Title: New approaches to Random Matrix Theory problems motivated by Number Theory

Abstract: Over the past few decades, our understanding of analytic number theory has been improved through modeling of L-functions (such as Riemann's zeta function) by random characteristic polynomials. I will introduce this mostly conjectural philosophy and then focus on a particular case where the probabilistic model is itself intractable. For instance, "What is, on average over the critical line, the norm of the derivative of zeta?" leads to the question "What is, on average among NxN unitary matrices, the norm of the derivative of the characteristic polynomial on the unit circle?" I will present very natural and effective techniques of algebraic combinatorics to attack this question (and others), and finally highlight connections with more classical tools of the trade, such as orthogonal polynomials, Selberg integrals, discrete Painleve equations or hypergeometric functions of matrix arguments.