Contact geometry is the overarching geometric setting for the study of dynamical systems because
contact manifolds are odd-dimensional and may be viewed as “phase-spacetime’’. Contact geometry is therefore particularly
suited to quantization independent of the choice of time. Applying BRST quantization to dynamics on a contact manifold
leads to a new formulation of quantum mechanics where quantum dynamics becomes parallel transport and
quantization is the problem of finding a flat connection on an associated Hilbert bundle. Contact manifolds also enjoy a
powerful version of the Darboux theorem that ensures that local classical dynamics are trivial. A quantum analog of the Darboux
theorem also holds and implies that Quantum Mechanics amounts to the study of contact topology.