Title: Zeta Values and Integral Representations in Characteristic p
Abstract: Many of the deepest and most far reaching results in number theory come from blending together algebraic and analytic techniques, for example, expressing a meaningful arithmetic quantity as an integral. This is called an integral representation. But what should you do when the analytic side is missing? In my talk, I will sketch a couple areas in classical number theory where this analytic theory is well developed, such as values of the Riemann zeta function and multiple zeta values. Then, I will describe a setting - characteristic p valued function fields and Drinfeld modules - where this integration theory is missing. Finally, I will describe my contributions towards developing an algebraic replacement for this missing integration theory in characteristic p and discuss what these results tell us.