Mathematics - Symplectic Geometry Publications (50)


Mathematics - Symplectic Geometry Publications

We study the generalization of quasipositive links from the three-sphere to arbitrary closed, orientable three-manifolds. As in the classical case, we see that this generalization of quasipositivity is intimately connected to contact and complex geometry. The main result is an essentially three-dimensional proof that every link in the boundary of a Stein domain which bounds a complex curve in the interior is quasipositive, generalizing a theorem of Boileau-Orevkov. Read More

We prove that the set of symmetrized conjugacy classes of the kernel of the Calabi homomorphism on the group of area-preserving diffeomorphisms of the $2$-disk is not quasi-isometric to the half line. Read More

Let X be a closed symplectic manifold equipped a Lagrangian torus fibration. A construction first considered by Kontsevich and Soibelman produces from this data a rigid analytic space, which can be considered as a variant of the T-dual introduced by Strominger, Yau, and Zaslow. We prove that the Fukaya category of X embeds fully faithfully in the derived category of coherent sheaves on the rigid analytic dual, under the technical assumption that the second homotopy group of X vanishes (all known examples satisfy this assumption). Read More

This note constructs sharp obstructions for stabilized symplectic embeddings of an ellipsoid into a ball, in the case when the initial four-dimensional ellipsoid has `eccentricity' of the form 3n-1 for some integer n. Read More

We extend the classical Fuller index, to prove what may be considered to be one "corrected" version of the original Seifert conjecture, on the existence of periodic orbits for vector fields on $S ^{2k+1} $. We also give some generalizations and extensions of this theorem for certain other smooth manifolds. As one application we show that for a given smooth non-singular vector field $X _{1} $ on $S^{2k+1} $ with no periodic orbits, and any homotopy $\{X _{t} \}$ of smooth non-singular vector fields, with $X _{0} $ the Hopf vector field, $\{X _{t}\}$ has a sky catastrophe, which is a kind bifurcation originally discovered by Fuller. Read More

We consider the stability of nonlinear traveling waves in a class of activator-inhibitor systems. The eigenvalue equation arising from linearizing about the wave is seen to preserve the manifold of Lagrangian planes for a nonstandard symplectic form. This allows us to define a Maslov index for the wave corresponding to the spatial evolution of the unstable bundle. Read More

Let $(M, \alpha)$ be a connected compact contact toric manifold of Reeb type. In this note, we give a proof that the fundamental group of $M$ is finite cyclic. We also point out the fundamental groups of certain related quotients of $M$. Read More

We describe a new method for combinatorially computing the transverse invariant in knot Floer homology. Previous work of the authors and Stone used braid diagrams to combinatorially compute knot Floer homology of braid closures. However, that approach was unable to explicitly identify the invariant of transverse links that naturally appears in braid diagrams. Read More

We inquire into the relation between the curl operators, the Poisson coboundary operators and contravariant derivatives on Poisson manifolds to study the theory of differential operators in Poisson geometry. Given an oriented Poisson manifold, we describe locally those two differential operators in terms of Poisson connection whose torsion is vanishing. Moreover, we introduce the notion of the modular operator for an oriented Poisson manifold. Read More

We study cosmetic contact surgeries along transverse knots in the standard contact 3-sphere, i.e. contact surgeries that yield again the standard contact 3-sphere. Read More

We study the Morse-Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz Condition. We consider solvmanifolds and Oeljeklaus-Toma manifolds. Read More

We prove two tropical gluing formulae for Gromov-Witten invariants of exploded manifolds, useful for calculating Gromov-Witten invariants of a symplectic manifold using a normal-crossing degeneration. The first formula generalizes the symplectic-sum formula for Gromov-Witten invariants. The second formula is stronger, and also generalizes Kontsevich and Manin's splitting and genus-reduction axioms. Read More

We construct a symplectic structure on a disc that admits a compactly supported symplectomorphism which is not smoothly isotopic to the identity. The symplectic structure has an overtwisted concave end; the construction of the symplectomorphism is based on a unitary version of the Milnor-Munkres pairing. En route, we introduce a symplectic analogue of the Gromoll filtration. Read More

