Classifies modules of surface-knots in terms of their properties.
problem Characterizing modules of surface-knots in terms of their properties.
method Using homology and covering spaces, the reduced first module is characterized.
result The reduced first module for every genus g is characterized in terms of properties of a finitely generated module.
New method learns both module structure and sequencing in neural networks.
problem Learning only the parameters and order of execution of neural modules.
method Expands the approach to learn the internal structure of modules, including the ordering and combination of arithmetic operators.
result Performance comparable to hand-designed modules achieved without extra supervisory signals.
Curvature defined for Hilbert modules and Kasparov modules.
problem Defining and studying curvature in Hilbert modules and Kasparov modules.
method Introduced curvature for densely defined universal connections on Hilbert C∗-modules relative to spectral triples. result Curvature only depends on the represented form of the universal connection modulo junk forms.
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.
Defines super projective modules and explores their properties.
problem Exploring the geometric-algebraic link in super geometry.
method Defined and explored super projective modules over supersmooth functions.
result Module of vector fields over a supersphere is a super projective module.
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 …
We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.
Paper compares skein modules to Kauffman bracket modules.
problem Comparing skein modules to Kauffman bracket modules.
method Using skein relations and Reshetikhin-Turaev model.
result Resolved the problem of comparing skein modules to Kauffman bracket modules.
Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills of homologies and homotopies, distant mountains - the Kauffman bracket skein mo…
Generalized Steinberg module presentation for Gaussian and Eisenstein integers.
problem Presenting Steinberg modules for specific number rings.
method Generalization of Bykovskii's presentation to Gaussian and Eisenstein integers.
result Generalization does not yield a presentation for all Euclidean number rings.
Enhanced Alexander module detects linking numbers in links.
problem Detecting linking numbers in links using Alexander modules.
method Defining and singling out meridians and longitudes in reduced Alexander modules.
result The enhanced Alexander module determines all linking numbers.
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…
Combinatorial approach to compute satellite knot invariants using graph theory.
problem Computing knot invariants for satellite knots using bordered Heegaard Floer homology.
method Construct weighted A∞-modules using decorated planar graphs and prove their isomorphism. result Combinatorial proof of A∞ structure relations for the constructed modules. New sl(2) action defined on a mathematical module.
problem No specific problem stated; focuses on mathematical construction.
method Construction of sl(2)-action on equivariant skein lasagna module.
result Infinitesimal sl(2)-symmetries constructed.
Studies modules over a category of Jacobi diagrams in handlebodies.
problem Understanding modules over a specific category of Jacobi diagrams.
method Generalizes adjunctions and studies subquotient modules.
result Generalizes adjunctions between modules and Casimir Lie algebra modules.
Enhances knot and link invariants using quandle modules.
problem Distinguishing knots and links using polynomial invariants.
method Integrates quandle modules into the quandle coloring quiver.
result The enhanced invariant distinguishes knots and links.
Introduces admissible skein modules for non-semisimple categories.
problem No specific problem stated; generalization of Kauffman skein algebra.
method Introduces admissible skein modules associated to ideals in pivotal categories.
result These modules generalize Kauffman skein algebra and relate to quantum invariants.
Let {T1,…,Tn} be a set of n commuting bounded linear operators on a Hilbert space H. Then the n-tuple (T1,…,Tn) turns H into a module over C[z1,…,zn] in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad …
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 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.
Formula for interleaving distance of rectangle persistence modules.
problem Calculating distances between rectangle persistence modules.
method Formulas based on rectangle geometry, extended to decomposable modules.
result Closed formulas for interleaving and bottleneck distances.
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.
New formula proves skein modules are finite for 3-manifolds.
problem Proving skein modules are finite for closed 3-manifolds.
method Using Heegaard splittings and algebraic computation.
result Skein modules are finite-dimensional, resolving a conjecture.
Researchers create a Fredholm module on fractal shapes like the Cantor set.
problem Constructing Fredholm modules on complex fractal structures.
method Combining combinatorial techniques with higher-dimensional analogues.
result Calculated Dixmier trace of operators induced by the module.
