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Cette thèse étudie deux problèmes de nature géométrique et topologique associés au crochet de Poisson sur les variétés symplectiques. Le premier problème porte sur la notion de submersion symplectique que nous introduisons dans le présent texte et qui généralise la notion de symplectomorphisme. Il s'avère qu'une submersion symplectique est un morphisme de Poisson : il s'agit d'une application entre variétés symplectiques qui préserve le crochet de Poisson. Notre intérêt pour ces fonctions réside dans le fait que le théorème de non-tassement de Gromov porte sur l'aire minimale possible pour les images des submersions symplectiques (allant d'une boule symplectique vers le plan symplectique) obtenues comme compositions d'un plongement symplectique dans l'espace symplectique euclidien de dimension 2n et de la projection standard vers le plan de coordonnées conjuguées (p_1, q_1). Nous investiguons le problème inverse dit « de représentabilité » : nous obtenons des conditions nécessaires et suffisantes pour qu'une submersion symplectique comme ci-dessus se factorise comme précédemment à travers un plongement ou une immersion symplectique dans l'espace euclidien. Nous montrons par ailleurs qu'il existe une submersion symplectique qui ne se factorise pas de la sorte à travers une immersion et qu'il existe aussi une submersion symplectique qui se factorise de la sorte à travers une immersion, mais pas à travers un plongement. Le deuxième problème porte sur la conjecture du crochet de Poisson de Polterovich. Étant donné une variété symplectique (M, omega) et un recouvrement U de M, nous pouvons définir l'invariant pb(F) associé à une partition de l'unité F subordonnée à U, qui est une sorte de norme sur les crochets de Poisson entre les paires de fonctions de la partition. En dénotant e(U) l'énergie de disjonction de Hofer maximale d'un ouvert du recouvrement U, la conjecture demande s'il existe une constante positive C indépendante de U et de F telle que le produit de pb(F) et de e(U) soit supérieur à C. Cette conjecture a été établie récemment par Buhovski-Logunov-Tanny dans le cas des surfaces ; en nous inspirant de travaux antérieurs de Buhovski-Tanny, nous avons aussi démontré la conjecture pour les surfaces de genre plus grand que 1. Nous exposons notre approche dans le second chapitre de cette thèse. À l'aide des submersions symplectiques, nous généralisons nos méthodes afin d'attaquer la conjecture en dimensions supérieures ; nous obtenons ainsi une nouvelle preuve d'un théorème de Polterovich et de Buhovski-Tanny concernant l'invariant pb pour des recouvrements formés de petits ouverts. Afin de rendre cette thèse aussi accessible et auto-suffisante que possible, nous débutons par une introduction à la topologie symplectique. Des annexes recueillent les faits plus particuliers que nous utilisons tout au long de ce travail. / This thesis studies two problems of geometric and topological nature associated to the Poisson bracket on symplectic manifolds. The first problem concerns the notion of "symplectic submersion" that we introduce here and which generalizes the concept of symplectomorphism. A symplectic submersion turns out to be a Poisson morphism, namely a map between symplectic manifolds which preserves the Poisson bracket. Our interest in those maps stems from the fact that Gromov's nonsqueezing theorem is a statement about the minimal area possible for the images of the symplectic submersions (going from a symplectic ball to a symplectic plane) which are compositions of a symplectic embedding into the Euclidean symplectic space and of the standard projection onto the plane of conjugated variables (p_1, q_1). We investigate the inverse "representability" problem: we give necessary and sufficient conditions for a symplectic submersionas above to factorize in the previous way either through a symplectic embedding or through a symplectic immersion into Euclidean space. We show moreover that there exists a symplectic submersion which does not factorize in this way through an immersion, and also that there exists a symplectic submersion which does factorize in this way through an immersion, but not through an embedding. The second problem concerns Polterovich's Poisson bracket conjecture. Given a symplectic manifold (M, omega) and an open cover U of M, we can define the invariantpb(F) of a partition of unity F subordinated to U, which is a sort of norm on the pairwise Poisson brackets of the functions in F. Denoting e(U) the maximal Hofer displacement energy of a set in U, the conjecture asks whether there exists a positive constant C independent of U and F such that the product of pb(F) and e(U) is greater than C. This conjecture was proved recently by Buhovsky-Logunov-Tanny in the case of surfaces; based on earlier work of Buhovsky-Tanny , we also proved the conjecture for surfaces of genus one and above. We present our approach in the second chapter of this thesis. Using symplectic submersions, we generalize our methods in order to tackle the conjecture in higher dimensions; in particular, we obtain a new proof of a theorem of Polterovich and Buhovsky-Tanny about the pb invariant of covers made up of small open sets. In order to make this thesis as accessible and self-contained as possible, we first give an introduction to symplectic topology. The appendices also collect the more specialized facts we use throughout this work.

