Square-tiled torimath.GT
Study of square-tiled tori and their SL(2,Z) orbits.
problem Counting and classifying square-tiled tori.
method Natural parametrization and SL(2,Z) action.
result Exact size of every SL(2,Z) orbit.
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48 results for “square-tiled tori”
Shear moves connect square-tiled surfaces in quadratic differentials.
problem Connecting square-tiled surfaces via specific moves.
method Shear moves corresponding to diagonal flips preserving square-tiled properties.
result Connected components of reconfiguration problem are in bijection with moduli space of quadratic differentials.
Arithmeticity of the Kontsevich--Zorich monodromies of certain families of square-tiled surfacesmath.DS
The study finds arithmetic groups often in square-tiled surface monodromies.
problem Understanding arithmetic properties of square-tiled surfaces.
method Analyzing variations of Hodge structures and Kontsevich-Zorich monodromies.
result Arithmetic groups are frequent in low genus square-tiled surfaces.
The paper studies counting problems on square-tiled surfaces.
problem Understanding the frequency of properties in square-tiled surfaces.
method Examining properties of the square torus and their implications in translation surfaces.
result Implications between properties and their frequency in translation surfaces.
Computes the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes.
problem Calculating the contribution of specific geometric structures to volume calculations.
method Explicit computation and asymptotic analysis of square-tiled surfaces.
result The relative contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes is asymptotically 1/d, where d is the dimension of the stratum.
Formulae count square-tiled surfaces in genus two.
problem Counting square-tiled surfaces in genus two.
method Parametrized classification into four diagrams, provided formulae for enumeration.
result Formulae for enumeration of square-tiled surfaces in four diagrams, completing the count for genus two.
The straight-line flow on almost every staircase and on almost every square tiled staircase is recurrent. For almost every square tiled staircase the set of periodic orbits is dense in the phase space.
Study of random multicurves and square-tiled surfaces on large genus surfaces.
problem Understanding the geometry and combinatorial properties of random multicurves and square-tiled surfaces on surfaces of large genus.
method Combination of combinatorial and geometric analysis, including large genus asymptotic analysis of moduli space volumes and intersection numbers.
result Random multicurves and square-tiled surfaces have well-approximated properties by random permutations, with specific expected values.
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.
Estimates surface count with prescribed foliations.
problem Counting square-tiled surfaces with specific foliations.
method Effective estimate with power saving error term.
result Strengthens asymptotic counting formulas.
Random square-tiled surfaces have normal genus distribution and cover all integer vectors.
problem Distribution and properties of random square-tiled surfaces.
method Randomizing model and local central limit theorem for genus.
result The distribution of the genus is asymptotically normal and contains all primitive integer vectors.
The study constructs surfaces to show ratio-optimizing pseudo-Anosovs are common in Abelian differentials.
problem Understanding ratio-optimizing pseudo-Anosovs in Abelian differentials.
method Constructing square-tiled surfaces to demonstrate the ubiquity of ratio-optimizing pseudo-Anosovs.
result Pseudo-Anosovs optimizing the ratio of Teichmüller to curve graph translation length are common in Abelian differentials.
Quadratis Puzzlesmath.GT
New puzzles from geometry and topology.
problem Exploring puzzles from geometric and topological concepts.
method Construction from square-tiled shapes, discussion of underlying mathematics.
result Puzzles naturally associated to puzzle spaces.
Study of -cylinder surfaces to calculate Masur-Veech volumes.
problem Calculating Masur-Veech volumes for hyperbolic surfaces.
method Combinatorial approach using metric ribbon graphs and plane trees.
result Found generating function for -cylinder contributions.
Square-tiled surfaces with fixed combinatorics become equidistributed in moduli spaces.
problem Equidistribution of square-tiled surfaces with specific combinatorics.
method Analyzing the asymptotic behavior of square-tiled surfaces and their contributions to moduli spaces.
result Square-tiled surfaces with fixed combinatorics become asymptotically equidistributed in moduli spaces.
Connected Prym eigenforms found in a specific mathematical space.
problem Understanding the structure of Prym eigenforms in a particular mathematical space.
method Proving the connectedness of Prym eigenform loci and classifying square-tiled surfaces.
result The projection of Prym eigenform loci is a single Teichmüller curve.
