Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Mostraremos que é possível aproximar um difeomorfismo simplético com derivada contínua por um difeomorfismo simplético, infinitamente diferenciáveis, sobre uma variedade simplética compacta. Além disso, provamos o Teorema de Darboux e Moser.

Le sujet de cette thèse est la question du nombre minimal d'orbites de Reeb distinctes sur une variété de contact qui est le bord d'une variété symplectique compacte.<p>L'homologie symplectique $S^1$-équivariante positive est un des outils principaux de cette thèse; elle est construite à partir d'orbites périodiques de champs de vecteurs hamiltoniens sur une variété symplectique<p>dont le bord est la variété de contact considérée.<p>Nous analysons la relation entre les différentes variantes d'homologie symplectique d'une variété symplectique exacte compacte (domaine de Liouville) et les orbites de Reeb de son bord.<p>Nous démontrons certaines propriétés de ces homologies.<p>Pour un domaine de Liouville plongé dans un autre, nous construisons un morphisme entre leurs homologies.<p>Nous étudions ensuite l'invariance de ces homologies par rapport au choix de la forme de contact sur le bord.<p>Nous utilisons l'homologie symplectique $S^1$-équivariante positive pour donner une nouvelle preuve d'un théorème de Ekeland et Lasry<p>sur le nombre minimal d'orbites de Reeb distinctes sur certaines hypersurfaces dans $R^{2n}$.<p>Nous indiquons comment étendre au cas de certaines hypersurfaces dans certains fibrés en droites complexes négatifs.<p>Nous donnons une caractérisation et une nouvelle façon de calculer l'indice de Conley-Zehnder généralisé, défini par Robbin et Salamon pour tout chemin de matrices symplectiques.<p>Ceci nous a mené à développer de nouvelles formes normales de matrices symplectiques.<p>/<p>This thesis deals with the question of the minimal number of distinct periodic Reeb orbits on a contact manifold which is the boundary of a compact symplectic manifold.<p>The positive $S^1$-equivariant symplectic homology is one of the main tools considered in this thesis.<p>It is built from periodic orbits of Hamiltonian vector fields in a symplectic manifold whose boundary is the given contact manifold.<p>Our first result describes the relation between the symplectic homologies of an exact compact symplectic manifold with contact type boundary (also called Liouville domain), and the periodic Reeb orbits on the boundary.<p>We then prove some properties of these homologies.<p>For a Liouville domain embedded into another one, we construct a morphism between their homologies.<p>We study the invariance of the homologies with respect to the choice of the contact form on the boundary.<p>We use the positive $S^1$-equivariant symplectic homology to give a new proof of a Theorem by Ekeland and Lasry about the minimal number of distinct periodic Reeb orbits on some hypersurfaces in $R^{2n}$.<p>We indicate how it extends to some hypersurfaces in some negative line bundles.<p>We also give a characterisation and a new way to compute the generalized Conley-Zehnder index defined by Robbin and Salamon for any path of symplectic matrices.<p>A tool for this is a new analysis of normal forms for symplectic matrices. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Résumé de la thèse :ce travail porte sur la notion d'espace symétrique symplectique extrinsèque. Ces espaces sont des espaces symétriques symplectiques dont la structure est induite par le plongement dans variété symplectique ambiante munie d'une connexion.<p><p>Par analogie à la théorie standard des espaces symétriques, nous démontrons un théorème d'équivalence entre les espaces symétriques symplectiques extrinsèques d'une variété qui est elle-même un espace symétrique symplectique.<p><p>La définition d'un espace symétrique symplectique extrinsèque fait intervenir l'existence d'affinités globales de la variété ambiante, les ``symétries extrinsèques', qui induisent la structure symétrique de la sous-variété ;ceci mène à poser une question du type :quelles sont les variétés possédant ``beaucoup' de ces affinités~? Une question précise ainsi qu'une réponse sont fournies dans un contexte où la variété ambiante est seulement supposée munie d'une structure<p>symplectique et d'une connexion symplectiques. Nous considérons également le cas où ces symétries commutent avec un champ $K$ d'endomorphismes symplectiques fixé de la variété, de carré $pmId$. Nous définissons une notion de courbure sectionnelle pour plans $K$-stables et montrons que les espaces à $K$-courbure sectionnelle constantes sont localement symétriques de type Ricci.<p><p>Par suite nous étudions les espaces symétriques symplectiques extrinsèques dans un espace vectoriel symplectique. Nous montrons par exemple qu'un tel espace, s'ils est de dimension deux, est forcément intrinsèquement plat (c.-à-d. à courbure intrinsèque nulle), mais que son image n'est pas forcément un plan affin de l'espace vectoriel ambiant. Nous décrivons en fait explicitement tous les espaces<p>symétriques symplectiques extrinsèques, dans un espace vectoriel, dont la courbure intrinsèque s'annule identiquement. Nous décrivons également une famille d'exemples d'espaces extrinsèques, dont nous montrons qu'elle fournit la totalité des espaces extrinsèques de codimension $2$, dans un espace vectoriel.<p><p>Enfin, nous décrivons quelques exemples d'espaces symétriques symplectiques extrinsèques qui sont totalement géodésiques, dans un espace de type Ricci particulier.<p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Quantum systems with finite Hilbert space are considered. Position and mo- mentum states and their relation through a Fourier transform, displacement in the position-momentum phase-space, and symplectic transformations are introduced and their properties are studied. Symplectic Sp(2l;Zp) trans- formations in l-partite finite system are explicit constructed. The general method is applied to bi-partite and tri-partite systems. The effect of these transformations on the correlations is discussed. Entanglement calculations between the subsystems in a bi-partite system and a tri-partite system are presented. The effect of measurements is also studied.

