Renormalization in Statistical Mechanics and Partial Differential Equations

各种各样的数学模型涉及多个尺度：对系统微观特性的明确描述是给定的，而一个挑战是在更大的尺度上描述由它引起的有效行为。为了从理论上预测正确的大规模有效行为，物理学家引入了“重整化”的概念，这是一种对感兴趣的系统进行逐步粗化的系统技术。该奖项支持的研究的广泛目标是扩大和加强对这一思想的数学理解。本项目为研究生提供研究训练机会。在过去的几年里，可以用随机系数的偏微分方程来表示的基本模型取得了很大的进展。在目前的项目中，研究者计划测试在新类型模型上开发的方法的多功能性和力量，包括朗之万动力学和相互作用中的粒子系统。该项目将从重整化思想中获得灵感，研究具有无序相互作用的平均场模型，如自旋玻璃。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.