## ABSTRACT -- Paul Seidel

Real and complex Morse theory in two variables

In the early seventies, singularity theorists
(A'Campo and Gusein-Zade) discovered a connection between real and
complex Morse theory in two variables. Real Morse theory means that
we consider, say, a real polynomial *p(x,y)* in two variables, such
that the critical points are nondegenerate. There is one additional
condition: all saddle points must lie on the same level set, let's
say *p(x,y) = 0* for any saddle point *(x,y)*. Then one can get a good
grip on p by just drawing the curve (with self-intersections) *p*^{-1}(0)
in **R**^{2}. Complex Morse theory means that we look at the same *p* as a
complex function: the fibres *p*^{-1}*(z)* are now a family of open
Riemann surfaces, and we have typically "complex" phenomena like
vanishing cycles and Picard-Lefschetz monodromy.

The correspondence as established by A'Campo and Gusein-Zade allows
one to compute certain topological (really, homological) invariants
of the complex family of Riemann surfaces, in terms of the real
picture; and vice versa. I will explain how recent progress in
Floer cohomology leads to deeper insight into this relation. To be
honest, that hasn't yet led to concrete progress on any of the
known open conjectures; but at least it serves to show where
"new" invariants like Floer cohomology stand with respect to the
more familiar topological ones.

### LITERATURE

A'Campo, Norbert: *Le groupe de monodromie du deploiement des
singularites isolees de courbes planes. II*. Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pp. 395--404.

A'Campo, Norbert: *Generic immersions of curves, knots, monodromy and Gordian number*. Inst. Hautes Etudes Sci. Publ. Math. No. 88, (1998), 151--169 (1999).

Gusein-Zade, S. M.: *Invariants of generic plane curves and invariants of singularities*. Algebraic geometry---Santa Cruz 1995, 423--433, Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, 1997.

Seidel, P.: *More about vanishing cycles and mutation*, Preprint, 2000 (available on http://xxx.lanl.gov)

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Last modified: February 28, 2001

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