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)

Quantum systems with finite Hilbert space where position x and momentum p take values in Z(d) (integers modulo d) are considered. Symplectic tranformations S(2ξ,Z(p)) in ξ-partite finite quantum systems are studied and constructed explicitly. Examples of applying such simple method is given for the case of bi-partite and tri-partite systems. The quantum correlations between the sub-systems after applying these transformations are discussed and quantified using various methods. An extended phase-space x-p-X-P where X, P ε Z(d) are position increment and momentum increment, is introduced. In this phase space the extended Wigner and Weyl functions are defined and their marginal properties are studied. The fourth order interference in the extended phase space is studied and verified using the extended Wigner function. It is seen that for both pure and mixed states the fourth order interference can be obtained.

Nous nous intéressons dans cette thèse, à l'étude de certaines méthodes numériques de type krylov dans le cas symplectique, en utilisant la technique de blocs. Ces méthodes, contrairement aux méthodes classiques, permettent à la matrice réduite de conserver la structure Hamiltonienne ou anti-Hamiltonienne ou encore symplectique d'une matrice donnée. Parmi ces méthodes, nous nous sommes intéressés à la méthodes d'Arnoldi symplectique par bloc que nous appelons aussi bloc J-Arnoldi. Notre but essentiel est d’étudier cette méthode de façon théorique et numérique, sur la nouvelle structure du K-module libre ℝ²nx²s avec K = ℝ²sx²s où s ≪ n désigne la taille des blocs utilisés. Un deuxième objectif est de chercher une approximation de l'epérateur exp(A)V, nous étudions en particulier le cas où A est une matrice réelle Hamiltonnienne et anti-symétrique de taille 2n x 2n et V est une matrice rectangulaire ortho-symplectique de taille 2n x 2s sur le sous-espace de Krylov par blocs Km(A,V) = blockspan {V,AV,...,Am-1V}, en conservant la structure de la matrice V. Cette approximation permet de résoudre plusieurs problèmes issus des équations différentielles dépendants d'un paramètre (EDP) et des systèmes d'équations différentielles ordinaires (EDO). Nous présentons également une méthode de Lanczos symplectique par bloc, que nous nommons bloc J-Lanczos. Cette méthode permet de réduire une matrice structurée sous la forme J-tridiagonale par bloc. Nous proposons des algorithmes basés sur deux types de normalisation : la factorisation S R et la factorisation Rj R. Dans une dernière partie, nous proposons un algorithme qui généralise la méthode de Greville afin de déterminer la pseudo inverse de Moore-Penros bloc de lignes par bloc de lignes d'une matrice rectangulaire de manière itérative. Nous proposons un algorithme qui utilise la technique de bloc. Pour toutes ces méthodes, nous proposons des exemples numériques qui montrent l'efficacité de nos approches. / We study, in this thesis, some numerical block Krylov subspace methods. These methods preserve geometric properties of the reduced matrix (Hamiltonian or skew-Hamiltonian or symplectic). Among these methods, we interest on block symplectic Arnoldi, namely block J-Arnoldi algorithm. Our main goal is to study this method, theoretically and numerically, on using ℝ²nx²s as free module on (ℝ²sx²s, +, x) with s ≪ n the size of block. A second aim is to study the approximation of exp (A)V, where A is a real Hamiltonian and skew-symmetric matrix of size 2n x 2n and V a rectangular matrix of size 2n x 2s on block Krylov subspace Km (A, V) = blockspan {V, AV,...Am-1V}, that preserve the structure of the initial matrix. this approximation is required in many applications. For example, this approximation is important for solving systems of ordinary differential equations (ODEs) or time-dependant partial differential equations (PDEs). We also present a block symplectic structure preserving Lanczos method, namely block J-Lanczos algorithm. Our approach is based on a block J-tridiagonalization procedure of a structured matrix. We propose algorithms based on two normalization methods : the SR factorization and the Rj R factorization. In the last part, we proposea generalized algorithm of Greville method for iteratively computing the Moore-Penrose inverse of a rectangular real matrix. our purpose is to give a block version of Greville's method. All methods are completed by many numerical examples.

