Generates moves for oriented singular links using algebraic structures.
problem Understanding and distinguishing oriented singular links.
method Introduced algebraic structures to study oriented singular knots and links.
result Colorings of singular knots by new algebraic structures are invariants and distinguish some links.
Study on 2D special Kaehler structures focusing on singularities.
problem Characterizing singularities in 2D special Kaehler structures.
method Analysis of isolated singularities and computation of monodromy.
result Construction of continuous families of special Kaehler structures with isolated singularities.
Defines algebraic structures from singular knot theory.
problem Distinguishing singular knots and links.
method Generalized Reidemeister moves and colorings.
result Set of colorings is an invariant of singular links.
Study non-degenerate singular points of Poisson-Nijenhuis structures.
problem Non-degenerate singular points of Poisson-Nijenhuis structures.
method Completely describe pairs of compatible Poisson structures near singular points.
result Pairs of compatible Poisson structures near singular points are completely described.
We call a singularity of a presymplectic form ω ω ω removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the removability of singularities is given in terms of regularizing functions for pure spinor…
Develops Chern-Weil theory for singular foliations.
problem Chern-Weil theory for Haefliger-singular foliations.
method Constructs explicit forms representing characteristic classes in de Rham cohomology.
result Theory applies to general smooth Haefliger structures up to homotopy.
In General Relativity the metric can be recovered from the structure of the lightcones and a measure giving the volume element. Since the causal structure seems to be simpler than the Lorentzian manifold structure, this suggests that it is more fundamental. But there are cases when seemingly healthy causal structure an…
Study of G 2 G_2 G 2 -structures with isolated singularities and bounded torsion.
problem Understanding G 2 G_2 G 2 -structures with special torsion and isolated singularities. method Revisiting known examples, describing symmetries, and analyzing collapsing of circle fibres.
result Collapsing circle fibres at isolated points cannot produce G 2 G_2 G 2 -structures with bounded torsion. 3D foliation study finds Poisson structure with specific singularities.
problem Characterizing Poisson structures on Bott-Morse foliations in 3D.
method Analyzes Bott-Morse foliations, computes Poisson bivectors and symplectic forms.
result Linear, singular Poisson structure of rank 2 with Bott-Morse singularities found.
Study on contact Hamiltonian functions for singular contact structures.
problem Understanding infinitesimal contact transformations on singular contact structures.
method Showed injectivity and provided an explicit local formula for the inverse map.
result Explicit local formula for the inverse map when contact structure has singularities of the first type.
Study reveals GAGA phenomenon in Poisson cohomology for plane structures with isolated singularities.
problem Understanding Poisson cohomology for plane structures with isolated singularities.
method Determined Gerstenhaber algebra structure over Poisson cohomology groups.
result GAGA type phenomenon observed in Poisson cohomology.
The study proves that certain surface singularities are planar only for A n A_n A n -singularities.
problem Characterizing planar links of normal surface singularities.
method Using contact structures and symplectic fillings, the study applies obstructions to canonical contact structures.
result Links of isolated singularities of surfaces in the complex 3-space are planar only for A n A_n A n -singularities. Local models for special Kähler structures in 2D computed without essential singularities.
problem Computing holonomy of special Kähler structures in 2D.
method Constructing local models assuming no essential singularities in the holomorphic cubic form.
result Computed holonomy of the flat symplectic connection.
Study focuses on classifying special geometric structures.
problem Classify singular affine structures of integrable systems.
method Classification through simple semitoric systems equivalence.
result Counterexamples exist for multiple pinched fibers.
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 V S B n VSB_n V S B n is the splittable extension of V S P n VSP_n V S P n by the symmetric group S n S_n S n . Study on contact structures of singularity links and existence of Stein cobordisms.
problem Existence problem of Stein cobordisms between contact structures of singularity links.
method Construction of explicit Stein cobordism and detection of contact Ozsvath-Szabo invariants.
result U-filtration depth obstructs the existence of Stein cobordism from proper almost rational to rational singularity.
