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Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

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Pith's one-line read The Masur–Veech volume and the area Siegel–Veech constant of the principal stratum are explicit polynomials in psi-class intersection numbers, summed over stable graphs.

desk verdict A substantial, careful paper that delivers explicit formulas for Masur-Veech volumes and a new bridge to hyperbolic counting; the main caveat is a normalization constant left to future work. read the letter →

arxiv 1908.08611 v1 pith:STGYVZ53 submitted 2019-08-22 math.GT math.AGmath.CO

Vincent Delecroix , Elise Goujard , Peter Zograf , Anton Zorich This is my paper · ORCID
classification math.GTmath.AGmath.CO MSC 14H1032G1530F30
keywords Masur-Veechvolumequadraticdifferentialssquare-tiledsurfacesintersectionnumberspsi-classesstablegraphssimpleclosedgeodesicslargegenusasymptotics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes an exact formula for the Masur–Veech volume of the moduli space of meromorphic quadratic differentials with simple poles: the volume is a sum over stable graphs of contributions built from the multivariate volume polynomials that also count trivalent ribbon graphs and appear in Weil–Petersson volume recursions. Each graph contributes a rational multiple of a power of $\pi$, obtained from a lattice-point count that turns sums over square-tiled surfaces into zeta values. The same machinery gives a graph-by-graph formula for the area Siegel–Veech constant. A second theorem shows that the flat density of square-tiled surfaces with a fixed cylinder decomposition and the hyperbolic frequency of simple closed geodesic multicurves of a fixed topological type are proportional by an explicit normalization constant. In large genus the paper proves that separating simple closed geodesics are exponentially rarer than nonseparating ones, and, under stated conjectures, describes a random multicurve as a Poisson-distributed collection of about half a logarithm of the genus many curves.

What carries the argument

The central object is the stable graph $\Gamma$ dual to a multicurve: vertices are the components of the surface cut along the curves, edges are the curves, and labeled legs are the poles. To each vertex $v$ the paper attaches the volume polynomial $N_{g_v,n_v}(b_v)$, with coefficients given by psi-class intersection numbers; multiplying over vertices and over edge variables $b_e$ with one combinatorial prefactor gives $P_\Gamma$. The operator $Z$ sends a monomial $\prod_i b_i^{m_i}$ to $\prod_i m_i!\,\zeta(m_i+1)$ and arises as the sum over positive cylinder heights $H_i$ of the operator $Y(H)$, which sends $b_i^{m_i}$ to $m_i!/H_i^{m_i+1}$. The proof counts square-tiled surfaces: a Jenkins–Strebel decomposition into horizontal cylinders is classified by $\Gamma$ and by a height vector $H$, the waist-length parameters obey parity conditions encoded in a sublattice of index $2^{|V(\Gamma)|-1}$, and a lattice-point lemma converts the weighted count into $Z(P_\Gamma)$. This is the mechanism that expresses flat volumes as intersection numbers and, after comparing term by term with the hyperbolic counting formula, fixes the constant in the flat-to-hyperbolic proportionality.

What would settle it

Evaluate the proportionality constant between the period-coordinate volume element used in Section 2.1 and the symplectic Masur–Veech volume element on one stratum such as $Q_{2,0}$; if it is not the value implicitly fixed by the cited conventions, every numerical volume in Theorem 1.6 is scaled by a $(g,n)$-dependent factor. Independently, compute the flat square-tiled count for a single multicurve class in $Q_{2,0}$ and compare it with the right-hand side of (1.30); a systematic mismatch would falsify the stated constant.

