Abstract

Given a compact Riemannian manifold $(M,g)$ with boundary, the geodesic ray transform is the mapping which takes a function on $M$ to its integrals over the maximally extended geodesics of $(M,g)$. We are interested primarily in two questions: whether this transform is injective, and whether there is a stability estimate between appropriate Sobolev spaces for its inversion. It is well known that for so-called “simple manifolds”, which in particular do not have caustics, the transform is injective, and there is a stability estimate. On the other hand, in the two dimensional case it has been proven that as soon as there are caustics no stability estimate between any Sobolev spaces is possible. This is the case even though there are two dimensional examples which have caustics, but for which the transform is injective. The question motivating this talk is whether the same phenomenon happens in three dimensions. The talk will examine recent results on the stability of the inversion of the geodesic ray transform in the presence of caustics in three dimensions, contrasting them with what is known on the injectivity.