EVENT DETAILS AND ABSTRACT

Title: Geometric analytic number theory
Speaker: Professor Jordan Ellenberg (U of Wisconsin)
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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.