Title: Tomography by Flat Tori Transform on Grassmannians and other Symmetric Spaces of Compact Type
Speaker: Eric Grinberg (U Massachusetts Boston)
Abstract: In a 1913 paper Paul Funk proved that a suitable function on the sphere S2 is odd if and only if its integrals over great circles (closed geodesics) vanish, and that an even function is determined by such integrals. His motivation came from problems in differential geometry. This context of integral geometry is similar to that of the Radon transform and its cousins, is used in medical imaging (CAT scanners, MRI), and overlaps a number of subjects in mathematics, including but not restricted to number theory. We replace the sphere S2 by a symmetric space of compact type, e.g., a Grassmann manifold, and great circles by maximal totally geodesic flat tori, and consider the transform that integrates over these. We show that, when the symmetric space is the "universal covered space" in its class, the torus transform is injective, and otherwise the transform is non- injective, with a kernel that is directly linked to deck transformations of the appropriate symmetric covering space. This gives one of the direct extensions of Funk's transform and its injectivity properties. We also discuss other extensions and generalizations of the great circle transform, and propose conjectures and open problems.