Watch

Extended reading notes

Core claim

The paper's central discovery is that the Masur–Veech volume of the moduli space $Q_{g,n}$ of meromorphic quadratic differentials with $n$ simple poles and no other poles is a finite sum over stable graphs, $\mathrm{Vol}\,Q_{g,n}=\sum_{\Gamma\in\mathcal G_{g,n}} Z(P_\Gamma)$. Each $P_\Gamma$ is an explicit polynomial built from the volume polynomials $N_{g_v,n_v}$ attached to the vertices of $\Gamma$, and the operator $Z$ sends $b^m$ to $m!\,\zeta(m+1)$. The same stable-graph machinery gives a formula for the area Siegel–Veech constant. The paper further proves that the volume contribution of square-tiled surfaces with horizontal cylinder decomposition of type $\gamma$ equals the hyperbolic frequency $c(\gamma)$ multiplied by an explicit constant depending only on $g$ and $n$, so the flat and hyperbolic counts carry the same information. All of these formulas are stated in the period-coordinate normalization of the volume element, and the paper explicitly postpones the comparison factor with the standard symplectic volume.

Load-bearing premise

The load-bearing premise is that the volume element fixed by the period-coordinate lattice in Section 2.1 is the Masur–Veech volume element up to a constant that the paper postpones to a later paper; all explicit numbers in Theorem 1.6, such as $\mathrm{Vol}\,Q_2=\pi^6/15$, are stated in that normalization.

Editorial extensions

If this is right

  • For every admissible $(g,n)$, the volume $\mathrm{Vol}\,Q_{g,n}$ and the area Siegel–Veech constant become finite explicit expressions in psi-class intersection numbers, so values such as $\mathrm{Vol}\,Q_2=\pi^6/15$ and $\mathrm{Vol}\,Q_3=115/33264\,\pi^{12}$ are produced by the formula rather than by a separate dynamical computation.
  • The flat and hyperbolic counts are equivalent: the density of square-tiled surfaces with cylinder type $\gamma$ equals the hyperbolic frequency $c(\gamma)$ up to the explicit factor in (1.31), and the averaged Thurston measure $b_{g,n}$ is obtained from the flat volume by formula (1.32).
  • In genus zero, the comparison yields $b_{0,n}=(\pi/2)^{2(n-3)}/(n-3)!$, and Stirling's formula gives the large-$n$ asymptotic $b_{0,n}\sim (\pi^2 e/4n)^{n-3}/\sqrt{2\pi n}$.
  • For large genus, the one-cylinder contribution to $\mathrm{Vol}\,Q_g$ is asymptotic to $\sqrt{2}/(3\pi g)\,(8/3)^{4g-4}$, and the ratio of separating to nonseparating simple closed geodesic frequencies is asymptotic to $\sqrt{2}/(3\pi g)\,4^{-g}$.
  • Under the paper's conjectures, the number of cylinders of a random square-tiled surface or integral multicurve in large genus converges in total variation to a Poisson distribution with parameter $(\log(6g-6)+\gamma)/2+\log 2-1$, and the probability that all cylinder heights are one tends to $\sqrt{2}/2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because Theorem 1.6 expresses the volume as a finite sum over stable graphs, it gives a direct algorithmic route to $\mathrm{Vol}\,Q_{g,n}$ using only the string and dilaton equations for psi-class intersection numbers; the authors do not emphasize this computational corollary, but it follows immediately.
  • The flat–hyperbolic proportionality in Theorem 1.20 suggests that the same lattice-counting dictionary should hold for strata of quadratic differentials whose zeros all have odd degree, where the paper notes one must subtract the contribution of squares of Abelian differentials; testing this in genus two would show whether the mechanism is special to the principal stratum.
  • The Poisson approximation with parameter $(\log(6g-6)+\gamma)/2+\log 2-1$ offers a concrete null model: a random reduced multicurve in large genus should look like a random subset of about $(1/2)\log g$ curves, with all weights one with probability $\sqrt{2}/2$; this is testable numerically before the analytic conjectures are settled.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper develops an explicit formula for the Masur-Veech volume and the area Siegel-Veech constant of the moduli space Qg,n of meromorphic quadratic differentials with n simple poles as a sum over stable graphs of expressions involving Kontsevich's volume polynomials and zeta values. The proof is based on counting square-tiled surfaces and on a lattice-point lemma. The paper also proves that the volume contribution of a multicurve equals Mirzakhani's frequency c(γ) up to an explicit factor, and it studies small-genus and large-genus consequences, including the statement that separating simple closed geodesics are exponentially rarer than non-separating ones in large genus. Several large-scale statements are conditional on conjectural asymptotics of intersection numbers and of multiple harmonic sums.

