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8 0. OverviewNote that the space of metric ribbon graphs MRG Γ (x) possesses a tractable system ofcoordinates, provided by the edge lengths. In particular, with respect to these coordinates,Ω is a constant 2-form.Furthermore, MRG Γ (x) has a simple description — it is thequotient of a polytope by the action of the finite group Aut Γ. Therefore, modulo someanalytical details, we have the equalitylimN→∞V Γ (Nx)N 6g−6+2n = ∫MRG Γ (x)Ω 3g−3+n(3g − 3 + n)! .From the previous discussion, this is essentially the integral of a constant volume formover a polytope. The integration is most easily evaluated after taking the Laplace transform,which results in the desired product over edges of Γ. Furthermore, the action of the1finite group Aut Γ on this polytope naturally introduces the factor of which appears|Aut Γ|on the right hand side of Kontsevich’s combinatorial formula.Part 3: Where does the combinatorial constant come from?Interestingly, the remaining factor of 2 2g−2+n on the right hand side of Kontsevich’s combinatorialformula is no simple matter to explain. In fact, its appearance boils down to thefollowing statement.Theorem 3.22. Let Γ be a trivalent ribbon graph of type (g, n) with n edges coloured white and theremaining 6g − 6 + 2n edges coloured black. Let A be the n × n adjacency matrix formed betweenthe faces and the white edges. Let B be the (6g − 6 + 2n) × (6g − 6 + 2n) oriented adjacencymatrix formed between the black edges. Thendet B = 2 2g−2 (det A) 2 .Perhaps surprisingly, there does not exist a purely combinatorial proof of this statementin the literature. Instead, we essentially follow the argument from Appendix C of Kontsevich’spaper [26], which uses the torsion of an acyclic chain complex associated to atrivalent ribbon graph. However, our exposition is greatly expanded to make the proofboth more thorough and more elementary.Of course, Kontsevich’s combinatorial formula in itself is not a new result. What is novel, in thispart of the thesis, is the hyperbolic geometric approach and the explicit connection between thework of Kontsevich and Mirzakhani. We believe that this proof of Kontsevich’s combinatorialformula is rather intuitive in nature, avoids the technical difficulties which are inherent in theoriginal proof, and may lead to further insights. Indeed, as a part of joint work with Safnuk [10],we have extended these ideas to produce a recursive formula à la Mirzakhani which computesthe asymptotics of V g,n (L). The differential version of this formula is the Virasoro constraintcondition, thereby providing a new path to Witten’s conjecture. We also believe that it will notbe difficult to extend these ideas to integration over the combinatorially defined Witten cycles.
Chapter 1A gentle introduction to modulispaces of curvesIn this chapter, we introduce the main characters of our story, moduli spaces of curves. The aimis to provide a concise exposition of the important results and ideas which form the backgroundto this thesis. Newcomers to the area will hopefully find this chapter a suitable point of entry tothe now vast body of knowledge concerning moduli spaces of curves. However, the selectionof material presented here is necessarily only a small subset, chosen to suit our specific needsand reflect our particular point of view. For example, a great deal of attention has been paidto intersection <strong>theory</strong> on moduli spaces of curves, to the interaction between algebraic andhyperbolic geometry, and to the recent results of Mirzakhani. Throughout the chapter, detailsand proofs have often been omitted for the sake of clarity and space. For those interested infurther information, there are numerous references to the relevant sources in the literature. 11.1 Moduli spaces of curvesFirst principlesInformally, the points of a moduli space classify objects of a certain type, while its geometryreflects the way in which these objects can vary in families. For example, consider the modulispaceM g = {C | C is a smooth algebraic curve of genus g} / ∼1 In particular, we start by mentioning the articles [43, 55, 56] which have influenced our exposition and which aresuitable for those wishing to discover this remarkable area of mathematics for the first time.9
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
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2.4. Applications and extensions 59
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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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