Mathematical Sciences: Loop Space Analysis

抽象驾驶员 首席研究员建议继续发展对固定和自由回路组的积分和微分分析。这个建议是围绕调查的“热核”措施的循环组，而不是更标准的钉扎维纳措施。 然而，P.I.确实计划调查 在这两个措施之间。 P.I.建议调查的具体问题是：（i）微分算子在loop群上的形式的解析性质（相对于热核测度）（ii）loop群的Hodge-de Rham定理，iii）loop群上的热核测度与pinned Wiener之间的关系 测度，iv）热核对数Sobolev不等式与Gross' pinned Wiener测度对数Sobolev不等式之间的关系，v）具有热核测度的环群上的Meyer型不等式，vi）P.I.还有私家侦探和L.恶心 私家侦探的工作集中在积分和微分分析的“无限”维空间组成的封闭曲线在有限维空间。 粗略地说，私家侦探研究的问题。类似于日常生活中的以下“更”熟悉的例子：当你拨动一个 吉他弦，弯曲的金属板在某个时间的温度是多少，给定早期的温度知识，物质如何在具有不同渗透性和形状的区域内扩散？研究这些问题的无限维版本的动机之一来自量子场论的研究， 物理学家用来描述亚原子粒子的一类数学理论。量子场论处理上面提到的相同问题，并且与P.I.研究的数学理论密切相关。 私家侦探事实上， 量子场论和粒子物理学的分支。

Abstract Driver The principal investigator proposes to continue the development of integral and differential analysis on pinned and free loop groups. This proposal is centered around the investigation of the "heat kernel" measure on the loop group rather than the more standard pinned Wiener measure. However, the P.I. does plan to investigate the relationship between these two measures. The specific issues proposed to investigate by the P.I. are: (i) analytic properties of the differential operator on forms on the loop group (relative to the heat kernel measure) (ii) the Hodge-de Rham theorem for the loop group, iii) the relationship between the heat kernel measures on the loop group and pinned Wiener measure, iv) the relationship between the heat kernel logarithmic Sobolev inequality and Gross' pinned Wiener measure logarithmic Sobolev inequality, v) Meyer type inequalities on the loop group with the heat kernel measure, and vi) the loop group analogue of the "Taylor series" isomorphism considered by the P.I. and the P.I. and L. Gross. The P.I.'s work is centered on integral and differential analysis of "infinite" dimensional spaces consisting of closed curves in a finite dimensional space. Roughly speaking, the sort of questions studied by the P.I. are analogous to the following "more" familiar examples in every day life: what are the frequencies you hear when plucking a guitar string, what is the temperature of a bent metal plate at some time given knowledge of the temperature at an earlier time, and how do substances diffuse inside regions with varying permeability and shape? One of the motivations for studying the infinite dimensional versions of these questions comes from the study of quantum field theories, a certain class of mathematical theories used by physicists to describe sub-atomic particles. Quantum field theories deal with the same issues alluded to above and are closely related to the mathematical theories studied by the P.I.. The P.I.'s work may in fact ha ve future ramifications for quantum field theories and hence particle physics.