Number Theory Seminar

Let X be a locally topologically noetherian scheme. In their paper on the proétale topology, Bhatt and Scholze defined the proétale fundamental group π1proét(X). The profinite completion of π1proét(X) recovers the usual étale fundamental group. Moreover, π1proét(X) agrees with π1ét(X) when X is normal, but π1proét(X) has the better property that it classifies Qp-local systems. In this talk, we'll explain how to use condensed mathematics to define a "condensed homotopy type" whose fundamental group refines the proétale fundamental group. In particular, this condensed fundamental group classifies local systems with values in any condensed ring. We'll also explain a number of computations of and foundational results about the condensed fundamental group. The ideas from this project originate in our work with Barwick and Glasman, and the paper we'll discuss is joint work with Holzschuh, Lara, Mair, Martini, and Wolf.