Algorithm finds periodic points on Veech surfaces.
problem Finding periodic points on non-square-tiled Veech surfaces.
method Developed an algorithm to compute periodic points.
result Proved that in low discriminant, non-square-tiled Veech surfaces have no periodic points, except for fixed points of the Prym involution.
This research classifies Teichmüller curves in genus 2, proving parity conjectures for specific cases.
problem Classifying imprimitive Teichmüller curves in $\M_2$ related to square-tiled surfaces and modular curves.
method Analyzing square-tiled surfaces and their modular curves, proving parity conjectures for specific cases.
result Established the parity conjecture for in three cases, showing number of components does not depend on .
Classifies Teichmüller curves in genus three from genus two.
problem Classifying Teichmüller curves in genus three from genus two.
method Action on homology modulo two of affine groups of translation surfaces.
result Classification of orbits of square-tiled surfaces.
We study the congruence problem for subgroups of the modular group that appear as Veech groups of square-tiled surfaces in the minimal stratum of abelian differentials of genus two.
We prove formulae for the countings by orbit of square-tiled surfaces of genus two with one singularity. These formulae were conjectured by Hubert & Lelièvre. We show that these countings admit quasimodular forms as generating functions.
Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length generically has quadratic asymptotics in this length, with a common coefficient c…
New model for STSs with restricted horizontal gluings, focusing on maximal horizontal cylinders.
problem Modeling STSs with specific horizontal restrictions.
method Modified model with conjugacy classes of permutations to restrict horizontal gluings.
result Asymptotic analysis of components, genus distribution, and saddle connections.
The paper calculates Veech groups and Galois invariants for general origamis.
problem Understanding the structure and symmetries of origamis and their Galois invariants.
method Developed an algorithm to calculate Veech groups and orbits of Galois invariants for general origamis.
result Calculated Veech groups and Galois invariants for all origamis of degree .
Origamis with specific groups have Veech groups that surject onto SL(2, Z/nZ).
problem Characterizing Veech groups of origamis as totally non-congruence groups.
method Using results on SL(2, Z/nZ) and properties of origamis' deck transformation groups.
result Origamis with certain triangle group quotients have Veech groups that surject onto SL(2, Z/nZ).
The paper constructs minimal coherent filling pairs on surfaces.
problem Finding minimal intersecting coherent filling pairs on surfaces.
method Geometric procedure starting from a torus filling pair.
result Construction of minimal intersecting coherent filling pairs on for .
Study of translation covers of platonic solids reveals monodromy group structures.
problem Understanding monodromy groups of translation covers of platonic solids.
method Computed Zariski closures using generators, constraints, and Lyapunov spectrum analysis.
result Zariski closures of monodromy groups are powers of SL(2, R).
We calculate volumes of quadratic differentials using topological recursion.
problem Calculating volumes of quadratic differentials on curves.
method Topological recursion and geometric recursion applied to hyperbolic lengths of multicurves.
result Formula for constant terms of polynomials in terms of stable graphs.
A cyclic cover over the Riemann sphere branched at four points inherits a natural flat structure from the "pillow" flat structure on the basic sphere. We give an explicit formula for all individual Lyapunov exponents of the Hodge bundle over the corresponding arithmetic Teichmuller curve. The key technical element is e…
We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the co…
This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.
problem Understanding the Hamiltonian stationarity of twisted Lagrangian tori in C^2.
method Investigation of differential geometry of twisted tori, including product and Chekanov's exotic tori.
result Only product tori are minimal under Hamiltonian deformations, indicating Chekanov's exotic tori are not area minimal.
Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.
problem Approximating smooth 2-tori in high-dimensional spaces.
method Polyhedral approximation using Lagrangian and isotropic tori.
result Smooth 2-tori can be approximated by polyhedral Lagrangian or isotropic tori in C0 or C1 sense.
Counting meanders on surfaces of arbitrary genus, with precise asymptotics.
problem Counting and understanding meanders on surfaces of arbitrary genus.
method Square-tiled surfaces, moduli spaces of Abelian and quadratic differentials, Witten-Kontsevich 2-correlators.
result Asymptotic probability and polynomial growth of meanders with intersections.
