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 R2. 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.
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Last modified: February 28, 2001
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