Stochastic Differential Equations, Heat Kernel Analysis, and Random Matrix Theory

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Random processes dominate our world: from the fluctuations of stock prices, to the large-scale behavior of queueing systems in computer or biological networks, to the core behavior of quantum systems. Most science and engineering problems in part boil down to separating random noise from structure, or sometimes utilizing random noise to amplify structure. One of the great triumphs of 20th Century science was the development of many robust tools for understanding and analyzing random noise in many kinds of systems. One such set of tools is stochastic analysis, which presents a rigorous mathematical treatment of ideas that first appeared in physics, adding random noise to the equations of state that describe our world. These so-called stochastic differential equations have playing a fundamental role in advancing our knowledge in many area: economics, systems engineering, biological dynamics, and within mathematics, with applications to fields like differential geometry and quantum information theory.This project addresses questions relating stochastic differential equations, heat kernel analysis, and random matrix theory. The central theme is understanding how differential equations with some randomness affect the evolution of eigenvalues of random matrices. These ideas connect with geometry, since the flow of heat on Lie groups (groups of continuous symmetries of geometric objects) can be characterized by such matrix stochastic differential equations. Herein, ten research projects are proposed which yield connections between these topics and applications to others. Generally, these problems can be described as studying the large-dimension asymptotic behavior of geometrically-motivated matrix-valued stochastic differential equations. Of particular note are questions related to the fine (edge) structure of Brownian motion on high dimensional Lie groups from the classical compact families. The intended research, upon completion, will settle several interesting open questions and present a major contribution both to the theory of stochastic differential equations and random matrix theory.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.