The proposed research lies on the intersection of integral geometry and harmonic analysis. The main focus is the so-called higher-rank problems, which are typical for multi-dimensional geometric and analytic objects in geometry, statistics, tomography, and other sciences. The project includes the following general topics: (a) Radon transforms and more general intertwining operators on Stiefel manifolds, spaces of matrices, and symmetric spaces of arbitrary rank; (b) Analytic families of fractional integrals of functions of matrix argument with kernels having a complicated geometric structure of singularities; (c) Problems in convex geometry and geometric tomography, that can be treated using higher-rank objects studied in (a) and (b). The work combines methods and ideas from several branches of mathematics: real and complex analysis, convex geometry, harmonic analysis on homogeneous spaces, Jordan algebras, group representations, function spaces, and approximation theory. The main objectives include derivation of new inversion formulas and the study of the algebraic structure of Radon-like transforms on spaces with complicated, but conceptually natural, geometry (matrix spaces, flag manifolds, symmetric spaces, and others). The PI plans to study analytic families of potential operators of the Riesz type and the Riemann-Liouville-Gindikin integrals associated to homogeneous cones in context of applications to higher-rank geometric problems. The aim is to understand new higher-rank phenomena in the Radon transform theory on the conceptual level, develop new effective methods based on group representations, wavelet transforms, and other tools of harmonic analysis. The project also deals with detailed investigation of newly introduced composite cosine transforms on Stiefel manifolds, totally geodesic Radon transforms on rank-one Riemannian symmetric spaces of the non-compact type and a series of open problems of the Busemann-Petty type, related to two- and three-dimensional sections of convex bodies in spaces of higher dimension.
The proposed research is closely related to contemporary trends in integral geometry and harmonic analysis. New higher-rank techniques can also be applied to multivariate statistics and tomography. The activities include development of research-based educational materials useful for teaching, mentoring early-career scientists and other effective pedagogic forms.
