CyberSec Research

We prove that the spectral selectors introduced by the author for closed strongly orderable contact manifolds satisfy algebraic properties analogous to those of the spectral selectors for lens spaces constructed by Allais, Sandon and the author using Givental's nonlinear Maslov index. As applications, first we establish a contact big fiber theorem for closed strongly orderable contact manifolds as well as for lens spaces. Second, when the Reeb flow is periodic, we construct a stably unbounded conjugation invariant norm on the contactomorphism group universal cover. Moreover, when all its orbits have the same period, we show that the Reeb flow is a geodesic for the discriminant and oscillation norms of Colin-Sandon.

We give a sufficient condition for an $\mathbb{S}^1$-bundle over a $3$-manifold to admit an immersion (or embedding) into $\mathbb{C}^3$ so that its complex tangencies define an Engel structure. In particular, every oriented $\mathbb{S}^1$-bundle over a closed, oriented $3$-manifold admits such an immersion. If the bundle is trivial, this immersion can be chosen to be an embedding and, moreover, infinitely many pairwise smoothly non-isotopic embeddings of this type can be constructed. These are the first examples of compact submanifolds of $\mathbb{C}^3$ whose complex tangencies are Engel, answering a question of Y. Eliashberg.

On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson $r$-matrix structure. The known $r$-matrices are defined on the Riemann sphere (rational), the cylinder (trigonometric), or the torus (elliptic). We extend the formalism to the case of a Riemann surface $\mathcal C$ of higher genus $g$: we consider the moduli space of framed vector bundles of rank $n$ and degree $ng$, where the framing consists in a choice of basis of $n$ independent holomorphic sections chosen to trivialize the fiber at a given point $\infty\in \mathcal C$. The co-tangent space is known to be identified with the set of Higgs fields, i.e., one-forms on $\mathcal C$ with values in the endomorphisms of the vector bundle, with an additional simple pole at $\infty$. The natural symplectic structure on the co-tangent bundle of the moduli space induces a Poisson structure on the Higgs fields. The result is then an explicit $r$--matrix that generalizes the known ones. A detailed discussion of the elliptic case with comparison to the literature is also provided.

In [LZ02], Long and Zhu established originally the common index jump theorem (CIJT) for symplectic paths in 2002, which was later generalized to its enhanced version (ECIJT) by Duan, Long and Wang in [DLW16] in 2016. Started from [GGM18] of 2018 and [GG20] of 2020, and finally in [CGG24] of 2024, a similar index theorem was obtained, i.e., Theorem 3.3 of [CGG24], which was called the index recurrence theorem there. In this short note, we give detailed proofs to show that the first 4 assertions dealing with index iterations in the total of 5 assertions in Theorem 3.3 of [CGG24] actually coincide completely with results in (ECIJT) of [DLW16]

We prove that any contact form on the standard contact 5-sphere with Zoll Reeb flow is strictly contactomorphic to a scaling of the standard Zoll contact form.

We develop an index theory for variational problems on noncompact quantum graphs. The main results are a spectral flow formula, relating the net change of eigenvalues to the Maslov index of boundary data, and a Morse index theorem, equating the negative directions of the Lagrangian action with the total multiplicity of conjugate instants along the edges. These results extend classical tools in global analysis and symplectic geometry to graph based models, with applications to nonlinear wave equations such as the nonlinear Schroedinger equation. The spectral flow formula is proved by constructing a Lagrangian intersection theory in the Gelfand-Robbin quotients of the second variation of the action. This approach also recovers, in a unified way, the known formulas for heteroclinic, halfclinic, homoclinic, and bounded orbits of (non)autonomous Lagrangian systems.

We construct a new family of skein exact triangles for link Floer homology. The skein triples are described by a triple of rational tangles $(R_0,R_1,R_{2n+1})$, where $R_0$ is the trivial tangle and $R_k$ is obtained from it by applying $k$ positive half-twists. We also set up an appropriate framework for potential construction of further skein exact triangles corresponding to arbitrary triples of rational tangles.

A manifold is said to be $n$-plectic if it is equipped with a closed, nondegenerate $(n+1)$-form. This thesis develops the theory of \emph{relative $n$-plectic structures}, where the classical condition is replaced by a closed, nondegenerate \emph{relative} $(n+1)$-form defined with respect to a smooth map. Analogous to how $n$-plectic manifolds give rise to $L_\infty$-algebras of observables, we show that relative $n$-plectic structures naturally induce corresponding $L_\infty$-algebras. These structures provide a conceptual bridge between the frameworks of quasi-Hamiltonian $G$-spaces and $2$-plectic geometry. As an application, we examine the relative $2$-plectic structure canonically associated to quasi-Hamiltonian $G$-spaces. We show that every quasi-Hamiltonian $G$-space defines a closed, nondegenerate relative $3$-form, and that the group action induces a Hamiltonian infinitesimal action compatible with this structure. We then construct explicit homotopy moment maps as $L_\infty$-morphisms from the Lie algebra $\mathfrak{g}$ into the Lie $2$-algebra of relative observables, extending the moment map formalism to the higher and relative geometric setting.

