3:00pm Nathanael Berestycki (Cambridge)
Random planar geometry, KPZ relation and Brownian motion.
Abstract: I will survey some recent progress on how to define natural notions of conformally invariant random metric in two dimensions. This is motivated by work of physicists in the 70s in the context of Liouville quantum gravity, and in particular by the so-called KPZ relation, which describes a way to relate geometric quantities associated with Euclidean models of statistical physics to their formulation in random geometry. I plan to discuss in an informal manner what is the problem and survey what is known rigorously. While the existence of a conformally invariant random metric is still open, I will explain a recent result of mine showing it is possible to construct a natural notion of Brownian motion in this geometry.
4:30pm Corinna Ulcigrai (Bristol)
On the spectrum of time-changes of horocycle flows.
Abstract: The horocycle flow on a compact hyperbolic surface is a classical dynamical flow whose ergodic and spectral properties have been studied in great detail. If one considers the simplest possible perturbations -time changes- though, it turns out that very little is known for generic smooth time-changes of the horocycle flow. After reviewing some properties of the classical horocycle flow, in this talk we will focus on the nature of the spectrum and the mixing properties of its time-changes. In particular, in joint work with G. Forni (Maryland) we proved a conjecture by Katok and Thouvenot, by showing that the spectrum of all smooth time-changes of horocycle flows is absolutely continuous (and in fact, Lebesgue).