Not a proof of the Schoenflies conjecture
The alternate title for the talk is “Stein trisections and homotopy 4-balls.” It’s well known that if X is a compact Stein surface and B is a homotopy 4-ball embedded in X with pseudoconvex boundary, then B must be smoothly standard. In this talk, we will introduce Stein trisections and describe a compelling reembedding construction for homotopy 4-balls in C^2. In particular, if B is a homotopy 4-ball smoothly embedded in C^2, there is a diffeomorphic homotopy 4-ball Z that is the union of three pseudoconvex domains in C^2. Finally, we give some analytic criteria to deduce when this homotopy 4-ball is standard.