Classical gravity can be formulated in terms of a theory, called Shape Dynamics (SD), of dynamic locally scale-invariant geometry. SD, discovered by myself and two other collaborators, can be proven to be equivalent to General Relativity (GR), despite having a different fundamental symmetry, because local foliation invariance is traded for local scale (or conformal) invariance. Broadly, the purpose of my proposed research is to try to understand the structure and implications, both classical and quantum, of SD. Towards this end, I have identified three distinct, but related, research directions. The first is an attempt to formulate SD, in the presence of a cosmological constant, in terms of conformally invariant connection variables. The hope is to generalize work done in 2+1 dimension to the physical case, which presents important new difficulties. Discovering such a formulation would: i) help to understand the mechanism behind the symmetry trading, and ii) allow for Loop Quantum Gravity-like methods of quantization to be utilized. The second direction would be to explore the effect of the unimodular condition (i.e., the condition that sets the local volume to a fixed density) on the quantization of SD. The unimodular condition is known not to modify the classical theory; but, because it excludes degenerate spacetime metrics that would otherwise be integrated over, the quantum theory could be very different. The last project is to understand the Hamiltonian of SD from knowledge only of conformal spatial geometry. Currently, the Hamiltonian is chosen uniquely to reproduce GR evolution. However, Machs principles suggest that physics should only depend upon scale-invariant information. Thus, we propose to search for a holographic definition of SD where Hamiltonian evolution is reproduced by the Renormalization Group flow in a Conformal Field Theory in one less dimension.