The paper studies Kähler groups and their virtual algebraic fibrations, finding implications for their fundamental groups.
problem Understanding the structure of Kähler groups and their finite index subgroups.
method Analyzing the virtual algebraic fibrations and Albanese dimension of Kähler groups.
result Groups with virtual Albanese dimension ≤ 1 have strong relations with surface groups and coincide with Green--Lazarsfeld sets.
New groups found that don't virtually algebraically fiber, related to mapping class group orbits.
problem Existence of finite orbits for higher Prym representations of the mapping class group.
method Study of surface-by-surface and surface-by-free groups.
result Existence of free-by-free and free-by-surface groups that do not algebraically fiber.
Study BNS invariants to prove algebraic fibrations of certain groups.
problem Proving algebraic fibrations of group extensions.
method Study of BNS invariants to analyze group extensions.
result Groups with specific BNS invariants algebraically fiber.
Invariants for trivalent graphs using algebraic colorings.
problem Creating an invariant for virtual trivalent spatial graphs.
method Colorings by virtual Niebrzydowski algebras.
result Generalization and computational implementation of invariants.
Paper explores algebraic, topological, and combinatorial properties of singular virtual braids.
problem Understanding singular virtual braids and their properties.
method Algebraic relations, topological and combinatorial bijections, presentations.
result A bijection between singular abstract braids and singular virtual braids, leading to a presentation of the singular pure virtual braid monoid.
New invariant for virtual links using multi-switches and algebraic systems.
problem Creating invariants for virtual links.
method Introducing a general approach to construct invariant algebraic systems from multi-switches and virtual links.
result Introduces a new quandle invariant for virtual links.
Fibers of abelian varieties over curves can be approximated algebraically.
problem Approximating fibers of abelian varieties over curves algebraically.
method Proving the existence of algebraic approximations for fibrations.
result Fibrations with abelian varieties over curves have algebraic approximations.
Projectors defined in virtual Temperley-Lieb algebra with properties similar to Jones-Wenzl projectors.
problem Defining projectors in the virtual Temperley-Lieb algebra.
method Generalizing Jones-Wenzl projectors and defining new projectors with specific properties.
result Uniqueness of the projector fn and explicit formula for fn. This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this…
Study on algebraic structures of virtual singular braid monoid and its splittable extension.
problem Algebraic structures of virtual singular braid monoid and its splittable extension.
method Exploration of algebraic structures, construction of representation.
result Monoid VSBn is the splittable extension of VSPn by the symmetric group Sn. The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.
problem Understanding the relationship between Kähler-Ricci shrinkers and Fano fibrations.
method Using birational algebraic geometry, the paper proves properties of Kähler-Ricci shrinkers and formulates conjectures relating them to Fano fibrations.
result The existence of Kähler-Ricci shrinkers is conjectured to be related to K-stability of polarized Fano fibrations.
Study singular fibers in genus 2 algebraic fibrations and their monodromy factorizations.
problem Classify singular fibers in genus 2 algebraic fibrations and relate them to Lefschetz fibrations.
method Analyze four families of hypersurface singularities in C^3, determine resolutions, and find flat deformations into simpler pieces.
result Establish a dictionary between configurations of curves and monodromy factorizations for some genus 2 fibrations.
Study algebraic invariants of multi-virtual links.
problem Algebraic invariants of multi-virtual links.
method Determining moves, introducing operator quandles, constructing invariants.
result Established new invariants for multi-virtual links.
Classifies surfaces of section for Seifert fibrations.
problem Classifying surfaces of section for Seifert fibrations.
method Discussing branched coverings and relating surfaces of section to algebraic curves.
result Relates surfaces of section to algebraic curves in weighted complex projective planes.
