March 15, 2022

**Title:** *Tame topology and a finiteness theorem for variations of Hodge structures*

**Speaker:** Thomas Grimm

About 20 years ago physicists found a remarkable set of solutions to some of the fundamental equations arising from string theory. It has subsequently been conjectured by Douglas and others that this set of solutions is finite. This conjecture has since shaped the view of string theory as a predictive theory leading to a finite number of possible universes.

In this talk, I will first describe the underlying mathematical statement of this conjecture as a finiteness condition on the set of self-dual integral classes in a variation of Hodge structure. I will then argue that the conjecture can be proved using a recent remarkable result of Bakker, Klingler, and Tsimerman that connects Hodge theory and tame topology. I will give a brief introduction to tame topologies built from o-minimal structures and describe the structure that tames the period mapping capturing the variation of a Hodge structure. The finiteness theorem for self-dual classes is a generalization of a theorem for the locus of Hodge classes by Cattani, Deligne, and Kaplan.

This talk is based on joint work with B. Bakker, C. Schnell, and J. Tsimerman, see 2112.06995 [math.AG].