  This paper proves universality of the distribution of the smallest and
largest gaps between eigenvalues of generalized Wigner matrices, under some
smoothness assumption for the density of the entries.
  The proof relies on the Erd{\H o}s-Schlein-Yau dynamic approach. We exhibit a
new observable that satisfies a stochastic advection equation and reduces local
relaxation of the Dyson Brownian motion to a maximum principle. This observable
also provides a simple and unified proof of universality in the bulk and at the
edge, which is quantitative. To illustrate this, we give the first explicit
rate of convergence to the Tracy-Widom distribution for generalized Wigner
matrices.
