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Fibred algebraic surfaces and commutators in the Symplectic groupJun 13 2019We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ ... More

Relative quantum cohomologyJun 11 2019We establish a system of PDE, called open WDVV, that constrains the bulk-deformed superpotential and associated open Gromov-Witten invariants of a Lagrangian submanifold $L \subset X$ with a bounding chain. Simultaneously, we define the quantum cohomology ... More

(Co)isotropic triples and poset representationsJun 11 2019We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; in particular, we classify indecomposable structures of this kind. The classification depends on the ground field, which we only assume to be perfect and not of characteristic ... More

Central extensions of Lie groups preserving a differential formJun 07 2019Let $M$ be a manifold with a closed, integral $(k+1)$-form $\omega$, and let $G$ be a Fr\'echet-Lie group acting on $(M,\omega)$. As a generalization of the Kostant-Souriau extension for symplectic manifolds, we consider a canonical class of central extensions ... More

Singular Improper Affine Spheres from a given Lagrangian SubmanifoldJun 07 2019Given a Lagrangian submanifold $L$ of the affine symplectic $2n$-space, one can canonically and uniquely define a center-chord and a special improper affine sphere of dimension $2n$, both of whose sets of singularities contain $L$. Although these improper ... More

Counting curves with local tangency constraintsJun 06 2019We construct invariants of any semipositive symplectic manifold which count rational curves satisfying multibranched tangency constraints to a local divisor. We also construct analogous invariants counting punctured curves with negative ends on a small ... More

ECH capacities, Ehrhart theory, and toric varietiesJun 05 2019ECH capacities were developed by Hutchings to study embedding problems for symplectic $4$-manifolds with boundary. They have found especial success in the case of certain toric symplectic manifolds where many of the computations resemble calculations ... More

Global mirrors and discrepant transformations for toric Deligne-Mumford stacksJun 03 2019We introduce a global Landau-Ginzburg model which is mirror to several toric Deligne-Mumford stacks and describe the change of the Gromov-Witten theories under discrepant transformations. We prove a formal decomposition of the quantum cohomology D-modules ... More

The geometry of graded cotangent bundlesMay 30 2019Given a vector bundle $A\to M$ we study the geometry of the graded manifolds $T^*[k]A[1]$, including their canonical symplectic structures, compatible Q-structures and Lagrangian Q-submanifolds. We relate these graded objects to classical structures, ... More

Rigid fibers of spinning topsMay 30 2019(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann ... More

On topological properties of positive complexity one spacesMay 30 2019Motivated by work of Fine and Panov, and of Lindsay and Panov, we prove that every closed symplectic complexity one space that is positive (e.g. positive monotone) enjoys topological properties that Fano varieties with a complexity one holomorphic torus ... More

On Action of $PSL(n+1, {\bf C})$ on Space of $L_k^p$-maps on ${\bf P}^n$May 29 2019In this paper, we prove properness of the action of the reparametrization group $PSL(n+1, {\bf C})$ on the space of $v$-stable $L_k^p$-maps on ${\bf P}^n$ as well as related results.

The convexity package for Hamiltonian actions on conformal symplectic manifoldsMay 29 2019Jun 02 2019Consider a Hamiltonian action of a compact connected Lie group on a conformal symplectic manifold. We prove a convexity theorem for the moment map under the assumption that the action is of Lee type, which establishes an analog of Kirwan's convexity theorem ... More

Towards the geography of bilinearized Legendrian contact homologyMay 28 2019We study the geography of bilinearized Legendrian contact homology for closed, connected Legendrian submanifolds with vanishing Maslov class in 1-jet spaces. We show that this invariant detects whether the two augmentations used to define it are DGA homotopic ... More

Getting a handle on contact manifoldsMay 28 2019We develop the details of a surgery theory for contact manifolds of arbitrary dimension via convex structures, extending the 3-dimensional theory developed by Giroux. The theory is analogous to that of Weinstein manifolds in symplectic geometry, with ... More

