Mathematical Sciences: Analysis Over Loop Spaces

9501238 PI：格罗斯将继续研究循环空间上与布朗桥测度相关的自然狄里克莱型算子。所使用的技巧涉及李群上的平方可积函数的调和分析，其依据的是李群的万能蕴涵代数的完备性。他希望将这种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。李群理论是以挪威数学家索菲斯·李的名字命名的，一直是20世纪数学的主要主题之一。作为利用系统固有对称性的数学工具，李群的表示理论对数学本身，特别是在分析和数论方面，以及对理论物理，特别是量子力学和基本粒子物理，都产生了深远的影响。

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.