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 F_{4}. In particular, we will see that a variety of type
F_{4}/P_{4}, 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" WS_{n} 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.

In this talk I will explain what a generic torus of a
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 G_{2} 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.

Given a fixed binary form f(u,v) of degree d over a field k, the associated Clifford algebra is the k-algebra C_{f}=k{u,v}/I,where $I$ is the two-sided ideal generated by elements of the form

(a u+b v)^^{d}-f(a,b),

as a and b range over k. All representations of C_{f} 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=U_{f,r} for the irreducible rd-dimensional representations of C_{f} for each r >= 2. Our construction starts with the projective curve C in P^{2}_{k} defined by the equation w^{d}=f(u,v), and produces U_{f,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.
------------------------------------------------------------

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 p³ 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.

## Note for Attendees

Tea & cookies afterwards!