Dirac geometry and integration of Poisson homogeneous spacesMay 27 2019Jun 06 2019Using tools from Dirac geometry, we show through an explicit construction that any Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for a principal ... More

Dirac geometry and integration of Poisson homogeneous spacesMay 27 2019Using tools from Dirac geometry and through an explicit construction, we show that any Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for ... More

Spin/Pin-Structures and Real Enumerative GeometryMay 27 2019The present, partly expository, monograph consists of three parts. The first part treats Spin- and Pin-structures from three different perspectives and shows them to be suitably equivalent. It also introduces an intrinsic perspective on the relative Spin- ... More

A family of 8-dimensional generalized complex nilmanifolds with infinitely many real homotopy typesMay 27 2019We prove that there are infinitely many real homotopy types of $8$-dimensional nilmanifolds admitting generalized complex structures of type $k$ for every $0 \leq k \leq 4$. This is in deep contrast to the $6$-dimensional case.

No homotopy 4-sphere invariants using ECH = SWFMay 27 2019In relation to the 4-dimensional smooth Poincar\'e conjecture we construct an invariant of smooth homotopy 4-spheres using embedded contact homology (and Seiberg-Witten theory). But they turn out to vanish, for good reason.

Constructing local models for Lagrangian torus fibrationsMay 22 2019We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find a Lagrangian torus fibration on the 3-fold negative vertex whose ... More

Functorial LCH for immersed Lagrangian cobordismsMay 21 2019For $1$-dimensional Legendrian submanifolds of $1$-jet spaces, we extend the functorality of the Legendrian contact homology DG-algebra (DGA) from embedded exact Lagrangian cobordisms, as in \cite{EHK}, to a class of immersed exact Lagrangian cobordisms ... More

Introduction to the BV-BFV formalismMay 20 2019These notes give an introduction to the mathematical framework of the Batalin-Vilkovisky and Batalin-Fradkin-Vilkovisky formalisms. Some of the presented content was given as a mini course by the first author at the 2018 QSPACE conference in Benasque. ... More

The flux homomorphism and central extensions of diffeomorphism groupsMay 20 2019Let $D$ be a 2-dimensional closed unit disk and $\rm{Symp}(D,0)_{\rm{rel}}$ the group of symplectomorphisms preserving the origin and the boundary $\partial D$ pointwise. We consider the $\mathbb{R}$-valued flux homomorphism on $\rm{Symp}(D,0)_{\rm{rel}}$ ... More

Lower bound for the Poisson bracket invariant on surfacesMay 20 2019Recently, L. Buhovsky, A. Logunov and S. Tanny proved the (strong) Poisson bracket conjecture by Leonid Polterovich in dimension 2. In this note, instead of open cover constituted of displaceable sets in their work, considering open cover constituted ... More

On $C^0$-continuity of the spectral norm on non-symplectically aspherical manifoldsMay 19 2019Our first main result states that on rational symplectic manifolds the spectral norm $\gamma$ of a Hamiltonian is close to $\lambda_0 \mathbb{Z}$ if the Hamiltonian generates a time-1 map that is $C^0$-close to $Id$ where $\lambda_0$ denotes the rationality ... More

On $C^0$-continuity of the spectral norm on non-symplectically aspherical manifoldsMay 19 2019May 24 2019Our first main result states that on rational symplectic manifolds the spectral norm $\gamma$ of a Hamiltonian is close to $\lambda_0 \mathbb{Z}$ if the Hamiltonian generates a time-1 map that is $C^0$-close to $Id$ where $\lambda_0$ denotes the rationality ... More

Pseudo-rotations and holomorphic curvesMay 18 2019We prove a variant of the Chance-McDuff conjecture for pseudo-rotations: under certain additional conditions, a closed symplectic manifold which admits a Hamiltonian pseudo-rotation must have deformed quantum product and, in particular, some non-zero ... More

