Georgia Institute of Technology College of Sciences

A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, the existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. In this talk we will consider the existence of such structures in the ambient settings, that is, manifolds/domains with various degree of convexity as open/compact subsets of a complex manifold, e.g. complex 2-space C^2. In particular, I will discuss the following question: Which homology spheres embed in C^2 as the boundary of a Stein domain? This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that no non-trivial Brieskorn homology sphere, with either orientation, embeds in C^2 as a Stein boundary. In this talk, I will survey what we know about this conjecture, and report on some closely related recent work in progress that ties to an interesting symplectic rigidity phenomena in low dimensions.