EVENT DETAILS AND ABSTRACT

Title: The local inverse problem for the geodesic X-ray transform
Speaker: Andras Vasy
Speaker Info: Stanford University
Brief Description:
Special Note: NOTE SPECIAL DAY, TIME, AND ROOM
Abstract:

In this talk, based on joint work with Gunther Uhlmann, I consider the geodesic X-ray transform on a Riemannian manifold with boundary. I will explain how, under a convexity assumption on the boundary, one can invert the local geodesic X-ray transform in a stable manner. Here the local transform means that one would like to recover a function in a suitable neighborhood of a point on the boundary of the manifold given its integral along geodesic segments that stay in this neighborhood (i.e. with both endpoints on the boundary of the manifold). Our method relies on the introduction of an artificial boundary at which the `microlocal normal operator' we construct is (essentially) a scattering pseudodifferential operator in the sense of Melrose's scattering calculus. I will then also explain how, under the assumption of the existence of a strictly convex family of hypersurfaces foliating the manifold, this gives immediately the solution of the global inverse problem by a stable `layer stripping' type construction.