We introduce a modified version of the necklace Lie bialgebra associated to a quiver, in which the bracket and cobracket insert (rather than remove) pairs of arrows in involution. This structure is then related to canonical quartic Poisson/Batalin-Vilkovisky structures on suitable representation varieties of the quiver. Constructions on the representation side take place in symmetric monoidal $\Pi$-categories, which prompts a discussion of graded differential operators on commutative monoids in any such category. The generality of the categorical approach allows us to fully recover necklace structures, showing how the modified and classical necklace operations are related via dualisability.

Given a Legendrian knot $\Lambda \subset \mathbb{R}^3$ and a vertical line dividing the front projection of $\Lambda$ into two halves, we construct a differential graded algebra associated to each half-knot. We then show that one may obtain the commutative algebra from Legendrian Rational Symplectic Field Theory as a pushout of the two bordered algebras. This construction extends the bordered Chekanov-Eliashberg differential graded algebra by incorporating disks with multiple positive punctures into the differential.

In this note we show that the Barutello-Ortega-Verzini regularization map is scale smooth.

We study equivariant Gromov-Witten invariants and quantum cohomology in GKM theory. Building on the localization formula, we prove that the resulting expression is independent of the choice of compatible connection, and provide an equivalent formulation without auxiliary choices. Motivated by this theoretical refinement, we develop a software package, $\texttt{GKMtools.jl}$, that implements the computation of equivariant GW invariants and quantum products directly from the GKM graph. We apply our framework to several geometric settings: Calabi-Yau rank two vector bundles on $\mathbb{P}^1$, where we obtain a new proof for a recent connection to Donaldson-Thomas theory of Kronecker quivers; twisted flag manifolds, which give symplectic but non-algebraic examples of GKM spaces; realizability questions for abstract GKM graphs; and classical enumerative problems involving curves in hyperplanes. These results demonstrate both the theoretical flexibility of GKM methods and the effectiveness of computational tools in exploring new phenomena.

We formulated a homological and computer-aided approach to study certain unions of symplectic surfaces, called symplectic configurations, in a rational $4$-manifold $X=CP^2\# N\overline{CP^2}$. We addressed several fundamental theoretical questions, and also as a technical device, developed a symplectic analog of the so-called quadratic Cremona transformations in complex algebraic geometry. As an application, we gave a new proof that a certain line arrangement in $CP^2$, called Fano planes, does not exist in the symplectic category. The nonexistence of Fano planes in the holomorphic category was due to Hirzebruch, and in the topological category, it was first proved by Ruberman and Starkston. Our proof in the symplectic category is independent to both.

We prove that the space of contractible simple loops of a given fixed area in any compact oriented surface has infinite diameter as a homogeneous space of the group of area-preserving diffeomorphisms endowed with the $L^p$-metric. As a special case, this resolves the $L^p$-metric analogue of the well-known question in symplectic topology regarding the space of equators on the two-sphere. Our methods involve a new class of functionals on a normed group, which are more general than quasi-morphisms.

We construct open-closed maps on various versions of Hochschild and cyclic homology of the Fukaya $A_\infty$ algebra of a Lagrangian submanifold modeled on differential forms. The $A_\infty$ algebra may be curved. Properties analogous to Gromov-Witten axioms are verified. The paper is written with applications in mind to gravitational descendants and obstruction theory.

We develop a categorical framework for simple homotopy theory in Fukaya categories, based on the fundamental group of the ambient symplectic manifold. When the first Chern class vanishes, we show that any isomorphism in the Fukaya category of a Weinstein manifold has trivial Whitehead torsion. As an application, we prove that any pair of closed connected Lagrangians that are isomorphic in the Fukaya category of such Weinstein manifolds are simple homotopy equivalent, provided one of the Lagrangians is homotopy equivalent to the ambient symplectic manifold and their fundamental groups are isomorphic.

We consider semi-free Hamiltonian $S^1$-manifolds of dimension six and establish when the equivariant cohomology and data on the fixed point set determine the isomorphism type. Gonzales listed conditions under which the isomorphism type of such spaces is determined by fixed point data. We pointed out in an earlier paper that this result as stated is erroneous, and proved a corrected version. However, that version relied on a certain distribution of fixed points that is not at all necessary. In this paper, we replace the latter assumption with a global assumption on equivariant cohomology that is necessary for an isomorphism. We also extend our result to the equivariant (non-symplectic) topological category. The variation in the earlier paper was tailored to suit the requirements of Cho's application of Gonzales' statement to classify semi-free monotone, Hamiltonian $S^1$-manifolds of dimension six. In the current paper, we aim to give the definitive statement relating fixed point data and equivariant cohomology to the isomorphism type of a semi-free Hamiltonian $S^1$-manifold.

We study the spherical pendulum system with an arbitrary potential function $V = V(z)$, which is an integrable system with a first integral whose Hamiltonian flow is periodic. We give explicit solutions to these integrable systems. In the special case when the potential function is symmetric quadratic like $V = z^2$, we compute its action-angle coordinates in terms of elliptic integrals.