Unusual Concentration Phenomena in Probability, Analysis, and Geometry

Many complex systems behave in a random fashion at the smallest scales. Why then does the world around us appear to be so predictable? This is explained in a very general context by a principle known as concentration of measure: smooth functions of many independent random variables behave in an essentially predictable manner.

Precise mathematical formulations of this phenomenon provide powerful tools for studying complex random structures. This project aims to study new concentration phenomena that arise from unexpected connections with several different areas of mathematics: from the study of embeddings (how well can data be represented in a particular space?); from the study of exotic shapes which date back to old problems in geometry from over a century ago; and from the study of non homogeneous random matrices, which are widely used in modern data science.

The unusual features of these problems motivate the development of the theory in new directions, as well as the introduction of new tools that may be applied to a wide range of random structures that arise in both pure and applied mathematics. The project includes educational, mentoring and outreach activities that are aimed at attracting and training the next generation of mathematicians, and at increasing participation and diversity in the mathematical sciences. It also provides research training opportunities for graduate students.

The aim of this project is to systematically develop novel concentration phenomena that arise from problems in probability, functional analysis, metric geometry, and convex geometry. The project is organized around three topics. The first topic aims to develop a general theory of concentration inequalities for functions taking values in normed spaces.

The second topic aims to understand certain long-standing questions in convex geometry that may be viewed as unusual analogues of the concentration phenomenon. The third topic is concerned with concentration inequalities for the norms of non-homogeneous random matrices. The investigation of concentration phenomena in nonstandard settings motivates new questions and the development of new tools that are both of direct probabilistic significance, and that provide new perspectives on problems in other areas of mathematics.

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.