There is a question in geophysics of determining the inner
structure of the Earth from the measurement of travel times of seismic waves at the surface. The mathematical formulation of the question consists of recovering a function or more generally a Riemannian metric from the distance or lens data, which is known as the boundary or lens rigidity problem. The linearization of the problem is concerned with an integral geometry question regarding the inversion of the geodesic X-ray transforms of scalar functions or tensor fields, and has important applications in medical imaging techniques. In this
talk, I will present recent progress on inverting X-ray transforms with a possible weight through a local-to-global approach. I will also discuss their direct applications to the rigidity problems, as well as to other non-linear inverse problems arising in tomography and mathematical physics. Part of the talk is based on joint work with Gabriel Paternain, Mikko Salo and Gunther Uhlmann.