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Series: PDE Seminar

The large-time behavior of solutions to Burgers equation with small viscosity isdescribed using invariant manifolds. In particular, a geometric explanation is provided for aphenomenon known as metastability, which in the present context means that solutions spend avery long time near the family of solutions known as diffusive N-waves before finallyconverging to a stable self-similar diffusion wave. More precisely, it is shown that in termsof similarity, or scaling, variables in an algebraically weighted L^2 space, theself-similar diffusion waves correspond to a one-dimensional global center manifold ofstationary solutions. Through each of these fixed points there exists a one-dimensional,global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus,metastability corresponds to a fast transient in which solutions approach this ``metastable"manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally,convergence to the self-similar diffusion wave. This is joint work with C. Eugene Wayne.

Series: Research Horizons Seminar

Hosted by: Huy Huynh and Yao Li

In the preceeding talk, I outlined a framework for variational problems

and some of the basic tools and results. In this talk I will attempt

describe several problems of current interest.

and some of the basic tools and results. In this talk I will attempt

describe several problems of current interest.

Series: Job Candidate Talk

In the lecture I will explain how various fundamental structures from group representation theory appear naturally in the context of discrete harmonic analysis and can be applied to solve concrete problems from digital signal processing. I will begin the lecture by describing our solution to the problem of finding a canonical orthonormal basis of eigenfunctions of the discrete Fourier transform (DFT). Then I will explain how to generalize the construction to obtain a larger collection of functions that we call "The oscillator dictionary". Functions in the oscillator dictionary admit many interesting pseudo-random properties, in particular, I will explain several of these properties which arise in the context of problems of current interest in communication theory.

Series: Graph Theory Seminar

A graph G contains a graph H as a minor if a graph isomorphic to H can be obtained from a subgraph of G bycontracting edges. One of the central results of the rich theory of graph minors developed by Robertson and Seymour is an approximate description of graphs that do not contain a fixed graph as a minor. An exact description is only known in a few cases when the excluded minor is quite small.In recent joint work with Robin Thomas we have proved a conjecture of his, giving an exact characterization of all large, t-connected graphs G that do not contain K_t, the complete graph on t vertices, as a minor. Namely, we have shown that for every integer t there exists an integer N=N(t) such that a t-connected graph G on at least N vertices has no K_t minor if and only if G contains a set of at most t- 5 vertices whose deletion makes G planar. In this talk I will describe the motivation behind this result, outline its proof and mention potential applications of our methods to other problems.

Series: Analysis Working Seminar

This is the first meeting of a weekly working seminar on two weight inequalities in Harmonic Analysis. James Curry will present the paper arXiv:0911.3437, which proves two-weight norm inequalities for a class of dyadic, positive operators.

Series: Combinatorics Seminar

Hajos' conjecture is false, and it seems that graphs without a

subdivision of a big complete graph do not behave as well as

those without a minor of a big complete graph.

subdivision of a big complete graph do not behave as well as

those without a minor of a big complete graph.

In fact, the graph minor theorem (a proof of Wagner's

conjecture) is not true if we replace the minor relation by the

subdivision relation. I.e, For every infinite sequence

G_1,G_2, ... of graphs, there exist distinct integers

i < j such that G_i is a minor of G_j, but if we replace

''minor" by ''subdivision", this is no longer true.

This is partially because we do not really know what the graphs

without a subdivision of a big complete graph look like.

In this talk, we shall discuss this issue. In particular,

assuming some moderate connectivity condition, we can say

something, which we will present in this talk.

Topics also include coloring graphs without a subdivision of a

large complete graph, and some algorithmic aspects. Some of the

results are joint work with Theo Muller.

Series: SIAM Student Seminar

The Fundamental Theorem of Algebra implies that a complex valued nxn matrix has n eigenvalues (including multiplicities). In this talk we introduce a general method for reducing the size of a square matrix while preserving this spectrum. This can then be used to improve on the classic eigenvalue estimates of Gershgorin, Brauer, and Brualdi. As this process has a natural graph theoretic interpretation this talk should be accessible to most anyone with a basic understanding of matrices and graphs. These results are based on joint work with Dr. Bunimovich.

Series: Job Candidate Talk

In this talk, I will first discuss several chemotaxis models includingthe classical Keller-Segel model.Chemotaxis is the phenomenon in which cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals (chemoattractants) in their environment. The mathematical models of chemotaxis are usually described by highly nonlinear time dependent systems of PDEs. Therefore, accurate and efficient numerical methods are very important for the validation and analysis of these systems. Furthermore, a common property of all existing chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions rapidly growing in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. In either case, capturing such solutions numerically is a challenging problem. In our work we propose a family of stable (even at times near blow up) and highly accurate numerical methods, based on interior penalty discontinuous Galerkin schemes (IPDG) for the Keller-Segel chemotaxis model with parabolic-parabolic coupling. This model is the basic step in the modeling of many real biological processes and it is described by a system of a convection-diffusion equation for the cell density, coupled with a reaction-diffusion equation for the chemoattractant concentration.We prove theoretical hp error estimates for the proposed discontinuous Galerkin schemes. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution.Numerical experiments to demonstrate the stability and accuracy of the proposed methods for chemotaxis models and comparison with other methods will be presented. Ongoing research projects will be discussed as well.

Thursday, January 21, 2010 - 11:05 ,
Location: Skiles 269 ,
Bruce Reed ,
McGill University ,
Organizer: Robin Thomas

The term Probabilistic Method refers to the proof of

deterministic statements using probabilistic tools. Two of the most famous

examples arise in number theory. these are: the first non-analytic proof

of the prime number theorem given by Erdos in the 1940s, and the recent

proof of the Hardy-Littlewood Conjecture (that there are arbitrarily long

arithmetic progressions of primes) by Green and Tao.

The method has also been succesfully applied in the field of graph

colouring. We survey some of the results thereby obtained.

The talk is targeted at a general audience. We will first define graph

colouring, explain the type of graph colouring problems which tend to

attract interest, and then explain the probabilistic tools which are

used

to solve them, and why we would expect the type of tools that are used to

be effective for solving the types of problems typically studied.

deterministic statements using probabilistic tools. Two of the most famous

examples arise in number theory. these are: the first non-analytic proof

of the prime number theorem given by Erdos in the 1940s, and the recent

proof of the Hardy-Littlewood Conjecture (that there are arbitrarily long

arithmetic progressions of primes) by Green and Tao.

The method has also been succesfully applied in the field of graph

colouring. We survey some of the results thereby obtained.

The talk is targeted at a general audience. We will first define graph

colouring, explain the type of graph colouring problems which tend to

attract interest, and then explain the probabilistic tools which are

used

to solve them, and why we would expect the type of tools that are used to

be effective for solving the types of problems typically studied.

Series: Analysis Seminar

Müntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful generalization of Weierstrass's Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials on the interval of orthogonality, and in particular obtain new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics on the interval, and the zero spacing behavior follows. This is the first time that such asymptotics have been obtained for general Müntz exponents. We also look at the asymptotic behavior outside the interval, and the asymptotic properties of the associated Christoffel functions.