In this note, we answer a question of Mirzakhani on asymptotic behavior of the one-point volume polynomial of moduli spaces of curves. We also present some applications of Mirzakhani's asymptotic formulae of Weil-Petersson volumes.
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48 results for “Mirzakhani's asymptotics”
Mirzakhani extended curve counting from simple to all curves.
problem Counting curves in the mapping class group orbit with length at most L.
method Low-tech argument showing general result from simple curves.
result Derived the asymptotic growth of curves for arbitrary curves.
Mirzakhani's thesis counts geodesics on hyperbolic surfaces, finding a specific asymptotic formula.
problem Counting simple closed geodesics on hyperbolic surfaces.
method Inspired by lattice point counting, uses principles of homogeneous dynamics.
result The number of simple closed geodesics of length ≤ L is asymptotic to L^(6g-6) times a constant.
This paper counts specific square-tiled surfaces related to hyperbolic geodesics.
problem Counting square-tiled surfaces with specific foliations.
method Conceptual and geometric methods inspired by Mirzakhani's work.
result The number of square-tiled surfaces is asymptotic to \(L^{6g-6+2n}\) times constants from Mirzakhani's count.
Maryam Mirzakhani (in her doctoral dissertation) has proved the author's conjecture that the number of simple curves of length bounded by L on a hyperbolic surface S is assymptotic to a constant times L to the power d, where d is the dimension of the Teichmuller space of S. In this note we clarify and simplify Mirzakha…
Study on hyperbolic surfaces' volumes, proving asymptotic expansion for high genus.
problem Analyzing the volume of moduli spaces of hyperbolic surfaces with varying genus.
method Topological recursion formula by Mirzakhani, asymptotic expansion for high genus.
result Explicit computation of the second term in the asymptotic expansion.
A 2008 general overview on Weil-Petersson geometry is offered. A preliminary plan for the subsequent CBMS lectures at Central Connecticut State University is included. Mirzakhani's solution of Witten-Kontsevich is not included - this work essentially requires its own lectures. Lectures on Mirzakhani's Witten-Kontsevich…
Mirzakhani studied Riemann surfaces and their spaces.
problem Understanding Riemann surfaces and their spaces.
method Survey of her 20 papers.
result Contributions to understanding Riemann surfaces and their spaces.
Lengths of simple closed geodesics on hyperbolic surfaces in prescribed homology classes
problem Estimating the number of simple closed geodesics of a fixed length and homology class on a hyperbolic surface
method Using asymptotic formulas and numerical evidence
result Proving an asymptotic lower bound for the number of such geodesics
Mirzakhani volumes of moduli spaces are polylogarithmic.
problem Understanding the volume of moduli spaces of hyperbolic surfaces.
method Expressed as a sum of polylogarithms evaluated at specific points.
result Mirzakhani volumes are polylogarithmic.
Counting hyperbolic multi-geodesics with individual component lengths.
problem Counting hyperbolic multi-geodesics with specific component lengths.
method Unified geometric and topological techniques, combining Mirzakhani's results and Margulis's ideas.
result Asymptotic polynomial counts of multi-geodesics in mapping class group orbits, generalizing Wolpert's conjecture.
The paper calculates super Weil-Petersson volumes for large genus.
problem Calculating super Weil-Petersson volumes for large genus.
method Analyzes super intersection numbers, proves coefficients are polynomials, and provides an algorithm to compute them.
result Proves existence of a complete asymptotic expansion of super Weil-Petersson volumes.
Counting subgroups of a surface using convex core lengths.
problem Counting conjugacy classes of subgroups of fundamental groups of surfaces.
method Using half the sum of the lengths of the boundaries of the convex core of a subgroup.
result The number of conjugacy classes of subgroups is asymptotic to .
Study counts geodesics on special projective planes, finding a noninteger growth rate.
problem Counting geodesics on specific projective planes with removed points.
method Proved true asymptotic formula for geodesics of length ≤ L, independent of hyperbolic structure.
result Growth exponent is noninteger, contrasting with results for orientable surfaces.
Mirzakhani connects earthquake and Teichmuller flows on surfaces.
problem Understanding the relationship between earthquake and Teichmuller flows.
method Geometric account of connections between flows, avoiding technical prerequisites.
result Mirzakhani's theorem relating earthquake and Teichmuller flows.
