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)	CANCELLED	Jean-Luc
Sept 21	Joyce McLaughlin (RPI)	Mathematics for Imaging Biomechanical Parameters in Dynamic Elastography	WIMAW
Sept 28	Eric Marberg (MIT)	Supercharacters for algebra groups: applications and extensions	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
Thur, Oct 11, 4pm	Irene Gamba (UT-Austin)	Analytical and numerical issues associated with the dynamics of the non-linear Boltzmann equation	WIMAW
Oct 12	Joachim Rosenthal (Univ. of Zurich)	Linear Random Network Codes, a Grassmannian Approach	Boston
Oct 19	Saverio E Spagnolie (Madison)	TBA	Roch
Oct 26	Luke Oeding (UC Berkeley)	The Trifocal Variety	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 5	John Jones (ASU)	TBA	Boston
April 19	Andrew Snowden (MIT)	TBA	Street
April 26	Davesh Maulik (Columbia)	TBA	Street

Abstracts

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.

Sept 21: Joyce R. McLaughlin (Rensselaer Polytechnic Institute)

Mathematics for Imaging Biomechanical Parameters in Dynamic Elastography

Elastography, the imaging of biomechanical parameters in tissue, is motivated by the doctor’s palpation exam where the doctor presses against the skin to detect stiff and abnormal tissue changes. In dynamic elastography experiments, the tissue is in motion with displacement amplitudes on the order of tens of microns. The displacement(s) are determined with sequences of MR data sets or sequences of ultrasound RF/IQ data sets within the tissue; and the data shows a dispersive effect indicating that tissue is viscoelastic. A choice of viscoelastic model must be made. For each model the biomechanical parameters satisfy a first order, linear or nonlinear, partial differential equation (system) with real or complex coefficients. We discuss the mathematical properties of these equations and how those properties lead to successful interpretation of the data, and to successful algorithms and images. We show biomechanical images of breast cancer and prostate cancer and compare those images to ultrasound images and histology slides with marked cancerous inclusions.

Sept 28: Eric Marberg (MIT)

Supercharacters for algebra groups: applications and extensions

The group U_n(F_q) of unipotent upper triangular matrices over a finite field belongs to the same list of fundamental examples as the symmetric or general linear groups. It comes as some surprise, therefore, that the group's irreducible characters are unknown, and considered in some sense unknowable. In order to tackle problems normally requiring knowledge of a group's irreducible characters, Diaconis and Isaacs developed the notion of the supercharacters of an algebra group, generalizing work of Andre and Yan. Algebra groups form a well-behaved class of p-groups including U_n(F_q) as a prototypical example, and supercharacters are certain reducible characters which form a useful approximation to the set of irreducible characters. In this talk I will survey several equivalent definitions of the supercharacters of an algebra group, and discuss some applications and extensions of these approaches. On one end of things, I intend briefly to introduce the recent discovery of how certain representation theoretic operations on the supercharacters of U_n(F_q) naturally define a Hopf algebra structure, which has been studied under a different name by combinatorialists. In another direction, I will explain how one can view the supercharacters of an algebra group as the first step in a more general reduction process, which can be used to shed light on some mysterious properties of U_n(F_q).

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.

Thur, Oct 11: Irene Gamba (University of Texas-Austin)

Analytical and numerical issues associated with the dynamics of the non-linear Boltzmann equation

The non-linear Boltzmann equation models the evolution of a statistical flow associated with particle systems in a rarefied or mesoscopic regimes. Its analytical local as well as long time behavior depends strongly on the growth conditions of the collisional kernels, as functions of the intermolecular potentials and scattering mechanisms. We will present recent analytical results, such as classical convolutional inequalities and sharp moments estimates, which imply propagation of $W^{k,p}$ norms and exponential decay of high energy tails, and their consequences on the existence, regularity, and stability of solutions for initial value problems as well as control of decay rates to equilibrium. In particular we present a numerical approximation to the non-linear Boltzmann problem by a conservative spectral scheme, and show spectral accuracy as well as error estimates.

Fri, Oct 26: Luke Oeding (UC Berkeley)

The Trifocal Variety

Abstract: In Computer Vision one considers many cameras looking at the same scene. From this setup many interesting geometric and algebraic questions arise. In this talk we will focus on the case of 3 cameras and study the so called trifocal tensors. Trifocal tensors are constructed from a bilinear map defined using the trifocal setup. A natural question is, given a particular tensor, how can one determine if it is a trifocal tensor? This question can be answered by finding implicit defining equations for the trifocal variety. From an algebraic standpoint, it is also interesting to know the minimal generators of the defining ideal of the trifocal variety.

In this talk I will explain our use of symbolic and numerical computations aided by Representation Theory and Numerical Algebraic Geometry to find the minimal generators of the ideal of the trifocal variety. This is joint work with Chris Aholt (Washington). Our work builds on the work of others (such as Hartley-Zisserman, Alzati-Tortora and Papadopoulo-Faugeras) who have already considered this problem set-theoretically.

My goal is to make most of the talk accessible to anyone with a modest background in Linear Algebra.