Pure Maths Colloquium: Combinatorics as Geometry
23 November 2009 16:15 in CM 107
We know, thanks to the work of A. Weil, that counting points of varieties over finite fields yields purely topological information about them. For example, an algebraic curve is topologically a certain number g of donuts glued together. The same number g, on the other hand, determines how the number of points it has over a finite field grows as the size of this field increases.
This interaction between complex geometry, the continuous, and finite field geometry, the discrete, has been a very fruitful two-way street that allows the transfer of results from one context to the other.
In this talk I will first describe how we may count the number of points over finite fields for certain character varieties, parameterizing representations of the fundamental group of a Riemann surface into GL_n. The calculation involves an array of techniques from combinatorics to the representation theory of finite groups of Lie type. I will then discuss the geometric implications of this computation and the conjectures it has led to.
This is joint work with T. Hausel and E. Letellier