The question is just whether they are only a nuisance, or whether they can possibly be harnessed. It is a function on the gauge and diffeomorphism invariant configuration space, whose gradient is orthogonal to the two physical degrees of freedom, in the metric defined by the Ashtekar formulation of general relativity. The imaginary part of the Chern-Simons form reduces in the limit of small cosmological constant, Lambda, and solutions close to DeSitter spacetime, to the York extrinsic time coordinate. We then propose this is the natural vacuum state of the theory for nonzero Lambda. Finally, forms for the physical hamil- tonian and inner product are suggested and a new action principle for general relativity, as a geodesic principle on the connection superspace, is found.
Symplectic geometry is the geometry of classical phase spaces.
That is, it's the geometry of spaces on which one can take Poisson brackets of functions in a manner given locally by the usual formulas. Gromov has really revolutionized the subject, and these lectures look like a good place to begin learning what is going on.