  In this thesis, I will present studies on the collective modes of the
fractional quantum Hall states, which are bulk neutral excitations reflecting
the incompressibility that defines the topological nature of these states. It
was first pointed out by Haldane that the non-commutative geometry of the
fractional quantum Hall effects (FQHE) plays an important role in the
intra-Landau-level dynamics. The geometrical aspects of the FQHE will be
illustrated by calculating the linear response to a spatially varying
electromagnetic field, and by a numerical scheme for constructing model
wavefunctions for the neutral bulk excitations. Compared to early studies of
the magneto-roton modes with single mode approximation (SMA), the scheme
presented in this thesis is good not only in the long wavelength limit, but
also for large momenta where the neutral excitations evolve into
quasihole-quasiparticle pair. It is also shown that in the long wavelength
limit, the SMA scheme produces exact model wavefunctions describing a
quadrupole excitation. The same scheme can also extend to describe the neutral
fermion mode in the Moore-Read state, reflecting its non-Abelian nature.
  The numerically generated model wavefunctions are then identified with a
family of analytic wavefunctions that describe both the magneto-roton modes and
the neutral fermion modes. Like the ground state wavefunction of the Laughlin
and Moore-Read state, ...
