we discuss a sum-product type result on points near the unit circle that are almost closed under additions. Later we apply the obtained result to establish several universal behavior of the roots of random eigenfunctions on flat tori.
The closest point method (CPM) is a numerical method that was originally developed to solve partial differential equations (PDEs) on smooth, static surfaces using standard finite difference and interpolation methods. In this talk, we describe a recent generalization of the CPM to evolving surfaces. In our approach, the desired surface motion is obtained by evolving the underlying surface representation via the grid based particle method. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for reaction-diffusion, advection-diffusion and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented.
Let ζ(s) :=∑∞n=11/ns for<(s)>1 denote the Riemann zeta-function. It is well known, due to Euler, that ζ(2k) is a rational multiple of π2k. However, the nature of ζ(2k+ 1) remains a mystery. On adopting a wider perspective, the valuesζ(n) seem intimately connected with the values of its multi-variable analog at positive integers, namely, multi-zeta values (MZVs). MZVs satisfy a plethora of interesting identities and enjoy a rich algebraic structure. In this talk, we discuss a similar connection in the context of the Lerch zeta-function (an exponential twist of the Hurwitzzeta-function). This is joint work with M. Ram Murty.
Prime numbers are central objects of study in number theory. In the 1730s, Euler gave a novel proof of the infinitude of primes by showing that∑p1/p diverges, where the sum runs over all prime numbers. Deriving inspiration from Euler’s idea, Dirichlet proved the infinitude of primes in arithmetic progressions in 1837. His proof relied on the fact that∑∞n=1χ(n)/n6= 0, whereχis a periodic multiplicative function taking values on the unit circle. Intrigued by this curious non-vanishing result, in the early 1960s, S. Chowla initiated a study of values of the L-function,L(s,f) :=∑∞n=1f(n)/ns, for any periodic function f on Z, at positive integer arguments. In this talk, we will discuss how methods from analytic, algebraic and transcendental number theory come together harmoniously, giving rise to a beautiful theory of these special values. Around the same time as Chowla, Erd ̋os conjectured that the series∑∞n=1f(n)/n6= 0 whenever it converges, for certain periodic functions f, taking values in{−1,0,1}. This conjecture was proved by M. R. Murty and N. Saradha when q, the period of f, satisfies q≡3 mod 4. This conjecture remains open in the case q≡1 mod 4. Using probabilistic techniques, we show that Erd ̋os’s conjecture holds with 'probability' one.
The UBC Gender Equity & STEM Conference 2020 is scheduled to take place on March 14th from 9:30 am to 3:30 pm at the UBC Irving K. Barber Learning Centre. The main goal of this conference is to advance dialogue around gender inclusion and equality in STEM fields, specifically with regards to women and other gender-oppressed communities, and equip attendees with the inspiration and motivation to tackle the STEM gender gap in their own spheres. Topics will include the stigma and bias women and gender-oppressed communities face when pursuing careers and education in STEM, how gender-oppressed communities have made a positive impact on the world through STEM, how the gender gap in STEM can best be addressed through policy and innovation, and the societal, health and economic impacts of addressing this gap.
Featured speakers:
Dr. Steph van Willigenburg, Professor of Mathematics, UBC
Dhalie Patara, Director of Finance & Paloma Corvalan, Director of Youth Engagement with Society for Canadian Women in Science and Technology (SCWIST)
Dr. Naznin Virji-Babul, Associate Professor, Senior Advisor to the Provost on Women and Gender-Diverse, UBC Faculty of Medicine & Physical Therapy
Selena Wilson, Principal Project Manager at Mott MacDonald
Cybele Negris, co-founder and CEO of Webnames.ca, the first .ca registrar
The Girl Guides of Canada - BC Council
UBC Young Women for STEM members
The conference will encourage collaboration across disciplines and experience levels. It is open to anyone! We look forward to having male and female STEM students, faculty, staff, and professionals at the conference. Attendees are invited to learn, share, and collaborate by providing the opportunity to partake in workshops and engage with speakers during the panel session.
It is conjectured that two cubic fourfolds are birational if their associated K3 categories are equivalent. We prove this conjecture for very general cubic fourfolds containing a Veronese surface, where the birational maps are induced from a Cremona transformation. Using these birational maps, we find new rational cubic fourfolds. This is joint work with Yu-Wei Fan.
In this talk I will give an overview of the notion of viscosity solutions for the crystalline mean curvature flow in an arbitrary dimension, introduced recently in joint work with Yoshikazu Giga from the University of Tokyo. This problem serves as a model of crystal growth but it also has applications in image processing and related fields. Its level set formulation leads to a nonlocal, very singular parabolic equation with non-smooth, faceted solutions to which the standard viscosity theory does not apply. We introduce a reduced class of faceted test functions and show that they are sufficient to establish the comparison principle as well as an existence result for a rather general class of problems with the crystalline mean curvature.
In this talk, we describe a half-space property of ancient solutions to the Mean Curvature Flow. In particular, we construct a family of compact ancient solutions to the curve shortening flow which spans the half-space.
The multiplicative weights method is an algorithm for the problem of prediction with expert advice. It achieves the minimax regret asymptotically if the number of experts is large, and the time horizon is known in advance. Optimal algorithms are also known if there are exactly two or three experts, and the time horizon is known in advance.
In the anytime setting, where the time horizon is not known in advance, algorithms can be obtained by the doubling trick, but they are not optimal, let alone practical. No minimax optimal algorithm was previously known in the anytime setting, regardless of the number of experts.
We design the first minimax optimal algorithm for minimizing regret in the anytime setting. We consider the case of two experts, and prove that the optimal regret is \gamma \sqrt{t} / 2 at all time steps t, where \gamma is a natural constant that arose 35 years ago in studying fundamental properties of Brownian motion. The algorithm is designed by considering a continuous analogue, which is solved using ideas from stochastic calculus.
This is joint work with Nick Harvey, Ed Perkins, and Sikander Randhawa.
Looming in the background of my project, like some spooky mountain, is the L-space conjecture. Let’s just say that L-spaces are homology Lens spaces (with respect to Heegaard Floer homology) and there is a difficult question concerning them. One way to get a handle on L-spaces is to construct them by doing Dehn surgery on knots. In this talk, I will explain what (1,1) knots are, how to compute their Heegaard Floer homology and I will present the simple method due to Greene, Lewallen and Vafaee for checking almost instantly whether a (1,1) knot admits a Dehn surgery to an L-space.
Turing instabilities in reaction diffusion systems describe potential mechanisms for pattern formation in qualitative models of microbiological processes. A recent direction of research has been to incorporate bulk-membrane coupling (BMC) into these models which introduces a process of attachment and detachment to and from the cell membrane. In these models chemical species can therefore undergo periods of bulk- and membrane-bound diffusion in addition to prescribed kinetics. Linear stability analysis and numerical simulations have revealed that differences between membrane and cytosol diffusivities can trigger Turing-like pattern-forming instabilities in BMC models. We further investigate the role of bulk-membrane coupling by analyzing its effect in a singularly perturbed model where the diffusivity of one membrane-bound species is asymptotically small. In this context, localized solutions are known to exist and can be approximated using asymptotic methods. Additionally, the linear stability and long time dynamics of these localized solutions leads to novel non-local eigenvalue problems and differential-algebraic systems. In this talk we will outline this asymptotic framework and highlight the role of bulk-membrane coupling in the stability properties of localized solutions.
Note for Attendees
Refreshments will be served in ESB 4148, the PIMS Lounge from 2:30-3:00 p.m.
In this talk we will discuss how bistability in a spatially extended system can lead to fascinating localized steady-state solutions. We will primarily focus on the Swift-Hohenberg equation, which is a is known to support a variety of spatially localized steady-states. In one spatial dimensional the Swift-Hohenberg equation exhibits spatially localized steady-state solutions which give way to a bifurcation structure known as snaking. That is, these solutions bounce between two different values of the bifurcation parameter while ascending in norm. The mechanism that drives snaking in one spatial dimension is now well-understood, but recent numerical investigations indicate that upon moving to two spatial dimensions, radially-symmetric and hexagonal spatially-localized solutions take on a significantly different snaking structure. This talk details my recent work on explaining the bifurcation structure of localized solutions in higher space dimensions as well as discussing a number of open problems related to the formation of localized structures in bistable systems.
In this talk, I will present a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws. The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility.
Despite this, many open problems remain in the analysis and approximation of such problems. In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations. Approximation of the resulting optimal control problem is computationally challenging, requiring massive amounts of computational time and memory storage.
The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement. An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally. Furthermore, parallelisation as well as an adaptive-step gradient update or the control are employed to further improve efficiency. Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and
efficiency.
A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.
Looming in the background of my project, like some spooky mountain, is the L-space conjecture. Let’s just say that L-spaces are homology Lens spaces (with respect to Heegaard Floer homology) and there is a difficult question concerning them. One way to get a handle on L-spaces is to construct them by doing Dehn surgery on knots. In this talk, I will explain what (1,1) knots are, how to compute their Heegaard Floer homology and I will present the simple method due to Greene, Lewallen and Vafaee for checking almost instantly whether a (1,1) knot admits a Dehn surgery to an L-space.
Critical layers are singularities of waves propagating in shear flows, and they play crucial roles in the mixing and transition to turbulence in ocean and atmosphere. Recently, much attention has given to the `baroclinic critical layers' which arise in stratified flows with horizontal shear. The recently discovered `zombie vortices' replicate themselves through forcing the baroclinic critical layers, and the self-replication has been suggested as a possible route for the accretion of protoplanetary disks, which is the essential process in star formation. In this talk, I will present a theoretical approach to understand the evolution of forced baroclinic critical layers. We use the method of matched asymptotic analysis to tackle the fine scale of the critical layers. In linear and weakly nonlinear analysis, we derive explicit asymptotic solutions to describe the excitation and evolution of the critical layer. Our results demonstrate that the vorticity field evolves from a pair of ellipses to a dipolar stripe. We then demonstrate the dipolar stripe is unstable and thus gives rise to the secondary instability, which can explain the rollup of the stripe and the excitation of new critical layers. We have found good agreements with previous numerical simulations.
In this talk I will present a "dynamical paradigm" for modeling networks of interacting genes and proteins that regulate every aspect of cell physiology. The paradigm is based on dynamical systems theory of nonlinear ODEs, especially one- and two-parameter bifurcation diagrams. I will show how we have used this paradigm to unravel the mechanisms controlling "multiple fission" cycles in the photosynthetic green alga Chlamydomonas. While most eukaryotic cells maintain a characteristic size by executing binary division after each mass doubling, Chlamydomonas cells can grow more than eight-fold during daytime before undergoing rapid cycles of DNA replication, mitosis and cell division at night, which produce up to 16 daughter cells. We propose that this unusual strategy of growth and division (which is clearly advantageous for a photosynthetic organism) can be governed by a size-dependent bistable switch that turns on and off a limit cycle oscillator that drives cells through rapid cycles of DNA synthesis and mitosis. We show that this simple ‘sizer-oscillator’ arrangement reproduces the experimentally observed features of multiple-fission cycles and the response of Chlamydomonas cells to different light-dark regimes. Our model makes unexpected predictions about the size-dependence of the time of onset of cell-cycle oscillations after cells are transferred from light to dark conditions, and these predictions are confirmed by single-cell experiments.
Collaborators: Stefan Heldt and Bela Novak (Oxford Univ) on the modeling; Fred Cross (Rockefeller Univ) on the experiments.
In this talk, I report on recent work with Nathan Ng. Assuming a conjectural formula for a certain family of additive divisor sums, we prove an asymptotic formula for mean values of long Dirichlet polynomials with higher order shifted divisor functions as coefficients. This establishes a conjecture of Coney-Keating under the assumption of an additive divisor conjecture. As a consequence, we prove a special case of a conjecture of Conrey-Gonek when the additive divisor conjecture is known.
Narnaviruses have been described as positive-sense RNA viruses with a remarkably simple genome of ∼ 3 kb, encoding only a highly conserved RNA-dependent RNA polymerase (RdRp). Many narnaviruses, however, are ‘ambigrammatic’ and harbor an additional uninterrupted open reading frame (ORF) covering almost the entire length of the reverse complement strand. No function has been described for this ORF, yet the absence of stops is conserved across diverse narnaviruses, and in every case the codons in the reverse ORF and the RdRp are aligned. The > 3 kb ORF overlap on opposite strands, unprecedented among RNA viruses, motivates an exploration of ambigrammatic sequences in general.
In the past four years, a number of immersed curve invariants have emerged in low-dimensional topology. Hanselman, Rasmussen and Watson used their immersed curve invariant for 3-manifolds with torus boundary to give an elegant proof of the fact that the space of L-space fillings of such 3-manifolds is always an interval. I will discuss some progress towards analogous results for 4-ended tangles in Heegaard Floer, Bar-Natan and Khovanov homology. This is joint work with Liam Watson and Artem Kotelskiy.
In this talk, we will use compartmental models to examine the power of age-targeted mitigation strategies for COVID-19. We will present evidence that, in the context of strategies which end with herd immunity, age-heterogeneous strategies are better for reducing direct mortalities across a wide parameter regime. And using a model which integrates empirical data on age-contact patterns in the United States and recent estimates of COVID-19 mortality and hospitalization rates, we will present evidence that age-targeted approaches have the potential to greatly reduce mortalities and ICU utilization for COVID-19, among strategies which ultimately end the epidemic by reaching herd immunity. This is joint work with Maria Chikina.
Plumbed three-manifolds are those three-manifolds that can be be realized as links of isolated complex surface singularities. Inspired by Heegaard Floer theory Nemethi introduced a combinatorial invariant of complex surface singularities (lattice cohomology) that is conjectured to be isomorphic to Heegaard Floer homology. I will expose some recent work in collaboration with John Baldwin, Irving Dai, and Steven Sivek showing that the lattice cohomology of an almost-rational singularity is isomorphic to the framed instanton Floer homology of its link. The proof goes through lattice cohomology and makes use of the decomposition along characteristic vectors of the instanton cobordism maps recently found by Baldwin and Sivek.
I will introduce deformations of lattice cohomology corresponding to the knot homologies found by Ozsváth, Stipsicz and Szabó in https://arxiv.org/abs/1407.1795. If time permits I will show that by means of holomorphic triangles counting, one can prove equivalence with the analytic theory in the case of algebraic knots sitting in graph manifolds with at most one bad point. This yields to some interesting combinatorial formulae for the upsilon invariant of the latter knots.
Until very recently most cancer biologists operated with the assumption that the most common route to metastasis involved cells of the primary tumor transforming to a motile single-cell phenotype via complete EMT (the epithelial-mesenchymal transition). This change allowed them to migrate individually to distant organs, eventually leading to clonal growths in other locations. But, a new more nuanced picture has been emerging, based on advanced measurements and on computational systems biology approaches. It has now been realized that cells can readily adopt states with hybrid properties, use these properties to move collectively and cooperatively, and reach distant niches as highly metastatic clusters. This talk will focus on the accumulating evidence for this revised perspective, the role of biological physics theory in instigating this whole line of investigation, and on open questions currently under investigation.
A variety of cut-and-paste techniques is being developed to study Khovanov, Heegaard Floer and instanton homologies. We will describe one of such techniques, centered around immersed curves in surfaces. First, we will give an overview of existing curve-invariants. Next, a criterion for when a bordered invariant can be viewed as an immersed curve will be given. Lastly, we will interpret knot Floer homology as an immersed curve in the twice-punctured disc, and describe how it is related to the immersed curve associated to the knot exterior. The talk is based on a joint work with Liam Watson and Claudius Zibrowius.
Event on Zoom: Seminar on Quadratic forms, linear algebraic groups and beyond
We prove the last open case of the conjecture on the possible values of the first isotropy index of an anisotropic quadratic form over a field. It was initially stated by Detlev Hoffmann for fields of characteristic not 2 and then extended to arbitrary characteristic by Burt Totaro. The initial statement was proven by the speaker in 2002. In characteristic 2, the case of a totally singular quadratic form was done by Stephen Scully in 2015 and the nonsingular case by Eric Primozic in early 2019.
May 11-15th at UBC. Due to the COVID-19 virus, CFB13 has been postponed to May10-14, 2021.
Mon 11 May 2020, 9:00am-4:30pm
Details
3th International Conference on Fluidized Bed Technology (CFB-13)
1) Early Bird Registration Ending Soon, 2) Pre-Conference Courses, 3) and Student Competition
1) Registration
Since 1985 the International Conference on Fluidized Bed Technology has been the forum for academia and industrial professionals to showcase the latest research and discoveries related to fluidized beds. The 13th International Conference on Fluidized Bed Technology (CFB-13) will be held at the University of British Columbia (UBC) in Vancouver, Canada, hosted by the UBC Clean Energy Research Centre (CERC), UBC Fluidization Research Centre (FRC), GLAB Reactor and Fluidization Technologies Inc., and the UBC Department of Chemical and Biological Engineering. Conference proceedings will be published in a volume with ISBN number.
Conference Topics
CFB-13 addresses fundamental and applied aspects of circulating fluidized beds and fluidized bed systems, covering topics in:
1. Fluid dynamics of gas-solid flow
2. Modeling and simulation
3. Heat and mass transfer
4. Fine particle and nano-particle systems
5. Process design and scale-up
6. Measurements and instrumentation
7. Fluid catalytic cracking and other catalytic processes
8. Combustion, pyrolysis and gasification
9. Looping processes
10. Industrial experience
For more information, visit the conference website: https://cfb13.org/
Register at: https://cfb13.org/register-now. Early bird registration ends on February 1st.
2) Pre-Conference Courses
As part of CFB-13, short courses will be offered by world-renowned fluidization experts on fundamentals of gas-solid fluidization design and operation as well as common commercial applications. This event is aimed at engineers and researchers from academia and industry who are new to the field, or would like to improve their knowledge of the field. The full day event on May 11th, covers topics related to fluidized bed hydrodynamics, scale-up, and modeling, as well as specific applications in the chemical/petrochemical and energy industries. Course and instructor information is at: https://cfb13.org/fluidization-short-course
13th International Conference on
Fluidized Bed Technology (CFB-13)
3) Student Competition - Proceedings Book Cover Page Design
Enter the competition to design the cover page for the CFB-13 Proceedings Book.
Criteria:
Text required:
Proceedings of the 13th International Conference on Fluidized Bed Technology (CFB-13)
May 11-15, 2020
Vancouver, Canada
The design may include graphics/images with consideration for Canadian and/or indigenous elements. The conference logo can be downloaded from the website and is optional.
Submit your design to cfb.13@ubc.ca by February 7th, 2020 in image format or pdf. The conference committee will select the cover page design and the winner will receive a $200 award.
Description: "Filmed in Canada, Iran, and the United States, Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani examines the life and mathematical work of Maryam Mirzakhani, an Iranian immigrant to the United States who became a superstar in her field. In 2014, she was both the first woman and the first Iranian to be honoured by mathematics' highest prize, the Fields Medal. Mirzakhani's contributions are explained by leading mathematicians and illustrated by animated sequences. Her mathematical colleagues from around the world, as well as former teachers, classmates, and students in Iran today, convey the deep impact of her achievements. The path of her education, success on Iran's Math Olympiad team, and her brilliant work, make Mirzakhani an ideal role model for girls looking toward careers in science and mathematics."-George Csicsery, Director.
Note for Attendees
Watch for the send out of the Zoom online link earlier on the day of Tuesday, May 12th.
The screening of the film starts at 3 pm. The film is 59 minutes in length. After the film, please feel free to join in our Zoom Breakout rooms where you can post comments about the film.
Event: CRG Novel Techniques in Low Dimension- 1 Day Virtual Conference (Online)
Date and Time: 05/13/2020 09:30
Location: Online
This event is being held virtually, below is the schedule:
9:30 to 10:30, Niny Arcila Maya (UBC)
Title: Decomposition of Azumaya Algebra
Abstract: An Azumaya algebra of degree n over X is a bundle of algebras over X that is locally isomorphic to the matrix algebra. The tensor product of algebras can be extended to Azumaya algebras by performing the operation fiberwise. In topology, we can give conditions for positive integers m and n and the space X such that a topological Azumaya algebra of degree mn can be decomposed as the tensor product of Azumaya algebras of degrees m and n, but no such conditions are known in algebra. We will explain the topological proof and indicate the difficulties in extending these to the algebraic context.
10:30 to 11:00, break
11:00 to 12:00, Onkar Gujral (Duke)
Title: Khovanov homology and the linking of component-preserving cobordisms
Abstract: In this paper we show that up to homotopy up to unit, the Khovanov functor is indifferent to the linking of component-preserving cobordisms between split links. As an application, this allows Levin-Zemke's result on Khovanov homology inducing injective maps on ribbon concordances to be extended to strongly homotopy ribbon concordances. This is joint work with Adam Levine.
12:00 to 13:45, break
13:45 to 14:45, Orsola Capovilla-Searle (Duke)
Title: Obstructions to immersed lagrangian fillings with double points of vanishing index and action.
Abstract: For immersed, exact Lagrangian fillings of a Legendrian knot where all double points have vanishing action and index, the genus, g, and the number of double points, p, satisfies a linear relationship determined by the Thurston-Bennequin invariant of the Legendrian boundary. It is always possible to decrease p at the cost of increasing g; we show that it is not always possible to decrease g at the cost of increasing p. To show this, we extend a result Pan and show that an embedded, exact Lagrangian cobordism between Legendrian links Λ± induces an injective map on associated augmentation categories. This is joint work with Noemi Legout, Maylis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor.
14:45 to 15:15, break
15:15 to 16:15, Keegan Boyle (UBC)
Title: Periodic Knots and Alexander Polynomials
Abstract: It was proved by Murasugi that the Alexander polynomial of a periodic knot is a multiple of the Alexander polynomial of its quotient knot. This is already an extremely strong restriction on the relationship between these polynomials. However using knot Floer homology, specifically a spectral sequence due to Hendricks, Lipshitz and Sarkar, we can prove (in certain cases) and conjecture (more generally) new relationships between these polynomials.
