A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. The similar phenomenon is valid for the families of parallel closed geodesics.
We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface S. We count the number of saddle connections of length less than L on a generic flat surface S; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of A.Eskin and H.Masur these numbers have quadratic asymptotics with respect to L. Here we explicitly compute the constant in this quqadratic asymptotics for a configuration of every type. The constant is found from a Siegel--Veech formula.
To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.
In the 80's H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel-Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymptotic behavior of quasi-modular forms, D. Chen, M. Moeller, and D. Zagier proved this conjecture for strata of differentials with simple zeros. Here, we prove that the conjecture holds for the other extreme case, i.e. for strata of differentials with a unique zero. Our main ingre…
Let $\mathcal{H}$ denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on $\mathbb{R}^2$ is in $L^2(\mathcal{H}, μ)$, where $μ$ is Lebesgue measure on $\mathcal{H}$, and give applications to bounding error terms for counting problems for saddle connections. We also propose a new invariant associated to $SL(2, \mathbb{R})$-invariant measures on strata satisfying certain integrability conditions.
We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.
We present an explicit formula relating volumes of strata of meromorphicquadratic differentials with at most simple poles on Riemann surfacesand counting functions of the number of flat cylinders filled by closedgeodesics in associated flat metric with singularities. This generalizes the resultof Athreya, Eskin and Zorich in genus 0 to higher genera.
An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders filled by parallel geodesics of the same length. The growth rate of the number of such regions counted with weights, as a function of the length, is quadratic with a coefficient, called Siegel-Veech constant, that is shared by almost all translation surfaces in the ambient stratum. We evaluate various Siegel-Veech constants associated to the geometry of confi…
In this paper we consider the large genus asymptotics for two classes of Siegel-Veech constants associated with an arbitrary connected stratum $\mathcal{H} (α)$ of Abelian differentials. The first is the saddle connection Siegel-Veech constant $c_{\text{sc}}^{m_i, m_j} \big( \mathcal{H} (α) \big)$ counting saddle connections between two distinct, fixed zeros of prescribed orders $m_i$ and $m_j$, and the second is the area Siegel-Veech constant $c_{\text{area}} \big( \mathcal{H}(α) \big)$ counting maximal cylinders weighted by area. By combining a combinatorial analysis of explicit formulas of …
We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We exhibit effective asymptotic formulas for the number of (isotopy classes of) periodic billiard trajectories (up to $\mathbb{Z}^2$-translations) on Veech wind-tree billiards, that is, wind-tree billiards whose underlying compact translation surfaces are Veech surfaces. We show that the error term depends on spectral properties of the Veech group and give explicit estimates in the case when obstacles are squares of side length $1/2$.
For a non-uniform lattice in SL(2,R), we consider excursions in cusp neighborhoods of a random geodesic on the corresponding finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. In the Teichmuller setting, we consider invariant measures for the SL(2,R) action on the moduli spaces of quadratic differentials. By the work of Eskin and Mirzakhani, these measures are supported on affine invariant submanifolds of a stratum of quadratic differentials. For a Teic…
In this paper we analyze the large genus asymptotics for intersection numbers between $ψ$-classes, also called correlators, on the moduli space of stable curves. Our proofs proceed through a combinatorial analysis of the recursive relations (Virasoro constraints) that uniquely determine these correlators, together with a comparison between the coefficients in these relations with the jump probabilities of a certain asymmetric simple random walk. As an application of this result, we provide the large genus limits for Masur-Veech volumes and area Siegel-Veech constants associated with principal …
We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $ψ$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though with…
We prove that the asymptotic number of pairs of saddle connections with length smaller than $L$ with bounded virtual area is quadratic for almost every translation surface with respect to any ergodic $SL(2,\mathbb{R})$-invariant measure. A key tool of the proof is that Siegel-Veech transforms of bounded functions with compact supports are in $L^{2+κ}$ for every $SL(2,\mathbb{R})$-invariant measure.
Consider the 1-dimensional Hurwitz space parameterizing covers of P^1 branched at four points. We study its intersection with divisor classes on the moduli space of curves. As an application, we calculate the slope of the Teichmuller curve parameterizing square-tiled cyclic covers and recover the sum of its Lyapunov exponents obtained by Forni, Matheus and Zorich. Motivated by the work of Eskin, Kontsevich and Zorich, we exhibit a relation among the slope of Hurwitz spaces, the sum of Lyapunov exponents and the Siegel-Veech constant for the moduli space of quadratic differentials.
We give effective estimates for the number of saddle connections on a translation surface that have length $\leq L$ and are in a prescribed homology class modulo $q$. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum.
In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volume $ν_1 \big( \mathcal{H}_1 (m) \big)$ of a stratum indexed by a partition $m = (m_1, m_2, \ldots , m_n)$ is $\big( 4 + o(1) \big) \prod_{i = 1}^n (m_i + 1)^{-1}$ as $2g - 2 = \sum_{i = 1}^n m_i$ tends to $\infty$. This confirms a prediction of Eskin-Zorich and generalizes some of the recent results of Chen-Moeller-Zagier and Sauva…
We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to $\mathbb{Z}^2$-translations) on the wind-tree billiard. We also compute explicitly the associated Siegel-Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.
We prove that square-tiled surfaces having fixed combinatorics of horizontal cylinder decomposition and tiled with smaller and smaller squares become asymptotically equidistributed in any ambient linear $GL(\mathbb R)$-invariant suborbifold defined over $\mathbb Q$ in the moduli space of Abelian differentials. Moreover, we prove that the combinatorics of the horizontal and of the vertical decompositions are asymptotically uncorrelated. As a consequence, we prove the existence of an asymptotic distribution for the combinatorics of a "random" interval exchange transformation with integer lengths…
We define Ratner-Marklof-Strombergsson measures. These are probability measures supported on cut-and-project sets in R^d (d > 1) which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project se…
In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral …
Representations of binary quadratic forms by quaternary quadratic forms
Korea institute for Advanced Study（韩国高等研究院）；Institute for Advanced Study（普林斯顿高等研究院）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）
We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on the Riemann sphere and counting functions of the number of (bands of) closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago. The proof is based on a formula for the Lyapunov exponents of the geodesic flow on the moduli space, which gives a recursive formula from which the volumes could be recovered. An appendix with an ergodic theorem proved by Jon Chaika …
In this paper we study counting functions representing the number of solutions of systems of linear inequalities which arise in the theory of Diophantine approximation. We develop a method that allows us to explain the random-like behavior that these functions exhibit and prove a Central Limit Theorem for them. Our approach is based on a quantitative study of higher-order correlations for functions defined on the space of lattices and a novel technique for estimating cumulants of Siegel transforms.
