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Collective Heavy Top DynamicsJul 18 2019We construct a Poisson map $\mathbf{M}\colon T^{*}\mathbb{C}^{2} \to \mathfrak{se}(3)^{*}$ with respect to the canonical Poisson bracket on $T^{*}\mathbb{C}^{2} \cong T^{*}\mathbb{R}^{4}$ and the $(-)$-Lie--Poisson bracket on the dual $\mathfrak{se}(3)^{*}$ ... More
A non commutative Kähler structure on the Poincaré disk of a C*-algebraJul 10 2019We study the Poincar\'e disk $\d=\{z\in\a: \|z\|<1\}$ of a C$^*$-algebra $\a$ as a homogeneous space under the action of an appropriate Banach-Lie group $\u(\theta)$ of $2\times 2$ matrices with entries in $\a$. We define on $\d$ a homogeneous K\"ahler ... More
New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetriesJul 05 2019We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact Hamiltonian and Lagrangian equations, and we review and compare the two Lagrangian formalisms ... More
New contributions to the Hamiltonian and Lagrangian contact formalisms for dissipative mechanical systems and their symmetriesJul 05 2019Jul 22 2019We provide new insights into the contact Hamiltonian and Lagrangian formulations of dissipative mechanical systems. In particular, we state a new form of the contact dynamical equations, and we review two recently presented Lagrangian formalisms, studying ... More
Curves in a spacelike hypersurface in the Minkowski space-timeJul 03 2019Submanifolds in Lorentz-Minkowski space are investigated from various mathematical viewpoints and are of interest also in relativity theory. We define the hyperbolic surface and the de Sitter surface of a curve in the spacelike hypersurface M in the Minkowski ... More
Formality morphism as the mechanism of $\star$-product associativity: how it worksJul 01 2019The formality morphism $\boldsymbol{\mathcal{F}}=\{\mathcal{F}_n$, $n\geqslant1\}$ in Kontsevich's deformation quantization is a collection of maps from tensor powers of the differential graded Lie algebra (dgLa) of multivector fields to the dgLa of polydifferential ... More
Multisymplectic actions of compact Lie groups on spheresJun 20 2019We give a geometric description of the obstruction to the existence of homotopy comoment maps in multisymplectic geometry. We apply this description to determine the existence of comoments for multisymplectic compact group actions on spheres and provide ... More
Random Čech Complexes on Manifolds with BoundaryJun 18 2019Let $M$ be a compact, unit volume, Riemannian manifold with boundary. In this paper we study the homology of a random \v{C}ech-complex generated by a homogeneous Poisson process in $M$. Our main results are two asymptotic threshold formulas, an upper ... More
Direct Characterization of Spectral Stability of Small Amplitude Periodic Waves in Scalar Hamiltonian Problems Via Dispersion RelationJun 11 2019Various approaches to studying the stability of solutions of nonlinear PDEs lead to explicit formulae determining the stability or instability of the wave for a wide range of classes of equations. However, these are typically specialized to a particular ... More
A local version of the Myers-Steenrod TheoremJun 07 2019We prove the Myers-Steenrod theorem for local topological groups of isometries acting on pointed $\mathcal{C}^{k,\alpha}$-Riemannian manifolds, with $k+\alpha>0$. As an application, we infer a new regularity result for a certain class of locally homogeneous ... More
On Poincaré lemma or Volterra theorem about differential forms and cohomology groupsMay 30 2019The Poincar\'{e} lemma (or Volterra theorem) is of utmost importance both in theory and in practice. It tells us every differential form which is closed, is locally exact. In other words, on a contractible manifold all closed forms are exact. The aim ... More
Quantization of Polysymplectic ManifoldsMay 30 2019We adapt the framework of geometric quantization to the polysymplectic setting. Considering prequantization as the extension of symmetries from an underlying polysymplectic manifold to the space of sections of a Hermitian vector bundle, a natural definition ... More
The 2-Calabi-Yau property for multiplicative preprojective algebrasMay 28 2019We prove that multiplicative preprojective algebras, defined by Crawley-Boevey and Shaw, are 2-Calabi-Yau algebras for quivers containing an unoriented cycle. We also prove that the dg versions of these algebras (arising in Fukaya categories of certain ... More
A family of 8-dimensional generalized complex nilmanifolds with infinitely many real homotopy typesMay 27 2019We prove that there are infinitely many real homotopy types of $8$-dimensional nilmanifolds admitting generalized complex structures of type $k$ for every $0 \leq k \leq 4$. This is in deep contrast to the $6$-dimensional case.
