Geometry and harmonic analysis related to symmetric spaces

与对称空间相关的几何和调和分析

• 批准号: 0801010
• 负责人: Gestur Olafsson
• 金额: $ 26.99万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801010&HistoricalAwards=false
• 关键词: Geometry; harmonic; analysis; related; symmetric

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.

PI 将致力于解决调和分析、几何和表示论中的几个相互关联的问题。主要工作是对称空间的调和分析以及相应的表示理论。这项工作包括黎曼对称空间和非黎曼对称空间、紧致空间和非紧致空间以及这些空间的无限维限制。重点是几何和调和分析/表示理论之间的相互作用。该作品结合了多个数学领域的方法和思想：复分析、实数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。其他问题涉及我们研究生的参与。第一组问题集中于紧致对称空间的局部 Paley-Wiener 型定理及其归纳极限。简而言之，问题是将光滑函数空间的图像描述为指数增长的全纯函数，并由足够小的测地线球支撑。我们还将研究这些空间的射影极限，以导出对称空间归纳极限的 Paley-Wiener 型定理。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图像有关。这里我们还将考虑无限维限制。其他计划的研究方向包括简单李群在紧流形上的几何作用以及黎曼和非黎曼对称空间作为齐次空间的紧致化的调和分析。这导致了将单一表示的限制分解为子群以及将复杂有界对称域的一些已知结果推广到其真实对应物的问题。我们的工作还将涉及欧几里得空间的调和分析问题，特别是Radon变换、再现核希尔伯特空间、小波分析、小波集、与有限Coxeter群相关的调和分析、与拓扑群表示相关的函数空间（广义Coorbit空间）以及与薛定谔算子相关的函数空间（贝索夫空间）。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广泛的深层次和广泛的问题。 我们的研究围绕对称空间调和分析的基本问题。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，特别是小波分析和氡​​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在更高维度上构造最小小波。 拟议的研究涉及 PI 的研究生，并帮助他们进行研究和数学推理方面的教育。