Applications of Random Matrix Theory to Analytic Number Theory

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This project explores several problems at the intersection of random matrix theory and analytic number theory. The first of these areas historically draws much of its impetus from mathematical physics and broadly concerns statistical patterns found in matrices where entries have been chosen at random; ideas from this area have found applications in a large number of subjects ranging from epidemiology to telecommunications. The second area makes use of mathematical analysis to study properties of the integers -- for instance to study the distribution of primes or the way large integers tend to factor; these are natural problems which mathematicians have been interested in for a long time and which have applications to data-security. The two areas were first linked by the surprising and still largely conjectural resemblance between the local distribution of zeros of L-functions -- these are certain functions which are of central interest in analytic number theory -- and the distribution of eigenvalues of a random matrix. In fact, distributions of this sort have been found in a diverse range of situations, and it is an important problem to understand why these distributions arise so universally. In more detail, the project consists of two related parts. The first is to study the link between zeros of L-functions and eigenvalues of random matrices by making use of combinatorial decompositions of arithmetic functions; analogous decompositions can be found in a function field setting and have shed light on related problems there. The second part is to investigate the extent to which certain pseudo-random walks on compact matrix groups equidistribute in the same fashion as classical random walks do; results pertaining to this second project have recently been used to resolve open questions about the distribution of certain trigonometric polynomials. The two components of the project share in common the use they make of combinatorial representation theory and also their study of sequences of random variables that are weakly dependent, with a dependence characterized by arithmetic/combinatorial considerations.