The paper explores scaling symmetries in symplectic geometry and their applications to central configurations.
problem Understanding scaling symmetries and their impact on central configurations in symplectic geometry.
method Introducing conformally symplectic maps, conformally Hamiltonian systems, and generalized momentum maps.
result Relative equilibria of scaling symmetries are solutions to specific equations involving the conformal momentum map and primitive one-form.
Study of Hamiltonian boundary value problems with symplectic symmetries.
problem Bifurcations of solutions in Hamiltonian boundary value problems.
method Convenient, coordinate-free framework for separated Lagrangian boundary value problems; analysis of conformal symplectic symmetries.
result Existence of obstructions due to conformal symplectic symmetries on bifurcation behavior.
We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant qu…
Study on symmetries and equilibria in Poisson manifolds, with applications to rigid body dynamics.
problem Characterizing conformal relative equilibria on Poisson manifolds.
method Introducing conformally Poisson actions and momentum maps, establishing algebraic criteria.
result Classification of nontrivial conformal relative equilibria in Lie algebras, with applications to rigid body dynamics.
This work takes place over a conformally flat spin manifold (M,g). We prove existence and uniqueness of the conformally equivariant quantization valued in spinor differential operators, and provide an explicit formula for it when restricted to first order operators. The Poisson algebra of symbols is realized as a space…
The geometry that is defined by the scalars in couplings of Einstein-Maxwell theories in N=2 supergravity in 4 dimensions is denoted as special Kaehler geometry. There are several equivalent definitions, the most elegant ones involve the symplectic duality group. The original construction used conformal symmetry, which…
Proves integrability of dispersionless Hirota type equations in 4D implies symplectic Monge-Ampere property.
problem Proving integrability of dispersionless Hirota type equations in 4D.
method Analyzing the symplectic Monge-Ampere property and deriving relations.
result Complete classification of integrable dispersionless PDEs of Hirota type in 4D.
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.
The paper introduces polarizations in symplectic and orthogonal settings.
problem Understanding polarizations in symplectic and orthogonal contexts.
method Exposition of symplectic and orthogonal polarizations, emphasizing symmetry.
result Grassmannians of polarizations in symplectic and orthogonal settings.
In this paper, we study the blow-up of a locally conformal symplectic manifold.We show that there exists a locally conformal symplectic structure on the blow-up of a locally conformal symplectic manifold along a compact induced symplectic submanifold.
The study constructs optimal tori on Fano manifolds and confirms mirror symmetry.
problem Constructing optimal tori on Fano manifolds and understanding their symplectic geometry.
method Using graph potentials and symplectic geometry of moduli spaces of vector bundles.
result Confirmation of mirror symmetry between A-model and B-model of graph potentials.
The study explores conformal symplectic foliations on closed manifolds, proving their existence in dimensions 5 and above.
problem Existence of conformal symplectic foliations on closed manifolds.
method Analysis of symplectic and conformal symplectic codimension-one foliations on closed manifolds of dimension at least 5.
result Construction of conformal symplectic foliations on closed, simply-connected, almost contact manifolds in dimension 5.
In this paper we will present Lagrangian and Hamiltonian k-symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will o…
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.
The study classifies and constructs symplectic structures on Lie algebras and solvmanifolds.
problem Classifying and constructing symplectic structures on Lie algebras and solvmanifolds.
method Analyzes locally conformally symplectic Lie algebras and solvmanifolds.
result Classifies and constructs symplectic structures on Lie algebras and solvmanifolds.
New formulae derived for conformal symmetry breaking operators.
problem Understanding conformal symmetry breaking operators.
method Bernstein-Sato identities for distribution kernels.
result New formulae for conformal symmetry breaking differential operators.
Convexity theorem for Hamiltonian actions on conformal symplectic manifolds.
problem Establishing convexity in conformal symplectic geometry.
method Proving a convexity theorem for moment maps under Lee type actions.
result Analog of Kirwan's convexity theorem in conformal symplectic geometry.
Explains conformal symmetry with examples in geometry and analysis.
problem None explicitly stated; focuses on introduction.
method Introduction based on examples of Yamabe operator and its applications.
result Illustrates conformal symmetry in geometry and analysis.
We study first and second order conformal symmetries of the Yamabe Laplacian on a general pseudo-Riemannian manifold and of the Paneitz operator on Einstein spaces. We show that first order conformal symmetries of the Yamabe operator induce second order conformal symmetries. We show that on an Einstein space every conf…
Extends Gromov non-squeezing to locally conformally symplectic structures.
problem Generalizing Gromov non-squeezing to new geometric structures.
method Deformation theory applied to locally conformally symplectic structures.
result Proves a new extension of the Gromov non-squeezing phenomenon.
Reduces cotangent bundles using symplectic methods.
problem Symplectic reduction of cotangent bundles.
method Locally conformally symplectic reduction at regular values.
result Proves theorem analogous to symplectic setting.
Reduces LCS manifolds with symplectic actions, preserving conformal structure.
problem Reduction of LCS manifolds with symplectic actions.
method Locally conformally symplectic reduction procedure.
result Compatibility with locally conformally Kähler structure and contact reduction.
Classifies symplectic phase space deformations preserving angular momentum symmetry.
problem Preserving symplectic o(n)-symmetry in phase space deformations. method Classifying linear deformations of phase space that preserve symplectic o(n)-symmetry. result Standard phase space and related manifolds as degenerations of a 3-dimensional family of coadjoint orbits.