Sheaves and symplectic geometry of cotangent bundlesMay 17 2019This paper is essentially made of the three preprints arXiv:1212.5818, arXiv:1311.0187, arXiv:1603.07876 gathered in a single text, with simplified proofs. We recall several results of the microlocal theory of sheaves of Kashiwara-Schapira and apply them ... More

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions IIIMay 17 2019In this paper, we complete the classification of six-dimensional closed monotone symplectic manifolds admitting semifree Hamiltonian $S^1$-actions. We also show that every such manifold is $S^1$-equivariantly symplectomorphic to some K\"{a}ahler Fano ... More

Symplectic dominationMay 14 2019Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is a theorem ... More

A symplectic embedding of the cube with minimal sections and a question by SchlenkMay 14 2019I prove that the open unit cube can be symplectically embedded into a longer polydisc in such a way that the area of each section satisfies a sharp bound and the complement of each section is path-connected. This answers a variant of a question by F. ... More

Complex Germen on invariant isotropic tori under the Hamiltonian phases flow with in involution Hamilton functionsMay 13 2019M. M. Nekhoroshev put forward the problem of to find the Complex Germ on a isotropic invariant torus with respect to Hamiltonian phases flows which come from k-functions in involution. This statement was partially solved in [9] establishing that if certain ... More

Pseudorotations and Steenrod squaresMay 13 2019In this note we prove that if a closed monotone symplectic manifold $M$ of dimension $2n,$ satisfying a homological condition, that holds in particular when the minimal Chern number is $N>n,$ admits a Hamiltonian pseudorotation, then the quantum Steenrod ... More

A wall-crossing formula and the invariance of GLSM correlation functionsMay 13 2019In this paper we prove a wall-crossing formula, a crucial ingredient needed to prove that the correlation function of gauged linear sigma model is independent of the choice of perturbations.

Feedback control of charged ideal fluidsMay 12 2019The theory of controlled mechanical systems of [6, 3, 4] is extended to the case of ideal incompressible fluids consisting of charged particles in the presence of an external magnetic field. The resulting control is of feedback type and depends on the ... More

On the Hofer-Zehnder conjectureMay 12 2019We prove that if a Hamiltonian diffeomorphism on a closed monotone symplectic manifold with semisimple quantum homology has a finite number of contractible periodic points then the sum of the ranks of the local Floer homologies at its contractible fixed ... More

A monotone Lagrangian casebookMay 10 2019We present an array of new calculations in Lagrangian Floer theory which demonstrate observations relating to symplectic reduction, grading periodicity, and the closed-open map. We also illustrate Perutz's symplectic Gysin sequence and the quilt theory ... More

The $\mathbb{Z}/(p)$-equivariant product-isomorphism in fixed point Floer cohomologyMay 09 2019Let $p \geq 2$ be a prime, and $\mathbb{F}_p$ be the field with $p$ elements. Extending a result of Seidel for $p=2,$ we construct an isomorphism between the Floer cohomology of an exact or Hamiltonian symplectomorphism $\phi,$ with $\mathbb{F}_p$ coefficients, ... More

Symplectic geometry of $p$-adic Teichmüller uniformization for ordinary nilpotent indigenous bundlesMay 08 2019The aim of the present paper is to provide a new aspect of the $p$-adic Teichm\"{u}ller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ (= the moduli classifying ... More

The contact structure induced by a line fibration of R^3 is standardMay 08 2019Building on the work of and answering a question by Michael Harrison, we show that any contact structure on Euclidean 3-space induced by a line fibration is diffeomorphic to the standard contact structure.

Kähler packings of projective complex manifoldsMay 08 2019In this note we show that the multipoint Seshadri constant determines the maximum possible radii of embeddings of K\"ahler balls and vice versa.

