The Local Inverse Problem for the Geodesic X-ray Transform and Boundary Rigidity

Abstract

In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, the speaker will discuss the geodesic X-ray transform on a Riemannian manifold with boundary. The geodesic X-ray transform associates to a function its integral along geodesic curves, so for instance in domains in Euclidean space along straight lines. He will explain how, under a convexity assumption on the boundary, one can invert the local geodesic X-ray transform on functions, i.e. determine the function from its 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). The method relies on microlocal analysis, in a form that was introduced by Melrose. The speaker 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. Finally, he will discuss the relationship with, and implications for, the boundary rigidity problem, i.e. determining a Riemannian metric from the restriction of its distance function to the boundary.

About the speaker

Prof. András Vasy received his PhD in Mathematics from the Massachusetts Institute of Technology (MIT) in 1997. He had been faculty members of Mathematics at the University of California at Berkeley and MIT later. He is currently Professor of Mathematics at Stanford University.

Prof. Vasy’s research interests include microlocal Analysis, partial differential equations, many-body scattering, symmetric spaces and analysis on manifolds.