We obtain some obstructions to existence of Legendrian surgeries between tight lens spaces. We also study Legendrian surgeries between overtwisted contact manifolds.

We prove a generalization of the Conley conjecture: every Hamiltonian diffeomorphism of a closed symplectic manifold has infinitely many periodic orbits if the first Chern class vanishes over the second fundamental group. In particular, this removes the rationality condition from similar theorems by Ginzburg and Gürel. The proof in the irrational case involves several new ingredients including the definition and the properties of the filtered Floer homology for Hamiltonians on irrational manifolds. We also develop a method of localizing the filtered Floer homology for short action intervals using a direct sum decomposition, where one of the summands only depends on the behavior of the Hamiltonian in a fixed open set.

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form $\partial\bar{\partial}$-closed, i.e., to strong Kähler with torsion (SKT) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed symplectic structure but non-admitting Kähler structures. We show some negative results for the existence of symplectic forms taming complex structures on compact quotients of Lie groups by discrete subgroups. In particular, we prove that if $M$ is a nilmanifold (not a torus) endowed with an invariant complex structure $J$, then $(M,J)$ does not admit any symplectic form taming $J$. Moreover, we show that if a nilmanifold $M$ endowed with an invariant complex structure $J$ admits an SKT metric, then $M$ is at most 2-step. As a consequence we classify eight-dimensional nilmanifolds endowed with an invariant complex structure admitting an SKT metric.

We prove that quasi-morphisms and quasi-states on a closed rational symplectic manifold descend under symplectic reduction to symplectic hyperplane sections. Along the way we show that quasi-morphisms that arise from spectral invariants are the Calabi homomorphism when restricted to Hamiltonians supported on stably displaceable sets.

We establish a product formula for Gromov–Witten invariants for closed relatively semi-positive Hamiltonian fibrations, with connected fiber, and over any connected symplectic base. Furthermore, we show that the fibration projection induces a locally trivial (orbi-)fibration map from the moduli space of pseudo-holomorphic maps with marked points in the total space of the Hamiltonian fibration to the corresponding moduli space of pseudo-holomorphic maps with marked points in the base. We use this induced map to recover the product formula by means of integration. Finally, we give applications to c-splitting and symplectic uniruledness.