A symplectic dynamics proof of the degree-genus formulaMay 08 2019We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere, defining the Hopf fibration. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process ... More

Geometric quantization of Hamiltonian flows and the Gutzwiller trace formulaMay 08 2019We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the quantum Hamiltonian dynamics associated with classical Hamiltonian flows over closed prequantized symplectic manifolds in the context of geometric quantization of Kostant ... More

Locally conformally symplectic reduction of the cotangent bundleMay 07 2019In a previous article, we introduced a reduction procedure for locally conformally symplectic manifolds at any regular value of the momentum mapping. We use this construction to prove an analogue of a well-known theorem in the symplectic setting about ... More

Factorization of symplectic matrices into elementary factorsMay 07 2019We prove that a symplectic matrix with entries in a ring with Bass stable rank one can be factored as a product of elementary symplectic matrices. This also holds for null-homotopic symplectic matrices with entries in a Banach algebra or in the ring of ... More

Factorization of symplectic matrices into elementary factorsMay 07 2019May 22 2019We prove that a symplectic matrix with entries in a ring with Bass stable rank one can be factored as a product of elementary symplectic matrices. This also holds for null-homotopic symplectic matrices with entries in a Banach algebra or in the ring of ... More

Momentum sections in Hamiltonian mechanics and sigma modelsMay 07 2019We show a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently introduced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic ... More

Momentum sections in Hamiltonian mechanics and sigma modelsMay 07 2019May 24 2019We show a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently introduced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic ... More

On Legendrian products and twist spunsMay 04 2019The Legendrian product of two Legendrian knots, as defined by Lambert-Cole, is a Legendrian torus. We show that this Legendrian torus is a twist spun whenever one of the Legendrian knot components is sufficiently large. We then study examples of Legendrian ... More

A Comparison between Hofer's metric and C^1-topologyMay 02 2019Hofer's metric is a bi-invariant metric on Hamiltonian diffeomorphism groups. Our main result shows that the topology induced from Hofer's metric is weaker than C^1-topology if the symplectic manifold is closed.

Toric topology of the Grassmannian of planes in $\mathbb{C}^5$ and the del Pezzo surface of degree fiveApr 30 2019We are complementing the work of Buchstaber and Terzi\'c by providing an alternative approach to determine the homology of the orbit space of a maximal compact torus action on the complex Grassmannian Gr(2,5). Our approach uses the well-known Geometric ... More

Toric topology of the Grassmannian of planes in $\mathbb{C}^5$ and the del Pezzo surface of degree $5$Apr 30 2019May 05 2019We are complementing the work of Buchstaber and Terzi\'c by providing an alternative approach to determine the homology of the orbit space of a maximal compact torus action on the complex Grassmannian Gr(2,5). Our approach uses the well-known Geometric ... More

The Kontsevich graph orientation morphism revisitedApr 30 2019The orientation morphism $Or\colon\Gamma\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ on finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $\Gamma$ with ordered sets ... More

Group Actions in Deformation QuantisationApr 29 2019This set of notes corresponds to a mini-course given in September 2018 in Bedlewo; it does not contain any new result; it complements -- with intersection -- the introduction to formal deformation quantization and group actions, corresponding to a course ... More

Definition of the cord algebra of knots using Morse TheoryApr 29 2019We redefine the cord algebra, which was introduced by Lenhard Ng as a topological knot invariant, in terms of Morse Theory. The determination of the cord algebra of the unknot and of the righthanded trefoil are given. We proove that the cord algebra in ... More

Reconstruction of $T_{\mathbb{P}^2}$ via tropical Lagrangian multi-sectionApr 29 2019In this paper, we study the reconstruction problem of the holomorphic tangent bundle $T_{\mathbb{P}^2}$ of the complex projective plane $\mathbb{P}^2$. We introduce the notion of tropical Lagrangian multi-section and cook up one from a family of Hermitian ... More

A note on Schwartzman-Fried-Sullivan Theory, with an applicationApr 29 2019May 13 2019This note presents the most basic definitions and techniques in the theory of asymptotic cycles. As an application we prove a theorem on the existence of global surfaces of section with prescribed spanning orbits and homology class. This result is a modification ... More

