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Hamiltonian Floer theory for nonlinear Schrödinger equations and the small divisor problemApr 18 2019We prove the existence of infinitely many time-periodic solutions of nonlinear Schr\"odinger equations using pseudo-holomorphic curve methods from Hamiltonian Floer theory. For the generalization of the Gromov-Floer compactness theorem to infinite dimensions, ... More

Special Lagrangian submanifolds of log Calabi-Yau manifoldsApr 17 2019We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat K\"ahler metric constructed by Tian-Yau. We prove that if $X$ is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at ... More

A computation of the ring structure in wrapped Floer homologyApr 15 2019We give an explicit computation of the ring structure in wrapped Floer homology of real Lagrangians in $A_k$-type Milnor fibers. In the $A_k$-type plumbing description, those Lagrangians correspond to cotangent fibers or diagonal Lagrangians. The main ... More

String topology and a conjecture of ViterboApr 15 2019We identify a class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk bundle. This class of manifolds is characterized in topological ... More

Stochastic differential equations for Lie group valued moment mapsApr 14 2019The celebrated result by Biane-Bougerol-O'Connell relates Duistermaat-Heckman (DH) measures for coadjoint orbits of a compact Lie group $G$ with the multi-dimensional Pitman transform of the Wiener process on its Cartan subalgebra. The DH theory admits ... More

Tropical Lagrangians and Homological Mirror SymmetryApr 12 2019We produce for each tropical hypersurface $V(\phi)\subset Q=\mathbb{R}^n$ a Lagrangian $L(\phi)\subset (\mathbb{C}^*)^n$ whose moment map projection is a tropical amoeba of $V(\phi)$. When these Lagrangians are admissible in the Fukaya-Seidel category, ... More

Generating functions of planar polygons from homological mirror symmetry of elliptic curvesApr 10 2019We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers' mock theta function. ... More

On symplectic fillings of small Seifert $3$-manifoldsApr 10 2019In this paper, we study a surgical description for the symplectic fillings of small Seifert $3$-manifolds with a canonical contact structure. As a result, we demonstrate that every minimal symplectic filling of small Seifert $3$-manifolds satisfying certain ... More

On auto-equivalences and complete derived invariants of gentle algebrasApr 09 2019We study triangulated categories which can be modeled by an oriented marked surface $\mathcal{S}$ and a line field $\eta$ on $\mathcal{S}$. This includes bounded derived categories of gentle algebras and -- conjecturally -- all partially wrapped Fukaya ... More

Lagrangian Floer homology on symplectic blow upsApr 08 2019We show how to compute the Lagrangian Floer homology in the one-point blow up of the proper transform of Lagrangians submanifolds, solely in terms of information of the base manifold. As an example we present an alternative computation of the Lagrangian ... More

WDVV-Type Relations for Disk Gromov-Witten Invariants in Dimension 6Apr 08 2019The first author's previous work established Solomon's WDVV-type relations for Welschinger's invariant curve counts in real symplectic fourfolds by lifting geometric relations over possibly unorientable morphisms. We apply her framework to obtain WDVV-style ... More

Adiabatic limits, Theta functions, and geometric quantizationApr 08 2019Let $\pi\colon (M,\omega)\to B$ be a (non-singular) Lagrangian torus fibration on a compact, complete base $B$ with prequantum line bundle $(L,\nabla^L)\to (M,\omega)$. For a positive integer $N$ and a compatible almost complex structure $J$ on $(M,\omega)$ ... More

Topological Persistence in Geometry and AnalysisApr 08 2019The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions with geometry ... More

Kostant-Toda lattices and the universal centralizerApr 05 2019To each complex semisimple Lie algebra $\mathfrak{g}$ decorated with appropriate data, one may associate two completely integrable systems. One is the well-studied Kostant-Toda lattice, while the second is an integrable system defined on the universal ... More

Meromorphic connections in filtered $A_{\infty}$ categoriesApr 04 2019In this note, introducing notions of CH module, CH morphism and CH connection, we define a meromorphic connection in the "$z$-direction" on periodic cyclic homology of an $A_\infty$ category as a connection on cohomology of a CH module. Moreover, we study ... More

A complete derived invariant for gentle algebras via winding numbers and Arf invariantsApr 04 2019Apr 11 2019Gentle algebras are in bijection with admissible dissections of marked oriented surfaces. In this paper, we further study the properties of admissible dissections and we show that silting objects for gentle algebras are given by admissible dissections ... More

