Contact type hypersurfaces in small symplectic 4-manifolds

Abstract: A codimension-1 submanifold embedded in a symplectic manifold is called “contact type” if it satisfies a certain convexity condition with respect to the symplectic structure. Given a symplectic manifold $X$ it is natural to ask which manifolds $Y$ can arise as contact type hypersurfaces. We consider this question in dimension 4, which appears much more constrained than higher dimensions; in particular we review evidence that no homology 3-sphere can arise as a contact type hypersurface in $\mathbb{R}^4$ except the 3-sphere. We exhibit an obstruction for a contact 3-manifold to embed in certain closed symplectic 4-manifolds as the boundary of a Weinstein domain — a slightly stronger condition than contact type — and explore consequences for the symplectic topology of small rational surfaces and potential applications to smooth 4-dimensional topology.

The morning introductory talk (at 11:00) will review symplectic structures, symplectic convexity, and the related notion of pseudoconvexity, together with some aspects of “embedding questions” for 3-manifolds in $\mathbb{R}^4$ or other 4-manifolds.