Symplectic Geometry

In a series of papers on Bohr-Sommerfeld-Heisenberg quantization of completely integrable systems we interpreted shifting operators as quantization of functions ${\mathrm{e}}^{ \pm i{\theta}_j}$ , where $(I_j , {\theta}_j )$ are action…

We show that a monotone Lagrangian $L$ in $\mathbb{CP}^n$ of minimal Maslov number $n + 1$ is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to $\mathbb{RP}^n$. To prove this we use Zapolsky's canonical pearl…

In a previous article, we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the…

We introduce a family of spaces, parametrized by positive real numbers, that includes all of the Hirzebruch surfaces. Each space is viewed from two distinct perspectives. First, as a leaf space of a compact, complex, foliated manifold,…

The Whitney immersion is a Lagrangian sphere inside the four-dimensional symplectic vector space which has a single transverse double point of Whitney self-intersection number $+1.$ This Lagrangian also arises as the Weinstein skeleton of…

We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it…

We build examples of Poisson structure whose Poisson diffeomorphism group is not locally path-connected.

Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, is a relatively new approach to resolving transversality issues that arise in the study of $J$-holomorphic curves in symplectic geometry. This approach has recently led to a…

It is possible to construct distinct polyfolds which model a given moduli space problem in subtly different ways. These distinct polyfolds yield invariants which, a priori, we cannot assume are equivalent. We provide a general framework for…

Let $(M,\omega_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that $(M,\omega_M)$ is $S^1$-equivariant symplectomorphic to some K\"{a}hler Fano manifold…

We construct the chain level $L_\infty$-structure that extends the Lie bracket on symplectic cohomology.

It is known that orbit reduction can be performed in one or two stages and it has been proven that the two processes are symplectically equivalent. In the context of orbit reduction by one stage we shall write an expression for the reduced…

In this paper we show how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures in Hamiltonian Floer theory. Using the SFT of Hamiltonian mapping tori we…

We introduce the concept of fittings to symplectic fillings of the unit cotangent bundle of odd-dimensional spheres. Assuming symplectic asphericity we show that all fittings are diffeomorphic to the respective unit co-disc bundle.

The Bourgeois construction associates to every contact open book on a manifold $V$ a contact structure on $V\times T^2$. We study in this article some of the properties of $V$ that are inherited by $V\times T^2$ and some that are not.…

We study the symplectic Bott-Chern cohomology by L.-S. Tseng and S.-T. Yau for solvmanifolds endowed with left-invariant symplectic structures. Our results are applicable to cohomology with values in local systems. Studying symplectic…

In this article, we study polygonal symplectic billiards. We provide new results, some of which are inspired by numerical investigations. In particular, we present several polygons for which all orbits are periodic. We demonstrate their…

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these…

We generalize the familiar notions of overtwistedness and Giroux torsion in 3-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k \ge 0$ can be…

We consider Reeb dynamics on the 3-sphere associated to a tight contact form. Our main result gives necessary and sufficient conditions for a periodic Reeb orbit to bound a disk-like global section for the Reeb flow, when the contact form…