arXiv research

Generalizes symplectic reduction to cosymplectic groupoid actions.

Extends symplectic reduction and theorem to Lie algebroids.

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

We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's splitting theorem. Our approach turns out to be essentially canonical, and as a…

The paper studies deformations of Lagrangian submanifolds using algebraic tools.

The paper proves symplectic neighbourhood theorems for stratified subspaces.

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.

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.

Two reduction schemes for symplectic manifolds are shown equivalent.

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 …

We establish the charged Penrose inequality for time symmetric initial data sets having an outermost minimal surface boundary and finitely many asymptotically cylindrical ends, with an appropriate rigidity statement. This is accomplished by a doubling argument based on the work of Weinstein and Yamada, and a subsequent…

We introduce the notion of a Hamiltonian action of an étale Lie group stack on an étale symplectic stack and establish versions of the Kirwan convexity theorem, the Meyer-Marsden-Weinstein symplectic reduction theorem, and the Duistermaat-Heckman theorem in this context.

We prove the following result, conjectured by Alan Weinstein: every smooth proper Lie groupoid near a fixed point is locally linearizable, i.e. it is locally isomorphic to the associated groupoid of a linear action of a compact Lie group. In combination with a slice theorem of Weinstein, our result implies the smooth l…

A symplectic groupoid $G.:=(G_1 \rightrightarrows G_0)$ determines a Poisson structure on $G_0$. In this case, we call $G.$ a symplectic groupoid of the Poisson manifold $G_0$. 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, as the general existence problem for periodic orbits of Hamiltonian or Reeb flows, has been among the central questions in symplectic topology for over two decades and its investigation has led to understanding of some fundamental properties of Hamiltonian flows. In this paper we survey some r…

The article proves Lie's third theorem for Lie algebroids via singular Lie groupoids.

In this paper we extend the well-know normal form theorem for Lagrangian submanifolds proved by A. Weinstein in symplectic geometry to the setting of k-symplectic manifolds.

Let $G_¶$ be a compact simple Poisson-Lie group equipped with a Poisson structure $¶$ and $(M, ø)$ be a symplectic manifold. Assume that $M$ carries a Poisson action of $G_¶$ and there is an equivariant moment map in the sense of Lu and Weinstein which acts to the dual Poisson-Lie group $G^*_¶$, $\m: M\rightarrow G^*_¶…

We give a detailed, self-contained proof of Geoffrey Martin's normal form theorem for Lagrangian submanifolds of standard multisymplectic manifolds (that generalises Alan Weinstein's famous normal form theorem in symplectic geometry), providing also complete proofs for the necessary results in foliated differential top…

Introduces generalized moment maps for almost Hermitian settings.

The paper explores similarities in even and odd-dimensional geometry.

This paper sets a lower limit for the size of Weinstein's Lagrangian tubular neighborhoods.

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.

This paper constructs a Weinstein trisection for a surface bundle.

We prove that under certain mild assumptions a Lie bialgebroid integrates to a Poisson groupoid. This includes, in particular, a new proof of the existence of local symplectic groupoids for any Poisson manifold, a theorem of Karasev and of Weinstein.

Study Weinstein structures on toric divisors' complements.

A general study of symmetries in optimal control theory is given, starting from the presymplectic description of this kind of system. Then, Noether's theorem, as well as the corresponding reduction procedure (based on the application of the Marsden-Weinstein theorem adapted to the presymplectic case) are stated both in…

We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zung's theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The passing to general orbits (Weinstein) is given a more conceptual inte…

The paper simplifies symmetries in complex geometric structures.

Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.

In this article, we provide an introduction to an algorithm for constructing Weinstein handlebodies for complements of certain smoothed toric divisors using explicit coordinates and a simple example. This article also serves to welcome newcomers to Weinstein handlebody diagrams and Weinstein Kirby calculus. Finally, we…

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

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

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

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the existence of relative perodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. In particular we prove that under appropriate hypothes…

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

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

According to the Weinstein splitting theorem, any Poisson manifold is locally, near any given point, a product of a symplectic manifold with another Poisson manifold whose Poisson structure vanishes at the point. Similar splitting results are known e.g. for Lie algebroids, Dirac structures and generalized complex struc…

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes (i.e., periodic orbits) near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. More specifically we signific…

The abstract proves a global splitting theorem for Poisson manifolds.

We give a construction to obtain canonically an ``isotropic average'' of given $C^1$-close isotropic submanifolds of a symplectic manifold. To do so we use an improvement of Weinstein's submanifold averaging theorem (obtained in collaboration with H. Karcher) and apply ``Moser's trick''. We also present an application …

We introduce, for every $\mathbb{Z}$-graded manifold, a formal exponential map defined in a purely algebraic way and study its properties. As an application, we give a simple new construction of a Fedosov type resolution of the algebra of smooth functions of $\mathbb{Z}$-graded manifolds and we extend the Emmrich--Wein…

Teaches Dirac operators for geometry and topology.

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.