Study knot invariants using automorphism groups of free nilpotent groups.
problem Developing knot invariants using automorphism groups.
method Nilpotently p-localization of knot groups and automorphism groups of free nilpotent groups. result Maps from outer automorphism groups yield knot invariants.
The study examines invariants of homology cylinders and their relations to free nilpotent groups.
problem Understanding invariants of homology cylinders and their connections to free nilpotent groups.
method Extensions of Johnson homomorphisms, Milnor invariants, and Orr invariants of links to homology cylinders; establishment of a combined filtration.
result Determination of the image of the filtration under the invariants and investigation of relations among the invariants.
Non-invariant complex structures on Lie groups are not biholomorphic to invariant ones.
problem Understanding non-invariant deformations of complex structures on Lie groups.
method Computed cohomologies to show non-biholomorphicity.
result Non-invariant complex structures are not biholomorphic to invariant ones.
The paper classifies invariant structures on complex almost Abelian groups.
problem Investigating invariant geometric structures on almost Abelian Lie groups.
method Explicit formulas for Haar measures, modular function, and generator fields were derived.
result All invariant tensor fields have constant coefficients in the invariant frame.
Invariants measure letter interleaving in groups, detecting group dimensions.
problem Detecting group dimensions in arbitrary groups.
method Defining letter-braiding invariants from cochain models of spaces with prescribed fundamental groups.
result Letter-braiding invariants are complete invariants of group dimension series.
Study convolution of invariant valuations on Lie groups.
problem Understanding convolution of valuations on Lie groups.
method Explicit formula for left-invariant valuations, showing existence of smooth bi-invariant valuations, defining convolution on arbitrary Lie groups.
result Unified convolution operations on Lie groups.
The paper explores left invariant k-symplectic structures on Lie groups with bi-invariant metrics.
problem Existence and properties of left invariant k-symplectic structures on Lie groups with bi-invariant metrics.
method Analyzing compact semi-simple Lie groups and specific Lie groups with bi-invariant pseudo-Riemannian metrics.
result Compact semi-simple Lie groups and many Lie groups do not carry left invariant k-symplectic structures, except for specific cases.
The BNS invariant is applied to Kähler groups in new proofs and results.
problem Understanding properties of Kähler groups through the BNS invariant.
method Applications of the Bieri-Neumann-Strebel invariant on Kähler groups.
result Amenable Kähler groups have an empty complement of the BNS invariant.
In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invarian…
New concept SB-generation helps classify transformation groups.
problem Classifying transformation groups through quasi-isometry invariants.
method Identifying SB-generated groups in specific transformation groups.
result SB-generation provides robust extension of finite generation.
The Hausmann-Weinberger invariant of a group G is the minimal Euler characteristic of a closed orientable 4-manifold M with fundamental group G. We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.
Paper proves Rohlin invariant's uniqueness and extends homology sphere invariants.
problem Proving the uniqueness of Rohlin invariant and extending homology sphere invariants.
method Using the Rohlin invariant's uniqueness, the paper extends invariants from trivial 2-cocycles to those with 2-torsion.
result Generalized invariants of homology spheres with 2-torsion values.
Classifies and computes cohomologies of complex structures on Lie groups.
problem Classifying and computing cohomologies of complex structures on Lie groups.
method Complete classification and computation of invariant cohomologies for left invariant structures.
result Computed invariant cohomologies for various generalized complex and Kähler structures.
Investigates BNSR invariants of link and knot groups, proving specific properties.
problem Characterizing finiteness properties of normal subgroups in link and knot groups.
method Analyzes BNSR invariants of link and knot groups, proving specific properties.
result Proves specific conditions for finiteness properties of link and knot groups.
This work provides statistical guarantees for GANs that are invariant to certain group symmetries.
problem Learning group-invariant distributions efficiently.
method Study of group-invariant GANs and their performance guarantees.
result Group-invariant GANs require fewer samples and have a reduced discriminator approximation error.
Group-invariant neural networks improve approximation accuracy for symmetric functions.
problem Improving approximation accuracy for symmetric functions using neural networks.
method Investigates the generalization error of group-invariant neural networks within the Barron framework.
result Group invariance introduces a factor δ that can significantly improve approximation accuracy when it is small.
The paper creates knot invariants using free groups.
problem Invariants of free knots (virtual knots).
method Constructing invariants valued in free groups.
result Series of invariants for free knots.
Survey on invariant conformal Killing forms on Lie groups.
problem Understanding invariant conformal Killing forms on Lie groups.
method Review of recent results and mention of open questions.
result Discussion of recent findings and open research areas.
We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in H3, the commensurability invariants known as the invariant trace field and invariant quaternion algebra. Our scripts also allow one to determine arithmeticity of such gr…
Defines invariants for reflection groups and connects them to Frobenius structures.
problem Understanding invariants for reflection groups and their relation to Frobenius structures.
method Defines good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants for reflection groups lead to Frobenius structure constants.
Finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of ea…
To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannia…
The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group then th…
This research studies affine invariance in continuous-domain convolutional neural networks.
problem Recognizing patterns and features under affine transformations in continuous domains.
method Introduces a new criterion for assessing affine invariance, embeds images into the affine Lie group, and analyzes convolution over this group.
result Extends the scope of geometrical transformations that deep-learning pipelines can handle.
New groups defined from knot diagrams, invariant under Reidemeister moves.
problem Classical knot groups are not invariant under all Reidemeister moves.
method Define quotient groups based on knot diagrams, invariant under Reidemeister moves.
result New groups include extended knot groups and are invariant under all Reidemeister moves.
