In joint work with S.-T. Yau, we construct new cohomologies of differential forms and elliptic operators on symplectic manifolds. Their construction can be described simply following a symplectic decomposition of the exterior derivative operator into two first-order differential operators, which are analogous to the Dolbeault operators in complex geometry. These first-order operators lead to new cohomologies which are finite-dimensional and associated elliptic operators that exhibit Hodge theoretical properties. The symplectic cohomologies give new invariants for non-K?hler symplectic manifolds.
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