Study of q-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.
problem Geometry of q-rationals and their properties. method Construction and analysis of deformed Farey triangulation and deformed modular surface; definition and study of Springborn operations.
result Derivation of a formula for the q-deformed midpoint and new q-deformation of Markov numbers. Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.
problem Understanding the boundary of K-moduli of prime Fano threefolds of genus twelve.
method Developed a modular relation between Fano threefolds and their anticanonical K3 surfaces, proving forgetful morphism is an open immersion.
result Proved the boundary of K-moduli of V22 is purely divisorial and consists of four irreducible components. Introduces q-transpose for q-deformed modular group matrices.
problem Understanding q-deformed rational numbers and their properties. method Introduces q-transpose and applies it to refine q-deformed modular group actions. result New proof and refinement of Leclere and Morier-Genoud's trace palindromicity theorem.
Study of knot complements yields quantum modularity insights.
problem Understanding quantum invariants of knot complements.
method Large-N analysis of q-series invariants, counts of holomorphic curves. result Closed-form expressions for a-deformed FK for (2,2p+1)-torus knots. User interfaces provide an interactive window between physical and virtual environments. A new concept in the field of human-computer interaction is a soft user interface; a compliant surface that facilitates touch interaction through deformation. Despite the potential of these interfaces, they currently lack a signal …
Study of Batalin-Vilkovisky algebra on Poisson manifolds with diagonalizable modular symmetry.
problem Exploring Batalin-Vilkovisky algebra structures on Poisson manifolds with specific symmetry conditions.
method Analysis of twisted Poincaré duality and mixed complex structure, combined with Kontsevich's deformation quantization and Koszul duality.
result Generalization of Batalin-Vilkovisky algebra structure to Poisson manifolds with diagonalizable modular symmetry.
Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.
problem Understanding the curvature properties of modular surfaces in Lorentz-Minkowski space.
method Analyzing the sign of Gaussian and mean curvature, classifying surfaces, and applying to conformal field theories.
result Complete classification of zero Gaussian curvature modular surfaces and non-existence of non-planar maximal modular surfaces.
Finite specializations of a q-deformed modular group at roots of unity.
problem Understanding the finiteness of specializations of a q-deformed modular group at roots of unity.
method Introduced a q-deformed modular group and studied its specializations at roots of unity.
result For ζn being a primitive nth root of unity, PSLq(2,Z)∣q=ζn is finite if and only if Gq(ζn) is finite. On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a C∗ action with compact fixed locus. Applying virtual localisation we define invariants constant …
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
Countable modular groups found on surfaces with infinite type.
problem Finding modular groups of infinite type surfaces.
method Proving countable modular groups for orientable infinite type surfaces.
result Every orientable infinite type surface has a countable modular group.
Every rack Q provides a set-theoretic solution cQ of the Yang-Baxter equation. This article examines the deformation theory of cQ within the space of Yang-Baxter operators over a ring $\A$, a problem initiated by Freyd and Yetter in 1989. As our main result we classify deformations in the modular case, which ha…
Geodesics on modular surface yield arithmetic 3-manifolds.
problem Understanding arithmetic properties of modular surfaces.
method Constructing geodesics and analyzing their lifts.
result Complements of canonical lifts are arithmetic 3-manifolds.
New patterns deform Farey triangulation in symmetric space.
problem Deforming Farey triangulation in symmetric space.
method Realized representations of modular group as isometry groups of geodesic patterns in SL3(R)/SO(3). result 2-parameter family of deformations of Farey triangulation.
The article is devoted to the qR-conformal modular functors, which being ``deformations'' of the conformal modular functor (the projective representation of the category Train(Diff+(S1)), the train of the group Diff+(S1) of all orientation preserving diffeomorphisms of a circle) in the class of all projectiv…
Study of parabolic-preserving deformations of hyperbolic lattices.
problem Preserving parabolic subgroups during lattice deformations.
method Analysis of deformations into SU(n,1) and SO(n+1,1).
result Existence of 1-parameter families of parabolic-preserving deformations.
A new method integrates forms on Riemann surfaces, leading to modular forms.
problem Integrating differential forms with poles on Riemann surfaces.
method Simple procedure to integrate differential forms with arbitrary holomorphic poles, establishing an analytic theory for integrals over configuration spaces.
result Regularized graph integrals on elliptic curves are almost-holomorphic modular forms.
Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…
We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N-cusped surface. We also …
Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.
problem Proving quantum modularity of SU(2) TQFT signature for genus 2 surfaces.
method Using quantum modularity of generalized Dedekind sums associated with modular forms and trigonometric sum expressions.
result Quantum modularity of SU(2) TQFT signature on genus 2 surfaces proved.
Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and deve…
Study counts geodesics on modular surface, linking to necklace counting.
problem Counting geodesics on modular surface with specific winding numbers.
method Asymptotic expansion, generating function analysis, correspondence to necklace counting.
result Obtained asymptotic growth rate of m low-lying geodesics in terms of word length.
