Zariski Topological Reconstruction Theorems via Model Theory

Abstract: A recent and quite surprising book of Kollár-Lieblich-Olsson-Sawin (KLOS) showed that in many cases, the full structure of an algebraic variety is determined by its underlying Zariski topological space. More precisely, this means there is a certain classification of all homeomorphisms between varieties in the relevant cases. The strongest version of the original KLOS theorem only works for certain varieties over uncountable fields of characteristic zero. Meanwhile, in a recent work with Ronan O’Gorman — working over uncountable algebraically closed fields — we prove the natural generalization to arbitrary varieties in arbitrary characteristic. Interestingly, the key idea in generalizing the proof comes from model theory. In this talk, I will try to explain what exactly the KLOS theorem says and why it is interesting; then I will give a sketch of why the original proof failed to generalize and why model theory ends up being relevant.

algebraic geometrylogicnumber theory

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic