Structured Randomness: Random Matrices and Geometry

Princeton University, Princeton NJ

Investigators

Abstract

Many complex problems in mathematics exhibit both random features and non-random structures. While different structures are often studied on a case-by-case basis, this project aims to develop general mathematical tools to explain the interplay between randomness and the underlying structure. The focus of the project is on two distinct areas, random matrix theory and metric geometry, where recent advances provide powerful new tools to understand essentially arbitrarily structured models. Such technology makes it possible to study a range of problems that are outside the reach of traditional methods, with applications both in pure mathematics and in areas such as data science and computer science. Graduate and undergraduate student researchers will participate in the project, there will be outreach to schools including middle-school math festivals, and substantial pedagogical materials will be developed and disseminated. Concretely, the project has two main themes. The first theme aims to develop a broadly applicable toolbox to investigate arbitrarily structured random matrices. The guiding principle behind this toolbox is that the behavior of structured random matrices can be modelled by the behavior of deterministic operators whose spectra are explicitly computable, building on recent advances in this area. The second theme aims to develop probabilistic methods that are motivated by the study of embeddings of metric spaces in normed spaces. Both topics leverage connections between probability theory, operator theory, functional analysis, and metric geometry. While the two themes are independent and involve different mathematical ideas, what they have in common is that they aim to develop general principles and tools to study structured problems. 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.