This event is part of the PIMS CRG in Novel Techniques in Low Dimension.
Other Information: If you would like to attend this meeting, please email the organizers to get the meeting link.
Online; the link is available from O. Angel, M. Murugan or E. Perkins
Mon 18 May 2020, 9:00am-10:10am
Abstract
We consider the problem of estimating large rank-one matrix, given noisy observations. This inference problem is known to have a phase transition, in the sense that partial recovery of the original matrix is only possible if the signal-to-noise ratio exceeds a (non-zero) value. We will present a new proof of this fact based on the study of a Hamilton-Jacobi equation. This alternative argument allows one to obtain better rates of convergence, and also seems more amenable to extensions to other models such as spin glasses.
Note for Attendees
The link for this lecture is available from one of the local faculty organizers (Angel, Murugan or Perkins)
Online; the link is available from O. Angel, M. Murugan or E. Perkins
Tue 19 May 2020, 9:00am-10:10am
Abstract
We consider the problem of estimating large rank-one matrix, given noisy observations. This inference problem is known to have a phase transition, in the sense that partial recovery of the original matrix is only possible if the signal-to-noise ratio exceeds a (non-zero) value. We will present a new proof of this fact based on the study of a Hamilton-Jacobi equation. This alternative argument allows one to obtain better rates of convergence, and also seems more amenable to extensions to other models such as spin glasses.
Binocular rivalry explores the question of how the brain copes with contradictory information. A subject is shown two different pictures – one to each eye. What images does the subject perceive? Results from rivalry experiments usually lead to alternation of percepts and are often surprising. Hugh Wilson proposed modeling rivalry in the brain by using structured networks of differential equations. We use Wilson networks as modeling devices and equivariant Hopf bifurcation as a tool to both post-dict and predict experimentally observed percepts.
This work is joint with Casey Diekman, Zhong-Lin Lu, Tyler McMillen, Ian Stewart, Yunjiao Wang, and Yukai Zhao.
We prove that the L-space conjecture holds for those L-spaces obtained from Dehn surgery on knots which are closures of iterated 1-bridge braids, i.e., the braids obtained from satellite operations on 1-bridge braids. This statement generalizes Tran's result on non-left-orderability surgeries on L-space twisted torus knots of form Tp,kp \pm 1l,m.
A separable algebra over a field k is a finite direct sum of central simple algebras over finite separable extensions of k. It is natural to attach separable algebras to k-forms of algebraic objects. The fundamental example is the central simple algebra corresponding to a Severi-Brauer variety. Blunk considered a pair of Azumaya algebras attached to a del Pezzo surface of degree 6. More generally, one can consider endomorphism algebras of exceptional objects in derived categories. Alternatively, one can view these constructions as cohomological invariants of degree 2 with values in quasitrivial tori. In the case of Severi-Brauer varieties and Blunk's example of del Pezzo surfaces of degree 6, these invariants suffice to completely determine the isomorphism classes of the underlying objects. However, in general they are not sufficient. We characterize which k-forms can be distinguished from one another using the theory of coflasque resolutions of reductive algebraic groups. Moreover, we discuss connections to rationality questions and to the Tate-Shafarevich group for number fields. This is based on joint work with Matthew Ballard, Alicia Lamarche, and Patrick McFaddin.
Online--See O. Angel, M. Murugan or E. Perkins for link
Mon 25 May 2020, 9:00am-10:00am
Abstract
Critical and near-critical percolation is well-understood in dimension 2 and in high dimensions. The behaviour in intermediate dimensions (in particular 3) is still largely not understood, but in recent years there was some progress in this field, with contributions by van den Berg, Cerf, Duminil-Copin, Tassion and others. We will survey this recent progress (and a few older but not sufficiently known results).
Multiscale multicellular models combine representations of subcellular biological networks, cell behaviors, tissue level effects and whole body effects to describe tissue outcomes during development, homeostasis and disease. I will briefly introduce these simulation methodologies, the CompuCell3D simulation environment and their applications, then focus on a multiscale simulation of an acute primary infection of an epithelial tissue infected by a virus like SARS-CoV-2, a simplified cellular immune response and viral and immune-induced tissue damage. The model exhibits four basic parameter regimes: where the immune response fails to contain or significantly slow the spread of viral infection, where it significantly slows but fails to stop the spread of infection, where it eliminates all infected epithelial cells, but reinfection occurs from residual extracellular virus and where it clears the both infected cells and extracellular virus, leaving a substantial fraction of epithelial cells uninfected. Even this simplified model can illustrate the effects of a number of drug therapy concepts. Inhibition of viral internalization and faster immune-cell recruitment promote containment of infection. Fast viral internalization and slower immune response lead to uncontrolled spread of infection. Existing antivirals, despite blocking viral replication, show reduced clinical benefit when given later during the course of infection. Simulation of a drug which reduces the replication rate of viral RNA, shows that a low dosage that provides only a relatively limited reduction of viral RNA replication greatly decreases the total tissue damage and extracellular virus when given near the beginning of infection. However, even a high dosage that greatly reduces the rate of RNA replication rapidly loses efficacy when given later after infection. Many combinations of dosage and treatment time lead to distinct stochastic outcomes, with some replicas showing clearance or control of the virus (treatment success), while others show rapid infection of all epithelial cells (treatment failure). This switch between a regime of frequent treatment success and frequent failure occurs is quite abrupt as the time of treatment increases. The model is open-source and modular, allowing rapid development and extension of its components by groups working in parallel.
We will describe several appearances of Milnor’s invariants in the link Floer complex. This will include a formula that expresses the Milnor triple linking number in terms of the h-function. We will also show that the triple linking number is involved in a structural property of the d-invariants of surgery on certain algebraically split links. We will apply the above properties toward new detection results for the Borromean and Whitehead links. This is joint work with Gorsky, Lidman and Liu.
Online--see O. Angel, M. Murugan or E. Perkins for link
Fri 29 May 2020, 8:00am-10:00am
Abstract
Many important models in integrable probability (e.g. the KPZ equation, solvable directed polymers, ASEP, stochastic six vertex model) can be embedded into Gibbisan line ensembles. This hidden probabilistic structure provides new tools to control the behaviour and asymptotics of these systems. In my third talk I will zoom out and discuss the origins of this hidden structure.
In my lectures I will try toward from first principles as much as possible. The TA sessions and problems sets will serve to reinforce and fill in the details from the lectures.
Online. See Omer Angel, Mathav Murugan or Ed Perkins for link
Mon 1 Jun 2020, 9:00am-10:00am
Abstract
We give an introduction to branching random walks and their continuous counterpart, branching Brownian motion.We explain some recent results on the maximum of a branching random walk and its relation to point processes, as well as a connection with fragmentations. The focus will be on open questions.
Let U_n^r denote the space of r-tuples of nxn complex matrices that generate the matrix algebra Mat_n(C) as an algebra over C. The projective general linear group acts on U_n^r by simultaneous conjugation, and one can form the quotient: B_n^r. These two families of spaces have many interesting homotopical properties. Notably, the spaces B_n^r approximate the classifying space BPGL_n(C); I will show that there is a map B_n^r->BPGL_n(C) that induces an isomorphism on homotopy groups in degrees <2(r-1)(n-1)-1. Lastly, I will sketch the computation of the singular cohomology groups of U_2^r and B_2^r in degrees <4r-6.
This talk will describe local-global principles for torsors for algebraic groups over a semiglobal field -- that is, a one variable function field over a complete discretely valued field.
In particular, I will describe recent joint work with Colliot-Thélčne, Harbater, Hartmann, Parimala and Suresh in which we connect this question in certain cases to questions of R-equivalence for the group, and in some cases are able to give finiteness results and combinatorial descriptions for the obstruction to local-global principles.
Online--see Omer Angel, M. Murugan or E. Perkins for the link
Mon 8 Jun 2020, 9:00am-10:30am
Abstract
Many important models in integrable probability (e.g. the KPZ equation, solvable directed polymers, ASEP, stochastic six vertex model) can be embedded into Gibbisan line ensembles. This hidden probabilistic structure provides new tools to control the behaviour and asymptotics of these
systems. In my first talk I will discuss the Airy line ensemble and its origins and properties.
In my lectures I will try toward from first principles as much as possible. The TA sessions and problems sets will serve to reinforce and fill in the details from the lectures.
Online--See O. Angel, M. Murugan or E. Perkins for the link.
Tue 9 Jun 2020, 9:00am-10:30am
Abstract
Many important models in integrable probability (e.g. the KPZ equation, solvable directed polymers, ASEP, stochastic six vertex model) can be embedded into Gibbisan line ensembles. This hidden probabilistic structure provides new tools to control the behaviour and asymptotics of these
systems. In my second talk I will discuss the KPZ line ensemble and explain how this structure is used to probe the temporal correlation structure of the KPZ equation.
In my lectures I will try toward from first principles as much as possible. The TA sessions and problems sets will serve to reinforce and fill in the details from the lectures.
Online. See O. Angel, M. Murugan or E. Perkins for link.
Wed 10 Jun 2020, 9:00am-10:30am
Abstract
Many important models in integrable probability (e.g. the KPZ equation, solvable directed polymers, ASEP, stochastic six vertex model) can be embedded into Gibbsian line ensembles. This hidden probabilistic structure provides new tools to control the behaviour and asymptotics of these equations. In this final talk, I will Zoom out and discuss the origins of this hidden structure.
I will try to work from first principles as much as possible.
Online. See O. Angel, M. Murugan or E. Perkins for link.
Wed 10 Jun 2020, 9:00am-10:10am
Details
Metastability is a wide-spread phenomenon in the dynamics of non-linear systems subject to noise. In the narrower perspective of statistical physics, metastable behaviour can be seen as the dynamical manifestation of a first-order phase transition.
A fruitful approach to metastability is via potential theory. The key point is the realization that most questions of interest can be reduced to the computation of capacities, and that these capacities in turn can be estimated by exploiting variational principles. In this way, the metastable dynamics of the system can essentially by understood via an analysis of its statics. This constitutes a major simplification, and acts as a guiding principle. The setting of potential theory relevant for interacting particle systems is that of reversible Markov processes.
Within this limitation, there is a wide range of models that are adequate to describe a variety of different systems. Our aim is to unveil the common universal features of these systems with respect to their metastable behaviour.
This fourth and final lecture will focus on Widom-Rowlinson dynamics on the continuum.
Given an algebraic variety X, the Brauer group of X is the group of Azumaya algebras over X, or equivalently the group of Severi-Brauer varieties over X, i.e. fibrations over X which are etale locally isomorphic to a projective space. It was first studied in the case where X is the spectrum of a field by Noether and Brauer, and has since became a central object in algebraic and arithmetic geometry, being for example one of the first obstructions to rationality used to produce counterexamples to Noether's problem of whether given a representation V of a finite group G the quotient V/G is rational. While the Brauer group has been widely studied for schemes, computations at the level of moduli stacks are relatively recent, the most prominent of them being the computations by Antieau and Meier of the Brauer group of the moduli stack of elliptic curves over a variety of bases, including Z, Q, and finite fields. In a recent joint work with A. Di Lorenzo, we use the theory of cohomological invariants, and its extension to algebraic stacks, to completely describe the Brauer group of the moduli stacks of hyperelliptic curves over fields of characteristic zero, and the prime-to-char(k) part in positive characteristic. It turns out that the (non-trivial part of the) group is generated by cyclic algebras, by an element coming from a map to the classifying stack of etale algebras of degree 2g+2, and when g is odd by the Brauer-Severi fibration induced by taking the quotient of the universal curve by the hyperelliptic involution. This paints a richer picture than in the case of elliptic curves, where all non-trivial elements come from cyclic algebras.
Online. See O. Angel, M. Murugan or E. Perkins for link.
Mon 15 Jun 2020, 9:00am-10:00am
Abstract
Mixing and hitting times are fundamental parameters of a Markov chain. In this mini-course I will discuss connections between them for reversible Markov chains.
Online. See O. Angel, M. Murugan or E. Perkins for link.
Tue 16 Jun 2020, 9:00am-10:00am
Abstract
Mixing and hitting times are fundamental parameters of a Markov chain. In this mini-course I will discuss connections between them for reversible Markov chains.
Throughout the lifespan of an organism, tissues are remodeled to shape organs and organisms and to maintain tissue integrity and homeostasis. Apical constriction is a ubiquitous cell shape change of epithelial tissues that promotes epithelia folding and cell/tissue invagination in a variety of contexts. Apical constriction promotes tissue bending by changing the shape of constituent cells from a columnar-shape to a wedge-shape. Drosophila gastrulation is one of the classic examples of apical constriction, where cells constrict to fold the primitive epithelial sheet and internalize cells that will give rise to internal organs.
The actin cytoskeleton is organized in both time and space to facilitate apical constriction. We found that upstream signals that regulate apical constriction and myosin II activity exhibit a radially polarized spatial organization within the apical domain, which is critical for cell shape change. Furthermore, the cytoskeleton undergoes pulsatile dynamics, which are required for force transmission between cells. Finally, tissue wide forces orient cytoskeletal fibers to promote anisotropic force generation that promotes correct fold orientation.
Many recent concordance invariants of knots come from perturbing the differential on a knot homology theory to get a complex with trivial homology but an interesting filtration. I will describe the invariant coming from Rasmussen's E(-1) spectral sequence from Khovanov homology in this way, and show that it gives a bound on the nonorientable slice genus.
Online. See O. Angel, M. Murugan or E. Perkins for link.
Thu 18 Jun 2020, 9:00am-10:00am
Abstract
Mixing and hitting times are fundamental parameters of a Markov chain. In this mini-course I will discuss connections between them for reversible Markov chains.
In this talk we discuss variations of Serre’s notion of complete reducibility. Let G be reductive algebraic group and K be a reductive subgroup. First we consider a relative version in the case of a subgroup of the G which normalizes the identity component K^0 of K. It turns that such a subgroup is relatively G-completely reducible with respect to K if and only if its image in the automorphism group of K^0 is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting.
By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We discuss that other variations of this notion, such as relative complete reducibility and σ-complete reducibility which can also be viewed as special cases of this building-theoretic definition.
This is based on joint work with A. Litterick and G. Röhrle.
Online. See O. Angel, M. Murugan or E. Perkins for link.
Mon 22 Jun 2020, 9:00am-10:00am
Abstract
Metastability is a wide-spread phenomenon in the dynamics of non-linear systems subject to noise. In the narrower perspective of statistical physics, metastable behaviour can be seen as the dynamical manifestation of a first-order phase transition.
A fruitful approach to metastability is via potential theory. The key point is the realization that most questions of interest can be reduced to the computation of capacities, and that these capacities in turn can be estimated by exploiting variational principles. In this way, the metastable dynamics of the system can essentially by understood via an analysis of its statics. This constitutes a major simplification, and acts as a guiding principle. The setting of potential theory relevant for interacting particle systems is that of reversible Markov processes.
Within this limitation, there is a wide range of models that are adequate to describe a variety of different systems. Our aim is to unveil the common universal features of these systems with respect to their metastable behaviour.
This first lecture will be an introduction to metastability.
Online. See O. Angel, M. Murugan or E. Perkins for link.
Tue 23 Jun 2020, 9:00am-10:10am
Abstract
Metastability is a wide-spread phenomenon in the dynamics of non-linear systems subject to noise. In the narrower perspective of statistical physics, metastable behaviour can be seen as the dynamical manifestation of a first-order phase transition.
A fruitful approach to metastability is via potential theory. The key point is the realization that most questions of interest can be reduced to the computation of capacities, and that these capacities in turn can be estimated by exploiting variational principles. In this way, the metastable dynamics of the system can essentially by understood via an analysis of its statics. This constitutes a major simplification, and acts as a guiding principle. The setting of potential theory relevant for interacting particle systems is that of reversible Markov processes.
Within this limitation, there is a wide range of models that are adequate to describe a variety of different systems. Our aim is to unveil the common universal features of these systems with respect to their metastable behaviour.
This second lecture will focus on Kawasaki dynamics on lattices.
Online. See O. Angel, M. Murugan or E. Perkins for link.
Thu 25 Jun 2020, 9:00am-10:10am
Abstract
Metastability is a wide-spread phenomenon in the dynamics of non-linear systems subject to noise. In the narrower perspective of statistical physics, metastable behaviour can be seen as the dynamical manifestation of a first-order phase transition.
A fruitful approach to metastability is via potential theory. The key point is the realization that most questions of interest can be reduced to the computation of capacities, and that these capacities in turn can be estimated by exploiting variational principles. In this way, the metastable dynamics of the system can essentially by understood via an analysis of its statics. This constitutes a major simplification, and acts as a guiding principle. The setting of potential theory relevant for interacting particle systems is that of reversible Markov processes.
Within this limitation, there is a wide range of models that are adequate to describe a variety of different systems. Our aim is to unveil the common universal features of these systems with respect to their metastable behaviour.
This third lecture will focus on Glauber dynamics on random graphs.
We determine the mod p cohomological invariants for several affine group schemes G in chararacteristic p. These are invariants of G-torsors with values in etale motivic cohomology, or equivalently in Kato's version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2.
Online. See O. Angel, M. Murugan or E. Perkins for link.
Fri 26 Jun 2020, 9:00am-10:10am
Abstract
Metastability is a wide-spread phenomenon in the dynamics of non-linear systems subject to noise. In the narrower perspective of statistical physics, metastable behaviour can be seen as the dynamical manifestation of a first-order phase transition.
A fruitful approach to metastability is via potential theory. The key point is the realization that most questions of interest can be reduced to the computation of capacities, and that these capacities in turn can be estimated by exploiting variational principles. In this way, the metastable dynamics of the system can essentially by understood via an analysis of its statics. This constitutes a major simplification, and acts as a guiding principle. The setting of potential theory relevant for interacting particle systems is that of reversible Markov processes.
Within this limitation, there is a wide range of models that are adequate to describe a variety of different systems. Our aim is to unveil the common universal features of these systems with respect to their metastable behaviour.
This fourth and final lecture will focus on Widom-Rowlinson dynamics on the continuum.
Online (see O. Angel, M. Murugan or E. Perkins for link)
Mon 29 Jun 2020, 9:00am-10:30am
Abstract
Historically, mean field spin glass models come from the study of statistical physics and have served as prototypical examples of complex energy landscapes. To tackle these questions statistical physicists developed a new class of tools, such as the cavity method and the replica symmetry breaking. Since their introduction, these methods have been applied to a wide variety of problems from statistical physics, to combinatorics, to data science.
This course will serve as a high-level introduction to the basics of mean field spin glasses and is intended to introduce the audience to the basic notions that will arise in other courses during the ensuing program. In this first lecture we plan to cover the random energy model, the ultrametric decomposition of Gibbs measures in p-spin glass models and the connection to Poisson-Dirichlet statistics.
Online (see O. Angel, M. Murugan or E. Perkins for link)
Tue 30 Jun 2020, 9:00am-10:30am
Abstract
In this second lecture we introduce the notions of free energy barriers and overlap gaps, and their connection to spectral gap inequalities and algorithmic hardness results.
Online (see O. Angel, M. Murugan or E. Perkins for link)
Wed 1 Jul 2020, 9:00am-10:00am
Abstract
Besides being a classical research topic at the juncture of combinatorics and probability with applications in several other disciplines, random graphs and their phase transitions have been attracting the interest of the statistical physics community. From this perspective, random graphs can be viewed as disordered systems, real-world examples of which include glasses and spin glasses. Physicists have thus brought to bear techniques centered around the notion of "replica symmetry breaking", thereby putting forward a multitude of predictions.
In this course we will learn about the current state of rigorising these predictions, and about the new mathematical tools developed over the recent years. In addition, we will look at applications, particularly in the area of Bayesian inference.
Online (see O. Angel, M. Murugan or E. Perkins for link)
Thu 2 Jul 2020, 10:30am-11:30am
Abstract
Besides being a classical research topic at the juncture of combinatorics and probability with applications in several other disciplines, random graphs and their phase transitions have been attracting the interest of the statistical physics community. From this perspective, random graphs can be viewed as disordered systems, real-world examples of which include glasses and spin glasses. Physicists have thus brought to bear techniques centered around the notion of "replica symmetry breaking", thereby putting forward a multitude of predictions.
In this course we will learn about the current state of rigorising these predictions, and about the new mathematical tools developed over the recent years. In addition, we will look at applications, particularly in the area of Bayesian inference.
T.A. Springer knew decades ago of the existence of a Group invariant commutative algebra structure on the 3875 dimensional representation of E8. It was recently shown by S. Garibaldi and R. Guralnick that the automorphism group of this unique commutative algebra coincides with E8. However a description of this algebra has been a lingering question, ever since it was noticed by T.A. Springer.
In this talk, based on joint work with Skip Garibaldi, we explain a correspondence which associates to each simple Lie algebra, a commutative non associative unital algebra, and provide an explicit closed form expression for the product. This correspondence encompasses the 3875 invariant algebra for E8 via the addition of a unit. These algebras turn out to be simple and are endowed with a non-degenerate “associative” bilinear invariant form. Unlike their closet cousins, the Jordan Algebras, these algebras are not power associative and share the unusual property of having the unit interval as part of their Pierce Spectrum.
Online (see O. Angel, M. Murugan or E. Perkins for link)
Fri 3 Jul 2020, 9:00am-10:00am
Abstract
Besides being a classical research topic at the juncture of combinatorics and probability with applications in several other disciplines, random graphs and their phase transitions have been attracting the interest of the statistical physics community. From this perspective, random graphs can be viewed as disordered systems, real-world examples of which include glasses and spin glasses. Physicists have thus brought to bear techniques centered around the notion of "replica symmetry breaking", thereby putting forward a multitude of predictions.