We show that the Kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus. Using this we derive nice formulas for the Jones-Wenzl idempotents and analyze the structure of the Kauffman bracket skein module of the unknot as a module over the Kauffman bracket skein modu…
New knot theory module shows torsion-ness in number theory.
problem Torsion-ness of Selmer modules in Galois representations.
method Introducing adjoint homological Selmer module for SL2-representations of knot groups. result Finitely generated torsion-ness of the new Selmer module.
The dualizing module of GL_n(O) varies, affecting cohomology vanishing and nonvanishing.
problem Understanding the dualizing module of GL_n(O) and its impact on cohomology.
method Analyzing the Steinberg module and a variant for GL_n(O), proving vanishing and nonvanishing theorems.
result The dualizing module of GL_n(O) is not always the Steinberg module, but a variant that accounts for orientation.
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.
Explains connections between monopoles and modules on elliptic curves.
problem Understanding relationships between different mathematical objects.
method Explains equivalences between monopoles and polystable bundles and modules.
result Monopoles and polystable difference modules on elliptic curves are equivalent.
The multivariate Alexander module of a link L has several subsets that admit quandle operations defined using the module operations. One of them, the fundamental multivariate Alexander quandle, determines the link module sequence of L.
A complex vector space V is a prehomogeneous G-module if G acts rationally on V with a Zariski-open orbit. The module is called etale if dimV=dimG. We study etale modules for reductive algebraic groups G with one-dimensional center. For such G, even though every etale module is a regular prehomogeneou…
A commuting n-tuple (T1,…,Tn) of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module H over C[z1,…,zn] in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h…
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.
This document contains tables with the classification of prehomogeneous modules for reductive algebraic groups with up to two simple factors due to Sato, Kimura and many others, as well as corresponding tables of the étale modules appearing in this list, determined by the author. It is intended as a convenient referenc…
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.
The study introduces a new equivalence for Poisson modules on complex projective varieties.
problem Understanding the structure of Poisson modules on complex projective varieties with mild singularities.
method Introducing a weak concept of Morita equivalence in the birational context for Poisson modules.
result Poisson modules are classified into three types based on their properties.
Some neural network modules are more critical to performance than others.
problem Understanding why some neural network architectures generalize better than others.
method Introduced module criticality, a measure based on the shape of loss valleys.
result Module criticality explains superior generalization performance of some architectures.
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.
The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry''. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohom…
Study meromorphic open-string vertex algebras and modules over Riemannian manifolds.
problem Characterize meromorphic open-string vertex algebras and their modules over Riemannian manifolds.
method Explicitly determine bases for meromorphic open-string vertex algebras and their modules, using parallel tensors and eigenfunctions of the Laplace-Beltrami operator.
result Every irreducible module of a specific type is completely reducible if every composition factor is generated by eigenfunctions of eigenvalue p(p−1)K for some p∈Z+. 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. New (co)homology theory for symmetric quandles developed.
problem Developing strong invariants for symmetric quandles.
method Introducing symmetric quandle modules and Beck modules, extending module theory, and constructing generalized (co)homology.
result Established an explicit isomorphism between symmetric quandle cohomology and group cohomology.
Classifies extended Abelian Chern-Simons theories using quadratic modules.
problem Classifying extended Abelian Chern-Simons theories.
method Using quadratic modules to classify theories.
result Finite quadratic modules classify extended Abelian Chern-Simons theories.
Enhanced bikei modules distinguish unoriented and non-orientable surface-links.
problem Distinguishing unoriented and non-orientable surface-links.
method Extending biquandle module invariants to unoriented surface-links using bikei modules.
result Enhanced bikei modules are more effective at distinguishing non-orientable surface-links than bikei homset cardinality alone.
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…
Corrects a paper on Alexander modules and answers a related question.
problem Peripheral elements in reduced Alexander modules
method Analyzes and corrects a paper on Alexander modules
result Answers a question and corrects a minor error in the original paper
The space of m-ary differential operators acting on weighted densities is a (m+1)-parameter family of modules over the Lie algebra of vector fields. For almost all the parameters, we construct a canonical isomorphism between this space and the corresponding space of symbols as sl(2)-modules. This yields to the notion o…
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…