Significance. If the normalization issue were resolved, the main formula would be a very useful explicit intersection-theoretic expression for Masur-Veech volumes and Siegel-Veech constants, with direct applications to counting problems and to statistics of square-tiled surfaces. The paper carefully checks its formula against previously known values in genus 2 and 3, and it provides a bridge between flat and hyperbolic counting via Theorem 1.20 and Corollary 1.22. The large-genus result on the relative frequency of separating curves is also valuable. The paper is ambitious and contains a large amount of correct and interesting computation, but its central claim is currently stated in a normalization that is not identified with the standard symplectic volume element.

major comments (2)
  1. [Section 2.1, after Eq. (1.4) and Theorem 1.6] The paper explicitly postpones the evaluation of the proportionality constant between the symplectic Masur-Veech volume element dVol_symplectic and the period-coordinate volume element dVol_period: 'We postpone evaluation of this constant factor to another paper. Throughout this paper we consider the normalization dVol = dVol_period.' All numerical content of the paper, including Theorem 1.6, Tables 1-5, and Corollary 1.22, is stated in the dVol_period normalization. Since Section 1.1 initially defines the Masur-Veech volume element as the one induced by the canonical symplectic structure, the paper does not, as written, establish the numerical values of the standard symplectic Masur-Veech volumes. The central theorem is internally consistent, but the unqualified identification of the computed quantities with the standard Masur-Veech volumes is load-bearing and needs to be fixed, either by computing the constant or by clearly and consistently qualifying all statements as period-normalized volumes.
  2. [Section 4.5, Lemma 4.13] The asymptotic evaluation of the hypergeometric sum S(g) is argued by saying that the normalized distributions of the binomial coefficients tend to normal distributions and that the product of two normal distributions is a normal distribution. This is not a rigorous derivation of (4.41); the product of two binomial coefficients is not jointly normal in the sense claimed, and the argument does not supply the necessary uniform error bounds. Since (4.41) is used in Proposition 4.12 and then in Theorem 1.27, this is a load-bearing step for the large-genus claim. A rigorous proof via Stirling's formula with uniform errors, or a precise citation to a standard local limit theorem with error rates, should be supplied.
minor comments (6)
  1. [Equation (1.31)] The displayed formula for constg,n ends with '24g−3+n·' and the expression is incomplete; the missing factor should be supplied.
  2. [Table 1] The table is difficult to read because the stable graphs are represented with boxes and the columns run together; a larger figure or a separate listing of the graphs would improve clarity.
  3. [Appendix D, Guess D.4] The item labeled 'Guess D.4' is a mathematical conjecture and should be called 'Conjecture' for consistency with the rest of the paper.
  4. [Section 4.2, proof of Lemma 4.2] The proof refers to 'bounds (4.3)' where the lemma states bounds (4.4); this cross-reference should be corrected.
  5. [Remark 1.23] The sentence 'However, Mirzakhani but does not give any close formula for the value of the normalization constant' contains a grammatical error and should read 'Mirzakhani does not give any closed formula for the value of the normalization constant.'
  6. [Section 2.1, discussion of square-tiled surfaces] The sentence 'The integer points in Qg,n are exactly those quadratic differentials for which the associated flat surface with the metric |q| can be tiled with 1/2 × 1/2 squares' is clear, but the subsequent link to the normalization of the lattice would benefit from a short example to make the factor of 2 in (1.4) transparent.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the volume formula is an independent lattice-point computation; self-citations are non-load-bearing, and the postponed period-normalization constant is a caveat, not a circular step.

