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Lagrangian pairs of pantsFeb 08 2018We construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of the differential of a smooth function defined on the real blow up of a Lagrangian coamoeba. Lagrangian ... More

The geometry of E-manifoldsFeb 08 2018Motivated by the study of symplectic Lie algebroids, we study a describe a type of algebroid (called an $E$-tangent bundle) which is particularly well-suited to study of singular differential forms and their cohomology. This setting generalizes the study ... More

Remarks on the self-shrinking Clifford torusFeb 05 2018On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian ... More

New example of modified moduli space of special Bohr - Sommerfeld lagrangian submanifoldsFeb 02 2018We present an example of modified moduli space of special Bohr - Sommerfeld lagrangian submanifolds for the case when the given algebraic variety is the full flag $F^3$ for $\mathbb{C}^3$ and the very ample bundle is $K^{- \frac{1}{2}}_{F^3}$

Invariant Whitney FunctionsFeb 01 2018A theorem of Gerald Schwarz [24, Thm. 1] says that for a linear action of a compact Lie group $G$ on a finite dimensional real vector space $V$ any smooth $G$-invariant function on $V$ can be written as a composite with the Hilbert map. We prove a similar ... More

Deformation Cohomology of Lie Algebroids and Morita EquivalenceJan 30 2018Let $A \Rightarrow M$ be a Lie algebroid. In this short note we prove that a pull-back of $A$ along a fibration with homologically $k$-connected fibers, shares the same deformation cohomology of $A$ up to degree $k$.

Kähler fibrations in quantum information theoryJan 29 2018We discuss the fibre bundle of co-adjoint orbits of compact Lie groups, and show how it admits a compatible K\"ahler structure. The case of the unitary group allows us to reformulate the geometric framework of quantum information theory. In particular, ... More

On Bott-Morse Foliations and their Poisson Structures in Dimension 3Jan 29 2018We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic ... More

The Kähler geometry of Bott manifoldsJan 29 2018We study the K\"ahler geometry of stage n Bott manifolds, which can be viewed as $n$-dimensional generalizations of Hirzebruch surfaces. We show, using a simple induction argument and the generalized Calabi construction from [ACGT04,ACGT11], that any ... More

Khovanov homology detects the trefoilsJan 23 2018We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact triangle in sutured ... More

Infinite staircases in the symplectic embedding problem for four-dimensional ellipsoids into polydisksJan 21 2018We study the symplectic embedding capacity function $C_{\beta}$ for ellipsoids $E(1,\alpha)\subset R^4$ into dilates of polydisks $P(1,\beta)$ as both $\alpha$ and $\beta$ vary through $[1,\infty)$. For $\beta=1$ Frenkel and Mueller showed that $C_{\beta}$ ... More

A note on the knot Floer homology of fibered knotsJan 19 2018We prove that the knot Floer homology of a fibered knot is nontrivial in its next-to-top Alexander grading. Immediate applications include new proofs of Krcatovich's result that knots with $L$-space surgeries are prime and Hedden and Watson's result that ... More

The Gromov width of generalized Bott manifoldsJan 19 2018By Delzant's theorem, closed symplectic toric manifolds are classified by the images of moment maps. In the case of a generalized Bott manifold, this image is a polytope $P$ combinatorially equivalent to the product of simplices. We compute the Gromov ... More

Computing rotation numbers in open booksJan 03 2018We give explicit formulas and algorithms for the computation of the rotation number of a nullhomologous Legendrian knot on a page of a contact open book. On the way, we derive new formulas for the computation of the Thurston-Bennequin invariant of such ... More

Some properties of the Bourgeois contact structuresJan 03 2018The Bourgeois construction associates to every contact open book on a manifold $V$ a contact structure on $V\times T^2$. We study in this article some of the properties of $V$ that are inherited by $V\times T^2$ and some that are not. Giroux has provided ... More

Variational order for forced Lagrangian systemsDec 26 2017We are able to derive the equations of motion for forced mechanical systems in a purely variational setting, both in the context of Lagrangian or Hamiltonian mechanics, by duplicating the variables of the system as introduced by Galley [2013], Galley, ... More

