Symplectic Geometry

We study homotopically non-trivial spheres of Legendrians in the standard contact R3 and S3. We prove that there is a homotopy injection of the contactomorphism group of S3 into some connected components of the space of Legendrians induced…

In this paper and its companion, we define a new filled-section in the splicing in the smooth model for GW theory, and prove that it is of class $C^1$ in the usual Banach analysis rather than in the frame work of polyfold theory.

We call a singularity of a presymplectic form $\omega$ removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the…

This paper presents a proof of the existence of standard symplectic coordinates near a set of smooth, orthogonally intersecting symplectic submanifolds. It is a generalization of the standard symplectic neighborhood theorem. Moreover, in…

We develop a method of gluing the local mirrors and functors constructed from immersed Lagrangians in the same deformation class. As a result, we obtain a global mirror geometry and a canonical mirror functor. We apply the method to…

The main theme of the paper is the dynamics of Hamiltonian diffeomorphisms of ${\mathbb C}{\mathbb P}^n$ with the minimal possible number of periodic points (equal to $n+1$ by Arnold's conjecture), called here Hamiltonian pseudo-rotations.…

Under natural restrictions it is known that a nonlinear Schr\"odinger equation is a Hamiltonian PDE which defines a symplectic flow on a symplectic Hilbert space preserving the Hilbert norm. When the potential is one-periodic in time and…

This paper is the continuation of the previous two papers with the same title.

For a symplectic manifold M without boundary (not necessarily compact), we prove Poincare type duality in filtered cohomology rings of differential forms on M.

It is shown that the Poisson structure related to $\kappa$-Poincar\'e group is dual to a certain Lie algebroid structure, the related Poisson structure on the (affine) Minkowski space is described in a geometric way.

We establish results concerning the existence and nonexistence of regular $J$-holomorphic cylinders between nested pairs of ellipsoids in $\mathbb{R}^4$.

Let $X$ be an $n$-dimensional smooth complex projective variety embedded in $\mathbb{C}\mathbb{P}^{N}$. We construct a smooth family $\mathcal{X}$ over $\mathbb{C}$ with an embedding in $\mathbb{C}\mathbb{P}^{N} \times \mathbb{C}$ whose…

A twin Lagrangian fibration, originally introduced by Yau and the first author, is roughly a geometric structure consisting of two Lagrangian fibrations whose fibers intersect with each other cleanly. In this paper, we show the existence of…

In this paper we generalize the higher-degree smoothness results in perturbation theory from the case that the stable maps have the fixed domain $S^2$ to the general genus zero case.

We consider Hamiltonian Floer cohomology groups associated to a Lefschetz fibration, and the structure of operations on them. As an application, we will (under an important additional assumption) equip those groups with connections, which…

In this paper we give a rigorous definition of cylindrical contact homology for contact $3$-manifolds that admit nondegenerate contact forms with no contractible Reeb orbits, and show that the cylindrical contact homology is an invariant of…

The Ligon-Schaaf regularization (LS mapping) was introduced in 1976 and has been used several times. However, we are not aware of any direct usage of the inverse mapping, perhaps since it appears at first sight to be quite complex and…

This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds…

We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive…

Given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a…