Study on frequencies of non-simple curves in surfaces of large genus.
problem Frequency of non-simple curves in surfaces of large genus.
method Expression for frequency, large genus asymptotics, comparison with previous work.
result Identify most common types of non-simple curves with K intersections.
Formulae for Masur-Veech volumes derived from intersection numbers of curves.
problem Computing volumes and densities of geodesics and surfaces.
method Intersection numbers of psi-classes and lattice point count.
result Formulae for Masur-Veech volumes as polynomials in intersection numbers.
Estimates point counts in Teichmüller space for mapping class groups.
problem Counting points in Teichmüller space under mapping class group actions.
method Quantitative estimates with power saving error terms for Teichmüller metric balls.
result Effectivizes asymptotic counting results of Athreya et al.
Counting curves on orbifoldsmath.GT
New theorem counts curves on orbifolds.
problem Counting curves on surfaces.
method Applied Mirzakhani's theorem to orbifolds.
result Curve counting theorem extends to orbifolds.
Experimental evidence for curve ratios on genus two surfaces.
problem Determining the ratio of topological curve types on surfaces.
method Experimental statistics applied to Mirzakhani's genus two surface results.
result Separating and non-separating curves occur in the ratio 1:48.
New growth rate for pseudo-Anosov conjugacy classes in Teichmüller space.
problem Understanding growth rates of conjugacy classes in Teichmüller space.
method Analyzing pseudo-Anosov mapping classes and their conjugacy classes in Teichmüller space.
result The number of lattice points of pseudo-Anosov conjugacy classes intersecting a closed ball of radius R is coarsely asymptotic to \(e^{\frac{h}{2}R}\).
Study counts orbits of mapping class group in shearing coordinates.
problem Counting orbits of mapping class group in shearing coordinates.
method Uses shearing coordinates and asymptotics of Teichmüller space.
result Asymptotic behavior of mapping class group orbits in shearing coordinates.
Some Generalizations of Mirzakhani's Recursion and Masur-Veech Volumes via Topological Recursionsmath-ph
The paper explores generalizations of Mirzakhani's recursion and computes volumes for physical gravity models.
problem Computing volumes for physical gravity models.
method Topological recursion and physical two-dimensional gravity models.
result Derivation of Virasoro constraints and cut-and-join equations for generalized Mirzakhani's recursions.
Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.
problem Counting minimal Lagrangians in hyperbolic surfaces.
method Uses Mirzakhani functions to show growth rate and proves rigidity of area spectrum.
result Number of minimal Lagrangians grows as .
Formula for integrating random variables on hyperbolic surfaces.
problem Integrating random variables on the moduli space of hyperbolic surfaces.
method Integration formula for lengths of closed geodesics.
result Integral of geometric random variables can be expressed as an integral over R.
Growth rate of Dehn twist lattice points in Teichmüller space is slower than mapping class group lattice points.
problem Analyzing the growth rate of Dehn twist lattice points in Teichmüller space.
method Comparing growth rates of Dehn twist, mapping class group, and multi-twist lattice points.
result The growth rate of Dehn twist lattice points is coarsely asymptotic to $e^{ rac{h}{2}R}$, slower than the mapping class group.
Formulae for Masur-Veech volumes and frequencies of geodesics derived from intersection numbers.
problem Calculating volumes and frequencies of geodesics in moduli spaces.
method Lattice point counts and intersection numbers of ψ-classes, with explicit rational coefficients.
result Formulae for Masur-Veech volumes and frequencies of simple closed geodesics.
Study curves and geodesics on a torus, finding Euler totient function.
problem Understanding the structure of curves and geodesics on a torus.
method Counting specific orbits of the mapping class group up to a certain length.
result Euler totient function emerges and is involved in formulae.
We give an overview of the proof for Mirzakhani's volume recursion for the Weil-Petersson volumes of the moduli spaces of genus hyperbolic surfaces with labeled geodesic boundary components, and her application of this recursion to Witten's conjecture and the study of simple geodesic length spectrum growth rate…
New recursion formula for non-orientable surfaces resolves divergences.
problem Computing volumes of moduli spaces for non-orientable surfaces.
method Generalization of Mirzakhani's recursion to non-orientable surfaces, handling divergences with integral kernels.
result Regularized volumes can be computed with a cutoff on crosscap size.