A note on Schwartzman-Fried-Sullivan Theory, with an applicationApr 29 2019This note presents the most basic definitions and techniques in the theory of asymptotic cycles. It also serves as an excuse to describe in full details the proof of a result on the existence of global surfaces of section with prescribed spanning orbits ... More

Tropically constructed Lagrangians in mirror quintic threefoldsApr 26 2019We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2875 lines on the quintic ... More

An asymptotic behavior of Vianna's exotic Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ as $\max\{a, b, c\} \to \infty$Apr 26 2019May 07 2019In this paper, we study asymptotic behavior of Vianna's infinite family of monotone Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ associated to Markov triples $(a,b,c)$. We prove that there exists a constant $\varepsilon_0 > 0$ independent of $(a, b, ... More

An asymptotic behavior of Vianna's exotic Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ as $\max\{a, b, c\} \to \infty$Apr 26 2019In this paper, we study asymptotic behavior of Vianna's infinite family of monotone Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ associated to Markov triples $(a,b,c)$. We prove that there exists a constant $\varepsilon_0 > 0$ independent of $(a, b, ... More

Deformation quantization and Kähler geometry with moment mapApr 26 2019In the first part of this paper we outline the constructions and properties of Fedosov star product and Berezin-Toeplitz star product. In the second part we outline the basic ideas and recent developments on Yau-Tian-Donaldson conjecture on the existence ... More

Singular Lagrangians and precontact Hamiltonian SystemsApr 25 2019In this paper we discuss singular Lagrangian systems on the framework of contact geometry. These systems exhibit a dissipative behavior in contrast with the symplectic scenario. We develop a constraint algorithm similar to the presymplectic one studied ... More

Shifted Coisotropic CorrespondencesApr 25 2019We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks where the $i$-morphisms are given by $i$-fold coisotropic correspondences. Assuming an expected ... More

Classification of six dimensional monotone symplectic manifolds admitting semifree circle actions IIApr 24 2019Let $(M,\omega_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the maximal and the minimal fixed component are both two dimensional, then $(M,\omega_M)$ is $S^1$-equivariantly ... More

The Rotation Number for Quantum Integrable SystemsApr 24 2019For a two degree of freedom quantum integrable system, a new spectral quantity is defined, the quantum rotation number. In the semiclassical limit, the quantum rotation number can be detected on a joint spectrum and is shown to converge to the well-known ... More

The Rotation Number for Quantum Integrable SystemsApr 24 2019Jun 07 2019For a two degree of freedom quantum integrable system, a new spectral quantity is defined, the quantum rotation number. In the semiclassical limit, the quantum rotation number can be detected on a joint spectrum and is shown to converge to the well-known ... More

Fixed-Depth Two-Qubit Circuits and the Monodromy PolytopeApr 23 2019For a native gate set which includes all single-qubit gates, we apply results from symplectic geometry to analyze the spaces of two-qubit programs accessible within a fixed number of gates. These techniques yield an explicit description of this subspace ... More

$A_\infty$-Minimal Model on Differential Graded AlgebrasApr 23 2019For a formal differential graded algebra, if extended by an odd degree element, we prove that the extended algebra has an $A_\infty$-minimal model with only $m_2$ and $m_3$ non-trivial. As an application, the $A_\infty$-algebras constructed by Tsai, Tseng ... More

Local normal forms of dynamical systems with a singular underlying geometric structureApr 22 2019In this paper we prove the existence of a simultaneous local normalization for couples $(X,\mathcal{G})$, where $X$ is a vector field which vanishes at a point and $\mathcal{G}$ is a singular underlying geometric structure which is invariant with respect ... More

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 2019May 09 2019In this paper, we investigate the minimal symplectic fillings of small Seifert 3-manifolds with a canonical contact structure. As a result, we classify all minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, ... 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

Adiabatic limits, Theta functions, and geometric quantizationApr 08 2019May 14 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