Researchers find higher symmetries in symplectic Dirac operator.
problem Understanding symmetries in symplectic Dirac operator.
method Constructing higher symmetry algebra of symplectic Dirac operator Daise0.22ex/s. result Higher symmetry algebra structure corresponds to a completely prime primitive ideal.
This work explores symplectic structures on graded manifolds and higher Lie groupoids.
problem Understanding symplectic structures on graded manifolds and their global counterparts.
method Introduction and study of graded manifolds, symplectic Q-manifolds, higher Lie groupoids, and their symplectic structures.
result Developed a graded analogue of Weinstein's tubular neighborhood theorem and explored its applications.
The paper studies quasimorphisms on groups and proves non-existence results for symplectic geometry.
problem Non-existence of sections of flux homomorphisms on higher genus surfaces.
method Analysis of G^-invariant quasimorphisms and symplectic geometry of surfaces. result Py's Calabi quasimorphism is the unique non-extendable quasimorphism to some group.
Defines a higher version of omni-Lie algebroid for new geometries.
problem No specific problem stated; focuses on definition.
method Proposes a new definition of higher omni-Lie algebroid.
result Studies isotropic and involutive subbundles of higher omni-Lie algebroid.
Symplectic structures on graded manifolds are explored.
problem Exploring symplectic structures on graded manifolds.
method Definition and analysis of symplectic Z2n-manifolds. result Generalization of symplectic geometry to higher graded settings.
This paper associates homotopy Poisson-n algebras to higher symplectic structures.
problem Generalizing symplectic Poisson algebras to higher symplectic structures.
method Introducing a homotopy Poisson-n algebra associated with higher symplectic structures.
result The exterior product does not close on Poisson cotensors, but valid computations remain.
Unique symplectic fillings found for specific cotangent bundles.
problem Symplectic fillings of unit cotangent bundles.
method Proved uniqueness up to diffeomorphism.
result Unique symplectically aspherical fillings found.
We study higher-degree generalizations of symplectic groupoids, referred to as {\em multisymplectic groupoids}. Recalling that Poisson structures may be viewed as infinitesimal counterparts of symplectic groupoids, we describe "higher'' versions of Poisson structures by identifying the infinitesimal counterparts of mul…
We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…
We give a classification of 1st order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so called higher symplectic, harmonic …
This note elaborates on Th. Voronov's construction [math/0304038,math/0412202] of L∞-structures via higher derived brackets with a Maurer-Cartan element. It is shown that gauge equivalent Maurer-Cartan elements induce L∞-isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are d…
Sturm theory applied to symplectic geometry and mechanics.
problem Detecting geometric properties of solutions in symplectic geometry and mechanics.
method Generalization of symplectic Sturm theory to Hamiltonians and application to semi-Riemannian manifolds and singular Lagrangian systems.
result Detection of conjugate and focal points on semi-Riemannian manifolds and geometrical properties of solutions space.
Study on 4D hypersurfaces with symplectic structure and shape operator rank.
problem Characterizing 4D Lorentzian affine hypersurfaces with an almost symplectic form.
method Analyzing hypersurfaces with a Lorentzian second fundamental form and an almost symplectic structure, proving rank constraints.
result Rank of shape operator is at most one under certain conditions.
Introduces derived Lie n-groupoids with shifted symplectic structures.
problem Defines structures for higher groupoids and their symplectic properties.
method Introduced derived Lie n-groupoids and their shifted symplectic structures, defining shifted lagrangian structures and proving composition well-defined.
result Shows that the framework includes various reduction procedures.
Novel multisymplectic framework for pseudo-Fueter curves in Hamiltonian field theory.
problem Generalizing Floer theory to multisymplectic geometry.
method Introducing pseudo-Fueter curves in a compatible almost hyperkähler structure.
result Gradient lines of multisymplectic action functional are pseudo-Fueter curves.
The paper classifies affine hypersurfaces with symplectic structures and constraints on their curvature.
problem Characterizing affine hypersurfaces with symplectic structures and curvature constraints.
method Analyzing hypersurfaces with non-degenerate second fundamental forms and almost symplectic structures.
result The rank of the shape operator is at most one under certain conditions on the almost symplectic form.
