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Real Lagrangians in Calabi-Yau ThreefoldsAug 19 2019We compute the mod $2$ cohomology groups of real Lagrangians in Calabi-Yau threefolds using well-behaved torus fibrations constructed by Gross. To do this we study a long exact sequence introduced by Casta\~{n}o-Bernard and Matessi, which relates the ... More

Irrationality and monodromy for cubic threefoldsAug 19 2019We show the cohomological monodromy for the universal family of smooth cubic threefolds does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds.

Bourgeois contact structures: tightness, fillability and applicationsAug 15 2019Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension ... More

On Riemann-Poisson Lie groupsAug 14 2019A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in C.R. Acad. Sci. Paris s\'er. {\bf I 333} (2001) 763-768. We study these Lie ... More

The Legendrian Whitney trickAug 13 2019In this article, we prove a Legendrian Whitney trick which allows for the removal of intersections between codimension-two contact submanifolds and Legendrian submanifolds, assuming such a smooth cancellation is possible. This technique is applied to ... More

Homological mirror symmetry for the genus 2 curve in an abelian variety and its generalized Strominger-Yau-Zaslow mirrorAug 12 2019Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs $X$ and $Y$ such that the complex geometry on $X$ mirrors the symplectic geometry on $Y$. It allows one to deduce symplectic information about $Y$ ... More

Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cellsAug 12 2019Given a standard complex semisimple Poisson Lie group $(G, \pi_{st})$, generalised double Bruhat cells $G^{u, v}$ and generalised Bruhat cells $O^u$ equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl ... More

Plasma in monopole background is not twisted PoissonAug 12 2019For a particle in the magnetic field of a cloud of monopoles, the naturally associated 2-form on phase space is not closed, and so the corresponding bracket operation on functions does not satisfy the Jacobi identity. Thus, it is not a Poisson bracket; ... More

Poisson--Kähler fibration I: curvature of the base manifoldAug 11 2019We start from a finite dimensional Higgs bundle description of a result of Burns on negative curvature property of the space of complex structures, then we apply the corresponding infinite dimensional Higgs bundle picture and obtain a precise curvature ... More

Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curvesAug 08 2019We prove the existence of a forced time-periodic solution to general nonlinear Hamiltonian PDEs by using pseudoholomorphic curves as in symplectic homology theory. With a number of assumptions on the nonlinearity, we prove that for a generic time period ... More

Rational Morita equivalence for holomorphic Poisson modulesAug 06 2019We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita ... More

Orlov and Viterbo functors in partially wrapped Fukaya categoriesAug 06 2019We study two functors between (partially) wrapped Fukaya categories. The first is the Orlov functor from the Fukaya category of a stop to the Fukaya category of the ambient sector. We give a geometric criterion for when this functor is spherical in the ... More

J-holomorphic curves and Dirac-harmonic mapsAug 06 2019Dirac-harmonic maps are critical points of a fermionic action functional, generalizing the Dirichlet energy for harmonic maps. We consider the case where the source manifold is a closed Riemann surface with the canonical Spin^c-structure determined by ... More

Scale Calculus and M-Polyfolds -- An IntroductionAug 04 2019These are lecture notes on scale calculus and M-polyfolds written for a graduate course at UNICAMP March-June 2018 and an advanced mini-course given during the biannual meeting of Brazilian mathematicians, CBM-32, at IMPA in August 2019.

The stabilized symplectic embedding problem for polydiscsJul 30 2019A stabilized polydisc is a product of a symplectic polydisc and several copies of the complex plane. This paper gives restrictions to symplectic embeddings of stabilized polydiscs. When the source polydisc is sufficiently "skinny" (eccentricity at least ... More

Admissible restrictions of irreducible representations of reductive Lie groups: symplectic geometry and discretely decomposabilityJul 30 2019Let $G$ be a real reductive Lie group, $L$ a compact subgroup, and $\pi$ an irreducible admissible representation of $G$. This paper proves a necessary and sufficient condition for the finiteness of the multiplicities of $L$-types occurring in $\pi$ based ... More

Quasipositive links and electromagnetismJul 24 2019For every link $L$ we construct a complex algebraic plane curve that intersects $S^3$ transversally in a link $\tilde{L}$ that contains $L$ as a sublink. This construction proves that every link $L$ is the sublink of a quasipositive link that is a satellite ... More

