While dilute atomic Bose gases are typically in a weakly interacting regime, they can be driven into regimes of strong correlations by various means. A very interesting possible way to do this is by the use of a rotating optical lattice. I shall discuss the nature of the groundstate of such systems, as described by the Bose-Hubbard model in a uniform effective magnetic field. The relevant physics involves the interplay between the fractional quantum Hall effect for bosons and the "Hofstadter butterfly" spectrum. I will explain how that this interplay can lead to strongly correlated phases that have no counterpart in the continuum.