Study centers of mapping-torus groups to define knot and mapping class invariants.
problem Understanding the center of mapping-torus groups.
method Determine the center of meta-nilpotent quotients of mapping-torus groups.
result Introduce two invariants of knots and mapping classes as quadratic forms.
New invariants found for mappings between non-symmetric affine spaces.
problem Finding new invariants for mappings between non-symmetric affine spaces.
method Obtained invariants using factored deformation tensor and novel Weyl type invariants.
result Novel Weyl type invariants for mappings between non-symmetric affine spaces.
The article defines and analyzes Clairaut anti-invariant maps between Riemannian and trans-Sasakian manifolds.
problem Characterizing Clairaut anti-invariant Riemannian maps between specific types of manifolds.
method Deriving conditions for Clairaut maps, discussing integrability, and establishing harmonicity.
result Necessary and sufficient conditions for Clairaut anti-invariant maps are derived.
New forms of symmetric shift-invariant subspaces found for harmonic maps.
problem Understanding harmonic maps into symmetric and k-symmetric spaces. method Imposing a symmetry condition on shift-invariant subspaces of a Hilbert space.
result Obtained new general forms for symmetric shift-invariant subspaces and extended solutions.
Algorithm calculates Hopf invariant for simplicial mappings.
problem Computing Hopf invariant for simplicial mappings.
method Proposed an algorithm based on Whitehead's integral formula.
result Algorithm successfully computes Hopf invariant.
As a generalization of anti-invariant Riemannian submersions, we introduce anti-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We give examples and investigate the geometry of foliations which are arisen from the definition of an anti-Riemannian map. Then we give a decomposition theo…
New invariants found for a specific type of mapping in generalized Riemannian space.
problem Understanding transformations of Christoffel symbols in generalized Riemannian space.
method Analysis of third type almost geodesic mappings.
result Obtained new invariants that are projective parameters and tensors.
Study beta function for convex billiard maps, linking spectral invariants.
problem Understanding spectral invariants of convex billiard maps.
method Birkhoff normal form via constructive generating functions, explicit beta function formula.
result Linked spectral invariants to beta function for convex billiard maps.
Innovates a three-component link homotopy invariant.
problem Classifying three-component link maps up to homotopy.
method Developed tools and invariants for distinguishing three-component link maps.
result Found three-component link maps that are not homotopic.
New invariant for CR maps from spheres discovered.
problem Identifying CR maps from spheres.
method Introducing a CR analogue of the Ahlfors derivative.
result The invariant distinguishes many sphere maps and vanishes for linear embeddings.
Subtle issues arise when extending homotopy invariants to spaces of functions having little regularity, e.g., Sobolev spaces containing discontinuous functions. Sometimes it is not possible to extend the invariant at all, and sometimes, even when the formulas defining the invariants make sense, they may not have expect…
Defines new link-homotopy invariants using Milnor's higher order link invariants.
problem Link-homotopy invariants for link maps of multiple components.
method Uses Milnor's higher order link invariants and combinatorial theory of cut-diagrams.
result Provides practical algorithms to compute these invariants and detects families of examples.
New descriptions of a subgroup in mapping class groups.
problem Understanding the normal subgroup generated by genus n bounding pair maps. method Using Chillingworth and Casson-Morita invariants.
result Two descriptions of the subgroup generated by genus n bounding pair maps. Study numerical invariants under retraction maps between topological spaces.
problem Understand behavior of invariants like cohomological dimensions under retractions.
method Introduced a notion of retraction and studied several numerical invariants.
result Proved inequalities between invariants hold under retractions.
Study links' flat-virtual diagrams to create link invariants.
problem Equivalence and invariants of flat-virtual diagrams.
method Maps from links in thickened surfaces to flat-virtual links.
result Investigation of flat-virtual diagrams' equivalence and invariants.
New formulas compute link invariants using configuration space degrees.
problem Computing specific link invariants without homotopy theory.
method Degrees of maps of configuration spaces, Jin suspension operations.
result Unexpected formulas for β and μ invariants. The paper studies Clairaut maps on Sasakian manifolds.
problem Investigating Clairaut maps on Sasakian manifolds.
method Analyzing necessary and sufficient conditions for geodesics and biharmonicity.
result Conditions for Clairaut anti-invariant Riemannian maps on Sasakian manifolds.
Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.
problem Understanding the relationship between Gaussian curvature and singularities of Gauss maps of cuspidal edges.
method Analyzes geometric invariants and types of singularities of Gauss maps to define and characterize positivity/negativity of cusps.
result Defines and characterizes positivity/negativity of cusps of Gauss maps by geometric invariants of cuspidal edges, and shows relation between sign of cusps and Gaussian curvature.
In this paper, we discuss relations among several invariants of 3-manifolds including Meyer's function, the eta-invariant, the von Neumann rho-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface.
New shifting chain map enhances quandle invariants for links.
problem Enhancing quandle invariants for links and surfaces.
method Introducing a shifting chain map σ and its pull-back σ# to transform cocycles.
result Shifting chain map σ# transforms 2-cocycles to 3-cocycles, enhancing invariants.
Study invariants of Z/p-homology 3-spheres from abelianization of mapping class groups.
problem Deciding and constructing invariants of Z/p-homology 3-spheres. method Formulating a criterion and using families of trivial 2-cocycles on the abelianization of the level-p mapping class group. result Disproved a conjectured extension of the Casson invariant for rational homology 3-spheres.
General invariants of a geometric mapping of a symmetric affine connection space are obtained in this paper. These invariants are generalizations of the previous obtained basic invariants (see [16]). Moreover, these invariants are related with the Thomas projective parameter and the Weyl projective tensor.
We study quantum moment maps of G-invariant star products, which are a quantum analogue of the moment map for classical Hamiltonian systems. Introducing an integral representation, we show that any quantum moment map for a G-invariant star product is differentiable. This property gives us a new method for the class…
Study non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
problem Non-linear Dirichlet-to-Neumann map for Poincaré-Einstein fillings.
method Description of range in terms of Dirichlet-to-Neumann tensor, construction of hypersurface invariants.
result Unique conformally invariant Dirichlet-to-Neumann hypersurface invariants for Poincaré-Einstein fillings.
Study links harmonic maps to shift-invariant subspaces in complex function spaces.
problem Understanding the relationship between harmonic maps and shift-invariant subspaces.
method Operator-theoretic methods to derive a criterion for the finiteness of the uniton number.
result Derives a criterion for the finiteness of the uniton number in harmonic maps.
The Hopf invariant is linked to null-homotopy properties of maps.
problem Understanding the relationship between the Hopf invariant and null-homotopy of maps.
method Using the generalized Hopf invariant and constructing smooth null-homotopies, the paper explores the relationship between the Hopf invariant and null-homotopy properties of maps.
result Sharp results on the relationship between the Hopf invariant and null-homotopy properties of maps, showing the necessity and sufficiency of certain conditions.
Extends knot invariant to filtered grid complexes.
problem Knot invariants and grid complexes.
method Combining Ozsváth-Szabó-Stipsicz crossing-change maps with Alishahi-Eftekhary l(K) invariant.
result Combinatorial formulation of knot invariant.
This study is motivated by the researches in the field of invariants of geodesic and conformal mappings presented in (T. Y. Thomas, [22]) and (H. Weyl, [25]). The Thomas projective parameter and the Weyl projective tensor are generalized in this article. Generators for vector spaces of invariants of geometric mappings …
Characterizes conformal Gauss maps into spheres.
problem Understanding conformal Gauss maps of surfaces.
method Invariant formulation and proof of characterisation of harmonic maps.
result Characterisation of harmonic maps as conformal Gauss maps of Willmore surfaces.
New invariants explain topological properties of pseudo-Anosov maps.
problem Understanding invariants of pseudo-Anosov maps at roots of unity.
method Numerical methods and quantum modularity conjecture.
result Descendant invariants related by Fourier transform.
This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed norm…
Researchers calculate spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.
problem Calculating spectral invariants from Dirichlet-to-Neumann map for Witten-Laplacian with potential.
method Established an effective procedure to calculate all coefficients of the spectral asymptotic formula of the Dirichlet-to-Neumann map.
result Explicitly provided the first four coefficients of the spectral asymptotic formula.
