Concentration of measure, large deviations, normal approximation and applications

建议研究概率论中的几个问题。其中一类问题涉及罕见事件发生时的局部化现象，应用于从非线性色散方程到随机图的各种问题。第二类问题涉及衡量集中度的前沿问题，特别是与被称为“超集中度”的主题有关的问题。PI建议就这些主题写一本书长的专著，汇编他在以前的工作中证明的一系列结果，以及一些新的结果和想法。最后，第三类问题围绕着提出者在现代问题中关于正态逼近的早期工作的继续。测量集中和大偏差是统计、计算机科学、工程、物理和许多其他领域中经常使用的基本数学工具。在这份提案中，PI概述了一项计划，该计划可能会阐明测量的大偏差和集中度中的一些基本问题，并应用于偏微分方程、随机图和网络中的开放问题，以及其他几个主题。这本拟议的书籍长度的专著的目的是向广大受众阐明这些想法。

It is proposed to study several problems in probability. One class of problems concerns the phenomenon of localization under the occurrence of a rare event, with applications to problems ranging from nonlinear dispersive equations to random graphs. A second class of problems involves cutting-edge issues in concentration of measure, particularly related to a topic called "superconcentration". The PI is proposing to writing a book-length monograph on these topics, compiling a bunch of results that he proved in previous work together with some new results and ideas. Finally, a third class of problems centers around a continuation of the proposer's earlier work of normal approximation in modern problems.Concentration of measure and large deviations are basic mathematical tools that are frequently used in statistics, computer science, engineering, physics, and many other fields. In this proposal, the PI has outlined a plan that may shed light on some fundamental issues in large deviations and concentration of measure, with applications to open questions in partial differential equations, random graphs and networks, and several other topics. The purpose of the proposed book-length monograph is to elucidate the ideas to a broad audience.