The Geodesic X-Ray Transform in a Nutshell

The X-ray transform is a linear map that takes a function and integrates it over geodesics ("shortest paths") on a manifold. This talk will address two questions: 

1.  Is the geodesic X-ray transform injective?

2.  If so, can we explicitly write down its inverse?

We will answer these questions in the Euclidean case (where geodesics are just straight lines) and more generally for simple manifolds. We will also briefly touch on some of the applications of this inverse problem to CT scans, a medical imaging technique. This talk is based on Geometric inverse problems, with emphasis on two dimensions by Paternain, Salo, and Uhlmann.