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)).

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.

Univalent Foundations of Mathematics

Vladimir Voevodsky
Institute for Advanced Study
December 10, 2010

The correspondence between homotopy types and higher categorical analogs of groupoids which was first conjectured by Alexander Grothendieck naturally leads to a view of mathematics where sets are used to parametrize collections of objects without "internal structure" while collections of objects with "internal structure" are parametrized by more general homotopy types. Univalent Foundations are based on the combination of this view with the discovery that it is possible to directly formalize reasoning about homotopy types using Martin-Lof type theories.

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

Parahoric Subgroups and Supercuspidal Representations of p-Adic groups

Dick Gross
Harvard University
December 9, 2010

This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is that this induced representation can (in certain situations) have finite length. I will describe the parahorics and characters for which this occurs, and what the Langlands parameters of the corresponding irreducible summands must be.

On Families of Filtered phi Modules and Crystalline Representations

Eugen Hellmann
University of Bonn
December 8, 2010

We study families of filtered phi-modules associated to families of p-adic Galois representations as considered by Berger and Colmez. We show that the weakly admissible locus in a family of filtered phi-modules is open and that the groupoid of weakly admissible modules is in fact an Artin stack. Working in the category of adic spaces instead of the category of rigid analytic spaces one can show that there is an open substack of the weakly admissible locus over which the filtered phi-modules is induced from a family of crystalline representations.