Mathematical Sciences: Mathematical Physics: Statistical Mechanics and Field Theory

9401520 Imbrie本方案用于凝聚态物理和高能物理中基本问题的数学研究。提出的研究包括临界现象（自避免行走和其他相互作用的聚合物系统，随机薛定谔算子）理论中的严格重整化群方法。计划进一步研究超对称场论模型的相结构及其受指标定理约束的方式。进一步的问题包括研究类ising系统在临界温度附近从晶格约束界面到连续界面的交叉。磁体或聚合物等物理系统在经历状态变化时表现出特有的行为。例如，当磁化开始时，原子相关的长度会无限制地增长。相关性的长度尺度在温度或其他参数中以幂律的形式增长。这项提议的目标是发展理解这些现象所需的数学。这些思想将在聚合物、粒子、场和表面的研究中得到应用。＊＊＊

9401520 Imbrie This proposal is for mathematical research on fundamental problems in condensed matter physics and high energy physics. The proposed research includes rigorous renormalization group methods in the theory of critical phenomena (self-avoiding walk and other interacting polymer systems, random Schroedinger operators). Additional research is planned on the phase structure of supersymmetric field theory models and the way it is constrained by index theorems. Further problems include the study of the crossover from lattice- constrained interfaces to continuum interfaces near the critical temperature in Ising-like systems. Physical systems such as magnets or polymers exhibit characteristic behavior as they undergo a change of state. For example, the length over which atoms are correlated grows without bound as a magnetization begins to set in. Length scales for correlations grow as a power law in the temperature or other parameters. The goal of this proposal is to develop the mathematics needed to understand these phenomena. Application of these ideas will be pursued in the study of polymers, particles and fields, and surfaces. ***