Psybrackets define invariants for complex knots and links.
problem Defining invariants for complex knots and links.
method Introduced algebraic structures called psybrackets and used them to define invariants of pseudoknots and singular knots and links.
result Examples and computations provided for the invariants defined.
New invariants for singular knots and links defined using shadow structures.
problem Defining invariants for singular knots and links.
method Introducing action of singquandles on sets and defining shadow counting and polynomial invariants.
result Enhanced shadow counting invariant for singular knots and links.
The paper proves stability of certain singularities in integrable systems.
problem Stability of singularities in integrable systems under perturbations.
method Analytic and smooth perturbations of completely integrable systems, connectedness condition.
result Non-degenerate singular fibers are structurally stable under small perturbations.
Survey of algebraic structures for singular knots.
problem Invariants of singular knots using quandle-like structures.
method Exploration of singquandles, psyquandles, and their invariants.
result Enhancements to the singquandle counting invariant and new polynomial invariants.
Study finite singularities in G2 structure flows using Shi-type estimates.
problem Finite time singularities in G2 structure flows.
method Extend Shi-type estimates to G2 structure flows and prove κ-non-collapsing theorem.
result Prove finite time singularities of G2 structure flows.
Study weightings from singular Lie filtrations.
problem Generalize constructions for singular Lie filtrations.
method Study weightings arising from singular Lie filtrations.
result Generalizes constructions for (regular) Lie filtrations.
Extends bundle structure in a three-manifold with removable singularities.
problem Point-singularity in Sasakian three-manifold.
method Complex analysis methods.
result Extends bundle structure across removable point-singularity.
Any singular level of a completely integrable system (c.i.s.) with non-degenerate singularities has a singular affine structure. We shall show how to construct a simple c.i.s. around the level, having the above affine structure. The cotangent budle of the desingularised level is used to perform the construction, and th…
Study wall singularities in spaces with upper curvature bounds.
problem Understanding singularities in spaces with curvature constraints.
method Geometric structure theorem and geometric characterization for codimension one and two.
result Necessary and sufficient conditions for singular sets to be of codimension at least two.
Study the pullbacks and blowups of Lie algebroids and related structures.
problem Understanding the relationship between Lie algebroids, singular foliations, and Dirac structures under maps.
method Examine pullbacks and blowups of Lie algebroids and related structures under maps with constant rank or transversality assumptions.
result Establish the relation between the blowup of a Lie algebroid and its singular foliation.
Study solutions and singularities of G2-structures flows on specific manifolds.
problem Investigate singularities and solutions of G2-structures flows.
method Explicit solutions and singularities of Ricci-harmonic flow, Ricci-like flows, and negative gradient flow of G2-structures on specific manifolds.
result First examples of Type I singularities of Ricci-harmonic flow and Type IIb and Type III singularities of Ricci-like flows.
Study calculates ring structure in instanton homology for a surface with points.
problem Calculating the ring structure in instanton homology for a surface with points.
method Used singular instanton Floer homology and excision formula.
result Proved an excision formula for instanton Floer homology when n=1.
Study on contact structures of surface singularity links using Heegaard Floer homology.
problem Understanding contact structures on singularity links using Heegaard Floer homology.
method Interplay between Heegaard Floer homology and Némethi's lattice cohomology.
result Heegaard Floer invariant cannot lie in the image of the U-action for canonical contact structures on singularity links.
The paper studies the structure of endpoint maps near nice singular curves in sub-Riemannian structures.
problem Understanding the structure of endpoint maps near nice singular curves in sub-Riemannian structures.
method Finding a normal form for the endpoint map and studying its restriction to level sets of the action functional.
result The endpoint map can be written as a sum of a linear map and a quadratic form locally around a nice singular curve.
Classifies neighborhoods around specific leaf structures.
problem Classifying singular foliations with given leaf and transverse singular foliation.
method Analyzes the structure of singular foliations and their leaves.
result Developed a method to classify neighborhoods around specific leaf structures.
This thesis studies geometric structures using Lie algebroids to simplify singular behavior.
problem Understanding and simplifying geometric structures with singularities.
method Developed a framework using Lie algebroids to lift and study geometric structures.
result Lifted structures to Lie algebroid versions, making them less singular and easier to study.
