Passcode: 304188




Title: Condensed mathematics.

Abstract: While topology is a fundamental notion that serves to formulate and study various continuity phenomena in mathematics, its classical implementation runs into several technical problems when applied to the context of derived and homological algebra. The theory of condensed mathematics, developed by Clausen and Scholze (and, independently, by Barwick and Haine under the name "pyknotic mathematics"), aims to remedy these problems by replacing the category of topological spaces with a close variant of condensed sets. Unlike the former, the latter is the category of shelves on a certain site (the pro-etale site of a geometric point). In particular, this allows a natural definition of condensed objects in any category (or even higher category) with excellent formal properties. For example, the category of condensed abelian groups is an abelian category, in contrast with the category of topological abelian groups, and hence amenable to methods of derived/homological algebra. Similarly, condensed rings, modules, algebras, sheaves, etc. are substitutes for their classical topological counterparts in which various foundational issues can be resolved in a satisfactory manner. The resulting theory admits many applications to algebraic/analytic geometry, representation theory, algebraic K-theory, and more. In this talk, we will introduce the basic definitions, properties, and examples of this theory.