A complete derived invariant for gentle algebras via winding numbers and Arf invariantsApr 04 2019Gentle algebras are in bijection with admissible dissections of marked oriented surfaces. In this paper, we further study the properties of admissible dissections and we show that silting objects for gentle algebras are given by admissible dissections ... More

The Steenrod problem for orbifolds and polyfold invariants as intersection numbersApr 03 2019The Steenrod problem for closed orientable manifolds was solved completely by Thom. Following this approach, we solve the Steenrod problem for closed orientable orbifolds, proving that the rational homology groups of a closed orientable orbifold have ... More

Monotonic lagrangian tori of standard and non standard types in toric and pseudotoric manifoldsApr 03 2019In recent papers, summarized in survey [1], we construct a number of examples of non standard lagrangian tori on compact toric varieties and as well on certain non toric varieties which admit pseudotoric structures. Using this pseudotoric technique we ... More

State dependent Hamiltonian delay equations and Neumann one-formsApr 03 2019In this note we study critical points of a variation of the action functional of classical mechanics, where the Hamiltonian term is retarded. Following a more than hundert and fifty year old paper by Carl Neumann we as well introduce Taylor approximations ... More

Symplectic manifolds and Hamiltonian dynamical systemsApr 02 2019This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete integrability of these ... More

Quantum $SL_2$, Infinite curvature and Pitman's 2M-X theoremApr 01 2019It is understood that Pitman's theorem in probability theory is intimately related to the representation theory of $\mathcal{U}_{q}(\mathfrak{sl}_2)$, in the so-called crystal regime $q \rightarrow 0$. This relationship has been explored by Biane and ... More

The $L_\infty$-structure on symplectic cohomologyMar 28 2019We construct the chain level $L_\infty$-structure that extends the Lie bracket on symplectic cohomology.

SFT computations and intersection theory in higher-dimensional contact manifoldsMar 28 2019We construct infinitely many non-diffeomorphic examples of $5$-dimensional contact manifolds which are tight, admit no strong fillings, and do not have Giroux torsion. We obtain obstruction results for symplectic cobordisms, for which we give a proof ... More

Algebraic and Giroux torsion in higher-dimensional contact manifoldsMar 28 2019We construct examples in any odd dimension of contact manifolds with finite and non-zero algebraic torsion (in the sense of Latschev-Wendl), which are therefore tight and do not admit strong symplectic fillings. We prove that Giroux torsion implies algebraic ... More

5-dimensional Bourgeois contact structures are tightMar 28 2019Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction for a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension ... More

GKM theory and Hamiltonian non-Kähler actions in dimension $6$Mar 27 2019Using the classification of $6$-dimensional manifolds by Wall, Jupp and \v{Z}ubr, we observe that the diffeomorphism type of simply-connected, compact $6$-dimensional integer GKM $T^2$-manifolds is encoded in their GKM graph. As an application, we show ... More

Homologically trivial symplectic cyclic actions need not extend to Hamiltonian circle actionsMar 27 2019We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods ... More

Cyclic actions on rational ruled symplectic four-manifoldsMar 27 2019Let $(M,\omega)$ be a ruled symplectic four-manifold. If $(M, \omega)$ is rational, then every homologically trivial symplectic cyclic action on $(M,\omega)$ is the restriction of a Hamiltonian circle action.

Mirror symmetry for perverse schobers from birational geometryMar 27 2019Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, ... More

A two-category of Hamiltonian manifolds, and a (1+1+1) field theoryMar 26 2019We define an extended field theory in dimensions $1+1+1$, that takes the form of a `quasi 2-functor' with values in a strict 2-category $\widehat{\mathcal{H}am}$, defined as the `completion of a partial 2-category' $\mathcal{H}am$, notions which we define. ... More

Almost complex manifolds are (almost) complexMar 24 2019We analyze the differential relation corresponding to integrability of almost complex structures, reformulated as a directed immersion relation by Demailly and Gaussier. We prove that the relation has formal solutions up to complex dimension 77, and that, ... More

Towards a higher-dimensional construction of stable/unstable Lagrangian laminationsMar 22 2019We generalize some properties of surface automorphisms of pseudo-Anosov type. First, we generalize the Penner construction of a pseudo-Anosov homeomorphism and show that a symplectic automorphism which is constructed by our generalized Penner construction ... More