The paper studies deformations of Lagrangian submanifolds using algebraic tools.
problem Deformation theory of Lagrangian submanifolds in symplectic geometry.
method Graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem, attaching an L∞-algebra to each submanifold. result Controls the deformation theory of Lagrangian NQ-submanifolds using an L∞-algebra. Constructs symplectic 6-manifolds using bifibration structures.
problem Creating symplectic 6-manifolds from combinatorial data.
method Using bifibration structures and compatible pairs of monodromies and braid group relations.
result Established methods for computing topological invariants of symplectic 6-manifolds.
Quantizes symplectic fibrations to analyze vector bundles and metrics.
problem Quantizing higher rank vector bundles and understanding their metrics.
method Relates Berezin-Toeplitz quantization to hybrid systems and symplectic fibrations.
result Established refined estimates for computing balanced metrics on Kähler manifolds.
New proof of mass formula for 4D Kaehler manifolds with weaker fall-off conditions.
problem Proving mass formula for 4D Kaehler manifolds with weak fall-off conditions.
method Symplectic geometry techniques to prove mass formula with weaker fall-off conditions.
result New proof of Penrose-type inequality for 4D Kaehler manifolds with weaker fall-off conditions.
Smooth actions of infinite groups linked to homotopy theory.
problem Connecting infinite-dimensional smooth groups to homotopy theory.
method Two computations: diffeological homotopy groups and localization of a strict category.
result Natural constructions yield homotopically coherent group actions of G.
Extends String structures to indefinite Lie groups O(p, q).
problem No new problem introduced.
method Extension of String structures from definite to indefinite signature orthogonal groups.
result Provides a new starting point for constructions in higher geometry and physics.
Method resolves 4D symplectic orbifolds using complex geometry.
problem Resolving symplectic orbifolds in 4 dimensions.
method Combining complex geometry techniques with symplectic form gluing.
result Examples of 4D symplectic orbifolds successfully resolved.
We consider generalizations of symplectic manifolds called n-plectic manifolds. A manifold is n-plectic if it is equipped with a closed, nondegenerate form of degree n+1. We show that higher structures arise on these manifolds which can be understood as the categorified or homotopy analogues of important structures stu…
The paper classifies symplectic Lie and L∞ algebroids and their applications.
problem Classifying symplectic Lie and L∞ algebroids and their higher gauge symmetries. method Classifying zero-, one-, and two-shifted symplectic algebroids using classical geometric higher structures.
result New examples of twisted Courant algebroids from codimension-two cycles and symplectic interpretations of higher structures.
New flows introduced for symplectic geometry.
problem No specific problem stated; focuses on new flows.
method Introduces several geometric flows on symplectic manifolds.
result Examples include the Hitchin gradient flow and dual Ricci flow.
Abstract collects open problems in billiards and symplectic geometry.
problem Open problems in billiards and symplectic geometry.
method Compilation of open problems from discussions.
result Compilation of open problems.
Study geometric mKdV flows for Legendrian curves in a 3-sphere.
problem Investigate geometric evolution equations for Legendrian curves.
method Define a symplectic structure and show mKdV and associated flows.
result Show mKdV equation as curvature evolution induced by Hamiltonian flows.
Madelung transform connects quantum and fluid dynamics via symplectic geometry.
problem Relating quantum mechanics and fluid dynamics equations.
method Proved Madelung transform as a Kähler map between wave functions and cotangent bundles.
result Madelung transform is a symplectomorphism and isometry between wave function space and density space.
This thesis extends Hamiltonian actions to multisymplectic geometry, classifying actions on spheres and constructing homotopy comomentum maps.
problem Extending Hamiltonian actions to multisymplectic geometry.
method Explicit constructions and concrete examples of homotopy comomentum maps.
result Complete classification of compact group actions on multisymplectic spheres and explicit construction of homotopy comomentum maps.
Higher-dimensional contact manifolds lack certain properties.
problem Characterizing properties of higher-dimensional contact manifolds.
method Symplectic handle attachment and alternative proof techniques.
result They do not arise as nonseparating weak contact-type hypersurfaces in closed symplectic manifolds.