Contact integral geometry and the Heisenberg algebraDec 26 2017Jan 03 2018Generalizing Weyl's tube formula and building on Chern's work, Alesker reinterpreted the Lipschitz-Killing curvature integrals as a family of valuations (finitely-additive measures with good analytic properties), attached canonically to any Riemannian ... More

$\mathcal{C}^0$-rigidity of Lagrangian submanifolds and punctured holomorphic discs in the cotangent bundleDec 18 2017Our main result is the $\mathcal{C}^0$-rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic discs in cotangent bundles with boundary on the zero section, whose boundaries ... More

On some examples and constructions of contact manifoldsNov 27 2017The first goal of this paper is to construct examples of higher dimensional contact manifolds with specific properties. Our main results in this direction are the existence of tight virtually overtwisted closed contact manifolds in all dimensions and ... More

The Local Structure of Generalized Contact BundlesNov 22 2017Generalized contact bundles are odd dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting ... More

The local structure of generalized complex branesNov 14 2017Nov 15 2017We show (modulo a parity condition) that, a generalized complex brane in a generalized complex manifold is locally equivalent to a holomorphic coisotropic submanifold of a holomorphic Poisson structure, with higher-rank branes corresponding to holomorphic ... More

Almost conformally almost Fedosov structuresOct 16 2017We study the relations between the projective and the almost conformally symplectic structures on a smooth even dimensional manifold. We describe these relations by a single almost conformally symplectic connection with totally trace--free torsion sharing ... More

Holomorphic Jacobi Manifolds and Complex Contact GroupoidsOct 09 2017This is the second part of a series of two papers dedicated to a systematic study of holomorphic Jacobi structures. In the first part, we introduced and study the concept of a holomorphic Jacobi manifold in a very natural way as well as various tools. ... More

Contact orderability up to conjugationSep 27 2017We study in this paper the remnants of the contact partial order on the orbits of the adjoint action of contactomorphism groups on their Lie algebras. Our main interest is a class of non-compact contact manifolds, called convex at infinity.

Toric contact geometry in arbitrary codimensionAug 16 2017Nov 23 2017We define toric contact manifolds in arbitrary codimension and give a description of such manifolds in terms of a kind of labelled polytope embedded into a grassmannian, analogous to the Delzant polytope of a toric symplectic manifold.

Weinstein manifolds revisitedJul 11 2017Aug 29 2017This is a very biased and incomplete survey of some basic notions, old and new results, as well as open problems concerning Weinstein symplectic manifolds.

Singularities and Semistable Degenerations for Symplectic TopologyJul 05 2017We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case ... More

Special homogeneous almost complex structures on symplectic manifoldsJun 20 2017Oct 19 2017Homogeneous compatible almost complex structures on symplectic manifolds are studied, focusing on those which are special, meaning that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be covered by ... More

Equivariant maps into Anti-de Sitter space and the symplectic geometry of $\mathbb H^2\times \mathbb H^2$May 31 2017Sep 28 2017Given two Fuchsian representations $\rho_l$ and $\rho_r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, we study the relation between Lagrangian submanifolds of $M_\rho=(\mathbb{H}^2/\rho_l(\pi_1(S)))\times (\mathbb{H}^2/\rho_r(\pi_1(S)))$ ... More

Deformations of coisotropic submanifolds in Jacobi manifoldsMay 24 2017In this thesis, we study the deformation problem of coisotropic submanifolds in Jacobi manifolds. In particular we attach two algebraic invariants to any coisotropic submanifold $S$ in a Jacobi manifold, namely the $L_\infty[1]$-algebra and the BFV-complex ... More

A classification of star log symplectic structures on a compact oriented surfaceMay 04 2017May 22 2017Given a compact oriented surface, we classify log Poisson bi-vectors whose degeneracy loci are locally modeled by a finite set of lines in the plane intersecting at a point. Further, we compute the Poisson cohomology of such structures and discuss the ... More

Differential algebra of cubic planar graphsMay 02 2017May 03 2017In this article we associate a combinatorial differential graded algebra to a cubic planar graph G. This algebra is defined combinatorially by counting binary sequences, which we introduce, and several explicit computations are provided. In addition, ... More

