Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces
By Maryam Mirzakhani. Inventiones Mathematicae, 2006
489 citations recorded by the source corpus.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2b22b53298ac486d7b8b120aa4d06200b88eba63
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In Applied Mathematics
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Citations in the corpus
- Cross ratios and identities for higher Teichmüller-Thurston theory (2006, methodology): “Mirzakhani gives in [19] a recursive formula for the volumes of the moduli space of hyperbolic structures identified with the quotient of Teichmüller space by the mapping class group.”
- Towards large genus asymptotics of intersection numbers on moduli spaces of curves (2011, methodology): “n as follows ([DN]): ∂Vg,n+1 ∂Ln+1 (L 1,...,Ln,2π √ −1) = 2π √ −1(2g−2+n)Vg,n(L 1,...,Ln). When n= 0, Vg,1(2π √ −1) = 0, and ∂Vg,1 ∂L (2π √ −1) = 2π √ −1(2g−2)Vg,0. (2.9) 7 • For a proof of (III) see [M2]; note that (III) applies only when n>0 (in case of n= 0,formula (2.9) gives the necessary es”
- Hurwitz numbers, matrix models and enumerative geometry (2007, methodology): “d in [EO] in the context of matrix models. This procedure has already found applications in two-dimensional topological gravity, and it has been shown in [EO1,E] that Mirzakhani’s recursion relations [Mir] can be regarded as particular cases of the general formalism of [EO]. The second part of the p”
- Generalized Markoff maps and McShane's identity (2005, background): “B,−BAas defined by (4) in §2. We omit the proof here. A detailed proof by cross-ratiocalculations can be found in [35]. For another proof, one may adapt Mirzakhani’s proof in the real case as given in [22] to let it work for the general case here by using Fenchel’s cosine rule for oriented right angl”
- Polynomial recursion formula for linear Hodge integrals (2009, background): “Each part µi determines a local description of the map f , which is given by z 7−→ zµi in terms of a local coordinate z of X around pi.”
- Low-dimensional de Sitter quantum gravity (2019, background): “hey are determined via the Eynard-Orantin topological recursion relations [54{56], which for the double-scaled density of states above are related to the Mirzakhani recursion relation for the V g;n’s [49,50]. We refer the reader to the nice summary of both in Section 2 of [21]. Indeed, the genus exp”
- Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves (2019, methodology): “lynomial Ng,n(b1,...,bn) coincides with the top homogeneous part of Mirzakhani’s volume polynomial Vg,n(b1,...,bn) providing the Weil–Petersson volume of the moduli space of bordered Riemann surfaces [Mi1]. Note that the classical Weil– Peterssonvolume of Mg,n correspondsto the constant term of Vg,n”
- Growth of the number of simple closed geodesics on hyperbolic surfaces (2008, methodology): “In [ Mirz2 ], by using an identity for the lengths of simple closed geodesics on hyperbolic surfaces and using Theorem 4:1, we obtain a recursive method for calculating volume polynomials.”
- Ergodic Theory of the Earthquake Flow (2010, methodology): “Using the results of this paper, we prove an equidistribution result for the level sets of the lengths of simple closed curves in the moduli space [25].”
- Lengths of closed geodesics on random surfaces of large genus (2017, methodology): “called the Weil-Petersson symplectic form [Gol84, Wol83], that descends to Mg,n(L). It turns out that the Weil-Petersson volume Vg,n(L) of Mg,n(L) is a polynomial in L2 1,...,L 2 n of degree 3g+ n− 3 [Mir07a]. Moreover, the coefficients of these polynomials can be computed in terms of integrals of the”