The study of curves in projective varieties has been central to algebraic geometry since the nineteenth century. It was reinvigorated through its introduction into symplectic topology in Gromov’s seminal work [4] and now plays prominent roles in symplectic topology and string theory as well. The foundations of (complex) Gromov-Witten invariants, i.e. counts of J-holomorphic curves in symplectic...

Donaldson’s work on Lefschetz pencils has shown that, after a slight perturbation of the symplectic form and a finite number of blow-ups, any closed symplectic manifold (M,ω) can be expressed as a singular fibration with generic fiber a smooth codimension 2 symplectic submanifold. Thus fibrations play a fundamental role in symplectic geometry. It is then natural to study smooth (nonsingular) ru...

B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flattening a given torsion free symplectic connection. In this paper, a classical a...

We study symplectic Laplacians on compact symplectic manifolds with boundary. These Laplacians are associated with symplectic cohomologies of differential forms and can be of fourth-order. We introduce several natural boundary conditions on differential forms and use them to establish Hodge theory by proving various form decomposition and also isomorphisms between the symplectic cohomologies an...

We introduce a symplectic surgery in six dimensions which collapses Lagrangian three-spheres and replaces them by symplectic two-spheres. Under mirror symmetry it corresponds to an operation on complex 3-folds studied by Clemens, Friedman and Tian. We describe several examples which show that there are either many more Calabi-Yau manifolds (e.g., rigid ones) than previously thought or there exi...

We investigate the Poisson geometry of the Marsden-Weinstein reductions of the moment map associated to the cotangent bundle of the space of representations of a quiver. We show that the stratification by representation type equals the stratification by symplectic leaves. The deformed symplectic quotient singularities spectra of centres of symplectic reflection algebras associated to a wreath p...

In this paper we classify symplectic Lefschetz fibrations (with empty base locus) on a four-manifold which is the product of a three-manifold with a circle. This result provides further evidence in support of the following conjecture regarding symplectic structures on such a four-manifold: if the product of a three-manifold with a circle admits a symplectic structure, then the three-manifold mu...

The multi-symplectic Runge-Kutta (MSRK) methods and multi-symplectic Fourier spectral (MSFS) methods will be employed to solve the fourth-order Schrödinger equations with trapped term. Using the idea of split-step numerical method and the MSRK methods, we devise a new kind of multi-symplectic integrators, which is called split-stepmulti-symplectic (SSMS)methods. The numerical experiments show t...

In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has in nitely many concave symplectic llings and that all overtwisted contact 3-manifolds are \symplectic cobordism equivalent".

In [15] the authors have introduced a new technique to produce symplectic manifolds. It consists on taking a symplectic non-free action of a finite group on a symplectic manifold and resolving symplectically the singularities of the quotient. This has allowed to produce the first example of a non-formal simply connected compact symplectic manifold of dimension 8. Here we present another descrip...