Symplectic Geometry

We discuss a quantum counterpart, in the sense of the Berezin-Toeplitz quantization, of certain constraints on Poisson brackets coming from "hard" symplectic geometry. It turns out that they can be interpreted in terms of the quantum noise…

Biinvariant diagonal classes give rise to right inverses of the Kirwan map. By means of multivalued perturbations of the gradient flow equation such classes are constructed explicitly for $S^1$-Hamiltonian spaces. Moreover, the notion of…

We construct absolute and relative versions of Hamiltonian Floer homology algebras for strongly semi-positive compact symplectic manifolds with convex boundary, where the ring structures are given by the appropriate versions of the…

We use quantum and Floer homology to construct (partial) quasi-morphisms on the universal cover of the group of compactly supported Hamiltonian diffeomorphisms for a certain class of non-closed strongly semi-positive symplectic manifolds…

We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in $\mathbb{R}^3$. In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms…

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain and target are ellipsoids…

We develop techniques for defining and working with virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves which are not necessarily cut out transversally. Such techniques have the potential for applications as foundations…

There are two types of $J$-holomorphic spheres in a symplectic manifold invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic…

Kuranishi atlases were introduced by McDuff and Wehrheim to build a virtual fundamental class on moduli spaces of J-holomorphic curves and resolve some of the challenges in this field. This paper considers Gromov-Witten moduli spaces and…

This paper consists of two parts. In the first part, we use symplectic homology to distinguish the contact structures on the Brieskorn manifolds $\Sigma(2l,2,2,2)$, which contact homology cannot distinguish. This answers a question from…

The following interesting quantity was introduced by K. Cieliebak and K. Mohnke for a Lagrangian submanifold $L$ of a symplectic manifold: the minimal positive symplectic area of a disc with boundary on $L$. They also showed that this…

We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to g-quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these g-quasi-Poisson…

We show that for $m>n\geq 2$, there are at least two exact isotropic $n$-tori in $\mathbb{C}^m$ which are not Hamiltonian isotopic in $\mathbb{C}^m$, even though they are smoothly isotopic as isotropic $n$-tori. We apply this discovery to…

We study periodic orbits on a nondegenerate dynamically convex starshaped hypersurface in $\mathbb{R}^{2n}$ along the lines of Long and Zhu, but using properties of the $S^1$-equivariant symplectic homology. We prove that there exist at…

Let $M$ be a closed Fano symplectic manifold with a semifree Hamiltonian circle action with isolated maximum. We compute the Gromov width and the Hofer-Zehnder capacity of $M$ using a moment map.

Let $(M, \omega)$ be a 6-dimensional closed symplectic manifold with a symplectic $S^1$-action with $M^{S^1} \neq \emptyset$ and $\dim M^{S^1} \leq 2$. Assume that $\omega$ is integral with a generalized moment map $\mu$. We first prove…

The Madelung transform relates the non-linear Schr\"odinger equation and a compressible Euler equation known as the quantum hydrodynamical system. We prove that the Madelung transform is a momentum map associated with an action of the…

The first result of this paper is that every contact form on $\mathbb{R} P^3$ sufficiently $C^\infty$-close to a dynamically convex contact form admits an elliptic-parabolic closed Reeb orbit which is $2$-unknotted, has self-linking number…

We prove a generalization of Gromov's symplectic non-squeezing theorem for the case of Hilbert spaces. Our approach is based on filling almost complex Hilbert spaces by complex discs partially extending Gromov's results on existence of…

Related to each degeneration from CP^2 to CP(a^2,b^2,c^2), for (a,b,c) a Markov triple - positive integers satisfying a^2 + b^2 + c^2 = 3abc - there is a monotone Lagrangian torus, which we call T(a^2,b^2,c^2). We employ techniques from…