Colloquia 2012-2013: Difference between revisions
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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. | 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 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). | |||
Revision as of 20:28, 26 August 2012
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) | 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 |
| 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.
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).
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.