RMT statistics in number theory and in quantum chaos

Montgomery's pair correlation conjecture ushered a new paradigm into the theory of the Riemann zeta function, that of the occurrence of Random Matrix Theory statistics, as developed in part by Dyson, into the theory. A parallel development was the introduction of the same paradigm into the study of quantum systems having a chaotic classical limit. In this talk I will survey these ideas, emphasizing problems in quantum chaos, and describe a new perspective from the theory of moduli spaces of Riemann surfaces, as developed by Mizrakhani.