A Remark on the Localization formulas about two Killing vector fieldsApr 28 2017In this article, we will discuss a localization formulas of equivariant cohomology about two Killing vector fields on the set of zero points ${\rm{Zero}}(X_{M}-\sqrt{-1}Y_{M})=\{x\in M \mid |Y_{M}(x)|=|X_{M}(x)|=0 \}.$ As application, we use it to get ... More

Symplectic stability on manifolds with cylindrical endsApr 27 2017Jan 26 2018A famous result of Jurgen Moser states that a symplectic form on a compact manifold cannot be deformed within its cohomology class to an inequivalent symplectic form. It is well known that this does not hold in general for noncompact symplectic manifolds. ... More

Examples of non-trivial contact mapping classes for overtwisted contact manifolds in all dimensionsApr 24 2017May 19 2017The aim of the article is to construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity. Hence, overtwisted contact structures ... More

Integrable systems, symmetries and quantizationApr 21 2017Oct 16 2017These notes correspond to a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularity and expose recent research in connection ... More

Givental's non-linear Maslov index on lens spacesApr 19 2017Aug 04 2017Givental's non-linear Maslov index, constructed in 1990, is a quasimorphism on the universal cover of the identity component of the contactomorphism group of real projective space. This invariant was used by several authors to prove contact rigidity phenomena ... More

$J$-holomorphic disks with pre-Lagrangian boundary conditionsApr 09 2017Apr 28 2017The purpose of this paper is to carry out a classical construction of a non-constant holomorphic disk with boundary on (the suspension of) a Lagrangian submanifold in $\mathbb{R}^{2 n}$ in the case the Lagrangian is the lift of a coisotropic (a.k.a. pre-Lagrangian) ... More

Structure of locally conformally symplectic Lie algebras and solvmanifoldsApr 04 2017We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients of all four-dimensional ... More

On the stabilized symplectic embedding problem for ellipsoidsMar 22 2017Jun 29 2017This 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.

The fundamental group of contact toric manifoldsMar 21 2017Jan 16 2018Let $M$ be a connected compact contact toric manifold. We explicitly compute $\pi_1(M)$. In particular, for those manifolds of Reeb type, we show that $\pi_1(M)$ is finite cyclic, and we describe how to read the order of $\pi_1(M)$ from the moment map ... More

Cosmetic contact surgeries along transverse knots and the knot complement problemMar 16 2017We 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. The main result is that we can exclude non-trivial cosmetic contact surgeries (in sufficiently ... More

Symplectomorphisms of exotic discsMar 15 2017We 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 ... More

Fibrations and stable generalized complex structuresMar 10 2017A 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 ... More

Deformations of pre-symplectic structures and the Koszul $L_\infty$-algebraMar 01 2017Jul 20 2017We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_\infty$-algebra, which we call the Koszul ... More

Morse structures on partial open books with extendable monodromyFeb 24 2017The 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 ... More

Legendrian Contact Homology in Closed Contact ManifoldsFeb 23 2017We 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, ... More

The ghost stairs stabilize to sharp symplectic embedding obstructionsFeb 13 2017In 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. ... More

Torsion contact forms in three dimensions have two or infinitely many Reeb orbitsJan 09 2017We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it follows that under ... More

A new approximation method for geodesics on the space of Kähler metrics using complexified symplectomorphisms and Gröbner Lie seriesJan 06 2017It has been shown that the Cauchy problem for geodesics in the space of K\"ahler metrics with a fixed cohomology class on a compact complex manifold $M$ can be effectively reduced to the problem of finding the flow of a related hamiltonian vector field ... More

Parabolic conformally symplectic structures III; Invariant differential operators and complexesJan 05 2017Jan 12 2018This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, ... More

Quaternionic toric manifoldsDec 12 2016In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate $m$-dimensional Delzant polytopes, we obtain manifolds of real dimension $4m$, acted ... More

Periodic Reeb orbits on prequantization bundlesDec 07 2016In this paper, we prove that every graphical hypersurface in a prequantization bundle over a symplectic manifold $M$, pinched between two circle bundles whose ratio of radii is less than $\sqrt{2}$ carries either one short simple periodic orbit or carries ... More