Le but de cette thèse est d'étudier les structures symplectiques dans la catégorie des variétés linéaires par morceaux (PL). La question centrale est de déterminer si toute variété symplectique lisse $(M,omega)$ peut être triangulée de manière symplectique, au sens où il existe une variété linéaire par morceaux $K$ et une triangulation $h :K -> M$ telle que $h^*omega$ est une forme symplectique constante par morceaux. Nous étudions d'abord un problème plus simple, qui consiste à trianguler les formes volumes lisses. Étant donnée une variété lisse $M$ munie d'une forme volume $Omega$, nous montrons qu'il existe une triangulation lisse $h :K -> M$ telle que $h^*Omega$ est une forme volume constante par morceaux. En particulier, les variétés symplectiques lisses de dimension 2 admettent donc des triangulations symplectiques. Étant donnée une variété symplectique fermée $(M,omega)$, nous montrons ensuite que pour certaines triangulations lisses $h :K -> M$, on peut, par une modification arbitrairement petite du complexe $K$, supposer que la forme $h^*omega$ est de rang maximal le long de tous les simplexes de $K$. Ce résultat permet d'approximer arbitrairement bien toute variété symplectique fermée par une variété symplectique PL. Nous nous intéressons finalement au cas d'une sous-variété symplectique $M$ d'un espace ambiant qui admet lui-même une triangulation symplectique. Nous montrons qu'il est possible de construire un cobordisme entre la sous-variété $M$ considérée et une approximation lisse par morceaux de celle-ci, triangulée par un complexe symplectique. / In this thesis, we study symplectic structures in a piecewise linear (PL) setting. The central question is to determine whether a smooth symplectic manifold can be triangulated symplectically, in the sense that there exists a triangulation $h :K -> M$ such that $h^*omega$ is a piecewise constant symplectic form on $K$. We first focus on a simpler related problem, and show that any smooth volume form $Omega$ on $M$ can be triangulated. This means that there always exists a triangulation $h :K -> M$ such that $h^*Omega$ is a piecewise constant volume form. In particular, symplectic surfaces admit symplectic triangulations. Given a closed symplectic manifold $(M,omega)$, we then prove that there exists triangulations $h :K -> M$ for which the piecewise smooth form $h^*omega$ has maximal rank along all the simplices of $K$. This result allows to approximate arbitrarily closely any closed symplectic manifold by a PL one. Finally, we investigate the case of a symplectic submanifold $M$ of an ambient space which is itself symplectically triangulated, and give the construction of a cobordism between $M$ and a piecewise smooth approximation of $M$, triangulated by a symplectic complex. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Este trabalho tem como objetivo a classificação simplética de germes de curvas parametrizadas e de estrelas lagrangianas por meio do método das restrições algébricas. Classificamos simpleticamente germes de curvas parametrizadas com semigrupos (4; 5; 6); (4; 5; 7) e (4; 5; 6; 7). Introduzimos um invariante para distinguir restrições algébricas a germes de curvas parametrizadas quase homogêneas: a parte de quase grau mínimo proporcional. Através do método das restrições algébricas, este invariante é capaz de distinguir diferentes órbitas de germes de curvas parametrizadas quase homogêneas sob a ação dos germes de simplectomorfismos. Classificamos estrelas lagrangianas duas a duas transversais com respeito ao grupo dos simplectomorfismos. / This work aims the symplectic classification of parametrized curve-germs and Lagrangian stars using the method of algebraic restrictions. We classify simplecticaly parametrized curve-germs with semigroups (4; 5; 6); (4; 5; 7) e (4; 5; 6; 7) We introduce an invariant for algebraic restrictions to quasi-homogeneous parametrized curve-germs: the proportional minimum quasi degree part. By the method of algebraic restrictions, this invariant is able to distinguish different orbits of parameterized quasi-homogeneous curve-germs under the action of symplectomorphisms. We classify Lagrangian stars two to two transversal with respect to the group of simplectomorphisms.

Symplectic geometry can be traced back to Lagrange and his work on celestial mechanics and has since then been a very active field in mathematics, partly because of the applications it offers but also because of the beauty of the objects it deals with. I this thesis we begin by the simplest fact of symplectic geometry. We give the definition of a symplectic space and of the symplectic group, Sp(n). A symplectic space is the data of an even-dimensional space and of a form which satisfies a number of properties. Having done this we give a definition of the Lagrangian Grassmannian Lag(n) which consists of all n-dimensional subspaces of the symplectic space on which the symplectic form vanishes. We carefully study the topology of these spaces and their universal coverings. It is of great interest to know how the elements of the Lagrangian Grassmannian intersect each other. A lot of efforts have therefore been made to construct intersection indices for elements of Lag(n). They have gone under many names but have had a sole purpose, namely to give us a way to determine how these elements intersect. We show how these elements are constructed and extend the definition to paths of elements of Lag(n) and Sp(n). We end this thesis by extending the definition of an index defined by Conley and Zehnder bu using the properties of the Leray index. Their index plays a significant role in the theory of periodic Hamiltonian orbit.

abstract: The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system, with a projective Hilbert space regarded as the phase space. This thesis extends the theory by including some aspects of the symplectic topology of the quantum phase space. It is shown that the quantum mechanical uncertainty principle is a special case of an inequality from J-holomorphic map theory, that is, J-holomorphic curves minimize the difference between the quantum covariance matrix determinant and a symplectic area. An immediate consequence is that a minimal determinant is a topological invariant, within a fixed homology class of the curve. Various choices of quantum operators are studied with reference to the implications of the J-holomorphic condition. The mean curvature vector field and Maslov class are calculated for a lagrangian torus of an integrable quantum system. The mean curvature one-form is simply related to the canonical connection which determines the geometric phases and polarization linear response. Adiabatic deformations of a quantum system are analyzed in terms of vector bundle classifying maps and related to the mean curvature flow of quantum states. The dielectric response function for a periodic solid is calculated to be the curvature of a connection on a vector bundle. / Dissertation/Thesis / Ph.D. Mathematics 2011

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)