  Cold atomic gases of interacting bosons subject to rapid rotation and
confined in anharmonic traps can theoretically exhibit analogues of the
fractional quantum Hall effect for electrons in strong magnetic fields. In this
setting the Coriolis force due to the rotation mimics the Lorentz force on
charged particles but artificial gauge fields can also be obtained by coupling
the internal structure of the atoms to light fields. The chapter discusses
mathematical aspects of transitions to different strongly correlated phases
that appear when the parameters of a model Hamiltonian are varied.
