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

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viContentsVolumes and symplectic reduction . . . . . . . . . . . . . . . . . . . . . . . . 362 Weil–Petersson volume relations and hyperbolic cone surfaces 412.1 Volume polynomial relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Generalised string and dilaton equations . . . . . . . . . . . . . . . . . . . . . 41Three viewpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2 Proofs via algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Pull-back relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Proof of the generalised string equation . . . . . . . . . . . . . . . . . . . . . 46Proof of the generalised dilaton equation . . . . . . . . . . . . . . . . . . . . . 472.3 Proofs via Mirzakhani’s recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Proof of the generalised dilaton equation . . . . . . . . . . . . . . . . . . . . . 50Proof of the generalised string equation . . . . . . . . . . . . . . . . . . . . . 532.4 Applications and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Small genus Weil–Petersson volumes . . . . . . . . . . . . . . . . . . . . . . . 58Closed hyperbolic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Further volume polynomial relations . . . . . . . . . . . . . . . . . . . . . . . 61Hyperbolic cone surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 A new approach to Kontsevich’s combinatorial formula 653.1 Kontsevich’s combinatorial formula . . . . . . . . . . . . . . . . . . . . . . . . . . 66Ribbon graphs and intersection numbers . . . . . . . . . . . . . . . . . . . . . 66From Witten to Kontsevich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68The proof of Kontsevich’s combinatorial formula: Part 0 . . . . . . . . . . . . 703.2 Hyperbolic surfaces and ribbon graphs . . . . . . . . . . . . . . . . . . . . . . . . 72Combinatorial moduli space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72The proof of Kontsevich’s combinatorial formula: Part 1 . . . . . . . . . . . . 763.3 Hyperbolic surfaces with long boundaries . . . . . . . . . . . . . . . . . . . . . . . 78
ContentsviiPreliminary geometric lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Asymptotic behaviour of the Weil–Petersson form . . . . . . . . . . . . . . . 81The proof of Kontsevich’s combinatorial formula: Part 2 . . . . . . . . . . . . 853.4 Calculating the combinatorial constant . . . . . . . . . . . . . . . . . . . . . . . . . 88Chain complexes associated to a ribbon graph . . . . . . . . . . . . . . . . . . 88Torsion of an acyclic complex . . . . . . . . . . . . . . . . . . . . . . . . . . . 92The proof of Kontsevich’s combinatorial formula: Part 3 . . . . . . . . . . . . 964 Concluding remarks 99Bibliography 101A Weil–Petersson volumes 107A.1 Table of Weil–Petersson volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107A.2 Program for genus 0 Weil–Petersson volumes . . . . . . . . . . . . . . . . . . . . . 108
Page 1: Intersection <stro
Page 4 and 5: iiDeclarationThis is to certify tha
Page 6 and 7: ivAcknowledgementsFirst and foremos
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 and 50: 1.4. Volumes of moduli spaces 39Pro
Page 51 and 52: Chapter 2Weil-Petersson volume rela
Page 53 and 54: 2.1. Volume polynomial relations 43
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2.3. Proofs via Mirzakhani’s recu
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Chapter 3A new approach to Kontsevi
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3.1. Kontsevich’s combinatorial f
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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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