The PI will continue his investigations into the relationship between asymptotic conjectures in number theory and stable cohomology of moduli spaces. The moduli spaces that arise from consideration of analytic number theory over function fields (Hurwitz spaces, moduli spaces of holomorphic rational curves on varieties) are spaces that have already attracted a great deal of attention from topologists; it turns out that topological theorems and conjectures about the rational cohomology of these spaces translates, via the Lefschetz trace formula, into very clean asymptotic formulas for things like "the average size of p-torsion in the class group of a quadratic imaginary field." Some of these formulas agree with function-field analogues of established conjectures in number theory, and thus prove those analogues; others suggest new conjectures. Beyond this main theme, the PI will study algebro-geometric methods for Kakeya problems, applications of expander graphs in arithmetic geometry, and the arithmetic of nilpotent quotients of fundamental groups.
The PI is carrying out research at the interface between two fields that are quite different on the surface. The first field is the classical subject of analytic number theory, whose central questions involve "counting." For example: what is the chance that a random number is not a multiple of any perfect square greater than 1? The second field, much newer, is that of topology, which asks questions about abstract shapes, like curvy high-dimensional surfaces. For instance, one can ask about the space of all sets of n different points in the plane. It turns out, thanks to fundamental insights developed by Grothendieck in the 1960s, that topological questions about high-dimensional spaces can give us very deep insights into counting questions about whole numbers! The PI and his collaborators are proving new topological theorems which prove some old conjectures in number theory, and suggest new conjectures; this work will help build new bridges between the two subjects. The PI will also continue outreach work as a mathematical expositor, maintaining a popular math blog and writing articles about mathematics for national publications.