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 Events
Samuel Bach
UBC
Mon 2 Oct 2017, 4:00pm
Algebraic Geometry Seminar
MATH 126
Derived algebraic geometry and L-theory
MATH 126
Mon 2 Oct 2017, 4:00pm-5:00pm

Abstract

 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.
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Eldad Haber
Earth and Ocean Sciences, UBC
Tue 3 Oct 2017, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS Lounge)
Deep Neural Networks meets PDE's
ESB 4133 (PIMS Lounge)
Tue 3 Oct 2017, 12:30pm-1:30pm

Abstract

 
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.
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University of Tennessee, Knoxville
Tue 3 Oct 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
(this talk is cancelled)
ESB 2012
Tue 3 Oct 2017, 3:30pm-4:30pm

Abstract

Please note this talk is cancelled.
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Imre Barany
UCL and Renyi Institute
Tue 3 Oct 2017, 4:00pm
Discrete Math Seminar
ESB 4127
Small subset sums
ESB 4127
Tue 3 Oct 2017, 4:00pm-5:00pm

Abstract

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.
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Tel Aviv
Wed 4 Oct 2017, 3:00pm
Probability Seminar
ESB 2012
A condition for long-range order in discrete spin systems
ESB 2012
Wed 4 Oct 2017, 3:00pm-4:15pm

Abstract


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.
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University of Minnesota
Wed 4 Oct 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Homology of braid groups via quantum shuffle algebras
ESB 4133 (PIMS Lounge)
Wed 4 Oct 2017, 3:15pm-4:15pm

Abstract

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.
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UBC
Thu 5 Oct 2017, 3:30pm
Number Theory Seminar
MATH 126
Decomposable polynomials in recursively defined sequences of polynomials
MATH 126
Thu 5 Oct 2017, 3:30pm-5:15pm

Abstract

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
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University of Minnesota
Fri 6 Oct 2017, 3:00pm
Department Colloquium
ESB 2012
Topological approaches to the distribution of Galois groups
ESB 2012
Fri 6 Oct 2017, 3:00pm-4:00pm

Abstract

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.
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Yong Liu
North China Electric University
Tue 10 Oct 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Nondegeneracy, Morse Index and Orbital Stability of KP-I Lump Solution
ESB 2012
Tue 10 Oct 2017, 3:30pm-4:30pm

Abstract

 We prove that the lump solution of the classical KP-I equation is nondegenerate and its Morex index is one. As a consequence, it is orbital stable. 
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UBC
Tue 10 Oct 2017, 4:00pm
Discrete Math Seminar
ESB 4127
Chromatic symmetric functions and e-positivity
ESB 4127
Tue 10 Oct 2017, 4:00pm-5:00pm

Abstract

 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.
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UBC
Wed 11 Oct 2017, 3:00pm
Probability Seminar
ESB 2012
Self-avoiding walk, spin systems, and renormalisation
ESB 2012
Wed 11 Oct 2017, 3:00pm-4:00pm

Abstract


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.
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UBC
Thu 12 Oct 2017, 3:30pm
Number Theory Seminar
MATH 126
Normally Distributed Arithmetic Functions
MATH 126
Thu 12 Oct 2017, 3:30pm-5:15pm

Abstract

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).
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Tel-Aviv University
Mon 16 Oct 2017, 3:00pm
Harmonic Analysis Seminar
GEOG 101
Interpolation sets and arithmetic progressions
GEOG 101
Mon 16 Oct 2017, 3:00pm-4:00pm

Abstract

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.
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FU Berlin
Mon 16 Oct 2017, 4:00pm
Algebraic Geometry Seminar
MATH 126
Infinitesimal qG-deformations of cyclic quotient singularities
MATH 126
Mon 16 Oct 2017, 4:00pm-5:00pm

Abstract

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)
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Bamdad Hosseini
Department of Mathematics, SFU
Tue 17 Oct 2017, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS Lounge)
Non-Gaussian priors in Bayesian inverse problems: from theory to applications
ESB 4133 (PIMS Lounge)
Tue 17 Oct 2017, 12:30pm-1:30pm

Abstract

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. 
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Yonsei University
Tue 17 Oct 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Global well-posedness and asymptotics of a type of Keller-Segel models coupled to fluid flow
ESB 2012
Tue 17 Oct 2017, 3:30pm-4:30pm

Abstract

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.
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Technical University of Berlin
Wed 18 Oct 2017, 3:00pm
Probability Seminar
ESB 2012
Harnack inequality for degenerate balanced random walks
ESB 2012
Wed 18 Oct 2017, 3:00pm-4:00pm

