Prismatic $F$-gauges

Abstract: Prismatic $F$-gauges are the natural coefficient systems for prismatic cohomology, analogous to variations
of Hodge structures in classical Hodge theory. In this talk, I will explain what they are, and why understanding them for
$\mathrm{Spf}(\mathbf{Z}_p)$ itself can be useful in both geometry and arithmetic. 

In geometry, we shall describe Drinfeld's recent proof of (a refinement of) the Deligne--Illusie theorem for algebraic de Rham cohomology in characteristic $p$. 

In arithmetic, we shall explain how $F$-gauges yield a meaningful notion of crystallinity for integral/torsion representations of the absolute Galois group of $\mathbf{Q}_p$; the cohomology of $F$-gauges then gives an analog of the local Bloch--Kato Selmer groups for such representations with favorable properties (e.g., it interacts quite well with (a refinement of) the local Tate duality theorem).

This is joint work in progress with Jacob Lurie, building on work of Drinfeld.