# University of Toronto's Symplectic Geometry Seminar

March 3, 2003, 2pm

SS5017A

## Marina Ville

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Boston

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Branch points and Milnor numbers of surfaces in 4-manifolds

** 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.