LEAPS-MPS: Analytic Approaches to Limit Theorems and Random Structures

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。大型随机结构是数学中的核心问题，在物理学、计算机科学、生物学和许多其他科学分支中被用作各种现象的模型。一些具体的例子包括磁、气体、肿瘤扩散和数据集中的噪声模型。这个项目的目标是通过新的分析技术推进对许多这些重要模型的研究。除了研究上述具体模型外，该项目还将建立可用于研究大型随机结构的一般数学工具。该项目将提供培训机会，让本科生和研究生参与一些拟议项目的研究。这个项目的中心是对离散概率问题的分析方法。该项目的一个中心目标是基于生成函数和特征函数的分析性质，创建用于演示随机变量的极限定理、反集中和相关衰减的工具。这些方法直接应用于组合学和数学物理中的模型，如伊辛模型、吉布斯点过程和硬核模型。PI将使用类似的复数和傅立叶分析技术来研究随机代数结构，如随机矩阵、随机多项式、随机划分和对称群的随机特征。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Large random structures are of central interest in mathematics and are used as models of various phenomena in physics, computer science, biology and many other branches of science. Some specific examples include models for magnetism, gases, the spread of tumors, and noise in data sets. The goal of this project is to advance the study of many of these important models through new analytic techniques. In addition to studying the specific models mentioned above, the project will build mathematical tools that can be used to study large random structures in general. The project will provide training opportunities that will involve undergraduate and graduate students in research on some of the proposed projects. This project centers on analytic approaches to problems in discrete probability. A central goal of the project is to create tools for demonstrating limit theorems, anti-concentration, and correlation decay for random variables based on analytic properties of generating and characteristic functions. These methods contain immediate applications to models in combinatorics and mathematical physics such as the Ising model, Gibbs point processes, and the hard core model. The PI will use similar complex- and Fourier-analytic techniques to study random algebraic structures such as random matrices, random polynomials, random partitions, and random characters of the symmetric group.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.