We consider the self-Floer theory of a monotone Lagrangian L in a closed symplectic manifold X in the presence of different kinds of symmetry. First, when (X, L) is K-homogeneous we show how to compute the image of low codimension K-invariant subvarieties under the length zero closed-open map. Then (for not necessarily homogeneous (X, L)) we extend the action of Symp(X, L) on the Oh spectral sequence to coefficients of arbitrary characteristic, working over an enriched Novikov ring, and use it to constrain the differentials. Read More

We study augmentations of a Legendrian surface $L$ in the $1$-jet space, $J^1M$, of a surface $M$. We introduce two types of algebraic/combinatorial structures related to the front projection of $L$ that we call chain homotopy diagrams (CHDs) and Morse complex $2$-families (MC2Fs), and show that the existence of either a $\rho$-graded CHD or MC2F is equivalent to the existence of a $\rho$-graded augmentation of the Legendrian contact homology DGA to $\mathbb{Z}/2$. A CHD is an assignment of chain complexes, chain maps, and homotopy operators to the $0$-, $1$-, and $2$-cells of a compatible polygonal decomposition of the base projection of $L$ with restrictions arising from the front projection of $L$. Read More

Given an arbitrary graph $\Gamma$, let $X_\Gamma$ be the Weinstein $4$-manifold obtained by plumbing copies of $T^*S^2$ according to this graph. We compute the wrapped Fukaya category of $X_\Gamma$ (with bulk parameters) using Legendrian surgery extending our previous work arXiv:1502.07922 where $\Gamma$ was assumed to be a tree. Read More

Using the wonderful compactification of a semisimple adjoint affine algebraic group G defined over an algebraically closed field k of arbitrary characteristic, we construct a natural compactification Y of the G-character variety of any finitely generated group F. When F is a free group, we show that this compactification is always simply connected with respect to the \'etale fundamental group, and when k=C it is also topologically simply connected. For other groups F, we describe conditions for the compactification of the moduli space to be simply connected and give examples when these conditions are satisfied, including closed surface groups and free abelian groups when G=PGL(n,C). Read More

We give a direct, explicit and self-contained construction of a local Lie groupoid integrating a given Lie algebroid which only depends on the choice of a connection. We also give a complete account of local Lie theory based on these explicit constructions. On the resulting local Lie groupoids, called spray groupoids, we obtain formulas for integrating infinitesimally multiplicative objects. Read More

We prove the full off-diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. As an application, we construct the algebra of Toeplitz operators on the symplectic manifold associated with the renormalized Bochner-Laplacian. Read More

We establish multiplicity results for geometrically distinct contractible closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles. The results hold under certain index requirements on the contact form and are sharp for unit cotangent bundles of CROSS's. In particular, we generalize and put in the symplectic-topological context a theorem of Duan, Liu, Long, and Wang for the standard contact sphere. Read More

We study Lagrangian correspondences between Liouville manifolds and construct functors between wrapped Fukaya categories. The study naturally brings up the question on comparing two versions of wrapped Fukaya categories of the product manifold, which we prove quasi-isomorphism on the level of wrapped Floer cohomology. To prove representability of these functors constructed from Lagrangian correspondences, we introduce the geometric compositions of Lagrangian correspondences under wrapping, as new classes of objects in the wrapped Fukaya category, which we prove to represent the functors by establishing a canonical isomorphism of quilted version of wrapped Floer cohomology under geometric composition. Read More

A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic techniques adapted to Lie algebroids, and use these to construct stable generalized complex structures out of log-symplectic structures. Read More

We associate to any integrable Poisson manifold a stack, i.e.~a category fibered in groupoids over a site. Read More

Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of math-ph/0702045, seeing it as a quantization of certain quadratic Lagrangians in $T^*V$ for some vector space $V$. KS topological recursion is a procedure which takes as initial data a quantum Airy structure -- a family of at most quadratic differential operators on $V$ satisfying some axioms -- and gives as outcome a formal series of functions in $V$ (the partition function) simultaneously annihilated by these operators. Finding and classifying quantum Airy structures modulo gauge group action, is by itself an interesting problem which we study here. Read More

