Mathematics Colloquia and Seminars

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Start with a two-variable complex polynomial f with an isolated critical point at the origin. One can associate to it a smooth Riemann surface with boundary: the Milnor fiber of f, M_f, given by a smoothing of f near the origin. This comes equipped with a distinguished collection of S^1s on M_f, called vanishing cycles. We explain how an algorithm of A'Campo allows one to encode all of this information in a planar graph. We explore consequences for the complement of the discriminant locus for f, and (symplectic) mapping class groups associated to it. No prior knowledge of singularity theory or symplectic geometry will be assumed.