Back to Departmental Colloquium: Fall 1999
Departmental Colloquium
Date: Tuesday, Nov 23, 1999
Time: 4:15PM
Location: JWB 335
Mark Kisin
Sydney, Australia)
Title |
A Riemann-Hilbert correpondence for unit F -crystals |
Abstract |
Let X be a complex manifold. It is a fundamental fact that there is a correspondence between linear differential equations on X (i.e. vector bundles with flat connection) and complex representations of the fundamental group of X (i.e. local systems of complex vector spaces on X ). This correspondence is an equivalence of categories respecting direct sums, tensor products etc. One deficiency is that if f: X –> Y is a map of complex manifolds, the direct image of a vector bundle (resp. local system) need not be a vector bundle (resp. local system). To remedy this one generalises both types of objects: Vector bundles are replaced by D -modules, and local systems are replaced by constructible sheaves. Also, one now gets a correspondence not between objects, but between complexes - so a single D -module corresponds to a complex of constructible sheaves. This is called the Riemann-Hilbert correspondence, and it respects tensor product, and direct and inverse image functors. After reviewing the classical complex situation, I will report on joint work with M. Emerton concerning an analogue of this correspondence in characteristic p. These ideas have lead to a proof of an old conjecture of Katz on p -adic L -functions. |
Representation Theory
Topology