Title: The inverse problem for the geodesic X-ray transform on asymptotically conic spaces
Speaker: Andras Vasy
Speaker Info: Stanford
On a Riemannian manifold with boundary, the geodesic X-ray transform associates to a function its integrals along geodesic segments connecting boundary points; there is also a generalization to tensors. On complete Riemannian manifolds one can instead integrate along all non-trapped geodesics, assuming suitable decay of the function. The basic question is if one can recover a function/tensor from these integrals.Date: Monday, November 06, 2023
In this talk I will explain recent results in joint work with Evangelie Zachos and Qiuye Jia on the geodesic X-ray transform on asymptotically conic spaces, asymptotic to the `large’ end of a cone, both on functions and on symmetric 2-tensors. This includes perturbations of Euclidean space and certain kinds of conjugate points are allowed. The key analytic tool, beyond the “artificial boundary” approach introduced by Uhlmann and the speaker in the compact setting, which localizes to a neighborhood of infinity in this case, is the introduction of a new pseudodifferential operator algebra, the 1-cusp algebra, and its semiclassical version. The semiclassical part of the argument in fact by itself sheds a new light on the X-ray transform in the compact case.