A flat Higgs bundle structure on the complexified Kähler coneDec 07 2016We shall construct a natural Higgs bundle structure on the complexified K\"ahler cone of a compact K\"ahler manifold, which can be seen as an analogy of the classical Higgs bundle structure associated to a variation of Hodge structure. In the proof of ... More

Fluxes, bundle gerbes and 2-Hilbert spacesDec 06 2016We elaborate on the construction of a prequantum 2-Hilbert space from a bundle gerbe over a 2-plectic manifold, providing the first steps in a program of higher geometric quantisation of closed strings in flux compactifications and of M5-branes in C-fields. ... More

Homologie Instanton Symplectique : somme connexe, chirurgie entière, et applications induites par cobordismesDec 05 2016Symplectic instanton homology is an invariant for closed oriented three-manifolds, defined by Manolescu and Woodward, which conjecturally corresponds to a symplectic version of a variant of Floer's instanton homology. In this thesis we study the behaviour ... More

Fragmented Hofer's geometry on Hameomorphism groups and its applicationDec 04 2016Stefan M$\ddot{\mathrm{u}}$ller posed the problem "Do Hofer's metrics on the group of Hamiltonian diffeomorphism and the one of Hamiltonian homeomorphisms (Hameomorphisms) correspond?". Let $(M,\omega)$ be a compact exact symplectic manifold. We prove ... More

Fragmented Hofer's geometryDec 04 2016Hofer's norm (metric) is an important and interesting topic in symplectic geometry. In the present paper, we define fragmented Hofer's norms which are Hofer's norms controlled by fragmentation norms and give some observations on fragmented Hofer's norms. ... More

Rigid constellations of closed Reeb orbitsDec 04 2016We use Hamiltonian Floer theory to recover and generalize a classic rigidity theorem of Ekelend and Lasry. That theorem can be rephrased as an assertion about the existence of multiple closed Reeb orbits for certain tight contact forms on the sphere that ... More

Lectures on Symplectic Field TheoryDec 03 2016Dec 08 2016This is the preliminary manuscript of a book on symplectic field theory based on a lecture course for PhD students given in 2015-16. It covers the essentials of the analytical theory of punctured pseudoholomorphic curves, taking the opportunity to fill ... More

Lectures on Symplectic Field TheoryDec 03 2016This is the preliminary manuscript of a book on symplectic field theory based on a lecture course for PhD students given in 2015-16. It covers the essentials of the analytical theory of punctured pseudoholomorphic curves, taking the opportunity to fill ... More

On the equatorial Dehn twist of a Lagrangian nodal sphereDec 01 2016Let $(M^4,\omega)$ be a geometrically bounded symplectic manifold, and $L\subset M$ a Lagrangian nodal sphere such that $\omega\mid_{\pi_2(M,L)}=0$. We show that an equatorial Dehn twist of $L$ does not extend to a Hamiltonian diffeomorphism of $M$. We ... More

On the moduli space of flat symplectic surface bundlesNov 30 2016In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. We construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology ... More

Toric Geometry of the Regular Convex PolyhedraNov 30 2016In this article, we describe symplectic and complex toric spaces associated to the five regular convex polyhedra. The regular tetrahedron and the cube are rational and simple, the regular octahedron is not simple, the regular dodecahedron is not rational ... More

Geometric and viscosity solutions for the Cauchy Problem of first OrderNov 30 2016There are two kinds of solutions of the Cauchy problem of first order, the viscosity solution and the more geometric minimax solution and in general they are different. The aim of this article is to show how they are related: iterating the minimax procedure ... More

Finsler geodesics, periodic Reeb orbits, and open booksNov 30 2016We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that ... More

A Functorial Symplectic Instanton Homology via Traceless Character VarietiesNov 29 2016Using ideas pioneered by Wehrheim and Woodward, we associate to any closed, oriented $3$-manifold $Y$ a finitely generated abelian group $\mathrm{SI}(Y)$ obtained from (quilted) Lagrangian Floer homology in a certain moduli space of $\mathrm{SU}(2)$-representations ... More

Braces and Poisson additivityNov 29 2016We relate the brace construction introduced by Calaque and Willwacher to an additivity functor. That is, we construct a functor from brace algebras associated to an operad $O$ to associative algebras in the category of homotopy $O$-algebras. As an example, ... More

