Prof. Dr. Dustin Clausen: Algebraic geometry from the solid perspective

Yesterday, Prof. Dr. Dustin Clausen (University of Copenhagen) gave an inspiring talk entitled "Algebraic geometry from the solid perspective" at this year's Friedrich Hirzebruch Colloquium at the Faculty of Mathematics and Computer Science.

Abstract:
Traditionally in scheme theory, one studies a commutative ring R by introducing a topological space Spec(R) such that many questions about R can be studied locally on Spec(R). The most fundamental example of this is that we have "descent for R-modules", meaning that linear algebra over R in some sense localizes on Spec(R). This is quite powerful, but for some purposes, usual R-modules are not sufficient. This led us, in joint work with Peter Scholze, to introduce a larger category of "Solid R-modules".

In this talk I'll explain what is a solid R-module, why we care to consider these more general modules, and finally, what kind of natural space the solid R-modules localize on. It turns out that this space is much more intricate and expressive than the usual Spec(R).