Criterion found for blowing down in 6D symplectic geometry.
problem Blowing down criterion in 6D symplectic geometry.
method Criterion for blowing down in 6D symplectic geometry.
result Criterion established for blowing down in 6D symplectic geometry.
We define higher genus Gromov-Witten invariants and establish a mathematical theory of sigma model coupled with gravity over any semi-positive symplectic manifolds. As applications, we verify the stablizing conjecture of symplectic 4-manifolds for simply connected elliptic surfaces and construct smooth 6-manifolds admi…
Abstract: Surveying aspects of complex dimension two anti-canonical pairs.
problem Smooth topology, algebraic geometry, symplectic geometry, and contact geometry of anti-canonical pairs.
method Survey and review of existing work.
result Survey of various geometric properties of anti-canonical pairs.
Introduces symplectic reduction in nonrational toric geometry.
problem Symplectic reduction in nonrational toric geometry.
method Specializes to nonrational toric geometry and rational case for symplectic reduction.
result Symplectic reduction for nonrational Lie subgroups.
A dictionary connects symplectic to contact geometry, with applications to complex and G-structures.
problem Formalizing the relationship between symplectic and contact geometry.
method Developing a Symplectic-to-Contact Dictionary.
result The dictionary can be applied to complex and G-structures, revealing new geometries.
Introduces systolic inequalities in Riemannian and symplectic geometry.
problem Exploring systolic inequalities in different geometric settings.
method Comparing classical Riemannian metrics to recent symplectic measurements.
result Illustrates connections between Riemannian and symplectic geometry.
Symplectic and Poisson structures proved for information geometry's Frobenius manifold.
problem Connecting disconnected theories in information geometry.
method Proving symplectic and Poisson structures on the Frobenius manifold.
result Established a bridge between Vinberg, Souriau, and Koszul's theories.
The paper explores symplectic geometry of Cartan-Hartogs domains.
problem Understanding the symplectic geometry of Cartan-Hartogs domains.
method Constructing a dual counterpart and computing symplectic capacity.
result A Cartan-Hartogs domain admits symplectic duality if and only if it reduces to a complex hyperbolic space.
The paper solves symplectic embedding problems in higher dimensions, proving new embedding conditions.
problem Symplectic embedding problems in higher dimensions.
method Symplectic blowup construction, h-principle for symplectic surfaces, stabilization of pseudoholomorphic curves.
result New embedding conditions for symplectic balls and surfaces in higher dimensions.
The study examines formality and Lefschetz properties in symplectic and cosymplectic geometry.
problem Exploring topological properties in symplectic and cosymplectic geometry.
method Review and analysis of Kähler, symplectic, coKähler, and cosymplectic manifolds.
result Formality and Lefschetz properties are distinguished in various cases.
In this paper we derive the symplectic framework for field theories defined by higher-order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher-order system of PDEs to a constrained first-order one, the symplectic structures natura…
Quantizes symplectic manifolds with bounded geometry using Berezin-Toeplitz method.
problem Quantization of symplectic manifolds with bounded geometry.
method Berezin-Toeplitz quantization theory.
result Correct semiclassical limit achieved.
Characterizes Anosov flows in 3D using symplectic and contact geometry.
problem Understanding Anosov flows in 3D.
method Purely contact and symplectic geometric methods.
result Characterization of Anosov flows based on Reeb flows and underlying (bi)-contact structures.
Develops a new deformation theory for Dirac structures.
problem Interpolating between twisted Dirac and Poisson geometries.
method Introduces a new deformation theory compatible with Dirac geometry operations.
result Uniform deformation theory recovering various special cases.
Proves a vanishing property for symplectic manifold cohomology.
problem Generalizing complex geometry results to symplectic geometry.
method Based on Tseng and Zhou's vanishing property under symplectic flatness.
result Establishes necessity of symplectic flatness for certain results.
Introduces locally conformally symplectic and Kähler geometry.
problem None explicitly stated, focuses on background and applications.
method Background introduction and demonstration of applications.
result Illustrates applications of these geometries in physics.
Symplectic structures simplified for compact manifolds.
problem Locally conformally symplectic structures on compact manifolds.
method Symplectic analogue of Vaisman's theorem.
result Locally conformally symplectic structures become globally symplectic.