Analytic geometry and p-adic automorphic forms

In the last few years Clausen and Scholze have been developing a new theory of Analytic Geometry that encloses all the classical theories of algebraic, rigid, adic, real and complex geometries in the same world. In this talk I will explain how this theory provides new tools for the study of p-adic automorphic forms, obtaining new geometric interactions between p-adic Hodge theory, p-adic differential equations, and p-adic representation theory.