A Compactness Theorem for $SO(3)$ Anti-Self-Dual Equation with Translation SymmetryJul 24 2019Motivated by the $SO(3)$ Atiyah-Floer conjecture, we consider anti-self-dual instantons on two types of noncompact four-manifolds: the product of the real line with a three-manifold having a cylindrical end and the product of the complex plane with a ... More

Symplectic Structures with Non-Isomorphic Primitive Cohomology on open 4-ManifoldsJul 23 2019We analyze two different fibrations of a link complement M constructed by McMullen-Taubes, and studied further by Vidussi. These examples lead to inequivalent symplectic forms on a 4-manifold X = S x M, which can be distinguished by the dimension of the ... More

Hofer's metric in compact Lie groupsJul 23 2019In this article we study the Hofer geometry of a compact Lie group $K$ which acts by Hamiltonian diffeomorphisms on a symplectic manifold $M$. Generalized Hofer norms on the Lie algebra of $K$ are introduced and analyzed with tools from group invariant ... More

Symplectic homology capacity of convex bodies and loop space homologyJul 23 2019Floer-Hofer-Wysocki introduced a capacity invariant for (open) subsets in the symplectic vector space using symplectic homology, which we call symplectic homology capacity. We prove that, for any convex body $K$ in the symplectic vector space, the symplectic ... More

Lagrangian cobordisms and Legendrian invariants in knot Floer homologyJul 23 2019We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our ... More

Symplectic fillings of asymptotically dynamically convex manifolds IJul 22 2019We consider exact fillings with vanishing first Chern class of asymptotically dynamically convex (ADC) manifolds. We construct two structure maps on positive symplectic cohomology and prove that they are independent of the filling for ADC manifolds. The ... More

Exact Calabi-Yau categories and q-intersection numbersJul 22 2019An exact Calabi-Yau structure, originally introduced by Keller, is a special kind of smooth Calabi-Yau structures in the sense of Kontsevich-Vlassopoulos. For a Weinstein manifold $M$, the existence of an exact Calabi-Yau structure on its wrapped Fukaya ... More

Coisotropic submanifolds in $b$-symplectic geometryJul 22 2019We study coisotropic submanifolds of $b$-symplectic manifolds. We prove that $b$-coisotropic submanifolds (those transverse to the degeneracy locus) determine the $b$-symplectic structure in a neighborhood, and provide a normal form theorem. This extends ... More

Collective Heavy Top DynamicsJul 18 2019We construct a Poisson map $\mathbf{M}\colon T^{*}\mathbb{C}^{2} \to \mathfrak{se}(3)^{*}$ with respect to the canonical Poisson bracket on $T^{*}\mathbb{C}^{2} \cong T^{*}\mathbb{R}^{4}$ and the $(-)$-Lie--Poisson bracket on the dual $\mathfrak{se}(3)^{*}$ ... More

Symplectic homology of convex domains and Clarke's dualityJul 17 2019We prove that the Floer complex that is associated with a convex Hamiltonian function on $\mathbb{R}^{2n}$ is isomorphic to the Morse complex of Clarke's dual action functional that is associated with the Fenchel-dual Hamiltonian. This isomorphism preserves ... More

Indefinite Stein fillings and Pin(2)-monopole Floer homologyJul 17 2019Given a spin$^c$ rational homology sphere $(Y,\mathfrak{s})$ with $\mathfrak{s}$ self-conjugate and for which the reduced monopole Floer homology $\mathit{HM}_{\bullet}(Y,\mathfrak{s})$ has rank one, we provide obstructions to the intersection forms of ... More

Fukaya-Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$Jul 16 2019We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation ... More

Fukaya-Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$Jul 16 2019Jul 18 2019We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation ... More

The symplectic isotopy problem for rational cuspidal curvesJul 15 2019We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible ... More

Legendrian Hopf linksJul 15 2019We completely classify Legendrian realisations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.