In this letter, first we give a decomposition for any Lie-Poisson structure πg associated to the modular vector. In particular, πg splits into two compatible Lie-Poisson structures if dimg≤3. As an application, we classified quadratic deformations of Lie-Poisson structures on R3 up to linear d…
Study geometrically measures to decide if modular companions are conformally equivalent.
problem Deciding if two modular companions are conformally equivalent under a given group action.
method Construct a moduli space and equivariant tilings to measure conformal equivalence.
result Presented a geometric measure to decide conformal equivalence of modular companions.
Erdős-Kac theorem applied to geodesics on modular surface.
problem Understanding the distribution of geodesics on modular surfaces.
method Analyzing the number of scattering geodesics with a fixed sojourn time.
result Gaussian behavior for the number of scattering geodesics on modular surface.
A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the foll…
We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive real…
Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.
problem Understanding the action of mapping class groups on cohomology of surfaces.
method Associate cochain complexes to surfaces, with mapping class groups acting projectively on cohomology.
result Projective action of mapping class groups on Hochschild cohomology of Hopf algebras.
Infinite families of quantum modular invariants for 3-manifolds are discovered.
problem Quantum modular forms and invariants of 3-manifolds.
method Lattice cohomology and BPS q-series of 3-manifolds, recently developed theory by AJK. result First calculation of AJK series for an infinite family of 3-manifolds.
In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynami…
The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-di…
Describes representations of modular group into SL(3,R)/SO(3).
problem Understanding representations of modular group into isometry group of symmetric space.
method Analyzes connected component of conjugacy classes of representations.
result Certain representations in the component are Anosov.
Study linking numbers in hyperbolic 3-folds, linking to Siegel modular forms.
problem Linking numbers between geodesics in arithmetic hyperbolic 3-folds.
method Analyzing integral of Kudla--Millson theta series over Seifert surfaces.
result Series converges to genus 2 Siegel modular forms of weight 2.
Our aim is to introduce and advocate non-Σ (non-symmetric) modular operads. While ordinary modular operads were inspired by the structure of the moduli space of stable complex curves, non-Σ modular operads model surfaces with open strings outputs. An immediate application of our theory is a short proof that the mod…
Study on scattering geodesics on modular surface and their sojourn times.
problem Distribution of scattering geodesics and their sojourn times on modular surface.
method Analysis of scattering geodesics in modular surface, establishing connection to prime divisors in arithmetic progression.
result Established a connection between scattering geodesics and prime divisors in arithmetic progression.
Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
Proves representability of complex semigroup systems.
problem Representability of systems of proportionally modular numerical semigroups.
method Canonical equivariant resolution of weighted homogeneous surface singularities.
result Every system of proportionally modular numerical semigroups is representable.
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…
We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmu…
We study geodesics on the modular surface, comparing WP and hyperbolic metrics.
problem Comparing geodesics on the modular surface under different metrics.
method Lift WP geodesics to the universal cover, analyze geometric properties, and compare deviations.
result WP and hyperbolic geodesics fellow-travel in the thick part of the universal cover.
Using the Weil-Brezin-Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely-generated projective modules over the algebra Aθ of the noncommutative torus. We show that such Aθ-modules have a natural interpretatio…
Holomorphic torsion invariant for log-Enriques surfaces derived from Borcherds products.
problem Holomorphic torsion invariant for log-Enriques surfaces.
method Introduced a holomorphic torsion invariant using Borcherds products.
result The invariant is given by the Petersson norm of an explicit Borcherds product.
Minimal surfaces can be transformed into others with unchanged bending content.
problem Understanding the deformation properties of minimal surfaces.
method Refined polar decomposition theorem to identify bending-neutral deformations.
result Every minimal surface can be transformed into another by a bending-neutral deformation.
We prove that the modular operad of diffeomorphism classes of Riemann surfaces with both `open' and `closed' boundary components, in the sense of string field theory, is the modular completion of its genus 0 part quotiented by the Cardy condition. We also provide a finitary presentation of a version of this modular two…
Unified approach to conformal and modular invariants on surfaces.
problem Constructing a general family of conformal invariants on surfaces.
method Using an identification of Teichmüller space and rigged moduli space, and analytic work on harmonic functions.
result Unified conformal and modular invariants can be viewed as generalized modular invariants and functions on the rigged moduli space.
Researchers describe isometric deformations of T-hedra and T-surfaces.
problem Understanding the isometric deformations of discrete and smooth T-surfaces.
method Synthetic and analytic descriptions of T-hedra and T-surfaces, providing parametrizations of isometric deformations.
result Explicit parametrization of isometric deformations of T-hedra and T-surfaces.
New Fuchsian groups found with special embedding properties.
problem Finding new Fuchsian groups with specific embedding properties.
method Using period domains and properties of complex hyperbolic surfaces.
result First cocompact nonarithmetic Fuchsian groups with modular embedding not commensurable with triangle groups.
Geometrically deforms L∞ algebras to Lie algebroids, revealing new invariants.
problem Classifying geometric invariants of L∞ algebras arising from vector bundles. method Define geometric deformations of curved L∞ algebras and show they correspond to Lie algebroid structures. result Geometric deformations of L∞ algebras classify new geometric invariants.