Abstract: Symplectic geometry and contact geometry are closely related geometries in even and odd dimensions respectively. And there are two types of symplectic manifolds with contact boundary, the convex ones and the concave ones, depending on the direction of the Reeb vector fields along the boundary. Convex symplectic manifolds, including the Stein manifolds, have been extensively studied in the past 30 years. The concave ones are more abundant and flexible than the convex ones. In this talk I will discuss several results/speculations which indicate that the concave ones also deserve a systematic study, at least in dimension 4. This is a joint work with Cheuk Yu Mak, and partly with Koichi Yasui.
