Wojciech J. Gajda

Abstract: Grothendieck's étale fundamental group is one of the central objects of study in arithmetic geometry. It has a "higher" refinement: the Artin-Mazur-Friedlander étale homotopy type, which allows one to speak about higher homotopy groups of a scheme. In recent years, the étale topology and its fundamental group were generalized by their "pro-étale" counterparts in the work of Bhatt–Scholze. In this talk, I will discuss a common refinement of the pro-étale fundamental group and the étale homotopy type that uses the "condensed mathematics" of Clausen–Scholze and Barwick–Haine. This is joint work with P. J. Haine, T. Holzschuh, C. Mair, L. Martini, and S. Wolf.