DG-MP-PDE Seminar: Gunther Uhlmann (Washington)

The boundary rigidity problem consists in determining the Riemannian metric of a compact Riemannian manifold with boundary by measuring the lengths of geodesics joining points of the boundary. The lens rigidity problem consists in determining the Riemannian metric of a compact Riemannian manifold with boundary by measuring the scattering relation or lens relation: We know the point of exit and direction of exit of a geodesic if we know its point of entrance and direction of entrance.

These two problems arise in travel time tomography in which one attempts to determine the index of refraction of a medium by measuring the travel times of waves going through the medium.

We will survey what is known about this problem and some recent results.