Mathematics Lecture Series: Radon Transforms for a Pair of Grassmannians

Abstract. The term Radon transform is commonly used for an integral operator that assigns to a function on a set X a collection of integrals of that function over certain subsets of X. For example, one can integrate a function on the plane over straight lines in that plane, or a function on the 2-sphere over great circles in that sphere. One of the basic problems is to reconstruct a function from such integrals.
In his 1911 thesis, Paul Funk, who studied the case of great circles on the sphere, suggested a remarkable method that reduces the reconstruction problem to solution of a simple Abel integral equation connecting the corresponding mean values. This method was later employed by Radon for lines on the plane and extended by Helgason to arbitrary constant curvature spaces.
Dr. Rubin is planning to explain Funk’s approach in the n-dimensional setting and show its generalization
to the higher-rank case, when the source space is the real Grassmann manifold of k-dimensional linear subspaces of Rn and the target space is a similar manifold of higher dimension. It turns out that Funk’s ideas extend nicely to this more general situation if the
one-dimensional Abel integrals are replaced by the corresponding G°arding-Gindikin integrals over the cone of positive definite matrices.
The talk is accessible to the audience with standard prerequisites in analysis and linear algebra. It is a joint work with Eric Grinberg.

Free