New High Dimensional Phenomena and Applications

The project's research lies at the crossroads of several themes in mathematics and is focused on the study of probabilistic, geometric, and information-theoretic aspects of concentration of measure and other high-dimensional phenomena. In complex systems where rules of randomness are well understood and sufficiently many underlying events are independent of each other, aggregate behavior at large scales tends to deviate very little from the median behavior. This phenomenon, known as concentration of measure, has been the subject of exciting developments, since concentration tools allow one to analyze many essential properties of rather general systems. Being strongly motivated by challenging purely mathematical questions, this research area also has a wide range of applications in disciplines such as information theory, statistics, computer science, and machine learning, among others. This research project is aimed in particular at a correct understanding of the role of the growing dimension, especially in the problems where high dimension serves as a unifying force. High-dimensional models are useful in practice for instance to understand the entire evolution of a phenomenon in time, not just the governing local rules, which may lead one's low-dimensional intuition astray. The project will have an important impact correcting such misconceptions in mathematics, and will have further impact when these ideas are applied to areas such as statistics and machine learning. The project will also have broader impact on educating undergraduate and graduate students in the mathematical sciences.

The project's themes refer either to long-standing open problems or to challenging questions related to recent developments. More specifically, the investigator plans to develop new concentration tools starting from the spherical concentration phenomenon and its extensions to Grassman and Stiefel manifolds. With new tools, one of the targets of investigation will be the circle of problems related to the K-L-S conjecture of Kannan, Lovasz, and Simonovits. The project will explore refined concentration properties of high dimensional projections of log-concave and more general convex measures; in particular, the work will investigate new integral geometric characteristics of convex measures on Euclidean spaces that are responsible for spectral gap and Cheeger isoperimetric constants. Part of the project is devoted to asymptotic expansions in the central limit theorem for the relative entropy and Fisher information, including Berry-Esseen bounds, and their applications to optimal transport for sums of independent random vectors. The project also deals with information-theoretic inequalities and transport problems about empirical distributions.