Time: 

May 8,10,11, 1:30-3:00pm

Abstract:

In analogy to Simpson's complex non-abelian Hodge correspondence, the conjectural p-adic Simpson correspondence aims to relate p-adic representations of the étale fundamental group of a smooth proper rigid space X over C_p to Higgs bundles on X. As I will discuss in the first talk, this "non-abelian Hodge theory" can be rephrased as the study of v-vector bundles on X.

The main theme of this series of talks will be to explain how one can use p-adic analytic moduli spaces to reorganise and extend various aspects of the theory. For this I will offer several different perspectives: First, I will explain the situation in the simpler case of rank one, which turns out to be a good prototype for the general case.

I will then explain how one can construct moduli spaces for Higgs bundles and v-vector bundles in general: This is the "local" aspect of the approach. We then discuss various ways in which moduli spaces can be used to globalise the theory: The first is a new approach to the global correspondence via the Hodge--Tate stack, joint with Johannes Anschütz and Arthur-César Le Bras. This also leads to generalisations of the non-abelian Hodge correspondence in various directions, including to varieties over local fields, and from vector bundles to perfect complexes. Finally, I will discuss how studying the geometry of moduli spaces can in turn help construct p-adic Simpson functors.