Applications of random matrix theory in analytic number theory

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This research proposal lies at the intersection of analytic number theory and random matrix theory. Analytic number theory is the part of number theory that makes use of mathematical analysis to study topics like the distribution of prime numbers. Topics like these are a part of pure math but have applications to cryptography for instance. Random matrix theory is the study of matrices with entries that have been chosen randomly. Many important questions in the area concern the distribution of eigenvalues of random matrices. Such questions were first motivated by mathematical physics, statistics, and population biology and answers provided by random matrix theory yield important insights in these fields. Analytic number theory and random matrix theory are quite disparate fields, but there exist remarkable connections between them. The first such link arose in work of H. Montgomery on the zeros of the Riemann zeta-function. These zeros are important because they characterize the distribution of primes. Remarkably, at least numerically, the spacings between the zeros seem to resemble the spacings between eigenvalues of a wide variety of random matrices - but no one can prove that this is actually so. Other complex systems also seem to display the same or related patterns, and why this pattern appears in such disparate contexts remains a mystery. (Another surprising example is the spacing between bus arrival times in the Mexican city of Cuernavaca.) One part of the research program outlined in this proposal seeks to better understand why spacings between zeta zeros resemble spacings between eigenvalues by 1) developing an illuminating combinatorial framework for understanding this fact, 2) building random models of the Riemann zeta-function, and 3) developing links to the theory of stochastic point processes. Aspects of these three points have already been used to resolve or shed light on old unresolved problems. A second part of this proposal involves the study of products of pseudo-random matrices - this has applications to the distribution of the famous Rudin-Shapiro polynomials, which are interesting for their own sake to analysts and number theorists, but which also have applications in signal processing. Again, ideas related to this second part have also been used to resolve old open problems in mathematics. Highly qualified personnel will be trained throughout this proposal by learning and developing aspects of probability (including random matrix theory and the theory of point processes), combinatorics (including combinatorial representation theory), and number theory, with an eventual goal of pursuing careers in academia or industry (in for instance data science, wireless communications, or data security).