Differential elliptic operators are powerful tools for
understanding geometrical properties of Kahler and complex manifolds.  In
joint work with S.-T. Yau, we asked on symplectic manifolds, what elliptic
operators are there that are intrinsically symplectic?  In this talk, I
will introduce a number of new symplectic elliptic operators.  Their
construction follows simply from a symplectic decomposition of the
exterior derivative operator into two linear differential operators, which
are analogous to the Dolbeault operators in complex geometry.  These
elliptic operators exhibit Hodge theoretical properties and encode new
symplectic invariants especially for non-Kahler symplectic manifolds.