I give an introductory talk about geometric inverse problems and ray transforms (no proofs are involved). I mainly focus on the Euclidean X-ray transform of scalar fields and vector fields, but also introduce the basic properties of the X-ray transform of higher order tensor fields. I give some unique continuation results for the normal operator of the X-ray transform which allow us to prove uniqueness results for partial data problems. I generalize the X-ray transform to Riemannian manifolds and use the boundary rigidity problem/travel time tomography problem as a motivation. Finally, if time is left, I introduce the definition of mixing ray transforms which are generalizations of the geodesic X-ray transform on Riemannian manifolds.

Host: Teemu Saksala tssaksal@ncsu.edu