University of Toronto's Symplectic Geometry Seminar

Abstract: We look at a sequence of surfaces $S_n$, which are immersed in a $4$-manifold $M$ and which degenerate to a surface $S_0$ with a branch point. The genus of $S_0$ is not larger than the genus of $S_n$; we lose topology but we gain singularity. We ask: \\ QUESTION. Can we read on the branch point of $S_0$ how much genus we have lost going from $S_n$ to $S_0$?\\ If $M$ is a complex surface and the $S_n$'s are complex curves, the answer is YES, and it is given by the {\it Milnor number}.\\ We try to generalize this to the non complex case and get 3 invariants instead of one (the normal index of a branch point; the tangent and normal Milnor numbers of a sequence of surfaces). If $M$ is Riemannian and the $S_n$'s are minimal surfaces, these invariants give us a partial answer to the QUESTION above.