## ENSPM2021 Talk

**Title:** Condensed Mathematics

**Abstract:** (joint with Dustin Clausen) It is a well-known problem that topological spaces have no good categorical properties — for example, topological abelian groups do not form an abelian category, and in a complex of topological vector spaces, differentials may not have closed image, leading to pathological cohomology groups. However, we realized that one can replace topological spaces by the closely related notion of condensed sets, resolving all of these foundational problems. This makes it possible to develop new foundations for both nonarchimedean and archimedean functional analysis, even allowing a very general formalism of analytic spaces — encompassing complex manifolds, real manifolds of all flavours, schemes, formal schemes, rigid-analytic varieties, and adic spaces into one unified framework. We will try to give an overview of these ideas.

## Peter Scholze (Bonn University, Germany)

German mathematician known for his work in algebraic geometry has been a professor at the University of Bonn since 2012 and director at the Max Planck Institute for Mathematics since 2018. In 2012, he was awarded the Prix and Cours Peccot. He was awarded the 2013 SASTRA Ramanujan Prize. In 2014, he received the Clay Research Award. In 2015, he was awarded the Frank Nelson Cole Prize in Algebra, and the Ostrowski Prize. He received the Fermat Prize 2015 from the Institut de Mathématiques de Toulouse. In 2016, he was awarded the Leibniz Prize 2016 by the German Research Foundation, and was awarded the Fields Medal in 2018, for “transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to Galois representations, and for the development of new cohomology theories”.