Brown’s Spectral Measure: New Computational Methods from Stochastics, Partial Differential Equations, and Operator Theory

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The modern world is built on data, which is usually represented in rectangular arrays of numbers. Mathematicians call those tables matrices, and try to find hidden structure within them. One such hidden signature is a list of numbers called eigenvalues that describes some of the properties of the larger matrix of numbers. Eigenvalues not only summarize the information in a large array; they can reveal information that was hidden from plain view, through ubiquitous tools like principal component analysis that statisticians have used to great effect for decades. The main purpose of this proposal is to explore promising new computational tools to understand the behavior of eigenvalues of very large matrices lacking any overall symmetry. The PI and his team have discovered new and surprising connections between several different mathematical fields of study that can be used to compute the large-scale behavior of the eigenvalues of such symmetry-free matrices, opening the door to solve problems that have remained inaccessible for decades. As a new methodology, there is much to explore including low-hanging fruit that is perfectly suited to research by Ph.D. students and postdoctoral researchers. Funds will be used to develop these computational tools in collaboration with colleagues and postdoctoral researchers, and will support the research of a diverse body of Ph.D. students – both developing their research skills to prepare them for academic or technical careers, and furthering the research goals of the project.This project will address questions relating operator theory, stochastic differential equations, diffusion on Lie groups, and random matrix theory. The central theme will be to deploy a promising new set of computational tools, based on stochastic and partial differential equations, to calculate and prove regularity of spectral measures of non-normal operators in von Neumann algebras. These arise naturally as the high-dimension limits of random matrix models that appear in wireless communication and information theory, as well as throughout geometry and analysis in pure mathematics. The project concerns eleven research directions, which yield connections between these topics and applications to others. The intended research, upon completion, will settle several interesting open questions and present a major contribution to the theory, as well as provide ample opportunity for the mentoring of Ph.D. students and postdoctoral researchers.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.