It is known that an arbitrary smooth, oriented four-manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken Lefschetz fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. The aim of this paper is to prove that these modifications are sufficient to obtain every broken Lefschetz fibration in a given homotopy class of smooth maps. One notable application is that adding an additional projection move generates all broken Lefschetz fibrations, regardless of homotopy class. The paper ends with further applications and open problems. / text

One of the most important tools to study the geometry of irreducible holomorphic symplectic manifolds is the monodromy group. The first part of this dissertation concerns the construction and studyof monodromy operators on irreducible holomorphic symplectic manifolds which are deformation equivalent to the 10-dimensional example constructed by O'Grady. The second part uses the knowledge of the monodromy group to compute the number of connected components of moduli spaces of bothmarked and polarised irreducible holomorphic symplectic manifolds which are deformationequivalent to generalised Kummer varieties.

In this thesis we study the self-Floer theory of a monotone Lagrangian submanifold $L$ of a closed symplectic manifold $X$ in the presence of various kinds of symmetry. First we consider the group $\mathrm{Symp}(X, L)$ of symplectomorphisms of $X$ preserving $L$ setwise, and extend its action on the Oh spectral sequence to coefficients of arbitrary characteristic, working over an enriched Novikov ring. This imposes constraints on the differentials in the spectral sequence which force them to vanish in certain situations. We then specialise to the case where $L$ is $K$-homogeneous for a compact Lie group $K$, meaning roughly that $X$ is Kaehler, $K$ acts on $X$ by holomorphic automorphisms, and $L$ is a Lagrangian orbit. By studying holomorphic discs with boundary on $L$ we compute the image of low codimension $K$-invariant subvarieties of $X$ under the length zero closed-open string map. This places restrictions on the self-Floer cohomology of $L$ which generalise and refine the Auroux-Kontsevich-Seidel criterion. These often result in the need to work over fields of specific positive characteristics in order to obtain non-zero cohomology. The disc analysis is then developed further, with the introduction of the notion of poles and a reflection mechanism for completing holomorphic discs into spheres. This theory is applied to two main families of examples. The first is the collection of four Platonic Lagrangians in quasihomogeneous threefolds of $\mathrm{SL}(2, \mathbb{C})$, starting with the Chiang Lagrangian in $\mathbb{CP}^3$. These were previously studied by Evans and Lekili, who computed the self-Floer cohomology of the latter. We simplify their argument, which is based on an explicit construction of the Biran-Cornea pearl complex, and deal with the remaining three cases. The second is a family of $\mathrm{PSU}(n)$-homogeneous Lagrangians in products of projective spaces. Here the presence of both discrete and continuous symmetries leads to some unusual properties: in particular we obtain non-displaceable monotone Lagrangians which are narrow in a strong sense. We also discuss related examples including applications of Perutz's symplectic Gysin sequence and quilt functors. The thesis concludes with a discussion of directions for further research and a collection of technical appendices.

