Disordered freezing: random matrix theory, statistical mechanics and the extreme values of ζ(1/2+it)
I shall sketch some recent ideas, developed with Yan Fyodorov, that connect the problems of determining (a) the extreme values of the Riemann zeta function on its critical line and (b) the characteristic polynomials of random unitary matrices with the freezing transition scenario observed in certain disordered systems. This encourages us to speculate on the possibility of there being an analogous freezing transition in the zeta function and in the random matrix polynomials, and to explore the consequences if there is one.