Special Lagrangian linear subspaces in product symplectic space

Małgorzata Mikosz

Displaying similar documents to “Special Lagrangian linear subspaces in product symplectic space”

Generalized Conley-Zehnder index

Jean Gutt (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit $1$ as an eigenvalue. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W, \bar{Ω})$ , having chosen a given...

Rabinowitz Floer homology and symplectic homology

Kai Cieliebak, Urs Frauenfelder, Alexandru Oancea (2010)

Annales scientifiques de l'École Normale Supérieure

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The first two authors have recently defined Rabinowitz Floer homology groups $R F H_{*} (M, W)$ associated to a separating exact embedding of a contact manifold $(M, ξ)$ into a symplectic manifold $(W, ω)$ . These depend only on the bounded component $V$ of $W ∖ M$ . We construct a long exact sequence in which symplectic cohomology of $V$ maps to symplectic homology of $V$ , which in turn maps to Rabinowitz Floer homology $R F H_{*} (M, W)$ , which then maps to symplectic cohomology of $V$ . We compute $R F H_{*} (S T^{*} L, T^{*} L)$ , where $S T^{*} L$ is the unit cosphere bundle of a closed...

Rational symplectic field theory over $ℤ^{2}$ for exact Lagrangian cobordisms

Tobias Ekholm (2008)

Journal of the European Mathematical Society

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We construct a version of rational Symplectic Field Theory for pairs $(X, L)$ , where $X$ is an exact symplectic manifold, where $L \subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X, L)$ a $ℤ$ -graded chain complex of vector spaces over $ℤ_{2}$ , filtered with $k$ filtration levels. The corresponding $k$ -level spectral sequence is invariant under deformations of $(X, L)$ and has the following property: if $(X, L)$ is obtained by...

A characterization of symplectic groups related to Fermat primes

Behnam Ebrahimzadeh, Alireza K. Asboei (2021)

Commentationes Mathematicae Universitatis Carolinae

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We proved that the symplectic groups $PSp (4, 2^{n})$ , where $2^{2 n} + 1$ is a Fermat prime number is uniquely determined by its order, the first largest element orders and the second largest element orders.

Symplectic periods of the continuous spectrum of $GL (2 n)$

Shunsuke Yamana (2014)

Annales de l’institut Fourier

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We provide a formula for the symplectic period of an Eisenstein series on $GL (2 n)$ and determine when it is not identically zero.

Symplectic critical surfaces in Kähler surfaces

Xiaoli Han, Jiayu Li (2010)

Journal of the European Mathematical Society

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Let $M$ be a Kähler surface and $Σ$ be a closed symplectic surface which is smoothly immersed in $M$ . Let $α$ be the Kähler angle of $Σ$ in $M$ . We first deduce the Euler-Lagrange equation of the functional $L = \int_{Σ} \frac{1}{cos α} d μ$ in the class of symplectic surfaces. It is ${cos}^{3} α H = {(J {(J \nabla cos α)}^{⊤})}^{⊥}$ , where $H$ is the mean curvature vector of $Σ$ in $M$ , $J$ is the complex structure compatible with the Kähler form $ω$ in $M$ , which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface...

Characterization of diffeomorphisms that are symplectomorphisms

Stanisław Janeczko, Zbigniew Jelonek (2009)

Fundamenta Mathematicae

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Let $(X, ω_{X})$ and $(Y, ω_{Y})$ be compact symplectic manifolds (resp. symplectic manifolds) of dimension 2n > 2. Fix 0 < s < n (resp. 0 < k ≤ n) and assume that a diffeomorphism Φ : X → Y maps all 2s-dimensional symplectic submanifolds of X to symplectic submanifolds of Y (resp. all isotropic k-dimensional tori of X to isotropic tori of Y). We prove that in both cases Φ is a conformal symplectomorphism, i.e., there is a constant c ≠0 such that $Φ * ω_{Y} = c ω_{X}$ .

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

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We show that if $(M, ω)$ is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer’s metric on the group of Hamiltonian diffeomorphisms of $(M, ω)$ has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer’s metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in $M \times M$ ...

Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville

Daniel Greb, Christian Lehn, Sönke Rollenske (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a compact hyperkähler manifold containing a complex torus $L$ as a Lagrangian subvariety. Beauville posed the question whether $X$ admits a Lagrangian fibration with fibre $L$ . We show that this is indeed the case if $X$ is not projective. If $X$ is projective we find an almost holomorphic Lagrangian fibration with fibre $L$ under additional assumptions on the pair $(X, L)$ , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic...

Exotic Deformations of Calabi-Yau Manifolds

Paolo de Bartolomeis, Adriano Tomassini (2013)

Annales de l’institut Fourier

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We introduce Quantum Inner State manifolds (QIS manifolds) as (compact) $2 n$ -dimensional symplectic manifolds $(M, κ)$ endowed with a $κ$ -tamed almost complex structure $J$ and with a nowhere vanishing and normalized section $ϵ$ of the bundle $Λ_{J}^{n, 0} (M)$ satisfying the condition ${\bar{\partial}}_{J} ϵ = 0$ . We study the moduli space $𝔐$ of QIS deformations of a given Calabi-Yau manifold, computing its tangent space...

Lagrangian fibrations on generalized Kummer varieties

Martin G. Gulbrandsen (2007)

Bulletin de la Société Mathématique de France

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We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface $A$ of Picard number one we find the following: The Kummer variety $K^{n} A$ is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if $n$ is a perfect square. And this is the case if and only if $K^{n} A$ carries a divisor with vanishing Beauville-Bogomolov square.

On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.

