The paper classifies singularities of plane congruences and affine distance functions.
problem Classifying singularities of plane congruences and affine distance functions.
method Classification through 2-parameter plane congruences in \(\mathbb{R^4}\) and affine normal plane congruences.
result Generic singularities of plane congruences and affine distance functions are classified.
The paper explores affine geometry of line congruences using singularity theory.
problem Understanding the affine geometry of line congruences and their focal sets.
method Use of singularity theory to describe generic phenomena and singularities.
result Identification of a key projective quadric in the tangent space.
Let S be a smooth affine algebraic curve, and let S' be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of the mapping class group Mod(S') by showing that its congruence kernel lies in the centralizer of every braid in Mod(S'). As a corollary, we obtain …
The paper studies elliptical surfaces in 3D affine space, classifying them based on curvature.
problem Classifying regular elliptical surfaces in affine space A3 based on curvature. method Defined a moving frame of minimal order for regular elliptical surfaces and derived differential invariants.
result Classified regular elliptical surfaces of constant curvatures up to affine congruence.
The paper classifies singularities of line congruences in 4D space.
problem Classifying singularities of line congruences in 4D space.
method Generic classification approach for 3-parameter line congruences and Blaschke normal congruences.
result Generic classification of singularities of 3-parameter line congruences in R4. The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by automorphisms of a finite index subgroup of the Artin group of type A2, which descends to a fait…
An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geom…
Study on braid group quotients by congruence subgroups.
problem Understanding the image of congruence subgroups in GL(n,Z).
method Characterization through symplectic congruence subgroups.
result Open problem solved: image of congruence subgroups in GL(n,Z).
Given an oriented Riemannian surface (Σ,g), its tangent bundle TΣ enjoys a natural pseudo-Kähler structure, that is the combination of a complex structure $\J$, a pseudo-metric $\G$ with neutral signature and a symplectic structure $\Om$. We give a local classification of those surfaces of TΣ which are both Lagr…
New metrics defined for full-rank correlation matrices, ensuring unique operations.
problem No suitable problem statement as the abstract does not describe a problem to be solved.
method New Riemannian metrics defined on full-rank correlation matrices, providing unique operations.
result Unique Riemannian logarithm and Fréchet mean defined for full-rank correlation matrices.
We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by const…
Enumerated all principal congruence link complements in 3D space.
problem Identifying all principal congruence link complements in S3. method Enumerated all principal congruence link complements in S3. result Answered a question posed by W. Thurston about principal congruence link complements.
The paper describes the geometric properties of line congruences' singularities.
problem Understanding the singularities of generic line congruences.
method Use of an equiaffine pair to define generic line congruences.
result Geometric description of folds, cusps, and swallowtails as singularities of generic line congruences.
Study on braid groups' congruence subgroups and their crystallographic quotients.
problem Understanding congruence subgroups and crystallographic quotients of braid groups.
method Investigation of lower central series of congruence braid groups related to B3. result Quotients of congruence braid groups are almost crystallographic.
Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3-manifolds generated by Dehn surgery with denominator f: weak f-congruence, f-congruence, and strong f-congruence. If f is odd, weak f-congruence preserves the ring structure on cohomology with Z_f-coefficients. We…
This paper explores geometric insights into discrete R-congruences and their envelopes.
problem Understanding the ambiguity in discrete R-congruences and their envelopes.
method Analyzes discrete R-congruences that are enveloped by specific types of surfaces and maps.
result Discovers a 2-parameter family of discrete enveloping surfaces for discrete R-congruences.
Paper explores relations between braid groups and their quotients.
problem Understanding relations between braid groups and their quotients.
method Recalling and introducing elements of congruence braid groups, establishing isomorphisms between crystallographic and congruence braid groups.
result Established isomorphisms between crystallographic braid groups and quotients of congruence braid groups.
Study investigates metrizability of Finsler spaces with specific metrics.
problem Local metrizability of Finsler spaces with m-Kropina metric. method Investigates the local metrizability of Finsler spaces with m-Kropina metric F=α1+mβ−m, where β is a closed null 1-form. result Proves that the affine connection on such an m-Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor is symmetric. New BDEs reveal singular surfaces from line congruences.
problem Understanding binary differential equations associated with line congruences.
method Applied pointwise to quadratic differential forms, studying quotients of quadratic forms and associated polar lines.
result Introduced a new singular surface in Euclidean 3-space.
Technical report classifies all principal congruence link groups.
problem Classifying all principal congruence link complements in S^3.
method Comprehensive case analysis, necessary computations, and computer programs.
result Complete classification of all principal congruence link groups.
The study explores congruence subgroups of braid groups and their quotients.
problem Understanding the structure of congruence subgroups of braid groups.
method Utilizing the integral Burau representation and results from integral matrices, the study examines quotients of these subgroups.
result Findings of quotients that are not isomorphic to symmetric groups.
Criterion for congruence RFRS towers in hyperbolic lattices.
problem Criteria for hyperbolic lattices to have congruence RFRS towers.
method Criterion based on congruence subgroups.
result Virtually fibered Bianchi groups and new RFRS Kähler groups.
Study of line congruences for Appell's rank-4 hypergeometric functions.
problem Understanding line congruences for Appell's rank-4 hypergeometric functions.
method Derived original formulae for Laplace transform of rank-4 system, applied to geometry of surfaces defined by these functions.
result Natural line congruences for Laplace transforms of Appell's rank-4 functions form a W-congruence.
