Constraints on Contact Type Hypersurfaces in Symplectic 4-Manifolds

In joint work with Bulent Tosun, it was shown that Heegaard Floer theory provides an obstruction for a contact 3-manifold to embed as a contact type hypersurface in standard symplectic 4-space. As one consequence, no Brieskorn homology sphere admits such an embedding (regardless of the contact structure). I will review the ideas that lead to these results, and discuss recent extensions that can obstruct suitably convex embeddings in closed symplectic 4-manifolds, particularly rational complex surfaces.