  This paper discusses the canonical quantization of 1+1-dimensional Yang-Mills
theory on a spacetime cylinder, from the point of view of coherent states, or
equivalently, the Segal-Bargmann transform. Before gauge symmetry is imposed,
the coherent states are simply ordinary coherent states labeled by points in an
infinite-dimensional linear phase space. Gauge symmetry is imposed by
projecting the original coherent states onto the gauge-invariant subspace,
using a suitable regularization procedure. We obtain in this way a new family
of "reduced" coherent states labeled by points in the reduced phase space,
which in this case is simply the cotangent bundle of the structure group K.
  The main result explained here, obtained originally in a joint work of the
author with B. Driver, is this: The reduced coherent states are precisely those
associated to the generalized Segal-Bargmann transform for K, as introduced by
the author from a different point of view. This result agrees with that of K.
Wren, who uses a different method of implementing the gauge symmetry. The
coherent states also provide a rigorous way of making sense out of the quantum
Hamiltonian for the unreduced system.
  Various related issues are discussed, including the complex structure on the
reduced phase space and the question of whether quantization commutes with
reduction.
