Geometry and harmonic analysis related to symmetric spaces

PI将致力于调和分析、几何和表示理论中几个相互关联的问题。主要工作是对称空间上的调和分析和相应的表示理论。这项工作包括黎曼对称空间和非黎曼对称空间，紧空间和非紧空间以及这些空间的无限维极限。重点是几何学和调和分析/表示理论之间的相互作用。这项工作结合了几个数学领域的方法和思想：复数分析、实和复流形上的群作用、经典调和分析和应用数学。我们项目的大部分将与美国、欧洲和墨西哥的专家合作进行。其他问题涉及到我们研究生的参与。第一组问题集中于紧对称空间的局部Paley-Wiener型定理及其归纳极限。简而言之，问题是将由足够小的测地球支撑的光滑函数空间的映象描述为指数增长的全纯函数。我们还将研究这些空间的射影极限，从而得到对称空间的归纳极限的Paley-Wiener定理。稍后，我们还将考虑更一般的交换空间的类似问题。第二类问题与冠上全纯函数空间中的热半群的像有关。在这里，我们还将考虑无限维极限。其他计划的研究方向包括紧致流形上单李群的几何作用和关于黎曼和非黎曼对称空间作为齐次空间紧化的调和分析。这就产生了将么正表示的约束分解为子群的问题，以及将复有界对称域的某些已知结果推广到其实对称域的问题。我们的工作还将涉及欧氏空间上的调和分析问题，特别是Radon变换、再生核Hilbert空间、小波分析、小波集、与有限Coxeter群有关的调和分析、与拓扑群表示有关的函数空间(广义余轨空间)以及与薛定谔算子有关的函数空间(Besov空间)。调和分析和几何是与物理学和应用科学密切相关的两门学科。这两个主题包括纯数学和应用数学以及基础科学中的广泛而深入的问题。我们的研究围绕着对称空间上调和分析的基本问题展开。这些空间可以作为我们生活的现实世界的模型或近似值。我们计划解决的其他问题涉及工程和科学中出现的问题，特别是小波分析和Radon变换。我们计划的项目之一包括使用对称性来构造函数空间以及构造更高维的极小波。拟议的研究涉及PI的研究生，并帮助他们在研究和数学推理方面进行教育。

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.