This paper explores dual relations between line congruences and surfaces in 3D and 4D.
problem Understanding the duality between line congruences in R3 and surfaces in R4. method Analyzes the correspondence between principal lines and asymptotic lines, ridge curves and flat ridge curves, and subparabolic curves.
result Discusses the behavior of subparabolic curves at the discriminant curve of the line congruence and the parabolic curve of the dual surface.
Abstract lists known link diagrams for all principal congruence link complements.
problem Identifying all known link diagrams for principal congruence link complements.
method Compilation of known link diagrams.
result Compilation of known link diagrams for principal congruence link complements.
DCL method improves congruency in machine learning tasks.
problem Varying task-specific semantics in training data.
method Direction Concentration Learning (DCL) method to enhance congruency.
result Improves performance in saliency prediction, continual learning, and classification tasks.
Characterizes W-congruences to study their stable umbilical points.
problem Understanding singularities of W-congruences.
method Characterization of W-congruences to study umbilical points.
result Examples of isolated stable umbilical points of type Am. We give a sufficient condition for isometric actions to have the congruency of orbits, that is, all orbits are isometrically congruent to each other. As applications, we give simple and unified proofs for some known congruence results, and also provide new examples of isometric actions on symmetric spaces of noncompact…
We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the …
Proves congruence subgroup property for mapping class groups of hyperbolic surfaces.
problem Residual finiteness of hyperbolic groups and congruence subgroup property for mapping class groups.
method Assumption of residual finiteness of hyperbolic groups leads to proof of congruence subgroup property.
result Congruence subgroup property for mapping class groups of hyperbolic surfaces.
Extends Kummer's theory to singular surfaces for line congruences.
problem Applying Kummer's theory to singular surfaces for line congruences.
method Analyzing the equation of principal surfaces and developable surfaces for normal congruences.
result The multiplicative factor for the principal surfaces is associated with the singular set of ξ. We study "how far away" a finite index subgroup G of SL(2,Z) is from being a congruence group. For this we define its deficiency of being a congruence group. We show that the index of the image of G in SL(2,Z/nZ) is biggest, if n is the general Wohlfahrt level. We furthermore show that the Veech groups of origamis (or …
Uniform mixing proved for hyperbolic manifolds using congruence covers.
problem Proving uniform exponential mixing for hyperbolic manifolds.
method Using Dolgopyat's method and spectral bounds for congruence transfer operators.
result Uniform exponential mixing for congruence covers of hyperbolic manifolds.
Lower bounds for systole growth in quaternionic hyperbolic manifolds.
problem Finding lower bounds for systole growth in quaternionic hyperbolic manifolds.
method Explicit lower bound calculation for systole in congruence covers.
result Optimal lower bound for systole growth proved.
We study Veech groups of covering surfaces of primitive translation surfaces. Therefore we define congruence subgroups in Veech groups of primitive translation surfaces using their action on the homology with entries in Z/aZ. We introduce a congruence level definition and a property of a primitive t…
It is known that the level 2 principal congruence subgroup of GL(n;Z) has a finite generating set. In this paper, we give a finite presentation of the level 2 principal congruence subgroup of GL(n;Z).
Survey on systole growth in congruence symmetric spaces.
problem Understanding systole growth in congruence symmetric spaces.
method Survey and simple proofs for best constants.
result Best possible constants for systole growth in symmetric spaces.
Study virtual braid groups, proving a key subgroup result.
problem Understanding congruence subgroups in virtual braid groups.
method Using an extension of the integral Burau representation.
result Proved level 2 congruence subgroup of virtual braid group is pure virtual braid group.
Proves Congruence Subgroup Property for two types of groups.
problem Proving Congruence Subgroup Property for specific groups.
method Elementary proof of Johnson filtration and geometric subsurface inclusions.
result Proves Congruence Subgroup Property for nilpotent quotients and subsurface subgroups.
The paper studies congruence subgroups and crystallographic quotients of small Coxeter groups.
problem The congruence subgroup property for small Coxeter groups.
method Analyzes the properties of small Coxeter groups, proving the failure of the congruence subgroup property for certain groups.
result Proves the failure of the congruence subgroup property for infinite small Coxeter groups which are not virtually abelian.
Improved estimate for remainder term in Weyl law for congruence subgroups of Chevalley groups.
problem Improving the remainder term estimate in the Weyl law for congruence subgroups of Chevalley groups.
method Sharpened the Weyl law remainder term estimate for simply connected Chevalley groups.
result Power saving estimate for the remainder term.
We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.
Study maximal orders to find surfaces without genus 212.
problem Which genera can be obtained by closed congruence surfaces?
method Investigate maximal orders in quaternion algebras to construct surfaces.
result No surface of genus 212 can be constructed by maximal orders.
Totally non congruence Veech groups found in translation surfaces.
problem Understanding Veech groups in translation surfaces.
method Analyzing origamis and their Veech groups in various strata of translation surfaces.
result Infinitely many origamis have totally non congruence Veech groups.
Theorem proves congruence for compact submanifolds in a sphere.
problem Understanding submanifolds in a sphere with specific embedding properties.
method Used a Reilly type formula for space forms.
result Proved a congruence theorem for compact embedded hypersurfaces.
Ng and Schauenburg proved that the kernel of a (2+1)-dimensional topological quantum field theory representation of SL(2,Z) is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group B3 with finite image enjoys a congruen…
Unified framework for studying Torelli group and congruence subgroup maps.
problem Understanding maps between Torelli group and congruence subgroup.
method Unified algorithms for computing Rochlin invariants and maps.
result Unified proofs and explicit evaluations of maps on Dehn twists.
The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Mouta…