Extends classification of invariant measures on geodesic currents.
problem Classifying invariant measures on geodesic currents.
method Decomposition of currents into measured laminations, multi-curves, and bound currents.
result Extension of classification to geodesic currents.
Study describes frequencies of geodesics on hyperbolic surfaces as genus grows.
problem Large genus asymptotic behaviors of geodesic frequencies on hyperbolic surfaces.
method Proof of conjecture involving separating and nonseparating geodesics.
result Explicit function $f( rac{n}{g})$ for frequency ratio given.
Study shows twist tori equidistribute in moduli space, with other families having singular distributions.
problem Statistical behavior of twist tori in moduli space of hyperbolic surfaces.
method Analyzing expanding families of twist tori and their limiting distributions.
result Equidistribution of twist tori to a Lebesgue measure, with other families having singular distributions.
Improved bounds on a function linking geodesics to moduli spaces.
problem Understanding geodesics on hyperbolic surfaces.
method Analyzing the Mirzakhani function and its behavior near cusps.
result The function is square-integrable with respect to the Weil-Petersson volume form.
The paper counts orbits of curves with up to K self-intersections on a surface.
problem Counting orbits of curves with self-intersections on a surface.
method Combinatorial approach to studying geodesics on surfaces.
result Developed a new combinatorial method to study geodesics on surfaces.
Study volumes of Klein surfaces, extending Mirzakhani's recursion.
problem Volumes of moduli spaces of bordered Klein surfaces.
method Generalization of Mirzakhani's recursion, integration over regularised moduli space.
result Explicit formula for Klein bottle moduli space volumes, recursion for arbitrary topologies.
This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.
problem Understanding relationships between geodesic and orthogeodesic lengths on hyperbolic surfaces.
method Investigates a broad family of identities involving lengths of all closed geodesics and orthogeodesics.
result Introduces new identities that include lengths of all closed geodesics, contrasting with previous identities.
GOE statistics emerge from surface moduli space averages.
problem Understanding spectral statistics on hyperbolic surfaces.
method Defined a smooth linear statistic, averaged over moduli space, and analyzed variance.
result GOE statistics are recovered in the large genus and high energy limits.
The dynamics of earthquake flow equidistributes geodesics on hyperbolic surfaces.
problem Equidistribution of geodesics on hyperbolic surfaces.
method Dynamics of the earthquake flow.
result The dynamics of the earthquake flow equidistributes geodesics on hyperbolic surfaces.
Study on lengths of random multicurves on hyperbolic surfaces.
problem Distribution of lengths of random multicurves on closed hyperbolic surfaces.
method Using Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres.
result Distribution of lengths admits a polynomial density, with coefficients expressible in terms of intersection numbers of psi-classes.
Convergence of Siegel-Veech constants for weakly convergent measures on translation surfaces.
problem Convergence of Siegel-Veech constants for weakly convergent measures on translation surfaces.
method Recurrence result related to Eskin-Masur techniques, measure equidistribution result.
result Convergence of sequences of Siegel-Veech constants associated to Teichmüller curves in genus two.
We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.
Study on random hyperbolic surfaces with many cusps, focusing on tight geodesics.
problem Understanding length statistics of geodesics on random hyperbolic surfaces with cusps.
method Recursion formula for tight Weil-Petersson volumes and generalization of Mirzakhani's integration formula.
result Recovery of Poisson point process in large genus limit for length statistics of tight geodesics.
We introduce a new method of calculating intersections on \bar{M}_{g,n}, using localization of equivariant cohomology. As an application, we give a proof of Mirzakhani's recursion relation for calculating intersections of mixed psi and kappa_1 classes.
We generalise in this article the Mc Shane-Mirzakhani identities in hyperbolic geometry to arbitrary cross ratios. We give an expression of them in the case of Hitchin representations of surface groups in PSL(n, R) in a suitable choice of Fock-Goncharov coordinates.
New bounds reveal double exponential growth in conjugacy classes of fully irreducibles.
problem Counting conjugacy classes of fully irreducibles in Out(F_r).
method Equivalence to pseudo-Anosovs and logarithmic dilatations.
result Double exponential growth in the number of conjugacy classes.
We derive an identity for Margulis invariants of affine deformations of a complete orientable one-ended hyperbolic sur- face following the identities of McShane, Mirzakhani and Tan- Wong-Zhang. As a corollary, a deformation of the surface which infinitesimally lengthens all interior simple closed curves must in- finite…
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. I…