RESUMEN: On compact Riemannian manifolds with boundary, the (anisotropic) Calderón’s problem asks to what extent the Dirichlet-to-Neumann map associated with the Laplace—Beltrami equation determines the metric (up to a natural obstruction).
In this talk, I will discuss a similar problem, known as inverse scattering for asymptotically hyperbolic manifolds (i.e., manifolds that, outside a compact region, behave like the hyperbolic space): fixed an “energy level”, and given the analog of the Neumann data for solutions to certain 0-elliptic PDE depending on the fixed energy level, determine the metric of the manifold.
I will show that, under some geometric conditions, one can determine the Taylor series of the metric at the boundary.