The paper studies deformations of cohesive modules on complex manifolds.
problem Deformation theory of cohesive modules on compact complex manifolds.
method Development of Kuranishi maps and obstructions for deformations of cohesive modules.
result Generalization of deformation theory for holomorphic vector bundles and coherent sheaves.
Proves finiteness and holonomicity of skein modules for 3-manifolds.
problem Finiteness and holonomicity of skein modules for 3-manifolds.
method Defining skein transfer bimodules and using q-analogues of D-module theory.
result Internal skein modules are holonomic modules over the internal skein algebra of the boundary.
We show that for the Kauffman bracket skein module over the field of rational functions in variable A, the module of a connected sum of 3-manifolds is the tensor product of modules of the individual manifolds.
Paper computes skein modules of 3-manifolds using braids.
problem Computing skein modules of 3-manifolds.
method Using braids and knot algebras.
result Braid approach to HOMFLYPT and Kauffman bracket skein modules of specific 3-manifolds.
A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is …
Study on skein modules and character varieties of Seifert manifolds.
problem Characterizing and analyzing skein modules and character varieties of Seifert manifolds.
method Analyzing character varieties and skein modules of Seifert manifolds, computing dimensions and showing their equivalence.
result Finitely generated skein modules and reduced character varieties for certain Seifert manifolds.
Researchers compute the Kauffman bracket skein module of a specific 3-manifold.
problem Understanding the structure of Kauffman bracket skein modules for non-prime manifolds.
method Analyzing handle sliding relations to compute the module over Z[A±1]. result The skein module of (S1imesS2) # (S1imesS2) does not split into free and torsion submodules. The paper develops residue currents for cohesive modules and proves a generalized Poincaré-Lelong formula.
problem Analyzing coherent sheaves on complex manifolds using global analytic methods.
method Developing residue currents for cohesive modules and proving their properties.
result Proves a generalized Poincaré-Lelong formula for cohesive modules.
Skein lasagna module calculates 4-manifold invariants using handle decompositions.
problem Calculating invariants of 4-manifolds and links.
method Express skein lasagna module in terms of handle decompositions.
result Skein lasagna module can be locally infinite dimensional.
New definition of skein lasagna module for specific 4-manifolds.
problem Defining a new homology theory for specific 4-manifolds.
method Elementary definition using diagrams in the 4-manifold.
result Explicit calculations for disk bundles over S^2.
Novel method computes Kauffman bracket skein modules of small 3-manifolds.
problem Computing Kauffman bracket skein modules of small 3-manifolds.
method Developed a novel method to compute these modules.
result Dimension of skein modules of 3-manifolds related to character varieties and Abouzaid-Manolescu invariants.
Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpret…
Formulae for 1-3 handle attachments in 4-manifolds.
problem Understanding handle attachments in 4-dimensional manifolds.
method Developed general formulae for 1-3 handle attachments using skein modules.
result Derived complete description of gluing homomorphism on skein modules.
Holomorphic structures on quantum flag manifolds uniquely defined.
problem Defining unique holomorphic structures on quantum flag manifolds.
method Constructing covariant q-deformed holomorphic structures. result Holomorphic structures are unique for simple relative Hopf modules.
Researchers computed Kauffman bracket skein modules of specific Seifert manifolds.
problem Computing Kauffman bracket skein modules of Seifert manifolds.
method Provided presentations to compute the modules and showed their properties.
result The modules of certain Seifert manifolds are free or finitely generated.
We introduce a new skein module for three manifolds based on properly embedded surfaces and their relations introduced by D.Bar-Natan, and modified by M.Khovanov. We compute the structure of the modules for some manifolds, including Seifert fibred manifolds.
Introduces Floer lasagna modules using link Floer homology.
problem No specific problem stated; focuses on new mathematical concept.
method Inspired by skein lasagna module, uses link Floer homology.
result Computes Floer lasagna modules for specific 4-manifolds.
This paper generalizes L2 cohomology theory for complex manifolds.
problem Developing a L2 cohomology theory for Hodge modules on infinite covering spaces.
method Formulating a conjectural generalization of L2-Mixed Hodge structures using Saito's Mixed Hodge Modules.
result Partial results in the conjectural generalization of L2-Mixed Hodge structures.
Study on skein module dimensions at irreducible representations.
problem Dimension of skein module at irreducible representations.
method Localization of skein module at maximal ideal corresponding to irreducible representation.
result Localization forms a one-dimensional free module over the unreduced coordinate ring.
Counterexample disproves conjecture about 3-manifold modules.
problem Disproving a conjecture about 3-manifold modules.
method Provided a specific counterexample.
result Disproved Marché's conjecture about 3-manifold modules.
Extends Kauffman's formula to 3-manifolds with markings.
problem Generalizing Jones polynomial to 3-manifolds with markings.
method Epimorphism between skein modules of tangles in 3-manifolds.
result Defines new skein modules measuring the difference between Jones and bracket skein modules.
Proves finiteness of Kauffman bracket skein modules for 3-manifolds.
problem Finiteness of Kauffman bracket skein modules for 3-manifolds.
method Constructive proof methods, alternative to Witten's original proof.
result Proves finiteness conjecture for 3-manifolds with boundary and closed 3-manifolds.
Skein modules over 3-manifolds are shown to form line bundles.
problem Understanding the structure of skein modules over 3-manifolds.
method Using the Frobenius morphism, the skein module is mapped to a coherent sheaf over the SL2 character scheme.
result When the character scheme is reduced, the sheaf is a line bundle.
We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten.
Invariants for 3D manifolds with boundaries using crossed modules.
problem Develop invariants for compact 3D manifolds with boundaries.
method Employ crossed modules to count correct colors over triangulations.
result Validated invariants for compact 3D manifolds with boundaries.
Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum g…
Dirac operator invertibility proven for specific manifolds.
problem Invertibility of twisted Dirac operator on manifolds.
method Closed connected spin manifold with non-negative scalar curvature, flat Hilbert module bundle.
result Dirac operator is invertible under given conditions.
New Langlands duality conjectures for 3-manifold skein modules.
problem Understanding Langlands duality in 3-manifold skein modules.
method Combining representation theory of double affine Hecke algebras and 1-form symmetry structure.
result Recent special cases confirmed, with detailed proofs.
New algebraic framework for studying surfaces in 3-manifolds.
problem Understanding incompressible surfaces in 3-manifolds.
method Defining Bar-Natan modules and functors from Frobenius algebras.
result Geometric content of Bar-Natan modules is presented via tunneling graphs.
Study of skein modules in 3-manifolds, showing non-injectivity results.
problem Understanding skein modules in 3-manifolds and their behavior under gluing.
method Extended Kauffman bracket skein modules to 3-manifolds with marking, introduced new maps and studied their properties.
result Non-injectivity of certain maps in stated skein modules, especially when quantum parameter is a root of 1.
We present a definition of indefinite Kasparov modules, a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov module we can associate a pair of (genuine) Kasparov modules, and that this process is reve…
Extend Kauffman bracket skein module to homology theory using Heegaard splittings
problem Extend Kauffman bracket skein module to homology theory
method Combinatorial approach using Heegaard splittings
result Homology theory depends on Heegaard splittings
Quantizes b-symplectic toric manifolds using T-modules.
problem Quantization of b-symplectic toric manifolds. method Bohr-Sommerfeld quantization via T-modules. result Dimension of quantization coincides with signed count of integral points in moment polytope.
Koszul duality for manifold modules proven.
problem Proving Koszul self duality of manifold modules.
method Using generalized Thom complexes and operads in Top.
result Koszul self duality of little disk modules proven.
We improve our previous results on indefinite Kasparov modules, which provide a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. In particular, we can weaken the assumptions that are imposed on indefinite Kasparov modules. Using a new theorem by Lesch an…
The paper shows compatibility between two quantum maps for surfaces and 3-manifolds.
problem Connecting quantum trace and UV-IR maps for surfaces and 3-manifolds.
method Analyzing compatibility under triangulation changes and using skein modules.
result Compatibility of quantum trace and UV-IR maps for surfaces and 3-manifolds.
Let k be an integral domain containing the invertible elements α, s and \frac{1}{s-s^{-1}}. If M is an oriented 3-manifold, let K(M) denote the Kauffman skein module of M over k. Based on the work on Birman-Murakami-Wenzl algebra by Beliakova and Blanchet, we give an ``idempotent-like'' basis for the Kauffman skein mod…
We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the Z_2 homology of the manifold, …
We define 2-crossed module bundle 2-gerbes related to general Lie 2-crossed modules and discuss their properties. A 2-crossed module bundle 2-gerbe over a manifold is defined in terms of a so called 2-crossed module bundle gerbe, which is a crossed module bundle gerbe equipped with an extra sructure. It is shown that s…
Study of cubic skein modules in 3-sphere and arbitrary 3-manifolds.
problem Lack of systematic study of higher degree skein modules.
method Investigation of cubic skein module structure and properties in 3-sphere and arbitrary 3-manifolds.
result Establishment of a foundational framework for higher skein modules.
3D quantum trace map connects 3-manifold quantizations.
problem Quantization of 3-manifold character varieties.
method Study of stated skein modules and face suspensions.
result Existence of 3D quantum trace map proved.
Geometrically proves twisted Poincaré duality for orientable Poisson manifolds.
problem Establishing twisted Poincaré duality for Poisson manifolds.
method Geometrically reinterprets algebraic constructions of twisted Poisson modules and Poisson chain complexes.
result Explicit chain isomorphism between Poisson cochain and chain complexes with coefficients in Poisson modules.
We describe in this chapter (Chapter IX) the idea of building an algebraic topology based on knots (or more generally on the position of embedded objects). That is, our basic building blocks are considered up to ambient isotopy (not homotopy or homology). For example, one should start from knots in 3-manifolds, surface…
Study quandle modules over geometric quandles and their relation to Lie-Yamaguti representations.
problem Understanding quandle modules and their connection to Lie-Yamaguti representations.
method Examine quandle modules over quandle spaces, focusing on geometric structures.
result Modules over quandle spaces are linked to representations of Lie-Yamaguti algebras.
Study q-series for 3-manifolds with line defects, proving homomorphism and conjecturing holomorphic modularity.
problem Understanding BPS q-series for 3-manifolds with line defects. method Proving homomorphism from skein module to space of q-series, conjecturing holomorphic modularity. result Holomorphic quantum modularity of q-series suggests new approach to Langlands duality. Quantum trace map defined for 3-manifolds with torus boundaries.
problem Quantifying topological structures of 3-manifolds with torus boundaries.
method Defining a quantum trace map from skein module to a quantum torus module.
result Established a 3D quantum trace map for 3-manifolds with torus boundaries.
Study torsion in Kauffman bracket skein modules of 3-manifolds.
problem Identify compact oriented 3-manifolds with torsion in their Kauffman bracket skein modules.
method New criteria based on SL2(C)-character varieties and effective criteria for manifolds with incompressible tori. result Many counterexamples to Kirby's list and new criteria for torsion presence.
Study Kauffman bracket skein modules of Seifert fibered spaces.
problem Understanding the structure of Kauffman bracket skein modules.
method Investigate spanning sets and module structure.
result Kauffman bracket skein modules are finitely generated.