辛几何

This paper introduces two constructions of Legendrian submanifolds, called the Legendrian product and spinning, and computes their classical invariants, the Thurston-Bennequin invariant and the Maslov class, in R^{2n+1}. These constructions…

We use the method of algebraic restrictions to classify symplectic $U_7$, $U_8$ and $U_9$ singularities. We use discrete symplectic invariants to distinguish symplectic singularities of the curves. We also give the geometric description of…

We prove an equivariant version of the local splitting theorem for tame Poisson structures and Poisson actions of compact Lie groups. As a consequence, we obtain an equivariant linearization result for Poisson structures whose transverse…

We prove a Poincare lemma for a set of r smooth functions on a 2n-dimensional smooth manifold satisfying a commutation relation determined by r singular vector fields associated to a Cartan subalgebra of $\frak{sp}(2r,\mathbb R)$. This…

We prove the action-angle theorem in the general, and most natural, context of integrable systems on Poisson manifolds, thereby generalizing the classical proof, which is given in the context of symplectic manifolds. The topological part of…

We introduce the notion of induced Maslov cycle, which describes and unifies geometrical and topological invariants of many apparently unrelated problems, from Real Algebraic Geometry to sub-Riemannian Geometry.

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact symplectic manifold $X$ and an exact…

We define a higher analogue of Dirac structures on a manifold M. Under a regularity assumption, higher Dirac structures can be described by a foliation and a (not necessarily closed, non-unique) differential form on M, and are equivalent to…

In this note we prove that the space of linear anti-symplectic involutions is the homogenous space $Gl(n,\R)\Sp(n)$. This result is motivated by the study of symmetric periodic orbits in the restricted 3-body problem.

This survey paper addresses uniqueness questions for symplectic forms on closed manifolds, explains what is known in several examples, and reviews some open problems.

We define objects made of marked complex disks connected by metric line segments and construct nonsymmetric and symmetric moduli spaces of these objects. This allows choices of coherent perturbations over the corresponding versions of the…

We introduce a notion of the noncommutative integrability within a framework of contact geometry.

The classical Hurwitz numbers which count coverings of a complex curve have an analog when the curve is endowed with a theta characteristic. These "spin Hurwitz numbers", recently studied by Eskin, Okounkov and Pandharipande, are…

In recent times a great amount of progress has been achieved in symplectic and contact geometry, leading to the development of powerful invariants of 3-manifolds such as Heegaard Floer homology and embedded contact homology. These…

We derive a formula for the the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular…

Motivated by questions from quantum group and field theories, we review structures on manifolds that are weaker versions of Poisson structures, and variants of the notion of Lie algebroid. We give a simple definition of the Courant…

We extend Perutz's Lagrangian matching invariants to 3-manifolds which are not necessarily fibred using the technology of holomorphic quilts. We prove an isomorphism of these invariants with Ozsvath-Szabo's Heegaard Floer invariants for…

Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of…

After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson…

We discuss symplectic cutting for Hamiltonian actions of non-Abelian compact groups. By using a degeneration based on the Vinberg monoid we give, in good cases, a global quotient description of a surgery construction introduced by Woodward…