Random Matrix Theory and Number Theory

Over the last 35 years, evidence has accumulated hinting  
at profound connections between random unitary matrices and the  
theory of the Riemann zeta function.  (The zeta function encodes  
information about the primes and is the subject of the Riemann  
Hypothesis, one of the central problems in Pure Mathematics.)  In  
recent years, a general understanding has developed which sets this  
in much wider story, linking a range of fundamental problems in  
number theory to properties of random matrices. This talk will be a  
survey of the some of the key ideas and developments.