Concentration of measure, large deviations, normal approximation and applications

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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.