New theory connects non-abelian bundle gerbes to abelian ones.
problem Challenges in extending higher gauge theory beyond fake-flat sector.
method Developed a comprehensive theory of adjusted connections on non-abelian bundle gerbes.
result Established a new coordinate-independent formulation of lifting theorem.
Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition functi…
Introduces semi-abelian generalized complex structures.
problem Deformation theory of abelian complex structures.
method Definition and examples of semi-abelian generalized complex structures.
result Illustration of new concept with examples.
We discuss a general framework for cutting constructions and reinterpret in this setting the work on non-Abelian symplectic cuts by Weitsman. We then introduce two analogous non-Abelian modification constructions for hyperkähler manifolds: one modifies the topology significantly, the other gives metric deformations. We…
Classifies SU(2)-abelian graph manifolds with a single JSJ torus.
problem Classifying SU(2)-abelian 3-manifolds.
method Using Seifert coefficients and Heegaard Floer homology.
result SU(2)-abelian graph manifolds are Heegaard Floer homology L-spaces.
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
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.
3-manifolds with two abelian-filled spaces.
problem Finding 3-manifolds with multiple abelian-filled spaces.
method Constructing 3-manifolds with specific embedding properties.
result 3-manifolds with at least two abelian-filled spaces.
Abstract: Proves non-abelian group of equivariant concordance.
problem Equivariant concordance group non-abelian
method Infinite family of nontrivial commutators
result Equivariant concordance group is not abelian
Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.
problem Characterizing harmonic almost complex structures on almost abelian Lie groups and solvmanifolds.
method Adapted Gray-Hervella classification to almost abelian Lie groups, characterized harmonic structures.
result Examples of harmonic almost complex structures in different Gray-Hervella classes on compact almost abelian solvmanifolds.
Characterizes quandles with abelian inner automorphisms.
problem Understanding quandles with specific automorphism properties.
method Generalizes previous work to construct new quandles.
result Homogeneous quandles with abelian inner automorphisms are abelian extensions of trivial quandles.
Extends Chern character to non-abelian cohomology, linking to physics.
problem Generalizing Chern character to non-abelian cohomology.
method Leveraging dg-algebraic rational homotopy theory and de Rham theorem.
result Generalizes Chern-Dold character, Chern-Weil homomorphism, and Cheeger-Simons homomorphism.
Study abelianization of Lie algebroids and groupoids, providing conditions for existence.
problem Existence conditions for abelianization of Lie algebroids and groupoids.
method Investigation of abelianization for Lie algebroids and groupoids, providing necessary and sufficient conditions.
result Necessary and sufficient conditions for the existence of abelianization in both Lie algebroids and groupoids.
Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.
problem Characterizing abelian structures on odd-dimensional Lie algebras.
method Introducing and analyzing abelian almost contact and almost 3-contact structures, and their compatibility conditions.
result Classification of 5-dimensional Sasakian Lie algebras and 7-dimensional abelian almost 3-contact Lie algebras.
The paper classifies fixed subgroups of endomorphisms in free-abelian times surface groups.
problem Characterizing fixed subgroups of endomorphisms in specific group structures.
method Study of endomorphisms, classification of fixed subgroups, and equivalent conditions for end-fixed subgroups.
result Complete classification of fixed subgroups in free-abelian times surface groups.
Study of complex and Hermitian structures on specific Lie groups.
problem Classifying Lie groups with specific geometric structures.
method Analysis of left-invariant structures on almost abelian Lie groups.
result Classification of six-dimensional generalized Kähler almost abelian Lie groups.
Study transcendence of abelian differential periods from bi-algebraic perspective.
problem Arithmetic and functional transcendence of periods of abelian differentials.
method Bi-algebraic structure on strata of abelian differentials.
result Characterization of arithmetic points and proof of linear bi-algebraic curves.
We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H whe…
Study pseudo-Kähler structures on almost abelian solvmanifolds.
problem Classify pseudo-Kähler structures on almost abelian Lie algebras.
method Analyzing invariant structures on solvmanifolds with specific Lie algebra properties.
result Classification of pseudo-Kähler structures on almost abelian Lie algebras.
Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.
problem Classify balanced Hermitian structures on almost abelian Lie algebras.
method Classify six-dimensional almost abelian Lie algebras with balanced structures, investigate flow of balanced metrics and anomaly flow.
result Prove conjecture for compact almost abelian solvmanifolds with left-invariant complex structures.
Complete Calabi-Yau metrics on abelian fibrations over complex space.
problem Constructing complete Calabi-Yau metrics on noncompact abelian fibrations.
method Using abelian fibrations over C, we construct complete Calabi-Yau metrics and provide compactification. result We provide a compactification for the abelian fibration X such that the compactified variety has a negative canonical bundle. Study on CKY forms on almost abelian Lie groups, proving parallelism and characterizing non-parallel cases.
problem Characterizing CKY forms on almost abelian Lie groups and proving parallelism.
method Analyzing almost abelian metric Lie algebras, proving parallelism for CKY forms, and classifying cases up to dimension 5.
result CKY forms are parallel on almost abelian Lie algebras, with exceptions for p=1 and p=n−1. Proves even degrees can be realized in Abelian differential strata.
problem Realizing even degrees as stretch factors in Abelian differentials.
method Analyzes Thurston-Veech stretch factors in moduli spaces.
