Invited Talk 4: An introduction to the Calderon and Steklov inverse problems on Riemannian manifolds with boundary

Given a compact Riemannian manifold with boundary, the Dirichlet-to-Neumann operator is a non-local map which assigns to data prescribed on the boundary of the manifold the normal derivative of the unique solution of the Laplace-Beltrami equation determined by the given boundary data. Physically, it can be thought of for example as a voltage to current map in an anisotropic medium in which the conductivity is modeled geometrically through a Riemannian metric. The Calderon problem is the inverse problem of recovering the Riemannian metric from the Dirichlet-to-Neumann operator, while the Steklov inverse problem is to recover the metric from the knowledge of the spectrum of the Dirichlet-to-Neumann operator. These inverse problems are both severely ill-posed . We will give an overview of some of the main results known about these questions, and time permitting, we will discuss the question of stability for the inverse Steklov problem.