In this course we will learn about the current state of rigorising these predictions, and about the new mathematical tools developed over the recent years. In addition, we will look at applications, particularly in the area of Bayesian inference.
Online (See O Angel M Murugan or E Perkins for link)
Mon 6 Jul 2020, 9:00am-10:00am
Abstract
Starting in the 70's, physicists have introduced a class of random energy functions and corresponding random probability distributions (Gibbs measures), the are known as mean-field spin glasses. Over the years, it has become increasingly clear that a broad array of canonical models in random combinatorics and (more recently) high-dimensional statistics are in fact examples of mean-field spin glasses, and can be studied using tools developed in that area.
Crucially, these new application domains have brought up a number of interesting new questions that were not central from the viewpoint of stat. physics. These lectures will focus on these new questions:
(i) Statistical questions: what is the accuracy or uncertainty associated to a certain statistical method?
(ii) Computational questions: can we efficiently compute marginals of a Gibbs measure? Can we generate low-energy configurations?
Lecture 1: High dimensional statistics. General setting and key questions. The role of sharp asymptotic. Examples and general phenomena.
Starting in the 70's, physicists have introduced a class of random energy functions and corresponding random probability distributions (Gibbs measures), the are known as mean-field spin glasses. Over the years, it has become increasingly clear that a broad array of canonical models in random combinatorics and (more recently) high-dimensional statistics are in fact examples of mean-field spin glasses, and can be studied using tools developed in that area.
Crucially, these new application domains have brought up a number of interesting new questions that were not central from the viewpoint of stat. physics. These lectures will focus on these new questions:
(i) Statistical questions: what is the accuracy or uncertainty associated to a certain statistical method?
(ii) Computational questions: can we efficiently compute marginals of a Gibbs measure? Can we generate low-energy configurations?
Lecture 2) Message passing algorithms and approximate message passing (AMP). Sharp analysis of AMP.
Starting in the 70's, physicists have introduced a class of random energy functions and corresponding random probability distributions (Gibbs measures), the are known as mean-field spin glasses. Over the years, it has become increasingly clear that a broad array of canonical models in random combinatorics and (more recently) high-dimensional statistics are in fact examples of mean-field spin glasses, and can be studied using tools developed in that area.
Crucially, these new application domains have brought up a number of interesting new questions that were not central from the viewpoint of stat. physics. These lectures will focus on these new questions: (i) Statistical questions: what is the accuracy or uncertainty associated to a certain statistical method? (ii) Computational questions: can we efficiently compute marginals of a Gibbs measure? Can we generate low-energy configurations?
Lecture 3) Optimal AMP algorithms. Connection with Bayes error. Connection with convex optimization.
Comparisons are constantly being made between the 1918 influenza pandemic and the present COVID-19 pandemic. We will discuss our previous work on influenza pandemics, and the tools we have used to understand the temporal patterns of those outbreaks. Applying similar tools to the COVID-19 pandemic is easier in some respects and harder in others. We will describe our current approach to modelling the spread of COVID-19, and some of the challenges and limitations of epidemic forecasting.
Starting in the 70's, physicists have introduced a class of random energy functions and corresponding random probability distributions (Gibbs measures), the are known as mean-field spin glasses. Over the years, it has become increasingly clear that a broad array of canonical models in random combinatorics and (more recently) high-dimensional statistics are in fact examples of mean-field spin glasses, and can be studied using tools developed in that area.
Crucially, these new application domains have brought up a number of interesting new questions that were not central from the viewpoint of stat. physics. These lectures will focus on these new questions: (i) Statistical questions: what is the accuracy or uncertainty associated to a certain statistical method? (ii) Computational questions: can we efficiently compute marginals of a Gibbs measure? Can we generate low-energy configurations?
A result of Forster says that if R is a noetherian ring of (Krull) dimension d, then a rank-n projective module over R can be generated by d+n elements, and results of Chase and Swan imply that this bound is sharp—there exist examples that cannot be generated by fewer than d+n elements. We view "projective modules" as forms of the most trivial kind of non-unital R-algebra, i.e., where the multiplication is identically 0. We take the results of Forster, Chase and Swan as a starting point for investigations into forms of other algebras.
Fix a field k and a k-algebra B, not assumed unital or commutative. Let G denote the automorphism group scheme of B as an algebra. Let U_r denote the variety of r-tuples of elements that generate B as a k-algebra. In favourable circumstances, Ur/G is a k-variety, generalizing the Grassmannian, that classifies forms of the algebra B equipped with r generators. In addition, as far as A1-invariant cohomology theories are concerned Ur/G approximates the classifying stack BG. By measuring the non-injectivity of the map of Chow rings CH(BG)->CH(Ur/G), we can produce examples of algebras (over a ring R) requiring many generators, generalizing the example of Chase and Swan. I will tell a fuller version of this story, with emphasis on the case where B is a matrix algebra, so that Ur/G classifies Azumaya algebras with r generators. The majority of the talk concerns joint work with Uriya First and Zinovy Reichstein, but I will mention some joint work with Taeuk Nam & Cindy Tan and some independent work of Sebastian Gant.
Starting in the 70's, physicists have introduced a class of random energy functions and corresponding random probability distributions (Gibbs measures), the are known as mean-field spin glasses. Over the years, it has become increasingly clear that a broad array of canonical models in random combinatorics and (more recently) high-dimensional statistics are in fact examples of mean-field spin glasses, and can be studied using tools developed in that area.
Crucially, these new application domains have brought up a number of interesting new questions that were not central from the viewpoint of stat. physics. These lectures will focus on these new questions: (i) Statistical questions: what is the accuracy or uncertainty associated to a certain statistical method? (ii) Computational questions: can we efficiently compute marginals of a Gibbs measure? Can we generate low-energy configurations?
Lecture 5) Optimization algorithms for mean field spin glasses.
There is a very simple algorithm for the inference of posteriors for probability Markov models on trees. Asymptotic properties of this algorithm were first studied in statistical physics and have later played a role in coding theory, in machine learning, and in evolutionary inference, among other areas. The lectures will highlight various phase transitions for this model and their connections to modern statistical inference. Finally we show that, perhaps unexpectedly, this ``simple algorithm" requires complex computation in a number of models.
There is a very simple algorithm for the inference of posteriors for probability Markov models on trees. Asymptotic properties of this algorithm were first studied in statistical physics and have later played a role in coding theory, in machine learning, and in evolutionary inference, among other areas. The lectures will highlight various phase transitions for this model and their connections to modern statistical inference. Finally we show that, perhaps unexpectedly, this ``simple algorithm" requires complex computation in a number of models.
There is a very simple algorithm for the inference of posteriors for probability Markov models on trees. Asymptotic properties of this algorithm were first studied in statistical physics and have later played a role in coding theory, in machine learning, and in evolutionary inference, among other areas. The lectures will highlight various phase transitions for this model and their connections to modern statistical inference. Finally we show that, perhaps unexpectedly, this ``simple algorithm" requires complex computation in a number of models.
The first response of epithelial cells to local wounds is a dramatic increase in cytosolic calcium. This increase occurs quickly – calcium floods into damaged cells within 0.1 s, moves into adjacent cells over ~20 s, and appears in a much larger set of surrounding cells via a delayed second expansion over 40-300 s – but calcium is nonetheless a reporter: cells must detect wounds even earlier. Using the calcium response as a proxy for wound detection, we have identified an upstream G-protein-coupled-receptor (GPCR) signaling pathway, including the receptor and its chemokine ligand. We present experimental and computational evidence that multiple proteases released during cell lysis/wounding serve as the instructive signal, proteolytically liberating active ligand to diffuse to GPCRs on surrounding epithelial cells. Epithelial wounds are thus detected by the activation of a protease bait. We will discuss the experimental evidence and a corresponding computational model developed to test the plausibility of these hypothesized mechanisms. The model includes calcium currents between each cell’s cytosol and its endoplasmic reticulum (ER), between cytosol and extracellular space, and between the cytosol of neighboring cells. These calcium currents are initiated in the model by cavitation-induced microtears in the plasma membranes of cells near the wound (initial influx), by flow through gap junctions into adjacent cells (first expansion), and by the activation of GPCRs via a proteolytically activated diffusible ligand (second expansion). We will discuss how the model matches experimental observations and makes experimentally testable predictions.
Systemic chemotherapy is one of the main anticancer treatments used for most kinds of clinically diagnosed tumors. However, the efficacy of these drugs can be hampered by the physical attributes of the tumor tissue, such as tortuous vasculature, dense and fibrous extracellular matrix, irregular cellular architecture, metabolic gradients, and non-uniform expression of the cell membrane receptors. This can impede the transport of therapeutic agents to tumor cells in quantities sufficient to exert the desired effect. In addition, tumor microenvironments undergo dynamic spatio-temporal changes during treatment, which can also obstruct the observed drug efficacy. To examine ways to improve drug delivery on a cell-to-tissue scale (single-cell pharmacology), we developed the microscale pharmacokinetics/pharmacodynamics modeling framework “microPKPD”. I will present how this framework can be used to design optimal schedules for various treatments and to investigate the development of drug-induced resistance.
Online (Ask Liam Watson or Ben Williams for the Zoom link)
Wed 29 Jul 2020, 10:00am-11:00am
Abstract
Joint work with Ajneet Dhillon. The classical Skolem--Noether Theorem by Giraud shows us:
(1) how we can assign to an Azumaya algebra A on a scheme X a class in H^2(X,\mathbf G_m) and
(2) how Azumaya algebras correspond to twisted vector bundles.
The Derived Skolem--Noether Theorem by Lieblich generalizes this to étale forms of derived endomorphism algebras of perfect coherent sheaves in the derived 1-category. We show that in general for a co-family of presentable monoidal quasi-categories with descent over a quasi-category with a Grothendieck topology, there is a fibre sequence giving the assignments (1) and (2). In the case of complexes over a qcqs scheme X, the long exact sequence on homotopy sheaves splits giving in particular the aforementioned versions. Further applications include complexes in Derived Algebraic Geometry, module spectra in Spectral Algebraic Geometry and IndCoh and crystals in Derived Algebraic Geometry in characteristic 0.
The Schramm-Loewner evolution (SLE) is a random fractal curve in the plane that describes the scaling limit of interfaces in several statistical physics models. It is uniquely characterized by two properties known as conformal invariance and the domain Markov property. The first two lectures of the course will be an introduction to SLE and its basic properties via classical Loewner chain theory.
Prerequisities: The course will require no prior knowledge except standard graduate probability courses. We will use small amount of stochastic calculus on a few occasions.
The Schramm-Loewner evolution (SLE) is a random fractal curve in the plane that describes the scaling limit of interfaces in several statistical physics models. It is uniquely characterized by two properties known as conformal invariance and the domain Markov property. The first two lectures of the course will be an introduction to SLE and its basic properties via classical Loewner chain theory.
Prerequisities: The course will require no prior knowledge except standard graduate probability courses. We will use small amount of stochastic calculus on a few occasions.
It has been well established that the mechanical stiffness of the substrate with which cells interact affects various intracellular processes, including cell spread areas, speeds at which motile cells translocate, and the number and strength of cell-substrate adhesions. This mechanosensitivity is modulated through conformational changes in cell-substrate adhesion proteins that in turn regulate downstream processes, including processes involving proteins required for motility, actin and myosin. The aim of this work is to better understand how substrate stiffness affects actin dynamics and myosin activity in cell spreading. We present an axisymmetric model of a flat cell spreading on a two-dimensional substrate. The actin network is modeled as a viscous gel, and actin spreading and contraction dynamics are incorporated into the model as a local active rate of deformation. The model also incorporates membrane tension and stress-dependent focal adhesion dynamics, which in turn modulate a cell’s protrusive activity and speed of actin retrograde flow, thereby controlling the spreading rate. Using this model, we are able to recapitulate the three phases of cell spreading dynamics described in Gianonne et al. (Cell, 2004), and we predict how the balance of protrusive activity, actin retrograde flow, adhesion strength, and local actomyosin contractions are affected by substrate stiffness.
The Schramm-Loewner evolution (SLE) is a random fractal curve in the plane that describes the scaling limit of interfaces in several statistical physics models. It is uniquely characterized by two properties known as conformal invariance and the domain Markov property. The third lecture will be about imaginary geometry which gives a very useful alternative perspective on SLE.
Prerequisities: The course will require no prior knowledge except standard graduate probability courses. We will use small amount of stochastic calculus on a few occasions.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 14 Sep 2020, 8:00am-9:00am
Abstract
Kneser proposed a conjecture that if G is a semi-simple simply connected linear algebraic group defined over a number field k and Y a principal homogeneous space under G, then Y satisfies Hasse principle, i.e., Y has a rational point over k if it does over completions of k at all its places. This is now a theorem due to Kneser for classical groups, Harder for exceptional groups of type other than E8 and Chernousov for groups of type E8. There were questions and conjectures on similar Hasse principles over function fields of p-adic curves and more generally, semi global fields, i.e., function fields of curves over complete discrete valued fields, with respect to all their discrete valuations. We shall discuss recent progress in this direction.
Consider three gamblers with initial capital A,B,C. Each time, a pair of them is chosen at random, a fair coin is flipped and $1 is transferred. Classically, the chance that the first gambler wins all the money is A/ (A + B + C). Consider the first time one of the three goes broke. How long does this take, how is the money split between the other two (and how do these things depend on A,B,C)? These problems are seen as facts about rates of convergence to quasi-stationarity for Markov chains. In joint work with Kelsey Houston-Edwards and Laurent Saloff-Coste we have identified geometric conditions (John Domains, Whitney covers, inner regular domains) which allow fairly sharp analysis. This borrows heavily from geometry,PDE and harmonic analysis(Carleson estimates, parabolic Harnack inequalities). Adapting the tools to discrete problems is interesting work. I will try to explain all of the above 'in English'.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 21 Sep 2020, 8:00am-9:00am
Abstract
Using ideas of Kac and Vinberg, we give a simple combinatorial method of computing the Galois cohomology of semisimple groups over the field R of real numbers. I will explain the method by the examples of simple groups of type E_7 (both adjoint and simply connected). This is joint work with Dmitry A. Timashev, Moscow. Preprint available at http://arxiv.org/abs/2008.11763
Abstract. This talk focuses on estimating temporal couplings of stochastic processes with optimal transport (OT), motivated by applications in developmental biology and cellular reprogramming. For nearly a century, the prevailing mathematical theory of developmental biology has been based on Waddington's `epigenetic landscape’ – a potential energy surface that determines trajectories of cellular development. Now, with the advent of high-throughput measurement technologies like single cell RNA-sequencing (scRNA-seq), the prospect of charting this landscape is within reach. This holds tremendous potential for diverse applications from regenerative medicine (e.g. cellular reprogramming) to agriculture (e.g. predicting impacts of climate change on crops or growing artificial meat). While the problem of recovering the landscape is inherently nonconvex, we demonstrate that the ‘laws on paths’ induced by this potential energy surface can be recovered using convex optimization. Our approach provides a general framework for investigating cellular differentiation.
Zoom - contact organizer (abuttens at math ubc ca) for meeting id
Wed 23 Sep 2020, 2:05pm-3:05pm
Abstract
Wild-type zebrafish (Danio rerio) are characterized by black and yellow stripes, which form on their body and fins due to the self-organization of thousands of pigment cells. Mutant zebrafish and sibling species in the Danio genus, on the other hand, feature altered, variable patterns, including spots and labyrinth curves. The longterm goal of my work is to better link genotype, cell behavior, and phenotype by helping to identify the specific alterations to cell interactions that lead to these different fish patterns. Using a phenomenological approach, we develop agent-based models for cell interactions and simulate pattern formation on growing domains. In this talk, I will overview our models and highlight some topological techniques that allow us to quantitatively compare our simulations to in vivo images. I will also discuss current directions and open questions related to taking a more mechanistic and quantitative approach to describing cell behavior in zebrafish.
Note for Attendees
Dr. Volkening will also deliver the Rising Stars colloquium on Oct 30, 2020.
Wilson's algorithm allows efficient sampling of the uniform spanning tree (UST) by using loop-erased random walks. This connection gives a tractable method to study the UST. The strategy has been fruitful for scaling limits of the UST in the planar case and high dimensions. However, three-dimensional scaling limits are far from understood. This talk is about recent advances in this problem when we describe the UST as a metric measure space. Our main result is on the existence of sub-sequential scaling limits and convergence under dyadic scalings with respect to a Gromov-Hausdorff-type topology. We will also discuss some properties of the limit tree.
This is joint work with Omer Angel, David Croydon, and Daisuke Shiraishi.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 28 Sep 2020, 8:00am-9:00am
Abstract
Our talk deals with the cohomological invariants of smooth and connected linear algebraic groups over an arbitrary field. The notion of cohomological invariants was formalized by Serre in the 90’s. It enables to study via Galois cohomology the geometry of linear algebraic groups or forms of algebraic stuctures (such as central simple algebras with involution).
We intend to introduce the general ideas of the theory and to present a generalization of a result by Blinstein and Merkurjev on degree 2 invariants with coefficients Q/Z(1), that is invariants taking values in the Brauer group. More precisely our result gives a description of these invariants for every smooth and connected linear groups, in particular for non reductive groups over an imperfect field (as pseudo-reductive or unipotent groups for instance).
Motivated by neuroscience phenomena, the study of convex neural codes seeks to characterize how convex sets may intersect and overlap one another in Euclidean space. We will see how certain arrangements of convex sets, called sunflowers and flexible sunflowers, can help answer this question. In particular, we will describe a new Helly-type theorem that constrains these arrangements, and highlight several applications to the theory of convex neural codes. We will briefly contextualize these results by examining minors of codes, a framework that is somewhat analogous to minors of graphs.
Knots in S^3 modulo cylindrical cobordisms form a group (the operation is connect sum) called the concordance group. This group has been studied extensively, and the difference between this group in the smooth setting and this group in the topological setting was used to construct non-standard smooth structures on R^4. A related group called the strongly invertible concordance group is defined for knots with a particular type of involution called a strong inversion up to equivariant cylindrical cobordism. Interestingly, the usual concordance group is a subgroup of the strongly invertible concordance group. In this talk I will give an introduction to the strongly invertible concordance group and discuss ongoing work with Ahmad Issa. In particular, I will discuss the equivariant genera of strongly invertible knots and two new group homomorphisms from the strongly invertible concordance group to the usual concordance group.
Zoom - contact organizer (abuttens at math ubc ca) for meeting id
Wed 30 Sep 2020, 2:05pm-3:05pm
Abstract
The mammalian hippocampus, comprised of serially connected subfields, participates in a variety of functions involving memory. It has been postulated that parallel circuitry embedded within hippocampal subfields may underlie such functional diversity. Here, I will present our work spanning experimental, computational, and mathematical neuroscience that shows distinct processes of memory that can emerge from parallel hippocampal circuits.
We study a natural random walk over the upper triangular matrices, with entries in Z/mZ, generated by steps which add or subtract row i+1 to row i. We show that the mixing time of the lazy random walk is O((nm)^2) which is optimal up to constants. This generalizes a result of Peres and Sly and answers a question of Stong and of Arias-Castro, Diaconis and Stanley. This is joint work with Allan Sly.
https://ubc.zoom.us/j/67916711780 (password:the number of lines on a generic quintic threefold)
Fri 2 Oct 2020, 8:30am-10:00am
Abstract
The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the SL_2 character variety of a topological surface. I will explain how to realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov-Witten invariants of a log Calabi-Yau cubic surface. This leads to a proof of a previously conjectured positivity property of the bracelets bases of the skein algebras of the 4-punctured sphere and of the 1-punctured torus.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 5 Oct 2020, 8:00am-9:00am
Abstract
Techniques involving reduction are very common in number theory and arithmetic geometry. In particular, elliptic curves and general abelian varieties having good reduction have been the subject of very intensive investigations over the years. The purpose of this talk is to report on recent work that focuses on good reduction in the context of reductive linear algebraic groups over finitely generated fields. In addition, we will highlight some applications to the study of local-global principles and the analysis of algebraic groups having the same maximal tori. (Parts of this work are joint with V. Chernousov and A. Rapinchuk.)
Abstract: Loosely speaking, an integer program (IP) becomes simpler if we relax some of the constraints that enforce variables be integer-valued. In this talk, we analyze IPs using mixed integer relaxations (MIPs). For integer linear programs, we consider structural questions such as 'When can an IP be recast as an MIP with fewer integer variables?' and 'What is the distance between solutions to an IP and solutions to a linear program (LP)?' If time permits, we will also consider how to extend the algorithmic principles of gradient descent, which is a convex optimization algorithm, to an algorithm for convex integer programs.
We construct three new Hopf algebras based on the Loday-Ronco operations on trees. The first involves a new visualisation of parking functions as decorated binary trees; it is bidendriform and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon. The second and third algebras are on planar trees and labelled planar trees, with connections to the algebra of packed words WQSym. We define a partial order on each of the three types of trees, generalising the m-Tamari order of Bergeron and Preville-Ratelle, through which we obtain a monomial basis on each algebra that exhibits its cofreeness. We give a positive product formula and a cancellation-free antipode formula for our monomial bases, through our new abstraction of the arguments of Aguiar and Sottile on the Malvenuto-Reutenauer algebra - that is, we list axioms for the partial order on a general combinatorial Hopf algebra that guarantee such formulas.
This is joint work with N. Bergeron, R. Gonzalez D'Leon, A. Pang, Y. Vargas.
Both Keegan and Ahmad have given talks recently about equivariant surfaces bounding knots with particular symmetries. Picking up this thread, I will focus on the case of strong inversions and, leaving aside the questions of equivariant surfaces, give an overview of the various invariants available for studying these symmetries in the context of strong inversions that can be derived from Khovanov homology. The talk will centre on examples.
