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Cohomologies on sympletic quotients of locally Euclidean Frolicher spacesTshilombo, Mukinayi Hermenegilde 08 1900 (has links)
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)
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Difusões em variedades de poisson / Poisson manifolds diffusionsCosta, Paulo Henrique Pereira da, 1983 08 July 2009 (has links)
Orientador: Paulo Regis Caron Ruffino / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T23:01:19Z (GMT). No. of bitstreams: 1
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Previous issue date: 2009 / Resumo: O objetivo desse trabalho é estudar as equações de Hamilton no contexto estocástico. Sendo necessário para tal um pouco de conhecimento a cerca dos seguintes assuntos: cálculo estocástico, geometria de segunda ordem, estruturas simpléticas e de Poisson. Abordamos importantes resultados, dentre eles o teorema de Darboux (coordenadas locais) em variedades simpléticas, teorema de Lie-Weinstein que de certa forma generaliza o teorema de Darboux em variedades de Poisson. Veremos que apesar de o ambiente natural para se estudar sistemas hamiltonianos ser variedades simpléticas, no caso estocástico esses sistemas se adaptam bem em variedades de Poisson. Além disso, para atingir a nossa meta, estudaremos equações diferenciais estocásticas em variedades de dimensão finita usando o operador de Stratonovich / Abstract: This dissertation deals with transfering Hamilton's equations in stochastic context. This requires some knowledge about the following: stochastic calculus, second order geometry and Poisson and simplectic structures. Important results that will be discussed in this theory are Darboux's theorem (local coordinates) for simplectic manifolds, and Lie-Weintein's theorem that is in a certain way of Darboux's theorem on Poisson manifolds. We will see that although the natural environment for studying hamiltonian systems is symplectic manifolds, if we have a Poisson structure we will still be able to study them. Moreover, to achieve our goal, we will study stochastic differential equations on finite dimensional manifolds using the Stratonovich operator / Mestrado / Geometria Estocastica / Mestre em Matemática
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Classificação simplética de germes de curvas parametrizadas e estrelas lagrangianas / Symplectic classification of parameterized curve-germs and Lagrangian starsFausto Assunção de Brito Lira 27 March 2015 (has links)
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.
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A dualidade Maxwell-Proca-Chern-Simons via Formalismo Simplético de ImersãoXavier, Luciana Miranda Vieira 27 February 2009 (has links)
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Previous issue date: 2009-02-27 / FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais / Nesta tese, revisa-se os principais métodos de quantização de sistemas vinculados a partir das técnicas Hamiltoniana de Dirac e Lagrangeana de Faddev-Jackiw ( sem vínculos) e sua extenção a de Barcelos Neto- Wotzasek (com vínculos), estes denominados simplesmente por Formalismo Simplético (FS). Em vista da correspondência entre os formalismos, eles serão aplicados ao Modelo de Skyrme SU(2) e ao Eletromagnetimo de Maxwell. Apresenta-se uma técnica contemporânea, que mergulha uma teoria de segunda classe em uma dual com invariância de calibre, a saber, o Formalismo Simplético de Imersão (FSI). Esse método baseia-se no FS e estende-se o espaço de configuração por meio das variáveis de Wess-Zumino. Para ilustrar esse FSI, constroi-se a eletrodinâmica de Maxwell como uma teoria de calibre, na qual as divergências clássicas não estejam presentes. Uma generalização relativística é a eletrodinâmica de Proca e de Chern-Simons, que consideram a possibilidade de existência de um fóton massivo e de um campo com alcance finito. A descrição dual reproduz o mesmo resultado encontrado na literatura através de outros métodos. Apesar da arbitrariedade dos geradores da simetria de calibre, os modos-zeros, mostram uma família de representações dinâmicas duais para o sistema em questão. / In this thesis, it will be revised the main quantization methods of constrained systems using the Dirac Hamiltonian method and Faddev-Jackiw Lagrangian techniques (without constrained), and its extension to the Barcelos Neto- Wotzasek Lagrangian method (with constrained), these known as Symplectic Formalism. Because of the correspondence among the formalisms, they will be applied of the Skyrme SU(2) model and Electromagnetism of Maxwell. It will be presented a contemporary technique that it embed a second class theory in a dual with gauge invariance, the Embedding Symplectic Formalism . This method is based on the Symplectic Formalism, it is extended the configuration space through Wess-Zumino variables. In order to illustrate this Embedding Symplectic Formalism, the Maxwell electrodynamics is built as a gauge theory, without the classic differences. A relativistic generalization is the Proca and Chern-Simons electrodynamics that consider the possibility of existence of a massive photon and a field with finite reach. The dual description reproduce the identic result reported in the literature using other methods. Although, the arbitrariness of the gauge symmetry generator, zero-mode, it reveals a family of dynamic dual representations to this system.
