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40 1. A gentle introduction to moduli spaces of curves
Chapter 2Weil–Petersson volume relations andhyperbolic cone surfacesIn this chapter, we prove the generalised string and dilaton equations, which take the form ofrelations between Weil–Petersson volumes. Two distinct proofs are supplied — one arising fromalgebraic geometry and the other from Mirzakhani’s recursion. One striking feature of theseresults is that they are highly suggestive of a third proof, involving the geometry of hyperboliccone surfaces. As applications of the generalised string and dilaton equations, we show howto efficiently calculate genus 0 and genus 1 Weil–Petersson volumes and give a formula for thevolume V g,0 , a case not dealt with by Mirzakhani’s recursion. The chapter concludes with someideas on how this work may be extended.2.1 Volume polynomial relationsGeneralised string and dilaton equationsMirzakhani’s theorem — Theorem 1.35 — asserts that the Weil–Petersson volumes are polynomialswhose coefficients store interesting information. In particular, any meaningful statementabout V g,n (L) gives a meaningful statement about the intersection <strong>theory</strong> on M g,n , and viceversa. Mirzakhani’s recursion — Theorem 1.29 — provides an effective algorithm to calculatethese Weil–Petersson volumes. Therefore, it is possible to adopt a numerological approach andsimply search for interesting patterns among the volume polynomials. These, in turn, shouldtranslate into relations concerning the intersection <strong>theory</strong> on moduli spaces of curves. The hopeis that packaging the intersection numbers on M g,n into the polynomial V g,n (L) may revealuseful information which is otherwise obscured.41
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
Page 10 and 11: viiiContents
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
Page 20 and 21: 10 1. A gentle introduction to modu
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Page 39 and 40: 1.3. Moduli spaces of hyperbolic su
Page 41 and 42: 1.4. Volumes of moduli spaces 31The
Page 43 and 44: 1.4. Volumes of moduli spaces 33vol
Page 45 and 46: 1.4. Volumes of moduli spaces 35In
Page 47 and 48: 1.4. Volumes of moduli spaces 37reg
Page 49: 1.4. Volumes of moduli spaces 39Pro
Page 53 and 54: 2.1. Volume polynomial relations 43
Page 55 and 56: 2.2. Proofs via algebraic geometry
Page 57 and 58: 2.2. Proofs via algebraic geometry
Page 59 and 60: 2.3. Proofs via Mirzakhani’s recu
Page 69 and 70: 2.4. Applications and extensions 59
Page 75 and 76: Chapter 3A new approach to Kontsevi
Page 77 and 78: 3.1. Kontsevich’s combinatorial f
Page 83 and 84: 3.2. Hyperbolic surfaces and ribbon
Page 89 and 90: 3.3. Hyperbolic surfaces with long
Page 99 and 100: 3.4. Calculating the combinatorial
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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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