Dustin Clausen and Peter Scholze are giving a course together this fall on Analytic Stacks, with Clausen lecturing at the IHES, Scholze from Bonn. Here’s the syllabus:
The purpose of this course is to propose new foundations for analytic geometry. The topics covered are as follows:
1. Light condensed abelian groups.
2. Analytic rings.
3. Analytic stacks.
Yesterday Clausen gave the first lecture (video here), explaining that the goal was to provide new foundations, encompassing several distinct possibilities currently in use (complex analytic spaces, locally analytic manifolds, rigid analytic geometry/adic spaces, Berkovich spaces). These new foundations in particular should work equally well for archimedean and non-archimedean geometry and hopefully will be the right language for bringing together the Fargues-Scholze geometrization of local Langlands at non-archimedean places with a new geometrization at the archimedean place. He describes as “(very) speculative” the possibility of a geometrization of global Langlands (with Scholze more optimistic about this than he is).
Tomorrow Scholze will take over, giving the next six lectures. Perhaps this characterization is a bit over-the-top, but seeing lectures of this sort and of this ambition taking place at the IHES brings to mind the glory days of Grothendieck’s years lecturing at the IHES on new foundations for algebraic geometry. I fear that keeping up on the details of this as it happens will require the energy of someone much younger than I am…