  We establish high probability estimates on the eigenvalue locations of
Brownian motion on the $N$-dimensional unitary group, as well as estimates on
the number of eigenvalues lying in any interval on the unit circle. These
estimates are optimal up to arbitrarily small polynomial factors in $N$. Our
results hold at the spectral edges (showing that the extremal eigenvalues are
within $\mathcal{O} (N^{-2/3+})$ of the edges of the limiting spectral
measure), in the spectral bulk, as well as for times near $4$ at which point
the limiting spectral measure forms a cusp. Our methods are dynamical and are
based on analyzing the evolution of the Borel transform of the empirical
spectral measure along the characteristics of the PDE satisfied by the limiting
spectral measure, that of the free unitary Brownian motion.
