András Vasy - Microlocal and Global Analysis, I...
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András Vasy - Microlocal and Global Analysis, Interactions with Geometry
Boundary rigidity and the local inverse problem for the geodesic X-ray transform on tensors.
In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the boundary
rigidity problem on manifolds with boundary (for instance, a domain in Euclidean space with a perturbed metric), i.e.
determining a Riemannian metric from the restriction of its distance function to the boundary. This corresponds to
travel time tomography, i.e. nding the Riemannian metric from the time it takes for solutions of the corresponding
wave equation to travel between boundary points. A version of this relates to nding the speed of seismic waves
inside the Earth from travel time data, which in turn permits a study of the structure of the inside of the Earth.
This non-linear problem in turn builds on the geodesic X-ray transform on such a Riemannian manifold with
boundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, so