p-adic analytic spaces have, from the perspective of étale cohomology, a more complicated local structure than complex analytic spaces. Things improve after one extracts a lot of pth power roots. This led to the theory of perfectoid spaces and diamonds, a very powerful language to analyze « topological » properties of rigid analytic spaces. However, recent developements in (non-abelian) p-adic Hodge theory and in the geometrization of the Langlands program, which I will briefly review, call for a formalism also able to retain some « differential » information about these spaces. I will explain a possible framework for doing this and highlight some applications. Based on joint work with Anschütz, Bosco, Rodriguez Camargo and Scholze.
Bao Le Hung
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