辛几何

We obtain families of non-isotopic closed exact Lagrangian submanifolds in quasi-projective holomorphic symplectic manifolds that admit contracting $\mathbb{C}^*$-actions. We show that the Floer cohomologies of these Lagrangians are…

We prove a version of Sandon's conjecture on the number of translated points of contactomorphisms for the case of prequantization bundles over certain closed monotone symplectic toric manifolds. Namely we show that any contactomorphism of…

This paper develops the analysis needed to set up a Morse-Bott version of embedded contact homology (ECH) of a contact three-manifold in certain cases. In particular we establish a correspondence between "cascades" of holomorphic curves in…

We give some properties of cosymplectic Lie algebras, we show, in particular, that they support a left symmetric product. We also give some constructions of cosymplectic Lie algebras, as well as a classification in three and…

We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution…

Suppose you have a nonorientable Lagrangian surface L in a symplectic 4-manifold. How far can you deform the symplectic form before the smooth isotopy class of L contains no Lagrangians? I solve this question for a particular Lagrangian…

The exterior algebra $E$ on a finite-rank free module $V$ carries a $\mathbb{Z}/2$-grading and an increasing filtration, and the $\mathbb{Z}/2$-graded filtered deformations of $E$ as an associative algebra are the familiar Clifford…

L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket conjecture by Leonid Polterovich in dimension $2$. In this note, instead of open cover consisting of displaceable sets in their work, we consider open cover…

We prove that the generalized symplectic capacities recognize objects in symplectic categories whose objects are of the form $(M, \omega)$, such that $M$ is a compact and 1-connected manifold, $\omega$ is an exact symplectic form on $M$,…

We prove a relationship between quantum Steenrod operations and the quantum connection. In particular there are operations extending the quantum Steenrod power operations that, when viewed as endomorphisms of equivariant quantum cohomology,…

In this paper we pose the question of whether the (generalized) Mukai inequalities hold for compact, positive monotone symplectic manifolds. We first provide a method that enables one to check whether the (generalized) Mukai inequalities…

We provide topological obstructions to the existence of orbit cylinders of symmetric orbits, for mechanical systems preserved by antisymplectic involutions (e.g. the restricted three-body problem). Such cylinders induce continuous paths…

We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of a circle family of global hypersurfaces of…

Let $\Sigma$ be a compact surface of genus $g \geq 1$ equipped with an area form. We construct eggbeater Hamiltonian diffeomorphisms which lie arbitrarily far in the Hofer metric from the set of autonomous Hamiltonians. This result is…

In this paper, we prove that if all closed characteristics on a compact non-degenerate star-shaped hypersurface $\Sigma$ in $\mathbf{R}^{2n}$ are elliptic, then either there exist exactly $n$ geometrically distinct closed characteristics,…

In the present article, we provide analytic foundation of the following nonlinear elliptic system, called the \emph{Hamiltonian-perturbed contact instanton equation}, $$ (du - X_H \otimes \gamma)^{\pi(0,1)} = 0, \quad d(e^{g_{H,…

In this article, we study conjectures of Sandon on the minimal number of translated points in the special case of the unit tangent bundle of a Riemannian manifold. We restrict ourselves to contactomorphisms of $SM$ that lift diffeomorphisms…

We establish computational results concerning the Lagrangian capacity from "Cieliebak and Mohnke - Punctured holomorphic curves and Lagrangian embeddings". More precisely, we show that the Lagrangian capacity of a 4-dimensional convex toric…

We already showed that Betti numbers are all zero when w is not equal to h for (w,h)-doubly weighted homology groups of some special pre Lie superalgebra and showed the first Betti number is 0 when w = h. In this note, we show that the…

Suppose that a compact $r$-dimensional torus $T^r$ acts in a holomorphic and Hamiltonian manner on polarized complex $d$-dimensional projective manifold $M$, with nowhere vanishing moment map $\Phi$. Assuming that $\Phi$ is transverse to…