The Hitchin--Kobayashi Correspondence for Quiver Bundles over Generalized Kähler ManifoldsMay 24 2019In this paper, we establish the Hitchin--Kobayashi correspondence for the $I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,\phi)$ over a compact generalized K\"{a}hler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon with respect to both $I_+$ ... More
Pontryagin algebras and representations up to homotopyMay 24 2019This paper defines the Pontryagin algebras of graded vector bundles of finite rank, with values in the cohomology vector spaces of a Lie algebroid over the same base. These Pontryagin algebras vanish if the graded vector bundle carries a representation ... More
Geometric model of the fracture as a manifold immersed in porous mediaMay 18 2019In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing man made fractures with complex geometries. We model the coupled fracture-porous media system where the linear Darcy flow is considered in ... More
A contact geometry framework for field theories with dissipationMay 17 2019We develop a new geometric framework suitable for the treatment of field theories with dissipation. To this end we define the notion of $k$-contact structure. With it, we introduce the so-called $k$-contact Hamiltonian systems, which are a generalization ... More
A contact geometry framework for field theories with dissipationMay 17 2019Jun 06 2019We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of $k$-contact structure and $k$-contact Hamiltonian system. This is a generalization of both the contact Hamiltonian ... More
Reconstruction of a Riemannian manifold from noisy intrinsic distancesMay 17 2019We consider reconstruction of a manifold, or, invariant manifold learning, where a smooth Riemannian manifold $M$ is determined from intrinsic distances (that is, geodesic distances) of points in a discrete subset of $M$. In the studied problem the Riemannian ... More
Symplectic dominationMay 14 2019Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is a theorem ... More
A symplectic embedding of the cube with minimal sections and a question by SchlenkMay 14 2019I prove that the open unit cube can be symplectically embedded into a longer polydisc in such a way that the area of each section satisfies a sharp bound and the complement of each section is path-connected. This answers a variant of a question by F. ... More
Swirling fluid flow in flexible, expandable elastic tubes: variational approach, reductions and integrabilityMay 11 2019Many engineering and physiological applications deal with situations when a fluid is moving in flexible tubes with elastic walls. In the real-life applications like blood flow, there is often an additional complexity of vorticity being present in the ... More
Swirling fluid flow in flexible, expandable elastic tubes: variational approach, reductions and integrabilityMay 11 2019May 24 2019Many engineering and physiological applications deal with situations when a fluid is moving in flexible tubes with elastic walls. In the real-life applications like blood flow, there is often an additional complexity of vorticity being present in the ... More
Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systemsMay 08 2019This work consists an introduction to the classical and quantum information theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical systems. Basic geometric and physical properties of the canonical nonholonomic deformations of G. Perelman ... More
A symplectic dynamics proof of the degree-genus formulaMay 08 2019We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere, defining the Hopf fibration. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process ... More
Locally conformally symplectic reduction of the cotangent bundleMay 07 2019In a previous article, we introduced a reduction procedure for locally conformally symplectic manifolds at any regular value of the momentum mapping. We use this construction to prove an analogue of a well-known theorem in the symplectic setting about ... More
The bijectivity of mirror functors on toriMay 02 2019By the SYZ construction, a mirror pair $(X,\check{X})$ of a complex torus $X$ and a mirror partner $\check{X}$ of the complex torus $X$ is described as the special Lagrangian torus fibrations $X \rightarrow B$ and $\check{X} \rightarrow B$ on the same ... More
The Gerrymandering Jumble: Map Projections Permute Districts' Compactness ScoresMay 01 2019In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well-established, yet competing, notions of geographic compactness are commonly used to evaluate the shapes of regions, including the Polsby-Popper ... More
The Gerrymandering Jumble: Map Projections Permute Districts' Compactness ScoresMay 01 2019May 13 2019In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well-established, yet competing, notions of geographic compactness are commonly used to evaluate the shapes of regions, including the Polsby-Popper ... More