We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the correspon…
The paper explores symmetries and conserved charges on pre-symplectic manifolds.
problem Analyzing conserved charges on solutions of Hamiltonian field theories.
method Using pre-symplectic structures and Gotay's coisotropic embedding theorem, the paper deals with gauge theories and examples like Electrodynamics and Klein-Gordon theory.
result Emergence of the energy-momentum tensor algebra of conserved currents.
A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…
Characterizes density-valued symplectic forms on multisymplectic manifolds.
problem Understanding density-valued symplectic forms on multisymplectic manifolds.
method Intrinsic characterization and Darboux-type theorems.
result Proves Darboux-type theorems for density-valued symplectic forms.
Develops a correspondence between symplectic orbits and Grassmannians.
problem Understanding the homotopy types of Grassmannians of linear subspaces in symplectic vector spaces.
method Uses orbit fibrations and linear symplectic reduction to compute homotopy types.
result Recover observations from Arnold, Oh-Park, and Lee-Leung in different cases.
Study symplectic cohomology of certain singularities using homological mirror symmetry.
problem Compute symplectic cohomology for specific singularities.
method Use homological mirror symmetry to compute symplectic cohomology.
result Suggests a new conjecture about the relationship between small resolutions and symplectic cohomology.
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.
Symmetries of Einstein-Weyl manifolds can be extended from boundary surfaces.
problem Extending symmetries from boundary surfaces to Einstein-Weyl manifolds.
method Starting from a symmetry of conformal Cartan connection on a boundary surface, proving symmetries can be extended.
result Symmetries of conformal Cartan connection on the boundary can be extended to symmetries of the Einstein-Weyl manifold.
On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential ope…
Solves symplectic and conformal symplectic group actions equivalence problem.
problem Equivalence problem for symplectic and conformal symplectic group actions.
method Computing differential invariants via the Lie-Tresse theorem.
result Solves equivalence problem for symplectic and conformal symplectic group actions.
Researchers develop Orlov-Schulman symmetries for self-dual conformal structures.
problem Developing symmetries for self-dual conformal structures.
method Explicit proof of compatibility with Lax-Sato flows, dressing scheme based on Riemann-Hilbert problem.
result Construction and proof of compatibility of Orlov-Schulman symmetries.
Study cohomologies on manifolds with locally conformally symplectic structures.
problem Understanding cohomologies on manifolds with locally conformally symplectic structures.
method Introduced lcs cohomologies, studied elliptic Hodge theory, dualities, and Hard Lefschetz Condition.
result Oeljeklaus-Toma manifolds with precisely one complex place and under an arithmetic condition satisfy the Mostow property.
Study characterizes toric LCK manifolds, proving conjecture and showing differences from symplectic case.
problem Characterizing compact toric locally conformally Kähler manifolds.
method Proves conjecture about toric LCK manifolds, constructs examples to show differences from symplectic case.
result Proves a conjecture about toric LCK manifolds and shows differences from symplectic case.
Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective g…
Two reduction schemes for symplectic manifolds are shown equivalent.
problem Reduction of Hamiltonian systems on exact symplectic manifolds.
method Modified Marsden-Meyer-Weinstein reduction theorem for exact symplectic manifolds and contact manifolds.
result Reduction schemes are equivalent for exact symplectic manifolds and energy hypersurfaces.
We obtain an example of a compact locally conformal symplectic nilmanifold which admits no locally conformal Kähler metrics. This gives a new positive answer to a question raised by L. Ornea and M. Verbitsky.
Abstract: Generalized reduction methods for symmetries in graded geometry.
problem Generalized reduction of symmetries in graded geometry.
method Graded symplectic reduction for Courant, Dirac, and generalized complex structures.
result Systematic recovery of reduction schemes for exact cases.
Locally symplectic structure found on Kerr space-time.
problem Understanding Kerr space-time using geodesics.
method Identifying locally conformally symplectic structure using characteristic classes and Kerr-Schild coordinates.
result Definition of cobordism category of contact 3-manifolds and locally conformally symplectic cobordisms.
In this article we study Ammann tilings from the perspective of symplectic geometry. Ammann tilings are nonperiodic tilings that are related to quasicrystals with icosahedral symmetry. We associate to each Ammann tiling two explicitly constructed highly singular symplectic spaces and we show that they are diffeomorphic…
We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of …
Paper proves h-principles for symplectic structures and foliations.
problem Existence of conformal symplectic structures and foliations.
method Application of h-principles and foliated Morse theory.
result Linear deformation of foliations to contact structures.
We show that a certain symmetry exists in the stable irreducible decomposition of the Lie algebra consisting of symplectic derivations of the free Lie algebra generated by the first homology group of compact oriented surfaces.
Study projective and almost conformally symplectic structures on manifolds.
problem Relations between projective and almost conformally symplectic structures.
method Single almost conformally symplectic connection with totally trace-free torsion.
result Generalizes Fedosov structures and encodes variability of connections in projective class.
We obtain universal models for several types of locally conformal symplectic manifolds via pullback or reduction. The relation with recent embedding results for locally conformal Kähler manifolds is discussed.
The paper discusses reducing Hamiltonian systems by scaling and standard symmetries, leading to Kirillov Hamiltonian systems.
problem Reduction of symplectic Hamiltonian systems by scaling and standard symmetries.
method Proof of Kirillov Hamiltonian systems and equivalence of reductions.
result Equivalent Kirillov Hamiltonian systems from different reduction orders.