Activities This Week

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

This is a report on joint work with V. Srinivas and Simon Pepin Lehalleur. Recently, with Arpon Raksit, we have shown that for a smooth projective variety X over a field k, the quadratic Euler characteristic of X, an element of the Grothendieck-Witt ring of quadratic forms over k, can be computed via the cup product on Hodge cohomology followed by the canonical trace map. Following work of Carlson-Griffiths, this leads to an explicit formula for the quadratic Euler characteristic of a smooth projective hypersurface defined by a homogeneous polynomial F in terms of the Jacobian ring of F, as well as a similar formula for a smooth hypersurface in a weighted projective space. In some special cases, this leads to quadratic versions of classical conductor formulas with some mysterious and unexpected correction terms, even in characteristic zero.