Concentration of measure, large deviations, normal approximation and applications

提出研究概率论中的几个问题。一类问题涉及罕见事件发生时的局部化现象，其应用范围从非线性色散方程到随机图。第二类问题涉及测量集中的前沿问题，特别是与“超集中”相关的话题。PI正提议就这些主题撰写一本书长的专著，将他在以前的工作中证明的一堆结果与一些新的结果和想法结合起来。最后，第三类问题围绕着提出者在现代问题中关于正态近似的早期工作的延续。度量集中和大偏差是基本的数学工具，经常用于统计学、计算机科学、工程、物理和许多其他领域。在这个提案中，PI概述了一个计划，该计划可能会揭示大偏差和集中测量中的一些基本问题，并应用于偏微分方程，随机图和网络以及其他几个主题中的开放问题。拟写这本书长的专著的目的是向广大读者阐明这些思想。

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.