Biharmonic tori in spheresmath.DG
Study on proper-biharmonic flat tori in spheres with CMC conditions.
problem Finding conditions for CMC proper-biharmonic immersions of tori in spheres.
method Analyzing rectangular and square tori, finding necessary and sufficient conditions, and explicit expressions.
result Explicit expressions of some CMC proper-biharmonic immersions of certain tori in spheres.
The paper classifies minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.
problem Classifying minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.
method General construction of homogeneous minimal flat n-tori in spheres, detailed investigations of shortest vectors in lattices.
result There exists a 2-parameter family of non-congruent λ1-minimal flat 4-tori.
Hopf tori and standard torimath.DG
Characterizes conformal classes of tori using differential geometry.
problem Classifying conformal classes of tori in complex dimension 1.
method Basic differential geometry methods, contrasting with Hopf tori.
result Complete characterization of conformal classes of product and standard flat tori.
Study finds non-isotopic transverse tori in Engel manifolds.
problem Identifying distinct transverse tori in Engel manifolds.
method Constructing an infinite family of non-isotopic transverse tori, introducing a homological invariant.
result Found an infinite family of non-isotopic transverse tori that are smoothly isotopic.
Study noncommutative coverings of irrational quantum tori.
problem Characterize noncommutative coverings of irrational quantum tori.
method Developed a framework for noncommutative coverings and studied irrational quantum tori.
result Characterized all connected noncommutative coverings of irrational quantum tori.
Isotropic tori in revisitedmath.SG
New isotropic tori found in complex space, not Hamiltonian isotopic.
problem Finding non-Hamiltonian isotopic isotropic tori in complex space.
method Analyzing isotropic tori in for .
result At least two exact isotropic -tori in are not Hamiltonian isotopic.
Constructs Riemannian foliations with exotic tori as leaves.
problem Creating Riemannian foliations with exotic tori.
method Smooth fiber bundles with exotic tori fibers and finite abelian fundamental group total space.
result Examples of Riemannian foliations with exotic tori leaves and finite abelian fundamental group total space.
New findings on isospectral tori and harmonic maps between flat tori.
problem Determining if isospectral tori are isometric using harmonic maps.
method Examined harmonic maps between flat tori, focusing on Milnor's isospectral tori.
result Milnor's isospectral tori cannot be distinguished by harmonic maps from lower-dimensional tori, but can be distinguished by higher-dimensional ones.
No complex analytic tori in complex deformations of Kummer varieties.
problem Existence of complex analytic tori in Kummer varieties.
method Proving non-existence through complex deformation analysis.
result Generic complex deformations of Kummer varieties contain no complex analytic tori.
The study of tiling homology on flat surfaces, proving impossibility of certain tilings.
problem Proving the non-existence of polyomino tilings on specific square-tiled surfaces.
method Study of homology groups for topological tilings, using coloring proofs.
result Several results about the non-existence of polyomino tilings on certain square-tiled surfaces.
Stable 2-lobed Delaunay tori found in 3-sphere.
problem Stability of 2-lobed Delaunay tori in the 3-sphere.
method Constrained Willmore surfaces in the 3-sphere.
result 2-lobed Delaunay tori are stable.
Constructs flows of tori in sphere perturbations for Morse homology.
problem Understanding tori in sphere perturbations.
method Constructs eternal mean curvature flows of tori.
result Constructs flows of tori in sphere perturbations.
We prove that the conformal immersions of complex two tori into which locally minimize their conformal volume in their conformal class all satisfy some elliptic PDE. We prove that they are either minimal tori, CMC flat tori, elliptic conformally constrained minimal tori or critical point of the area under some fi…
Ellipsoids host infinitely many minimal tori, bifurcating from a 2-torus orbit.
problem Minimal tori in ellipsoids.
method Analyzing 3D ellipsoids invariant under a 2-torus action.
result Infinitely many distinct minimal tori bifurcate from a 2-torus orbit.
Researchers identify only two types of tori with specific energy constraints.
problem Finding constrained Willmore tori in 3-space with specific energy limits.
method Analyzing isothermic constrained Willmore tori in the 3-sphere.
result Homogeneous and 2-lobe Delaunay tori are the only isothermic constrained Willmore tori with Willmore energy below 8π.