The Kontsevich graph orientation morphism revisitedApr 30 2019The orientation morphism $Or\colon\Gamma\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ on finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $\Gamma$ with ordered sets ... More
The positive scalar curvature cobordism categoryApr 29 2019We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve that goal, ... More
Tropically constructed Lagrangians in mirror quintic threefoldsApr 26 2019We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2875 lines on the quintic ... More
Almost product structures on statistical manifolds and para-Kähler-like statistical submersionsApr 20 2019The main purpose of the present work is to investigate statistical manifolds endowed with almost product structures. We prove that the statistical structure of a para-K\"{a}hler-like statistical manifold of constant curvature in the Kurose's sense is ... More
Double-Graded Supersymmetric Quantum MechanicsApr 15 2019May 04 2019A quantum mechanical model that realizes the $ \mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric ... More
Double-Graded Supersymmetric Quantum MechanicsApr 15 2019A quantum mechanical model that realizes the $ \mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric ... More
Symplectic surfaces and bridge positionApr 10 2019We give a new characterization of symplectic surfaces in CP^2 via bridge trisections. Specifically, a minimal genus surface in CP^2 is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in transverse bridge position. ... More
Mirror curve of orbifold Hurwitz numbersApr 09 2019Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting problems can be solved by a mechanism known as topological ... More
Adiabatic limits, Theta functions, and geometric quantizationApr 08 2019Let $\pi\colon (M,\omega)\to B$ be a (non-singular) Lagrangian torus fibration on a compact, complete base $B$ with prequantum line bundle $(L,\nabla^L)\to (M,\omega)$. For a positive integer $N$ and a compatible almost complex structure $J$ on $(M,\omega)$ ... More
Adiabatic limits, Theta functions, and geometric quantizationApr 08 2019May 14 2019Let $\pi\colon (M,\omega)\to B$ be a (non-singular) Lagrangian torus fibration on a compact, complete base $B$ with prequantum line bundle $(L,\nabla^L)\to (M,\omega)$. For a positive integer $N$ and a compatible almost complex structure $J$ on $(M,\omega)$ ... More
On the controllability and Stabilization of the Benjamin EquationApr 06 2019The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain $\mathbb{T}$. We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in $H_{p}^{s}(\mathbb{T}),$ ... More
Symplectic manifolds and Hamiltonian dynamical systemsApr 02 2019This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete integrability of these ... More
Lie groupoids in information geometryApr 01 2019We demonstrate that the proper general setting for contrast (potential) functions in statistical and information geometry is the one provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms ... More
A diameter gap for quotients of the unit sphereMar 29 2019We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.
Mirror symmetry for perverse schobers from birational geometryMar 27 2019Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, ... More
Towards a higher-dimensional construction of stable/unstable Lagrangian laminationsMar 22 2019We generalize some properties of surface automorphisms of pseudo-Anosov type. First, we generalize the Penner construction of a pseudo-Anosov homeomorphism and show that a symplectic automorphism which is constructed by our generalized Penner construction ... More
Towards a higher-dimensional construction of stable/unstable Lagrangian laminationsMar 22 2019Jun 07 2019We generalize some properties of surface automorphisms of pseudo-Anosov type. First, we generalize the Penner construction of a pseudo-Anosov homeomorphism and show that a symplectic automorphism which is constructed by our generalized Penner construction ... More
Parametric finite element approximations of curvature driven interface evolutionsMar 22 2019Parametric finite elements lead to very efficient numerical methods for surface evolution equations. We introduce several computational techniques for curvature driven evolution equations based on a weak formulation for the mean curvature. The approaches ... More
Logarithmic Gromov-Witten theory with expansionsMar 21 2019We construct a version of relative Gromov--Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we construct virtually ... More
Essential tori in spaces of symplectic embeddingsMar 20 2019Given two $2n$--dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, we show that a certain map from the $n$--torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map on singular ... More
