The X-ray transform and geometric inverse problems (Gabriel Paternain and Mikko Salo)

Abstract

This course will focus on the geodesic X-ray transform and related geometric inverse problems. The Euclidean X-ray transform, which encodes the integrals of a function over straight lines, is a classical topic (already going back to J. Radon in 1917) and forms the basis of imaging methods such as X-ray computed tomography. The geodesic X-ray transform encodes the integrals of a function over more general families of curves, such as the geodesics of a Riemannian metric. It has applications in seismic imaging and in geometric inverse problems including the boundary rigidity problem, the inverse conductivity problem posed by Calderó n, and inverse spectral problems.

There have been several advances in the study of geodesic X-ray transforms and geometric inverse problems recently, and this course will discuss selected results together with the necessary background. We will begin by covering the PDE approach to ray transforms and related energy estimates, where the central tool is the Pestov identity, following recent works of G.P. Paternain, M. Salo, and G. Uhlmann.

Schedule

• Monday April 27th 9:30-12:00 (Salo)
• Tuesday April 28th 9:15-12:15 (Salo)
• Wednesday April 29th 9:15-12:15 (Salo)
• Thursday April 30th 9:30-12:00 (Salo)
• Monday June 22nd 10:00-12:00 (Paternain)
• Monday June 23rd 10:00-12:00 (Paternain)
• Tuesday June 24th 14:00-16:00 (Paternain)