Mathematical Sciences: Research in Harmonic Analysis

格罗斯将继续他对非对易谐波分析的研究。这将包括对矩阵空间、正域和厄米对称空间上的广义超几何函数的研究，重点是发展算子值理论，将上下文扩展到包括II型西格尔域，推导梅林-巴恩斯表示，探索无限维表示理论的应用。格罗斯研究的数学基础是李群理论。这些群是为了纪念挪威数学家索菲斯·李而命名的，它们一直是20世纪数学的主要主题之一。作为利用系统中固有对称性的数学工具，李群表示理论对数学本身，特别是分析和数论，以及理论物理，特别是量子力学和基本粒子物理产生了深远的影响。Gross Gross将研究与布朗桥测量相关的环空间上的自然狄利克雷形式算子。特别感兴趣的是在同伦类上证明相关薛定谔算子基态的存在性和唯一性。这是Gross最近关于循环群上的对数Sobolev不等式的工作的延续，该工作对李群的表示理论有许多启示。李群理论是为了纪念挪威数学家索夫斯·李而命名的，它一直是20世纪数学的主要主题之一。李群表示理论作为探索系统对称性的数学工具，对数学本身和理论物理，特别是量子力学和基本粒子物理产生了深远的影响。

Gross will continue his study of noncommutative harmonic analysis. This will include an investigation of generalized hypergeometric functions on matrix spaces, domains of positivity, and Hermitian symmetric spaces, concentrating on developing the operator-valued theory, extending the context to include Siegel domains of type II, deriving Mellin-Barnes representations, and exploring applications to infinite-dimensional representation theory. The mathematical bedrock of Gross's research is the theory of Lie groups. Named in honor of the Norwegian mathematician Sophus Lie, these groups have been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.DMS-9200774 Gross Gross will investigate the natural Dirichlet form operator over loop spaces associated to Brownian bridge measure. The particular interest is in proving existence and uniqueness of ground states of the associated Schroedinger operators over homotopy classes. This is a continuation of Gross's recent work on logarithmic Sobolev inequalities on loop groups which has many implications for the representation theory of Lie groups. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself and theoretical physics, especially quantum mechanics and elementary particle physics.