Joint IAS/PU Arithmetic Geometry

A natural problem in the study of local systems on complex varieties is to characterize those that arise in a family of varieties. We refer to such local systems as motivic. Simpson conjectured that for a reductive group G, rigid G-local systems with suitable conditions at infinity are motivic. This was proven for curves when G = GL_n by Katz who classified such rigid local systems. In this talk, we prove Simpson's conjecture for curves for an arbitrary reductive group G. Our proof goes through the (tamely ramified) geometric Langlands program in characteristic zero. If time permits, we state a generalization of Simpson's conjecture to an arbitrary smooth projective variety.