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Mapas momento em teoria de calibre / Moment maps in gauge theoryBranco, Lucas Magalhães Pereira Castello, 1988- 22 August 2018 (has links)
Orientador: Marcos Benevenuto Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-22T22:29:57Z (GMT). No. of bitstreams: 1
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Previous issue date: 2013 / Resumo: Neste trabalho os aspectos básicos da teoria de calibre são abordados, incluindo as noções de conexão e curvatura em fibrados principais e vetoriais, considerações sobre o grupo de transformações de calibre e o espaço de moduli de soluções para a equação anti-auto-dual em dimensão quatro (o espaço de moduli de instantons). Posteriormente, mapas momento e redução são introduzidos. Primeiramente, no contexto clássico de geometria simplética e depois no contexto de geometria hyperkähler. Por fim, são apresentadas aplicações da teoria de mapas momento e redução em teoria de calibre. As equações ADHM são introduzidas e mostra se que estas podem ser dadas como o conjunto de zeros de um mapa momento hyperkähler. Além disso, considerações são feitas acerca da construção ADHM de instantons, que relaciona soluções dessas equações com as soluções da equação de anti-auto-dualidade. O espaço de moduli de conexões planas é também abordado. Neste caso, a curvatura é vista como um mapa momento e os cálculos podem ser generalizados para o espaço de moduli de conexões planas sobre variedades Kähler de dimensões mais altas e para o espaço de moduli de instantons sobre variedades hyperkähler de dimensão quatro / Abstract: In this work it is developed the basic concepts of gauge theory, including the notions of connections and curvature on principal bundles and vector bundles, considerations on the group of gauge transformations and the moduli space of anti-self-dual connections in dimension four (the instanton moduli space). After, moment maps and reduction are introduced. First in the classical context of symplectic geometry, then in hyperkähler geometry. At last, applications to the theory of moment maps and reduction in gauge theory are given. The ADHM equations are introduced and it is shown that solutions to these equations can be given by the zeros of a hyperkähler moment map. Furthermore, the ADHM construction, that relates the ADHM equations to instanton solutions, is discussed. The moduli space of flat connections over a Riemann surface is also treated. In this case, the curvature is seen as a moment map and the calculations can be generalized to flat connections over higher-dimensional Kähler manifolds and to the instanton moduli space over four dimensional hyperkähler manifolds / Mestrado / Matematica / Mestre em Matemática
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Effect of Legendrian surgery and an exact sequence for Legendrian links / Effet de chirurgies Legendriennes et une suite exacte de entrelacements LegendriensEslami Rad, Anahita 31 August 2012 (has links)
This thesis is devoted to the study of the effect of Legendrian surgery on contact manifolds. In particular, we study the effect of this surgery on the Reeb dynamics of the contact manifold on which we perform such a surgery along Legendrian links. We obtain an exact sequence of cyclic Legendrian homology for the Legendrian links. Then we present the applications in 3-dimension and higher dimensions. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished
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Géométrie complexe globale et infinitésimale de l'espace des twisteurs d'une variété hyperkählérienne / Global and infinitesimal complex geometry of twistor spaces of hyperkähler manifoldsPillet, Basile 13 June 2017 (has links)
L'objet de cette thèse est la construction d'objets géométriques sur une variété C paramétrant des courbes rationnelles dans l'espace des twisteurs d'une variété hyperkählérienne. On établira une correspondance entre la géométrie complexe de l'espace des twisteurs et des propriétés différentielles sur C (opérateurs différentiels et courbure de la structure riemanienne complexe héritée de la variété hyperkählérienne). Les premiers chapitres précisent le cadre et les résultats connus. Dans les chapitres 4, 5 et 6 on établit une équivalence de catégories entre fibrés triviaux en restriction à chaque droite de l'espace des twisteurs et les fibrés à connexion sur C satisfaisant une condition de courbure. Le chapitre 7 prolonge cette correspondance sur le plan cohomologique tandis que le chapitre 8 en fait l'étude infinitésimale en reliant la courbure de la connexion avec les épaississements infinitésimaux des fibrés le long des droites. / The purpose of this thesis is to construct geometric objects on a manifold C parametrizing rational curves in the twistor space of a hyperkähler manifold. We shall establish a correspondence between the complex geometry of the twistor space and some differential properties of C (differential operators and curvature of a complex riemannian structure inherited from the base hyperkähler manifold). The first chapters gather some classical results of the theory of hyperkähler manifolds and their twistor spaces. In the chapters 4, 5 and 6, we construct an equivalence of categories between bundles on the twistor space which are trivial on each line and bundles with a connexion of C satisfying certain curvature conditions. The chapter 7 extends this correspondence on the cohomological level whereas the chapter 8 explores its infinitesimal version ; it links curvature of the connexion with thickening (in the sense of LeBrun) of the bundle along the lines.
