Colloquia 2012-2013

Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Fall 2012

date	speaker	title	host(s)
Sept 14	Jordan Ellenberg (Madison)	FI-modules: an introduction	local
Sept 20, 4pm	Persi Diaconis (Stanford)	Spatial mixing: problems and progress	Jean-Luc
Sept 21	Joyce McLaughlin (RPI)	TBA	WIMAW
Sept 28	Eric Marberg (MIT)	TBA	Isaacs
Oct 5	Howard Masur (Chicago)	TBA	Dymarz
Wed, Oct 10, 4pm	Bas Lemmens (Univ. of Kent)	From hyperbolic geometry to nonlinear Perron-Frobenius theory	LAA lecture
Oct 12	Joachim Rosenthal (Univ. of Zurich)	TBA	Boston
Oct 19	Irene Gamba (Univ. of Texas)	TBA	WIMAW
Oct 26	Luke Oeding (UC Berkeley)	TBA	Gurevich
Tues, Oct 30	Andrew Majda (Courant)	TBA	Smith, Stechmann
Thurs, Nov 1	Peter Constantin (Princeton)	TBA	Distinguished Lecture Series
Nov 2	Peter Constantin (Princeton)	TBA	Distinguished Lecture Series
Nov 9 and later	Reserved for potential interviews

Spring 2013

date	speaker	title	host(s)
Feb 15	Eric Lauga (UCSD)	TBA	Spagnolie
Mar 1	Kirsten Wickelgren (Harvard)	TBA	Street
March 22	Neil O'Connell (Warwick)	TBA	Timo Seppalainen
March 29	Spring Break	No Colloquium
April 19	Andrew Snowden (MIT)	TBA	Street
April 26	Davesh Maulik (Columbia)	TBA	Street

Abstracts

Thu, Sept 14: Jordan Ellenberg (UW-Madison)

FI-modules: an introduction (joint work with T Church, B Farb, R Nagpal)

In topology and algebraic geometry one often encounters phenomena of _stability_. A famous example is the cohomology of the moduli space of curves M_g; Harer proved in the 1980s that the sequence of vector spaces H_i(M_g,Q), with g growing and i fixed, has dimension which is eventually constant as g grows with i fixed.

In many similar situations one is presented with a sequence {V_n}, where the V_n are not merely vector spaces, but come with an action of S_n. In many such situations the dimension of V_n does not become constant as n grows -- but there is still a sense in which it is eventually "always the same representation of S_n" as n grows. The preprint

http://arxiv.org/abs/1204.4533

shows how to interpret this kind of "representation stability" as a statement of finite generation in an appropriate category; we'll discuss this set-up and some applications to the topology of configuration spaces, the representation theory of the symmetric group, and diagonal coinvariant algebras. Finally, we'll discuss recent developments in the theory of FI-modules over general rings, which is joint work with (UW grad student) Rohit Nagpal.

Thu, Sept 20: Persi Diaconis (Stanford)

Spatial mixing: problems and progress

One standard way of mixing (cards, dominos, Mahjong tiles) is to 'smoosh' them around on the table with two hands. I will introduce some models for this, present data (it's surprisingly effective) and some first theorems. The math involved is related to fluid flow and Baxendale-Harris random homeomorphisims.

Wed, Oct 10: Bas Lemmens (University of Kent)

From hyperbolic geometry to nonlinear Perron-Frobenius theory

In a letter to Klein Hilbert remarked that the logarithm of the cross-ratio is a metric on any open, bounded, convex set in Euclidean space. These metric spaces are nowadays called Hilbert geometries. They are a natural non-Riemannian generalization of Klein's model of the hyperbolic plane, and play a role in the solution of Hilbert's fourth problem.

In the nineteen fifties Garrett Birkhoff and Hans Samelson independently discovered that one can use Hilbert's metric and the contraction mapping principle to prove the existence and uniqueness of a positive eigenvector for a variety of linear operators that leave a closed cone in a Banach space invariant. Their results are a direct extension of the classical Perron-Frobenius theorem concerning the eigenvectors and eigenvalues of nonnegative matrices. In the past decades this idea has been further developed and resulted in strikingly detailed nonlinear extensions of the Perron-Frobenius theorem. In this talk I will discuss the synergy between metric geometry and (nonlinear) operator theory and some of the recent results and open problems in this area.