Applications of random matrix theory to analytic number theory

This research proposal concerns several projects at the interface of analytic number theory and random matrix theory. As such the proposal involves ideas from number theory, combinatorics, and analysis. The proposal is divided into two sections, related by the perhaps surprising use they make of combinatorial representation theory, but also by the following theme: both sections will study a special family of weakly dependent random variables, usually with an arithmetic origin, and have as a goal to say how their behavior resembles and differs from a collection of entirely independent random variables. The first section pertains to the arithmetic meaning of the appearance of random matrix statistics among the zeros of L-functions. The second has its start in questions concerning certain trigonometric polynomials and pertains to noncommutative and compact variants of the classical central limit theorems of Salem and Zygmund.