We derive a fixed-point formula for integrals on moduli spaces of stable maps to projective spaces of even dimension. This gives a formula for the equivariant open Gromov-Witten invariants of (RP^{2m},CP^{2m}) and the structure constants of the equivariant Fukaya A-infinity algebra of RP^2m in CP^{2m} with bulk deformations. The formula involves contributions from Givental's correlators for the closed theory and the descendent integrals of discs, and specializes to give a new expression for the Welschinger count of real rational curves in the plane passing through some real and conjugation invariant pairs of points. Read More

We show that the presence of a non-contractible one-periodic trajectory in a Hamiltonian dynamics on a connected closed symplectic manifold $(M,\omega)$ implies the existence of infinitely many non-contractible simple periodic trajectories, provided that the symplectic form $\omega$ is aspherical and the fundamental group $\pi_1(M)$ is either a virtually abelian group or an $\mathrm{R}$-group. We also show that a similar statement holds for Hamiltonian dynamics on closed monotone or negative monotone symplectic manifolds under the same conditions on their fundamental groups. These results generalize some works by Ginzburg and G\"urel. Read More

The first author introduced a relative symplectic capacity $C$ for a symplectic manifold $(N,\omega_N)$ and its subset $X$ which measures the existence of non-contractible periodic trajectories of Hamiltonian isotopies on the product of $N$ with the annulus $A_R=(R,R)\times\mathbb{R}/\mathbb{Z}$. In the present paper, we give an exact computation of the capacity $C$ of the $2n$-torus $\mathbb{T}^{2n}$ relative to a Lagrangian submanifold $\mathbb{T}^n$ which implies the existence of non-contractible Hamiltonian periodic trajectories on $A_R\times\mathbb{T}^{2n}$. Moreover, we give a lower bound on the number of such trajectories. Read More

We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along with a few other geometric situations. We provide sample applications to the $C^0$-geometry of Morse functions and to Hofer's geometry of Hamiltonian diffeomorphisms, that go beyond spectral invariants and traditional persistent homology. Read More

Let $(\Sigma,\tau)$ denote a Riemann surface of genus $g \geq 2$ equipped with an anti-holomorphic involution $\tau$. In this paper we study the topology of the moduli space $M(r,\xi)^\tau$ of stable Real vector bundles over $(\Sigma,\tau)$ of rank $r$ and fixed determinant $\xi$ of degree coprime to $r$. We prove that $M(r,\xi)^{\tau}$ is an orientable and monotone Lagrangian submanifold of the complex moduli space $M(r,\xi)$ so it determines an object in the appropriate Fukaya category. Read More

We provide a short review of Courant algebroids and graded symplectic manifolds and show how different Courant algebroids emerge from double field theory. Fluxes of double field theory and their Bianchi identities are formulated by using pre-QP-manifolds. We stress the relation of the Poisson Courant algebroid with R-flux as a solution of the double field theory section condition. Read More

This is a mixture of survey article and research anouncement. We discuss Instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds. Read More

We prove that the open Gromov-Witten invariants on K3 surfaces satisfy the Kontsevich-Soibelman wall-crossing formula. One application is that the open Gromov-Witten invariants coincide with the weighted counting of tropical discs. This generalizes the correspondence theorem of Nishinou-Siebert on toric varieties to K3 surfaces. Read More

In this paper, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclid spaces. We also construct Lagrangian mean curvature flows in non-flat Calabi-Yau manifolds. Read More

The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra which depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty$ algebra instead. We develop a simplified method for describing this $L_\infty$ algebra and use it to prove that the $L_\infty$ algebras corresponding to different transversals are canonically $L_\infty$-isomorphic. In some cases, this isomorphism provides a formality map, as we show in several examples including (quasi)-Poisson geometry, Dirac structures on Lie groups, and Lie bialgebras. Read More

We introduce a new critical value $c_\infty(L)$ for Tonelli Lagrangians $L$ on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that $c_\infty(L)$ is strictly larger than the Ma\~n\'e critical value $c(L)$, and on every energy level $e\in(c(L),c_\infty(L))$ there exist infinitely many periodic orbits of the Lagrangian system of $L$, one of which is a local minimizer of the free-period action functional. This has applications to Finsler metrics of Randers type on the 2-sphere. Read More

In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the invariant is a determinant. Read More

The first author in recent work with D. Gay developed the notion of a Morse structure on an open book as a tool for studying closed contact 3-manifolds. We extend the notion of Morse structure to extendable partial open books in order to study contact 3-manifolds with convex boundary. Read More

