We review recent results about rotating Bose-Einstein condensates
confined to a two-dimensional disc in the framework of the Gross-Pitaevskii
theory and focus on the energy asymptotics, vorticity and qualitative
properties of the minimizers for large angular velocity and coupling
parameter (Thomas-Fermi regime). We identify three critical speeds:
At Ωc1 vortices start to appear and as long as the angular
velocity stays below Ωc2 the vorticity is uniformly distributed over
the disc. Above Ωc2 the centrifugal forces create a hole around the
center of the trap with strongly depleted density but the vorticity is still
uniformly distributed in an annulus containing the bulk of the density.
When Ωc3 is reached a transition to a giant vortex state takes
place and the vorticity disappears from the bulk of the condensate.