In a pair of highly influential works, Fyodorov, Hiary and Keating formulated a precise conjecture describing the maximum of the Riemann zeta function in short intervals of the critical line Re(z)=1/2. This conjecture has since seen much progress, owing in part to a connection with the theory of branching random walks.
In this talk, I will outline this connection and give a more general introduction to the zeta function from a probabilistic point of view. I will then discuss recent progress towards the Fyodorov-Hiary-Keating conjecture achieved in a joint work with Louis-Pierre Arguin.

(No prior knowledge of number theory will be assumed)