Basic Notions: Frank Calegari (Chicago) - The Cohomology of Arithmetic Groups and the Langlands Program

Thu, 25/01/201816:30-17:45
Hall 2, Manchester Building
We discuss recent progress in the Langlands Program (in particular, work of Scholze, Caraiani-Scholze, and Allen-Calegari-Caraiani-Gee-Helm-LeHung-Newton-Scholze-Taylor-Thorne) through the lens of the cohomology of arithmetic groups such as SL_n(Z) and their congruence subgroups (both integrally and over the real numbers). In particular, we try to explain what should be true in an idealized world (in practice, tricks are often employed to avoid technical difficulties) in terms of three phenomena: Hecke operators (whose existence, as shown by Margulis, precisely reflects arithmeticity), The Borel-Serre compactification, and the cohomology of Lie algebras ((g,K)-cohomology).