full rationale

The central derivation of Theorem 1.6 is self-contained: the Masur–Veech volume is computed by counting square-tiled surfaces in period coordinates, and the resulting lattice-point sums are evaluated using Kontsevich's ribbon-graph theorem together with the elementary limit in Lemma 2.3. The output is an explicit polynomial in psi-class intersection numbers, not a restatement of the input count: the stable-graph decomposition and the operators Y and Z are used to evaluate the leading term of the count, not to define the volume as the formula being proved. The comparison with Mirzakhani's hyperbolic frequencies in Theorem 1.20 is also a genuine equivalence proof: it algebraically identifies the independently derived flat count with Mirzakhani's independent formula and evaluates the proportionality constant, rather than assuming the equality. The paper does contain self-citations, notably [DGZZ2] for the existence of the limit in (1.4)–(1.5) and [Zog] for the 2-correlator recurrence used in the large-genus asymptotics. These are prior results used as inputs, and neither reduces to the target theorem or to a fitted parameter. The explicit postponement in Section 2.1 of the proportionality constant between the canonical symplectic volume element and the period-coordinate volume element is a genuine normalization caveat: all numerical values in the paper are stated for Vol = Vol_period, and if that constant is not the one implicitly assumed by the cited conventions, the numbers would differ from the standard symplectic normalization by a (g,n)-dependent factor. This is a limitation on interpretation of the central claim, not a circular derivation, because the paper is internally consistent in using the period-coordinate normalization throughout. No circular step was found, and the score is low because the main theorem is independently verified by agreement with prior computations, e.g., Vol Q_2 = pi^6/15 matching [G2], and with Mirzakhani's independent results.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central formula depends on standard results (Kontsevich's theorem, string equation, Mirzakhani's frequency theorem, Veech's Siegel-Veech formula) and on the uncomputed proportionality constant between volume elements. No free parameters are fitted and no new entities are introduced.

assumptions (6)
  • standard math Kontsevich's theorem: the weighted count of trivalent metric ribbon graphs with integer edge lengths is Ng,n(b) up to lower order terms.
    Used to identify ribbon graph counts with intersection number polynomials in Section 1.3 and Section 2.4. Proved by Kontsevich (Witten's conjecture).
  • standard math String equation for psi-class intersection numbers.
    Used in the proof of Proposition 2.5 to extend Kontsevich's count to ribbon graphs with leaves (univalent vertices).
  • domain assumption Mirzakhani's theorem giving the asymptotic frequency c(gamma) of simple closed geodesic multicurves, as in Theorem 1.2 of [Mi3].
    Used in Section 3 as the hyperbolic counterpart in Theorem 1.20, where flat square-tiled densities are compared to c(gamma).
  • domain assumption Veech and Vorobets' Siegel-Veech formula for area Siegel-Veech constants.
    Used to state and prove Theorem 1.16 and to relate the Siegel-Veech constant to flat geometry.
  • domain assumption Proportionality of the symplectic and period-coordinate volume elements on Qg,n.
    The paper uses dVol = dVolperiod and postpones the evaluation of the proportionality constant (Section 2.1). The existence of proportionality is due to Masur and Veech.
  • domain assumption Existence of the limit in (1.4) defining VolQg,n from square-tiled surface counts, based on the lattice structure of period coordinates.
    The limit's existence is assumed from prior work [DGZZ2] and is used to define the volume and the per-graph contributions Vol(Gamma).

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Pith. "Pith review of Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves." pith.science (2026). https://pith.science/paper/STGYVZ53

@misc{pith2026190808611,
  author       = {Pith},
  title        = {Pith review of: Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/STGYVZ53}},
  note         = {Machine review of arXiv:1908.08611}
}
read the original abstract

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi-classes supported on the boundary cycles of the Deligne-Mumford compactification of the moduli space of curves. Our formulae are derived from lattice point count involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli space of bordered hyperbolic Riemann surfaces. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani through completely different approach. We prove further result: up to an explicit normalization factor depending only on the genus and on the number of cusps, the density of the orbit of any simple closed multicurve computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to the simple closed multicurve. We study the resulting densities in more detail in the special case when there are no cusps. In particular, we compute explicitly the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are exponentially less frequent. We conclude with detailed conjectural description of combinatorial geometry of a random simple closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. This description is conditional to the conjectural asymptotic formula for the Masur-Veech volume in large genera and to the conjectural uniform asymptotic formula for certain sums of intersection numbers of psi-classes in large genera.