Quasi-polynomials and the singular $[Q,R]=0$ theoremJul 13 2019In this short note we revisit the `shift-desingularization' version of the $[Q,R]=0$ theorem for possibly singular symplectic quotients. We take as starting point an elegant proof due to Szenes-Vergne of the quasi-polynomial behavior of the multiplicity ... More

Convex hypersurface theory in contact topologyJul 13 2019We lay the foundations of convex hypersurface theory (CHT) in contact topology, extending the work of Giroux in dimension three. Specifically, we prove any that closed hypersurface in a contact manifold can be $C^0$-approximated by a convex one. We also ... More

Convex hypersurface theory in contact topologyJul 13 2019Jul 23 2019We lay the foundations of convex hypersurface theory (CHT) in contact topology, extending the work of Giroux in dimension three. Specifically, we prove that any closed hypersurface in a contact manifold can be $C^0$-approximated by a convex one. We also ... More

Novikov's theorem in higher dimensions?Jul 12 2019Novikov's theorem is a rigidity result on the class of taut foliations on three-manifolds. For higher dimensional manifolds, the existence of a strong symplectic form has been proposed as an analogue for tautness in order to achieve similar rigidity. ... More

Deformations of Vector Bundles over Lie GroupoidsJul 12 2019VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy of ... More

Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroidsJul 11 2019In this paper, we develop the theory of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids introduced by Cai, Liu and Sheng. Specifically, we introduce the notions of Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on ... More

On the fibres of Mishchenko-Fomenko systemsJul 09 2019This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra $\mathfrak{g}$. Their theory associates a maximal Poisson-commutative subalgebra of $\mathbb{C}[\mathfrak{g}]$ ... More

Large fronts in nonlocally coupled systems using Conley-Floer homologyJul 08 2019In this paper we study travelling front solutions for nonlocal equations of the type \begin{equation} \partial_t u = N * S(u) + \nabla F(u), \qquad u(t,x) \in \mathbf{R}^d. \end{equation} Here $N *$ denotes a convolution-type operator in the spatial variable ... More

Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerationsJul 08 2019We give a simple expression for the integral of the canonical holomorphic volume form in degenerating families of varieties constructed from wall structures and with central fiber a union of toric varieties. The cycles to integrate over are constructed ... More

Cyclic structures and broken cyclesJul 07 2019We introduce a new way to encode semicyclic structures using a stack of broken cycles. (We also prove an analogue for paracyclic structures.) This was motivated not only by higher algebra but also by Fukaya-categorical considerations. We also openly speculate ... More

Immersed Lagrangian Floer cohomology via pearly trajectoriesJul 06 2019We define Lagrangian Floer cohomology by counting pearly trajectories for graded, exact Lagrangian immersions that satisfy certain positivity condition on the index of the non-embedded points, and show that it is an invariant of the Lagrangian immersion ... More

Quasi-parabolic Higgs bundles and null hyperpolygon spacesJul 03 2019We introduce the moduli space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over a compact Riemann surface $\Sigma$ and consider a natural involution, studying its fixed point locus when $\Sigma$ is $\mathbb{C} \mathbb{P}^1$ and establishing an ... More

Semi-classical analysis of piecewise quasi-polynomial functionsJul 02 2019Motivated by applications to multiplicity formulas in index theory, we study a family of distributions $\Theta(m;k)$ associated to a piecewise quasi-polynomial function $m$. The family is indexed by an integer $k>0$, and admits an asymptotic expansion ... More

Distinguishing open symplectic mapping tori via their wrapped Fukaya categoriesJul 02 2019In this paper, we present partial results towards a classification of symplectic mapping tori using dynamical properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold $T_\phi$ associated to a Weinstein domain $M$, and ... More

Formality morphism as the mechanism of $\star$-product associativity: how it worksJul 01 2019The formality morphism $\boldsymbol{\mathcal{F}}=\{\mathcal{F}_n$, $n\geqslant1\}$ in Kontsevich's deformation quantization is a collection of maps from tensor powers of the differential graded Lie algebra (dgLa) of multivector fields to the dgLa of polydifferential ... More

Engel structures on complex surfacesJun 28 2019We classify complex surfaces $(M,\,J)$ admitting Engel structures $\mathcal{D}$ which are complex line bundles. Namely we prove that this happens if and only if $(M,\,J)$ has trivial Chern classes. We construct examples of such Engel structures by adapting ... More

