CAREER: New Techniques for High Dimensional Systems

Princeton University, Princeton NJ

Investigators

Abstract

The onset of "big data" has led to an increased appreciation of the value of large data sets in both industrial and academic settings. Typically, the analysis of such data sets begins by modeling the data as a high dimensional system, and so an understanding of the behavior of such systems (and techniques for dealing with them) has become equally important. This award supports a mathematical investigation of the behavior of such systems using techniques introduced by the researcher in previous work with collaborators. These techniques have led to the resolution of a number of open problems across various mathematical fields, including the Kadison-Singer problem and the existence of Ramanujan graphs of all sizes and degrees. This project aims to further develop substantial connections with research across numerous fields of science, including convex optimization, real algebraic geometry, functional analysis, combinatorics, and probability. The award also supports plans to focus interdisciplinary activities of particular benefit to students associated with the project. On a technical level, the project focuses on two areas of research: (A) to develop a theory of ''finite free probability,'' a collection of ideas lying in the intersection of random matrix theory, convex optimization, real algebraic geometry, and polynomial geometry, and (B) to generalize the previously mentioned techniques so as to widen their potential application. This includes extending the ideas to bivariate polynomials as well as a non-Hermitian setting. In the direction of (B), the project aims to address some of the obstacles that hinder the application of the current techniques (for example, the need for real rootedness of associated polynomials) by investigating analogous behaviors in more general settings. This would be a necessary part of extending (A) to non-Hermitian settings, for example, as such objects no longer satisfy the conditions necessary to apply the current techniques. Extensions in this direction would also open the possibility of application to new areas such as quantitative geometry.