Homotopy equivalence of cotangent bundles' function algebras is shown.
problem Understanding homotopy equivalence in cotangent bundles and their function algebras.
method Using shifted Poisson algebras and homotopy equivalence of bundles.
result Homotopy equivalent bundles have equivalent Poisson algebras.
Abstract: Explains translating derived algebraic geometry results to derived differential geometry.
problem Existence and classification of deformation quantizations on NQ-manifolds.
method Translation of results from derived algebraic geometry to derived differential geometry.
result Existence and classification of various deformation quantizations.
The paper integrates quasi-Poisson manifolds into multiplicative D-valued moment maps.
problem Integrating quasi-Poisson manifolds into a broader geometric framework.
method Develops new aspects of shifted symplectic and Poisson geometry, establishing Lie-type correspondences and systematic constructions.
result Identifies multiplicative D-valued moment maps integrating quasi-Poisson manifolds, extending known constructions.
The paper explores Poisson structures on differentiable stacks, developing new mathematical tools.
problem Investigating shifted Poisson structures on differentiable stacks.
method Developed new mathematical tools including Morita equivalence of quasi-Poisson groupoids and tangent/cotangent complexes.
result Shifted (+1) Poisson structures on differentiable stacks correspond to elements of the Maurer-Cartan moduli set of a specific Lie 2-algebra. Constructs integrable systems for Lie-Poisson structures at nilpotent elements.
problem Integrability of transverse Lie-Poisson structures at nilpotent elements.
method Using the argument shift method to construct families of functions in involution.
result Provides a uniform construction of completely integrable systems for an infinite family of nilpotent elements.
Derived Poisson structures from Lie pairs are studied and their algebraic properties are explored.
problem Exploring derived Poisson structures from Lie pairs.
method Algebraic and homotopy transfer theorems for derived Poisson algebras.
result Derived Poisson algebra structure on totΩA∙(Λ∙(L/A)) is unique up to isomorphism. New concept of coisotropic structures for differentiable stacks defined.
problem Defining coisotropic structures for differentiable stacks.
method Using twisted Dirac structures and Morita equivalences.
result 1-shifted coisotropic structures transfer through Morita equivalences.
Study flat connections with logarithmic singularities on complex plane curves.
problem Modeling flat connections with logarithmic singularities.
method Explicit finite-dimensional model construction and detailed investigation of specific cases.
result Construction of shifted Poisson structure on moduli spaces.
Defines projective limits of local shift morphisms and applies to KdV equation.
problem Defining and studying projective limits of local shift morphisms.
method Introducing projective limits, defining shift Poisson tensors, and applying to KdV equation.
result Illustrates the application of projective limits to the KdV equation on the circle.
Quantization of (-1)-shifted derived Poisson manifolds via BV-infinity operators.
problem Quantizing (−1)-shifted derived Poisson manifolds. method Using BV-infinity operators on the space of Berezinian half-densities, proving quantization via lifting of Maurer-Cartan elements.
result Quantization of (−1)-shifted derived Poisson manifolds is equivalent to the vanishing of the second Poisson cohomology group. 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.
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.
We show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrang…
PRGDS models count tensors with sparsity and burstiness.
problem Modeling sequential count data with sparsity and burstiness.
method Poisson-randomized gamma dynamical system with alternating Poisson and gamma latent states.
result Sparse PRGDS often outperforms other models in predicting count data.
New model fits term structures with positivity constraints.
problem Calibrating term structure models to market curves with positivity constraints.
method Time-changed approach to fit term structures.
result Model generates larger volatility and covariance effects under positivity constraints.
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.
Paper extends Poisson-Lichnerowicz cohomology to scalar difference Hamiltonian operators.
problem Classify and understand the deformations of scalar difference Hamiltonian operators.
method Extend Poisson-Lichnerowicz cohomology to difference case, study K0=S−S−1. result Triviality of cohomology for K0 with Hp(K0)=0 for p>1. Formalizes quantum path integrals using groupoids and differential forms.
problem Formalizing Feynman's path integral in quantum mechanics.
method Shifted focus to pair groupoid, using van Est map and piecewise linear structures.
result Developed a coordinate-free approach to integration of differential forms.
