András Vasy

Stanford University

The geodesic X-ray transform of a function on a manifold, possibly with boundary, such as a domain in an ambient Riemannian space, is the map associating to each geodesic within the manifold the integral of the function along the geodesic, defined under appropriate conditions, such as decay at infinity if the domain is non-compact. A basic question is the injectivity of this transform, together with estimates for the left inverse, and possible algorithms for constructing such a left inverse. There is also a tensorial version of this question; in this one takes a symmetric tensor and evaluates it on the tangent vector in each of its slots prior to integration, and one has to take into account the “obvious” nullspace: tensors which are the symmetric gradients of one lower rank tensors vanishing at the boundary/at infinity, called potential tensors. On the other hand, these problems are closely related to boundary rigidity: can one recover a Riemannian metric on a manifold with boundary from its distance function restricted to the boundary (in both slots)? The analogue of potential tensors here is the diffeomorphism invariance of the problem.

Dimensions three and higher have a different flavor from the two dimensional case; I will concentrate on three or higher dimensional manifolds. Much progress has been made on these problems in recent years, typically under some sort of convexity assumptions, such as the convexity of the boundary (for manifolds with boundary), and the existence of a potentially singular convex foliation. Analytically the heart of the approach is a geometric form of microlocal analysis, with the precise type adapted to the geometric setting.

In this minicourse I will explain the setting and the recent progress, in particular joint works with Gunther Uhlmann, Plamen Stefanov, Evangelie Zachos and Qiuye Jia. I will also explain the background analytic machinery, such as semiclassical foliation and scattering pseudodifferential operators, and how these are employed in these settings. I will in particular emphasize the role played by a smallness parameter, such as the semiclassical parameter in the first case and a domain thinness parameter in the second case, to deal with the a priori finite but potentially large dimensional nullspace that is typically given by microlocal tools.