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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

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

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

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

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

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

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

Toric generalized Kaehler structuresNov 14 2018Anti-diagonal toric generalized K$\ddot{a}$hler structures of symplectic type on a compact toric symplectic manifold were investigated in \cite{Wang2} . In this article, we consider \emph{general} toric generalized K$\ddot{a}$hler structures of symplectic ... More

Complex symplectic structures on Lie algebrasNov 14 2018We investigate Lie algebras endowed with a complex symplectic structure and develop a method, called \emph{complex symplectic oxidation}, to construct certain complex symplectic Lie algebras of dimension $4n+4$ from those of dimension $4n$. We specialize ... More

Obstructions for symplectic Lie algebroidsNov 13 2018Several generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, $b^k$-, scattering and elliptic-log Poisson structures. ... More

Surface area deviation between smooth convex bodies and polytopesNov 12 2018The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the surface areas of ... More

Contact Hamiltonian SystemsNov 08 2018In this paper we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We prove a coisotropic reduction theorem similar to the one in symplectic mechanics.

The Symmetric Representation of the Generalized Rigid Body Equations and Symplectic ReductionNov 07 2018We show that a symplectic reduction of the symmetric representation of the generalized $n$-dimensional rigid body equations yields the $n$-dimensional Euler equation. This result provides an alternative to the more elaborate relationship between these ... More

On an Extension of the Mean Index to a large subset of Linear Canonical RelationsNov 07 2018In this paper, viewing the symplectic linear group as a subset of the Lagrangian Grassmannian we extend the mean index to the complement of a codimension-two subset of the Grassmannian. This extension retains many of the desirable properties of the mean ... More

The CR geometry of weighted extremal Kahler and Sasaki metricsOct 24 2018Jan 04 2019We establish an equivalence between conformally Einstein--Maxwell Kahler 4-manifolds (recently studied in many works) and extremal Kahler 4-manifolds (in the sense of Calabi) with nowhere vanishing scalar curvature. The corresponding pairs of Kahler metrics ... More

Toric generalized K$\ddot{a}$hler structures. IOct 18 2018This is a sequel of \cite{Wang}, which provides a general formalism for this paper. We mainly investigate thoroughly a subclass of toric generalized K$\ddot{a}$hler manifolds of symplectic type introduced by Boulanger in \cite{Bou}. We find torus actions ... More

A weak notion of visibility, a family of examples, and Wolff--Denjoy theoremsOct 18 2018We investigate a form of visibility introduced recently by Bharali and Zimmer -- and shown to be possessed by a class of domains called Goldilocks domains. The range of theorems established for these domains stem from this form of visibility together ... More

Effective global generation on varieties with numerically trivial canonical classOct 16 2018Dec 16 2018We prove a Fujita-type theorem for varieties with numerically trivial canonical bundle. We deduce our result via a combination of algebraic and analytic methods, including the Kobayashi--Hitchin correspondence and positivity of direct image bundles. As ... More

On symplectic lifts of actions for complete Lagrangian fibrationsOct 12 2018In this note we discuss symplectic lifts of actions for a complete Lagrangian fibration. Firstly, we describe the symplectic cotangent lifts of a G-action on a manifold Q in terms of 1-cocycles in the cohomology of G induced by the action with values ... More

Rabinowitz Floer homology for tentacular HamiltoniansOct 12 2018This paper extends the definition of Rabinowitz Floer homology to non- compact hypersurfaces. We present a general framework for the construction of Rabi- nowitz Floer homology in the non-compact setting under suitable compactness assump- tions on the ... More

Rabinowitz Floer homology for tentacular HamiltoniansOct 12 2018Mar 27 2019This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic ... More

Reduction of a Hamilton-Jacobi equation for nonholonomic systemsOct 11 2018Oct 15 2018Nonholonomic mechanical systems have been attracting more interest in recent years because of their rich geometric properties and their applications in Engineering. In all generality, we discuss the reduction of a Hamilton-Jacobi theory for systems subject ... More

Polysymplectic Reduction and the Moduli Space of Flat ConnectionsOct 11 2018A polysymplectic structure is a vector-valued symplectic form, that is, a closed nondegenerate $2$-form with values in a vector space. We first outline the polysymplectic Hamiltonian formalism with coefficients in a vector space $V$, then apply this framework ... More

Symplectic integration of PDEs using Clebsch variablesOct 03 2018Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations,...) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected ... More

Lagrangian submanifolds of standard multisymplectic manifoldsSep 28 2018We give a detailed, self-contained proof of Geoffrey Martin's normal form theorem for Lagrangian submanifolds of standard multisymplectic manifolds (that generalises Alan Weinstein's famous normal form theorem in symplectic geometry), providing also complete ... More

On the Ekeland-Hofer symplectic capacities of the real bidiscSep 22 2018Jan 13 2019In $\mathbb{C}^2$ with the standard symplectic structure we consider the bidisc $D^2$ constructed as the product of two open real discs of radius $1$. We compute explicit values for the first, second and third Ekeland-Hofer symplectic capacity of $D^2$. ... More

Hilbert series associated to symplectic quotients by $\operatorname{SU}_2$Sep 20 2018We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an $\operatorname{SU}_2$-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert ... More

Generalized quasi-statistical structuresSep 13 2018Given a non-degenerate $(0,2)$-tensor field $h$ on a smooth manifold $M$, we consider a natural generalized complex and a generalized product structure on the generalized tangent bundle $TM\oplus T^*M$ of $M$ and we show that they are $\nabla$-integrable, ... More

Lectures on controlled Reeb dynamicsSep 10 2018These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian ... More

Degenerations, transitions and quantum cohomologySep 08 2018Sep 11 2018Given a singular variety I discuss the relations between quantum cohomology of its resolution and smoothing. In particular, I explain how toric degenerations helps with computing Gromov--Witten invariants, and the role of this story in Fanosearch programme. ... More