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
Special Colloquium in Honor of Clint McCrory's Retirement
Characteristic Classes on Singular Spaces
Abstract: The theory of characteristic classes of manifolds and vector bundles was developed by Stiefel, Chern, Whitney, Wu, Pontrjagin and many others, starting in the 1930’s with obstruction theory, continuing in the 1940’s (as cohomology classes) and the 1950’s (with Chern-Weil theory, classifying spaces and their topology). In the 1970’s it was discovered that many of these characteristic classes made sense for singular spaces, the most well-known examples being Sullivan’s Whitney classes for Euler spaces and the MacPherson-Schwarz Chern classes for singular varieties. There began a program to duplicate as much of the theory as possible, for singular spaces. Even into the 1990’s basic properties of these classes were still being described, and it is unlikely that the story ends there. I will try to give a picture of some of the main threads in the development of this theory, mentioning a few of Clint’s important contributions along the way.
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
The Non-Noetherianity of the Denjoy-Carleman Classes
Abstract: The Denjoy-Carleman classes are subrings of the ring of smooth functions defined by bounds on the growth of their derivatives. Initially studied by analysts, these functions also have fascinating algebraic properties: they fail to satisfy Weierstrass division, yet resolution of singularities holds. The failure of Weierstrass division makes it impossible to prove or disprove Noetherianity by the usual algebraic methods. I will discuss work in progress with Liat Kessler (MIT) using model theory that shows these Denjoy-Carleman rings are not Noetherian. Therefore, these rings constitute examples of non-excellent rings on which resolution of singularities holds. No background is assumed. I will explain the analysis, algebra, and model theory required for the proof.
January 14, 2010:
Julia Wolf, Rutgers University
An Introduction to Quadratic Fourier Analysis
Abstract: How large can a subset of the integers 1, 2, ..., N be
before it is guaranteed to contain a k-term arithmetic progression?
When k=3, a good bound can be given using Fourier analysis. Originally
developed by Gowers to give a strong quantitative answer to the case
k=4, "quadratic" Fourier analysis has found numerous applications to
related problems in number theory, notably in the work of Green and
Tao on long progressions in the primes. It turns out that quadratic
Fourier analysis has deep connections with ergodic theory as well as
the theory of hypergraphs. We intend to illustrate some of these
connections as we examine a very general class of linear patterns.
March 25, 2010:
Gavril Farkas, Humboldt University
Green's Conjecture for equations of canonical curves
Abstract: Mark Green's Conjecture on syzygies of canonical curves,
has been by far the most studied question in the theory of Riemann
surfaces in the last few decades. Formulated in 1984 and still wide
open, it is a deceptively simple statement which predicts that the
intrinsic geometry of the curve can be recovered in a precise way from
the extrinsic geometry of the canonical embedding (in the form of
syzygies). I will review the history of the problem, including Claire
Voisin's proof of Green's Conjecture for general curves, then I will
explain how one can use Voisin's work together with the geometry of
the moduli space of curves, to prove Green's Conjecture for arbitrary
curves lying on K3 surfaces. This is joint work with M. Aprodu.
April 22, 2010:
Maria Skopina, St. Petersburg University