arXiv research

This paper constructs a Weinstein trisection for a surface bundle.

New method to encode Weinstein 4-manifolds using multisections with divides.

Symplectic 4-manifolds can be divided into three parts with a special structure.

We prove symplectic hypersurfaces in Weinstein domains and give obstructions for manifold boundaries.

Study symplectic embeddings of 4-manifolds using Lefschetz fibrations.

Constructs Lagrangian skeleta for curve singularities.

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…

Study Weinstein structures on toric divisors' complements.

New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.

We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, 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…

We established existence of periodic Reeb orbits for a large class of tight contact structures on closed 3-manifolds, notably the Stein fillable structures, based on a fundamental theorem of Cliff Taubes on symplectic 4-manifolds.

The paper finds symplectic mapping class relations using pencil pairs.

We study Weinstein 4-manifolds which admit Lagrangian skeleta given by attaching disks to a surface along a collection of simple closed curves. In terms of the curves describing one such skeleton, we describe surgeries that preserve the ambient Weinstein manifold, but change the skeleton. The surgeries can be iterated …

Introduces algorithm for Weinstein handlebodies of certain divisors.

We 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. We discuss several potential applications, including the cla…

Generalizes symplectic reduction to cosymplectic groupoid actions.

Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.

Extends symplectic reduction and theorem to Lie algebroids.

We prove generic fibre of Painlevé moduli spaces are Weinstein handlebodies.

Develops Marsden-Meyer-Weinstein reduction for $k$-contact field theories.

Given a Legendrian link in $\#^k(S^1\times S^2)$, we extend the definition of a normal ruling from $J^1(S^1)$ given by Lavrov and Rutherford and show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over $\mathbb{Z}[t,t^{-1}]$ is equivalent to the existence of a …

The paper extends Marsden-Weinstein reduction to mechanical presymplectic structures for time-dependent Hamiltonian systems.

Extending work of Chen, we prove the Weinstein conjecture in dimension three for strongly fillable contact structures with either non-vanishing first Chern class or with strong and exact filling having non-trivial canonical bundle. This implies the Weinstein conjecture for certain Stein fillable contact structures obta…

The existence of a "Plastikstufe" for a contact structure implies the Weinstein conjecture for all supporting contact forms.

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 al…

Study non-Weinstein Liouville geometry via hyperbolic dynamics, proving rigidity results.

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…

This is a very biased and incomplete survey of some basic notions, old and new results, as well as open problems concerning Weinstein symplectic manifolds.

In this paper, we introduce the notions of an iterated planar Lefschetz fibration and an iterated planar open book decomposition and prove the Weinstein conjecture for contact manifolds supporting an open book that has iterated planar pages. For $n\geq 1$, we show that a $(2n+1)$-dimensional contact manifold $M$ suppor…

Improves Marsden-Weinstein reduction theory for k-polysymplectic manifolds.

We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.

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 $P_1\times P_2$ of two strongly geometrically bounded symplectic manifolds under some conditions with $P_1$. In particular, if $N$ is a closed manifold or a noncompact manifold of finite topological type, our result…

Local coordinates for non-integrable Lie algebroids constructed.

Authors provide counterexamples to Weinstein conjecture in 3D.

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.

In this paper, we establish a general relationship between the nonvanishing of GW invariants with the existence of the closed orbits of a Hamiltonian system. As an application, we completely solved the stabilized Weinstein conjecture.

We prove that the wrapped Fukaya category of any $2n$-dimensional Weinstein manifold (or, more generally, Weinstein sector) $W$ is generated by the unstable manifolds of the index $n$ critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer cohomology and t…

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 …

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 standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching 2-handles to convex boundaries of…

This is an introduction to Taubes's proof of the Weinstein conjecture, written for the AMS Current Events Bulletin. It is intended to be accessible to nonspecialists, so much of the article is devoted to background and context.

Proves Weinstein's and Arnold's conjectures using contact instantons.

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.

The paper studies deformations of Lagrangian submanifolds using algebraic tools.

The study finds tight contact structures without fillings in high dimensions.

Theorems adapted for prequantum systems, showing symplectomorphism and gauge transformation.

A locally conformally symplectic (LCS) form is an almost symplectic form $ω$ such that a closed one-form $θ$ exists with $dω= θ\wedge ω$. We present a version of the well-known result of Darboux and Weinstein in the LCS setting and give an application concerning Lagrangian submanifolds.