Abstract: Cohomologies of differential forms provide some of the most basic invariants on manifolds. In this talk, I will discuss some new cohomologies of forms on symplectic manifolds. These cohomologies can be
defined simply in terms of linear differential operators that are intrinsically symplectic. Of interest, they have standard Hodge theoretical properties and provide useful invariants especially for non-Kahler symplectic manifolds. Moreover, they also encode the data of Lefschetz maps between de Rham cohomologies. This is joint work with C.-J.Tsai and S.-T. Yau.