Loop and Path Space Analysis

9971036 Driver本研究主要致力于研究黎曼流形的路径和回路空间上的热核和维纳测度的几何分析。 私家侦探将继续他的研究有限维逼近路径积分公式的道路和循环空间紧凑黎曼流形。 希望这些近似将导致：1）Atiyah-Singer指标定理的启发式路径积分证明的严格解释，2）在路径和回路空间上构造布朗运动，3）在回路群上构造非平凡调和微分形式，以及4）在回路空间上构造Dirac算子。 这些目标是一致的主要研究人员正在进行的项目，以了解指数定理和霍奇-德拉姆定理循环空间。 本文还试图推广L.格罗斯和私家侦探从复有限维群到复循环群。在曲面和高维流形的数学研究中，回路空间自然出现。 例如，如果在一个曲面上有一个环，如果不把它撕开，它就不能变形成一个点，那么这个曲面就必须包含一个“洞”。“（想想甜甜圈的表面和球体的表面。） 物理学家的弦论描述的基本粒子的性质也使用环或“弦”。“弦理论的量子力学处理导致了类似于P.I. 这个建议致力于回答基本的数学问题的（费曼）路径积分公式的量子力学的有限维曲面和在这样的表面上的循环。 作为这项研究的副产品，P.I.希望为所谓的Atiyah-Singer指数定理做一个严谨的很漂亮的物理论证。 这个指数定理可以说是这个世纪最引人注目的数学发展之一，它在数学和物理学中都有深远的影响。

9971036 Driver This research is primarily devoted to the study of the geometric analysis associated to heat kernel and Wiener measures on path and loop spaces of Riemannian manifolds. The P.I. will continue his study of finite dimensional approximations to path integral formulas on path and loop spaces over compact Riemannian manifolds. It is hoped that these approximations will lead to: 1) a rigorous interpretation of the heuristic path integral proofs of the Atiyah--Singer index theorem, 2) construction of Brownian motions on path and loop spaces, 3) construction of non-trivial harmonic differential forms on loop groups, and 4) construction of Dirac operators on loop spaces. These goals are consistent with the principal investigators ongoing project to understand index theorems and the Hodge -- de Rham theorem for loop spaces. It is also proposed to try to generalize a result (related to the Segal Bargmann transform) of L. Gross and the P.I. from complex finite dimensional groups to complex loop groups. Spaces of loops naturally arise in the mathematical study of surfaces and of higher dimensional manifolds. For example, if there is a loop on a surface that can not be deformed to a point without tearing the loop apart, then the surface must contain a "hole." (Think of the surface of a doughnut versus the surface of a sphere.) The physicists' string theory description of the fundamental particles of nature also use loops or "strings." The quantum mechanical treatment of string theory leads to problems similar to those proposed for study by the P.I. This proposal is devoted to answering basic mathematical questions about the (Feynman) path integral formulation of quantum mechanics on both finite dimensional curved surfaces and on the loops on such surfaces. As a by-product of this research, the P.I. hopes to make rigorous the very beautiful physics argument for the so-called Atiyah-Singer Index theorem. This index theorem is arguably one of the most striking mathematical developments of this century and it has had far reaching implications in both mathematics and in physics.