## EVENT DETAILS AND ABSTRACT

**Colloquium**
**Title:** Geometric analytic number theory

**Speaker:** Professor Jordan Ellenberg (U of Wisconsin)

**Speaker Info:**

**Brief Description:**

**Special Note**:

**Abstract:**

Many problems in number theory are very productively studied by replacing the number theorist's
favorite field, Q (more generally, a number field) with the field F_q(t) for some finite field q
(more generally, the function field of an algebraic curve over a finite field.) This change allows
one to bring to bear the whole apparatus of arithmetic geometry on the study of problems closely
analogous to the original problems in number theory. These geometric methods often involve
changing field once again, replacing F_q(t) with the field C(t) of rational functions in a complex
variable. Remarkably, problems in analytic number theory, once transported all the way over to
this geometric setting, often reveal themselves as being analogous to topological problems of
independent interest. The Geometric Langlands program is, of course, the most well-known traversal
of this route, but the method reveals much of interest about more "classical" analytic number
theory as well. In the Monday number theory seminar, I will give a more technical talk focusing on
conjectures of Cohen-Lenstra and Bhargava and their topological analogs; in this talk, I will start
with the problem of counting pairs of relatively prime integers, which turns out to be analogous to
an old theorem of Graeme Segal and the computation of the cohomology of Artin's braid group. This
is the first case of the Batyrev-Manin heuristic for the asymptotic behavior of rational points of
bounded height on Fano varieties -- at the end I will make some speculative remarks about what
topology has to say to the Batyrev-Manin conjectures, and vice versa.

**Date:** Wednesday, May 20, 2009

**Time:** 4:10pm

**Where:** Lunt 105

**Contact Person:** Prof. Jeff Xia

**Contact email:** xia@math.northwestern.edu

**Contact Phone:** 847-491-5487

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