Mathematical Sciences: Analysis Over Loop Spaces

DMS-9501238 PI：大体 格罗斯将继续他的研究自然狄利克雷形式运营商循环空间相关的布朗桥措施。 所使用的技术涉及调和分析的平方可积函数的李群方面完成其普遍enetloping代数。 他希望将这种调和分析扩展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Sophus Lie的荣誉命名的，是世纪数学的重要课题之一。 李群表示论作为利用系统中固有对称性的数学工具，对数学本身，特别是分析和数论，以及理论物理学，特别是量子力学和基本粒子物理学产生了深远的影响。

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.