Mathematical Sciences: Analysis of Fermion Fields

伦纳德·格罗斯教授是数学物理学的领军人物之一。他在希尔伯特空间分析、概率论、对数索波列夫不等式和量子场论方面做出了重大贡献。当前提议中的最优先事项是建设量子化的杨-米尔场。他希望将他之前在阿贝尔规范群上的工作与他之前在非阿贝尔路径相关场强方面的工作结合起来，并结合随机微分方程的技术来获得欧几里得测度的紧性。其次，格罗斯教授计划寻求他早期关于费米子Clifford代数超收缩的工作的欧几里得扩展。这样的推广有望与欧几里德量子场论的Osterwalder-Schrader公式密切相关。第三，Gross教授计划通过群的中心极限型定理的方法来研究非紧Lie群和双曲n-空间上的对数Sobolev不等式的存在性。

Professor Leonard Gross is one of the leading researchers in mathematical physics. He has made major contributions to analysis on Hilbert space, probability theory, logarithmic Sobolev inequalities, and quantum field theory. Of highest priority in the current proposal is the construction of quantized Yang-Mills fields. He hopes to combine his previous work on abelian gauge goups with his previous work on nonabelian path dependent field strengths and with techniques from stochastic differential equations to obtain compactness of Euclidean measures. Second, Professor Gross plans to seek a Euclidean extension of his earlier work on Clifford algebra hypercontractivity for Fermions. Such an extension is expected to be closely related to the Osterwalder-Schrader formulation of Euclidean quantum field theory, Third, Professor Gross plans to explore the existence of logarithmic Sobolev inequalities on noncompact Lie groups and on hyperbolic n-space via the method of central limit type theorems on groups.