MATH 7510 - Berstein Seminar: Sheaves on Manifolds

Allen Knutson, fall 2016.

We'll be following Kashiwara and Schapira's book "Sheaves on Manifolds", available electronically through the Cornell library. One of the main theorems to get to is the Riemann-Hilbert correspondence. Our goal is to head toward applications to geometric representation theory.

The first of the objects being corresponded is D-modules on a manifold M, which is an algebraist's way of studying PDEs. Since the algebra of differential operators is noncommutative, this is a bedrock tool of geometric representation theory; for example, the Beilinson-Bernstein theorem (stated loosely) identifies the category of representations of a Lie algebra with the category of D-modules on the corresponding flag manifold.

The other objects are perverse sheaves on M, the most basic being the sheaf that takes an open set to its intersection homology. So we'll have to build up that technology as well.

Perverse sheaves are examples of (complexes of) constructible sheaves, with a pale shadow being the theory of constructible functions. D-modules, perverse sheaves, and constructible functions all have "singular support" living on a Lagrangian inside the cotangent bundle T^* M. As much as possible, we'll work with these shadows, a simpler version of the theory that seems to have gone neglected since the invention of perverse sheaves.

Prerequisites

Differential topology and, ideally, Lie groups.