In this talk I will discuss activated random walk, an interacting particle system that exhibits a phase transition on infinite domains and self-organized criticality on finite domains. In the infinite version, the system is initialized with density μ of particles which perform independent simple random walk, fall asleep at rate λ, and are woken up if another particle moves to the same site. There are two possible limiting behaviors: local fixation, where each site is visited finitely many times a.s., or non-fixation, where each site is visited infinitely many times a.s. Current research is focused on determining where the transition between these two phenomena occurs in terms of μ and λ. I will present some recent results and discuss the novel tools involved.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Fri 9 Oct 2020, 8:30am-10:00am
Abstract
We conjecture that any connected component $Q$ of the moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ where $X$ is a smooth projective variety, $E$ is a normal crossing anti-canonical divisor with a 0-stratum, every $E_i$ is smooth, and $\Theta$ is an ample divisor not containing any 0-stratum of $E$, is \emph{unirational}. More precisely: note that $Q$ has a natural embedding into the Kollár-Shepherd-Barron-Alexeev moduli space of stable pairs, we conjecture that its closure admits a finite cover by a complete toric variety. We construct the associated complete toric fan, generalizing the Gelfand-Kapranov-Zelevinski secondary fan for reflexive polytopes. Inspired by mirror symmetry, we speculate a synthetic construction of the universal family over this toric variety, as the Proj of a sheaf of graded algebras with a canonical basis, whose structure constants are given by counts of non-archimedean analytic disks. In the Fano case and under the assumption that the mirror contains a Zariski open torus, we construct the conjectural universal family, generalizing the families of Kapranov-Sturmfels-Zelevinski and Alexeev in the toric case. In the case of del Pezzo surfaces with an anti-canonical cycle of $(-1)$-curves, we prove the full conjecture. The reference is arXiv:2008.02299 joint with Hacking and Keel.
The Restriction and Kakeya conjectures are two central open problems in harmonic analysis. Incidence geometry is a branch of combinatorics that helps us understand extremal arrangements of simple geometric objects such as points and lines. In this talk I will discuss some recent advances in incidence geometry, and how discretized versions of these results can be used to attack the Restriction and Kakeya conjectures.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 12 Oct 2020, 8:00am-9:00am
Abstract
This is a joint work with A.Vishik. A relation between Chow theory and algebraic K-theory of smooth algebraic varieties is given by a ring homomorphism from the Chow ring to the graded Grothendieck ring of a variety associated with the topological filtration. A much better relation can be established via connective K-theory that maps to both Chow theory and K-theory, so the connective K-theory deserves detailed study.
Steenrod operations (mod p) and Adams operations are essentially all additive operations in Chow theory and K-theory respectively. In the talk we describe the ring of additive operations in connective K-theory.
Large scale dynamics of the oceans and the atmosphere are commonly governed by the primitive equations (PEs), also known as Hydrostatic Euler Equations. It is well-known that the 3D viscous primitive equations are globally well-posed in Sobolev spaces. In this talk, first, I will briefly discuss the ill-posedness in Sobolev spaces, the local well-posedness in the space of analytic functions, and the finite-time blowup of solution to the 3D inviscid PEs with rotation (Coriolis force). Moreover, I will demonstrate, in the case of “well-prepared” analytic initial data, the regularizing effect of the Coriolis force by providing a lower bound for the life-span of the solutions that grows toward infinity with the rotation rate. These are joint works with T. E. Ghoul (NYUAD), Quyuan Lin (Texas A&M) and Edriss S. Titi (Texas A&M and University of Cambridge).
In this talk I will describe some current efforts to understand the high-degree rational cohomology of SL_n(Z), or more generally the cohomology of SL_n(O) when O is a number ring. Although the groups SL_n(O) do not satisfy Poincare duality, they do satisfy a twisted form of duality, called Bieri-Eckmann duality. Consequently, their high-degree rational cohomology groups are governed by an SL_n(O)-representation called the Steinberg module. I will overview some results and ongoing work concerning these representations.
In this talk, we'll present a detailed analysis of the set of weighted homomorphisms from the discrete torus Z_m^n, where m is even, to any fixed graph. We'll show that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some dominant phase. We'll discuss some applications of this result which include solutions to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. We'll also introduce some classical tools from statistical physics which form the basis of our proofs. This is joint work with Peter Keevash.
Zoom - contact organizer (abuttens at math ubc ca) for meeting id
Wed 14 Oct 2020, 2:05pm-3:05pm
Abstract
Developing effective cancer treatment presents a unique challenge due to the overwhelming variability in tumour cell behaviour and spatial heterogeneity. Virotherapy is a type of cancer treatment that uses genetically engineered viruses to infect and lyse cancerous cells. When these viruses are administered with immune cells or immunostimulatory cytokines, an antitumour immune response is instigated. Developing a hybrid PDE/agent-based modelling for the treatment of glioblastoma (a type brain cancer), we predicted the variability in glioblastoma cells that hinders the efficacy of oncolytic virotherapy. We then show how this treatment could be improved for the majority of patients. Recently, gel-based mediums have been used to improve the efficacy of oncolytic virotherapy by providing a sustained therapeutic delivery of the vectors . Using a system of ODEs and a genetic algorithm, we show how this treatment could be further optimised by changing the gel-material to reduce the immune cell release rate. Overall, this talk aims to demonstrate complementing mathematical models and their applications in oncolytic virotherapy.
https://ubc.zoom.us/j/67916711780 (password:the number of lines on a generic quintic threefold)
Fri 16 Oct 2020, 8:30am-10:00am
Abstract
Holomorphic Floer theory is a joint project with Maxim Kontsevich, which is devoted to various aspects of the Floer theory in the framework of complex symplectic manifolds.
In my talk I will consider an important special case of the general story. Exponential integrals in finite and infinite dimension can be thought of generalization of the theory of periods (i.e variations of Hodge structure). In particular, there are comparison isomorphisms between Betti and de Rham cohomology in the exponential setting. These isomorphisms are corollaries of categorical equivalences which are incarnations of our generalized Riemann-Hilbert correspondence for complex symplectic manifolds.
Furthermore, fomal series which appear e.g. in the stationary phase method or Feynman expansions (in infinite dimensions) are Borel re-summable (resurgent). If time permits I will explain the underlying mathematical structure which we call analytic wall-crossing structure. From the perspective of Holomorphic Floer theory it is related to the estimates for the number of pseudo-holomorphic discs with boundaries on two given complex Lagrangian submanifolds.
I will explain what pi_3^{A1}(A^3 - {0}) means, and I will explain part of a presentation of this sheaf of groups and how to derive it. I will indicate how the result implies one case of a conjecture by Suslin from the 1980s in the K-theory of fields. This represents joint work with Aravind Asok and Jean Fasel.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 19 Oct 2020, 8:00am-9:00am
Abstract
Let k be a field and A a finite discrete Galois module. For any integer n >1, let x be a cohomology class in H^n(k, A). We show that there exists a countable familiy of (smooth, geometrically integral) k-varieties, such that the following holds: for any field extension K/k, the restriction of x vanishes in H^n(K, A) if and only if one of the varieties has an K-point. In the case n= 2, we note that one variety (called a splitting variety for x) is enough. The question of the existence of splitting varieties (or splitting families) is insprired by the construction of norm varieties for symbols by Rost. This is joint work with Mathieu Florence.
Asymptotically conical Calabi-Yau manifolds are a special class of complete Ricci-flat Kähler manifolds that are asymptotic to a cone at infinity. They can be constructed analytically by solving a complex Monge-Ampere equation, and the first construction goes back to the work of Tian and Yau in the 90s. In this talk, I will discuss some work with Tristan Collins and Bin Guo, where we study the degenerations of asymptotically conical Calabi-Yau metrics. As an application, we construct many examples of asymptotically conical Calabi-Yau metrics with singularities on quasi-projective varieties.
We discuss the large deviation theory in the context of a sequence of measures on the graphon space that is obtained by sampling from a fixed graphon. In the case of constant graphons, the existence of LDP with speed n^2 was proved by Chatterjee and Varadhan and it was then used to investigate the upper tail problem of triangle counts for homogeneous Erdős–Rényi random graph. In this talk, we show that sampling from a general graphon may lead to two non-trivial LDPs that correspond to speed n and n^2.
The L-space conjecture posits a connection between the Heegaard-Floer homology of a $3$-manifold, whether or not the manifold supports a coorientable taut foliation, and whether or not the fundamental group acts on the real line by orientation-preserving homeomorphisms (i.e. is left-orderable). This talk will focus on the last of these three conditions, and will introduce new tools for investigating it.
More specifically, in the context of low-dimensional topology it turns out that actions on the circle (i.e. circular orderings) are easier to find in nature than actions on the real line. Motivated by this, I'll introduce new theorems that show how you can determine whether or not a given circularly orderable group is left-orderable, so long as you're very good at finding circular orderings on direct products. As every circular ordering of a group is, by definition, a very particular kind of 2-cocycle, the answer naturally involves a detailed analysis of the second cohomology of the group. This is joint work with Jason Bell and Ty Ghaswala.
Zoom - contact organizer (abuttens at math ubc ca) for meeting id
Wed 21 Oct 2020, 2:05pm-3:05pm
Abstract
Nature has evolved many mechanisms for achieving directed motion on the subcellular level. The burnt-bridges ratchet (BBR) is one mechanism used to achieve superdiffusive molecular motion over long distances through the successive cleavage of surface-bound energy-rich substrate sites. The BBR mechanism is utilized throughout Nature: it can be found in bacteria, plants, humans, as well as non-life forms such as influenza. Motivated to understand how fundamental design principles alter BBR kinetics, we have built both computer models as well as synthetic experimental systems to understand BBR kinetics. In this talk I will present the results of our modelling work where we explore how multivalency, leg length, hub topology, landscape dimension, and landscape elasticity affect BBR kinetics. I will also present the preliminary results of our experimental work where we have created a micron-sized BBR that has achieved superdiffusive motion on a two-dimensional landscape. Our work provides insight into the mechanistic origin for the observed velocities and persistence found in both synthetic and biological (eg. Influenza and ParA/ParB) systems.
https://ubc.zoom.us/j/67916711780 (password:the number of lines on a generic quintic threefold)
Fri 23 Oct 2020, 8:30am-10:00am
Abstract
The Quot schemes of surfaces parametrizing quotients of dimension at most 1 of the trivial sheaf carry 2-term perfect obstruction theories. Several generating series of associated virtual invariants are conjectured to be given by rational functions. We show this is the case for several geometries including all smooth projective surfaces with p_g>0. This talk is based on joint work with Noah Arbesfeld, Drew Johnson, Woonam Lim and Rahul Pandharipande.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 26 Oct 2020, 8:00am-9:00am
Abstract
Let G be a linear algebraic group over a field F. As introduced by Serre, degree n cohomological invariants of G with coefficients in a group A, where A is equipped with an action of the absolute Galois group of F, are natural transformations of Galois cohomology functors H^1(-,G) -> H^n(-,A). Commonly studied are the degree three invariants with coefficients in Q/Z tensor Q/Z. These invariants were recently described by Merkurjev for the semisimple adjoint case, and by Bermudez and Ruozzi for semisimple G which are neither simply connected nor adjoint. In particular, they described the structure of the normalized degree three invariants (those which send the trivial object to zero) of the half-spin group HSpin_16. By generalizing a technique of Garibaldi we construct new injections into HSpin induced by the Kronecker tensor product map. In particular we construct an injection PSp_2n X PSp_2m -> HSpin_4nm which we use to describe the normalized invariants of HSpin_4k for any k, generalizing the result of Bermudez and Ruozzi.
The directions determined by a point set are the slopes of lines passing through at least two points of the set. A seminal result of R\'edei tells us that at least $(p+3)/2$ directions are determined by $p$ points in $AG(2,p)$. We consider cartesian product point sets, i.e. a set of the form $A \times B \subset AG(2,p)$, where $p$ is prime, $A$ and $B$ are subsets of $GF(p)$ each with at least two elements and $|A||B| <p$. In this case, we show that the number of directions determined is at least $|A||B| - \min\{|A|,|B|\} + 2$. We'll see that this gives an upper bound of about $\sqrt{p/2}$ on the clique number of Paley graphs, matching a bound obtained by Hanson and Petridis last year. Our main tool is the use of the R\'edei polynomial with Sz\H{o}nyi's extension. This is a joint work with J\'ozsef Solymosi and Daniel Di Benedetto
In joint work with Vladimir Sosnilo we proved a version of the
Dundas-Goodwillie-McCarthy theorem, relating K theory and topological
cyclic homology (TC), for perfect complexes over suitable algebraic stacks.
The same setup also works to say something about the K-theory and TC of
Voevodsky motives versus Chow motives. I will explain the characters
involved, the proof and what the future might hold.
Zoom - contact organizer (abuttens at math ubc ca) for meeting id
Wed 28 Oct 2020, 2:05pm-3:05pm
Abstract
In general, the mechanisms by which walled cells specify their shapes are not fully understood. This project aims to understand the cell-shape formation during tip-cell elongation at the early developmental stage of moss. To do so, we have developed a mathematical method that decomposes cell wall “active” growth and cell wall “passive” stretching due to turgor pressure. We demonstrate that it is possible to map the active and passive physical effects using cell outline data and cell-wall marker techniques such as quantum dots.
Joint work with: Luis Vidali, Giulia Galotto, Danush Chelladurai, Yaqi Deng, Chaozhen Wei, and Kamryn Spinelli.
https://ubc.zoom.us/j/67916711780 (password:the number of lines on a generic quintic threefold)
Fri 30 Oct 2020, 8:30am-10:00am
Abstract
The D-equivalence conjecture predicts that birationally equivalent projective Calabi-Yau manifolds have equivalent derived categories of coherent sheaves. It is motivated by homological mirror symmetry, and has inspired a lot of recent work on connections between birational geometry and derived categories. In dimension 3, the conjecture is settled, but little is known in higher dimensions. I will discuss a proof of this conjecture for the class of Calab-Yau manifolds that are birationally equivalent to some moduli space of stable sheaves on a K3 surface. This is the only class for which the conjecture is known in dimension >3. The proof uses a more general structure theory for the derived category of an algebraic stack equipped with a Theta-stratification, which we apply in this case to the Harder-Narasimhan stratification of the moduli of sheaves.
Election dynamics are a rich complex system, and forecasting next month's U.S. elections is an exciting, high-stakes problem with many sources of subjectivity and uncertainty. In this talk, we take a dynamical-systems perspective on election forecasting, with the goal of helping to shed light on the forecast process and raise questions for future work. By adapting a well-studied model from epidemiology, we show how to combine a compartmental approach with polling data to produce forecasts of presidential, senatorial, and gubernatorial elections at the state level. Our results for the last 16 years of U.S. elections are largely in agreement with those of popular analysts, and we apply our model to forecast the upcoming U.S. elections on 3 November 2020. We also use our modeling framework to explore how different methods for handling polling data and accounting for uncertainty affect forecasts.
This is joint work with Samuel Chian, William He, Christopher Lee, Daniel Linder, Mason Porter, and Grzegorz Rempala.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 2 Nov 2020, 8:00am-9:00am
Abstract
Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x1, x1-1, ... , xn, xn-1]. We prove that G has positive isotropic rank over R iff it has positive isotropic rank over the field of fractions k(x1, ... ,xn) of R, and if this is the case, then the natural map H1et(R,G) --> H1et( k(x1,... ,xn), G ) has trivial kernel and G is loop reductive, i.e. G contains a maximal R-torus. We also deduce that if G is a reductive group over R of isotropic rank >= 2, then the natural map of non-stable K1-functors K1G(R) --> K1G( k((x1)) ... ((xn)) ) is injective, and an isomorphism if G is moreover semisimple.
We analyze a model problem based on highly disparate elastic constants that we propose in order to understand the formation of tactoids--that is domains of phase in nematic/isotropic samples whose boundaries contain singularities in the form of corners or cusps. For a bounded planar domain $\Omega$ we investigate the $\epsilon \to 0$ asymptotics of the variational problem
\[
\inf \frac{1}{2}\int_\Omega \left( \frac{1}{\epsilon} W(u)+\epsilon |\nabla u|^2 + L_\e({\rm{div}} u)^2 \right) \,dx
\]
within various parameter regimes for $L_\epsilon > 0.$ Here $u:\Omega\to R^2$ and $W$ is a potential vanishing on the unit circle and at the origin. When $\epsilon\ll L_\epsilon\to 0$, we show that these functionals $\Gamma-$converge to a constant multiple of the perimeter of the phase boundary and the divergence penalty is not felt. However, when $L_\epsilon \equiv L > 0$, we find that a tangency requirement along the phase boundary for competitors in the conjectured $\Gamma$-limit becomes a mechanism for development of singularities. We establish criticality conditions for this limit and under a non-degeneracy assumption on the potential we prove compactness of energy bounded sequences in $L^2$. The role played by this tangency condition on the formation of interfacial singularities is investigated through several examples: each of these examples involves analytically rigorous reasoning motivated by numerical experiments. We argue that generically, ``elastic wall'' singularities between $S^1$-valued states are expected near the defects along the phase boundary. This is joint work with Dmitry Golovaty (Akron), Michael Novack (UT Austin)and Raghav Venkatraman (Carnegie Mellon).
Finding reasonably good upper and lower bounds for the clique number of Paley graphs is an old and open problem in additive combinatorics. For a (generalized) Paley graph over the finite field \F_q, the trivial upper bound on its clique number is \sqrt{q}. In this talk, I will discuss how to improve the trivial upper bound. I will also talk about its connection to the number of directions determined by a Cartesian product in an affine Galois plane.
Decoupling theory studies the following problem. Given finitely many functions $f_j$ defined on the plane with disjoint Fourier supports, how do they interact in the physical space? More precisely, give a Lebesgue exponent $p$, how does the $L^p$-norm of the sum of $f_j$'s compare with the $l^2$-sum of the $L^p$-norm of $f_j$? In 2015, Bourgain and Demeter proved a remarkable result when each $f_j$ has Fourier support on a neighborhood of the graph of the unit parabola over $I_j$ where $\{I_j\}$ is a uniform partition of the unit interval. In my work, I generalized this theorem to the case where each $f_j$ is Fourier supported on a neighborhood of the graph of a general polynomial curve over $I'_j$ where $\{I'_j\}$ is a partition of the unit interval that is determined by the curvature of each polynomial.
We will discuss a classification of which positive integral surgeries on positive torus knots bound rational homology balls. Additionally, for a given knot K we consider which cables K(p,q) admit integral surgeries that bound rational homology balls. For such cables, let S(K) be the set of corresponding rational numbers q/p. We show that if n-surgery on K bounds a rational homology ball then n is an accumulation point for S(K).
This is joint work with Paolo Aceto, Marco Golla, and Ana G. Lecuona.
In long-range percolation on Z^d, each potential edge {x,y} is included independently at random with probability roughly beta ||x-y||^{-d-alpha}, where alpha > 0 controls how long-range the model is and beta > 0 is an intensity parameter. The smaller alpha is, the easier it is for very long edges to appear. We are normally interested in fixing alpha and studying the phase transition that occurs as beta is increased and an infinite cluster emerges. Perhaps surprisingly, the phase transition for long-range percolation is much better understood than that of nearest neighbour percolation, at least when alpha is small: It is a theorem of Noam Berger that if alpha < d then the phase transition is continuous, meaning that there are no infinite clusters at the critical value of beta. (Proving the analogous result for nearest neighbour percolation is a notorious open problem!) In my talk I will describe a new, quantitative proof of Berger's theorem that yields power-law upper bounds on the distribution of the cluster of the origin at criticality.
As a part of this proof, I will describe a new universal inequality stating that on any graph, the maximum size of a percolation cluster is of the same order as its median with high probability. This inequality can also be used to give streamlined new proofs of various classical results on e.g. Erdos-Renyi random graphs, which I will hopefully have time to talk a little bit about also.
https://ubc.zoom.us/j/67916711780 (password:the number of lines on a generic quintic threefold)
Fri 6 Nov 2020, 9:00am-10:30am
Abstract
Given a smooth complex variety X (not necessarily compact), consider the unordered configuration space Conf^n(X) defined as {(x_1,...,x_n)\in X^n: x_i \neq x_j for i\neq j} / S_n. The singular cohomology of Conf^n(X) has long been an active area of research. In this talk, we investigate the following phenomenon: "puncturing once more" seems to have a very predictable effect on the cohomology of configuration spaces when the variety we start with is noncompact. In specific, a formula of Napolitano determines the Betti numbers of Conf^n(X - {P}) from the Betti numbers of Conf^m(X) (m \leq n) if X is a smooth *noncompact* algebraic curve and P is a point. We present a new proof using an explicit algebraic method, which also upgrades this formula about Betti numbers into a formula about mixed Hodge numbers and generalizes this formula to certain cases where X is of higher dimension.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 9 Nov 2020, 8:00am-9:00am
Abstract
In this talk, we will discuss almost homogeneous varieties for reductive groups over a perfect field k. Let K be an algebraic closure of k, and let G be a connected reductive K-group with a fixed k-form F. A normal G-variety over K is almost homogeneous if it has an open dense orbit. Given an almost homogeneous G-variety X, the goal of this talk will be to determine k-forms of X which are compatible with the k-form F of G. In order to do this, we describe an action of the Galois group on the combinatorics developed in Luna-Vust theory. This is joint work with Ronan Terpereau.
Quantum spin systems are many-body models which are of wide interest in modern physics and at the same time amenable to rigorous mathematical analysis. A central question about a quantum spin system is whether its Hamiltonian exhibits a spectral gap above the ground state. The existence of such a spectral gap has far-reaching consequences, e.g., for the ground state complexity. In this talk, we survey recent progress on deriving spectral gaps for frustration-free quantum spin systems in dimensions greater than 1, including in the antiferromagnetic models of Affleck-Kennedy-Lieb-Tasaki (AKLT).
