Renormalization in Statistical Mechanics and Partial Differential Equations

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A great variety of mathematical models involve multiple scales: an explicit description of a system's microscopic properties is given, and a challenge is to describe the effective behavior induced by it on much larger scales. In order to predict the correct large-scale effective behavior theoretically, physicists introduced the idea of "renormalization," a systematic technique for implementing a progressive coarsening of the system of interest. The broad goal of research supported by this award is to widen and strengthen the mathematical understanding of this idea. The project provides research training opportunities for graduate students. The past few years have seen great progress on fundamental models that can be represented as partial differential equations with random coefficients. In the present project, the investigator plans to test the versatility and power of the methods developed there on new classes of models, including Langevin dynamics and systems of particles in interaction. The project will take inspiration from renormalization ideas to study mean-field models with disordered interactions, such as spin glasses.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.