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26 1. A gentle introduction to moduli spaces of curvesrelatively straightforward to verify that they satisfy the relation [V m , V n ] = (m − n)V m+n for allm and n. One can state Witten’s conjecture in terms of the Virasoro operators in a rather succinctfashion.Theorem 1.21 (Witten’s conjecture — Virasoro version). For every integer n ≥ −1,V n (exp F) = 0.Witten originally provided evidence for his conjecture in the form of the string and dilatonequations as well as low genus results [57]. In particular, the string and dilaton equations correspondprecisely to the annihilation of exp F by the operators V −1 and V 0 , respectively. Sincethen, several proofs of Witten’s conjecture have emerged, three of which we briefly discuss here.Kontsevich [26]The year after Witten announced his conjecture, Kontsevich produced a proof as part of hisdoctoral thesis. He used a cell decomposition of M g,n × R n + arising from results concerningquadratic differentials on a Riemann surface known as Jenkins–Strebel differentials.This allowed him to deduce a combinatorial formula which equates a generating functionfor the psi-class intersection numbers with a particular enumeration of combinatorialobjects known as trivalent ribbon graphs. Kontsevich carried out this enumeration usingFeynman diagram techniques and a certain matrix model, from which Witten’s conjecturefollowed. His proof will be discussed in greater detail in Section 3.1.Okounkov and Pandharipande [44]The main tool in their proof of Witten’s conjecture was the ELSV formula. Originallyproven by Ekedahl, Lando, Shapiro and Vainshtein [11], this formula relates intersectionnumbers of psi-classes and Hodge classes — also known as Hodge integrals — withHurwitz numbers. Hurwitz numbers enumerate topological types of branched covers ofthe sphere or, equivalently, factorisations of permutations into transpositions. Okounkovand Pandharipande reproduced the ELSV formula using a technique known as virtuallocalisation and proceeded to show that Kontsevich’s combinatorial formula was a consequence,using asymptotic combinatorial methods.Mirzakhani [33, 34]More recently, Mirzakhani has produced a proof of Witten’s conjecture quite distinct fromthose before her. In particular, she adopts a hyperbolic geometric approach and considersthe symplectic geometry of moduli spaces of hyperbolic surfaces. We will have a lot moreto say about Mirzakhani’s work in Section 1.4.We point out that there are also other proofs — such as that by Kazarian and Lando [24] or byKim and Liu [25] — but these have a lesser bearing on the work contained in this thesis.
1.3. Moduli spaces of hyperbolic surfaces 271.3 Moduli spaces of hyperbolic surfacesThe uniformisation theoremOne of the most important foundational results in algebraic geometry asserts that the categoryof irreducible projective algebraic curves and the category of compact Riemann surfaces areequivalent. Due to this equivalence, the boundary between these two fields is rather porous,with techniques from complex analysis flowing into algebraic geometry and vice versa. Inaddition, the following theorem allows us to adopt a geometric viewpoint when dealing withalgebraic curves or Riemann surfaces.Theorem 1.22 (The uniformisation theorem). Every metric on a surface is conformally equivalent toa complete constant curvature metric. Furthermore, the sign of the curvature is equal to the sign of theEuler characteristic of the surface.From the previous discussion, a smooth genus g algebraic curve with n marked points correspondsto a genus g Riemann surface with n marked points, which we think of as punctures.{} {smooth algebraic curves with←→genus g and n marked points}Riemann surfaces withgenus g and n puncturesThe complex structure defines a conformal class of metrics which, by the uniformisation theorem,contains a hyperbolic metric when 2g − 2 + n > 0. Furthermore, if we demand that the resultingsurface has finite area, then this hyperbolic metric is unique and endows each puncturewith the structure of a hyperbolic cusp. So we have the following one-to-one correspondence.{} {}smooth algebraic curves withhyperbolic surfaces with←→genus g and n marked pointsgenus g and n cuspsModuli spaces of hyperbolic surfaces can be given a natural topology. The correspondencedescribed above defines a map from the moduli space of smooth algebraic curves to the modulispace of hyperbolic surfaces which respects not only this topology, but also the structurepreservingautomorphism group of the surface. In short, the map is a homeomorphism of orbifolds.Therefore, we can and will use the notation M g,n to denote the moduli space of smoothgenus g curves with n marked points as well as the moduli space of genus g hyperbolic surfaceswith n cusps — the particular meaning should be clear from the context.If the finite area condition is relaxed, then there exist hyperbolic metrics which endow eachpuncture with the structure of a hyperbolic flare. Each such flare has a unique geodesic waistcurve which, after being cut along, leaves a compact hyperbolic surface with geodesic boundaries.Furthermore, the only moduli of the removed flare is the length of the geodesic waist
Page 1: Intersection <stro
Page 4 and 5: iiDeclarationThis is to certify tha
Page 6 and 7: ivAcknowledgementsFirst and foremos
Page 8 and 9: viContentsVolumes and symplectic re
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Page 12 and 13: 2 0. Overviewwhich is far more trac
Page 14 and 15: 4 0. Overviewcurves. In this thesis
Page 16 and 17: 6 0. Overvieware defined to be grap
Page 18 and 19: 8 0. OverviewNote that the space of
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3.2. Hyperbolic surfaces and ribbon
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3.3. Hyperbolic surfaces with long
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3.4. Calculating the combinatorial
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Chapter 4Concluding remarksIn this
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Bibliography[1] ABIKOFF, W. The rea
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Bibliography 103[27] KRONHEIMER, P.
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Bibliography 105[58] WOLPERT, S. On
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Appendix AWeil-Petersson volumesA.1
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A.2. Program for genus 0 Weil-Peter
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