Pre-talk for graduate students: 2pm
Moduli of G-local systems

Abstract: In this talk, I introduce the category of G-local systems on a complex variety X, for a reductive group G; and study the moduli space of G-local systems on X. I will show that cohomological rigidity of a G-local system \rho is equivalent to the vanishing of the tangent space of the corresponding point on the moduli space.

Main talk: 3-3:50pm

Simpson conjectured that for a reductive group G, rigid G-local systems on a smooth projective complex variety are integral. I will discuss a proof of integrality for cohomologically rigid G-local systems. This generalizes and is inspired by work of Esnault and Groechenig for GL_n. Surprisingly, the main tools used in the proof (for general G and GL_n) are the work of L. Lafforgue on the Langlands program for curves over function fields, and work of Drinfeld on companions of \ell-adic sheaves. The major differences between general G and GL_n are first to make sense of companions for G-local systems, and second to show that the monodromy group of a rigid G-local system is semisimple. All work is joint with Stefan Patrikis. Speaker(s): Christian Klevdal (University of Utah)