We study Legendrian embeddings of a compact Legendrian submanifold $L$ sitting in a closed contact manifold $(M,\xi)$ whose contact structure is supported by a (contact) open book $\mathcal{OB}$ on $M$. We prove that if $\mathcal{OB}$ has Weinstein pages, then there exist a contact structure $\xi'$ on $M$, isotopic to $\xi$ and supported by $\mathcal{OB}$, and a contactomorphism $f:(M,\xi) \to (M,\xi')$ such that the image $f(L)$ of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of $\mathcal{OB}$. As a consequence, we conclude that for every compact embedded Legendrian submanifold of a closed contact manifold, there exists a well-defined Legendrian contact homology which is invariant under Legendrian isotopies. Read More

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Nemethi. Read More

In this paper, we give a correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the torus $ \mathbb{T}^2$. To achieve this, we use the correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the complex plane and a relation between the Berezin-Toeplitz quantization of a periodic symbol on the real phase space $\mathbb{R}^2$ and the Berezin-Toeplitz quantization of a symbol on the torus $ \mathbb{T}^2 $. Read More

We show that the main theorem of Morse theory holds for a large class of functions on singular spaces. The function must satisfy certain conditions extending the usual requirements on a manifold that Condition C holds and the gradient flow around the critical sets is well-behaved, and the singular space must satisfy a local deformation retract condition. We then show that these conditions are satisfied when the function is the norm-square of a moment map on an affine variety. Read More

We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive perturbation process indexed by the strata of the isotropy set. A global existence theorem for symmetric Morse-Smale pairs is also proved. Read More

We show that two Hamiltonian isotopic Lagrangians in (CP^2,\omega_\textup{FS}) induce two Lagrangian submanifolds in the one-point blow-up (\widetilde{CP}^2,\widetilde{\omega}_\rho) that are not Hamiltonian isotopic. Furthermore, we show that for any integer k>1 there are k Hamiltonian isotopic Lagrangians in (CP^2,\omega_\textup{FS}) that induce k Lagrangian submanifolds in the one-point blow-up such that no two of them are Hamiltonian isotopic. Read More

The subject of this dissertation is the Gysin homomorphism in equivariant cohomology for spaces with torus action. We consider spaces which are quotients of classical semisimple complex linear algebraic groups by a parabolic subgroup with the natural action of a maximal torus, the so-called partial flag varieties. We derive formulas for the Gysin homomorphism for the projection to a point, of the form \[\int_X \alpha = Res_{\mathbf{z}=\infty} \mathcal{Z}(\mathbf{z}, \mathbf{t}) \cdot \alpha(\mathbf{t}),\] for a certain residue operation and a map $\mathcal{Z}(\mathbf{z}, \mathbf{t})$, associated to the root system. Read More

Floer theory was originally devised to estimate the number of 1-periodic orbits of Hamiltonian systems. In earlier works, we constructed Floer homology for homoclinic orbits on two dimensional manifolds using combinatorial techniques. In the present paper, we study theoretic aspects of computational complexity of homoclinic Floer homology. Read More

This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by K\'alm\'an, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extends into knot theory, 3-manifold topology and Floer homology. Read More

In determining when a four-dimensional ellipsoid can be symplectically embedded into a ball, McDuff and Schlenk found an infinite sequence of "ghost" obstructions that generate an infinite "ghost staircase" determined by the even index Fibonacci numbers. The ghost obstructions are not visible for the four-dimensional embedding problem because strictly stronger obstructions also exist. We show that in contrast, the embedding constraints associated to the ghost obstructions are sharp for the stabilized problem; moreover, the corresponding optimal embeddings are given by symplectic folding. Read More

We will consider the 3-point blow-up of the manifold $ (S^2 \times S^2, \sigma \oplus \sigma)$ where $\sigma$ is the standard symplectic form which gives area 1 to the sphere $S^2$, and study its group of symplectomorphisms $\rm{Symp} ( S^2 \times S^2 \#\, 3\overline{\mathbb C\mathbb P}\,\!^2, \omega)$. So far, the monotone case was studied by J. Evans and he proved that this group is contractible. Read More