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Graduate Commutative Algebra & Algebraic Geometry Seminar: Condensation and the Algebraic Geometry of Topological Algebras

Event Type
241 Altgeld Hall
May 3, 2022   4:00 pm  
Brian Shin
Likun Xie

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.

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