High-Dimensional Phenomena, Limit Theorems, and Applications

The project's research covers several topics in mathematics and is focused on the study of probabilistic, geometric, and information-theoretic aspects of high dimensional phenomena, including concentration of measure and asymptotic behavior of various functions of a growing number of random variables. The concentration tools are the subject of many exciting developments, since they help explore most essential properties of general complex systems where randomness of their numerous small parts results in a stable limit behavior. Being connected with challenging mathematical problems, this research area has proved to be very useful for applications in other fields such as statistics, information theory, computer science, machine learning. One of the objectives of the project is to clarify the role of growing dimension as a unifying source in high-dimensional models and, in particular, its influence on the entire evolution in time as opposed to local rules. Proposed research will have a broader impact by creating new connections between different mathematical fields and providing them with powerful interdisciplinary tools. The project will also have an important impact on educating in mathematical sciences.

More specifically, the investigator intends to develop new advanced concentration tools for spaces with sufficiently many symmetries including Grassmanian manifolds. It is planned to apply them in the study of global properties of multidimensional projections for log-concave and more general hyperbolic measures, that are related to the thin shell (variance) problem and the K-L-S conjecture of Kannan, Lovasz and Simonovits. Another sort of applications deals with randomized models of summation for dependent data under correlation conditions. Part of the project is devoted to limit theorems and asymptotic expansions in the central limit theorem for information-theoretic distances such as the relative entropy (Kullback-Leibler distance) and relative Fisher information, which

will be accompanied by proper Berry-Esseen bounds. The project also deals with Edgeworth-type expansions and informational bounds in the problem of Poisson approximation. The proposed themes refer either to long-standing open problems or to challenging questions related to recent developments.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.