$T$-equivariant disc potential and SYZ mirror constructionJun 27 2019We develop a $G$-equivariant Lagrangian Floer theory by counting pearly trees in the Borel construction $L_G$. We apply the construction to smooth moment-map fibers of toric semi-Fano varieties and obtain the $T$-equivariant Landau-Ginzburg mirrors. We ... More

The Hamiltonian approach to the problem of derivation of production functions in economic growth theoryJun 26 2019We introduce a general Hamiltonian framework that appears to be a natural setting for the derivation of various production functions in economic growth theory, starting with the celebrated Cobb-Douglas function. Employing our method, we investigate some ... More

Variational order for forced Lagrangian systems II: Euler-Poincaré equations with forcingJun 24 2019In this paper we provide a variational derivation of the Euler-Poincar\'e equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others. Moreover, we study in detail the underlying geometry which ... More

Simulating Maxwell-Schrödinger Equations by High-Order Symplectic FDTD AlgorithmJun 23 2019A novel symplectic algorithm is proposed to solve the Maxwell-Schr\"odinger (M-S) system for investigating light-matter interaction. Using the fourth-order symplectic integration and fourth-order collocated differences, Maxwell-Schr\"odinger equations ... More

Multisymplectic actions of compact Lie groups on spheresJun 20 2019We give a geometric description of the obstruction to the existence of homotopy comoment maps in multisymplectic geometry. We apply this description to determine the existence of comoments for multisymplectic compact group actions on spheres and provide ... More

Quasimorphisms on surfaces and continuity in the Hofer normJun 20 2019There is a number of known constructions of quasimorphisms on Hamiltonian groups. We show that on surfaces many of these quasimorphisms are not compatible with the Hofer norm in a sense they are not continuous and not Lipschitz. The only exception known ... More

Exact Bohr-Sommerfeld Conditions for the Quantum Periodic Benjamin-Ono EquationJun 19 2019In this paper we describe the spectrum of the quantum periodic Benjamin-Ono equation in terms of the multi-phase solutions of the underlying classical system (the periodic multi-solitons). To do so, we show that the semi-classical quantization of this ... More

Non-autonomous curves on surfacesJun 19 2019Consider a symplectic surface $\Sigma$ with two properly embedded Hamiltonian isotopic curves $L$ and $L'$. Suppose $g \in Ham (\Sigma)$ is a Hamiltonian diffeomorphism which sends $L$ to $L'$. Which dynamical properties of $g$ can be detected by the ... More

Holomorphic curves for Legendrian surgeryJun 17 2019Let $X$ be a Weinstein manifold with ideal contact boundary $Y$. If $\Lambda\subset Y$ is a link of Legendrian spheres in $Y$ then by attaching Weinstein handles to $X$ along $\Lambda$ we get a Weinstein cobordism $X_{\Lambda}$ with a collection of Lagrangian ... More

Optimal transport on completely integrable toric manifoldsJun 17 2019We show that existence and uniqueness of solutions to transported Monge-Ampere problem on complex compact toric manifold follows easily from the real theory of optimal transportation.

The beginnings of symplectic topology in Bochum in the early eightiesJun 14 2019I outline the history and the original proof of the Arnold conjecture on fixed points of Hamiltonian maps for the special case of the torus, leading to a sketch of the proof for general symplectic manifolds and to Floer homology. This is the written version ... More

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

Relative quantum cohomologyJun 11 2019Jul 25 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

S^1-equivariant contact homology for hypertight contact formsJun 08 2019In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three-manifold with a dynamically convex contact form. However we did not show that this cylindrical contact ... 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

ECH capacities, Ehrhart theory, and toric varietiesJun 05 2019Jun 19 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

Geography of bilinearized Legendrian contact homologyMay 28 2019Jul 15 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 2019Jul 16 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.

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

Products and connected sums of spheres as monotone Lagrangian submanifoldsMay 21 2019Jun 18 2019We obtain topological restrictions on Maslov classes of monotone Lagrangian submanifolds of $\mathbb{C}^n$. We also construct families of new examples of monotone Lagrangian submanifolds, which show that the restrictions on Maslov classes are sharp in ... 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 2019Jun 11 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

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