Seminar - Steffen Marcus

On the Piecewise Polynomality of Double Hurwitz Numbers

Abstract: 

Hurwitz numbers count degree d branched covers of the Riemann sphere by a genus g Riemann surface with prescribed ramification over one branch point and simple ramification over the others. They are intimately related to the geometry of the moduli space of curves through the famous ELSV formula. Double Hurwitz numbers similarly count covers with prescribed ramification over two points. In this talk I'm going to explain how we can describe double Hurwitz numbers as intersection numbers on the moduli space of curves using the geometry of the moduli space of relative stable maps. This helps explain geometrically the chamber/wall-crossing piecewise polynomial structure of double Hurwitz numbers. This is joint work with Renzo Cavalieri.