  Let T(x,r) denote the first hitting time of the disc of radius r centered at
x for Brownian motion on the two dimensional torus. We prove that sup_{x}
T(x,r)/|log r|^2 --> 2/pi as r --> 0. The same applies to Brownian motion on
any smooth, compact connected, two-dimensional, Riemannian manifold with unit
area and no boundary. As a consequence, we prove a conjecture, due to Aldous
(1989), that the number of steps it takes a simple random walk to cover all
points of the lattice torus Z_n^2 is asymptotic to (2n log n)^2/pi. Determining
these asymptotics is an essential step toward analyzing the fractal structure
of the set of uncovered sites before coverage is complete; so far, this
structure was only studied non-rigorously in the physics literature. We also
establish a conjecture, due to Kesten and Revesz, that describes the
asymptotics for the number of steps needed by simple random walk in Z^2 to
cover the disc of radius n.
