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Completed cycles and singularities of stable maps

SPEAKER: Dimitri Zvonkine

TITLE: Completed cycles and singularities of stable maps

ABSTRACT For every family of genus 0 stable maps to CP^1 we can consider the locus of points where the map presents some given kind of singularity (chosen from the list of all possible singularities for genus 0 stable maps). We provide an effective method to compute the cohomology class PoincarĂ© dual to this locus in terms of several ”basic” classes. These expressions are universal, i.e., they do not depend on the family. They are called (generalized) Thom polynomials. In the second part of the talk, as time permits, we will present several conjectures concerning Thom polynomials for stable maps of arbitrary genus. They are related to the representation theory of the symmetric group (more precisely, to the so-called completed cycles), to the Gromov-Witten invariants of curves and to a conjectural ELSV formula for the space of r-spin structures.