Figures

Figures reproduced from arXiv: 1908.08611 by the authors.

Figure 1
Figure 1. Square-tiled surface in Q0,7, and associated multic￾urve and stable graph Suppose that the resulting closed square-tiled surface has genus g and n conical singularities with angle π, i.e. n vertices adjacent to only two squares. For example, the square-tiled surfaces in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Simple closed curves on a six-punctured sphere M. Mirzakhani found an explicit expression for the coefficient c(γ) and for the global normalization constant bg,n in terms of the intersection numbers of ψ-classes. Example 1.18. For any hyperbolic metric X on a sphere with 6 cusps as in [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. This ribbon graph G has genus g(G) = 1; it has n(G) = 2 boundary components and p(G) = 2 univalent vertices. with boundary, which has some genus gv. The genus decoration g = (gv)v associates to each v ∈ V the nonnegative integer gv. Any maximal horizontal cylinder of q has two boundary components which are canonically identified with appropriate boundary components of tubular neighborhoods of appropriate singular la… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Graph of f(x) It is curious to note that the sequence {ag,2, ag,5, . . . , ag,3jmax−1}, where jmax is the maximum integer satisfying 3jmax−1 ≤ [PITH_FULL_IMAGE:figures/full_fig_p051_4.png]
Figure 5
Figure 5. Figure 5: Graph Γ1(g) on the left represents non-separating simple closed curves on a surface of genus g (as in the middle) and the boundary divisor δ0,g of irreducible stable curves with single node as on the right. Theorem 4.9. The contribution of the graph Γ1(g) having single…
Figure 6
Figure 6. Figure 6: Graph ∆(g1, g2) represents simple closed curves on a surface of genus g separating the surface into subsurfaces of genera g1 and g2, where g1 + g2 = g (in the middle). The same graph represents the boundary divisor δg1,g ⊂ Mg − Mg for which a generic stable curve has t…
Figure 7
Figure 7. Figure 7: Graph Γk(g) (on the left) representing k pairwise non￾homotopic non-separating simple closed curves on a surface of genus g (on the right). Notation A(g, k)  B(g, k) used in this section for asymptotic inequalities be￾tween positive quantities means that for any ε > 0…
Figure 8
Figure 8. Figure 8: Experimental distribution of contribution of k￾cylinder surfaces (in orange) and the Poisson distribution with λ as in (F.17) (in blue) distribution choosing λ(26) slighly larger. However, the optimal approximation of the experimental data by all possible Poisson distr…
Figure 9
Figure 9. Figure 9: Experimental distribution of frequencies of k-cylinder square-tiled surfaces and the normalized distribution given by ex￾pression (F.2) for Vol Γk(g) in genus g = 26 . The experimental numerical data for frequencies of k-cylinder square-tiled sur￾faces and the normaliz…

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Works this paper leans on

54 extracted references · 51 canonical work pages

  1. [1]

    Large Genus Asymptotics for Volumes of Strata of Abelian Differentials

    A. Aggarwal, Large Genus Asymptotics for Volumes of Strata of Abelian Differentials, arXiv:1804.05431

    work page Pith review arXiv

  2. [2]

    Large Genus Asymptotics for Siegel-Veech Constants

    A. Aggarwal, Large Genus Asymptotics for Siegel--Veech Constants, to appear in GAFA, arXiv:1810.05227

    work page Pith review arXiv

  3. [3]

    J. E. Andersen, G. Borot, S. Charbonniery, V. Delecroix, A. Giacchetto, D. Lewa\'nski, C. Wheeler, Topological recursion for Masur--Veech volumes, arXiv:1905.10352

    arXiv 1905

  4. [4]

