Organized by:

Prof. Fernando Rodriguez Villegas (Abdus Salam International Centre for Theoretical Physics)

Combinatorics as geometry

May 10, 2016 - 4:00pm

Combinatorics as geometry

May 10, 2016 - 4:00pm

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 quantity 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

This is joint work with T. Hausel and E. Letellier

Short bio

Prof. Villegas got his Ph.D. under the direction of W. Sinnot and K. Rubin in 1990 from Ohio State University. Since then, he worked in many different topics in number theory, expecially on arithmetic aspects of L-functions of number fields and modular forms and, more recently, on quivers. He is currently Head of the Mathematics Section at ICTP; previously he was full professor in University of Texas at Austin. He is author of about 50 publications, he publishes regularly in the most influential mathematical journals, and had about 20 Ph.D. students.

More info available at http://users.ictp.it/~villegas/ and http://users.ictp.it/~villegas/frv-cv.pdf

More info available at http://users.ictp.it/~villegas/ and http://users.ictp.it/~villegas/frv-cv.pdf