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\textsc{N16}
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\textsc{Colloquium}
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\textit{Combinatorics as Geometry}
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Dr. Fernando Rodriguez Villegas\par
University of Texas (Austin)
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\textbf{Abstract:} 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\vspace{\fill}
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Thursday, October 30, 2008, 4:00 pm\par
Mathematics and Science Center: W301
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\textsc{Mathematics and Computer Science}\par
\textsc{Emory University}
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