March 26, 2021
Non-uniqueness for the anisotropic Calderon problem.
The anisotropic Calderon problem consists in recovering, up to some natural gauge equivalences, the metric of a compact Riemannian manifold with boundary from the knowledge of the Dirichlet-to-Neumann map for the Laplacian, at fixed energy. The Calderon problem has been the object of a significant amount of research activity in geometric analysis since it was first formulated by Calderon in 1980, and is still open in its most general form. After giving a motivated introduction to the problem, we shall review its current status and present some recently obtained counter-examples to uniqueness. The latter results, obtained in collaboration with Thierry Daude (Cergy-Pontoise) and Francois Nicoleau (Nantes), involve an unexpected mixture of conformal geometry and classical analysis.