"High-dimensional random phenomena and rare events"

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Applications in diverse fields, including data analysis, convex geometry, biology, physics, economics, engineering, operations research, and computer science, give rise to questions about random phenomena in high dimensions. For example, given data that lives in a high-dimensional space, what information can be obtained by studying lower-dimensional projections of the data? Given a large number of interacting agents, who strategically make choices based only on their own state and the distribution of states of the other agents, what do their equilibria (or optimal strategies) look like? This award supports the development of diverse mathematical techniques for the analysis of such questions. The focus will be on characterizing large deviations from typical behavior, which though rare, are often of crucial importance in applications. The investigator will help train new mathematics researchers, mentor early career researchers, and be involved in outreach efforts. The research will address three topics concerning high-dimensional random phenomena. The first involves the study of random projections of probability measures in high-dimensional Euclidean spaces. This is relevant both for the statistical analysis of high-dimensional data, as well as for asymptotic convex geometry and asymptotic functional analysis. The focus is on characterizing the large deviation behavior of such random projections, as the dimension goes to infinity. This is of interest because, unlike fluctuations, the large deviation behavior is non-universal and thus distinguishes between different high-dimensional distributions. The second topic considers properties of high-dimensional Gibbs distributions on a class of hard-core configurations associated with a graph. Whereas classical models assume discrete spins, the focus of this research is on versions with continuous spins, which exhibit quite different behavior and require new techniques for their analysis. The last topic concerns the analysis of fluctuations, large deviations and concentration inequalities for Nash equilibria in both static and dynamic games of mean-field type with a large number of rational agents. This has implications for a broad range of applications and entails the development of new mathematical techniques, including large deviations principles for set-valued random elements, and analysis of solutions to a master equation for mean-field games.