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Explore advanced topics in p-adic arithmetic geometry through this comprehensive workshop featuring sixteen specialized lectures from leading mathematicians at the Institute for Advanced Study. Delve into cutting-edge research areas including algebraic Frobenius structures, p-adic Simpson correspondence for torsors, crystallinity of rigid flat connections, and log prismatic cohomology. Examine the intricate connections between representation theory and p-adic groups through discussions of Hecke algebras, generic character sheaves for parahoric subgroups, and the mod p Langlands program for GL_2. Investigate geometric aspects through lectures on p-adic hyperbolicity of Shimura varieties, stacks in p-adic Hodge theory of rigid spaces, and quasicoherent sheaves on the relative Fargues-Fontaine curve. Study cohomological methods including essential dimension via prismatic cohomology, prismatic F-gauges and Fontaine-Laffaille modules, and failures of vanishing theorems. Discover applications to algebraic topology through explorations of K(n)-local sphere homotopy groups and exotic spheres derived from p-adic cohomology theories, while also examining the reduction modulo p of crystalline Breuil-Kisin modules.
