Differential Topology for Diamonds

Explore the construction of explicit Banach-Colmez tangent spaces for diamonds in smooth rigid analytic varieties and their cohomology in this advanced mathematics lecture. Delve into the principles behind these tangent spaces and their potential role in a theoretical analytic structure for diamonds. Examine conjectures in differential topology that describe underlying properties of diamonds using this differential data, focusing on preperfectoid loci and cohomological smoothness of morphisms. Engage with explicit examples and computations of tangent spaces and derivatives in this context, comparing the proposed conjectures with known results and related work. Gain insights into geometric Sen theory and the p-adic Simpson and Riemann-Hilbert correspondence, with the option to focus on or disregard these technical aspects as desired.