A symplectic action G : X of an algebraic group S on a symplectic algebraic variety X is called coisotropic if a generic orbit of this action is a coisotropic submanifold of X. In this article a classification of coisotropic symplectic linear actions G : V is given in the case where G is a reductive group.

Periodic cyclic homology group associated to a mixed complex was introduced by Goodwillie. In this talk, I will explain how to apply this construction to the symplectic cochain complex of a Liouville domain and obtain two periodic symplectic cohomology theories, which are called periodic symplectic cohomology and finitely supported periodic symplectic cohomology, respectively. The main result i...

In this article, we construct simply connected symplectic Calabi-Yau 6-manifold by applying Gompf’s symplectic fiber sum operation along T. Using our method, we also construct symplectic nonKähler Calabi-Yau 6-manifolds with fundamental group Z. We also produce the first examples of simply connected symplectic Calabi-Yau and non-Calabi-Yau 6-manifolds via coisotropic Luttinger surgery on non si...

In this work we define a relative version of the flux homomorphism, introduced by Calabi in 1969, for a symplectic manifold. We use it to study (the universal cover of) the group of symplectomorphisms of a symplectic manifold leaving a Lagrangian submanifold invariant. We show that some quotients of this group are stable under symplectic reduction.

In this paper we consider  (extended) metaplectic representation of the  semidirect product  $G_{mathbb{J}}=mathbb{R}^{2d}timesmathbb{J}$  where $mathbb{J}$ is a closed subgroup of $Sp(d,mathbb{R})$, the symplectic group. We will investigate continuous representation frame on $G_{mathbb{J}}$. We also discuss the existence of duals for such frames and give several characterization for them. Fina...

We develop differential and symplectic geometry of differentiable Deligne-Mumford stacks (orbifolds) including Hamiltonian group actions and symplectic reduction. As an application we construct new examples of symplectic toric DM stacks.

It is shown how one can do symplectic reduction for locally conformal symplectic manifolds, especially with an action of a Lie group. This generalizes well known procedures for symplectic manifolds to the slightly larger class of locally conformal symplectic manifolds. The whole setting is very conformally invariant.

1. Week 1 1 1.1. The cotangent bundle 1 1.2. Geodesic flow as Hamiltonian flow 4 2. Week 2 7 2.1. Darboux’s theorem 7 3. Week 3 10 3.1. Submanifolds of symplectic manifolds 10 3.2. Contact manifolds 12 4. Week 4 15 4.1. Symplectic linear group and linear complex structures 15 4.2. Symplectic vector bundles 18 5. Week 5 21 5.1. Almost complex manifolds 21 5.2. Kähler manifolds 24 6. Week 6 26 6....

An interesting question in symplectic geometry concerns whether or not a closed symplectic manifold can have a free symplectic circle action with orbits contractible in the manifold. Here we present a c-symplectic example, thus showing that the problem is truly geometric as opposed to topological. Furthermore, we see that our example is the only known example of a c-symplectic manifold having n...

We classify all homogeneous symplectic manifolds with a torsion free connection of special symplectic holonomy, i.e. a connection whose holonomy is an absolutely irreducible proper subgroup of the full symplectic group. Thereby, we obtain many new explicit descriptions of manifolds with special symplectic holonomies. We also show that manifolds with such a connection are homogeneous iff they co...