Abstract: In this talk we introduce some geometric rigidity problems that ask if you can determine a compact Riemannian manifold with boundary from measurements taken from the outside. The problems includes the boundary rigidity problem: Can you determine the metric inside if you know the distances (measured through the inside) between all pairs of boundary points? The lens rigidity problem: Can you determine the metric if you know how each entering geodesic exits and how long it takes to exit? We will present some counterexamples and some theorems as well as mention some relations to other problems and potential applications. Speaker(s): Christopher Croke (University of Pennsylvania)