New Poisson structures on algebras linked to derivatives.
problem Understanding Poisson structures on trivial extension algebras.
method Introducing a correspondence between Poisson structures and derivatives, and formulating results in terms of Lie algebroids.
result One-to-one correspondence between Poisson structures and data involving derivatives.
We reformulate the Poisson structure discovered by Fock and Rosly on moduli spaces of flat connections over marked surfaces in the framework of Poisson structures defined by Lie algebra actions and quasitriangular r-matrices, and we show that it is an example of a mixed product Poisson structure associated to pairs o…
This paper studies Poisson structures defined by divisor ideals.
problem Understanding Poisson structures with degeneracy captured by divisor ideals.
method Developed a framework using divisor ideals and Lie algebroids.
result Effective methods for studying Poisson structures of divisor-type.
Cohomology of 'book' Lie algebra Poisson structure computed.
problem Computing the cohomology of a specific Lie algebra structure.
method Direct computation of cohomology for the given Lie algebra structure.
result Explicit formula for Poisson cohomology of the 'book' Lie algebra.
A new Poisson bracket defined on Poisson structures with applications to fixed points and cohomology.
problem Defining a Poisson bracket on the space of Poisson structures.
method Constructing a Poisson bracket on P(M) depending on a volume form, and defining invariant of Poisson structures. result Invariant of Poisson structures detects unimodularity and related Poisson bracket for symplectic structures.
Study non-degenerate singular points of Poisson-Nijenhuis structures.
problem Non-degenerate singular points of Poisson-Nijenhuis structures.
method Completely describe pairs of compatible Poisson structures near singular points.
result Pairs of compatible Poisson structures near singular points are completely described.
Local formulas for Poisson structures on wrinkled fibrations are derived.
problem Understanding Poisson structures on specific geometric fibrations.
method Local formulæ for Poisson bivectors and symplectic forms on wrinkled fibrations.
result Local formulas for Poisson structures on wrinkled fibrations are derived.
A Poisson structure is represented by a bivector whose Schouten bracket vanishes. We study a global Poisson structure on S4 associated with a holomorphic Poisson structure on CP3. The space of the Poisson structures on S4 is a real algebraic variety in the space of holomorphic Poisson structures on $\…
We construct a Poisson isomorphism between the formal Poisson manifolds g^* and G^*, where g is a finite dimensional quasitriangular Lie bialgebra. Here g^* is equipped with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G^* with its Poisson-Lie structure. We also quantize Poisson-Lie dynamical r-matrices…
Character varieties get a natural Poisson structure.
problem Character varieties of surfaces.
method Natural Poisson structure.
result Character varieties have a Poisson structure.
Defines a new Poisson structure for generalized Sasakian spaces.
problem No specific problem stated; focuses on new structure definition.
method Defines a canonical Poisson structure on generalized contact metric spaces.
result Shows distinction between generalized Sasakian and coKähler structures.
In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts polysymplectically on a polysymplectic manifold then, under certain regularity conditio…
Survey on Nambu-Poisson structures in infinite dimensions.
problem Generalization of Poisson and Nambu-Poisson structures in infinite dimensions.
method Study properties of associated characteristic distribution and projective/direct limits.
result Properties and limits of Nambu-Poisson structures in convenient setting.
Deep analysis of KV-Poisson structures and new invariant computed.
problem Understanding KV-Poisson structures and their properties.
method Classification, properties analysis, and computation of Cohomological groups.
result KV-Poisson structures give rise to a new Cohomological invariant.
Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.
problem Understanding geometric structures related to geodesic foliations and dynamics.
method Introducing symmetric Poisson structures, proving correspondences with geodesic foliations and Jordan algebras.
result Symmetric Poisson structures correspond to totally geodesic foliations and Jacobi-Jordan algebras.
