Mathematical Sciences: Mathematical Physics: Statistical Mechanics and Field Theory

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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. ***