Constructing symplectomorphisms between symplectic torus quotientsMar 22 2019We identify a family of torus representations such that the corresponding singular symplectic quotients at the $0$-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a ... More

Unfolding of the unramified irregular singular generalized isomonodromic deformationMar 20 2019We introduce an unfolded moduli space of connections, which is an algebraic relative moduli space of connections on complex smooth projective curves, whose generic fiber is a moduli space of regular singular connections and whose special fiber is a moduli ... More

Comparison of mirror functors of elliptic curves via LG/CY correspondenceMar 20 2019Polishchuk-Zaslow explained the homological mirror symmetry between Fukaya category of symplectic torus and the derived category of coherent sheaves of elliptic curves via Lagrangian torus fibration. Recently, Cho-Hong-Lau found another proof of homological ... More

Essential tori in spaces of symplectic embeddingsMar 20 2019Given two $2n$--dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, we show that a certain map from the $n$--torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map on singular ... More

Homological mirror symmetry for hypertoric varieties IIMar 19 2019In this paper, we prove a homological mirror symmetry equivalence for pairs of multiplicative hypertoric varieties. We prove our equivalence by matching holomorphic Lagrangian skeleta, on the A-model side, with non-commutative resolutions on the B-model ... More

On the complex structure of symplectic quotientsMar 18 2019Let $K$ be a compact group. For a symplectic quotient of a compact Hamiltonian K\"ahler $K$-manifold $M_{\lambda}$, we show that the induced complex structure on $M_{\lambda}$ is locally invariant with respect to the parameter $\lambda\in \mathrm{Lie}(K)$ ... More

Two closed orbits for non-degenerate Reeb flowsMar 15 2019We prove that every non-degenerate Reeb flow on a closed contact manifold $M$ admitting a strong symplectic filling $W$ with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic ... More

An Introduction to Nonassociative PhysicsMar 13 2019Mar 31 2019We give a pedagogical introduction to the nonassociative structures arising from recent developments in quantum mechanics with magnetic monopoles, in string theory and M-theory with non-geometric fluxes, and in M-theory with non-geometric Kaluza-Klein ... More

An Introduction to Nonassociative PhysicsMar 13 2019We give a pedagogical introduction to the nonassociative structures arising from recent developments in quantum mechanics with magnetic monopoles, in string theory and M-theory with non-geometric fluxes, and in M-theory with non-geometric Kaluza-Klein ... More

Toric Symplectic StacksMar 13 2019We give an intrinsic definition of toric symplectic stacks, and show that they are classified by simple convex polytopes equipped with some additional combinatorial data. This generalizes Delzant's classification of toric symplectic manifolds. As an application, ... More

Toric Symplectic StacksMar 13 2019Apr 03 2019We give an intrinsic definition of toric symplectic stacks, and show that they are classified by simple convex polytopes equipped with some additional combinatorial data. This generalizes Delzant's classification of toric symplectic manifolds. As an application, ... More

Partly-local domain-dependent almost complex structuresMar 13 2019We fill a gap pointed out by N. Sheridan in the proof of independence of genus zero Gromov-Witten invariants from the choice of divisor in the Cieliebak-Mohnke perturbation scheme.

Gamma II for toric varieties from integrals on T-dual branes and homological mirror symmetryMar 13 2019In this paper we consider the oscillatory integrals on Lefschetz thimbles in the Landau-Ginzburg model as the mirror of a toric Fano manifold. We show these thimbles represent the same relative homology classes as the characteristic cycles of the corresponding ... More

Contact Dual PairsMar 12 2019We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condition. ... More

Smoothly non-isotopic Lagrangian disk fillings of Legendrian knotsMar 12 2019In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.

Infinitesimally Tight Lagrangian OrbitsMar 09 2019We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag ... More

Spin Calogero-Moser models on symmetric spacesMar 08 2019In this paper we construct and prove superintegrability of spin Calogero-Moser type systems on symplectic leaves of $K_1\backslash T^*G/K_2$ where $K_1,K_2\subset G$ are subgroups. We call them two sided spin Calogero-Moser systems. One important type ... More

Unchaining surgery and topology of symplectic 4-manifoldsMar 07 2019Mar 12 2019We study a symplectic surgery operation we call unchaining, which effectively reduces the second Betti number and the symplectic Kodaira dimension at the same time. Using unchaining, we give novel constructions of symplectic Calabi-Yau surfaces from complex ... More

