# Travel Time Tomography and Boundary Rigidity

We will consider the inverse problem of determining the speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.

We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary.

*Uhlmann studied mathematics as an undergraduate at the Universidad de Chile in Santiago, gaining his Licenciatura degree in 1973. He continued his studies at MIT where he received a PhD in 1976. He held postdoctoral positions at MIT, Harvard and NYU, including a Courant Instructorship at the Courant Institute in 1977–1978. In 1980, he became Assistant Professor at MIT and then moved in 1985 to the University of Washington. He has been the Walker Family Professor at the University of Washington since 2006. During 2010-2012 he was on leave at the University of California, Irvine, as the Excellence in Teaching Endowed Chair. Uhlmann was Finnish Distinguished Professor 2012-2017. He is currently also the Si-Yuan Professor at the Institute for Advanced Studies of the Hong Kong University of Science and Technology since 2014.*