Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.
problem Understanding the equivalence between stabilized convex symplectic manifolds and flexible Weinstein manifolds.
method Analyzing the homotopy type and symplectic properties of the manifolds.
result Stabilized convex symplectic manifolds are symplectomorphic to flexible Weinstein manifolds.
We introduce and discuss notions of regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains. There is a surprising abundance of flexible Lagrangians. In turn, this leads to new constructions of Legendrians submanifolds and Weinstein manifolds. For instance, many closed n-mani…
Study shows non-existence of certain contact structures on odd-dimensional manifolds.
problem Existence of contact structures with flexible pages on odd-dimensional manifolds.
method Proved a topological obstruction and used symplectic actions to show non-existence.
result Non-existence of contact open books with flexible pages for certain manifolds.
Groups of contact transformations are uniformly simple if supporting open books are flexible.
problem Understanding the algebraic and geometric properties of contact transformation groups.
method Exploring the connection between contact transformations, open book decompositions, and flexible Weinstein manifolds.
result The connected component of the identity in the automorphism group of a closed contact manifold is a uniformly simple group.
In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topol…
This paper constructs a Weinstein trisection for a surface bundle.
problem Constructing a trisection for a surface bundle.
method Weinstein trisection approach applied to ΣgimesΣh. result Explicit construction of a Weinstein trisection for S2imesS2. Survey of Weinstein symplectic manifolds, old and new.
problem Understanding Weinstein symplectic manifolds.
method Survey of basic notions and results.
result Discussion of open problems in the field.
New method to encode Weinstein 4-manifolds using multisections with divides.
problem Encoding and manipulating Weinstein 4-manifolds.
method Using multisection diagrams with divides and algorithms based on Kirby-Weinstein handle decompositions and PALFs.
result Definition and symplectic implementation of monodromy on multisections with divides.
Proves Weinstein conjecture for specific contact manifolds.
problem Proving the Weinstein conjecture for a specific class of contact manifolds.
method Introduced iterated planar Lefschetz fibrations and open book decompositions to prove the conjecture.
result Contact manifolds supporting iterated planar open book decompositions satisfy the Weinstein conjecture.
We prove symplectic hypersurfaces in Weinstein domains and give obstructions for manifold boundaries.
problem Obstructions for 3-manifolds to bound Weinstein domains in certain symplectic 4-manifolds.
method Symplectic embedding and deformation techniques.
result Obstructions for 3-manifolds to bound Weinstein domains in rational surfaces.
Study Weinstein structures on toric divisors' complements.
problem Understanding Weinstein structures on toric divisors' complements.
method Define a partially-centered condition on Delzant polytopes, develop an algorithm for Weinstein handlebody diagrams.
result Explicit Weinstein structures for complements of smoothed toric divisors.
Geometrically generates Fukaya categories of Weinstein manifolds.
problem Generating Fukaya categories of Weinstein manifolds.
method Geometric approach using surgery formula and Floer cohomology.
result Wrapped Fukaya category generated by index n critical points.
Generalizes symplectic reduction to cosymplectic groupoid actions.
problem Symplectic reduction for cosymplectic groupoid actions.
method Introduced cosymplectic groupoid actions and proved a theorem.
result Proved a theorem analogous to Mikami-Weinstein theorem.
In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.
In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product P1×P2 of two strongly geometrically bounded symplectic manifolds under some conditions with P1. In particular, if N is a closed manifold or a noncompact manifold of finite topological type, our result…
We prove generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
problem Understanding the generic fibre of Painlevé moduli spaces.
method Attach Weinstein handles to Stokes Legendrian.
result Generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
Symplectic 4-manifolds can be divided into three parts with a special structure.
problem Understanding the structure of symplectic 4-manifolds.
method Proved the existence of a trisection compatible with the symplectic structure.
result Symplectic 4-manifolds admit a trisection compatible with the symplectic structure.
Constructs Lagrangian skeleta for curve singularities.
problem Understanding Lagrangian skeleta of curve singularities.
method Constructs closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities.
result Provides computations of Legendrian and Weinstein invariants.
Improves Marsden-Weinstein reduction theory for k-polysymplectic manifolds.
problem Improving Marsden-Weinstein reduction theory for k-polysymplectic manifolds.
method Developed a theory of affine Lie group actions for k-polysymplectic momentum maps, removing technical conditions.
result Devise a k-polycosymplectic Marsden-Weinstein reduction theory.
Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid, and every Lie a…
A symplectic groupoid G.:=(G1⇉G0) determines a Poisson structure on G0. In this case, we call G. a symplectic groupoid of the Poisson manifold G0. However, not every Poisson manifold M has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing…
The Weinstein conjecture is extended to a new class of manifolds.
problem Existence of Reeb 2-curves in locally conformally symplectic manifolds.
method Extended Gromov-Witten theory and elliptic curve counts.
result Partial verification of the conjecture in higher dimensions.
New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.
problem Defining a symplectic invariant for Weinstein domains.
method Contact cut graph, Lefschetz fibrations, multisections with divides.
result Definition of Weinstein L-invariant and its relation to Kirby-Thompson's invariant. We present a generalized Weinstein Tubular Neighbourhood theorem for Lagrangian subbundles of Symplectic fibrations. This is then used to study the space of Lagrangian fibrations of symplectic manifolds.
Defines a new category structure on Weinstein manifolds using h-principle.
problem Defining a category structure on Weinstein manifolds.
method Defines a co/sheaf of categories on the skeleton of a Weinstein manifold using h-principle.
result Existence and uniqueness of the co/sheaf of categories up to isotopy.
Study singularities of Weinstein skeleta and arboreal singularities.
problem Understanding singularities in Weinstein skeleta and their relation to arboreal singularities.
method Deforming the skeleton via homotopies, Morse-Bott representative, generic perturbation, localization procedure.
result Singularities of Weinstein skeleta can be classified into arboreal singularities or tangency singularities.
Authors provide counterexamples to Weinstein conjecture in 3D.
problem Weinstein conjecture in 3D contact geometry.
method Construction of b-contact manifolds with specific orbits.
result Counterexamples to Weinstein conjecture in 3D.
Let (M,ω) be a symplectic manifold endowed with a agrangian foliation L, it has been shown by Weinstein [16] hat the symplectic structure of M defines on each leaf of L, connection which curvature and torsion forms vanish identically. uppose that L0 is a compact leaf which Weinstein connection …
We link Ginzburg algebras to Weinstein manifolds and Legendrian knots.
problem Understanding the relationship between Ginzburg algebras and Weinstein manifolds.
method Associated a stopped Weinstein manifold to a quiver and subquiver, proving quasi-isomorphism of relative Ginzburg algebra and Chekanov-Eliashberg dg-algebra.
result Relative Ginzburg algebra is quasi-isomorphic to Chekanov-Eliashberg dg-algebra of a singular Legendrian unknot link.
We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in S1×S2 or any connected sum #k(S1×S2), viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplec…
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. The study finds tight contact structures without fillings in high dimensions.
problem Finding tight contact structures that cannot be filled by symplectic forms.
method Construction of specific contact structures on manifolds of various dimensions.
result Existence of tight contact structures without fillings in all dimensions n≥3 and for n=2 under certain conditions. Theorems adapted for prequantum systems, showing symplectomorphism and gauge transformation.
problem Adapting classical theorems to prequantum systems.
method Establishing analogs of the Darboux, Moser, and Weinstein theorems.
result Prequantum systems with vanishing first cohomology are equivalent up to symplectomorphism and gauge transformation.
We use the minimal coupling procedure of Sternberg and Weinstein and our pseudo-symplectic capacity theory to prove that every closed symplectic submanifold in any symplectic manifold has an open neighborhood with finite (π1-sensitive) Hofer-Zehnder symplectic capacity. Consequently, the Weinstein conjecture holds n…
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.
Study invariants of elliptic curves in LCS manifolds, leading to new phenomena in Riemann-Finsler geometry.
problem Defining and studying invariants of elliptic curves in locally conformally symplectic manifolds.
method Using J-holomorphic curves and Gromov-Witten theory to define and study invariants. result Found new phenomena in Riemann-Finsler geometry and an analogue of the Weinstein conjecture.
We describe a new algebraic multisymplectic formulation of the classical BRST symmetry. The analogue of Marsden-Weinstein reduction for multisymplectic manifolds is described. We then give a homological description of Multisymplectic Marsden-Weinstein reduction.
A locally conformally symplectic (LCS) form is an almost symplectic form ω such that a closed one-form θ exists with dω=θ∧ω. We present a version of the well-known result of Darboux and Weinstein in the LCS setting and give an application concerning Lagrangian submanifolds.
Let U(n) be the unitary group, and u(n)∗ the dual of its Lie algebra, equipped with the Kirillov Poisson structure. In their 1983 paper, Guillemin-Sternberg introduced a densely defined Hamiltonian action of a torus of dimension (n−1)n/2 on u(n)∗, with moment map given by the Gelfand-Zeitlin coordinates. A few …
In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurf…
Introduces algorithm for Weinstein handlebodies of certain divisors.
problem Constructing Weinstein handlebodies for complements of smoothed toric divisors.
method Explicit coordinates and simple example for algorithm.
result Produces Weinstein Kirby diagrams for complements of smoothed toric divisors.
New disks found with similar outer shapes.
problem Finding similar outer shapes for Lagrangian disks.
method Modified construction of Abe and Tange.
result Arbitrarily large families of disks found.
In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincaré and Lagrange-Poincaré reduction. In fact, we show that solutions of the Hamilton-Poincaré equations and of the Lagrange-Poincaré equations are in one-to-one correspondence with dis…
This paper sets a lower limit for the size of Weinstein's Lagrangian tubular neighborhoods.
problem Finding the minimum size of Weinstein's Lagrangian tubular neighborhoods.
method Using the curvature tensor and second fundamental form of the submanifold.
result Explicit lower bounds for the radii of tubular neighborhoods are derived.
Functorial properties of knot invariants under specific concordances.
problem Computing obstructions to regular Lagrangian concordances.
method Functoriality of EH class and LOSS invariant under Lagrangian concordances in Weinstein cobordisms.
result Computable obstructions to regular Lagrangian concordances.
Study symplectic embeddings of 4-manifolds using Lefschetz fibrations.
problem Proper symplectic and iso-symplectic embeddings of 4-manifolds in 6-manifolds.
method Use Lefschetz fibrations to study symplectic embeddings.
result Closed orientable smooth 4-manifolds admitting Lefschetz fibrations over CP^1 can be embedded symplectically in (CP^1 × CP^1 × CP^1, ω_pr).
Classifies Kähler structures with special foliations using symplectic techniques.
problem Classifying Kähler structures with specific foliations.
method Weinstein's method of constructing symplectic bundles applied to Kähler data.
result New Kähler structures and metrics are obtained.
In this note, we study the Koszul-Brylinski homology of holomorphic Poisson manifolds. We show that it is isomorphic to the cohomology of a certain smooth complex Lie algebroid with values in the Evens-Lu-Weinstein duality module. As a consequence, we prove that the Evens-Lu-Weinstein pairing on Koszul-Brylinski homolo…