An introduction to condensed mathematics

Speaker: Chris Brav (SIMIS)

Time: 2025-12-03 13:00-18:00
Location: 1310, SIMIS
Zoom Meeting ID: 871 6008 9240 Passcode: SIMIS

摘要：

In various problems in representation theory and geometry, it is natural and necessary to consider topological rings together with topological modules over them. Technically, however, this is very awkward, as topological abelian groups do not form an abelian category. For example, the tautological map from the real numbers with the discrete topology to the real numbers with the Euclidean topology has trivial kernel and cokernel but is not an isomorphism. It is therefore not possible to do traditional homological algebra in the category of topological abelian groups. As an alternative, we introduce Clausen and Scholze’s “condensed abelian groups”, an abelian tensor category with excellent formal properties that faithfully enlarges the category of nice topological abelian groups to contain well-behaved cokernels. If time permits, we then explain how both non-Archimedean and real functional analysis are done in this category. In a future talk, we shall use the category of condensed abelian groups to build a very general analytic geometry, following Clausen and Scholze.