Symmetric products of a real curve and the moduli space of Higgs bundlesNov 29 2016Consider a Riemann surface $X$ of genus $g \geq 2$ equipped with an antiholomorphic involution $\tau$. This induces a natural involution on the moduli space $M(r,d)$ of semistable Higgs bundles of rank $r$ and degree $d$. If $D$ is a divisor such that ... More

Symmetric products of a real curve and the moduli space of Higgs bundlesNov 29 2016Nov 30 2016Consider a Riemann surface $X$ of genus $g \geq 2$ equipped with an antiholomorphic involution $\tau$. This induces a natural involution on the moduli space $M(r,d)$ of semistable Higgs bundles of rank $r$ and degree $d$. If $D$ is a divisor such that ... More

More on the admissible condition on differentiable maps $\varphi: (X^{\!A\!z},E;\nabla)\rightarrow Y$ in the construction of the non-Abelian Dirac-Born-Infeld action $S_{DBI}(\varphi,\nabla)$Nov 29 2016In D(13.1) (arXiv:1606.08529 [hep-th]), we introduced an admissible condition on differentiable maps $\varphi: (X^{\!A\!z}, E;\nabla)\rightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ (with the fundamental module $E$) with a connection $\nabla$ ... More

Graded manifolds of type $Δ$ and $n$-fold vector bundlesNov 28 2016Vector bundles and double vector bundles or $2$-fold vector bundles arise naturally for instance as base spaces for such algebraic structures as Lie algebroids, Courant algebroids and double Lie algebroids. All these structures possess a unified description ... More

Canonical symplectic structure and structure-preserving geometric algorithms for Schrödinger-Maxwell systemsNov 28 2016An infinite dimensional canonical symplectic structure and structure-preserving geometric algorithms are developed for the photon-matter interactions described by the Schr\"odinger-Maxwell equations. The algorithms preserve the symplectic structure of ... More

Poisson cohomology of holomorphic toric Poisson manifoldsNov 25 2016A holomorphic toric Poisson manifold is a nonsingular toric variety equipped with a holomorphic Poisson structure, which is invariant under the torus action. In this paper, we compute the Poisson cohomology for holomorphic toric Poisson structures on ... More

Perturbed gradient flow trees and $A_\infty$-algebra structures on Morse cochain complexesNov 23 2016We elaborate on an idea of M. Abouzaid of equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an $A_\infty$-algebra. This is a variation on K. Fukaya's definition of Morse-$A_\infty$-categories ... More

Symplectic $-2$ spheres and the symplectomorphism group of small rational 4-manifoldsNov 22 2016Let $(X,\omega)$ be a symplectic rational 4 manifold. We study the space of tamed almost complex structures $\mathcal{J}_{\omega}$ using a fine decomposition via smooth rational curves and a relative version of the infinite dimensional Alexander duality. ... More

The Horn inequalities from a geometric point of viewNov 21 2016We give an exposition of the Horn inequalities and their triple role characterizing tensor product invariants, eigenvalues of sums of Hermitian matrices, and intersections of Schubert varieties. We follow Belkale's geometric method, but assume only basic ... More

Infinitesimal Automorphisms of VB-groupoids and algebroidsNov 21 2016VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. In ... More

A Log PSS morphism with applications to Lagrangian embeddingsNov 21 2016Let $M$ be a smooth projective variety and $\mathbf{D}$ an ample normal crossings divisor. From topological data associated to the pair $(M, \mathbf{D})$, we construct, under assumptions on Gromov-Witten invariants, a series of distinguished classes in ... More

Operations on Legendrian submanifoldsNov 21 2016This paper deals with Legendrian submanifolds in the space of one jets of functions $J^1(R^n;R)$. We are interested in processes - operations - that build new Legendrians submanifolds from old ones. We introduce in particular two operations, namely the ... More

Relative Morse Categorification TheoryNov 20 2016In this paper, we define a relative Morse complex for manifold with boundary using the handlebody decomposition of the manifold. We prove that the homology of the relative Morse complex is isomorphic to the relative singular homology. Furthermore, we ... More

Exponential Networks and Representations of QuiversNov 18 2016We study the geometric description of BPS states in supersymmetric theories with eight supercharges in terms of geodesic networks on suitable spectral curves. We lift and extend several constructions of Gaiotto-Moore-Neitzke from gauge theory to local ... More

Stein fillings and SU(2) representationsNov 17 2016We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are induced by Stein structures on a single 4-manifold ... More

Relational Symplectic Groupoid Quantization for Constant Poisson StructuresNov 17 2016This note describes a quantization of the relational symplectic groupoid (RSG) for a constant Poisson structure using the BV-BFV formalism and shows how this induces the Moyal deformation quantization for the underlying Poisson manifold.

