Mathematical Sciences: Research in Harmonic Analysis

格罗斯将继续他对非对易调和的研究 分析. 这将包括对广义 矩阵空间上的超几何函数，正性域， 和埃尔米特对称空间，专注于开发 算子值理论，将上下文扩展到包括西格尔 域的类型II，推导梅林-巴恩斯表示，和 探索无限维表示的应用 理论 格罗斯研究的数学基础是 李群 以挪威数学家的荣誉命名 Sophus Lie，这些团体一直是 世纪数学。 作为数学工具， 利用系统中固有的对称性， 李群的表示理论对 数学本身，特别是在分析和数字方面， 理论和理论物理学，特别是量子 力学和基本粒子物理学DMS-9200774 毛 格罗斯将研究自然狄利克雷形式算子 环空间上的布朗桥测度。 的 特别感兴趣的是证明存在性和唯一性， 相关薛定谔算符的基态， 同伦类 这是格罗斯近期工作的延续 循环群上的对数Sobolev不等式 李群表示论的含义。 李群理论，以挪威人的荣誉命名 数学家Sophus Lie，一直是 世纪数学。 作为数学工具， 利用系统中固有的对称性， 李群的表示理论对 数学本身和理论物理学，特别是 量子力学和基本粒子物理学。

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.