In this paper we study two types of fibrations associated with a 3-dimensional unital associative irreducible algebra and their basic properties. We investigate trivial principal fibrations of degenerate semi-Euclidean sphere and their semi-conformal and projective models. We use Norden normalization method for constru…
Study on virtual singular braid groups with algebraic properties and homomorphisms.
problem Algebraic properties and homomorphisms of virtual singular braid groups.
method Numerical invariants, homomorphisms, semi-direct product decompositions, presentations, and quotients.
result Determined all group homomorphisms from VSGn to Sn and obtained corresponding semi-direct product decompositions. New proof of Genauer fibration sequence in cobordism categories.
problem Relating cobordism categories of closed and boundary-manifolds.
method Inspired from algebraic K-theory, using cobordism categories and delooping.
result Generalization of Waldhausen's Additivity theorem to cobordism categories.
New algebraic structure for distinguishing twisted virtual handlebody-links.
problem Distinguishing twisted virtual handlebody-links.
method Introducing twisted virtual bikeigebras and using them to define invariants.
result New invariants distinguish some twisted virtual handlebody-links.
Triality connects three polynomial bases in Lie algebra studies.
problem Understanding relationships between invariant polynomials.
method Triality of mSO(8,C) and Hitchin fibrations. result Explicit automorphisms linking polynomial bases identified.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Study shows crossing numbers for algebraic knots can differ by arbitrarily large amounts.
problem Comparing two crossing number definitions for algebraic knots.
method Analyzed Hopf fibration and complex singularities to compare crossing numbers.
result Difference between crossing numbers can be arbitrarily large.
Study on Milnor fibrations of arrangements with trivial algebraic monodromy.
problem Explicit formulas for Milnor fiber Betti numbers in complex hyperplane arrangements.
method Analysis of cohomology jump loci and lower central series quotients of π1(F).
result Found arrangements with same Betti numbers but different fundamental groups.
We study properly discontinuous and cocompact actions of a discrete subgroup Γ of an algebraic group G on a contractible algebraic manifold X. We suppose that this action comes from an algebraic action of G on X such that a maximal reductive subgroup of G fixes a point. When the real rank of any simple subg…
Study calculates Poisson cohomology of broken Lefschetz fibrations.
problem Computing Poisson cohomology of broken Lefschetz fibrations.
method Calculates cohomology at fold and Lefschetz singularities, adapting techniques for Sklyanin algebra.
result Compact formulas for Poisson coboundary operator in 4 dimensions.
We construct an explicit diffeomorphism taking any fibration of a sphere by great circles into the Hopf fibration, using elementary geometry--indeed the diffeomorphism is a local (differential) invariant, algebraic in derivatives.
This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. The paper sets up the background virtual knot theory, defines rotational virtual knot theory, stu…
This paper extends previous work on genus two fibrations by studying and resolving singular fibers.
problem Analyzing and resolving singular fibers in genus two fibrations.
method Using perturbations and carefully chosen polynomials to transform singular fibers into Lefschetz fibrations.
result Recovering the monodromy factorization and understanding the compactification of central fibers.
Introduces XC-tangles for quantum tangle invariants.
problem Quantum invariants of tangles and virtual tangles.
method Introduces XC-tangles and their equivalence to virtual upwards tangles.
result Extension of knot isotopy invariants to XC-tangles.
Study on p-Kähler structures on fibrations and Lie groups.
problem Existence of p-Kähler structures on complex manifolds. method Investigation of quasi-regular fibrations and reductive Lie groups with invariant complex structures.
result Construction of non-regular complex structures on Lie algebras sl(2m−1,R) for m≥2. Paper constructs representations for virtual braids and flat braids.
problem Calculating hyperbolic volumes of knot complements.
method Cluster algebra approach for virtual braid group.
result Forbidden relations do not hold in virtual braid group representation.
The paper defines and analyzes configuration Lie groupoids and orbifold braid groups.
problem Understanding the structure and properties of orbifold braid groups.
method Definitions and proofs of fibration theorems, short exact sequences, and poly-virtually free structures.
result The pure orbifold braid groups have poly-virtually free structure, generalizing classical braid groups.
