On a closed Riemannian manifold, the marked length spectrum rigidity problem consists in recovering the metric from the knowledge of the lengths of its closed geodesics (marked by the free homotopy classes of the manifold). I will present a solution to this problem for Anosov surfaces namely, surfaces with uniformly hyperbolic geodesic flow (in particular, all negatively-curved surfaces are Anosov). This generalizes to the Anosov setting the celebrated rigidity results by Croke and Otal from the 90s.

In this talk we will discuss the non-diffusive Westervelt equation, which describes the time evolution of pressure in a medium relative to an equilibrium position. It is a second order quasilinear hyperbolic equation, involving a space dependent parameter which multiplies the nonlinear term. Given a medium with compactly supported but unknown nonlinearity, we would like to recover the latter by probing the medium from different directions with high frequency waves and measuring the exiting wave. To do so, we construct approximate solutions for the forward problem via nonlinear geometric optics and discuss its well posedness. We then explain how the X-ray transform of the nonlinearity can be recovered from the measurements, which allows for it to be reconstructed. Based on joint work with Plamen Stefanov.