result Every even degree 2d≤2g is realized in Abelian differentials. The paper constructs toric vector bundles using spectral networks and non-abelianization.
problem Understanding how holomorphic vector bundles arise from spectral networks and non-abelianization.
method Constructing toric vector bundles on complete toric surfaces via spectral networks and non-abelianization.
result The moduli space of rank 2 toric vector bundles over toric surfaces admits an A-type X-cluster structure. We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form ω(ω-invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form ω, up to a linear isomorphism (ω-equivariant diffeomorphisms). Every abelian extensio…
We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting gro…
The paper constructs a new Ricci-flat metric on almost abelian Lie groups.
problem Finding Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups.
method Constructing left-invariant metrics on almost abelian Lie groups, focusing on dimensions four or higher.
result A new Ricci-flat metric that generalizes the Petrov solution to higher dimensions.
Classifies complex symplectic structures on Lie algebras with large abelian ideals.
problem Classifying complex symplectic structures on Lie algebras with large abelian ideals.
method Two constructions of complex symplectic structures on Lie algebras with large abelian ideals, considering compact quotients of Lie groups.
result Complete classification of complex symplectic structures on almost abelian Lie algebras.
We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of ran…
We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the …
This paper shows how to construct Abelian differentials with any prescribed singularities.
problem Constructing Abelian differentials with specific orders and residues.
method Flat representation of Abelian differentials.
result Every pattern of orders and residues can be realized in Abelian differentials, except for two families in genus zero.
Study shows virtually abelian subgroups have commensurable counterparts in mapping class groups.
problem Understanding virtually abelian subgroups in mapping class groups.
method Proving commensurability and normalizer relationships for virtually abelian subgroups.
result Upper bounds for geometric dimension of mapping class groups for abelian subgroups of bounded rank.
We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear ho…
The paper studies a flow on complex Lie groups, showing convergence to solitons.
problem The study of curvature flows on complex Lie groups.
method Positive Hermitian curvature flow on left-invariant metrics.
result The flow converges to solitons in both nilpotent and almost-abelian cases.
Paper confirms conjecture for specific Lie algebras.
problem Fino-Vezzoni conjecture on Lie algebras with abelian ideals of codimension two.
method Analyzes unimodular Lie algebras with abelian ideals of codimension two.
result Confirms the Fino-Vezzoni conjecture for this specific class of Lie algebras.
Quantum codes linked to abelian varieties, providing mathematical rigor.
problem Quantum error correction through complex abelian varieties.
method Mathematical formulation of Gottesman-Kitaev-Preskill codes using abelian varieties.
result Asymptotic isometry of encoding, precise gate realizations, and failure probability optimization.
New abelian cohomology theory applied to rigidity of flows.
problem Rigidity of Anosov flows and negatively curved surfaces.
method Development of abelian Livshits theory for transitive Anosov flows.
result Abelian Livshits theorem for homologically full Anosov flows.
Study abelian varieties' Weil-Petersson metric asymptotics.
problem Asymptotic behavior of Weil-Petersson metric on abelian varieties.
method Linking asymptotic with multi-scale collapsing limits of parametrized flat tori.
result Refined description of Weil-Petersson metric on abelian varieties.
The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.
problem Generalizing Kodaira vanishing theorems for non-abelian settings.
method Non-abelian Hodge theory and Mixed Twistor D-modules.
result Generalized Kodaira vanishing theorems for various settings.
The paper defines angle structures on 3-manifolds using abelian groups.
problem Finding conditions for labelings of tetrahedra in 3-manifolds.
method Non-trivial conditions on labelings of tetrahedra in a triangulated 3-manifold using an abelian group.
result Angle structures can be used to define a non-trivial condition on labelings of tetrahedra.
We establish a Penrose-Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly N=(1,0) and N=(2,0) supersymmetric non-Abeli…
Let Λ be a finite abelian group. A dynamical system with transformation group Λ is a triple (A,Λ,α), consisting of a unital locally convex algebra A, the finite abelian group Λ and a group homomorphism $α:Λ\rightarrow\Aut(A)$, which induces an action of Λ on A. In this paper we present a new, geometricall…
Study abelian factors in Lie algebras from graph edge labels.
problem Understanding abelian factors in Lie algebras from graph edge labels.
method Analyzing 2-step nilpotent Lie algebras constructed from graphs, computing abelian factors, and studying singularity properties.
result Explicit computation of abelian factors for various graph families.
Study non-Abelian gauge theories using Poisson bracket structures.
problem Defining a Poisson bracket structure on solution spaces.
method Using coisotropic embedding theorem.
result Defined Poisson bracket structure for non-Abelian gauge theories.
In this paper, we consider the concept of connection cochain of central extensions introduced by Moriyoshi and apply it to the abelian case. We will show the relationship between connection cochain and connection 1-form of a principal bundle whose structure group is abelian.
We determine the abelianization of the symmetric mapping class group of a double unbranched cover using the Riemann theta constant, Schottky theta constant, and the theta multiplier. We also give lower bounds of the abelianizations of some finite index subgroups of the mapping class group.
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.
The paper finds minimal generating sets and abelianizes the quasitoric braid group.
problem Understanding the structure of quasitoric braids and their subgroup properties.
method Provided two minimal generating sets and determined the abelianization.
result Minimal generating sets and abelianization of the quasitoric braid group were determined.