Generalized Conley-Zehnder index

Jean Gutt

Displaying similar documents to “Generalized Conley-Zehnder index”

Special Lagrangian linear subspaces in product symplectic space

Małgorzata Mikosz (2004)

Banach Center Publications

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The notes consist of a study of special Lagrangian linear subspaces. We will give a condition for the graph of a linear symplectomorphism $f : (ℝ^{2 n}, σ = \sum_{i = 1}^{n} d x_{i} \land d y_{i}) \to (ℝ^{2 n}, σ)$ to be a special Lagrangian linear subspace in $(ℝ^{2 n} \times ℝ^{2 n}, ω = π * ₂ σ - π * ₁ σ)$ . This way a special symplectic subset in the symplectic group is introduced. A stratification of special Lagrangian Grassmannian $S Λ_{2 n} ≃ S U (2 n) / S O (2 n)$ is defined.

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...

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...

Linking and the Morse complex

Michael Usher (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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For a Morse function $f$ on a compact oriented manifold $M$ , we show that $f$ has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in $M$ whose components have nontrivial linking number, such that the minimal value of $f$ on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of $f$ in terms of the Betti numbers of $M$ and the behavior of $f$ with respect...

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...

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...

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.

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,...

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...

Controllable and tolerable generalized eigenvectors of interval max-plus matrices

Matej Gazda, Ján Plavka (2021)

Kybernetika

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By max-plus algebra we mean the set of reals $ℝ$ equipped with the operations $a \oplus b = max {a, b}$ and $a \otimes b = a + b$ for $a, b \in ℝ .$ A vector $x$ is said to be a generalized eigenvector of max-plus matrices $A, B \in ℝ (m, n)$ if $A \otimes x = λ \otimes B \otimes x$ for some $λ \in ℝ$ . The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval)...

-simplicity of interval max-min matrices

Ján Plavka, Štefan Berežný (2018)

Kybernetika

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A matrix $A$ is said to have $𝐗$ -simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗 = {x : \underset{̲}{x} \leq x \leq \bar{x}}$ containing a constant vector is the unique solution of the system $A \otimes y = x$ in $𝐗$ . The main result of this paper is an extension of $𝐗$ -simplicity to interval max-min matrix $𝐀 = {A : \underset{̲}{A} \leq A \leq \bar{A}}$ distinguishing two possibilities, that at least one matrix or all matrices from a given interval have $𝐗$ -simple image eigenspace. $𝐗$ -simplicity of interval matrices in max-min algebra are studied and equivalent conditions for...

$(0, 1)$ -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

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Let $m$ and $n$ be positive integers, and let $R = (r_{1}, ..., r_{m})$ and $S = (s_{1}, ..., s_{n})$ be nonnegative integral vectors. Let $A (R, S)$ be the set of all $m \times n$ $(0, 1)$ -matrices with row sum vector $R$ and column vector $S$ . Let $R$ and $S$ be nonincreasing, and let $F (R)$ be the $m \times n$ $(0, 1)$ -matrix, where for each $i$ , the $i$ th row of $F (R, S)$ consists of $r_{i}$ 1’s followed by $(n - r_{i})$ 0’s. Let $A \in A (R, S)$ . The discrepancy of A, $disc (A)$ , is the number of positions in which $F (R)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m \times n$ matrices over...

Linear preservers of rc-majorization on matrices

Mohammad Soleymani (2024)

Czechoslovak Mathematical Journal

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Let $A$ , $B$ be $n \times m$ matrices. The concept of matrix majorization means the $j$ th column of $A$ is majorized by the $j$ th column of $B$ and this is done for all $j$ by a doubly stochastic matrix $D$ . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

The real symmetric matrices of odd order with a P-set of maximum size

Zhibin Du, Carlos Martins da Fonseca (2016)

Czechoslovak Mathematical Journal

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Suppose that $A$ is a real symmetric matrix of order $n$ . Denote by $m_{A} (0)$ the nullity of $A$ . For a nonempty subset $α$ of ${1, 2, ..., n}$ , let $A (α)$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $α$ . When $m_{A (α)} (0) = m_{A} (0) + | α |$ , we call $α$ a P-set of $A$ . It is known that every P-set of $A$ contains at most $⌊ n / 2 ⌋$ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As...

Maps on upper triangular matrices preserving zero products

Roksana Słowik (2017)

Czechoslovak Mathematical Journal

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Consider $𝒯_{n} (F)$ —the ring of all $n \times n$ upper triangular matrices defined over some field $F$ . A map $φ$ is called a zero product preserver on $𝒯_{n} (F)$ in both directions if for all $x, y \in 𝒯_{n} (F)$ the condition $x y = 0$ is satisfied if and only if $φ (x) φ (y) = 0$ . In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map $φ$ may act in any bijective way, whereas for the zero divisors and zero matrix one can write $φ$ as a...