On the controllability and stabilization of the linearized Benjamin equation on a periodic domainMar 12 2019In this work we study the controllability and stabilization of the linearized Benjamin equation which models the unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface ... More
Nonlinear expectations of random setsMar 12 2019Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions ... More
Bigraded cochain complexes and Poisson cohomologyMar 05 2019We present an algebraic framework for the computation of low-degree cohomology of a class of bigraded complexes which arise in Poisson geometry around (pre)symplectic leaves. We also show that this framework can be applied to the more general context ... More
Einstein Metrics, Harmonic Forms, and Conformally Kaehler GeometryMar 03 2019Mar 24 2019The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are ... More
Einstein Metrics, Harmonic Forms, and Conformally Kaehler GeometryMar 03 2019The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are ... More
Structure of Gauge-Invariant LagrangiansMar 01 2019The theory of gauge fields in Theoretical Physics poses several mathematical problems of interest in Differential Geometry and in Field Theory. Below we tackle one of these problems: The existence of a finite system of generators of gauge-invariant Lagrangians ... More
Geometric automorphism groups of symplectic 4-manifoldsFeb 26 2019Let $M$ be a closed, oriented, smooth $4-$manifold with intersection form $\Gamma$, $A(\Gamma)$ the automorphism group of $\Gamma$ and $D(M)$ the subgroup induced by orientation-preserving diffeomorphisms of $M$. In this note we study the question when ... More
Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curvesFeb 19 2019We define a symmetric monoidal category Trop which, roughly, has degrees of tropical curves as its objects and types of tropical curves as its morphisms. A symmetric monoidal functor with domain Trop is what we call a (2D) tropical quantum field theory ... More
A note on the total curvature of confined equilateral quadrilateralsFeb 17 2019In this note, we prove that the total expected curvature for random spatial equilateral quadrilaterals with diameter at most $r$ decreases as $r$ increases. To do so, we prove several curvature monotonicity inequalities and stochastic ordering lemmas ... More
On The Expected Total Curvature of Confined Equilateral QuadrilateralsFeb 17 2019Mar 05 2019In this paper, we prove that the total expected curvature for random spatial equilateral quadrilaterals with diameter at most $r$ decreases as $r$ increases. To do so, we prove several curvature monotonicity inequalities and stochastic ordering lemmas ... More
Metric Curvatures and their Applications 2: Metric Ricci Curvature and FlowFeb 09 2019In this second part of our overview of the different metric curvatures and their various applications, we concentrate on the Ricci curvature and flow for polyhedral surfaces and higher dimensional manifolds, and we largely review our previous studies ... More
Flat affine symplectic Lie groupsFeb 05 2019We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We prove that a left invariant flat affine symplectic connection on a connected ... More
Flat affine symplectic Lie groupsFeb 05 2019Mar 02 2019We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We prove that a left invariant flat affine symplectic connection on a connected ... More
On a systolic inequality for closed magnetic geodesics on surfacesFeb 04 2019Feb 06 2019We apply a local systolic-diastolic inequality for contact forms and odd-symplectic forms on three-manifolds to bound the magnetic length of closed curves with prescribed geodesic curvature (also known as magnetic geodesics) on an oriented closed surface. ... More
On a local systolic inequality for odd-symplectic formsFeb 04 2019Feb 06 2019The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let $\Omega$ be an odd-symplectic form on an oriented closed manifold $\Sigma$ of ... More
A local contact systolic inequality in dimension threeFeb 04 2019Feb 06 2019Let $\alpha$ be a contact form on a connected closed three-manifold $\Sigma$. The systolic ratio of $\alpha$ is defined as $\rho_{\mathrm{sys}}(\alpha):=\tfrac{1}{\mathrm{Vol}(\alpha)}T_{\min}(\alpha)^2$, where $T_{\min}(\alpha)$ and $\mathrm{Vol}(\alpha)$ ... More
Toric generalized Kaehler structures. IIIJan 30 2019The paper clarifies some subtle points surrounding the definition of scalar curvature in generalized K$\ddot{a}$hler (GK) geometry. We have solved an open problem in GK geometry of symplectic type posed by R. Goto \cite{Go1} on relating the scalar curvature ... More
Survey on recent developments in semitoric systemsJan 29 2019Semitoric systems are a special class of completely integrable systems in four dimensions for which one of the first integrals generates an $\mathbb{S}^1$-action. They were classified by Pelayo & Vu Ngoc in terms of five symplectic invariants about a ... More
