Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

Andras Stipsicz and I proved that any configuration of symplectic surfaces intersecting symplectically orthogonally with a negative definite intersection matrix, inside a symplectic 4-manifold, has a neighborhood with strongly convex boundary. Previously it was known (thanks to Grauert) that one could make the boundary weakly convex (which means that there is a contact structure on the boundary on which the symplectic form is positive); strong convexity says that the symplectic form on a collar neighborhood of the boundary is completely controlled by a contact structure on the boundary, and thus strong convexity is what is needed when cutting and glueing in the symplectic world. The purpose of this talk is to discuss the proof of our result (which uses a little bit of toric geometry) and some speculative ideas relating this to ideas in tropical geometry, as well as some potential applications.