Injectivity and Stability of the Geodesic X-Ray Transform

Loosely speaking, the geodesic X-ray transform takes a function and integrates it along geodesics of a Riemannian manifold with boundary. The Euclidean version (where geodesics are just lines) is the mathematical basis for CT scans. After touching on this application to medical imaging, we will focus on two questions:

1. If a function integrates to zero along all geodesics, is it zero?
2. If a function (roughly) integrates to zero along all geodesics, is it (roughly) zero?

We will answer these questions on "simple" manifolds, discuss the impact of conjugate points, and conclude with some open problems.