Structures in p-adic Hodge Theory and Representation Theory, with applications to Number Theory

Seminar in Mathematics

Abstract: Complex Hodge Theory is a method for studying the cohomology groups of a smooth manifold using partial differential equations. Developed by Hodge in the 1930s, it uses methods from complex analysis, complex geometry, Riemannian geometry, differential geometry, PDE and algebraic geometry. A key object in play here is the notion of a Hodge structure. 

In number theory, similar structures exist. The area of p-adic Hodge Theory can be thought of as "complex Hodge theory adapted to number-theoretic applications". Such techniques feature heavily in the Langlands Program, which threads through different fields such as representation theory (of p-adic reductive groups), number theory, algebraic geometry, and even the (algebraic) theory of differential equations. 

In this talk, I will start with motivations coming from classical number theory questions, and explain how representation theory and p-adic Hodge theory have proven to be powerful tools for attacking such questions. I will discuss my various results in this framework, and give a brief preview of some current and future projects.