In this talk, we will discuss how to improve the trivial upper bound on the clique number of Paley graphs and generalized Paley graphs using a different method. I will revisit the direction set determined by a Cartesian product in an affine Galois plane $AG(2,q)$ and show that most Cartesian products determine many directions. The main result we obtained is: for any positive function $h$ such that $h(x)=o(x)$ as $x \to \infty$, $\omega(P(q,d)) \leq \sqrt{q}-h(p)$ for almost all non-squares $q$.
https://ubc.zoom.us/j/67916711780 (password:the number of lines on a generic quintic threefold)
Fri 13 Nov 2020, 8:30am-9:30am
Abstract
(w E. Gonzalez) I will discuss a generalization of the Kirwan map to quantum K-theory, a presentation of quantum K-theory of toric varieties, and some open questions.
https://ubc.zoom.us/j/67916711780 (password:the number of lines on a generic quintic threefold)
Fri 13 Nov 2020, 9:45am-10:45am
Abstract
In the 90s', Witten gave a physical derivation of an isomorphism between the Verlinde algebra of GL(n) of level l and the quantum cohomology ring of the Grassmannian Gr(n,n+l). In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten's work by relating the GL_n Verlinde numbers to the level l quantum K-invariants of the Grassmannian Gr(n,n+l), and refer to it as the Verlinde/Grassmannian correspondence. The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there.
In this talk, I will first explain the background of this correspondence and its interpretation in physics. Then I will discuss the main idea of the proof for arbitrary rank. A new technical ingredient is the virtual nonabelian localization formula developed by Daniel Halpern-Leistner.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 16 Nov 2020, 8:00am-9:00am
Abstract
The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is split and R is unramified. To overcome obstacles that have so far kept the mixed characteristic case out of reach, we rely on the recently-established Cohen–Macaulay version of the resolution of singularities.
We discuss the existence problem of constant Chern scalar curvature metrics on a compact complex manifold. We prove a priori estimates for these metrics conditional on an upper bound on the entropy, extending a recent result by Chen-Cheng in the K\"ahler setting.
Zarankiewicz's problem in extremal graph theory asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain a copy of $K_{k,k}$, the complete bipartite with $k$ vertices in both classes. We will consider this question for incidence graphs of geometric objects. Significantly better bounds are known in this setting, in particular when the geometric objects are defined by systems of algebraic inequalities. We show even stronger bounds under the additional constraint that the defining inequalities are linear. We will also discuss connections of these results to combinatorial geometry and model theory. Joint work with Artem Chernikov, Sergei Starchenko, Terence Tao, and Chieu-Minh Tran.
In this talk, I will discuss recent work on the Fourier restriction problem for hyperboloids, focusing primarily on the one-sheeted hyperboloid in three dimensions. This surface exhibits both negative curvature and asymptotically conic behavior, making it interesting from the viewpoint of restriction theory. Local estimates via polynomial partitioning and global estimates via the bilinear method will be presented. This is partially joint work with Diogo Oliveira e Silva and Betsy Stovall.
Zoom - contact organizer (abuttens at math ubc ca) for meeting id
Wed 18 Nov 2020, 11:00am-12:00pm
Abstract
In this talk, we will discuss two studies performed by the SMILE group (Collčge de France/Sorbonne Université) in the beginning of the pandemic.
(1) We study a model where infected individuals can be symptomatic or asymptomatic, use a contact tracing mobile app or not. We investigate the effect of non-digital interventions (voluntary isolation upon symptom onset, quarantining private contacts) and of digital interventions (contact tracing thanks to the app), depending on the willingness to quarantine, parameterized by four cooperating probabilities. We show that moderate rates of adoption of a contact tracing app can reduce R0 but are by no means sufficient to reduce it below 1 unless it is already very close to 1 thanks to non-digital interventions.
(2) We present a general and tractable framework for modeling and "nowcasting" the epidemic at a national scale. Our approach is based on a fairly general stochastic model for complex diseases using an arbitrarily large number of types (e.g., infective stage, clinical state, risk factor class). We show how structuring the infected population by its infection age, i.e., time elapsed since infection, allows us to decouple dependencies between stages and to time. In the large population limit (obtained either by assumption or as a spontaneous effect of the outbreak), the global scale of the epidemic is well captured by a deterministic McKendrick-Von Foerster 1-d PDE, and such an approximation allows us to make robust predictions on the fate of the epidemic.
In low-dimensional topology we are often interested in determining 3-manifolds that arise as surgery along a knot, and investigating the surfaces they contain. In this talk, we study gluings X of the twisted I-bundle over the Klein bottle to knot complements, and investigate which gluings can be realized as integral Dehn surgery along a knot in S^3. All closed, orientable 3-manifolds containing a Klein bottle can be presented as such a gluing, and Heegaard Floer homology provides a way to study surgery obstructions. Using recent immersed curves techniques, we prove that if X is 8-surgery along a genus two knot and arises as such a gluing with an S^3 knot complement, then it is an L-space and the surgery knot has the same knot Floer homology as the (2,5)-torus knot.
In multi-particle Diffusion Limited Aggregation (DLA) a sea of particles performs independent random walks until they run into the aggregate and are absorbed. In dimension 1, the rate of growth of the aggregate depends on \lambda, the density of the particles. Kesten and Sidoravicius proved that when \lambda < 1 the aggregate grows like t^{1/2}. They furthermore predicted linear growth when \lambda > 1 (subsequently confirmed by Sly) and t^{2/3} growth at the critical density \lambda = 1. We address the critical case, confirming the $t^{2/3}$ growth and show that aggregate has a scaling limit whose derivative is a self-similar diffusion. Surprisingly, this contradicts conjectures on the speed in the mildly supercritical regime when \lambda = 1 + \epsilon.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Fri 20 Nov 2020, 9:00am-10:30am
Abstract
Although the moduli space of genus-zero curves is not a toric variety, it shares an intriguing amount of the combinatorial structure that a toric variety would enjoy. In fact, by adjusting the moduli problem slightly, one finds a moduli space that is indeed toric, known as Losev-Manin space. The associated polytope is the permutohedron, which also encodes the group-theoretic structure of the symmetric group. Batyrev and Blume generalized this story by constructing a "type-B" version of Losev-Manin space, whose associated polytope is a signed permutohedron that relates to the group of signed permutations. In joint work in progress with C. Damiolini, D. Huang, S. Li, and R. Ramadas, we carry out the next stage of generalization, defining a family of moduli space of curves with Z_r action encoded by an associated "permutohedral complex" for a more general complex reflection group, which specializes when r=2 to Batyrev and Blume's moduli space.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 23 Nov 2020, 8:00am-9:00am
Abstract
Tits construction produces a Lie algebra out of a composition algebra and an exceptional Jordan algebra. The type of the result is F4, 2E6, E7 or E8 when the composition algebra has dimension 1,2,4 or 8 respectively. Garibaldi and Petersson noted that the Tits index 2E635 cannot occur as a result of Tits construction. Recently Alex Henke proved that the Tits index E766 is also not possible. We push the analogy further and show that Lie algebras of Tits index E8133 don’t lie in the image of the Tits construction. The proof relies on basic facts about symmetric spaces and our joint result with Garibaldi and Semenov about isotropy of groups of type E7 in terms of the Rost invariant. This is a part of a work in progress joint with Simon Rigby.
The purpose of this talk is to explain how to describe the propagation of topological currents along magnetic interfaces (also known as magnetic walls) of a two-dimensional material using K-theory. We initially consider tight-binding magnetic models associated with generic magnetic multi-interfaces and we will describe the K-theoretical setting in which a bulk-interface duality can be derived. Then, we will focus on the (non trivial) case of the Iwatsuka magnetic field. The exposition is intended to be pedagogical and aimed at a non-specialist audience.
In 1958, Fine and Herstein proved that the an n by n matrix over the finite field F_q has a probability of q^{-n} to be nilpotent. A clever application of this result can lead to the formula \sum_{H: abelian p-group} 1/|Aut(H)| = 1/((1-p^-1)(1-p^-2)...), which is fundamental in building the Cohen--Lenstra distribution of abelian p-groups. There are other matrix enumeration results, including the counting of pairs of commuting matrices (Feit and Fine) and the counting of pairs of commuting nilpotent matrices (Fulman), all presented as generating functions that can be expressed as infinite products of rational functions. I will explain why all these above are the special cases of one general problem related to a moduli space in algebraic geometry, and why the following is the next unknown case of the problem: count the number of pairs of n by n matrices (A,B) such that AB=BA=0 (hence the word "mutually annihilating" in the title). In my recent work, I gave a generating function that answers this question, and factorized it into the form 1/((1-x)(1-q^-1 x)(1-q^-2 x)...)^2 H(x), where H(x) is an entire holomorphic function given explicitly by an infinite sum. Interesting analytic properties of H(x) will be discussed.
Zoom - contact organizer (abuttens at math ubc ca) for meeting id
Wed 25 Nov 2020, 11:00am-12:00pm
Abstract
The advances in bioimaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, statistical analyses in biomedical research are poised to incorporate more shape data. This leads to the question: how do we define quantitative descriptions of shape variability from images?
Mathematically, landmarks’ shapes, curve shapes, surface shapes, or shapes of objects in images are data that belong to non-Euclidean spaces, for example to Lie groups or quotient spaces. In this context, we introduce “Geometric statistics”, a statistical theory on non-Euclidean spaces. We present several studies showing the theory and applications of Geometric Statistics to the analysis of biomedical shape data.
A classical problem in algebraic geometry asks what rational cuspidal curves can be realised in the complex projective plane. In the last few years some substantial advancement regarding this problem have been made, also based on the methods of Heegaard Floer homology. I will discuss some open problems, conjectures, and constructions. This is joint work with Paolo Aceto (Oxford University).
We consider the restriction to the unit circle of random degree-n polynomials with iid normalized coefficients (Kac polynomials). Recent work of Yakir and Zeitouni shows that for Gaussian coefficients, the minimum modulus (suitably rescaled) follows a limiting exponential distribution. We show this is a universal phenomenon, extending their result to arbitrary sub-Gaussian coefficients, such as Rademacher signs. For discrete distributions we must now deal with possible arithmetic structure in the polynomial evaluated at different points of the circle. Our proof divides the circle into major arcs that are well approximated by rationals, which we handle by crude arguments, and complementary minor arcs, for which we obtain strong comparisons with the Gaussian model via sharp decay estimates on characteristic functions. Based on joint work with Hoi Nguyen.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Fri 27 Nov 2020, 9:00am-10:30am
Abstract
It is a strange and wonderful fact that Chow rings of matroids behave in many ways similarly to Chow rings of smooth projective varieties. Because of this, the Chow ring of a matroid is determined by a homogeneous polynomial called its volume polynomial, whose coefficients record the degrees of all possible top products of divisors. The use of the word "volume" is motivated by the fact that the volume polynomial of a smooth projective toric variety actually measures the volumes of certain polytopes associated to the variety. In the matroid setting, on the other hand, no such polytopes exist, and the goal of our work was to find more general polyhedral objects whose volume is measured by the volume polynomial of matroids. In this talk, I will motivate and describe these polyhedral objects. This is joint work with Anastasia Nathanson.
Motivated by the spatial segregation of intracellular proteins between the cytoplasm and the cellular membrane, we investigate the spatio-temporal dynamics of bulk-surface coupled models. For such models, a passive diffusion process occurring inside a bounded domain is coupled to a nonlinear reaction-diffusion process restricted to the boundary. A variety of idealized bulk geometries are considered, that consist of 1-D intervals and 2-D circular domains. Our emphasis is on the analysis of bifurcations, which characterize the onset of qualitative changes in the overall dynamics resulting from parameters crossing through critical values. A combination of analytical and numerical methods are employed to determine the stability of bifurcating solutions.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 30 Nov 2020, 8:00am-9:00am
Abstract
Many important algebraic objects can be viewed as G-torsors over a field F, where G is an algebraic group over F. For example, there is a natural bijection between F-isomorphism classes of central simple F-algebras of degree n and PGLn(F)-torsors over Spec(F). Much as one may study principal bundles on a manifold via characteristic classes, one may likewise study G-torsors over a field via certain associated Galois cohomology classes. This principle is made precise by the notion of a cohomological invariant, which was first introduced by Serre.
In this talk, we will determine the cohomological invariants for algebraic groups of multiplicative type with values in H1(-, Q/Z(1)). Our main technical analysis will center around a careful examination of mun-torsors over a smooth, connected, reductive algebraic group. Along the way, we will compute a related group of invariants for smooth, connected, reductive groups.
A neural code C on n neurons is a collection of subsets of the set of integers {1,2,...,n}. Usually, C is paired with a collection of n open subsets of some Euclidean space, with C encoding how those open sets intersect. A central problem concerning neural codes is determining convexity; that is, if the code can encode the intersections of n convex open subsets.
In this talk, I will generalize an example of Lienkaemper, Shiu, and Woodstock (2017) into a new criterion for nonconvexity called a 'wheel'. I will show why it forbids convexity, explain how one can find it combinatorially, and provide examples of previously unclassified codes that we now know to be nonconvex due to containing a wheel.
I will discuss some recent progress on the Falconer distance set problem for Borel sets of dimension either equal to or slightly larger than $d/2$ in $\mathbb{R}^d$. Parts of the talk will be based on joint work with Tamas Keleti and with Hong Wang.
For k a perfect fi eld of characteristic p > 0 and G a split reductive group over k with p a non-torsion prime for G, we compute the mod p motivic cohomology of the geometric classifying space BG_(r), where G_(r) is the rth Frobenius kernel of G. Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna.
There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et al. there were two nontrivial stationary distributions
https://ubc.zoom.us/j/67916711780 (password:the number of lines on a generic quintic threefold)
Fri 4 Dec 2020, 8:30am-9:30am
Abstract
In general, the topology of the moduli space of semistable sheaves on an algebraic variety relies heavily on the choice of the Euler characteristic of the sheaves. We show a striking phenomenon that, for the moduli of semistable sheaves on a toric del Pezzo surface (e.g. P^2) or the moduli of semistable Higgs bundles with respect to a divisor of degree > 2g-2 on a curve, the intersection cohomology of the moduli space is independent of the choice of the Euler characteristic. This confirms a conjecture of Bousseau for P^2, and proves a conjecture of Toda in the case of local toric Calabi-Yau 3-folds. In the proof, a generalized version of Ngô's support theorem plays a crucial role. Based on joint work in progress with Davesh Maulik.
https://ubc.zoom.us/j/67916711780 (password:the number of lines on a generic quintic threefold)
Fri 4 Dec 2020, 9:45am-10:45am
Abstract
Nilpotent orbits are important algebraic varieties arising in a number of different areas of mathematics. In spite of their simple definition, it is an open problem to give the defining equations of nilpotent orbit closures over a field of positive characteristic. For certain nilpotent orbits, Pappas and Rapoport conjectured a simple and explicit characteristic-free answer to this problem, with applications to their work in number theory. In this talk I will discuss a proof of their conjecture. It is part of a larger program describing the defining equations of an infinite-dimensional variety called the affine Grassmannian. I will also give an overview of the basics of nilpotent orbits, a subject which doesn't require much more than a background in basic linear algebra. This is based on joint work with Dinakar Muthiah and Oded Yacobi.
A random planar map is a canonical model for a discrete random surface which is studied in probability, combinatorics, mathematical physics, and geometry. Liouville quantum gravity is a canonical model for a random 2d Riemannian manifold with roots in the physics literature. After introducing these objects, I will present a joint work with Xin Sun where we prove convergence of random planar maps to Liouville quantum gravity under a discrete conformal embedding which we call the Cardy embedding.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Mon 7 Dec 2020, 8:00am-9:00am
Abstract
Morava K-theories K(n) are cohomology theories that have graded fields of positive characteristic as coefficient rings and that are obtained from algebraic cobordism of Levine-Morel by change of coefficients.Pure motives with respect to K(n) fit in-between Chow motives and K_0-motives (with p-localized or p-torsion coefficients),e.g. allowing to transfer K(n)-decompositions to K(m)-decompositions whenever m<n. Thus, it might be a reasonable approach in the study of motivic decompositions to start with K(1)-motives (i.e. more or less K_0-motives)and continue to K(2)-, K(3)-motives and so on, eventually arriving to Chow-motives. On the other hand we formulate a conjectural principle that connects the splitting of K(n)-motive. with the triviality of cohomological invariants of degrees less than n+1. I plan to outline the proof of this principle for quadrics and explain its consequences for Chow groups of quadrics lying in powers of the fundamental ideal in the Witt ring. The talk is mostly based on the joint work with Nikita Semenov.
As a disease spreads through a population, the pool of people still susceptible to the disease decreases. Eventually, it becomes unlikely that an infectious person will contact a susceptible person and as a result the number of infectious people starts to decrease. At this point, the fraction of people who are not susceptible is called the herd immunity threshold. The herd immunity threshold can be calculated for simple models and approximated or determined via simulation for more complex models. There has been an ongoing debate about where the herd immunity threshold might lie for the current pandemic. In this talk I will try to summarize the debate, and supply various other opinions.
Decoupling theory studies the following problem. Given finitely many functions $f_j$ defined on the plane with disjoint Fourier supports, how do they interact in the physical space? More precisely, give a Lebesgue exponent $p$, how does the $L^p$-norm of the sum of $f_j$'s compare with the $l^2$-sum of the $L^p$-norm of $f_j$? In 2015, Bourgain and Demeter proved a remarkable result when each $f_j$ has Fourier support on a neighbourhood of the graph of the unit parabola over $I_j$ where $\{I_j\}$ is a uniform partition of the unit interval. In my work, I generalised this theorem to the case where each $f_j$ is Fourier supported on a neighbourhood of the graph of a general polynomial curve over $I'_j$ where $\{I'_j\}$ is a partition of the unit interval that is determined by the curvature of each polynomial.
The work deals with the studies of the existence of solutions
of an integro-differential equation in the case of the anomalous diffusion
with the negative Laplace operator in a fractional power in R^d, d=4,5.
The proof of the existence of solutions relies on a fixed point technique.
Solvability conditions for non Fredholm elliptic operators in unbounded
domains are used.
We will discuss recent bounds for a triangular Hilbert transform with curvature. A main ingredient for the proof is a trilinear smoothing inequality, which also has other applications such as a nonlinear Roth-type theorem in the plane and a pointwise ergodic theorem for continuous bilinear averages of two commuting transformations.
The talk is based on joint works with Michael Christ and Polona Durcik, and with Michael Christ, Polona Durcik and Vjeko Kovac.
The more we come to understand collective cell behaviors, the more we realize how central they are to multicellular life. This fundamental importance makes such behaviors as collective migration potent targets for strategies that would allow us to harness and direct collective cell behaviors for practical purposes. However, realizing such control requires approaching cellular collectives from an engineering and swarm theoretic perspective to first define relevant, fundamental rules governing a given collective process, and then build experimental tools to program these collective behaviors. Here, we will look at several case studies of this approach from my group: tissue tessellation based on multi-tissue interactions; bioelectric 'sheepdogs' to literally herd and program collective migration; and cell-mimetic materials to reprogram tissue dynamics from the inside-out.
Recently, Chatterjee and Diaconis showed that most bijections, if applied between steps of a Markov chain, cause the resulting chain to mix much faster. However, explicit examples of this speedup phenomenon are rare. I will discuss recent work studying such walks on finite fields where the bijection is algebraically defined, and give a near-linear mixing time. This work gives a large collection of examples where this speedup phenomenon occurs. These walks can be seen as a non-linear analogue of the Chung-Diaconis-Graham process, where the bijection is multiplication by a non-zero element of the finite field.
We'll discuss three examples (lattices, translation surfaces, hyperbolic surfaces) of computing the variance of natural counting problems for random geometric structures. All of them can be viewed as generalizations of the space of flat structures on the two-dimensional torus. We'll carefully discuss this base example, and the three different generalizations. In the first part of the talk, we'll discuss lattices, and some joint work with G. Margulis; in the second, translation surfaces, and joint work with Y. Cheung and H. Masur, and in the third, hyperbolic surfaces, and joint work with F. Arana-Herrera.
NOTE: This will be an online event. More details to follow.
For more information on Qualifying Exams, please visit http://www.math.ubc.ca/Grad/QualifyingExams/index.shtml
In this talk we will present an area of analysis that is concerned with the relationship between differential operators, and the properties of their solutions, and the geometry of the domain on which they are considered. We will initially describe the case where the differential operator sees the domain as a homogeneous medium. We will contrast this with several inhomogeneous cases and mention several recent results in that direction. The tools used come from analysis of partial differential equations, harmonic analysis and geometric measure theory.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Mon 18 Jan 2021, 3:00pm-4:00pm
Abstract
The Kontsevich-Soibelman wall-crossing formula describes the wall-crossing behavior of BPS invariants in Donaldson-Thomas theory. It can be formulated as an identity between (possibly infinite) products of automorphisms of a formal power series ring. In this talk, I will explain how these same products also appear in the exact WKB analysis of Schrödinger's equation. In this context, they describe the Stokes phenomenon for objects known as Voros symbols.
Given a finite set A of integers, finding sharp upper bounds for the number of solutions J_{s,k}(A) to this system, when all the variables lie in the set A, has been an important topic of work. Recently, two major approaches have been developed to tackle this problem - the efficient congruencing method of Wooley, and the decoupling techniques of Bourgain-Demeter-Guth. Both these methods give upper bounds for J_{s,k}(A) in terms of s,k, and the cardinality |A| of A, and the diameter X of A. In particular, when X is large in terms of |A|, say when X >> exp(exp(|A|)), these bounds perform worse than the trivial estimates.