    Arbarello, M

    E. Arbarello, M. Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5:4 (1996), 705--749

    work page 1996

  5. [5]

    Athreya, A

    J. Athreya, A. Eskin, and A. Zorich, Counting generalized Jenkins--Strebel differentials, Geometriae Dedicata, 170:1 (2014), 195--217

    work page 2014

  6. [6]

    F. Arana--Herrera, Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics, arXiv:1902.05626

    work page Pith review arXiv 1902

  7. [7]

    Athreya, A

    J. Athreya, A. Eskin, and A. Zorich, Right-angled billiards and volumes of moduli spaces of quadratic differentials on , Ann. Scient. ENS, 4\`eme s\'erie, 49 (2016), 1307--1381

    work page 2016

  8. [8]

    Bonahon, Y

    F. Bonahon, Y. S\"ozen, The Weil--Petersson and Thurston Symplectic forms, Duke Math. J. 108 (2001), no. 3, 581--597

    work page 2001

Show all 54 references
  1. [9]

    Chapman, M

    K. Chapman, M. Mulase, B. Safnuk, The Kontsevich constants for the volume of the moduli of curves and topological recursion, Commun. in Number Theory and Phys. 5 (2011), no. 3, 643--698

    2011

  2. [10]

    Chen, Square-tiled surfaces and rigid curves on moduli spaces, Advances in Mathematics, 228 (2011), 1135--1162

    D. Chen, Square-tiled surfaces and rigid curves on moduli spaces, Advances in Mathematics, 228 (2011), 1135--1162

    2011

  3. [11]

    D. Chen, M. M\"oller, D. Zagier, Quasimodularity and large genus limits of Siegel--Veech constants, J. Amer. Math. Soc. 31 (2018), no. 4, 1059--1163

    2018

  4. [12]

    D. Chen, M. M\"oller, A. Sauvaget, D. Zagier, Masur--Veech volumes and intersection theory on moduli spaces of abelian differentials; arXiv:1901.01785

    1901 arXiv

  5. [13]

    Delecroix, E

    V. Delecroix, E. Goujard, P. Zograf, A. Zorich, Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes of the ambiant strata, arXiv:1612.08374 (2016) 60pp

    2016 arXiv

  6. [14]

    Delecroix, E

    V. Delecroix, E. Goujard, P. Zograf, A. Zorich, Enumeration of meanders and Masur-Veech volumes, arXiv:1705.05190

    arXiv

  7. [15]

    Delecroix, E

    V. Delecroix, E. Goujard, P. Zograf, A. Zorich, with appendix of Ph. Engel, Contribution of one-cylinder square-tiled surfaces to Masur--Veech volumes, arXiv:1903.10904 , to appear in Asterisque

    1903 arXiv

  8. [16]

    NIST Digital Library of Mathematical Functions. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds., http://dlmf.nist.gov/, Release 1.0.22 of 2019-03-15

    2019

  9. [17]

    Engel, Hurwitz Theory of Elliptic Orbifolds, I, arXiv:1706.06738

    Ph. Engel, Hurwitz Theory of Elliptic Orbifolds, I, arXiv:1706.06738

    arXiv

  10. [18]

    Engel, Hurwitz Theory of Elliptic Orbifolds, II, arXiv::1809.07434

    Ph. Engel, Hurwitz Theory of Elliptic Orbifolds, II, arXiv::1809.07434

    arXiv

  11. [19]

    Eskin, H

    A. Eskin, H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory and Dynamical Systems, 21 (2) (2001), 443--478

    2001

  12. [20]

    Eskin, M

    A. Eskin, M. Kontsevich, A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichm\"uller geodesic flow, Publications de l'IHES, 120:1 (2014), 207--333

    2014

  13. [21]

