4 problems

GUE Tracy–Widom fluctuation conjecture. There are functions $J:\mathbb{R}\to\mathbb{R}$ and $\sigma:\mathbb{R}\to\mathbb{R}$ such that

KPZ–Gumbel crossover conjecture. The timescale $(\log N)^2$ is the unique timescale at which the maximum exhibits a mixture of a Gumbel term and the KPZ equation.

Universality conjecture. These fluctuations should have a universal Tracy–Widom distribution after the appropriate centering and scaling.

Let $u_t(x,y)$ solve the backward equation for a system of sticky Brownian motions in the ordered Weyl chamber, with diffusion equation … The boundary conditions are indexed by pai…