In this talk, we present new upper bounds for J_{s,2}(A) which depend only on |A| and s. These improve upon, and generalise, a previous result of Bourgain and Demeter.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 20 Jan 2021, 8:30am-9:30am
Abstract
A famous conjecture of Grothendieck and Serre predicts that if G is a reductive group scheme over a semilocal regular domain R and X is a G-torsor, then X has a point over the fraction field of R if and only if it has an R-point. I will discuss recent work with Eva Bayer-Fluckiger and Raman Parimala in which we prove the conjecture for all forms of GL_n, Sp_n and SO_n when R is 2-dimensional, and all forms of GL_{2n+1} when R is 4-dimensional. (Here the ring R is not required to contain a field.) We approach the problem using the hermitian Gersten-Witt complex associated to an Azumaya algebra with involution (A, s) over a semilocal regular ring R. Specifically, we show that it is exact when the Krull dimension of R or the index of A are sufficiently small.
ODE models are widely used, and often derived or interpreted as a mean field approximation of some (often unspecified) continuous time stochastic model. Such ODE models often implicitly assume that, under the corresponding stochastic model, the time individuals spend in a given state is exponentially distributed. The linear chain trick (LCT) is a well-known technique for replacing exponentially distributed dwell times with Erlang distributions (i.e., gamma distributions with integer shape parameters). We have recently extended this technique beyond Erlang distributions to the much broader family of univariate, matrix exponential distributions known as phase-type distributions. These are the absorption time distributions for continuous time Markov chains, and include exponential, Erlang, and Coxian distributions, among others. This generalized linear chain trick (GLCT) helps clarify connections between individual-level stochastic model assumptions and the structure of corresponding mean field ODE models, and serves as a bridge allowing for the application of tools and concepts from Markov chain theory in the analysis and interpretation of mean field ODE models.
In this talk, I will (1) introduce the GLCT framework and some related concepts from Markov chain theory; (2) describe a procedure for using the GLCT to quickly generalize or approximate some existing ODE, DDE, or distributed delay equation models; and (3) illustrate some benefits of viewing ODE models from the perspective of the GLCT.
Consider a (continuous-time) symmetric Markovian jump process on a metric measure space. If the underlying metric measure space satisfies the volume-doubling and reverse-volume-doubling properties, then it is known that two-sided heat kernel estimates and the parabolic Harnack inequality are both stable under bounded perturbations of the jumping measure. However, the reverse-volume-doubling condition fails if the metric measure space is a graph (or more generally, if it contains any atoms). We generalize these previously known stability results to spaces that satisfy what may be thought of as "reverse-volume-doubling at sufficiently large scales". In particular, we show that heat kernel estimates and the parabolic Harnack inequality are both stable for symmetric jump processes on graphs (with the usual graph metric) that have infinite diameter and satisfy the volume-doubling property.
Zoom; Register before the talk at https://ubc.zoom.us/meeting/register/u50tceuqrzovG9aEPR_EGAcXzZ86CpaAA9uc
Fri 22 Jan 2021, 3:00pm-4:00pm
Abstract
Khovanov homology, though relatively young, is difficult to survey in an hour. This talk will nevertheless attempt to do so, by focussing on the problem of characterizing thin links—those links with simplest-possible Khovanov homology. This is a story that is still unfolding; I will describe some progress that is part of a joint project with Artem Kotelskiy and Claudius Zibrowius.
For more information and to register for the talk see the PIMS announcement.
In this talk, I will present a recent work on the existence of CMC 2-spheres in an arbitrary Riemannian 3-sphere. This is a joint work with Da Rong Cheng.
There is a long tradition of categorifying combinatorial Hopf algebras by the modules of a tower of algebras (or even better via the representation theory of a tower of groups). From the point of view of combinatorics, such a categorification supplies canonical bases, inner products, and a natural avenue to prove positivity results. Recent ideas in supercharacter theory have made fashioning the representation theory of a tower of groups into a Hopf structure more tractable. As a demonstration, this talk reports on the results of the following challenge: (1) Pick a well-known combinatorial Hopf algebra, (2) Find a way to categorify the structure via a tower of groups. In this case, we show how to find the Malvenuto Reutenauer Hopf algebra in the representation theory of a tower of elementary abelian p-groups.(With Nat Thiem.)
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 27 Jan 2021, 8:30am-9:30am
Abstract
In the talk I will explain a construction of a full exceptional collection of vector bundles on the homogeneous variety of the simple algebraic group of Dynkin type F_4 corresponding to its maximal parabolic subgroup P_4. The construction is based on the relation of this homogeneous variety to a homogeneous variety of type E_6 / P_1 and uses an exceptional collection constructed by Faenzi and Manivel. This is joint work with Pieter Belmans and Maxim Smirnov.
Migratory cells are known to adapt to environments that contain wide-ranging levels of chemoattractant. While biochemical models of adaptation have been previously proposed, here we discuss a different mechanism based on mechanosensing, where the interaction between biochemical signaling and cell tension facilitates adaptation. In this talk, we develop and analyze a model of mechanochemical-based adaptation consisting of a mechanics-based physical model coupled with the wave-pinning reaction-diffusion model for Rac GTPase activity. We use Local Perturbation Analysis to predict how cells adapt signaling parameters via feedback from mechanics to maintain polarity in response to chemoattractant levels. To confirm this prediction, we simulate the mechanochemical model in moving cells, demonstrating how mechanosensing results in persistent cell polarity when cells are stimulated with wide-ranging levels of chemoattractant in silico. These results demonstrate how mechanosensing may help cells adapt to maintain polarity in variable environments.
In this talk we consider the coalescing random walk model on general graphs G=(V,E). We set up a unified framework to study the leading order of decay rate of P(t), the fraction of occupied sites at time t, and we are particularly interested in the ‘Big Bang’ regime, where t<<t_{coal}:=E[inf{s: There is only one particle at time s}]. Our results show that P(t) satisfies certain ‘mean field behavior’ if the graphs satisfy a certain ‘transience-like’ condition.
We apply this framework to two families of graphs: (1) graphs given by the configuration model with minimal degree at least 3, and (2) vertex-transitive graphs. In the first case, we show that for 1<<t<< |V|, P(t) decays in the order of 1/t. We also determine the constant by showing that tP(t) is close to the probability that two particles starting from the root of the corresponding unimodular Galton-Watson tree never collide after one of them leaves the root. For the second family of graphs, taking any growing sequence of finite vertex-transitive graphs G(n)=(V(n), E(n)) such that the mean meeting time t_{meet} is O(|V(n)|) and the inverse of the spectral gap t_{rel} is o(|V(n)|), we show that for t_{rel}<<t<<t_{coal}, tP(t) =(1+o(1))/P(Two random walks never meet before time t)= (2+o(1))t_{meet}/|V(n)|. The first equality is also shown to hold for all infinite transient transitive unimodular graphs, in particular, all transient transitive amenable graphs. In addition, we define a certain natural ‘uniform transience’ condition for a sequence of finite graphs, and show that in the transitive setup it implies the above equality for all 1<<t<< t_{coal}.
A standard measure of the complexity of a knot K in S^3 is its 3-genus (or 4-genus) - the minimal genus of a surface in S^3 (or B^4) with boundary K. Since a larger genus surface may be found by attaching handles to a minimal genus surface, the genus completely characterizes which surfaces (up to homeomorphism) can have boundary K. It is more complicated to classify equivariant surfaces (up to equivariant homeomorphism) bounding a knot with an involution since surfaces with involutions are not classified simply by their genus. For example, in this talk I will give obstructions to a strongly invertible knot bounding an equivariant surface with a separating fixed arc, even though every strongly invertible knot bounds an equivariant surface. This talk will be very elementary in nature and will include many examples.
ZOOM. Please sign up in the mailing list https://kantorovich.org/#mailinglist by Wednesday Jan. 27.
Fri 29 Jan 2021, 8:00am-5:00pm
Details
The kickoff event for the Pacific Interdisciplinary hub on Optimal Transport (PIHOT)collaborative research groupwill take place Jan 29-30, 2021. Here is a tentative schedule for the event
ZOOM. Please sign up in the mailing list https://kantorovich.org/#mailinglist by Wednesday Jan. 27.
Sat 30 Jan 2021, 9:00am-12:00pm
Details
The kickoff event for the Pacific Interdisciplinary hub on Optimal Transport (PIHOT)collaborative research group will take place Jan 29-30, 2021. Here is a tentative schedule for the event
Connection details for the meeting will be provided via our mailing list. Please make sure you have subscribed to the list for information on this event and others.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Mon 1 Feb 2021, 3:00pm-4:00pm
Abstract
I will explain an equivalence between a flavor of Donaldson-Thomas theory (due to Bryan and Steinberg) on ADE surface fibrations and quasimaps to Hilbert schemes of ADE surfaces. The proof involves an explicit combinatorial description of vertices. The equivalence can be used to relate machinery from both sides, notably an equivariant K-theoretic DT crepant resolution conjecture and 3d mirror symmetry.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 3 Feb 2021, 8:30am-9:30am
Abstract
Over number fields, there is a classic obstruction to the local-global principle for the existence of rational points, known as the Brauer-Manin obstruction, which is conjectured to explain all failures of this principle for homogeneous spaces of connected linear groups. In the last few years, there has been an increasing interest in fields of a more geometric nature that are amenable to local-global principles and Brauer-Manin obstructions as well. These include, for instance, function fields of curves over discretely valued fields, by analogy with the case of global fields of positive characteristic. It is in this context that I will present recent work with Diego Izquierdo on local-global principles for homogeneous spaces with connected stabilizers. We will see that, although some of the known results for number fields have direct analogs (that can be obtained in the same way), the particularities of these new fields bring up new counterexamples that cannot be explained by the Brauer-Manin obstruction, contrary to the number field case.
During development, cellular self-organization by cell segregation leads to boundary formation and is critical for the organization of morphogenetic movement and tissue patterning. Signaling between membrane-bound EPHRINS and EPH receptor tyrosine kinases is essential in boundary formation, driving segregation between EPHRIN-expressing and EPH-expressing cells. Here we examine the basic cellular mechanistic drivers of EPH/EPHRIN cellular self-organization and boundary formation. Using a cell culture system to model EPH/EPHRIN cell segregation we analyzed the contact angle of cells to estimate the interfacial tension between EPHB2- and EPHRIN-B1-expressing cells. Heterotypic cell pairs exhibited increased interfacial tension relative to homotypic cell pairs. Inhibitors of actomyosin contractility significantly diminished this increase, suggesting that actomyosin contractility drives heterotypic interfacial tension. Cell segregation assays revealed that EPH/EPHRIN driven segregation is actomyosin contractility dependent. Further, atomic force microscopy showed that EPH/EPHRIN signaling results in increased cortical tension during cell segregation. Actomyosin contractility also drives increased EPHB2:EPHB2 homotypic contacts through an increase in tension away from the cell contact. Using a mouse model we demonstrated that actomyosin contractility is critical for EPH/EPHRIN cell segregation in vivo as well. Finally, we demonstrated that tissue-wide changes in cellular organization and tissue shape are driven by minimization of heterotypic contact. These data suggest a model for cell segregation and tissue organization in which Eph/ephrin signaling results in a cortical actin differential that prevents cells from making stable contacts and drives cell segregation to affect tissue morphology by modulating interfacial tension.
I will consider random walks in finite subsets of the square lattice that approximate nice bounded continuum planar domains in the scaling limit. The walk moves as the simple random walk inside the domain and, whenever it exits, it returns back via a uniformly-chosen boundary edge in the next step. Running the walk up to a positive multiple of the cover time, I will show that the scaling limits of various exceptional sets of the local time — specifically, the sets of suitably defined thick and thin points as well as the set of avoided (a.k.a. late) points — are distributed according to versions of the Liouville Quantum Gravity in the underlying continuum domain. The results are cleanest when the walk is parametrized by the local time spent at the “boundary vertex” with non-trivial corrections to the limit law arising in the conversion to the actual time. Based on joint papers with Yoshihiro Abe and Sangchul Lee.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Mon 8 Feb 2021, 3:00pm-4:00pm
Abstract
Moduli spaces of points on n-spheres carry natural actions of braid groups. For n=0,1, and 3, we prove that these symmetries extend to actions of mapping class groups of positive genus surfaces, through exceptional isomorphisms with certain character varieties. We also apply the exceptional isomorphisms to the study of Stokes matrices and exceptional collections of triangulated categories. Joint work with Junho Peter Whang.
Dyck paths and standard Young tableaux (SYT) are two of the most central sets in combinatorics. There is a well-known bijection between Dyck paths with 2n steps and SYT of shape (n,n). In recent work, we found nine other bijections between classes of Dyck paths and classes of SYT. I will present my favorites. Along the way, we will encounter connected matchings, non-crossing partitions and non-nesting partitions.
This is joint work with Juan Gil, Jordan Tirrell and Michael Weiner.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 10 Feb 2021, 8:30am-9:30am
Abstract
Essential dimension is a numerical measure of the complexity of algebraic objects invented by J. Buhler and Z. Reichstein in the 90s. Roughly speaking, the essential dimension of an algebraic object is the number of parameters it takes to define the object over a field. For example, by Kummer theory, it takes one parameter to define a mu_n torsor, so the essential dimension of the functor of mu_n torsors (or the essential dimension of mu_n for short) is 1. In a preprint from 2019, Farb, Kisin and Wolfson (FKW) prove theorems about the essential dimension of congruence covers of Shimura varieties using arithmetic methods. In many cases, they are able to prove that the congruence covers are incompressible, that is, they are not obtainable by base change from varieties of strictly smaller dimension.
In my talk, I will discuss recent work with Najmuddin Fakhruddin, where we recover many (but definitely not all) of the results of FKW, by geometric arguments using a new fixed point theorem. This also allows us to extend the incompressibility results of FKW to Shimura varieties of exceptional type to which the arithmetic methods of FKW do not apply. I will also discuss a general conjecture we make on the essential dimension of congruence covers arising from Hodge theory. (With some caveats, we conjecture that it is equal to the dimension of the image of the period map.)
The form and function of tissues and organs emerge out of cell-to-cell interactions. Cell interaction dynamics take place on a variety of temporal and spatial scales, and reflect processes—diffusion, migration, force production/sensing, growth, and proliferation. In this talk, we develop a multiscale framework where directly measurable quantities at the discrete cell-scale inform the model parameters at the continuum tissue scale
through upscaling. In principle, this enables the model to be truly predictive because the data used for calibration (e.g., at the cell scale) is distinct from that used for validation (e.g., at the tissue scale). This model borrows ideas from statistical physics, materials science and applied mathematics and follows the framework of dynamic density functional theory. This approach provides a strategy for coarse-graining systems of stochastically interacting particles. By appropriately accounting for cell size and shape variability, we obtain a system of continuum equations that are able to capture plastic, viscoelastic and elastic deformations in the clusters while providing single-cell resolution. We validate this approach by comparisons with recent in vitro studies of epithelial cell colonies using Madin-Darby canine kidney cells. We then use this framework to develop a new continuum elastic model for tissues that contains microscale information, including cell-cell correlations. The governing equations are obtained by using a 1-mode approximation and coarse-graining. We simulate the system numerically and analyze the system using matched asymptotic expansions to relate the new model with previously developed approaches.
We consider the Glauber dynamics (also called Gibbs sampling) for sampling from a discrete high-dimensional space, where in each step one variable is chosen uniformly at random and gets updated conditional on all other variables. We show an optimal mixing time bound for the Glauber dynamics in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows O(nlogn) mixing time for graphical models/spin systems on any n-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. Our proof approach combines classic tools of entropy tensorization/factorization and recent developments of high-dimensional expanders.
As an application of our results, for the hard-core model on independent sets weighted by a fugacity lambda, we establish O(nlogn) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree D when lambda<lambda_c(D) where lambda_c(D) is the critical point for the uniqueness/non-uniqueness phase transition on the D-regular tree. More generally, for any antiferromagnetic 2-spin system (e.g., Ising model) we prove O(nlogn) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(nlogn) mixing for sampling random q-colorings of triangle-free graphs of maximum degree D when the number of colors satisfies q > aD where a = 1.763…, and O(mlogn) mixing for generating random matchings of any graph with bounded degree and m edges.
Based on joint work with Kuikui Liu and Eric Vigoda.
The dimer model is the study of the set of dimer configurations (or perfect matchings) of a graph. In this talk, I will begin with an overview of the combinatorics of the dimer model, highlighting surprising connections between the dimer model and other areas of math such as algebraic geometry.
I will then present joint work with Ben Young and Gautam Webb which uses the dimer model and the less well-studied double-dimer model to resolve an open conjecture from enumerative geometry. To do so, we prove that two generating functions for plane partition-like objects (the "box-counting" formulae for the Calabi-Yau topological vertices in Donaldson-Thomas theory and Pandharipande-Thomas theory) are equal up to a factor of MacMahon's generating function for plane partitions.Our proof is combinatorial, and no prior knowledge of enumerative geometry (or the dimer model) is required to understand the talk.
(joint work with Jason Lotay) A standing conjecture of Richard Thomas, motivated by mirror symmetry, gives a stability condition supposed to control the existence of a special Lagrangian submanifold in a given Hamiltonian isotopy class of Lagrangians. Later, Thomas and Yau conjectured that a similar stability condition controls the long-time existence of the Lagrangian mean-curvature flow. In this talk I will explain how Jason Lotay and myself have recently proved versions of these conjectures on all circle symmetric hyperKahler 4-manifolds.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 17 Feb 2021, 8:30am-9:30am
Abstract
A well-known conjecture of Colliot-Thélčne asserts that a rationally isotropic quadratic space over a regular local ring is isotropic. If the ring contains a field, then this conjecture was proved by the efforts of the speaker, Pimenov and Scully. In the talk we will present new results in the mixed characteristic case.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Mon 22 Feb 2021, 3:00pm-4:00pm
Abstract
Let G be a finite group acting on a smooth complex variety M. Let X —> M/G be a crepant resolution by a smooth variety X. The Crepant Resolution Conjecture predicts a complicated relationship between the Gromov—Witten invariants of X and the orbifold Gromov—Witten invariants of the stack [M/G].
In this talk I will describe an analogous conjecture involving Landau—Ginzburg (LG) models. An LG model is, roughly, a smooth complex variety Y together with a regular function w: Y—> \CC. LG models can be used to give alternate “resolutions” of hypersurface singularities in a certain sense and are related to so-called noncommutative resolutions. I will briefly discuss the gauged linear sigma model, which is used to define curve counting invariants for LG models, and describe a new technique for computing these invariants.
In this talk, we consider infinite-volume gapped ground states of frustration-free quantum spin models. We will show that for models satisfying a local topological order condition, which captures that the finite volume ground states are effectively indistinguishable by local operators, the gap remains open in the presence of uniformly small short-ranged perturbations. In contrast to earlier results, we do not require a positive lower bound for a sequence of finite-volume Hamiltonians uniform in the system size. This result is proved by adapting the Bravyi-Hastings-Michalakis strategy to the GNS representation of the infinite-system ground state, which we discuss. This talk is based on joint work with Bruno Nachtergaele and Robert Sims.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 24 Feb 2021, 8:30am-9:30am
Abstract
Arithmetic function fields are one variable function fields over complete discretely valued fields. They naturally admit several collections of overfields with respect to which one can study local-global principles. We will focus on studying local-global principles for torsors under reductive groups that are defined over the underlying discrete valuation ring, reporting on joint work with J.L.-Colliot-Thélčne, D. Harbater, D. Krashen, R. Parimala, and V. Suresh.
Actin filaments are polymers that interact with myosin motor proteins inside cells and play important roles in cell motility, shape, and development. Depending on its function, this dynamic network of interacting proteins reshapes and organizes in a variety of structures, including bundles, clusters, and contractile rings. Motivated by observations from the reproductive system of the roundworm C. elegans, we use an agent-based modeling framework to simulate interactions between actin filaments and myosin motor proteins inside cells. We also develop tools based on topological data analysis to understand time-series data extracted from these filamentous network interactions. We use these tools to compare the filament organization resulting from myosin motors with different properties. Moving forward, we are interested in gaining insights into myosin motor regulation and the resulting actin architectures during cell cycle progression. This work also raises questions about how to assess the significance of topological features in common topological summary visualizations.
We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them. Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to ne_1. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity. The key tool is a new variational formula for the time constant. In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.
Zoom: Sign up for the mailing list in (https://kantorovich.org/talk/ki-seminar-palmer/) to receive the connection details
Thu 25 Feb 2021, 10:00am-11:00am
Abstract
The optimal transport problem provides a fundamental and quantitative way to measure the distance between probability distributions. Recently, it has been successfully used to analyze the evolutionary dynamics in physics and biology. Motivated by questions of pricing in financial mathematics and control of distributed agents, stochastic variants of optimal transport have been developed. Over the past few years, my postdoc supervisors at the University of British Columbia (Nassif Ghoussoub and Young-Heon Kim) and I have developed a robust method to analyze these problems using convex duality, stochastic optimal control theory, and partial differential equation analysis.
This talk will focus on these variants of optimal transport, their applications, and our methods of analysis. Particular attention will be paid to the connections with mean field games and to a new direction of research that incorporates the practical limitation of partial information.
Note for Attendees
To get the Zoom link, please sign up for the mailing list through the webpage: https://kantorovich.org/talk/ki-seminar-palmer/
The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds. One measure of the non-triviality of a three-manifold is the existence of non-trivial SU(2)-representations. In this talk I will show that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)-representations. This is joint work with Tye Lidman and Raphael Zentner.
