Mathematical Sciences: Mathematical Physics: Statistical Mechanics and Field Theory

小行星9401520 这一建议是为了对凝聚态物理和高能物理中的基本问题进行数学研究。拟议的研究包括严格的重整化群方法在理论的临界现象（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）。 计划对超对称场论模型的相结构及其受指数定理约束的方式进行进一步研究。 进一步的问题包括研究在类伊辛系统中，在临界温度附近从晶格约束界面到连续界面的交叉。 物理系统，如磁铁或聚合物，在经历状态变化时表现出特征行为。例如，当磁化开始时，原子关联的长度就无限增长。相关性的长度尺度随着温度或其他参数的幂律增长。 这个提议的目标是发展理解这些现象所需的数学。 这些思想的应用将在聚合物，粒子和场，以及表面的研究中进行。 ***

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. ***