The Fyodorov-Hiary-Keating conjecture describes the distribution of the local maximum of the Riemann zeta-function. In other words, pick a typical t between 0 and T, then the conjecture describes the fluctuations of the local maximum of the Riemann zeta-function \zeta(1/2 + i u) in a unit interval around t. The choice of a unit interval is immaterial and this can be enlarged or shrunk as needed.

In recent work with Arguin and Bourgade we establish tightness, thus for 99% of t's in [0, T] the local maximum is of size (\log T) (\log \log T)^{-3/4}. I will describe the broader context in which this work fits which is the study of log-correlated systems and the main ideas from the proof. Those are motivated by work of Bramson on branching random walks (and those in turn are motivated by the KPP equation in PDE's). In particular the proof shows that whenever the local maximum is achieved at 1/2 + iu, say, then the partial sums of \log \zeta at that point have to evolve in a very specific and rigid way.