    Eskin, H

    A. Eskin, H. Masur, A. Zorich, Moduli spaces of Abelian differentials: the principal boundary, counting problems and the Siegel--Veech constants, Publications Math\'ematiques de l'IHES, 97 (1) (2003), 61--179

    2003

  14. [22]

    Eskin, M

    A. Eskin, M. Mirzakhani, A. Mohammadi, Effective counting of simple closed geodesics on hyperbolic surfaces, arXiv:1905.04435

    1905 arXiv

  15. [23]

    Eskin, A

    A. Eskin, A. Okounkov. Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001), no. 1, 59--103

    2001

  16. [24]

    Eskin, A

    A. Eskin, A. Okounkov. Pillowcases and quasimodular forms, Algebraic Geometry and Number Theory, Progress in Mathematics 253 (2006), 1--25

    2006

  17. [25]

    Eskin, A

    A. Eskin, A. Okounkov, R. Pandharipande, The theta characteristic of a branched covering, Adv. Math., 217 no. 3 (2008), 873--888

    2008

  18. [26]

    Eskin, A

    A. Eskin, A. Zorich, Volumes of strata of Abelian differentials and Siegel--Veech constants in large genera, Arnold Mathematical Journal, 1:4 (2015), 481--488

    2015

  19. [27]

    Flajolet, R

    Ph. Flajolet, R. Sedgewick, Analytic Combinatorics. Cambridge University Press, 2009

    2009

  20. [28]

    Havil, Gamma: Exploring Euler's Constant

    J. Havil, Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003

    2003

  21. [29]

    Goujard, Siegel--Veech constants for strata of moduli spaces of quadratic differentials, GAFA, 25:5 (2015), 1440--1492

    E. Goujard, Siegel--Veech constants for strata of moduli spaces of quadratic differentials, GAFA, 25:5 (2015), 1440--1492

    2015

  22. [30]

    Goujard, Volumes of strata of moduli spaces of quadratic differentials: getting explicit values, Ann

    E. Goujard, Volumes of strata of moduli spaces of quadratic differentials: getting explicit values, Ann. Inst. Fourier, 66 no. 6 (2016), 2203--2251

    2016

  23. [31]

    Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm

    M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1--23

    1992

  24. [32]

    Lagarias, Euler's constant: Euler's work and modern developments Bull

    J. Lagarias, Euler's constant: Euler's work and modern developments Bull. Amer. Math. Soc. 50 (2013), no. 4, 527--628

    2013

  25. [33]

    Lanneau, Parity of the spin structure defined by a quadratic differential, Geometry and Topology 8, (2004) 511--538

    E. Lanneau, Parity of the spin structure defined by a quadratic differential, Geometry and Topology 8, (2004) 511--538

    2004

  26. [34]

    Masur, Interval exchange transformations and measured foliations, Ann

    H. Masur, Interval exchange transformations and measured foliations, Ann. of Math., 115 (1982), 169--200

    1982

  27. [35]

    Masur, The Teichm\"uller Flow is Hamiltonian, Proceedings of the AMS 123, No

    H. Masur, The Teichm\"uller Flow is Hamiltonian, Proceedings of the AMS 123, No. 12 (1995), 3739--3747

    1995

  28. [36]

    Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent

    M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007), no. 1, 179--222

    2007

  29. [37]

    Mirzakhani, Weil--Petersson volumes and intersection theory on the moduli space of curves, Journal of Amer

    M. Mirzakhani, Weil--Petersson volumes and intersection theory on the moduli space of curves, Journal of Amer. Math. Soc. 20 (2007), no. 1, 1--23

    2007

  30. [38]

    Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Annals of Math

    M. Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces, Annals of Math. (2) 168 (2008), no. 1, 97--125

    2008

  31. [39]

    Mirzakhani, Ergodic theory of the earthquake flow, IMRN 3 (2008), 1--39

    M. Mirzakhani, Ergodic theory of the earthquake flow, IMRN 3 (2008), 1--39

    2008

  32. [40]

    Mirzakhani, Counting mapping class group orbits on hyperbolic surfaces, (2016) arXiv:1601.03342

