Geometry and harmonic analysis related to symmetric spaces

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The PI will work on several interrelated problems in harmonic analysis, geometry, and representation theory. The main work will be on harmonic analysis on symmetric spaces and the corresponding representation theory. This work includes both Riemannian and non-Riemannian symmetric spaces, compact and non-compact spaces as well as the infinite dimensional limits of those spaces. The focus is on the interplay between geometry and harmonic analysis/representation theory. The work combines methods and ideas from several areas of mathematics: complex analysis, group action on real and complex manifolds, classical harmonic analysis, and applied mathematics. Most parts of our projects will be carried out in collaboration with specialists in the USA, Europe and Mexico. Other problems involve participation of our graduate students. The first set of problems centers about local Paley-Wiener type theorems for compact symmetric spaces and their inductive limits. In short, the problem is to describe the image of the space of smooth functions, supported in a sufficiently small geodesic ball, as holomorphic functions of exponential growth. We will also study the projective limit of those spaces to derive a Paley-Wiener type theorem for inductive limit of symmetric spaces. Later we will also consider similar problems for more general commutative spaces. A second class of problems is related to the image of the heat semigroup in the space of holomorphic functions on the crown. Here we will also consider infinite dimensional limit. Other planned research directions involve geometric action of simple Lie groups on compact manifolds and harmonic analysis on compactification of Riemannian and non-Riemannian symmetric spaces as homogeneous spaces. This leads to questions of decomposing restricitions of unitary representations to subgroups and generalization of some will known results for complex bounded symmetric domains to their real counterparts. Our work will also involve problems from harmonic analysis on Euclidean space, in particular Radon transforms, reproducing kernel Hilbert spaces, wavelet analysis, wavelet sets, harmonic analysis related to finite Coxeter groups, and function spaces associated to representations of topological groups (generalized Coorbit spaces) as well as function spaces related to Schrodinger operators (Besov spaces). Harmonic analysis and geometry are two subjects closely related to physics and applied sciences. Those two topics include wide spectrum of deep and wide-ranging problems in pure and applied mathematics as well as basic sciences. Our research is centered around fundamental questions in harmonic analysis on symmetric spaces. Those spaces can serve as models or approximation for the real world that we live in. Other problems that we plan to work on involve questions arising in engineering and sciences, in particular wavelet analysis and Radon transforms. One of our planned projects includes the use of symmetries to construct function spaces as well as constructing minimal wavelets in higher dimension. The proposed research involves the graduate student of the PI and helps educate them in research and mathematical reasoning.