Some recent advances on the $p$-adic Hodge theory of integral models of Shimura varieties

The geometry of Shimura varieties has played a central role in the development of, and progress in, the Langlands program. Complicating the study of these objects is the lack of moduli interpretations of general Shimura varieties which ought to involve the theory of motives -- a theory that remains largely conjectural. That said, recent advances in the $p$-adic Hodge theory of (formal) schemes over $\mathbb{Z}_p$ provide reasonable approximations to the theory of motives over such objects, and thus open up new avenues to the study of integral models of Shimura varieties. In this talk I will survey some of these advances in the case of Shimura varieties of abelian type, focusing on the resolution of the Pappas--Rapoport conjecture and a more refined version of this using Bhatt--Lurie's theory of prismatic F-gauges in the case of abelian type and hyperspecial level.