In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time t ∈ [ 0 , ∞ ) t \in [0, \infty) t ∈ [ 0 , ∞ ) with unif…
We identify the canonical contact structure on the link of a simple elliptic or cusp singularity by drawing a Legendrian handlebody diagram of one of its Stein fillings. We also show that the canonical contact structure on the link of a numerically Gorenstein surface singularity is trivial considered as a real plane bu…
New proof shows unique symplectic fillings for certain surface singularity links.
problem Uniqueness of symplectic fillings for specific rational surface singularity links.
method Analysis of positive monodromy factorizations for planar open books.
result Unique symplectic fillings proven for specified contact structures.
The paper proves infinitely many strong symplectic fillings for cusp singularity links.
problem Proving the existence of infinitely many strong symplectic fillings for specific types of singularity links.
method Analyzing S o l 3 Sol^3 S o l 3 -manifolds and S L ~ ( 2 ; R ) \widetilde{SL}(2;\mathbb{R}) S L ( 2 ; R ) -manifolds with canonical contact structures. result Links of cusp, unimodal, and hyperbolic Brieskorn singularities admit infinitely many non-diffeomorphic strong symplectic fillings.
Research describes all possible gradient vector fields on a sphere with up to ten singular points.
problem Characterizing gradient vector fields on a sphere with limited singular points.
method Using a graph to represent one-dimensional stable manifolds, specifying singularities and connections.
result Identified all topological structures of codimension one gradient vector fields on a sphere with up to ten singular points.
The paper studies complex affine structures near irregular singularities.
problem Understanding complex affine structures near irregular singularities.
method Introducing local invariants and a Delaunay decomposition.
result Upper bounds on the complexity of the Delaunay decomposition.
The article investigates conditions for isomorphism of singular tangent bundles.
problem Conditions for isomorphism of singular tangent bundles.
method Logarithmic and b b b -tangent bundles approach to resolve singularities. result Established a Poincaré-Hopf theorem for b m b^m b m -tangent bundles. The study explores conditions for realizing lens spaces as boundaries of Milnor fibers of hypersurface singularities.
problem Realizing other lens spaces as boundaries of Milnor fibers of hypersurface singularities.
method Examining necessary conditions for the realization of ( L ( p , q ) , ξ ) (L(p,q),ξ) ( L ( p , q ) , ξ ) as boundaries of Milnor fibers. result Obtained a series of necessary conditions for realizing other lens spaces as boundaries of Milnor fibers.
A new geometric structure for singular models is introduced.
problem Degenerate metrics in dually flat structures.
method Introducing quasi-Hessian manifolds with degenerate metrics and symmetric cubic tensors.
result Established Amari-Nagaoka's extended Pythagorean and projection theorems for singular models.
The paper explores hidden torus symmetries in integrable systems and their stability.
problem Structural stability of singularities in integrable systems.
method Use of hidden torus actions near singular orbits and integrable perturbations.
result Persistence of toric symmetries and structural stability of Kalashnikov's parabolic orbits.
We classify linear Nambu structures (which are generalized Poisson structures in Hamiltonian dynamics and which give rise to integrable differential forms and singular foliations), then give a linearization for Nambu structures anf integrable differential forms near a nondegenerate singular point.
We show that every closed oriented smooth 4-manifold admits a complete singular Poisson structure in each homotopy class of maps to the 2-sphere. The rank of this structure is 2 outside a small singularity set, which consists of finitely many circles and isolated points. The Poisson bivector has rank 0 on the singulari…
Method finds hidden structures in noisy data.
problem Finding unexpected or unknown structures in noisy point clouds.
method First find ridges in estimated density, then filter using Hessian eigenvalues.
result Outputs well-defined sets of dimensions lower than ambient space.
We describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R^2 x R^2. We introduce the notion of a finite type singularity analogous to CR geometry and, al…
We show that the contact structure on the link of a cusp singularity is contactomorphic to a Sol-manifold with the positive contact structure arising from the Anosov flow.
We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than π π π : any first-order deformation changes either one of those angles or the conformal …