p-adic twistors and applications

I will explain a construction of p-adic variations of twistor structure and how it relates to other recent developments in relative p-adic Hodge theory including: geometric Sen theory, the p-adic Simpson correspondence, and analytic prismatization. After introducing the basic theory and examples, I will discuss applications to the study of transversality and unlikely intersections in local Shimura varieties and related moduli spaces.