On a Riemannian manifold with boundary, the X-ray transform integrates a function or a tensor field along all geodesics through the manifold. The reconstruction of the integrand of interest from its X-ray transform is the basis of important inverse problems in seismology and medical imaging.

For this transform, statements of injectivity and microlocal stability in the interior of the domain were established some decades ago. However, boundary issues have only received increased attention over the past few years, in an attempt to find suitable invertibility spaces for the X-ray transform and its normal operator(s) (composition of the X-ray transform and its adjoint), as these issues became crucial in the context of Bayesian inversions from noisy X-ray data.

Notably, non-standard Sobolev scales (i.e., transmission spaces a la Hormander, others modeled after degenerate elliptic operators) and interesting boundary phenomena enter the picture.

This talk will attempt to survey recent progress on these topics. This is based on recent joint works with Rafe Mazzeo, Rohit Mishra, Joey Zou and Nikolas Eptaminitakis.