Unchaining surgery and topology of symplectic 4-manifoldsMar 07 2019We study a symplectic surgery operation we call, which effectively reduces the second Betti number and the symplectic Kodaira dimension at the same time. Using unchaining, we give novel constructions of symplectic Calabi-Yau surfaces from complex surfaces ... More

Quantization of Magnetic Poisson StructuresMar 07 2019We describe three perspectives on higher quantization, using the example of magnetic Poisson structures which embody recent discussions of nonassociativity in quantum mechanics with magnetic monopoles and string theory with non-geometric fluxes. We survey ... More

Non-covering Development Maps and Engel AutomorphismsMar 06 2019The development maps for Engel manifolds are very important tools to classify Engel manifolds. In case of that the development map is a covering, classification of these is algebraic. However, in case of that the development map is {\it not} a covering, ... More

Bulky Hamiltonian isotopy of Lagrangian tori with applicationsMar 05 2019We exhibit an example of a monotone Lagrangian torus inside the standard symplectic four dimensional unit ball which becomes Hamiltonian isotopic to a standard product torus only when considered inside a strictly larger ball (it is not even not symplectomorphic ... More

Invariance of immersed Floer cohomology under Lagrangian surgeryMar 05 2019We show that cellular Floer cohomology of an immersed Lagrangian brane is invariant under smoothing of a self-intersection point if the quantum valuation of the weakly bounding cochain vanishes and the Lagrangian has dimension at least two. The chain-level ... More

Intersection pairings in the N-fold reduced product of adjoint orbitsMar 05 2019In previous work we computed the symplectic volume of the symplectic reduced space of the product of N adjoint orbits of a compact Lie group. In this paper we compute the intersection pairings of the same object.

The Algebroid Structure of Double Field TheoryMar 05 2019By doubling the target space of a canonical Courant algebroid and subsequently projecting down to a specific subbundle, we identify the data of double field theory (DFT) and hence define its algebroid structure. We specify the properties of the DFT algebroid. ... More

The Algebroid Structure of Double Field TheoryMar 05 2019Mar 12 2019By doubling the target space of a canonical Courant algebroid and subsequently projecting down to a specific subbundle, we identify the data of double field theory (DFT) and hence define its algebroid structure. We specify the properties of the DFT algebroid. ... More

Bigraded cochain complexes and Poisson cohomologyMar 05 2019We present an algebraic framework for the computation of low-degree cohomology of a class of bigraded complexes which arise in Poisson geometry around (pre)symplectic leaves. We also show that this framework can be applied to the more general context ... More

Homological Berglund-Hübsch mirror symmetry for curve singularitiesMar 04 2019Given a two-variable invertible polynomial, we show that its category of maximally-graded matrix factorisations is quasi-equivalent to the Fukaya-Seidel category of its Berglund-H\"ubsch transpose. This was previously shown for Brieskorn-Pham and $D$-type ... More

Homological Berglund-Hübsch mirror symmetry for curve singularitiesMar 04 2019Apr 03 2019Given a two-variable invertible polynomial, we show that its category of maximally-graded matrix factorisations is quasi-equivalent to the Fukaya-Seidel category of its Berglund-H\"ubsch transpose. This was previously shown for Brieskorn-Pham and $D$-type ... More

Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applicationsMar 04 2019This is the first installment in a series of papers aimed at generalizing symplectic capacities and homologies. The main purposes of this paper are to construct analogues of Ekeland-Hofer and Hofer-Zehnder symplectic capacities based on a class of Hamiltonian ... More

Augmentations and link group representationsMar 03 2019We construct the augmentation representation. It is a representation of the fundamental group of the link complement associated to an augmentation of the framed cord algebra. This construction connects representations of two link invariants of different ... More

Representation formula for symmetric symplectic capacity and applicationsMar 02 2019This is the second installment in a series of papers aimed at generalizing symplectic capacities and homologies. We study symmetric versions of symplectic capacities for real symplectic manifolds, and obtain corresponding results for them to those of ... More

The Hall Algebras of AnnuliFeb 27 2019We refine and prove the central conjecture of our first paper for annuli with at least two marked intervals on each boundary component by computing the derived Hall algebras of their Fukaya categories.

Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varietiesFeb 25 2019The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra $\mathfrak g$, we obtain several results on completeness of homogeneous Poisson-commutative subalgebras of ${\mathcal S}(\mathfrak ... More

The second Betti number of doubly weighted homology groups of some pre Lie superalgebraFeb 25 2019We already showed that Betti numbers are all zero when w is not equal to h for (w,h)-doubly weighted homology groups of some special pre Lie superalgebra and showed the first Betti number is 0 when w = h. In this note, we show that the second Betti number ... More

Mahler's conjecture for some hyperplane sectionsFeb 24 2019We establish Mahler's conjecture for hyperplane sections of $\ell_p$-balls and of the Hanner polytopes.

Non-orientable Lagrangian surfaces in rational 4-manifoldsFeb 24 2019We show that for any nonzero class $A$ in $H_2(X; \mathbb{Z}_2)$ in a rational 4-manifold $X$, $A$ is represented by a nonorientable embedded Lagrangian surface L (for some symplectic structure) if and only if $P(A)\equiv (L) (mod\ 4)$; where $P(A)$ denotes ... More

Segal-Bargmann transforms from hyperbolic HamiltoniansFeb 23 2019We consider the imaginary time flow of a quadratic hyperbolic Hamiltonian on the symplectic plane, apply it to the Schr\"odinger polarization and study the corresponding evolution of polarized sections. The flow is periodic in imaginary time and the evolution ... More

A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphismFeb 18 2019Apr 06 2019We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n BO(n)$-action on $\varinjlim\limits_{N}T^*\mathbf{R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed in [GaRo] to give an enrichment ... More

A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphismFeb 18 2019We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n BO(n)$-action on $\varinjlim\limits_{N}T^*\mathbf{R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed in [GaRo] to give an enrichment ... More

A Hamiltonian $\coprod\limits_n BO(n)$-action, stratified Morse theory and the $J$-homomorphismFeb 18 2019Feb 19 2019We use sheaves of spectra to quantize a Hamiltonian $\coprod\limits_n BO(n)$-action on $\varinjlim\limits_{N}T^*\mathbf{R}^N$ that naturally arises from Bott periodicity. We employ the category of correspondences developed in [GaRo] to give an enrichment ... More

Morse-Bott Cohomology from Homological Perturbation TheoryFeb 18 2019In this paper, we construct cochain complexes generated by cohomology of critical manifolds for Morse-Bott theory under minimum transversality assumptions. We discuss the relations between different constructions of cochain complexes for Morse-Bott theory. ... More

Moduli spaces of framed $G$--Higgs bundles and symplectic geometryFeb 18 2019Let $X$ be a compact connected Riemann surface, $D\, \subset\, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x\, \subsetneq\, G_x$ a Zariski closed subgroup for every $x\, \in\, D$. A framed principal ... More

On the iterated Hamiltonian Floer homologyFeb 18 2019The focus of the paper is the behavior under iterations of the filtered and local Floer homology of a Hamiltonian on a symplectically aspherical manifold. The Floer homology of an iterated Hamiltonian comes with a natural cyclic group action. In the filtered ... More

On the Cohomology Ring of Symplectic FillingsFeb 18 2019Let $Y$ be the contact boundary of a $2n$-dimensional flexible Weinstein domain $W$ with vanishing first Chern class. We show that for any Liouville filling $W'$ of $Y$ with vanishing first Chern class, there is a linear isomorphism $\phi:H^*(W;\mathbb{R}) ... More

Physicists' $d=3+1$, $N=1$ superspace-time and supersymmetric QFTs from a tower construction in complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry and a purge-evaluation/index-contracting mapFeb 17 2019The complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry aspect of a superspace(-time) $\widehat{X}$ in Sec.\,1 of D(14.1) (arXiv:1808.05011 [math.DG]) together with the Spin-Statistics Theorem in Quantum Field Theory, which requires fermionic ... More

Basic Kirwan injectivity and its applicationsFeb 17 2019Feb 19 2019Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study ... More

Basic Kirwan infectivity and its applicationsFeb 17 2019Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study ... More

An application of spherical geometry to hyperkähler slicesFeb 14 2019This work is concerned with Bielawski's hyperk\"ahler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice to the data of a complex semisimple Lie group $G$, a reductive subgroup $H\subseteq G$, and a Slodowy ... More

On Lagrangian embeddings of closed non-orientable 3-manifoldsFeb 14 2019We prove that for any compact orientable connected 3-manifold with torus boundary, a concatenation of it and the direct product of the circle and the Klein bottle with an open 2-disk removed admits a Lagrangian embedding into the standard symplectic 6-space. ... More

A note on disk counting in toric orbifoldsFeb 14 2019Feb 20 2019We compute orbi-disk invariants of compact Gorenstein semi-Fano toric orbifolds by extending the method used for toric Calabi-Yau orbifolds. As a consequence the orbi-disc potential is analytic over complex numbers.

