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In this thesis, I will present studies on the collective modes of thefractional quantum Hall states, which are bulk neutral excitations reflectingthe incompressibility that defines the topological nature of these states. Itwas first pointed out by Haldane that the non-commutative geometry of thefractional quantum Hall effects (FQHE) plays an important role in theintra-Landau-level dynamics. The geometrical aspects of the FQHE will beillustrated by calculating the linear response to a spatially varyingelectromagnetic field, and by a numerical scheme for constructing modelwavefunctions for the neutral bulk excitations. Compared to early studies ofthe magneto-roton modes with single mode approximation (SMA), the schemepresented in this thesis is good not only in the long wavelength limit, butalso for large momenta where the neutral excitations evolve intoquasihole-quasiparticle pair. It is also shown that in the long wavelengthlimit, the SMA scheme produces exact model wavefunctions describing aquadrupole excitation. The same scheme can also extend to describe the neutralfermion mode in the Moore-Read state, reflecting its non-Abelian nature.  The numerically generated model wavefunctions are then identified with afamily of analytic wavefunctions that describe both the magneto-roton modes andthe neutral fermion modes. Like the ground state wavefunction of the Laughlinand Moore-Read state, ...

Geometric Aspects and Neutral Excitations in the Fractional Quantum Hall Effect