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将在谐波分析，几何学和表示理论中处理几个相互关联的问题。主要工作将是关于对称空间和相应表示理论的谐波分析。这项工作包括riemannian和非帝国对称空间，紧凑和非紧密的空间以及这些空间的无限尺寸限制。重点是几何和谐波分析/表示理论之间的相互作用。这项工作结合了数学几个领域的方法和思想：复杂分析，对真实和复杂流形的小组行动，经典的谐波分析和应用数学。我们项目的大多数部分将与美国，欧洲和墨西哥的专家合作进行。其他问题涉及我们的研究生参与。第一组问题集中于局部佩奇型定理，用于紧凑的对称空间及其感应限制。简而言之，问题是将平滑功能空间的图像描述为在足够小的大地球球中支持的平滑功能空间，这是指数级生长的霍明型函数。我们还将研究这些空间的投影限制，以得出对称空间的电感限制的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。 第二类问题与冠上霍明型功能空间中的热半群的图像有关。在这里，我们还将考虑无限的尺寸限制。其他计划的研究方向涉及简单谎言基团对紧凑型歧管的几何作用以及对riemannian和非里曼尼亚对称空间的紧凑型的谐波分析作为同质空间。这导致了将统一表示的修复到亚组的问题，并将某些意志的概括为复杂的界面对称域与其真实对应物的概括。我们的工作还将涉及对欧几里得空间的谐波分析，尤其是rad骨转换，再现内核希尔伯特空间，小波群分析，小波集，小波集，与有限的Coxeter组相关的谐波分析以及与拓扑组的表现相关的功能空间以及与拓扑组的表现相关的功能空间（广义的COORBIT SPACES）以及与Schrodinger Spaces相关的功能空间（schrodingerspaces）（schrodings spaces）。谐波分析和几何学是与物理和应用科学密切相关的两个主题。这两个主题包括纯数学和应用数学以及基本科学的广泛和广泛的问题。 我们的研究集中在对称空间的谐波分析中的基本问题上。这些空间可以用作我们所生活的现实世界的模型或近似。我们计划解决的其他问题涉及在工程和科学中引起的问题，尤其是小波分析和ra radon变换。我们计划的项目之一包括使用对称性来构建功能空间以及在更高维度中构造最小的小波。 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。