Arboreal singularities and loose Legendrians IFeb 13 2019Arboreal singularities are an important class of Lagrangian singularities. They are conical, meaning that they can be understood by studying their links, which are singular Legendrian spaces in $S^{2n-1}_{\text{std}}$. Loose Legendrians are a class of ... More

Categorical Saito theory, I: A comparison resultFeb 12 2019In this paper, we present an explicit cyclic minimal $A_\infty$ model for the category of matrix factorizations $\MF(W)$ of an isolated hypersurface singularity. The key observation is to use Kontsevich's deformation quantization technique. Pushing this ... More

The fine structure of Weber's hydrogen atom -- Bohr-Sommerfeld approachFeb 12 2019In this paper we determine in second order in the fine structure constant the energy levels of Weber's Hamiltonian admitting a quantized torus. Our formula coincides with the formula obtained by Wesley using the Schr\"odinger equation for Weber's Hamiltonian. ... More

Categorification of Legendrian knotsFeb 12 2019Feb 17 2019Perverse schober defined by Kapranov--Schechtman is a categorification of the notion of perverse sheaf. In their definition, a key ingredient is certain purity property of perverse sheaves. In this short note, we attempt to describe a real analogue of ... More

Categorification of Legendrian knotsFeb 12 2019Perverse schober defined by Kapranov--Schechtman is a categorification of the notion of perverse sheaf. In their definition, a key ingredient is certain purity property of perverse sheaves. In this short note, we attempt to describe a real analogue of ... More

Embedded contact knot homology and a surgery formulaFeb 11 2019Embedded contact knot homology (ECK) is a variation on Embedded contact homology (ECH), defined with respect to an open book decomposition compatible with a contact structure on some 3-manifold, M. The knot in question is given by the (null-homologous) ... More

Holomorphic curves in the presence of holomorphic hypersurface foliationsFeb 07 2019We prove a result which establishes restrictions on the pseudoholomorphic curves which can exist in a stable Hamiltonian manifold in the presence of certain $\mathbb{R}$-invariant foliations of the symplectization by holomorphic hypersurfaces. This result ... More

Quasi-morphisms on contactomorphism groups and Grassmannians of 2-planesFeb 06 2019We construct a natural prequantization space over a monotone product of a toric manifold and an arbitrary number of complex Grassmannians of 2-planes in even-dimensional complex spaces, and prove that the universal cover of the identity component of the ... More

Flat affine symplectic Lie groupsFeb 05 2019We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We prove that a left invariant flat affine symplectic connection on a connected ... More

Flat affine symplectic Lie groupsFeb 05 2019Mar 02 2019We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We prove that a left invariant flat affine symplectic connection on a connected ... More

Higher symplectic capacitiesFeb 04 2019We construct a new family of symplectic capacities indexed by certain symmetric polynomials, defined using rational symplectic field theory. We prove various structural properties of the capacities and discuss the connections with the equivariant L-infinity ... More

Higher symplectic capacitiesFeb 04 2019Feb 21 2019We construct a new family of symplectic capacities indexed by certain symmetric polynomials, defined using rational symplectic field theory. We prove various structural properties of the capacities and discuss the connections with the equivariant L-infinity ... More

Spine removal surgery and the geography of symplectic fillingsFeb 04 2019We prove that for any contact 3-manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof is an application of the spine ... More

Uniqueness of real Lagrangians up to cobordismFeb 04 2019Feb 13 2019We prove that a real Lagrangian submanifold in a closed symplectic manifold is unique up to cobordism. We then discuss the classification of real Lagrangians in $\mathbb{C} P^2$ and $S^2\times S^2$. In particular, we show that a real Lagrangian in $\mathbb{C} ... More

On a systolic inequality for closed magnetic geodesics on surfacesFeb 04 2019Feb 06 2019We apply a local systolic-diastolic inequality for contact forms and odd-symplectic forms on three-manifolds to bound the magnetic length of closed curves with prescribed geodesic curvature (also known as magnetic geodesics) on an oriented closed surface. ... More

On a local systolic inequality for odd-symplectic formsFeb 04 2019Feb 06 2019The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let $\Omega$ be an odd-symplectic form on an oriented closed manifold $\Sigma$ of ... More