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Dynamique hors équilibre des théories classiques des champs et des modèles de spin d’Ising / Out-of-equilibrium dynamics in classical field theories and Ising spin modelsRicateau, Hugo 29 September 2017 (has links)
Cette thèse est constituée de deux parties indépendantes. Dans le premier chapitre, nous introduisons une méthode numérique permettant d'intégrer des équations aux dérivées partielles représentant la dynamique Hamiltonienne de théories des champs. Cette méthode est un intégrateur multi-symplectique qui préserve localement le tenseur énergie-impulsion sur de très longues périodes de temps et avec précision. Son principal avantage est d'être extrêmement simple tout en restant bien définie localement. Nous la mettons à l'épreuve sur le cas particulier du modèle phi^4 en 1+1 dimensions; nous expliquons également comment l'implémenter en dimensions supérieures. De plus, nous faisons une présentation géométrique de la structure multi-symplectique et nous introduisons une construction permettant de résoudre le problème de dégénérescence pouvant l'affecter.Le second chapitre traite d'aspects hors équilibre dans les systèmes statistiques: nous nous intéressons en particulier à la question de l'impact d'un taux de refroidissement fini lors d'une trempe à travers une transition de phase du second ordre. Pour décrire plus fidèlement le régime hors équilibre qui se produit avant la transition de phase, nous étendons le mécanisme dit de Kibble-Zurek. Nous décrivons comment la taille caractéristique des objets géométriques présents dans le système dépend du temps et du taux de refroidissement; ceci, avant et une fois le point critique atteint. Ces prédictions théoriques sont mises à l'épreuve sur l'exemple du modèle d'Ising ferromagnétique. Nous décrivons également les propriétés géométriques des domaines qui apparaissent dans le système au cours de la dynamique de refroidissement. / This thesis is made up of two independent parts. In the first chapter, we introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent precision over very long periods. Its major advantage is that it is extremely simple (it is basically a centered box scheme) while remaining locally well defined. We put it to the test in the case of the non-linear wave equation (with quartic potential) in one spatial dimension, and we explain how to implement it in higher dimensions. A formal geometric presentation of the multi-symplectic structure is also given as well as a technical trick allowing to solve the degeneracy problem that potentially accompanies the multi-symplectic structure. In the second chapter, we address the issue of the influence of a finite cooling rate while performing a quench across a second order phase transition. We extend the Kibble-Zurek mechanism to describe in a more faithfully way the out-of-equilibrium regime of the dynamics before crossing the transition. We describe the time and cooling rate dependence of the typical growing size of the geometric objects, before and when reaching the critical point. These theoretical predictions are demonstrated through a numerical study of the emblematic kinetic ferromagnetic Ising model on the square lattice. A description of the geometric properties of the domains present in the system in the course of the annealing and when reaching the transition is also given.
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Sur les relations entre la topologie de contact et la dynamique de champs de Reeb / On the relationship between contact topology and the dynamics of Reeb flowsAlves, Marcelo Ribeiro de Resende 19 November 2015 (has links)
L'objectif de cette thèse est d'investiguer les relations entre les propriétés topologiques d'une variété de contact et la dynamique des flots de Reeb dans la variété de contact en question. Dans la première partie de la thèse, nous établissons une relation entre la croissance de l’homologie de contact cylindrique d'une variété de contact et l'entropie topologique des flots de Reeb dans cette variété de contact. Nous utilisons ce résultat dans les chapitres 8 et 9 pour montrer l'existence d'un grand nombre des nouvelles variétés de contact de dimension 3 dans lesquelles tous les flots de Reeb ont entropie topologique positive. Dans le chapitre 10, nous prouvons un résultat obtenu en collaboration avec Chris Wendl qui donne une obstruction dynamique pour qu'une variété de contact de dimension 3 soit planaire. Cette obstruction est utilisée pour montrer que, si une variété de contact de dimension 3 possède un flot de Reeb qui est uniformément hyperbolique (Anosov) avec variétés invariantes traversalement orientables, alors cette variété de contact n'est pas planaire. Dans le chapitre 11, nous étudions l'entropie topologique des flots de Reeb dans les fibrés unitaires des surfaces de genre plus grand que 1. Nous montrons que la restriction de chaque flot de Reeb en au ensemble limite de presque toute fibre unitaire a une entropie topologique positive. / In this thesis we study the relations between the contact topological properties of contact manifolds and the dynamics of Reeb flows. On the first part of the thesis, we establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We build on this to show in Chapter 6 that if a contact manifold M admits a hypertight contact form A for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on M has positive topological entropy. Using this result, we exhibit in Chapter 8 and 9 numerous new examples of contact 3-manifolds on which every Reeb flow has positive topological entropy. On Chapter 10 we present a joint result with Chris Wendl that gives a dynamical obstruction for contact 3-manifold to be planar. We then use the obstruction to show that a contact 3-manifold that possesses a Reeb flow that is a transversely orientable Anosov flow, cannot be planar. On Chapter 11 we study the topological entropy for Reeb flows on spherizations. The result we obtain is a refinement of a result of Macarini and Schlenk, that states that every Reeb flow on the unit tangent bundle U of a high genus surface S has positive topological entropy. We show that for any Reeb flow on U, the omega-limit of almost every Legendrian fiber is a compact invariant set on which the dynamics has positive topological entropy.
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A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencilsBenner, P., Mehrmann, V., Xu, H. 30 October 1998 (has links)
A new method is presented for the numerical computation of the generalized eigen- values of real Hamiltonian or symplectic pencils and matrices. The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order sqrt(epsilon), where epsilon is the machine precision, the new method computes the eigenvalues to full possible accuracy.
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