Abstract


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.
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Cornell University
Wed 18 Oct 2017, 3:15pm
Topology and related seminars
ESB 4133 (PIMS Lounge)
Deriving zeta functions
ESB 4133 (PIMS Lounge)
Wed 18 Oct 2017, 3:15pm-4:15pm

Abstract

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.
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Aaron Berk
Mathematics, UBC
Fri 20 Oct 2017, 12:00pm
Graduate Student Seminar
MATH 203
From Hoeffding's inequality to PAC learning
MATH 203
Fri 20 Oct 2017, 12:00pm-1:00pm

Abstract

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).
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Cambridge
Fri 20 Oct 2017, 3:00pm
Department Colloquium
ESB 2012
Universality for the dimer model
ESB 2012
Fri 20 Oct 2017, 3:00pm-4:00pm

Abstract

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.
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Courant Institute, NYU
Mon 23 Oct 2017, 3:00pm
Institute of Applied Mathematics / PIMS Seminars and PDF Colloquiums
ESB 2012
Numerical Investigation of Crouzeix’s Conjecture
ESB 2012
Mon 23 Oct 2017, 3:00pm-4:00pm

Abstract

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).
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SNS Pisa
Mon 23 Oct 2017, 4:00pm
Algebraic Geometry Seminar
MATH 126
Chow rings of some stacks of smooth curves
MATH 126
Mon 23 Oct 2017, 4:00pm-5:00pm

Abstract

 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.
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McGill
Tue 24 Oct 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
An isometric embedding problem and related geometric inequalities
ESB 2012
Tue 24 Oct 2017, 3:30pm-4:30pm

Abstract

Solutions to the classical Weyl problem by Nirenberg and Pogorelov
play fundamental role in the notion of quasi local masses and positive quasi
local mass theorems in general relativity. An interesting question in
differential geometry is whether one can isometrically embed compact surfaces
with positive Gauss curvature to a general 3 dimensional ambient space. Of
particular importance is the anti de Sitter Schwarzchild space in general
relativity.  We discuss some recent progress in this direction, the a priori
estimates for embedded surfaces in a joint work with Lu, the openness and
non-rigidity results of Li -Wang, and a new quasi local type inequality of
Lu-Miao. We will also discuss open problem related to isometric embeddings to
ambient spaces with horizons.
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UBC
Tue 24 Oct 2017, 4:00pm
Discrete Math Seminar
ESB 4127
Disjoint Cycles and Equitable Colorings in Graphs
ESB 4127
Tue 24 Oct 2017, 4:00pm-5:00pm

Abstract

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.

 

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Krishanu Sankar
UBC
Wed 25 Oct 2017, 3:15pm
Topology and related seminars
ESB 4133
Symmetric Powers and the Dual Steenrod Algebra - Part 1
ESB 4133
Wed 25 Oct 2017, 3:15pm-4:15pm

Abstract

The mod p Steenrod algebra is the (Hopf) algebra of stable operations on mod p cohomology, 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 infinite symmetric power of the n-dimensional sphere is the Eilenberg-Maclane space K(Z, n),and one can use an appropriate modification of this construction to compute the dual Steenrod algebra. The infinite symmetric power of the sphere spectrum has a filtration whose k-th cofiber miraculously turns out to be the Steinberg summand (from modular representation theory of GL_k(F_p)) of the classifying space of (Z/p)^k. This opens the door for slick computations - for example, the Milnor indecomposables can be picked out as explicit cells.

 

In this talk, I will introduce the concepts and results chronologically. I will also include hands-on homotopy theory computations as time permits.

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Alain Prat
UBC Math
Thu 26 Oct 2017, 12:30pm
Lunch Series on Teaching & Learning
Math 126
Working against the WeBWork clock: What are the behaviour patterns of students who struggle to complete online calculus assignments?
Math 126
Thu 26 Oct 2017, 12:30pm-1:30pm

Abstract

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.
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Nikhil Kher
Fri 27 Oct 2017, 12:00pm
Graduate Student Seminar
MATH 203
Operator Semigroups and Hille Yosida Theorem
MATH 203
Fri 27 Oct 2017, 12:00pm-1:00pm

Abstract

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.
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Jim Bryan
UBC
Mon 30 Oct 2017, 4:00pm
Algebraic Geometry Seminar
MATH 126
Donaldson-Thomas invariants of the banana manifold and elliptic genera.
MATH 126
Mon 30 Oct 2017, 4:00pm-5:00pm

Abstract

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.
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Tom Eaves
Mathematics, UBC
Tue 31 Oct 2017, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS Lounge)
What is turbulence, and how do we find it?
ESB 4133 (PIMS Lounge)
Tue 31 Oct 2017, 12:30pm-1:30pm

Abstract

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. 
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