School of Mathematics

Moment-Angle Complexes, Spaces of Hard-Disks and Their Associated Stable Decompositions

Fred Cohen
University of Rochester; Member, School of Mathematics
January 10, 2011

Topological spaces given by either (1) complements of coordinate planes in Euclidean space or (2) spaces of non-overlapping hard-disks in a fixed disk have several features in common. The main results, in joint work with many people, give decompositions for the so-called "stable structure" of these spaces as well as consequences of these decompositions.

This talk will present definitions as well as basic properties.

Two Dimensional Galois Representations Over Imaginary Quadratic Fields

Andrei Jorza
Institute for Advanced Study
December 16, 2010

To a regular algebraic cuspidal representation of GL(2) over a quadratic imaginary field, whose central character is conjugation invariant, Taylor et al. associated a two dimensional Galois representation which is unramified at l different from p outside a finite set of places. The first half of this talk concerns the crystallinity of the Galois representation at p , under a technical assumption. The second half of the talk is on recent work towards local-global compatibility (on GSp(4) and its implication for GL(2)).

Ramification in Iwasawa Modules

Chandrashekar Khare
Institute for Advanced Study
December 15, 2010

Iwasawa developed his theory for class groups in towers of cyclotomic fields partly in analogy with Weil's theory of curves over finite fields. In this talk, we present another such conjectural analogy. It seems intertwined with Leopoldt's conjecture. This talk is related to J-P.Wintenberger's talk here earlier this year.

Uniform Well-Posedness and Inviscid Limit for the Benjamin-Ono-Burgers Equation

Zihua Guo
Institute for Advanced Study
December 13, 2010

We prove that the Cauchy problem for the Benjamin-Ono-Burgers equation is uniformly globally well-posed in H^1 for all "\epsilon\in [0,1]". Moreover, we show that for any T>0 the solution converges in C([0,T]:H^1) to that of Benjamin-Ono equation as "\epsilon --> 0". Our results give a new proof for the global well-posedness of the BO equation in H^1(R) without using gauge transform, which was first obtained by Tao using gauge transform, and also solve the problem about the inviscid limit behavior in H1.

Colouring Tournaments

Paul Seymour
Princeton University
December 13, 2010

A ``tournament'' is a digraph obtained from a complete graph by directing its edges, and ``colouring'' a tournament means partitioning its vertex set into acyclic subsets (``acyclic'' means the subdigraph induced on the subset has no directed cycles). This concept is quite like that for graph-colouring, but different. For instance, there are some tournaments H such that every tournament not containing H as a subdigraph has bounded chromatic number. We call them ``heroes''; for example, all tournaments with at most four vertices are heroes.

A Classical Approximation Point of View on Some Results in the Spectral Theory of Jacobi Matrices

Mira Shamis
Institute for Advanced Study
December 10, 2010

Deift--Simon and Poltoratskii--Remling proved upper bounds on the measure of the absolutely continuous spectrum of Jacobi matrices. Using methods of classical approximation theory, we give a new proof of their results, and generalize them in several ways. First, we prove a sharper inequality taking the distribution of the values of the potential into account. Second, we prove a generalization of a "local" inequality of Deift--Simon to the non-ergodic setting. Based on joint work with Sasha Sodin