Kaimanovich-Masur showed that for a non-elementary finitely generated random walk on the mapping class group of a surface, almost every sample path in the Teichmüller space converges to the boundary. This gives the hitting measure on the boundary of the Teichmüller space, which is shown to be the unique stationary measure. They conjectured that the stationary measure is singular to all Patterson-Sullivan (or, conformal) measures for the group generated by the random walk. In this talk, I will present an affirmative answer to this conjecture for a certain class of random walks, showing the singularity with all Patterson-Sullivan measures. The proof is based on our generalization of Tukia's rigidity theorem for Kleinian groups to a wide class of group actions, which also generalizes Mostow's rigidity. If time permits, we will also discuss analogous singularity results for any finitely generated Kleinian groups and some discrete subgroups of higher rank Lie groups.

This is joint work with Andrew Zimmer.

We discuss how chaos, i.e., sensitivity to initial conditions, arises in the setting of polygonal billiards. In particular, we give a complete classification of the rational polygons whose billiard flow is weak mixing in almost every direction, proving a longstanding conjecture of Gutkin. This is joint work with Jon Chaika and Giovanni Forni. No previous knowledge on the subject will be assumed.

We consider the free boundary problem for the irrotational compressible Euler equation in a vacuum setting. By using the irrotationality condition in the Eulerian formulation of Ifrim and Tataru, we derive a formulation of the problem in terms of the velocity potential function, which turns out to be an acoustic wave equation that is widely used in solar seismology. This paper is a first step towards understanding what Strichartz estimates are achievable for the aforementioned equation. Our object of study is the corresponding linearized problem in a model case, in which our domain is represented by the upper half-space. For this, we investigate the geodesics corresponding to the resulting acoustic metric, which have multiple periodic reflections next to the boundary. Inspired by their dynamics, we define a class of whispering gallery type modes associated to our problem, and prove Strichartz estimates for them. By using a construction akin to a wave packet, we also prove that one necessarily has a loss of derivatives in the Strichartz estimates for the acoustic wave equation satisfied by the potential function. In particular, this suggests that the low regularity well-posedness result obtained by Ifrim and Tataru might be optimal, at least in a certain frequency regime. To the best of our knowledge, these are the first results of this kind for the irrotational vacuum compressible Euler equations.