Mathematics Colloquium

The solution to the one dimensional Fisher-KPP equation (1937) $u_t=\Delta u+u(1-u)$ starting from a step initial condition, converges after centering by $2t - 3/2 \log t$ to a traveling wave. The logarithmic correction term, and in particular the coefficient $3/2$, was computed by Bramson (1978), through a connection with the maximum of branching Brownian motion. Recently, this computation proved crucial in the solution of a variety of problems: the law of the maximum of the critical Gaussian free field, the cover time of the 2-sphere by Brownian sausage, the maxima of the characteristic polynomials of random matrices, and even the values of the Riemann zeta function on the critical line. These problems all share a hidden logarithmic (i.e., multiscale) correlation.  In the talk I will describe these development and will emphasize the common philosophy in studying these very different models.