Integral geometry is the study of transforms obtained by integration along certain submanifolds (called cycles) in a given space. The best known example is the X-ray transform, which starts with a function defined on Euclidean space and considers its integrals over all straight lines. Assuming the function decays so that these integrals all make sense, we obtain a new function on the space of straight lines. Another transform is the Radon transform obtained by integrating over hyperplanes. A typical question in integral geometry is the invertibility of such a transform. One can formulate a similar transform starting with a differential form or tensor field instead of a function. Similar problems can also be formulated when we integrate a function along other curves such as geodesics on a manifold. Such inverse problems arise naturally in many physical situations and in differential geometry. The solutions of some of these problems have had important applications such as the development of CAT scans. This seminar will formulate and study such transforms in integral geometry.

In the first half of the quarter Eastwood will consider complex integral geometry which studies inverse problems in the holomorphic setting or, more precisely, when there is some holomorphic structure to be exploited. In this setting the transform is often called the Penrose transform. Strangely, the complex case is somewhat easier to deal with than the real case. This part of the seminar will define the Penrose transform and study its properties in several different cases. Though there will be some representation theory lurking in the background, the approach will be differential geometric. The emphasis will be on examples. These will include the classical Penrose transform as it occurs in physics but also look at several examples whose motivation is purely mathematical.

In the second half of the quarter Sharafutdinov will consider the real case. He will consider the X-ray transform, first in the case of Euclidean space and then for integration of a symmetric tensor field along geodesics of some Riemannian metric. This arises in several physical situations since light propagates along geodesics. Another instructive example is the nonlinear inverse problem of recovering a metric from its boundary distance function. Linearization of the latter problem leads to the problem of inverting the geodesic X-ray transform for a symmetric tensor field of rank two. The problem of inverting such a transform is quite complicated and has only been solved under some restrictions on the metric. Sharafutdinov will lecture on what is known about this problem. A reference for this part of the course is Integral geometry of tensor fields, by Vladimir Sharafutdinov, VSP, Utrecht, The Netherlands (1994).