The Witten-Kontsevich Theorem

openThe Witten-Kontsevich theorem relates intersection products of certain cohomology classes in the tautological ring of the moduli space of stable curves, to the KdV hierarchy of partial differential equations. In this thesis, a recent proof of this theorem is presented. Firstly, the ELSV formula relates such intersection products to simple Hurwitz numbers, which count branched covers of algebraic curves. Subsequently, the link between Hurwitz theory and integrable systems is made via the Sato Grassmannian construction for the KP hierarchy.The Witten-Kontsevich theorem relates intersection products of certain cohomology classes in the tautological ring of the moduli space of stable curves, to the KdV hierarchy of partial differential equations. In this thesis, a recent proof of this theorem is presented. Firstly, the ELSV formula relates such intersection products to simple Hurwitz numbers, which count branched covers of algebraic curves. Subsequently, the link between Hurwitz theory and integrable systems is made via the Sato Grassmannian construction for the KP hierarchy