Symplectic Geometry

This paper is devoted to studying a notion of Bott integrability for Reeb flows on contact 3-manifolds. We show, in analogy with work of Fomenko-Zieschang on Hamiltonian flows in dimension 4, that Bott-integrable Reeb flows exist precisely…

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally…

We show that if the complement of a Donaldson hypersurface in a closed, integral symplectic manifold has the homology of a subcritical Stein manifold, then the hypersurface is of degree one. In particular, this demonstrates a conjecture by…

This note provides an affirmative answer to a question of Viterbo concerning the existence of nondiffeomorphic contact forms that share the same Reeb vector field. Starting from an observation by Croke-Kleiner and Abbondandolo that such…

We completely classify Legendrian realisations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.

We prove that for any compact toric symplectic manifold, if a Hamiltonian diffeomorphism admits more fixed points, counted homologically, than the total Betti number, then it has infinitely many simple periodic points. This provides a vast…

Fix a suitable link on the disk. Recently, F. Morabito associates each Hamiltonian symplecticmorphism preserving the link to a braid type. Based on this construction, Morabito defines a family of pseudometrics on the braid groups by using…

We present a new and simpler proof of the fact that any Lagrangian $\mathbb{R}P^2$ in $T^*\mathbb{R}P^2$ is Hamiltonian isotopic to the zero section. Our proof mirrors the one given by Li and Wu for the Hamiltonian uniqueness of Lagrangians…

We present a geometric realization of the duality between skeleta in $T^*\mathbb P^n$ and collars of local surfaces. Such duality is predicted by combining two auxiliary types of duality: on one side, symplectic duality between $T^*\mathbb…

In this paper we study a natural generalization of symplectic toric manifolds in the context of regular Poisson manifolds of compact types. To be more precise, we consider a class of multiplicity-free Hamiltonian actions by regular proper…

In this work we study the integrability of quotients of quasi-Poisson manifolds. Our approach allows us to put several classical results about the integrability of Poisson quotients in a common framework. By categorifying one of the already…

We show the existence of quasi-cluster $\mathcal{A}$-structures and cluster Poisson structures on moduli stacks of sheaves with singular support in the alternating strand diagram of grid plabic graphs by studying the microlocal parallel…

We formulate elementary SFT spectral invariants of a large class of symplectic cobordisms and stable Hamiltonian manifolds, in any dimension. We give criteria for the strong closing property using these invariants, and verify these criteria…

We analyze \emph{submersions with Poisson fibres}. These are submersions whose total space carries a Poisson structure, on which the ambient Poisson structure pulls back, as a Dirac structure, to Poisson structures on each individual fibre.…

We define a class of non-compact Fano toric manifolds, called admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed…

In this paper we study dually flat spaces arising from Delzant polytopes equipped with a symplectic potential together with their corresponding toric K\"ahler manifolds as their torifications.We introduce a dually flat structure and the…

Given a compact Lagrangian $L$ in a semipositive convex-at-infinity symplectic manifold $W$, we establish a cup-length estimate for the action values of $L$ associated to a Hamiltonian isotopy whose spectral norm is smaller than some…

Let $B^{2n}(R)$ denote the closed $2n$-dimensional symplectic ball of area $R$, and let $\Sigma_g(L)$ be a closed symplectic surface of genus $g$ and area $L$. We prove that there is a symplectic embedding $\bigsqcup_{i=1}^k B^4(R_i) \times…

Hamiltonian Floer theory plays an important role for finding periodic solutions of Hamilton's equation, which can be seen as a generalization of Newton's equation. Generalizing Newton's equation to Laplace's equation with non-linearity, we…

In previous work of the second- and third-named authors with Linhui Shen, cluster theory was used to construct wavefunctions for branes in threespace and conjecturally relate them to open Gromov-Witten invariants. This was done by defining…