- 6635 South Hall
Abstract: We study Hodge theory for symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are novel as they can be associated with symplectic cohomologies and be of fourth-order. We prove various Hodge decompositions and use them to obtain the isomorphisms between the cohomologies and the spaces of harmonic fields with certain prescribed boundary conditions. In order to establish Hodge theory in the non-vanishing boundary case, we are required to introduce some new boundary conditions. As an application, our results can be used to solve boundary value problems of differential forms. This is a joint work with Li-Sheng Tseng.