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.
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…
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.
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.
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.
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.
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.
Study properties of null string congruences in complex and real pseudo-Riemannian spaces.
problem Analyze properties of null string congruences in 4D spaces.
method Investigate the geometry of null strings and their congruences in complex and real pseudo-Riemannian spaces.
result Relations between null string congruences, Petrov-Penrose types, and Ricci tensor algebraic types are established.
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.
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 …
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.
We use quantum invariants to define a 3-manifold invariant j_p which lies in the non-negative integers. We relate j_p to the Heegard genus, and the cut number. We show that j_$ is an invariant of weak p-congruence.
Sharp upper bound for FI-module regularity, refining stability of congruence subgroup homology.
problem Stability of homology of congruence subgroups in various characteristics.
method Explicit upper bound on Castelnuovo-Mumford regularity using degrees of generators and relations.
result Refined stability theorem for homology of congruence subgroups, extending to small characteristics and integral homology.
New criterion for periodic links from Kauffman polynomial.
problem Determining periodicity of links.
method Using a one-variable specialization of the Kauffman polynomial and its mirror image.
result A new criterion for periodicity of links.
We give two proofs that the 3-torus is not weakly d-congruent to the connected sum of three S^1xS^2's, if d>2. We study how cohomology ring structure relates to weak congruence. We give an example of three 3--manifolds which are weakly 5-congruent but are not 5-congruent.
We generalize the "miraculous cancellation" formulas of Alvarez-Gaumé, Witten and Kefeng Liu to a twisted version where an extra complex line bundle is involved. We also apply our result to discuss intrinsic relations between the higher dimensional Rokhlin type congruence formulas of Ochanine [O1], Finashin [F] and Zha…
By calculating the Fermat limit of certain q-Fermat functions, we get explicit surgery formulae for the second and third Ohtsuki invariants for homology 3-spheres. The surgery formula of the second Ohtsuki invariant λ_2, combined with an argument using the general theory of finite type invariants of homology 3-spheres,…
We study the geodesic flow on the normal line congruence of a minimal surface in R3 induced by the neutral Kähler metric on the space of oriented lines. The metric is lorentz with isolated degenerate points and the flow is shown to be completely integrable. In addition, we give a new holomorphic description …
Study examines how Lorentz transformations affect foliations in spacetime.
problem Understanding foliations and their relation to observers under Lorentz transformations.
method Exploration of kinematic characteristics of timelike congruences (observers) and their relation to foliations.
result Local Lorentz transformations impact the existence of transversal foliations.
Conditions, related to the so-called bending problem are considered for hypersurfaces of a pseudo-Euclidean space. Corresponding theorems are proved.
In a previous paper, the author (together with Matthew Emerton) proved that the completed cohomology groups of SL_N(Z) are stable in fixed degree as N goes to infinity (Z may be replaced by the ring O_F of integers of any number field). In this paper, we relate these completed cohomology groups to K-theory and Galois c…
To every n-dimensional lens space L, we associate a congruence lattice L in Zm, with n=2m−1 and we prove a formula relating the multiplicities of Hodge-Laplace eigenvalues on L with the number of lattice elements of a given ∥⋅∥1-length in L. As a consequence, we show th…
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).
We describe natural Kähler or para-Kähler structures of the spaces of geodesics of pseudo-Riemannian space forms and relate the local geometry of hypersurfaces of space forms to that of their normal congruences, or Gauss maps, which are Lagrangian submanifolds. The space of geodesics L(S^{n+1}_{p,1}) of a pseudo-Rieman…
The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.
problem Characterizing and understanding the singularities and geometric properties of surfaces formed by the singular loci of normal congruences of frontals with pure-frontal singular points.
method Characterizations of singularities in terms of geometric invariants of the initial frontal are provided for the normal ruled surface. Relations between certain singularities of focal surfaces and geometric properties of the frontal are also explored.
result Behavior of Gaussian curvature of focal surfaces of frontals with a 5/2-cuspidal edge is considered. Virtual knots can be transformed by 2k-moves, affecting their writhes.
problem Understanding transformations between virtual knots using 2k-moves. method Analyzing the effect of 2k-moves on writhes and congruence classes. result Two virtual knots related by 2k-moves have congruent odd writhes modulo 2k. We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large…
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.
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. We develop the theory of maximal representations of the fundamental group of a compact connected oriented surface with boundary, into a group of Hermitian type. For any such representation we define the Toledo invariant, for which we establish properties such as uniform boundedness on the representation variety, additi…
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.
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.
A Clifford algebra model for M"obius geometry is presented. The notion of Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced, and the structure equations for adapted frames are derived. These equations are discretized and the geometry of the occuring discrete nets and sphere congruences is disc…
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.
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.
Study on almost Robinson geometries, focusing on their intrinsic torsion and leaf space properties.
problem Investigating the geometry of almost Robinson manifolds, a Lorentzian analog of almost Hermitian manifolds.
method Classification based on intrinsic torsion, analysis of leaf space properties, and conformal invariants.
result Comprehensive classification of almost Robinson manifolds based on their intrinsic torsion.
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 study the geometry of fanning curves in the Grassmann manifold of n-dimensional subspaces of Rkn; we construct a complete system of invariants which solve the congruence problem. The geometry of the invariants themselves and their relation with classical invariants is also studied.
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.
Study automorphisms and Torelli groups of right-angled Artin groups.
problem Characterize automorphisms and Torelli groups of right-angled Artin groups.
method Introduce palindromic automorphism and Torelli groups, relate to GL(n,Z) and mapping class groups, find generating sets.
result Finite generating sets for Pi A_G and centralizer, determine when groups coincide.
New congruence subgroups found from braid group representations.
problem Deciding when braid group representations yield congruence subgroups.
method Classification of low-dimensional braid group representations and application of number theory.
result Construction of non-congruence subgroups for specific braid group representations.
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.