All talks, unless otherwise mentioned, are on Thursday afternoon at 3:30 pm in Room 304 of
the Boyd Graduate Studies Building.
The colloquium talk is pitched at a general mathematical audience, including graduate students. All are warmly encouraged to attend. Typically, there is a post-colloquium dinner (time and place TBA at the colloquium), for which a substantial subsidy is available for graduate students.
September 17, 2009: Allen Knutson, Cornell
University
Why do matrices commute? Algebraic geometry meets
statistical mechanics
Abstract: The matrix equations M2 = 0 are quadratic, so to derive the linear equation Trace(M)=0 from them requires nonalgebraic operations. Are there corresponding "surprising" equations implied by the matrix equation XY=YX? This question was posed in the '60s, and still nobody knows. Even the (normalized) volume of this space {(X,Y) : XY=YX} is very difficult to compute for large matrices, and until recently was only known to start 1,3,31,1145. I'll talk about a bunch of related spaces of matrices, some of which are provably harder and some easier to understand than the commuting scheme {(X,Y) : XY=YX}, and the volumes of these spaces. Then I'll explain how physicists came up with the same set of numbers from a statistical mechanical model (making them much easier to compute), and why they are indeed the same. Some of this work is joint with Paul Zinn-Justin.
October 22, 2009: Keith Conrad, University of Connecticut
Why is the Riemann hypothesis so important?
Abstract: All mathematicians have heard that the Riemann Hypothesis is a significant
open problem, and perhaps also that it is equivalent to an error bound in the prime number theorem,
but why is it such a big deal? There are plenty of older unsolved problems in number theory which
get far less publicity. The reason it matters so much is that it is connected to so many other questions.
I will discuss the history, scope, and range of consequences of the Riemann Hypothesis,
so mathematicians outside number theory can see why it deserves to be counted as one of the most
important unsolved problems in mathematics.
November 5, 2009
Mark Goresky, Institute for Advanced Studies
Colloquium in Honor of Clinton McCrory's Retirement
Congruence subgroups of braid groups and mapping class groups: group theory, algebraic geometry, topology, arithmetic
Abstract: The Artin braid group B_n on n strands acts by automorphisms on the free group F_n on n letters, and thus on the finite set Hom(F_n,G) for any finite group G. The finite-index subgroup of B_n stabilizing an element of Hom(F_n,G) is called a "congruence subgroup" of B_n; more generally, we have a notion of congruence subgroup for mapping class groups in any genus. I'll survey some of the many mathematical contexts in which this action and its stabilizers have appeared, starting with Severi's original proof of the connectedness of the moduli space of curves. I will concentrate especially on "congruence subgroup properties" for mapping class groups (joint work with D.B. McReynolds, following results of Diaz-Donagi-Harbater, Asada, and Thurston) and the cohomology of congruence subgroups of the braid group (joint work with Akshay Venkatesh and Craig Westerland), with an eye towards applications to analytic number theory over function fields.
December 3, 2009:
Andreea C. Nicoara, University of Pennsylvania