Seminars and Colloquia by Series

The Geometric Weil Representation

Series
Algebra Seminar
Time
Wednesday, January 27, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Shamgar Gurevich – Institute for Advanced Study, Princeton –
This is a sequel to my first talk on "group representation patterns in digital signal processing". It will be slightly more specialized. The finite Weil representation is the algebra object that governs the symmetries of Fourier analysis of the Hilbert space L^2(F_q). The main objective of my talk is to introduce the geometric Weil representation---developed in a joint work with Ronny Hadani---which is an algebra-geometric (l-adic perverse Weil sheaf) counterpart of the finite Weil representation. Then, I will explain how the geometric Weil representation is used to prove the main results stated in my first talk. In the course, I will explain the Grothendieck geometrization procedure by which sets are replaced by algebraic varieties and functions by sheaf theoretic objects.

Using global invariant manifolds to understand metastability in Burgers equation with small viscosity

Series
PDE Seminar
Time
Tuesday, January 26, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Margaret Beck – Boston University
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.

Variational problems involving area (continued)

Series
Research Horizons Seminar
Time
Tuesday, January 26, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
John McCuan – School of Math, Georgia Tech

Please Note: 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.

Group Representation Patterns in Digital Signal Processing

Series
Job Candidate Talk
Time
Tuesday, January 26, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Shamgar Gurevich – Institute for Advanced Study, Princeton –
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.

K_t minors in large t-connected graphs

Series
Graph Theory Seminar
Time
Tuesday, January 26, 2010 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Sergey Norin – Princeton University
 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.

Graphs without subdivisions

Series
Combinatorics Seminar
Time
Friday, January 22, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Keni-chi Kawarabayashi – National Institute of Informatics
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. 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.

Reducing the Size of a Matrix While Maintaining its Spectrum

Series
SIAM Student Seminar
Time
Friday, January 22, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Benjamin Webb – School of Mathematics, Georgia Tech
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.

Chemotaxis and Numerical Methods for Chemotaxis Models

Series
Job Candidate Talk
Time
Thursday, January 21, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Yekaterina Epshteyn – Carnegie Mellon University
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.

Pages