Representation Theory and Harmonic Analysis on Homogeneous Spaces

拟议的研究涵盖了广泛的问题和思想，从抽象的对称空间，几何和无穷维类似物的调和分析，在阿贝尔调和分析的经典问题。拟研究的基本课题是对称空间G/H上的调和分析，它将分析、群论（对称性）和几何学联系起来。该项目包括H-球面分布的全纯延拓、交织算子的应用及其谱的估计、无穷维空间的分析、函数的表示理论和G-不变Banach空间。物理部分包括反射正性和转移算子。这项工作结合了方法和思想，从几个领域的数学：复杂的分析，群行动复杂的流形，经典谐波分析，应用数学，和量子场theory.Harmonic分析和表示论是两个中心学科的数学和发挥重要作用，在纯数学和应用数学以及理论物理。众所周知的应用实例是断层扫描和信号分析，但所提出的研究涉及更抽象和基本的问题。这个提议涉及数学的几个部分，包括纯数学和应用数学，并包括与物理学的几个联系，特别是反射正性形式的量子场论和应用科学。该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.