What can we tell about the interior structure of our planet, if we observe the travel time of a large number of earthquakes? This is the time it takes for a seismic wave to travel from the epicenter of the  earthquake to the seismic sensor. In the geometric literature, the boundary rigidity problem on a compact Riemannian manifold is an  analog of the previous physical setup. This asks the following: If two  Riemannian manifolds have a common boundary and the distances between any pairs of boundary points agree, does this imply that the manifolds are Riemannian isometric?  The problem belongs to the field of inverse problem, in which the goal is to determine an unknown quantity using indirect measurements.

In this talk I will survey some geometric inverse problems and my research related to the boundary rigidity problem. The problems I will introduce share a common goal: To determine topological, smooth and Riemannian structures from geometric boundary data, that are motivated by seismic measurements. In a geophysical literature, seismic waves are modeled with elastic systems in the 3-dimensional Euclidean space. I will explain why a compact Riemannian manifold is a good generalization for an isotropic medium, and why a compact Finsler manifold is a natural coordinate invariant analog of certain anisotropic medium. I will also introduce a linearization process that relates travel time measurements to integral geometry.