https://sites.uci.edu/inverse/

https://sites.uci.edu/inverse/

In this talk I will survey the classical Boundary Control method, originally developed by Belishev and Kurylev, which can be used to reduce an inverse problem for a hyperbolic equation, on a complete Riemannian manifold, to a purely geometric problem involving the so-called travel time data. For each point in the manifold the travel time data contains the distance function from this point to any point in a fixed a priori known closed observation set. If the Riemannian manifold is closed then the observation set is a closure of an open and bounded set, and in the case of a manifold with boundary the observation set is an open subset of the boundary. We will survey many known uniqueness and stability results related to the travel time data.

https://sites.uci.edu/inverse/

We discuss recent works studying sharp mapping properties of weighted X-ray transforms on the Euclidean disk and hyperbolic disk. These include a C^\infty isomorphism result (joint with R. Mishra and F. Monard) for certain weighted normal operators on the Euclidean disk, whose proof involves studying the spectrum of a distinguished Keldysh-type degenerate elliptic differential operator. We then discuss how to transfer these results to the hyperbolic disk (joint with N. Eptaminitakis and F. Monard), by using a projective equivalence between the Euclidean and hyperbolic disks via the Beltrami-Klein model.