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For any $N \geq 5$ nonformal simply connected symplectic manifolds of dimension $2N$ are constructed. This disproves the formality conjecture for simply connected symplectic manifolds which was introduced by Lupton and Oprea.
We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this…
We show that there exist infinite-dimensional quasi-flats in the compactly supported Hamiltonian diffeomorphism group of the Liouville domain, with respect to the spectral norm, if and only if the symplectic cohomology of this Liouville…
Let $\alpha$ be a contact form on a manifold $M$, and $L\subseteq M$ a closed Legendrian submanifold. I prove that $L$ intersects some characteristic for $\alpha$ at least twice if all characteristics are closed and of the same period, and…
Let (m,b) a pair of natural numbers. For m even (resp. m odd and b greater than or equal to 2) we show that if there is an m-dimensional non-formal compact oriented manifold whose first Betti number equals b, there is also a symplectic…
The general framework of Legendre transformation is extended to the case of symplectic groupoids, using an appropriate generalization of the notion of generating function (of a Lagrangian submanifold).
Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two…
We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact…
Examples of nonformal simply connected symplectic manifolds are constructed.
We provide an infinite family of diffeomorphic symplectic forms on ruled surfaces, which are pairwise non-isotopic. This answers a uniqueness question regarding symplectic structures up to isotopy on closed symplectic four-manifolds.
Let $L \subset Y$ be a Legendrian submanifold of a contact manifold, $S\subset L$ a framed embedded sphere bounding an isotropic disc $D_S \subset Y \setminus L$, and use $L_S$ to denote the manifold obtained from $L$ by a surgery on $S$.…
We consider an entropy-type invariant which measures the polynomial volume growth of submanifolds under the iterates of a map, and we establish sharp uniform lower bounds of this invariant for the following classes of symplectomorphisms of…
In this note we define the notion of collarable slices of Lagrangian submanifolds. Those are slices of Lagrangian submanifolds which can be isotoped through Lagrangian submanifolds to a cylinder over a Legendrian embedding near a contact…
A non-formal simply connected compact symplectic manifold of dimension 8 is constructed.
Assume $(M, \omega)$ is a connected, compact 6 dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict…
While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between…
While the Hamiltonian group actions on closed symplectic manifolds have been widely explored throughout the last couple of decades, the study on Hamiltonian group actions on symplectic manifolds with a contact type boundary has started only…
In this work we construct Calabi quasi-morphisms on the universal cover of the group Ham(M) of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the…
We construct using Lefschetz fibrations a large family of contact manifolds with the following properties: Any bounding contact embedding into an exact symplectic manifold satisfying a mild topological assumption is non-displaceable and…
We prove that all maximal-tb Legendrian torus links (n,m) in the standard contact 3-sphere, except for (2,m),(3,3),(3,4) and (3,5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing…