CyberSec Research

The tight versus overtwisted dichotomy has been an essential organizing principle and driving force in 3-dimensional contact geometry since its inception around 1990. In this article, we will discuss the genesis of this dichotomy in Eliashberg's seminal work and his influential contributions to the theory.

Let $L$ be a closed Lagrangian submanifold of a symplectic manifold $(X,\omega)$. Cieliebak and Mohnke define the symplectic area of $L$ as the minimal positive symplectic area of a smooth $2$-disk in $X$ with boundary on $L$. An extremal Lagrangian torus in $(X,\omega)$ is a Lagrangian torus that maximizes the symplectic area among the Lagrangian tori in $(X,\omega)$. We prove that every extremal Lagrangian torus in the symplectic unit ball $(\bar{B}^{2n}(1),\omega_{\mathrm{std}})$ is contained entirely in the boundary $\partial B^{2n}(1)$. This answers a question attributed to Lazzarini and completely settles a conjecture of Cieliebak and Mohnke in the affirmative. In addition, we prove the conjecture for a class of toric domains in $(\mathbb{C}^n, \omega_{\mathrm{std}})$, which includes all compact strictly convex four-dimensional toric domains. We explain with counterexamples that the general conjecture does not hold for non-convex domains.

The present article provides an overview of Yakov Eliashberg's seminal contributions to the concepts of orderability and contact non-squeezing. It also examines subsequent research by various authors, highlighting the significance of these notions and offering a detailed account of the current state of the field.

Complex (affine) lines are a major object of study in complex geometry, but their symplectic aspects are not well understood. We perform a systematic study based on their associated Ahlfors currents. In particular, we generalize (by a different method) a result of Bangert on the existence of complex lines. We show that Ahlfors currents control the asymptotic behavior of families of pseudoholomorphic curves, refining a result of Demailly. Lastly, we show that the space of Ahlfors currents is convex.

We present novel Hamiltonian descriptions of some three-dimensional systems including two well-known systems describing the three-wave-interaction problem and some well-known chaotic systems, namely, the Chen, L\"u, and Qi systems. We show that all of these systems can be described in a Hamiltonian framework in which the Poisson matrix $\mathcal{J}$ is supplemented by a resistance matrix $\mathcal{R}$. While such resistive-Hamiltonian systems are manifestly non-conservative, we construct higher-degree Poisson matrices via the Jordan product as $\mathcal{N} = \mathcal{J} \mathcal{R} + \mathcal{R} \mathcal{J}$, thereby leading to new bi-Hamiltonian systems. Finally, we discuss conformal Hamiltonian dynamics on Poisson manifolds and demonstrate that by appropriately choosing the underlying parameters, the reduced three-wave-interaction model as well as the Chen and L\"u systems can be described in this manner where the concomitant non-conservative part of the dynamics is described with the aid of the Euler vector field.

In this paper, we compare the symplectic (co)homology capacity with the spectral capacity in the relative case. This result establishes a chain of inequalities of relative symplectic capacities, which is an analogue of the non-relative case. This comparison gives us a criterion for the relative almost existence theorem in terms of heaviness. Also, we investigate a sufficient condition under which the symplectic (co)homology capacity and the first Gutt-Hutchings capacity are equal in both non-relative and relative cases. This condition is less restrictive than the dynamical convexity.

Let $G$ be a connected complex semisimple group with Lie algebra $\mathfrak{g}$ and fixed Kostant slice $\mathrm{Kos}\subseteq\mathfrak{g}^*$. In a previous work, we show that $((T^*G)_{\text{reg}}\rightrightarrows\mathfrak{g}^*_{\text{reg}},\mathrm{Kos})$ yields the open Moore-Tachikawa TQFT. Morphisms in the image of this TQFT are called open Moore-Tachikawa varieties. By replacing $T^*G\rightrightarrows\mathfrak{g}^*$ and $\mathrm{Kos}\subseteq\mathfrak{g}^*$ with the double $\mathrm{D}(G)\rightrightarrows G$ and a Steinberg slice $\mathrm{Ste}\subseteq G$, respectively, one obtains quasi-Hamiltonian analogues of the open Moore-Tachikawa TQFT and varieties. We consider a conjugacy class $\mathcal{C}$ of parabolic subalgebras of $\mathfrak{g}$. This class determines partial Grothendieck-Springer resolutions $\mu_{\mathcal{C}}:\mathfrak{g}_{\mathcal{C}}\longrightarrow\mathfrak{g}^*=\mathfrak{g}$ and $\nu_{\mathcal{C}}:G_{\mathcal{C}}\longrightarrow G$. We construct a canonical symplectic groupoid $(T^*G)_{\mathcal{C}}\rightrightarrows\mathfrak{g}_{\mathcal{C}}$ and quasi-symplectic groupoid $\mathrm{D}(G)_{\mathcal{C}}\rightrightarrows G_{\mathcal{C}}$. In addition, we prove that the pairs $(((T^*G)_{\mathcal{C}})_{\text{reg}}\rightrightarrows(\mathfrak{g}_{\mathcal{C}})_{\text{reg}},\mu_{\mathcal{C}}^{-1}(\mathrm{Kos}))$ and $((\mathrm{D}(G)_{\mathcal{C}})_{\text{reg}}\rightrightarrows(G_{\mathcal{C}})_{\text{reg}},\nu_{\mathcal{C}}^{-1}(\mathrm{Ste}))$ determine TQFTs in a $1$-shifted Weinstein symplectic category. Our main result is about the Hamiltonian symplectic varieties arising from the former TQFT; we show that these have canonical Lagrangian relations to the open Moore-Tachikawa varieties. Pertinent specializations of our results to the full Grothendieck-Springer resolution are discussed throughout this manuscript.

In these memos, we define a pregeometry $\mathcal{T}_{\mathbb{S}} ^{alg}$ and a geometry $\mathcal{G}_{\mathbb{S}} ^{alg}$ which integrate symplectic manifolds with $E_{\infty}$-ring sheaves, enabling the construction of $\mathcal{G}_{\mathbb{S}} ^{alg}$-schemes as structured $\infty$-topoi. Our framework and results establish a profound connection between algebraic invariants and homological properties, opening new pathways for exploring symplectic phenomena through the lens of higher category theory and derived geometry.

Double Bruhat cells in a complex semisimple Lie group $G$ emerged as a crucial concept in the work of S. Fomin and A. Zelevinsky on total positivity and cluster algebras. Double Bruhat cells are special instances of a broader class of cluster varieties known as generalized double Bruhat cells. These can be studied collectively as Poisson subvarieties of $\widetilde{F}_{2n} = G \times \mathcal{B}^{2n-1}$, where $\mathcal{B}$ is the flag variety of $G$. The spaces $\widetilde{F}_{2n}$ are Poisson groupoids over $\mathcal{B}^n$, and they were introduced in the study of configuration Poisson groupoids of flags by J.-H. Lu, V. Mouquin, and S. Yu. In this work, we describe the spaces $\widetilde{F}_{2n}$ as decorated moduli spaces of flat $G$-bundles over a disc. As a consequence, we obtain the following results. (1) We explicitly integrate the Poisson groupoids $\widetilde{F}_{2n}$ to double symplectic groupoids, which are complex algebraic varieties. Moreover, we show that these integrations are symplectically Morita equivalent for all $n$, thereby recovering the Poisson bimodule structures on double Bruhat cells via restriction. (2) Using the previous construction, we integrate the Poisson subgroupoids of $\widetilde{F}_{2n}$ given by unions of generalized double Bruhat cells to explicit double symplectic groupoids. As a corollary, we obtain integrations of the top-dimensional generalized double Bruhat cells inside them. (3) Finally, we relate our integration with the work of P. Boalch on meromorphic connections. We lift to the groupoids the torus actions that give rise to such cluster varieties and show that they correspond to the quasi-Hamiltonian actions on the fission spaces of irregular singularities.

In the author's previous paper, the author constructed holomorphic quilts from the bigons of the Lagrangian Floer chain group after performing Lagrangian composition. This paper proves the uniqueness of such holomorphic quilts. As a consequence, it provides a combinatorial method for computing the boundary map of immersed Lagrangian Floer chain groups when the symplectic manifolds are closed surfaces. One outcome is the construction of many examples exhibiting figure eight bubbling, which also confirms a conjecture of Cazassus Herald Kirk Kotelskiy.

We define an invariant of three-manifolds with an involution with non-empty fixed point set of codimension $2$; in particular, this applies to double branched covers over knots. Our construction gives the Heegaard Floer analogue of Li's real monopole Floer homology. It is a special case of a real version of Lagrangian Floer homology, which may be of independent interest to symplectic geometers. The Euler characteristic of the real Heegaard Floer homology is the analogue of Miyazawa's invariant, and can be computed combinatorially for all knots.

We introduce and develop the theory of spectral networks in real contact and symplectic topology. First, we establish the existence and pseudoholomorphic characterization of spectral networks for Lagrangian fillings in the cotangent bundle of a smooth surface. These are proven via analytic results on the adiabatic degeneration of Floer trajectories and the explicit computation of continuation strips. Second, we construct a Family Floer functor for Lagrangian fillings endowed with a spectral network and prove its equivalence to the non-abelianization functor. In particular, this implies that both the framed 2d-4d BPS states and the Gaiotto-Moore-Neitzke non-abelianized parallel transport are realized as part of the $A_\infty$-operations of the associated 4d partially wrapped Fukaya categories. To conclude, we present a new construction relating spectral networks and Lagrangian fillings using Demazure weaves, and show the precise relation between spectral networks and augmentations of the Legendrian contact dg-algebra.

In this article we investigate the fragility of invariant Lagrangian graphs for dissipative maps, focusing on their destruction under small perturbations. Inspired by Herman's work on conservative systems, we prove that all $C^0$-invariant Lagrangian graphs for an integrable dissipative twist maps can be destroyed by perturbations that are arbitrarily small in the $C^{1-\varepsilon}$-topology. This result is sharp, as evidenced by the persistence of $C^1$-invariant graphs under $C^1$-perturbations guaranteed by the normally hyperbolic invariant manifold theorem.

The purpose of this article is to prove sharp $L^p$ bounds for quasimodes of Berezin-Toeplitz operators. We consider examples with explicit computations and a general situation on compact spaces and $\Cm^n$. In both cases the eigenvalue is a regular value of the operator symbol. We then use the link between pseudodifferential and Berezin-Toeplitz operators to obtain an $L^p$ bound of the FBI transform of quasimodes of pseudodifferential operators.

We develop the theory of Poisson and Dirac manifolds of compact types, a broad generalization in Poisson and Dirac geometry of compact Lie algebras and Lie groups. We establish key structural results, including local normal forms, canonical stratifications, and a Weyl type resolution, which provides a way to resolve the singularities of the original structure. These tools allow us to show that the leaf space of such manifolds is an integral affine orbifold and to define their Weyl group. This group is a Coxeter group acting on the orbifold universal cover of the leaf space by integral affine transformations, and one can associate to it Weyl chambers, reflection hyperplanes, etc. We further develop a Duistermaat-Heckman theory for Poisson manifolds of s-proper type, proving the linear variation of cohomology of leafwise symplectic form and establishing a Weyl integration formula. As an application, we show that every Poisson manifold of compact type is necessarily regular. We conclude the paper with a list of open problems.

A 3-dimensional contact group is a 3-dimensional Lie group endowed with a left-invariant contact structure. Making use of techniques from Riemannian geometry, we prove that any simply connected 3-dimensional contact group not isomorphic to SU(2) satisfies a unique factorization property. As an application, we develop a method to construct embeddings of 3-dimensional simply connected contact groups into one among two model tight contact manifolds.

We introduce the notions of contact round surgery of index 1 and 2, respectively, on Legendrian knots in $\left(\mathbb{S}^3, \xi_{st}\right)$ and associate diagrams to them. We realize Jiro Adachi's contact round surgeries as special cases. We show that every closed connected contact 3-manifold can be obtained by performing a sequence of contact round surgeries on some Legendrian link in $\left(\mathbb{S}^3, \xi_{st}\right)$, thus obtaining a contact round surgery diagram for each contact 3-manifold. This is analogous to a similar result of Ding and Geiges for contact Dehn surgeries. We discuss a bridge between certain pairs of contact round surgery diagrams of index 1 and 2 and contact $\pm1$-surgery diagrams. We use this bridge to establish the result mentioned above.

We classify all contact projective spaces with contact surgery number one. In particular, this implies that there exist infinitely many non-isotopic contact structures on the real projective 3-space which cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere. Large parts of our proofs deal with a detailed analysis of Gompf's $\Gamma$-invariant of tangential 2-plane fields on 3-manifolds. From our main result we also deduce that the $\Gamma$-invariant of a tangential 2-plane field on the real projective 3-space only depends on its $d_3$-invariant.