  We prove that the Gross-Pitaevskii equation correctly describes the ground
state energy and corresponding one-particle density matrix of rotating, dilute,
trapped Bose gases with repulsive two-body interactions. We also show that
there is 100% Bose-Einstein condensation. While a proof that the GP equation
correctly describes non-rotating or slowly rotating gases was known for some
time, the rapidly rotating case was unclear because the Bose (i.e., symmetric)
ground state is not the lowest eigenstate of the Hamiltonian in this case. We
have been able to overcome this difficulty with the aid of coherent states. Our
proof also conceptually simplifies the previous proof for the slowly rotating
case. In the case of axially symmetric traps, our results show that the
appearance of quantized vortices causes spontaneous symmetry breaking in the
ground state.