Starting from considering deeper relationship between conjugacy classes and irreducible representations of a finite group G, we find some quite simple R−matrice defined by using finite groups. This construction produces many sets (or topological spaces) admitting braid group actions. We introduce conceptions "exten…
Study joint invariants on symplectic spaces, extending group and space variations.
problem Computing joint invariants on linear symplectic spaces.
method Review and extend previous work on group and space variations, relate to differential invariants.
result New computations and extensions of joint invariants.
Study describes isometry groups of specific Lie groups.
problem Understanding isometry groups of compact Lie groups.
method Analyzing bi-invariant metrics on SO(4) and U(n).
result Full groups of isometries identified for specific Lie groups.
This paper studies moduli spaces of statistical structures on Lie groups.
problem Understanding statistical structures on Lie groups.
method Introduced and studied moduli spaces for left-invariant statistical structures on Lie groups.
result Moduli spaces of left-invariant Riemannian metrics are singletons for certain Lie groups.
Study finds index of symmetry for solvable 3D Lie groups with left-invariant metrics.
problem Determining the index of symmetry for solvable 3D Lie groups with left-invariant metrics.
method Examined all solvable three-dimensional Lie groups, combined with previous work on unimodular groups.
result Index of symmetry is positive for every solvable 3D Lie algebra with a left-invariant metric, and is never 2.
Study classifies Riemann solitons on specific 3D Lorentzian groups.
problem Classifying Riemann solitons on three-dimensional Lorentzian Lie groups.
method Complete classification through analysis of left-invariant structures.
result Comprehensive classification of Riemann solitons on these groups.
Proves non-solvability of concordance groups using Milnor invariants.
problem Non-solvability of concordance groups of 2-string links and strongly invertible knots.
method Using Milnor invariants to prove non-solvability.
result Proves non-solvability of C(2) and equivariant concordance groups of strongly invertible knots. New method computes knot invariants using free group automorphisms.
problem Computing knot invariants efficiently and accurately.
method Using representations of braid groups by automorphisms of a free group.
result Compared isotopic invariants to Alexander polynomials.
We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. M…
Direct formula found for ADO invariants from homological representations.
problem Computing ADO invariants from quantum group representations.
method Direct homological formula for ADO invariants using partial traces of homological representations.
result Direct formula for ADO invariants without further truncations.
New approach to electric group for knots and links.
problem No previous publication of electric invariant for knots and links.
method Simple and general approach to electric group for oriented knots and links, using proper colouring of knot diagrams.
result Each homomorphism from the electric group to an arbitrary finite group can be described by a proper colouring of the diagram.
The paper extends the classification of bi-invariant 2-forms to infinite-dimensional Lie groups.
problem Classifying bi-invariant 2-forms on infinite-dimensional Lie groups.
method Generalized the classification from compact Lie groups to arbitrary finite-dimensional Lie groups and then to Milnor regular infinite-dimensional Lie groups.
result The classification of bi-invariant 2-forms extends to all Milnor regular infinite-dimensional Lie groups.
The study lists low-dimensional stratified groups and their properties.
problem Understanding the algebraic structure of stratified groups.
method Explicitly provided a list of low-dimensional stratified groups and their properties.
result All stratified groups in dimensions up to 7 and some free-nilpotent groups in dimensions up to 14 were studied.
New method prevents classifiers from relying on spurious correlations.
problem Group invariant learning fails to prevent classifiers from depending on spurious correlations.
method Statistical independence tests to construct groups and reweight samples by group label proportion.
result New method significantly outperforms existing group invariant learning methods in generalizing to spurious correlation shifts.
Positive braids linked to knot invariants and geometric monodromy groups.
problem Understanding knot invariants and geometric monodromy groups for positive braids.
method Associate braid monodromy groups to positive braids, identify these groups with framed mapping class groups for knots, and use these to determine knot invariants.
result Geometric monodromy groups of irreducible singularities are determined by genus and Arf invariant of associated knots.
Paper describes invariants of slice regular functions' automorphism group.
problem Understanding invariants of slice regular functions' automorphism group.
method Analyzes automorphism group of slice regular functions over Clifford algebras.
result Describes invariants of the automorphism group of slice regular functions.
Researchers compute BNSR-invariants for surface Houghton groups.
problem Computing BNSR-invariants for surface Houghton groups.
method Proved Stein--Farley cube complex is CAT(0), adapted Zaremsky's method.
result Computed BNSR-invariants of surface Houghton groups.
The paper classifies left-invariant pseudo-Riemannian metrics on specific Lie groups.
problem Classifying left-invariant pseudo-Riemannian metrics on Lie groups.
method Analyzing left-invariant metrics on specific Lie groups with n≥4. result A complete classification of left-invariant pseudo-Riemannian metrics for Lie groups of dimension n≥4. Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.
problem Defining invariants for elliptic Weyl groups.
method Defines a set of good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants give flat invariants and structure constants of Frobenius structures.
Vector fields invariant under Lie group action are finitely generated by polynomial fields.
problem Understanding invariant vector fields under Lie group actions.
method Analyzing the module of smooth vector fields invariant under a linear action of a compact Lie group.
result The module of invariant vector fields is finitely generated by polynomial fields.
Left-invariant metrics force 2-step nilpotent groups, preserving Kähler-like conditions.
problem Existence of left-invariant pluriclosed Hermitian metrics on Lie groups.
method Analyzing left-invariant metrics on unimodular Lie groups with abelian complex structures.
result Pluriclosed flow preserves Strominger Kähler-like conditions on 2-step nilpotent Lie groups.
New method classifies symplectic structures on Lie groups.
problem Classifying left-invariant symplectic structures on Lie groups.
method Using moduli space of left-invariant nondegenerate 2-forms.
result Classified left-invariant symplectic structures on specific Lie groups.
In 1987 Bieri, Neumann and Strebel introduced a geometric invariant for discrete groups. In this article we compute and explicitly describe the BNS-invariant for the pure braid groups.