Character Varieties

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The main focus of the research project concerns the moduli space of representations of the fundamental group of a smooth projective algebraic curve to a reductive algebraic group G. This moduli space is the Betti version of a cohomology group in non-abelian Hodge theory. Its other versions are, Dolbeault: the moduli space of semistable Higgs G-bundles on the surface and, de Rham: the moduli space of flat G-connections on it. In the Dolbeault and de Rham versions this space has been central to recent important work in the Langlands program, both the arithmetic by work of Ngo and Laumon and the geometric by work of Kapustin and Witten. The basic idea of the project is to use tools of Number Theory, Combinatorics and the Representation Theory of finite groups of Lie type to count points on the Betti space over finite fields. The Weil conjectures then yield cohomological and geometrical information about it (and hence also about its other two flavors).In the long run, Mathematics always has a way of making itself useful outside its own discipline. Who would have thought 20 years ago, say, that something as apparently removed from everyday life as the concept of an "elliptic curve" could end up as a crucial tool for the security of online shopping? An elliptic curve is a geometric construct with a very rich structure. Two points on the curve can be "added" by geometric means to produce a third point. This fact goes back more than 300 years. In more modern times we have learned how do to "geometry" (work with points, lines, curves, etc.) in a purely finite context, where the real or complex numbers are replaced by their finite counterparts, finite fields. This provides one of the most useful ways to apply abstract mathematical concepts from Geometry and Number Theory to real life situations. The research project uses finite fields in a different way: to probe the geometry, in the usual sense of the word, of certain spaces of great interest to both Mathematics and Physics. The PI finds the interplay between the discrete (counting points over finite fields) and the continuous (geometry over the complex numbers) in this project, as well as the fact that involves in a substantial way fairly distant areas of Mathematics (Number Theory, Combinatorics, Group Theory and Differential Geometry), fascinating.