arXiv research

We call a singularity of a presymplectic form $ω$ 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.

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 $G_2$-structures with isolated singularities and bounded torsion.

Study reveals GAGA phenomenon in Poisson cohomology for plane structures with isolated singularities.

Study on contact Hamiltonian functions for singular contact structures.

Study focuses on classifying special geometric structures.

We review properties of affine special Kaehler structures focusing on singularities of such structures in the simplest case of real dimension two. We describe all possible isolated singularities and compute the monodromy of the flat symplectic connection, which is a part of a special Kaehler structure, near a singulari…

We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of c…

Psybrackets define invariants for complex knots and links.

New invariants for singular knots and links defined using shadow structures.

The paper proves stability of certain singularities in integrable systems.

Survey of algebraic structures for singular knots.

Study weightings from singular Lie filtrations.

The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application we distinguish several singular knots and links.

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.

Study the pullbacks and blowups of Lie algebroids and related structures.

Study solutions and singularities of G2-structures flows on specific manifolds.

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic surfaces of positive genus. Applying these obstructions to canonical contact struct…

Study calculates ring structure in instanton homology for a surface with points.

Classifies neighborhoods around specific leaf structures.

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 $t \in [0, \infty)$ 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…

The paper proves infinitely many strong symplectic fillings for cusp singularity links.

Research describes all possible gradient vector fields on a sphere with up to ten singular points.

We construct local models of isolated singularities for special Kähler structures in real dimension two assuming that the associated holomorphic cubic form does not have essential singularities. As an application we compute the holonomy of the flat symplectic connection, which is a part of the special Kähler structure.

New proof shows unique symplectic fillings for certain surface singularity links.

The paper studies complex affine structures near irregular singularities.

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any co…

The article investigates conditions for isomorphism of singular tangent bundles.

A new geometric structure for singular models is introduced.

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…

The paper explores hidden torus symmetries in integrable systems and their stability.

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 show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.

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 $π$: any first-order deformation changes either one of those angles or the conformal …

Study 2D spaces with curvature, finding a graph structure.

The paper studies weak singular Hermite-Einstein structures on homogeneous vector bundles.

We study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

We introduce thermodynamic response functions for singular Bayesian models.

In this article we classify up to isotopy tight contact structures on Seifert manifolds over the torus with one singular fibre.

Maps with boundary definite fold points restrict manifold structure.

Study contact resolutions for Jacobi structures, providing examples and impossibility results.

We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular foliations, singular distributions, and Poisson structures. We show some basic p…

We show a duality which arises from distributions of Cartan type, having growth (2, 3, 5), from the view point of geometric control theory. In fact we consider the space of singular (or abnormal) paths on a given five dimensional space endowed with a Cartan distribution, which form another five dimensional space with a…