Differential aspects of p-adic moduli spaces

We will explain some differential computations on the moduli spaces arising in Fargues–Scholze's geometrization of the local Langlands correspondence and related moduli spaces in p-adic and real Hodge theory, discuss some applications of these computations to the geometry of period maps, and then pose some questions about the relation between these computations, geometric representation theory, and the p-adic and real local Langlands correspondences.