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Explore advanced techniques in the cohomology of Shimura varieties through this mathematical lecture focusing on Igusa stacks as innovative tools for studying both characteristic 0 and torsion coefficients. Delve into the Igusa stack diagram, a Cartesian construction that expresses Shimura varieties with infinite level at p as fiber products of Igusa stacks with purely local objects over the moduli stack of G-bundles on the Fargues-Fontaine curve. Examine how this framework enables the application of categorical local Langlands program techniques to Shimura variety cohomology. Learn about significant applications including torsion-vanishing results, Eichler-Shimura relations, and Ihara's lemma, drawing from research by Koshikawa, Hamann-Lee, Daniels-van-Hoften-Kim-Zhang, Yang-Zhu, and Yang. Discover ongoing work on the relative intersection cohomology of Igusa stacks, representing collaborative research between the speaker and colleagues including Linus Hamann. Gain insights into constructions developed by Zhang, Daniels-van Hoften-Kim-Zhang, and Kim across various levels of generality, providing a comprehensive foundation for understanding these sophisticated mathematical structures and their applications in modern algebraic geometry and number theory.