Aldo Bressan (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In [1] I and II various equivalence theorems are proved; e.g. an ODE $(\mathcal{E})\dot{z}=F(t,z,u,\dot{u})\,(\in\mathbb{R}^{m})$ with a scalar control $u=u(\cdot)$ is linear w.r.t. $\dot{u}$ iff $(\alpha)$ its solution $z(u,\cdot)$ with given initial conditions (chosen arbitrarily) is continuous w.r.t. $u$ in a certain sense, or iff $(\beta)$ $z(u,\cdot)$ satisfies certain conditions by which $1^{st}$ -order discontinuities of $u$ and $\dot{u}$ can be treated satisfactorily. In the case when, for $z=(q,p)$ equation $(\mathcal{E})$ is a semi-Hamiltonian system, equivalent to a system of Lagrangian equations of a general type,...

On sets of discontinuities of functions continuous on all lines

Luděk Zajíček (2022)

Commentationes Mathematicae Universitatis Carolinae

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Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a $C^{1}$ -smooth function $f$ on $[0, 1]$ and a closed set $M \subset graph f$ nowhere dense in $graph f$ such that there does not exist any linearly continuous function on $ℝ^{2}$ (i.e., function continuous on all lines) which is discontinuous at each point of $M$ . We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on $ℝ^{n}$ proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence...

The inverse problem in the calculus of variations: new developments

Thoan Do, Geoff Prince (2021)

Communications in Mathematics

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We deal with the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. A number of recent theorems are presented, using exterior differential systems theory (EDS). In particular, we indicate how to generalise Jesse Douglas’s famous solution for $n = 2$ . We then examine a new class of solutions in arbitrary dimension $n$ and give some non-trivial examples in dimension 3.

Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant

Gwénaël Massuyeau (2012)

Bulletin de la Société Mathématique de France

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Let $Σ$ be a compact connected oriented surface with one boundary component, and let $π$ be the fundamental group of $Σ$ . The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of $Σ$ , whose $k$ -th term consists of the self-homeomorphisms of $Σ$ that act trivially at the level of the $k$ -th nilpotent quotient of $π$ . Morita defined a homomorphism from the $k$ -th term of the Johnson filtration to the third homology group of the $k$ -th nilpotent quotient of $π$ . In this paper, we...

Some results on the co-intersection graph of submodules of a module

Lotf Ali Mahdavi, Yahya Talebi (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a ring with identity and $M$ be a unitary left $R$ -module. The co-intersection graph of proper submodules of $M$ , denoted by $Ω (M)$ , is an undirected simple graph whose vertex set $V (Ω)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N + K \neq M$ . We study the connectivity, the core and the clique number of $Ω (M)$ . Also, we provide some conditions on the module $M$ , under which the clique number of $Ω (M)$ is infinite and $Ω (M)$ is a planar graph. Moreover, we give...

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair $(𝒳, 𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $Γ$ . The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p : 𝒜 \to 𝒳$ . The space $𝒜$ is equipped with a closed $2$ -form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

Saturation numbers for linear forests $P_{6} + t P_{2}$

Jingru Yan (2023)

Czechoslovak Mathematical Journal

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A graph $G$ is $H$ -saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$ . The saturation number, $sat (n, H)$ , is the minimum number of edges of a graph in the set of all $H$ -saturated graphs of order $n$ . We determine the saturation number $sat (n, P_{6} + t P_{2})$ for $n \geq \frac{10}{3} t + 10$ and characterize the extremal graphs for $n > \frac{10}{3} t + 20$ .

Complete pairs of coanalytic sets

Jean Saint Raymond (2007)

Fundamenta Mathematicae

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Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of $ω^{ω}$ there exists a continuous function $f : ω^{ω} \to X$ such that $f^{- 1} (C ₀) = D ₀$ and $f^{- 1} (C ₁) = D ₁$ . We give several explicit examples of complete pairs of coanalytic sets.

Matchings in complete bipartite graphs and the $r$ -Lah numbers

Gábor Nyul, Gabriella Rácz (2021)

Czechoslovak Mathematical Journal

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We give a graph theoretic interpretation of $r$ -Lah numbers, namely, we show that the $r$ -Lah number ${⌊\binom{n}{k}⌋}_{r}$ counting the number of $r$ -partitions of an $(n + r)$ -element set into $k + r$ ordered blocks is just equal to the number of matchings consisting of $n - k$ edges in the complete bipartite graph with partite sets of cardinality $n$ and $n + 2 r - 1$ ( $0 \leq k \leq n$ , $r \geq 1$ ). We present five independent proofs including a direct, bijective one. Finally, we close our work with a similar result for $r$ -Stirling numbers of the second kind. ...

The Turán number of the graph $3 P_{4}$

Halina Bielak, Sebastian Kieliszek (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $e x (n, G)$ denote the maximum number of edges in a graph on $n$ vertices which does not contain $G$ as a subgraph. Let $P_{i}$ denote a path consisting of $i$ vertices and let $m P_{i}$ denote $m$ disjoint copies of $P_{i}$ . In this paper we count $e x (n, 3 P_{4})$ .

Some lagrangian invariants of symplectic manifolds

Michel Nguiffo Boyom (2007)

Banach Center Publications

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The KV-homology theory is a new framework which yields interesting properties of lagrangian foliations. This short note is devoted to relationships between the KV-homology and the KV-cohomology of a lagrangian foliation. Let us denote by $_{F}$ (resp. $V^{F}$ ) the KV-algebra (resp. the space of basic functions) of a lagrangian foliation F. We show that there exists a pairing of cohomology and homology to $V^{F}$ . That is to say, there is a bilinear map $H^{q} (_{F}, V^{F}) \times H_{q} (_{F}, V^{F}) \to V^{F}$ , which is invariant under F-preserving symplectic...