  Ultracold neutral bosons in a rapidly rotating atomic trap have been
predicted to exhibit fractional quantum Hall-like states. We describe how the
composite fermion theory, used in the description of the fractional quantum
Hall effect for electrons, can be applied to interacting bosons. Numerical
evidence supporting the formation of composite fermions, each being the bound
state of a boson and one flux quantum, is shown for filling fractions of the
type nu=p/(p+1), both by spectral analysis and by direct comparison with trial
wave functions. The rapidly rotating system of two-dimensional bosons thus
constitutes an interesting example of "statistical transmutation," with bosons
behaving like composite fermions. We also describe the difference between the
electronic and the bosonic cases when p approaches infinity. Residual
interactions between composite fermions are attractive in this limit, resulting
in a paired composite-fermion state described by the Moore-Read wave function.