Study shows symplectic mapping groups of K3 surfaces are infinitely generated.
problem Understanding symplectic mapping class groups of K3 surfaces.
method Uses Kronheimer's approach and Seiberg-Witten invariants.
result Symplectic mapping class groups of many K3 surfaces are infinitely generated.
Conditions for pre-quantizability of G-invariant forms are derived using moment maps.
problem Conditions for equivariant prequantizability of G-invariant forms.
method Conditions derived using moment maps and obstructions computed.
result Necessary and sufficient conditions for equivariant pre-quantizability are computed.
The paper studies bi-slant Riemannian maps to Kenmotsu manifolds and derives inequalities.
problem Investigating bi-slant Riemannian maps and their properties.
method Introducing and studying bi-slant Riemannian maps, deriving curvature relations and inequalities.
result Construction of Chen-Ricci inequalities, DDVV inequalities, and optimal inequalities involving Casorati curvatures.
Study on Casson invariant and its variants in mapping class groups.
problem Understanding finite type invariants and their vanishing properties in Johnson filtration.
method Analyzing Casson invariant, λ, λ2, and their morphisms on Johnson filtration levels. result Invariant λ2−18λ2+3λ vanishes on the fifth level of Johnson filtration, Mg,1(5). The paper studies quandles over a hyperboloid and computes a knot invariant.
problem Computing knot invariants for specific algebraic structures.
method Defined quandles over a hyperboloid and computed a longitudinal mapping invariant.
result Computed a new knot invariant for SL(2,R). Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.
problem Quantifying uniqueness of conformal-harmonic maps on 4-manifolds.
method Proves a quantitative uniqueness result using convexity and second order Hardy inequality.
result Proves a version of second order Hardy inequality on manifolds.
Study on stable maps from 3-sphere to 3-space, focusing on singularities and invariants.
problem Understanding the behavior of stable maps from S3 to R3. method Analysis of codimension-one transitions, singular set behavior, and global invariants.
result Effects of decompositions on global invariants with prescribed branch sets.
We affirmatively address the question of whether the proposed link homotopy invariant ω of Li is well-defined. It is also shown that if one wishes to adapt the homotopy invariant τ of Schneiderman-Teichner to a link homotopy invariant of link maps, the result coincides with ω.
We compare two different types of mapping class invariants: the Hochschild homology of an A∞ bimodule coming from bordered Heegaard Floer homology, and fixed point Floer cohomology. We first compute the bimodule invariants and their Hochschild homology in the genus two case. We then compare the resulting comput…
New link invariants derived from L2-Burau maps of braids.
problem Developing new link invariants from L2-Burau maps. method Generalizing L2-Burau maps to all quotients of the group of the braid closure and proving corresponding L2-Alexander torsions. result Obtained twisted L2-Alexander torsions of the braid closure, recovering topological information like hyperbolic volumes. In this paper, we study spacelike rotational surfaces which are called boost invariant surfaces in Minkowski 4-space E41. We give necessary and sufficient condition for flat spacelike rotational surface to have pointwise 1-type Gauss map. Also, we obtain a characterization for boost invariant marginally trapped surface…
Parity mappings from the chords of a Gauss diagram to the integers is defined. The parity of the chords is used to construct families of invariants of Gauss diagrams and virtual knots. One family consists of degree n Vassiliev invariants.
We show that an orientable pseudo-Anosov homeomorphism has vanishing Sah-Arnoux-Fathi invariant if and only if the minimal polynomial of its dilatation is not reciprocal. We relate this to works of Margalit-Spallone and Birman, Brinkmann and Kawamuro. Mainly, we use Veech's construction of pseudo-Anosov maps to give ex…
We introduce slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of slant immersions, invariant Riemannian maps and anti-invariant Riemannian maps. We give examples, obtain characterizations and investigate the harmonicity of such maps. We also obtain necessary and sufficie…
We outline the construction of invariants of Hamiltonian group actions on symplectic manifolds. These invariants can be viewed as an equivariant version of Gromov-Witten invariants. They are derived from solutions of a PDE involving the Cauchy-Riemann operator, the curvature of a connection, and the moment map.