Bay Area Microlocal Analysis Seminar: X-ray transforms and tensor tomography on surfaces

Seminar | May 15 | 4-5 p.m. | Stanford University, Room 384H

 Francois Monard, UC Santa Cruz

 Department of Mathematics

On $(M,g)$ a non-trapping Riemannian surface with boundary, the tensor tomography problem consists of inferring (i) what is reconstructible of a symmetric tensor field from knowledge of its integrals along geodesics through that surface, and (ii) how to reconstruct it. In the Euclidean case and zero-th order tensors (i.e., functions), this is the well-known X-Ray (or Radon) transform and it serves as the theoretical backbone of Computerized Tomography.

In a geometric setting, the answer to questions (i) and (ii) depends on the order of tensors considered, the underlying geometry, and what functional setting for the X-ray transform is chosen. In this talk I will review recent results on these aspects, and will discuss reconstruction approaches for functions and tensor fields, some valid in rather general settings, others requiring more Euclidean explicitness.