On two isomorphic Lie algebroids for Feedback LinearizationJan 27 2019Two Lie algebroids are presented that are linked to the construction of the linearizing output of an affine in the input nonlinear system. The algorithmic construction of the linearizing output proceeds inductively, and each stage has two structures, ... More
A generalization of pde's from a Krylov point of viewJan 21 2019We introduce and investigate the notion of a `generalized equation' of the form $f(D^2 u)=0$, based on the notions of subequations and Dirichlet duality. Precisely, a subset ${{\mathbb H}}\subset {\rm Sym}^2({\mathbb R}^n)$ is a generalized equation if ... More
Geometry of statistical submanifolds of statistical warped product manifolds by optimization techniquesJan 17 2019This paper deals with the applications of an optimization method on submanifolds, that is, geometric inequalities can be considered as optimization problems. In this regard, we obtain optimal Casorati inequalities and Chen-Ricci inequality for a statistical ... More
A family of integrable perturbed Kepler systemsJan 13 2019Jan 18 2019In the framework of the Poisson geometry of twistor space we consider a family of perturbed 3-dimensional Kepler systems. We show that Hamilton equations of this systems are integrated by quadratures. Their solutions for some subcases are given explicitly ... More
The Area Blow Up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionalsJan 11 2019In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in (J. Differential Geom., 2016), ... More
Duality of gauges and symplectic forms in vector spacesJan 10 2019A gauge $\gamma$ in a vector space $X$ is a distance function given by the Minkowski functional associated to a convex body $K$ containing the origin in its interior. Thus, the outcoming concept of gauge spaces $(X, \gamma)$ extends that of finite dimensional ... More
Poincare-Lovelock metrics on conformally compact manifoldsJan 08 2019An important tool in the study of conformal geometry, and the AdS/CFT correspondence in physics, is the Fefferman-Graham expansion of conformally compact Einstein metrics. We show that conformally compact metrics satisfying a generalization of the Einstein ... More
${\rm SL}_2$ quantum trace in quantum Teichmüller theory via writheDec 30 2018Quantization of Teichm\"uller space of a punctured Riemann surface $S$ is an approach to three dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop $\gamma$ in $S$ gives rise to a natural function ... More
Toric degenerations in symplectic geometryDec 28 2018A toric degeneration in algebraic geometry is a process where a given projective variety is being degenerated into a toric one. Then one can obtain information about the original variety via analyzing the toric one, which is a much easier object to study. ... More
Bulk-deformed potentials for toric Fano surfaces, wall-crossing and periodDec 20 2018Feb 27 2019We provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces via wall-crossing techniques and a tropical-holomorphic correspondence theorem for holomorphic discs. As an application of the correspondence theorem, we ... More
Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinityDec 20 2018We consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at ... More
Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinityDec 20 2018Jul 16 2019We consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at ... More
Constraint algorithm for singular field theories in the $k$-cosymplectic frameworkDec 20 2018The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of $k$-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of $k$-precosymplectic structure, ... More
Polarized vectorial Poisson structuresDec 20 2018We study various properties of polarized vectorial Poisson structures subordinate to polarized k-symplectic manifolds, and also, we study the notion of polarized vectorial Poisson manifold. Some properties and examples are given.
First-order invariants of differential 2-formsDec 18 2018Let $M$ be a smooth manifold of dimension $2n$, and let $O_{M}$ be the dense open subbundle in $\wedge^{2}T^{\ast}M$ of $2$-covectors of maximal rank. The algebra of $\operatorname*{Diff}M$-invariant smooth functions of first order on $O_{M}$ is proved ... More
Instantons on hyperkähler manifoldsDec 16 2018An instanton $(E, D)$ on a (pseudo-)hyperk\"ahler manifold $M$ is a vector bundle $E$ associated to a principal $G$-bundle with a connection $D$ whose curvature is pointwise invariant under the quaternionic structures of $T_x M, \ x\in M$, and thus satisfies ... More
Singular symplectic cotangent bundle reduction of gauge field theoryDec 11 2018We prove a theorem on singular symplectic cotangent bundle reduction in the Fr\'echet setting and apply it to Yang-Mills-Higgs theory with special emphasis on the Higgs sector of the Glashow-Weinberg-Salam model. For the latter model we give a detailed ... More