Tamagawa numbers were introduced nearly 60 years ago to associate some canonical number-theoretic volume to an algebraic group defined over a global field. This invariant is linked to Hasse's local-global principle and computation of mass formulae. In particular, established mass formulae for the isogeny classes of some (principally polarized) abelian varieties over finite fields relies on Tamagawa numbers of specific maximal algebraic tori in GSp. I will present current results on the latter as well as an implementation of algebraic tori in sagemath.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Mon 1 Mar 2021, 3:00pm-4:00pm
Abstract
In this talk, we introduce Virasoro operators in quantum singularity theories for nondegenerate quasi-homogeneous polynomials with nontrivial diagonal symmetries. Using Givental's quantization formula of quadratic Hamiltonians, these operators satisfy the Virasoro relations. Inspired by the famous Virasoro conjecture in Gromov-Witten theory, we conjecture that the genus g generating functions arise in quantum singularity theories are annihilated by the Virasoro operators. We verify the conjecture in various examples and discuss the connections to mirror symmetry of LG models and LG/CY correspondence. This talk is based on work joint with Weiqiang He.
In this talk, we will discuss a priori interior estimates for the Lagrangian mean curvature equation under certain natural restrictions on the Lagrangian phase. As an application, we will use these estimates to solve the Dirichlet problem for the Lagrangian mean curvature equation with continuous boundary data on a uniformly convex, bounded domain.
Let $\alpha=(a,b,\ldots)$ be a composition. Consider the associated poset $F(\alpha)$, called a fence, whose covering relations are
$$x_1\lhd x_2 \lhd \ldots\lhd x_{a+1}\rhd x_{a+2}\rhd \ldots\rhd x_{a+b+1}\lhd x_{a+b+2}\lhd \ldots\ .$$
We study the associated distributive lattice $L(\alpha)$ consisting of all lower order ideals of $F(\alpha)$. These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define $q$-analogues of rational numbers. In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that $L(\alpha)$ is rank unimodal. We show that if one of the parts of $\alpha$ is greater than the sum of the others, then the conjecture is true. We conjecture that $L(\alpha)$ enjoys the stronger properties of having a nested chain decomposition and having a rank sequence which is either top or bottom interlacing, the latter being a recently defined property of sequences. We verify that these properties hold for compositions with at most three parts and for what we call $d$-divided posets, generalizing work of Claussen and simplifying a construction of Gansner. This is joint work with Thomas McConville and Clifford Smyth.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 3 Mar 2021, 8:30am-9:30am
Abstract
Artin and Pfister have shown that a nonnegative real polynomial in n variables is a sum of 2^n squares of rational functions. In other words, the Pythagoras number of the field R(x1,...,xn) is at most 2n. In this talk, I will consider local variants of this statement. In particular, I will give a proof of a conjecture of Choi, Dai, Lam and Reznick: the Pythagoras number of the field of Laurent series R((x1,...,xn)) is at most 2n-1.
In 2015, Qingtao Chen and I conjectured that at the root of unity exp(2πi/r) instead of the usually considered root exp(πi/r), the Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially with growth rate the complex volume of the manifold. In this talk, I will review some known results about this conjecture and present a possible approach of solving it including a recent joint work with Ka Ho Wong.
In the trace reconstruction problem, there is an unknown binary string, and we observe noisy samples of this string after it has gone through a deletion channel. This deletion channel independently deletes each bit with constant probability q and concatenates the remaining bits. The goal is to learn the original string with high probability using as few traces as possible, where the sample complexity is characterized by the length of the string, n.
Researchers are stuck at bridging the gap between the exponential upper bound and polynomiallower bound. In the first part of this talk, I will explain the string trace reconstruction problem and the current best upper bound techniques. In the latter part, I will discuss several generalizations and variants of trace reconstruction, including our work on approximate trace reconstruction, which asks for a string “close” to the original unknown string with high probability.
Some of this talk is based on joint work with Miklos Z. Racz, Cyrus Rashtchian, and Benjamin G. Schiffer.
In this talk, we look at a novel way of finding roots of a scalar polynomial of the form c(z) = z a(z)b(z) + c_0 using eigenvalue techniques. This companion matrix construction can generate matrices with smaller entries than the Fiedler or Frobenius forms. We then look at the extension of this novel method to the polynomial eigenvalue problem (PEP). PEP have been used in many science and engineering applications such vibrations of structures, computer-aided geometric design, robotics, and machine learning. We show how to construct linearizations of matrix polynomials za(z)d_0 + c_0, a(z)b(z), a(z) + b(z) (when deg(b(z)) < deg(a(z))), and za(z)d_0b(z) + c_0 from linearizations of the component parts, matrix polynomials a(z) and b(z). Lastly, we investigate the numerical stability of algebraic linearizations experimentally, which re-uses the linearizations of matrix polynomials a(z) and b(z) to make a linearization for the matrix polynomial P(z) = z a(z)b(z) + c. Such a re-use seems more likely to produce a well-conditioned linearization, and thus the implied algorithm for finding the eigenvalues of P(z) seems likely to be more numerically stable than expanding out the product a(z)b(z) (in whatever polynomial basis one is using). We investigate this question experimentally by using pseudospectra.
https://ubc.zoom.us/j/67916711780 (password:the number of lines on a generic quintic threefold)
Mon 8 Mar 2021, 3:00pm-4:00pm
Abstract
In this talk, I will discuss some results on the structure of the cohomology of the moduli space of stable SL_n Higgs bundles on a curve. One consequence is a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration. We will also discuss connections to the P=W conjecture if time permits. Based on joint work with Junliang Shen.
The Kerr-Newman spacetime is the most general explicit black hole solution, and represents a stationary rotating charged black hole. Its stability to gravitational and electromagnetic perturbations has eluded a proof since the 80s in the black hole perturbation community, because of the "apparent indissolubility of the coupling between the spin-1 and spin-2 fields in the perturbed spacetime", as put by Chandrasekhar. We will present a derivation of the relevant equations which can be analyzed in physical-space to obtain a quantitative proof of stability.
Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The vanishing problem for Schubert polynomials asks if a coefficient of a Schubert polynomial is zero. We give a tableau criterion to solve this problem, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the n x n grid. In contrast, we show that computing these coefficients explicitly is #P-complete. This is joint work with Anshul Adve and Alexander Yong.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 10 Mar 2021, 8:30am-9:30am
Abstract
The following construction of a root system of type G_2 is given in J.-P. Serre’s book “Complex Semisimple Lie algebras” (Chapter V, Section 16): “It can be described as the set of algebraic integers of a cyclotomic field generated by a cubic root of unity, with norm 1 and 3”. This talk, based on joint work with Yu. G. Zarhin, concerns the problem of realization of root systems, their Weyl groups and their root lattices in the form of groups and lattices naturally associated with number fields.
Locally checkable labeling (LCL) problems are graph problems where the validity of a solution can be checked locally. Examples include proper vertex or edge colorings, perfect matching etc.
Such problems have been studied from different points of view. Most important for our investigation is the perspective of distributed computing and random processes. The connection between the two fields has been suspected for some time [Holroyd, Schramm, Wilson. Annals of Prob. 2017, Brandt et al. 2017 ] but in our work we give probably the first precise translation between the two worlds. Among others, this almost automatically answers 3 out of 4 open questions from [Holroyd, Schramm, Wilson. Annals of Prob. 2017].
In the first part of the talk we introduce the basic notions from distributed computing. In the second part we use this theory to answer aforementioned questions from [Holroyd, Schramm, Wilson. Annals of Prob. 2017].
Let C_2 denote the cyclic group of order two. Given a manifold with a C_2-action, we can consider its equivariant Bredon RO(C_2)-graded cohomology. In this talk, we give an overview of RO(C_2)-graded cohomology in constant Z/2 coefficients, and then explain how a version Thom isomorphism theorem in this setting can be used to develop a theory of fundamental classes for equivariant submanifolds.
A classical problem in representation theory is how a representation of a group decomposes when restricted to a subgroup. In the 1990s, Gross-Prasad formulated a conjecture regarding the restriction of representations, also known as branching laws, of special orthogonal groups.Gan, Gross and Prasad extended this conjecture, now known as the Gan-Gross-Prasad (GGP) conjecture, to the remaining classical groups. In this talk, we will discuss the GGP conjecture and the connection with period integrals.In addition, we will discuss the first step in provingthe GGP conjecture: a multiplicity at most one theorem. Aizenbud, Gourevitch, Rallis and Schiffmann proved a multiplicity at most one theorem for restrictions of irreducible representations of certain classical groups and Waldspurger proved the same theorem for the special orthogonal groups. We will discuss work that establishes a multiplicity at most one theorem for restrictions of irreducible representations for a non-classical group, the general spin group. This is joint work with Shuichiro Takeda.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Mon 15 Mar 2021, 3:00pm-4:00pm
Abstract
We use modular forms to study the birational geometry of some moduli spaces of hyperkähler fourfolds. I'll review a bit of the algebraic geometry of these moduli spaces, and then I'll explain some methods, due to Gritsenko, Hulek, and Sankaran, for computing their Kodaira dimensions. These methods make use of special modular forms defined on high rank orthogonal groups. I'll also report on an ongoing project with Jen Berg applying related techniques to certain moduli spaces of Enriques surfaces.
The Bernstein problem asks whether entire minimal graphs in dimension n+1 are necessarily hyperplanes. This problem was solved in combined works of Bernstein, Fleming, De Giorgi, Almgren, and Simons ("yes" if n < 8), and Bombieri-De Giorgi-Giusti ("no" otherwise). We will discuss the analogue of this problem for minimizers of general elliptic functionals. In particular, we will discuss new examples of nonlinear entire graphical minimizers in dimension n = 6, and recent work with Y. Yang towards constructing such examples in the lowest possible dimension n = 4.
In this work, we prove the CDE property of the trapezoidal shifted shapes by counting standard barely set-valued tableaux via q-integral method.
A standard barely set-valued tableau of shape $\lambda$ is a filling of the Young diagram $\lambda$ with integers $1,2,\dots,|\lambda|+1$ such that the integers are increasing in each row and column, and every cell contains one integer except one cell that contains two integers. Counting standard barely set-valued tableaux is closely related to proving Young's lattice has the coincidental down-degree expectations (CDE) property. Using $q$-integral techniques we give a formula for the number of standard barely set-valued tableaux of arbitrary shifted shapes. We then prove a conjecture of Reiner, Tenner and Yong on the CDE property of trapezoidal shifted shape $(n,n-2,\dots,n-2k)$.
This is joint work with Jang Soo Kim and Michael Schlosser.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 17 Mar 2021, 8:30am-9:30am
Abstract
We provide a geometric construction for the equivalence between the category of smooth affine group schemes over the ring of dual numbers k[ε] and the category of extensions 1 → Lie(G) → E → G → 1 where G is a smooth affine group scheme over k. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. As an application, we establish a Dieudonné classification for smooth, commutative, unipotent group schemes over k[ε] when k is a perfect field.
It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle. Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei. We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle. Based on joint work with Julian Sahasrabudhe.
Let R^n denote n-dimensional Euclidean space. A bilinear map F: R^r X R^s --> R^n is said to be nonsingular if F(x,y)=0 implies x = 0 or y = 0. In the 1840's Hamilton looked for such an F when r=s=n=3 without success. He did, however, find one such F for r=s=n=4: the multiplication map of "quaternions". In a sense it signifies the birth of sympletic geometry. The general question is: for what triples (r,s,n) will there exist a nonsingular bilinear map F? This remains unanswered up to the present day. In this talk I shall present some new examples of nonsingular bilinear maps, constructed via factorization theory of polynomials whose coefficients come from a non-commutative ring. I shall then discuss the geometry and topology behind these new examples, and explain their potential use for constructing elements in the stable homotopy groups of spheres. This is joint work with Carlos Dominguez.
A celebrated theorem of Jorgens-Calabi-Pogorelov says that global convex solutions to the Monge-Ampere equation det(D^2u) = 1 are quadratic polynomials. On the other hand, an example of Pogorelov shows that local solutions can have line singularities. It is natural to ask what kinds of singular structures can appear in functions that solve the Monge-Ampere equation outside of a small set. We will discuss examples of functions that solve the equation away from finitely many points but exhibit polyhedral and Y-shaped singularities. Along the way we will discuss geometric and applied motivations for constructing such examples, as well as their connection to a certain obstacle problem for the Monge-Ampere equation.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Mon 22 Mar 2021, 3:00pm-4:00pm
Abstract
I will introduce some recent developments and work in progress on studying Gauged Linear Sigma Models using logarithmic compactifications.
These logarithmic compactifications admit two types of virtual cycles --- the reduced virtual cycles that recover Gromov-Witten invariants of complete intersections, and the canonical virtual cycles that depend only on the geometry of ambient spaces. These two types of virtual cycles differ only by a third virtual cycle of the boundary of the logarithmic compactifications. Using the punctured logarithmic maps of Abramovich-Chen-Gross-Siebert, these virtual cycles can be computed via the tropical and equivariant geometry of the logarithmic compactifications. This leads to a new method for computing Gromov-Witten invariants of complete intersections.
The talk consists of joint work with Felix Janda, Yongbin Ruan, Adrien Sauvaget and Rachel Webb.
I will show that any closed connected hypersurface in four-dimensional Euclidean space with entropy less than or equal to that of the round cylinder is smoothly isotopic to the standard three-sphere. This is joint work with Jacob Bernstein.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 24 Mar 2021, 8:30am-9:30am
Abstract
The talk will address the following questions: given two algebraic groups G, H over a field, is the functor of group homomorphisms from G to H representable by a scheme M, locally of finite type? If so, how to describe the orbits of H acting on M via conjugation of homomorphisms? The representability question has a positive answer when G is reductive and H is smooth and affine, by a result of Demazure in SGA3 (which holds over an arbitary base). The talk will present an extension of this result to the class of “semi-reductive” algebraic groups, which includes reductive groups, finite groups and abelian varieties. In characteristic 0, we will also see that all the H-orbits in M are open. This rigidity property gives back results of Vinberg and Margaux.
Construction of Yang-Mills measures is considered to be an important step in the construction of quantum Yang-Mills theories. Yang-Mills measures have been constructed in 2D, but the problem remains open in dimensions three and higher. In this talk, I will present two kinds of abstract constructions of 3D Yang-Mills measures. One is easier, but with less desirable properties than the other. The easier construction is possible for essentially any 3D Yang-Mills theory. The more difficult construction is on a better state space with a richer class of measurable functions. We have been able to complete the second construction for U(1) theory, and we believe that it should also work for non-Abelian theories (this is work in progress). Both constructions build on a program of using the Yang-Mills heat flow to construct state spaces for Yang-Mills measures, initiated by Charalambous and Gross about ten years ago. All of this is based on ongoing joint work with Sky Cao. A preprint will be available soon.
The theory of large deviations provides with a way to compute asymptotically the probability that an interacting particle system moves from a given configuration to another one over a fixed time interval. The problem of finding the most likely evolution realising the desired transition can be seen as a prototype of stochastic optimal transport problem, whose specific formulation depends on the choice of interaction mechanism. The first goal of this talk is to present some notable examples of this family of transport problems such as the Schrödinger problem and its mean field and kinetic counterparts. The second goal of the talk is to discuss some (possibly open) questions on the ergodic behaviour of optimal solutions and how their answer relies upon a combination of tools coming from Riemannian geometry, functional inequalities and stochastic control.
A knot is doubly slice if it is the cross section of an unknotted sphere in S^4. We discuss various generalizations of this concept to links. The main result is to show that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly slice and having doubly slice components. We also give a variety of constructions for doubly slice links.
Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots in of a different kind: its construction is combinatorial, and connected to ideas from representation theory. There is hope that it can tell us more about smooth 4-manifolds; for example, Freedman, Gompf, Morrison and Walker suggested a strategy to disprove the 4D Poincare conjecture using Rasmussen's invariant from Khovanov homology. It is yet unclear whether their strategy can work, and I will explain some of its challenges, as well as a new attempt to pursue it (joint work with Lisa Piccirillo). I will also review other topological applications of Khovanov homology, with regard to smoothly embedded surfaces in 4-manifolds.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Mon 29 Mar 2021, 3:00pm-4:00pm
Abstract
I'll explain how to construct K3 surfaces as hyperkahler quotients and, as time permits, our expected application to counting open GW invariants. This is all joint work with M. Zimet.
We introduce the extended chromatic symmetric function of a weighted graph, which generalizes the chromatic polynomial. We present a multitude of properties including a general inclusion-exclusion relation. Intriguingly, we discover that two weighted paths have equal extended chromatic symmetric functions precisely when two ribbon Schur functions are equal.
This is joint work with Farid Aliniaeifard and Steph van Willigenburg and needs no prior knowledge.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 31 Mar 2021, 8:30am-9:30am
Abstract
The category of Chow motives defined by Grothendieck has plenty of various applications to quadratic forms, and, more generally, to projective homogeneous varieties. However, there are many open questions about the behaviour of Chow motives. In contrast, if we change the Chow group by Grothendieck’s K0 in the definition of motives, the resulting category behaves much more simply. One can define the category of motives corresponding to any oriented cohomology theory A and hopefully obtain invariants that are simpler than Chow motives but keep more information than K0-motives. In the talk, I will consider categories of motives constructed with respect to Morava K-theories, and describe the motives of generic quadrics. The talk is based on joint results with Nikita Semenov, Victor Petrov, and Pavel Sechin.
Why do organelles have their particular sizes, and how does the cell maintain them given the constant turnover of proteins and biomolecules? To address these fundamental biological questions, we formulate and study mathematical models of organelle size control rooted in the physicochemical principles of transport, chemical kinetics, and force balance. By studying the mathematical symmetries of competing models, we arrive at a hypothesis describing general principles of organelle size control. In particular, we consider flagellar length control in the unicellular green algaeChlamydomonas reinhardtii, and develop a minimal model in which diffusion gives rise to a length-dependent concentration of depolymerase at the flagellar tip. We explain how the same principles may be applied to other examples of organelle size and scaling such as the ratio of nucleus to cell volume.
We consider a family of random growth models in n-dimensional space. These models capture certain features expected to manifest at the mesoscopic level for certain self-interacting microscopic dynamics (such as once-reinforced random walk with strong reinforcement and origin-excited random walk). In a joint work with Pablo Groisman, Ruojun Huang and Vladas Sidoravicius, we establish for such models an averagingprincipleand deduce from it the convergence of the normalized domain boundary, to a limiting shape. The latter is expressed in terms of the invariant measure of an associated Markov chain.
Suppose $u\in D^{1,2} (R^n)$. In a seminal work in 1984, Struwe proved that if $||\Delta u+u^{\frac{n+2}{n-2}}||_{H^{-1}}:=\Gamma(u)\to 0$ then $\delta(u)\to 0$, where $\delta(u)$ denotes the $D^{1,2} (R^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. In 2020, Figalli and Glaudo obtained the first quantitative version of Struwe's decomposition in lower dimensions, namely $\delta(u)\lesssim \Gamma(u)$ when $3\leq n\leq 5$.
In this talk, we completely solve the remaining higher dimension cases and show that
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 7 Apr 2021, 8:30am-9:30am
Abstract
Every central simple algebra of p-power degree over a field of characteristic p is Brauer equivalent to a cyclic algebra by a result of Albert. The proof of this and other similar p-algebra results rely on the interplay between purely inseparable splitting fields and cyclic splitting fields of p-algebras. This talk on joint work with Adam Chapman and Mathieu Florence looks at new results on common splitting fields of symbol p-algebras with applications to symbol length.
The circular maximal function is a singular variant of the familiar Hardy--Littlewood maximal function. Rather than take maximal averages over concentric balls, we take maximal averages over concentric circles in the plane. The study of this operator is closely related to certain GMT packing problems for circles, as well as the theory of the Euclidean wave propagator. A celebrated result of Bourgain from the mid 80s showed that the circular maximal function is bounded on Lp if and only if p > 2. In this talk I will discuss a higher dimensional variant of Bourgain's theorem, in which the circles are replaced with space curves (such as helices) in R^3. Our main theorem is that the resulting helical maximal operator is bounded on Lp if and only if p > 3. The proof combines a number of recently developed Fourier analytic tools, and in particular a variant of the Littlewood--Paley theory for functions frequency supported in a neighbourhood of a cone. Joint work with David Beltran, Shaoming Guo and Andreas Seeger.
At the cortex of Dictyostelium, the actin cytoskeleton localizes in discrete patches which have been shown to oscillate at different phases. Recent findings suggest that the spatial coordination of actin oscillators is regulated by the G protein subunit G$\beta$, which diffuses rapidly throughout the cell. Upon G$\beta$ sequestration, the following phenomena are observed: (1) higher fraction of actin patches becomes oscillatory; (2) phase difference between different sectors becomes smaller. To understand these observations, we model each actin patch as a conditional oscillator, which is governed by an excitable activator-inhibitor model coupled by bulk diffusion of G$\beta$. Assuming that G$\beta$ promotes the actin activator Arp2/3 in each actin patch, we find that actin oscillations can emerge when the G$\beta$ concentration is low. We show that spatial heterogeneity of G$\beta$ can lead to phase differences in actin oscillators. We consider how additional spatial coupling by Arp2/3 can influence spatial patterning in this system.
The lace expansion is a flexible method that has been used since the 1980s to analyse the critical behaviour of high-dimensional random systems, including self-avoiding walk, percolation, and spin systems. It originated in work of Brydges and Spencer on weakly self-avoiding walk in dimensions above 4, and since then several different approaches have been developed to prove convergence of the expansion. I will explain what the lace expansion for self-avoiding walk is, and will then present a new and relatively simple method for proving convergence of the lace expansion for weakly self-avoiding walk. The talk is based on a paper to appear in Ann. Inst. H. Poincaré Probab. Statist., available at: https://arxiv.org/abs/2006.11241.
https://ubc.zoom.us/j/67916711780(password:the number of lines on a generic quintic threefold)
Mon 12 Apr 2021, 3:00pm-4:00pm
Abstract
3d mirror symmetry is a duality in physics, where Higgs and Coulomb branches of certain pairs of 3d N=4 SUSY gauge theories are exchanged with each other. Motivated from this, M. Aganagic and A. Okounkov introduced the enumerative geometric conjecture that the vertex functions of the mirror theories are related to each other. The two sets of q-difference equations satisfied by the vertex functions, in terms of the K\"ahler and equivariant parameters, are expected to exchange with each other. The conjecture therefore leads to a nontrivial relation between their monodromy matrices, the so-called elliptic stable envelopes. In this talk, I will discuss the proof in several cases of the conjecture for both vertex functions and elliptic stable envelopes. This is based on joint works with R. Rim\'anyi, A. Smirnov, and A. Varchenko.