Symplectic cohomology is an algebraic invariant of filled symplectic cobordisms that encodes dynamical information. In this thesis we define a modified symplectic cohomology theory, called action-completed symplectic cohomology, that exhibits quantitative behavior. We illustrate the non-trivial nature of this invariant by computing it for annulus subbundles of line bundles over complex projective space. The proof relies on understanding the symplectic cohomology of the complex fibers and the quantum cohomology of the projective base. We connect this result to mirror symmetry and prove a non-vanishing result in the presence of Lagrangian submanifolds with non-vanishing Floer homology. The proof uses Lagrangian quantum cohomology in conjunction with a closed-open map.

This thesis explores aspects of generalized geometry, a geometric framework introduced by Hitchin and Gualtieri in the early 2000s. In the first part, we introduce a new class of submanifolds in stable generalized complex manifolds, so-called Lagrangian branes with boundary. We establish a correspondence between stable generalized complex geometry and log symplectic geometry, which allows us to prove results on local neighbourhoods and small deformations of this new type of submanifold. We further investigate Lefschetz thimbles in stable generalized complex Lefschetz fibrations and show that Lagrangian branes with boundary arise in this context. Stable generalized complex geometry provides the simplest examples of generalized complex manifolds which are neither complex nor symplectic, but it is sufficiently similar to symplectic geometry for a multitude of symplectic results to generalize. Our results on Lefschetz thimbles in stable generalized complex geometry indicate that Lagrangian branes with boundary are part of a potential generalisation of the Wrapped Fukaya category to stable generalized complex manifolds. The work presented in this thesis should be seen as a first step towards the extension of Floer theory techniques to stable generalized complex geometry, which we hope to develop in future work. The second part of this thesis studies Dorfman brackets, a generalisation of the Courant- Dorfman bracket, within the framework of double vector bundles. We prove that every Dorfman bracket can be viewed as a restriction of the Courant-Dorfman bracket on the standard VB-Courant algebroid, which is in this sense universal. Dorfman brackets have previously not been considered in this context, but the results presented here are reminiscent of similar results on Lie and Dull algebroids: All three structures seem to fit into a more general duality between subspaces of sections of the standard VB-Courant algebroid and brackets on vector bundles of the form T M ⊕ E ∗ , E → M a vector bundle. We establish a correspondence between certain properties of the brackets on one, and the subspaces on the other side.

A flag of a finite dimensional vector space V is a nested sequence of subspaces of V . The symplectic group of V acts on the set of flags of V . We classify the orbits of this action by defining the incidence matrix of a flag of V and show- ing that two flags are in the same orbit precisely when they have the same incidence matrix. We give a formula for the number of orbits of a certain type and discuss how to list the incidence matrices of all orbits. In the case in which V is a vector space over a finite field, we discuss the permutation representations of the symplectic group of V corresponding to these orbits. For the case in which V = (F_q)^4 , we compute the conjugacy classes of the sym- plectic group of V and the values of the characters of the previously discussed permutation representations. / Mathematics