Paper presents a nearly invertible mapping between high-dimensional and lower-dimensional spheres.
problem Lack of rigorous mathematical foundation in deep learning algorithms.
method Utilizes Multicomplex rotation groups and polyspherical coordinates to define two maps and a composite LG Fibration.
result Derives a distance difference function to determine invariant inner products under the transformation.
Introduces virtual tribrackets for knot coloring.
problem Coloring regions in virtual knot diagrams.
method Algebraic structure (virtual tribrackets) for knot coloring.
result Invariant can distinguish certain virtual knots.
Lie ∞-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie ∞-groupoids called `Lie ∞-groups' by integrating finite type Lie n-algebras. In order to study the compatibility between this…
The paper studies algebraic fibre spaces with specific properties and proves key results about their structure.
problem Analyzing properties of algebraic fibre spaces with strictly nef relative anti-log canonical divisors.
method Using projective klt pairs and fibration techniques, the paper proves locally constant fibration properties and rational connectedness.
result The fibration is locally constant with rationally connected fibers, and the base is a canonically polarized hyperbolic projective manifold.
We solve a long-standing question about the monodromy of certain complex surfaces.
problem When is the monodromy group of an algebraic family of complex varieties arithmetic?
method Topological analysis of the 'geometric' monodromy, valued in the mapping class group of the fiber.
result We resolve the question affirmatively for Atiyah-Kodaira manifolds.
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.
In this paper we investigate the virtual string links via a probabilistic interpretation. This representation can be used to distinguish some virtual string links from classical string links. In order to study the algebraic structure behind this probabilistic interpretation we introduce the notion of virtual flat biqua…
Defines new algebras for virtual link invariants, matching known polynomials.
problem Developing new mathematical structures for virtual link invariants.
method Introducing two towers of algebras, VTL and ATL, and determining their presentations and Markov traces.
result The invariants derived from the Markov traces match known polynomials for virtual links.
Real algebraic structures help classify overtwisted contact 3-spheres.
problem Classifying overtwisted contact structures on 3-spheres.
method Using real algebraic functions and open book decompositions.
result Most overtwisted contact structures are real algebraic.
In this note we show that if a compact Kahler manifold with trivial canonical bundle is the total space of a holomorphic fibration without singular fibers, then the fibration is a holomorphic fiber bundle. In the algebraic case, the fibration becomes trivial after a finite base change.
The paper discusses colorings and doubled colorings of virtual doodles.
problem Coloring virtual doodles using a new algebraic structure.
method Introduced a new algebra called a doodle switch and defined an invariant for virtual doodles.
result Introduced doubled colorings and defined an invariant for virtual doodles.
The study enumerates virtual quandles up to isomorphism.
problem Classifying virtual quandles up to isomorphism.
method Computer search and classification based on conjugacy class structures of rack automorphism groups.
result Classifications of virtual racks and quandles up to order 8.
This paper studies an algebraic invariant of virtual knots called the biquandle. The biquandle generalizes the fundamental group and the quandle of virtual knots. The approach taken in this paper to the biquandle emphasizes understanding its structure in terms of compositions of morphisms, where elementary morphisms ar…
Study Vassiliev invariants for virtual knots, expanding quantum theory.
problem Understanding Vassiliev invariants for virtual knots.
method Define chord diagrams, weight systems, and Lie algebra weight systems for rotational virtual knots.
result Extended quantum invariants capture more information than standard invariants.
New groups algebraically fibre with high-dimensional hyperbolic groups.
problem Finding new quasi-isometry classes of hyperbolic groups.
method Constructing infinitely many hyperbolic groups as finite-index subgroups of right-angled Coxeter groups.
result Groups algebraically fibre with finitely presented kernels, expanding finiteness properties.
Proves Alexander- and Markov-type theorems for virtual trivalent braids.
problem Classifying virtual trivalent braids and graphs.
method Two versions of the Markov-type theorem: algebraic and L-move based.
result Established new theorems for virtual trivalent braids.