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Series: Stochastics Seminar

Finding ground states of spin glasses, a model of disordered materials,
has a deep connection to many hard combinatorial optimization problems,
such as satisfiability, maxcut, graph-bipartitioning, and coloring.
Much insight has been gained for the combinatorial problems from the
intuitive approaches developed in physics (such as replica theory and
the cavity method), some of which have been proven rigorously recently.
I present a treasure trove of numerical data obtained with heuristic
methods that suggest a number conjectures, such as an equivalence
between maxcut and bipartitioning for r-regular graphs, a simple
relation for their optimal configurations as a function of degree r,
and anomalous extreme-value fluctuations in a variety of models, hotly
debated in physics currently. For some, such as those related to
finite-size effects, not even a physics theory exists, for others
theory exists that calls for rigorous methods.

Series: Analysis Seminar

Given a set of complex exponential e^{i \lambda_n x} how large do you have to take r so that the sequence is independent in L^2[-r,r] ? The answer is given in terms of the Beurling-Mallivan density.

Series: Other Talks

Series: Algebra Seminar

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.

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.

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.
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.