Intersection theory on moduli spaces of curves ... - User Web Pages

iAbstractThis thesis explores the intersection <strong>theory</strong> on M g,n , the moduli space of genus g stable curveswith n marked points. Our approach will be via hyperbolic geometry and our starting pointwill be the recent work of Mirzakhani.One of the landmark results concerning the intersection <strong>theory</strong> on M g,n is Witten’s conjecture.Kontsevich was the first to provide a proof, the crux of which is a formula involving combinatorialobjects known as ribbon graphs. A subsequent proof, due to Mirzakhani, arises fromconsidering M g,n (L), the moduli space of genus g hyperbolic surfaces with n marked geodesicboundaries whose lengths are prescribed by L = (L 1 , L 2 , . . . , L n ). Through the Weil–Peterssonsymplectic structure on this space, one can associate to it a volume V g,n (L). Mirzakhani produceda recursion which can be used to effectively calculate these volumes. Furthermore, sheproved that V g,n (L) is a polynomial whose coefficients store intersection numbers on M g,n .Her work allows us to adopt the philosophy that any meaningful statement about the volumeV g,n (L) gives a meaningful statement about the intersection <strong>theory</strong> on M g,n , and vice versa.Two new results, known as the generalised string and dilaton equations, are introduced inthis thesis. These take the form of relations between the Weil–Petersson volumes V g,n (L) andV g,n+1 (L, L n+1 ). Two distinct proofs are supplied — one arising from algebraic geometry andthe other from Mirzakhani’s recursion. However, the particular form of the generalised stringand dilaton equations is highly suggestive of a third proof, using the geometry of hyperboliccone surfaces. We briefly discuss ideas related to such an approach, although this largely remainswork in progress. Applications of these relations include fast, effective algorithms tocalculate the Weil–Petersson volumes in genus 0 and 1. We also deduce a formula for the volumeV g,0 , a case not dealt with by Mirzakhani.In this thesis, we also give a new proof of Kontsevich’s combinatorial formula, relating the intersection<strong>theory</strong> on M g,n to the combinatorics of ribbon graphs. Mirzakhani’s theorem suggeststhat the asymptotics of V g,n (L) store valuable information. We demonstrate that this informationis precisely Kontsevich’s combinatorial formula. Our proof involves using hyperbolic geometryto develop a combinatorial model for M g,n (L) and to analyse the asymptotic behaviourof the Weil–Petersson symplectic form. The key geometric intuition involved is the fact that, asthe boundary lengths of a hyperbolic surface approach infinity, the surface resembles a ribbongraph after appropriate rescaling of the metric. This work draws together Kontsevich’s combinatorialapproach and Mirzakhani’s hyperbolic approach to Witten’s conjecture into a coherentnarrative.