    M. Mirzakhani, Counting mapping class group orbits on hyperbolic surfaces, (2016) arXiv:1601.03342

    2016 arXiv

  33. [41]

    Mirzakhani, P

    M. Mirzakhani, P. Zograf, Towards large genus asymtotics of intersection numbers on moduli spaces of curves, GAFA 25 (2015), 1258--1289

    2015

  34. [42]

    Monin, V

    L. Monin, V. Telpukhovskiy, On normalizations of Thurston measure on the space of measured laminations, arXiv:1902.04533

    1902 arXiv

  35. [43]

    Norbury, Counting lattice points in the moduli space of curves, Math

    P. Norbury, Counting lattice points in the moduli space of curves, Math. Res. Lett. 17 (2010), no. 3, 467--481

    2010

  36. [44]

    Okounkov and R

    A. Okounkov and R. Pandharipande, Gromov--Witten theory, Hurwitz theory, and matrix models, Algebraic geometry --- Seattle 2005. Part 1, 325--414, Proc. Sympos. Pure Math., 80, Part 1, Amer. Math. Soc., Providence, RI, 2009

    2005

  37. [45]

    K. Rafi, J. Souto, Geodesics Currents and Counting Problems, arXiv:1709.06834

    arXiv

  38. [46]

    Sauvaget, Volumes and Siegel--Veech constants of H (2g-2) and Hodge integrals , arXiv:1801.01744

    A. Sauvaget, Volumes and Siegel--Veech constants of H (2g-2) and Hodge integrals , arXiv:1801.01744

    arXiv

  39. [47]

    Sauvaget, The Large genus asymptotic expansion of Masur-Veech volumes, (2019) arXiv:1903.04454

    A. Sauvaget, The Large genus asymptotic expansion of Masur-Veech volumes, (2019) arXiv:1903.04454

    2019 arXiv

  40. [48]

    Vakil, The Moduli Space of Curves and Its Tautological Ring, Notices of the AMS (2003), 50:6, 647--658

    R. Vakil, The Moduli Space of Curves and Its Tautological Ring, Notices of the AMS (2003), 50:6, 647--658

    2003

  41. [49]

    Veech, Gauss measures for transformations on the space of interval exchange maps, Annals of Math., 115 (1982), 201--242

    W. Veech, Gauss measures for transformations on the space of interval exchange maps, Annals of Math., 115 (1982), 201--242

    1982

  42. [50]

    W. A. Veech, Siegel measures, Annals of Math., 148 (1998), 895--944

    1998

  43. [51]

    Vorobets, Periodic geodesics on generic translation surfaces

    Ya. Vorobets, Periodic geodesics on generic translation surfaces. In: ``Algebraic and Topological Dynamics'', S. Kolyada, Yu. I. Manin and T. Ward (Editors), Contemporary Math., Vol. 385, Amer. Math. Soc., Providence, RI, 205--258 (2005)

    2005

  44. [52]

    Witten, Two-dimensional gravity and intersection theory on moduli space, in Surveys in differential geometry (Cambridge, MA, 1990), 243--310, Lehigh Univ., Bethlehem, PA, 1991

    E. Witten, Two-dimensional gravity and intersection theory on moduli space, in Surveys in differential geometry (Cambridge, MA, 1990), 243--310, Lehigh Univ., Bethlehem, PA, 1991

    1990

  45. [53]

    Zograf, An explicit formula for Witten's 2 -correlators, Zap

    P. Zograf, An explicit formula for Witten's 2 -correlators, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 468 (2018), 53--57

    2018

  46. [54]

    Zorich, Square tiled surfaces and Teichm\"uller volumes of the moduli spaces of abelian differentials

    A. Zorich, Square tiled surfaces and Teichm\"uller volumes of the moduli spaces of abelian differentials. Rigidity in dynamics and geometry (Cambridge, 2000), 459--471, Springer, Berlin, 2002

    2000

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