Mathematical Sciences: Analysis Over Loop Spaces

DMS-9501238 PI: Gross Gross将继续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。所使用的技术涉及到李群上的平方可积函数的调和分析，根据其全称包络代数的完成。他期望将这个调和分析推广到非紧群和对称空间。这是证明循环空间的Hodge-deRham定理的长期目标的一部分。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。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。

DMS-9501238 PI: Gross Gross will continue his study of the natural Dirichlet form operator over loop spaces associated to Brownian bridge measure. The techniques used involve harmonic analysis of the square integrable functions on a Lie group in terms of a completion of its universal enteloping algebra. He expects to extend this harmonic analysis to noncompact groups and symmetric spaces. This is part of a long range goal of proving a Hodge-deRham theorem for loop spaces. 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, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.