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.

The classification of totally geodesic (i.e., rigid) submanifolds is a fundamental question in Riemannian geometry. Following work of Eskin, McMullen, Mukamel, and Wright, these submanifolds have played a fundamental role in the discovery of new phenomena in Teichmüller dynamics. I will discuss recent progress towards a complete classification of these submanifolds. This is joint work with Alex Wright. No previous familiarity with the subject will be assumed.

Moduli spaces of Higgs bundles were introduced by Hitchin in 1987 and are now ubiquitous in geometry. They are closely related to other fundamental spaces such as character varieties, and they carry an incredibly rich structure, including a hyperkähler metric (a Riemannian metric compatible with three complex structures). In this talk I will give a gentle introduction to all of these objects, and describe some recent progress towards describing the metric near the singular locus of the Hitchin fibration.