Clebsch-Lagrange variational principle and geometric constraint analysis of relativistic field theoriesDec 11 2018Inspired by the Clebsch optimal control problem, we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry. Its special feature is that the Lagrange ... More
On the homogeneity of non-uniform material bodiesDec 11 2018Dec 17 2018A groupoid $\Omega \left( \mathcal{B} \right)$ called material groupoid is naturally associated to any simple body $\mathcal{B}$. The material distribution is introduced due to the (possible) lack of differentiability of the material groupoid. Thus, the ... More
Black Holes with MDRs and Bekenstein-Hawking and Perelman Entropies for Finsler-Lagrange-Hamilton SpacesDec 04 2018Feb 19 2019New geometric and analytic methods for generating exact and parametric solutions in generalized Einstein-Finsler like gravity theories and nonholonomic Ricci soliton models are reviewed and developed. We show how generalizations of the Schwarzschild - ... More
Black Holes with MDRs and Bekenstein-Hawking and Perelman Entropies for Finsler-Lagrange-Hamilton SpacesDec 04 2018New geometric and analytic methods for generating exact and parametric solutions in generalized Einstein-Finsler like gravity theories and nonholonomic Ricci soliton models are reviewed and developed. We show how generalizations of the Schwarzschild - ... More
Projections of Poisson cut-outs in the Heisenberg group and the visual $3$-sphereDec 03 2018We study projectional properties of Poisson cut-out sets $E$ in non-Euclidean spaces. In the first Heisenbeg group, endowed with the Kor\'anyi metric, we show that the Hausdorff dimension of the vertical projection $\pi(E)$ (projection along the center ... More
Removing a ray from a noncompact symplectic manifoldDec 02 2018Mar 10 2019We prove that any noncompact symplectic manifold which admits a properly embedded ray with a wide neighborhood is symplectomorphic to the complement of the ray by constructing an explicit symplectomorphism in the case of the standard Euclidean space. ... More
Removing a ray from a noncompact symplectic manifoldDec 02 2018We prove that any noncompact symplectic manifold which admits a properly embedded ray with a wide neighborhood is symplectomorphic to the complement of the ray by constructing an explicit symplectomorphism between the standard Euclidean space and the ... More
On certain class of locally conformal symplectic structures of the second kindNov 29 2018May 28 2019We study locally conformal symplectic (LCS) structures of the second kind on a Lie algebra. We show a method to build new examples of Lie algebras admitting LCS structures of the second kind starting with a lower dimensional Lie algebra endowed with a ... More
The defining properties of the Kontsevich unoriented graph complexNov 26 2018Consider the real vector space of formal sums of non-empty, finite unoriented graphs without multiple edges and loops. Let the vertices of graphs be unlabelled but let every graph $\gamma$ be endowed with an ordered set of edges $\mathsf{E}(\gamma)$. ... More
Contact between Lagrangian manifoldsNov 26 2018Tangential intersections of Lagrangian manifolds up to contact equivalence correspond to smooth function germs (generating functions) up to right equivalence locally around the intersection point. We extend this result of Golubitsky and Guillemin for ... More
Local intersections of Lagrangian manifolds correspond to catastrophe theoryNov 26 2018Feb 11 2019Two smooth map germs are right-equivalent if and only if they generate two Lagrangian submanifolds in a cotangent bundle which have the same contact with the zero-section. In this paper we provide a reverse direction to this classical result of Golubitsky ... More
Equivariant wrapped Floer homology and symmetric periodic Reeb orbitsNov 20 2018The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a ... More
The orientation morphism: from graph cocycles to deformations of Poisson structuresNov 19 2018We recall the construction of the Kontsevich graph orientation morphism $\gamma \mapsto {\rm O\vec{r}}(\gamma)$ which maps cocycles $\gamma$ in the non-oriented graph complex to infinitesimal symmetries $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma)(\mathcal{P})$ ... More
The orientation morphism: from graph cocycles to deformations of Poisson structuresNov 19 2018Jul 01 2019We recall the construction of the Kontsevich graph orientation morphism $\gamma \mapsto {\rm O\vec{r}}(\gamma)$ which maps cocycles $\gamma$ in the non-oriented graph complex to infinitesimal symmetries $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma)(\mathcal{P})$ ... More
Almost Kaehler geometry of adjoint orbits of semisimple Lie groupsNov 16 2018Nov 26 2018We study the almost Kaehler geometry of adjoint orbits of non-compact real semisimple Lie groups endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almost complex structure. We give explicit formulas for the Chern-Ricci ... More