The Szemeredi-Trotter theorem asserts that m points and n lines in R^2 have at most O(m^{2/3}n^{2/3}+m+n) point-line incidences. In addition to answering a natural conjecture posed by Erdos and Purdy, this theorem and its variants have had many applications in combinatorial geometry and additive combinatorics. In 1999, Wolff noticed that certain problems in harmonic analysis and geometric measure theory were closely related to discretised versions of such geometric incidence theorems, where given a small number \delta>0 — the scale at which we can distinguish objects — a \delta-incidence occurs when two objects are within distance \delta. Most algebraic and topological arguments used for discrete incidence theorems break down in this regime, and despite partial progress the majority of these problems remain wide open.
In this talk, we present a quantitative discretised point-line incidence theorem in the plane. This theorem leads to progress on versions of the Furstenberg set problem and the Kakeya problem, and we discuss these applications. This is based on joint work with Joshua Zahl.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 14 Apr 2021, 8:30am-9:30am
Abstract
I will provide an elementary definition of the cobordism ring of involutions of smooth projective varieties over a field (of characteristic not 2). I will describe its structure, and give explicit “stable” polynomial generators. I will draw some concrete consequences concerning the geometry of fixed loci of involutions, in terms of Chern numbers. I will in particular mention an algebraic version of Boardman’s five halves theorem.
Cellular polarization plays a critical during cellular differentiation, development, and cellular migration through the establishment of a long-lived cell-front and cell-rear. Although mechanisms of polarization vary across cells types, some common biochemical players have emerged, namely the RhoGTPases Rac and Rho. The low diffusion coefficient of the active form of these molecules combined with their mutual inhibitory interaction dynamics have led to a prototypical pattern-formation system that can polarizes cell through a non-Turing pattern formation mechanism termed wave-pinning. We investigate the effects of paxillin, a master regulator of adhesion dynamics, on the Rac-Rho system through a positive feedback loop that amplifies Rac activation. We find that paxillin feedback onto the Rac-Rho system produces cells that (i) self-polarize in the absence of any input signal (i.e., paxllin feedback causes a Turing instability) and (ii) become arrested due to the development of multiple protrusive regions. The former effect is a positive finding that can be related to certain cell-types, while the latter outcome is likely an artefact of the model. In order to minimize the effects of this artefact and produce cells that can both self-polarize as well as migrate for extended periods of time, we revisit some of model's parameter values and use lessons from previous models of polarization. This approach allows us to draw conclusions about the biophysical properties and spatiotemporal dynamics of molecular systems required for autonomous decision making during cellular migration.
When does a system undergo a continuous phase transition, and when does a system undergo a first-order (i.e., discontinuous) phase transition? This is the question in some of the most central problems in discrete probability and statistical physics, like whether bond percolation occurs at criticality on the lattice in dimensions 3 to 10. Though physicists have many nonrigorous thoughts about this, not much is known in general. We look at the question for branching process events satisfying recursive properties. For example, let T_1 be the event that a Galton-Watson tree is infinite, and let T_2 be the event that it contains an infinite binary tree starting at the root. The event T_1 holds if and only if T_1 holds for at least one of the trees initiated by children of the root, and T_2 holds if and only if T_2 holds for at least two of these trees. The probability of T_1 has a continuous phase transition, increasing from 0 when the mean of the tree’s child distribution increases above 1. On the other hand, the probability of T_2 has a first-order phase transition, jumping discontinuously to a nonzero value at criticality. We give some explanation of why, explaining the connection between the recursive property satisfied by the event and the phase transition the event experiences.
Algebraic structure is at the heart of many problems in statistics and data analysis. We aim to fit data to a model, or to approximate data by a point on some locus of interest. I will discuss how algebraic structure can be used to capture the existence and uniqueness of a solution to these problems, as well as to suggest suitable algorithms. I will first consider parameter estimation in statistical models via maximum likelihood estimation. We will see connections between maximum likelihood estimation and invariant theory. I will then discuss tensors, the higher dimensional analogues of matrices. The loci of tensors that are of interest in applications often define semi-algebraic sets, given by polynomial equations and inequalities. One example is the signature, tensors that can be used to encode a path of time series data. We will see the algebraic structure that relates a path to its signature.
It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered "standard" tiling complement. It is not known whether this behavior persists for all tilings with no restrictions on the number of prime factors of M. In an ongoing collaboration with Itay Londner, we proved that this is true when M=(pqr)^2. In my talk I will discuss this problem and introduce the main ingredients in the proof.
Cell division is a vital mechanism for cell proliferation, but it often breaks its symmetry during animal development. Symmetry-breaking of cell division, such as the orientation of the cell division axis and asymmetry of daughter cell sizes, regulates morphogenesis and cell fate decision during embryogenesis, organogenesis, and stem cell division in a range of organisms. Despite its significance in development and disease, the mechanisms of symmetry-breaking of cell division remain unclear. Previous studies heavily focused on the mechanism of symmetry-breaking at metaphase of mitosis, wherein a localized microtubule-motor protein activity pulls the mitotic spindle. Recent studies found that cortical flow, the collective migration of the cell surface actin-myosin network, plays an independent role in the symmetry-breaking of cell division after anaphase. Using nematode C. elegans embryos, we identified extrinsic and intrinsic cues that pattern cortical flow during early embryogenesis. Each cue specifies distinct cellular arrangements and is involved in a critical developmental event such as the establishment of the left-right body axis, the dorsal-ventral body axis, and the formation of endoderm. Our research started to uncover the regulatory mechanisms underlying the cortical flow patterning during early embryogenesis.
The UBC- PIMS Mathematical Sciences Young Faculty Award prize was created by two founding donors, Anton Kuipers and Darrell Duffie, to recognize UBC researchers for their leading edge work in mathematics or its applications in the sciences. Dr Elina Robeva is the 2020 winner and will give her colloquium on Thursday April 21, 2021.
Dr. Robeva is an Assistant Professor with the Department of Mathematics at the University of British Columbia. From 2016 – 2019, Dr. Robeva was a Statistics Instructor and an NSF Postdoctoral Fellow in the Department of Mathematics and the Institute for Data, Systems, and Society, at the Massachusetts Institute of Technology. Dr. Robeva completed her PhD in 2016 from UC Berkeley, and won the Bernard Friedman Memorial Prize in Applied Mathematics, for her thesis.
Abstract: Nonparametric density estimation is a challenging problem in theoretical statistics -- in general a maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints allows a path forward.
In this talk I will first discuss non-parametric density estimation under totalpositivity (i.e. log-supermodularity) and log-concavity. Although they possess very special structure, totally positive random variables are quite common in real world data and have appealing mathematical properties. Given i.i.d. samples from a totally positive and log-concave distribution, we prove that the MLE exists with probability one assuming there are at least 3 samples. We characterize the domain of the MLE and if the observations are 2-dimensional, we show that the logarithm of the MLE is a tent function (i.e. a piecewise linear function) with "poles" at the observations, and we show that a certain convex program can find it.
I will finish by discussing density estimation for log-concave graphical models. As before, we show that the MLE exists and is unique with probability 1. We also characterize the domain of the MLE, and show how to find it if the graphical model corresponds to a chordal graph. I will conclude by discussing some future directions.
Recently, a formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ was given in terms of objects called multiline queues, which also compute probabilities of a statistical mechanics model called the multispecies asymmetric simple exclusion process (ASEP) on a ring. It is natural to ask whether the modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ can be obtained using a combinatorial gadget for some other statistical mechanics model. We answer this question in the affirmative. In this talk, we will give a new formula for $\widetilde{H}_{\lambda}(X;q,t)$ in terms of fillings of tableaux called polyqueue tableaux. We define a multispecies totally asymmetric zero range process (TAZRP) on a ring with parameter $t$, whose (unnormalized) stationary probabilities are computed by polyqueue tableaux, and whose partition function is equal to $\widetilde{H}_{\lambda}(X;1,t)$. This talk is based on joint work with Arvind Ayyer and James Martin.
To initiate movement, cells need to form a well-defined "front" and "rear" through the process of cellular polarization. Polarization is a crucial process involved in embryonic development and cell motility and it is not yet well understood. Mathematical models that have been developed to study the onset of polarization have explored either biochemical or mechanical pathways, yet few have proposed a combined mechano-chemical mechanism. However, experimental evidence suggests that most motile cells rely on both biochemical and mechanical components to break symmetry.I will describe a mechano-chemical mathematical model for emergent organization driven by both cytoskeletal dynamics and biochemical reactions. We have identified one of the simplest quantitative frameworks for a possible mechanism for spontaneous symmetry breaking for initiation of cell movement. The framework relies on local, linear coupling between minimal biochemical stochastic and mechanical deterministic systems; this coupling between mechanics and biochemistry has been speculated biologically, yet through our model, we demonstrate it is a necessary and sufficient condition for a cell to achieve a polarized state.
Cells in tissue can communicate short-range via direct contact, and long-range via diffusive signals. In addition, another class of cell-cell communication is by long, thin cellular protrusions that are ~100 microns in length and ~100 nanometers in width. These so-called non-canonical protrusions include cytonemes, nanotubes, and airinemes. But, before establishing communication, they must find their target cell. Here we demonstrate airinemes in zebrafish are consistent with a finite persistent random walk model. We study this model by stochastic simulation, and by numerically solving the survival probability equation using Strang splitting. The probability of contacting the target cell is maximized for a balance between ballistic search (straight) and diffusive (highly curved, random) search. We find that the curvature of airinemes in zebrafish, extracted from live cell microscopy, is approximately the same value as the optimum in the simple persistent random walk model. We also explore the ability of the target cell to infer direction of the airineme’s source, finding the experimentally observed parameters to be at a Pareto optimum balancing directional sensing with contact initiation.
Entropic optimal transport has received a lot of attention in recent years and has become a popular framework for computational optimal transport thanks to the Sinkhorn scaling algorithm. In this talk, I will discuss the multi-marginal case which arises in different applied contexts in physics, economics and machine learning. I will show in particular that the multi-marginal Schrödinger system is well posed (joint work with Maxime Laborde) and that the multi-marginal Sinkhorn algorithm converges linearly.
PIMS workshop on New Trends in Localized Patterns in PDEs: Mathematical Theory and Applications to Physics, Biology and the Social Sciences- a Conference in Honour of the 60th Birthday of Michael J. Ward.
Localized spatial-temporal patterns commonly occur for various classes of linear and nonlinear diffusive processes. In recent years, we have witnessed an explosion of many new and exciting developments in the theory and applications of localized solutions in PDEs. Many new frontiers have opened while new connections between old fields have been discovered.
This workshop will highlight Prof.Ward's influential contributions to the mathematics and applications of localized solutions to PDEs. It brings together his students, colleagues and collaborators in a celebration of Prof.Ward's 60th birthday. Over his long career, Prof. Ward has made numerous invaluable contributions to many areas of applied mathematics, including fluid dynamics, metastability, reaction-diffusion equations, cell biology and pattern formation.
Download the program here: https://www.pims.math.ca/files/2021_PDE_Conference_Program-_Full_04_30_0.pdf
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 12 May 2021, 8:30am-9:30am
Abstract
Consider an algebraic group G acting on an irreducible variety X. We say there exists a generic stabilizer for this action if there exists a nonempty open subset Y of X such that the stabilizers of any y in Y are all conjugate in G. In characteristic 0, there are general results of Richardson proving the existence of a generic stabilizer in many cases. We especially consider the case that G is a simple algebraic group in positive characteristic and X is an irreducible G-module. We show that a generic stabilizer always exists and determine the generic stabilizer in all cases. This fails for semisimple groups. This is joint work with Skip Garibaldi and Ross Lawther.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 19 May 2021, 8:30am-9:30am
Abstract
Which linear algebraic groups act birationally on a rational surface? And which are these actions, up to conjugacy by a birational map? The classification history is quite long over the field of complex numbers and cumulates in the works of Blanc and Dolgachev-Iskovskikh. Over non-closed fields, the classification is not complete yet, but there are many partial results. In this talk, I would like to present the way to attack the classification in general, as well as explain the complete list of actions (up to conjugacy) when the linear algebraic group is infinite.
Online (register for event at https://www.pims.math.ca/scientific-event/210519-tppfscr)
Wed 19 May 2021, 9:30am-10:30am
Abstract
The decade of the 2010s was the hottest yet in more than 150 years of global mean temperature measurements. The key climate change signatures include intensifying extreme events such as widespread droughts, flooding and heatwaves, severe impacts on human health, food security, ecology, and species biodiversity. Climate has been changing from ice-age and is expected to change in future, yet the rate of change is alarming. Data plays a crucial role in developing risk management, mitigation and adaptation strategies under changing climate conditions. This talk focuses on uncertainties in hydrological data and the subsequent effect on extreme events like floods, droughts and heatwaves. Projected changes along with apparent biases in the global climate models, tools available for understanding future climate, are discussed. Importance of understanding uncertainties in observations and simulations and the need to probabilistically evaluate simulations to identify those that agree with observations is emphasized. Finally, the effect of data accuracy and incorporating uncertainty in informed decisions and risk management strategies is highlighted through a case study.
For cells to divide, they must undergo mitosis: the process of spatially organizing their copied DNA (chromosomes) to precise locations in the cell. This procedure is carried out by stochastic components that manage to accomplish the task with astonishing speed and accuracy. New advances from our collaborators in the New York Dept of Health provide 3D spatial trajectories of every chromosome in a cell during mitosis. Can these trajectories tell us anything about the mechanisms driving them? The structure and context of this cutting-edge data makes utilizing classical tools from data science or particle tracking challenging. I will discuss my progress with Alex Mogilner on developing analysis for this data and mathematical modeling of emergent phenomena.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 26 May 2021, 8:30am-9:30am
Abstract
Torsion motives are Chow motives which disappear with rational coefficients. Surprisingly, such objects exist - examples were constructed by Gorchinsky-Orlov. Hypothetically, such motives should generate the kernel of the family of "isotropic realization" functors. I will discuss some invariants of torsion motives which, in particular, shed light on their size.
The classical Pontryagin--Thom isomorphism equates manifold bordism groups with corresponding stable homotopy groups. This construction moreover generalizes to the equivariant context. I will discuss work which establishes a Pontryagin--Thom isomorphism for orbispaces (an orbispace is a "space" which is locally modelled on Y/G for Y a space and G a finite group; examples of orbispaces include orbifolds and moduli spaces of pseudo-holomorphic curves). This involves defining a category of orbispectra and an involution of this category extending Spanier--Whitehead duality. Global homotopy theory also plays a key role.
Epithelial-mesenchymal transition (EMT), a process in which immotile cells that line surfaces in the body become motile mesenchymal cells, play a crucial role in major processes such as wound healing, embryo development, and cancer growth; therefore, examining the dynamics behind individual and collective cell migration would allow for a better understanding of these processes. It has been previously observed that the protein YAP is activated by external mechanical stimuli and affects the expression and activation of the proteins E-cadherin and Rac1, which are involved in intercellular adhesion and migratory ability respectively. It has also been demonstrated that the mechanical stimulation of expanding cell sheets leads to the formation of finger-like projections and EMT, as well as quantitative differences in properties between cells near the sheet edge and cells away from it. Such cell sheets can be simulated using Morpheus, an agent-based modelling and simulation environment. In this talk, I will propose an ODE model for YAP/Rac1/E-cadherin dynamics and demonstrate that the resulting Morpheus simulation gives results consistent with several observations seen in vitro.
Abstract: The crux of a card trick performed with a deck of cards usually depends on understanding how shuffles of the deck change the order of the cards. By understanding which permutations are possible, one knows if a given card may be brought into a certain position. The mathematics of shuffling a deck of 2n cards with two ``perfect shuffles'' was studied thoroughly by Diaconis, Graham and Kantor in 1983. I will report on our efforts to understand a generalisation of this problem, with a so-called ``many handed dealer'' shuffling kn cards by cutting into k piles with n cards in each pile and using k! possible shuffles. A conjecture of Medvedoff and Morrison suggests that all possible permutations of the deck of cards are achieved, as long as k is not 4 and n is not a power of k. We confirm this conjecture for three doubly infinite families of integers, but the conjecture remains open. We initiate a more general study of shuffle groups, which admit an arbitrary subgroup of shuffles. This is joint work with Carmen Amarra and Luke Morgan. About the Niven Lectures: Ivan Niven was a famous number theorist and expositor; his textbooks won numerous awards, have been translated into many languages and are widely used to this day. Niven was born in Vancouver in 1915, earned his Bachelor's and Master's degrees at UBC in 1934 and 1936 and his Ph.D. at the University of Chicago in 1938. He was a faculty member at the University of Oregon from 1947 until his retirement in 1982. The annual Niven Lecture Series, held at UBC since 2005, is funded in part through a generous bequest from Ivan and Betty Niven to the UBC Mathematics Department.
The COVID-19 pandemic has passed its initial peak in most countries in the world, making it ripe to assess whether the basic reproduction number (R0) is different across countries and what demographic, social, and environmental factors other than interventions characterize vulnerability to the virus. In this talk, I will show the association (linear and non-linear) between COVID-19 R0 across countries and 17 demographic, social and environmental variables obtained using a generalized additive model. Moreover, I will present a mathematical model of COVID-19 that we designed and used to explore the effects of adopting various vaccination and relaxation strategies on the COVID-19 epidemiological long-term projections in Ontario. Our findings are able to provide public health bodies with important insights on the effect of adopting various mitigation strategies, thereby guiding them in the decision-making process.
Online: link available at https://researchseminars.org/ or from Zinovy Reichstein
Wed 9 Jun 2021, 8:30am-9:30am
Abstract
A fundamental result in the theory of abelian varieties is the so-called rigidity lemma, which implies that any scheme morphism between abelian varieties which preserves identities is a homomorphism. The analogous assertion for affine group schemes is totally false in general. Nevertheless, we will discuss a somewhat analogous rigidity result for maps from open subsets of the projective line into wound unipotent groups, and give several applications, especially that unirationality for algebraic groups descends through separable extensions.
online: registration via https://ubc.zoom.us/meeting/register/u5Yrfu2sqTkoH9AqIzq7m7896a2yg2A6BlSe
Wed 15 Sep 2021, 3:00pm-4:00pm
Abstract
We give an asymptotic lower bound on the number of field extensions generated by algebraic points on superelliptic curves over $\mathbb{Q}$ with fixed degree $n$, discriminant bounded by $X$, and Galois closure $S_n$. For $C$ a fixed curve given by an affine equation $y^m = f(x)$ where $m \geq 2$ and $deg f(x) = d \geq m$, we find that for all degrees $n$ divisible by $gcd(m, d)$ and sufficiently large, the number of such fields is asymptotically bounded below by $X^{c_n}$ , where $c_n \mapsto 1/m^2$ as $n \to \infty$. This bound is determined explicitly by parameterizing $x$ and $y$ by rational functions, counting specializations, and accounting for multiplicity. We then give geometric heuristics suggesting that for $n$ not divisible by $\gcd(m, d)$, degree $n$ points may be less abundant than those for which $n$ is divisible by $\gcd(m, d)$. Namely, we discuss the obvious geometric sources from which we expect to find points on $C$ and discuss the relationship between these sources and our parametrization. When one a priori has a point on $C$ of degree not divisible by $\gcd(m, d)$, we argue that a similar counting argument applies. As a proof of concept we show in the case that $C$ has a rational point that our methods can be extended to bound the number of fields generated by a degree $n$ point of $C$, regardless of divisibility of $n$ by $\gcd(m, d)$. This talk is based on joint work with Christopher Keyes.
Note: The talk will be followed by 30 minutes of informal discussion with the attendees.
I will begin from two different motivations: 1) (the mathematical physics motivation) trying to derive a macroscopic, deterministic "mean-curvature-type" evolution for the boundary between two coexisting phases, starting from a microscopic stochastic dynamics and 2) (the computer science motivation) analyzing the running time of Monte Carlo Markov Chain (MCMC) algorithms that sample uniformly among lozenge tilings of a large region of the plane. I will discuss the link between these two problems and try to emphasize their connections with dimer covering problems and with interacting particle systems. Our results include: a) bounds on the disappearence time of droplets for the three-dimensional Ising model at zero temperature b) derivation of mean-curvature evolution for the two-dimensional Ising model at zero temperature and c) (in some situations) "almost-optimal" bounds on the running time of the lozenge tiling MCMC. 
minimization (SPGL1) algorithm in which the sequence of LASSO subproblems are replaced by a sequence of weighted LASSO subproblems with constant weights applied to a support estimate. The support estimate is derived from the data and is updated at every iteration. The algorithm also modifies the Pareto curve at every iteration to reflect the new weighted .gif){kind=spacer}
elements lie at distance [R-o(R),R+o(R)] from the origin, where R is the minimal ball in Z^k of size at least |G|), provided k-d(G) >> 1 where d(G) is the size of the smallest generating set of G. As conjectured by Aldous and Diaconis, the cutoff time is typically independent of the algebraic structure (it is given by the time at which the entropy of a random walk on Z^k is log |G|). 
Note for Attendees
There will be tea and cookies in the math lounge at approximately 2:45pm.