P-Adic Cohomologies Using Homotopy Theory (March 13, 2025)

In this lecture, Alberto Vezzani explores how non-Archimedean motivic homotopy theory can be applied to define and redefine rational p-adic cohomology theories. Drawing parallels to classical work by Monsky–Washnitzer, Elkik, Arabia, and others, Vezzani demonstrates how this approach yields new results in the field. The talk covers the definition of relative rigid cohomology and its finiteness properties (joint work with V. Ertl), as well as the definition of Hyodo–Kato and limit Hodge cohomology using motivic nearby cycles, resulting in an associated "motivic" Clemens–Schmid chain complex (joint work with F. Binda and M. Gallauer). This presentation is part of the 2025 Simons Collaboration on Perfection in Algebra, Geometry and Topology Annual Meeting.