Representation Theory and Harmonic Analysis on Homogeneous Spaces

提出的研究涵盖了广泛的问题和思想，从对称空间，几何和无限维类似物的抽象调和分析到阿贝尔调和分析中的经典问题。本研究的基础课题是对称空间G/H的谐波分析，它将分析、群论（对称）和几何联系在一起。项目内容包括h球分布的全纯推广，交织算子的应用及其谱的求值，无限维空间的分析，表示理论和函数的g不变巴拿赫空间。物理部分包括反射正性和传递算子。这项工作结合了几个数学领域的方法和思想：复分析，复流形上的群作用，经典谐波分析，应用数学和量子。古人的理论。调和分析和表示理论是数学的两个核心学科，在纯数学和应用数学以及理论物理中都占有重要地位。众所周知的应用例子是断层扫描和信号分析，但提出的研究涉及更抽象和基本的问题。该建议涉及数学的几个部分，包括纯粹的和应用的，并包括一些连接到物理学，特别是量子场论的形式反射的积极性，和应用科学。这项研究涉及广泛的合作，包括研究生和美国、墨西哥和欧洲的其他研究人员组成的网络。

The proposed research covers a broad spectrum of problems and ideas reaching from abstract harmonic analysis on symmetric spaces, geometry, and infinite dimensional analogues, to classical questions in abelian harmonic analysis. The basic subject of the proposed research is harmonic analysis on symmetric spaces G/H which connects analysis, group theory (symmetry) and geometry. The project includes holomorphic extension of H-spherical distributions, applications of intertwining operators and evaluations of their spectrum, analysis on infinite dimensional spaces, representation theory and G-invariant Banach spaces of functions. The physical parts include reflection positivity and transfer operators. This work combines methods and ideas from several areas of mathematics: complex analysis, group actions on complex manifolds, classical harmonic analysis, applied mathematics, and quantum field theory.Harmonic analysis and representation theory are two central subjects of mathematics and play an important role in pure and applied mathematics as well as theoretical physics. Well known examples of applications are tomography and signal analysis, but the proposed research deals with more abstract and fundamental questions. The proposal is related to several parts of mathematics, both pure and applied, and includes several connections to physics, in particular quantum field theory in form of reflection positivity, and applied sciences. The research involves wide spectrum of collaboration including graduate students and a network of other researchers in the USA, Mexico, and Europe.