This paper investigates ways to enlarge the Hamiltonian subgroup {\rm Ham} of the symplectomorphism group {\rm Symp} of a symplectic manifold $(M, \omega)$ to a group that both intersects every connected component of {\rm Symp} and characterizes symplectic bundles with fiber $M$ and closed connection form. As a consequence, it is shown that bundles with closed connection form are stable under appropriate small perturbations of the symplectic form. Further, the manifold $(M,\omega)$ has the property that every symplectic $M$-bundle has a closed connection form if and only if the flux group vanishes and the flux homomorphism extends to a crossed homomorphism defined on the whole group {\rm Symp}. The latter condition is equivalent to saying that a connected component of the commutator subgroup [{\rm Symp}, {\rm Symp}] intersects the identity component of {\rm Symp} only if it also intersects {\rm Ham}. It is not yet clear when this condition is satisfied. We show that if the symplectic form vanishes on 2-tori, the flux homomorphism extends to the subgroup of {\rm Symp} acting trivially on $\pi_1(M)$. We also give an explicit formula for the Kotschick--Morita extension of the flux homomorphism in the monotone case. The results in this paper belong to the realm of soft symplectic topology, but raise some questions that may need hard methods to answer.

We generalize a theorem of Delzant classifying compact connected symplectic manifolds with completely integrable torus actions to certain singular symplectic spaces. The assumption on singularities is that if they are not finite quotient then they are isolated.

We introduce variants of Floer-Novikov cohomology and give a lower bound for the number of fixed points of time-one maps of time-dependent symplectic flows under the non-degeneracy condition.

An open subset $U$ of a complex surface can be topologically perturbed to yield an open subset whose inherited complex structure is Stein, if and only if $U$ is homeomorphic to the interior of a handlebody whose handles all have index ${\le}2$.

Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder recently proved a general compactness result for moduli spaces of punctured holomorphic curves arising in symplectic field theory. In this paper we present an alternative proof of this result. The main idea is to determine a priori the levels at which holomorphic curves split, thus reducing the proof to two separate cases: long cylinders of small area, and regions with compact image. The second case requires a generalization of Gromov compactness for holomorphic curves with free boundary.

In this paper, we apply Donaldson's general moment map framework for the action of a symplectomorphism group on the corresponding space of compatible (almost) complex structures to the case of rational ruled surfaces. This gives a new approach to understanding the topology of their symplectomorphism groups, based on a result of independent interest: the space of compatible integrable complex structures on any symplectic rational ruled surface is (weakly) contractible. We also explain how in general, under this condition, there is a direct relationship between the topology of a symplectomorphism group, the deformation theory of compatible complex structures and the groups of complex automorphisms of these complex structures.

Consider the cotangent bundle of a closed Riemannian manifold and an almost complex structure close to the one induced by the Riemannian metric. For Hamiltonians which grow, for instance, quadratically in the fibers outside a compact set, one can define Floer homology and show that it is naturally isomorphic to singular homology of the free loop space. We review the three isomorphisms constructed by Viterbo [16], Salamon--Weber [18] and Abbondandolo--Schwarz [14]. The theory is illustrated by calculating Morse and Floer homology in case of the Euclidean \textit{n}-torus. Applications include existence of noncontractible periodic orbits of compactly supported Hamiltonians on open unit disc cotangent bundles which are sufficiently large over the zero section.

The purpose of this survey is to explain an approach to the Atiyah--Floer conjecture via a new instanton Floer homology with Lagrangian boundary conditions. This is a joint project with Dietmar Salamon; see [14] for an earlier exposition. This paper also provides a rough guide to the analysis of anti-self-dual instantons with Lagrangian boundary conditions in [21, 22], which is the crucial ingredient of our approach.

We discuss our recent results on the existence and classification problem of complex and Kähler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact solvmanifolds (and compact homogeneous manifolds in general).