Activities This Week

I will discuss expectations and results around the following question: If $X$ and $Y$ are two smooth projective varieties with equivalent derived categories, when can one conclude that $X$ and $Y$ are birational? The study of Fourier-Mukai equivalences yields many examples of non-birational varieties with equivalent derived categories. On the other hand, it appears that by considering slightly more structure than just the derived categories one can conclude birationality in many cases. This is joint work with Max Lieblich.

Recording available here: https://us02web.zoom.us/rec/share/U7Zp8zsHQrL4WGyhHAx9sSLRNwEPoFAp2AnvK5_lvC4M0_5aByj6YMYM00_zdsiG.Pr97s-6WdDsEx0qM

Imagine you have a group, with a discrete subgroup. Wouldn’t that be nice to relate random walks, and Poisson boundaries of the group and of the subgroup, in a meaningful way? This was done by Furstenberg for lattices in semisimple Lie groups as an essential tool in an important rigidity result. We are concerned with dense subgroups. We develop a technique for doing it that allows us to exhibit some new interesting phenomena in Poisson boundary theory. I’ll explain the setting in which we work, and will focus mainly on our construction (leaving the applications as “further reading”). Joint work with Michael Björklund and Hanna Oppelmayer