CAREER: Beyond Independence: Random Matrices and Applications

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project is dedicated to the study of Random Matrix Theory and its applications. Random matrices arise naturally in many diverse fields including statistics, data science, computer science, and physics. For example, random matrices were originally introduced in physics to study the nuclei of heavy atoms. The aim of this project is to understand the properties of certain random matrix models that arise in several diverse domains including control theory, statistical genetics, and the study of neural networks. This research opens the door to a deeper understanding of applications in these domains and has the potential to create avenues of future fundamental research in Random Matrix Theory and related fields. This project also features a number of educational components that integrate research and teaching. These components include a summer academy for high school students interested in advanced mathematics; participation in organized activities designed to allow high school students and scientists to meet and interact in an informal setting using interactive demonstrations; and undergraduate and graduate student mentoring. The overarching research goal of this project is to understand the behavior of the eigenvalues and eigenvectors of random matrices with dependent entries. The research program is divided into three themes, which are quite disparate in background, application, and tools. The first theme concerns the eigenvalues and eigenvectors of matrices arising in the study of random networks and graphs, including matrices that appear in synchronization problems and network control theory. Motivated by open questions in statistical genetics, the second theme concerns the spectral properties of sample covariance matrices constructed from dependent random samples. The third theme is inspired by cutting-edge results in the theoretical study of neural networks and involves random matrix products. This interdisciplinary research program is integrated with student research and exploration projects, designed for students ranging from the high school to graduate level. 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.