Mathematics Colloquium Winter 2011

January 11, 2011

The bounded denominator conjecture for vector-valued modular forms

Chris Marks
University of Edmonton, Canada

A well-known consequence of the theory of Hecke operators is that modular forms for congruence subgroups of the modular group $PSL_2(Z)$ have "bounded denominators", e.g. if $f$ is a modular form for a congruence subgroup whose Fourier coefficients are rational numbers, then there is some large integer $M$ such that the Fourier coefficients of $Mf$ are all rational integers. Atkin and Swinnerton-Dyer were the first to observe (in the early 1970s) that this property does not necessarily hold for noncongruence modular forms, and it has since been conjectured that having bounded denominators is in fact equivalent to being a congruence modular form. Since the original paper of ASD much work has been done in this area, but few techniques have emerged which lend themselves to a broad, systematic study of the conjecture. In the talk I will explain how the theory of vector-valued modular forms and modular differential equations provides a nice technology for studying noncongruence modular forms, and I will present some recent results I have obtained, where the noncongruence subgroups arise as kernels of irreducible three-dimensional representations of $SL_2(Z)$.

January 18, 2011

No Colloquium - After Martin Luther King's Birthday

January 25, 2011

Derived categories of Calabi-Yau manifolds and Mirror-Symmetry

Cristina Martínez
Universidad Autónoma de Madrid, Spain

The derived category of a variety encodes a lot of information about the variety. According to Kontsevich, there should be an equivalence of categories behind mirror duality, one category being the derived category of coherent sheaves on a Calabi-Yau manifold X and the other one being the Fukaya category of the mirror manifold X'. We will study here the derived category of a particular case of Calabi-Yau manifolds and its consequences from the point of view of Mirror-Symmetry.

February 1, 2011

Resonance and the periodic points of area-preserving diffeomorphisms of the sphere

Ely Kerman
University of Illinois, Champaign

A remarkable theorem by John Franks asserts that every area-preserving diffeomorphism of the sphere has either two or infinitely many periodic points. In this talk I will describe a Floer theoretic proof of this result. More precisely, Franks' theorem is recovered under some additional nondegeneracy assumptions, and it is strengthened through the detection of new restrictions on the periodic points which can occur in area-preserving diffeomorphisms with only two such points. Every previously known proof of Franks' theorem uses results from dynamical systems, such as Brouwer's translation theorem, which are unique to dimension two. The Floer theoretic tools introduced here are not unique to dimension two and we are hopeful that they can be used to address some of the conjectured generalizations of Franks' theorem for Hamiltonian diffeomorphisms on higher dimension symplectic manifolds. This talk is based on joint works with B. Collier, B. Reiniger, B. Turmunkh, and A. Zimmer, as well as with V.L. Ginzburg.

February 8, 2011

On Blocks of Finite Groups Whose Defect Groups are Small

Shigeo Koshitani
Chiba, Japan

One of the most interesting and important problems in representation theory of finite groups is to know which kind of structure a block of a finite group has when its defect group is given. The problem essentially goes back to Richard Brauer (1901-77) who was almost a unique pioneer in "modular" representation theory of finite groups. We will be discussing the problem.

February 15, 2011

Regularity of Area Minimizing Surfaces in Plateau's Problem

Anthony Tromba
University of California, Santa Cruz

We discuss a new approach to the classical question of whether or not area minimizing surfaces are immersed, including a resolution to the problem, unsolved in the last century, of whether they are immersed up to and including smooth boundaries. The history of this elementary question will be discussed as well.

February 22, 2011

Fusion Systems and Simple Groups

Michael Aschbacher

Fusion systems were introduced by Luis Puig as a tool in modular representation theory. Later homotopy theorists became interested in the work; in particular Broto, Levi, and Oliver defined the notion of a p-local finite group (consisting of a saturated fusion system and a linking system), and Levi and Oliver proved the existence of the exotic Solomon-Benson 2-local finite groups, which in some ways are analogous to sporadic finite simple groups. Most of the talk will consist of an introduction to fusion systems, with technicalities suppressed whenever possible. Near the end of the talk, I'll speculate about the possibility of simplifying the proof of the classification of the finite simple groups using fusion systems.

March 1, 2011

Generalized Modular Functions

Winfried Kohnen

Generalized modular functions (GMF) are holomorphic functions on the complex upper half-plane, meromorphic at the cusps, that satisfy the usual transformation law of a classical modular function of weight zero, however with the important exception that the character need not necessarily be unitary. The theory has been partly motivated from Conformal Field Theory in Physics. The aim of this talk is to report on recent joint work with G. Mason on arithmetic properties of the Fourier coefficients of GMF.

March 8, 2011

Regularity estimates and long-time existence results for nonlinear parabolic systems in two space dimensions

Maria Specovius-Neugebauer
University of Kassel, Germany

The subject of this talk is nonlinear parabolic systems in two space dimensions with coefficient functions that possess a potential and linear growth in “grad u,” a new hole-filling technique leads to Morrey estimates for “grad u” which imply “C^{alpha}-regularity in the case of two space dimensions. Thereby a coerciveness and natural entropy condition, but no monotonicity condition is assumed. If we allow also dependence on “u” of the coefficient functions and a lower order Coefficient “a_0 (t,x,u, grad u)$ in the system, existence of a regular long time solution can be obtained by verifying a regularity criterion of Arkhipova even without smallness assumptions on the data. In this case, among other technical assumptions coerciveness and monotonicity for the elliptic part are needed.