Mathematics

An introduction to D-modules and crystals

Emily Cliff (UIUC Math)

Abstract: We will introduce the notion of a D-module on a variety X, a generalization of the concept of a vector bundle with flat connection. We know that over a smooth manifold, a vector bundle with flat connection is equivalent to a local system, a family of vector spaces over the manifold related to each other by parallel transport along paths in the manifold. It seems hard to translate this picture back to algebraic geometry, for example because we have no good notion of paths in a variety, but luckily Grothendieck introduced the idea of crystals of sheaves: these are sheaves with parallel transport between infinitesimally close points. We will explain this definition, and sketch the proof of the equivalence between crystals and D-modules. We will consider advantages of each approach, and along the way will see naturally occurring (and not-too-scary) examples of stacks and their categories of sheaves.

Combinatorics, Dynamics, Integrability and Entropy

Nicholas Ercolani (University of Arizona)

Abstract: The purpose of this talk is to advertise a program that brings some new approaches to combinatorial problems in mathematical physics and probability theory. The origins of these approaches lie in older ideas of dynamical systems theory. Examples of the problems are calculating geodesic distance on random surfaces, random walks in random environments, and continuum limits of map enumeration. Sources of the new approaches include Moser's classical approach to billiard dynamics and Arnold's notion of algebraic entropy, generalizing the classical concepts of Kolmogorov-Sinai and topological entropy. This program is an outgrowth of the thesis of recent Arizona PhD Tova Brown.