Central charges of T-dual branes for toric varietiesNov 16 2016Given any equivariant coherent sheaf $\mathcal L$ on a compact semi-positive toric orbifold $\mathcal X$, its SYZ T-dual mirror dual is a Lagrangian brane in the Landau-Ginzburg mirror. We prove the oscillatory integral of the equivariant superpotential ... More

Quasimap Wall-crossings and Mirror SymmetryNov 15 2016We state a wall-crossing formula for the virtual classes of epsilon-stable quasimaps to GIT quotients and prove it for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing ... More

Quasimap Wall-crossings and Mirror SymmetryNov 15 2016Nov 22 2016We state a wall-crossing formula for the virtual classes of epsilon-stable quasimaps to GIT quotients and prove it for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing ... More

Groupoid equivariant prequantizationNov 15 2016In their 2005 paper, C. Laurent-Gengoux and P. Xu define prequantization for pre-Hamiltonian actions of quasi-presymplectic Lie groupoids in terms of central extensions of Lie groupoids. The definition requires that the quasi-presymplectic structure be ... More

Geometry and Dynamics of Gaussian Wave Packets and their Wigner TransformsNov 13 2016We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian and symplectic-geometric point of view. The main result states that the momentum map corresponding ... More

Integration of generalized complex structuresNov 11 2016We solve the integration problem for generalized complex mani- folds, obtaining as the natural integrating object a weakly holomorphic sym- plectic groupoid, which is a real symplectic groupoid with a compatible complex structure defined only on the associated ... More

Integration of generalized complex structuresNov 11 2016Nov 15 2016We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a weakly holomorphic symplectic groupoid, which is a real symplectic groupoid with a compatible complex structure defined only on the associated ... More

From Dynamics to Contact and Symplectic Topology and BackNov 08 2016This is a light survey article about the origins of contact and symplectic topology in dynamics and the more recent developments in the field. In lieu of formulas, numerous anecdotes are given.

Symplectic neighborhood of crossing divisorsNov 08 2016This paper presents a proof of the existence of standard symplectic coordinates near a set of smooth, orthogonally intersecting symplectic submanifolds. It is a generalization of the standard symplectic neighborhood theorem. Moreover, in the presence ... More

On higher Dirac structuresNov 07 2016Nov 09 2016We study higher-order analogues of Dirac structures, extending the multisymplectic structures that arise in field theory. We define higher Dirac structures as involutive subbundles of $TM+\wedge^k TM^*$ satisfying a weak version of the usual lagrangian ... More

The Topology and Geometry of Hyperkähler QuotientsNov 07 2016In this thesis we study the topology and geometry of hyperk\"ahler quotients, as well as some related non-compact K\"ahler quotients, from the point of view of Hamiltonian group actions. The main technical tool we employ is Morse theory with moment maps. ... More

On the mean Euler characteristic of Gorenstein toric contact manifoldsNov 02 2016We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, i.e. a good toric contact manifold with zero first Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A ... More

Hessenberg varieties and hyperplane arrangementsNov 01 2016Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $\mathcal{A}_I$, the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$, and the regular semisimple Hessenberg ... More

Hessenberg varieties and hyperplane arrangementsNov 01 2016Dec 05 2016Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $\mathcal{A}_I$, the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$, and the regular semisimple Hessenberg ... More

$G$-Invariant deformations of almost-coupling Poisson structuresOct 31 2016On a foliated manifold equipped with an action of a compact Lie group $G$, we study a class of almost-coupling Poisson and Dirac structures, in the context of the deformation theory and the method of averaging.

Gluing affine vorticesOct 31 2016We provide an analytical construction of the gluing map for stable affine vortices over the upper half plane with the Lagrangian boundary condition.