Mathematical Sciences: Loop Space Analysis

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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.