New Poisson structures defined on surface moduli spaces.
problem Generalizing Poisson brackets to quasi-surfaces.
method Defined quasi-Poisson brackets on quasi-surfaces.
result New Poisson structures on quasi-surfaces.
Study reveals GAGA phenomenon in Poisson cohomology for plane structures with isolated singularities.
problem Understanding Poisson cohomology for plane structures with isolated singularities.
method Determined Gerstenhaber algebra structure over Poisson cohomology groups.
result GAGA type phenomenon observed in Poisson cohomology.
The paper explores complex Poisson structures on smooth functions in complex manifolds.
problem Exploring complex Poisson structures on smooth functions in complex manifolds.
method Considering structures of complex Poisson brackets generated by a (1,1)-form. result Examples of complex Poisson structures are provided in $\C^\ast$.
The paper develops structures on Hom-Lie algebroids and Hom-Courant algebroids.
problem Exploring new algebraic structures on Hom-Lie algebroids and Hom-Courant algebroids.
method Introducing Hom-Poisson, Hom-Nijenhuis, and Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroids.
result Established properties and relationships among these structures, including a hierarchy and correspondence.
The paper defines and explores Poisson-Nijenhuis structures on Lie groupoids.
problem Defining and understanding Poisson-Nijenhuis structures on Lie groupoids.
method Introducing and studying right-invariant Poisson-Nijenhuis structures on Lie groupoids and their infinitesimal counterparts.
result A mutual correspondence between (Λ,n)−structures on Lie algebroids and Poisson-Nijenhuis structures on Lie groupoids. The paper studies quadratic Poisson structures on Lie algebras, finding a 10-parametric family.
problem Compatibility of quadratic Poisson structures with linear structures on Lie algebras.
method Developed general theory and studied families of functions in involution.
result Found a 10-parametric family of quadratic Poisson structures on $\gl(3)^*$.
We make a study of Poisson structures of T*M which are graded structures when restricted to the fiberwise polynomial algebra, and give examples. A class of more general graded bivector fields which induce a given Poisson structure w on the base manifold M is constructed. In particular, the horizontal lifting of a Poiss…
Non-exact Poisson structures found on toric varieties.
problem Existence of exact Poisson structures on toric varieties.
method Geometric criterion for non-exactness of Poisson structures with finite symplectic leaves.
result Non-exactness of Poisson structures on projective toric varieties.
Constructs Poisson structures with compact support on manifolds.
problem Creating Poisson structures with compact support on manifolds.
method Explicit construction of Poisson structures with polynomial coefficients and modification outside open balls.
result Even-dimensional manifolds can be equipped with Poisson structures that vanish to infinite order at codimension one subsets.
Constructs Poisson structure on Banach Lie algebroid predual.
problem Linear Poisson structure on Banach Lie algebroid predual.
method Alternative approach to existing sub-Poisson structure on dual bundle.
result Existence of queer Banach Lie algebroids.
Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler stru…
Proves local solvability for G2-structures with Poisson equations.
problem Local solvability of Poisson equations for G2-structures. method Proves local solvability for G2-structures with Poisson equations. result Local solvability of Poisson equations for closed G2-structures. Introduces θ-almost twisted Poisson structures and their cohomology.
problem Characterizing and understanding θ-almost twisted Poisson structures. method Definition and construction of θ-almost twisted Poisson structures, Lie-Rinehart algebra, cochain complex, and cohomology. result Definition and construction of θ-almost twisted Poisson cohomology. Constructs new Poisson structures on tangent bundles of Poisson manifolds.
problem Generating new Poisson structures on tangent bundles.
method Using algebroid brackets of differential forms to create Poisson structures.
result Found Casimir functions for constructed Poisson structures.
It is known that the computation of the Poisson cohomology is closely related to the classification of singularities of Poisson structures. In this paper, we will first look for the normal forms of germs at (0,0) of Poisson structures on the real (or complex) plane and recall a result given by Arnold. Then, we will com…