A key foundational input to the success of modern algebraic geometry is the niceness of the category of abelian groups. In particular, it is Grothendieck Abelian with a very nice tensor product. Seeing as many of our favorite algebras come with interesting topologies (e.g. the fields of real and p-adic numbers), one might hope these nice properties persist upon passing to topological abelian groups. Unfortunately, what is typically one's first guess at a definition fails dramatically: the category of topological abelian groups is *terrible*, and there are now many tensor products, each having its own set of pros and cons. These failures make the category of topological abelian groups unsuitable for a foundation of topologo-algebraic geometry. How are we to study algebraic geometry while incorporating the topologies on our rings? In this expository talk, we'll give a brief introduction to Clausen--Scholze's answer to this question: condensed mathematics and analytic geometry.
