New High Dimensional Phenomena and Related Questions

The project covers a number of topics in mathematics with focus on the study of new high dimensional phenomena, in probabilistic, geometric, and information-theoretic aspects. The core part of this investigation deals with the concentration of measure phenomena for probability distributions on multidimensional Euclidean spaces, including their relationship with the asymptotic behavior of smooth functionals of a growing number of dependent random variables. Being the subject of many exciting developments over the last three decades, the concentration tools help explore most essential properties of general complex systems where the randomness of a large number of small parts results in a stable limit behavior. This research area is deeply connected to many challenging mathematical problems and has numerous applications in other fields such as statistics, information theory, computer science, machine learning.

As a general far-reaching goal, the project aims to explore the role of the growing dimension as a unifying source in high-dimensional models and its influence on the entire time evolution of random processes. The project will have a broader impact by creating new promising connections between different mathematical fields, and providing them with powerful interdisciplinary tools. The project will also have an important impact on education in mathematical sciences.

More specifically, the project deals with advanced concentration techniques that will be developed in the framework of spaces with many symmetries such as Grassmanian manifolds. With new tools, one of the targets of this investigation is the circle of problems about the concentration of distributions of weighted sums and quadratic forms of dependent random variables under correlation-type conditions. The PI intends to explore refined concentration properties of high dimensional projections of convex measures, including new integral geometric characteristics that are responsible for spectral gap and isoperimetric constants. Part of the project is devoted to the central limit theorem in terms of the Renyi divergence and relative Fisher information, and to the Edgeworth-type expansions in the area of Poisson approximation. The project also deals with transport inequalities and their applications to matching problems. The themes of this project concern either long-standing open problems or 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.