Provamos o Teorema do Limite Central para transformações expansoras por pedaços em um intervalo e observáveis com variação limitada. Utilizamos a abordagem desenvolvida por R. Rousseau-Egele, como apresentada por A. Broise. O método da demonstração se baseia no estudo de pertubações do operador de transferência de Ruelle-Perron-Frobenius. Uma contribuição original é dada no último capítulo, onde provamos que, para transformações markovianas expansoras, todos os observáveis não constantes, contínuos e com variação limitada não são infinitamente cohomólogos à zero, generalizando um resultado de Bamón, Rivera-Letelier, Urzúa and Kiwi para observáveis lipschitzianos e transformações \'z POT. n\' . A demonstração se baseia na teoria dos operadores de Ruelle-Perron-Frobenius desenvolvida nos capítulos anteriores / We prove the Central Limit Theorem for piecewise expanding interval transformations and observables with bounded variation, using the approach of J.Rousseau-Egele as described by A. Broise. This approach makes use of pertubations of the so-called Ruelle-Perron-Frobenius transfer operator. An original contribution is given in the last chapter, where we prove that for Markovian expanding interval maps all observables which are non constant, continuous and have bounded variation are not infinitely cohomologous with zero, generalizing a result by Bamón, Rivera-Letelier, Urzúa and Kiwi for Lipschitzian observables and the transformations \'z POT. n\' . Our demosntration uses the theory of Ruelle-Perron-Frobenius operators developed in the previos chapters

A Cohomologia de Hochschild é um invariante associado a álgebras o qual pode nos fornecer propiedades homologicas das álgebras e suas categorias de módulos. Além disso tem aplicações em Geometria Algébrica e Teoria de Representações, entre outras áreas. Para álgebras A sobre um corpo, o i-ésimo grupo de cohomologia de Hochschild HH^i(A,M) de A, com coeficientes no bimódulo M, coincide com Ext^i_{A^e}(A,M). Logo, este pode ser calculado usando uma resolução projetiva da álgebra como A-bimódulo. Diferentes autores como Dieter Happel, Claude Cibils, Edward Green, David Anick, Michael Bardzell e Andrea Solotar desenvolveram ferramentas para a construção destas resoluções em casos específicos. Um resultado recente e muito importante é apresentado por Andrea Solotar e Sergio Chohuy, onde se mostra a construção de uma resolução projetiva de bimódulos para álgebras associativas generalizando o resultado para álgebras monomiais feito por Bardzell. Nesta dissertação pretendemos introduzir ao leitor no conceito de Cohomologia de Hochschild mostrando a importância da mesma mediante resultados conhecidos para álgebras de dimensão finita. Além disso, apresentamos os conceitos e resultados do trabalho de Chohuy e Solotar mencionado acima. No decorrer deste trabalho complementamos algumas demonstrações dos resultados enunciados com o fim de propiciar uma ferramenta para o melhor entendimento dos tópicos trabalhados aqui. / The Hochschild Cohomology is an invariant attached to associative algebras which may provide us some homological aspects of the algebras and its category of modules. Moreover, it has applications to Algebraic Geometry and Representation Theory, among others areas. For algebras A over a field the Hochschild cohomology group HH^i(A,M) of A with coeficients in a bimodule M coincides with Ext^i_{A^e}(A,M). So it can be computed using a projective resolution of the algebra, as a bimodule over itself. Therefore different authors like Dieter Happel, Claude Cibils, Edward Green, David Anick, Michael Bardzell, Sergio Chohuy and Andrea Solotar developed tools for the construction of these resolutions in particular cases. A recent and very important result was introduced by Andrea Solotar and Sergio Chohuy, where they show a construction of a projective bimodule resolution for associative algebras generalizing the result for monomial algebras made by Bardzell. In this dissertation we intend to introduce the reader in the cohomology Hochschild concept, showing its importance through known results for finite dimensional algebras. Besides, we exhibit the concepts and results of Chohuy and Solotar mentioned before. During this text, we complement some demonstrations with the purpose of giving a tool for the a better understanding.

La théorie des pavages apériodiques a connu des développements rapides depuis les années 1980, avec la découvertes d'alliages métalliques cristallisant dans une structure quasi-périodique.Dans cette thèse, on étudie particulièrement deux méthodes de construction de pavages : par coupe et projection, et par substitution. Deux angles d'approche sont développés : l'étude de la fonction de complexité, et l'étude métrique de l'espace de pavages.Dans une première partie, on calcule l'asymptotique de la fonction de complexité pour des pavages coupe et projection, généralisant ainsi des résultats connus en dynamiques symbolique pour la dimension 1. On montre que pour un pavage coupe et projection canonique N sur d sans période, la complexité croît (à des constantes près) comme n à la puissance a, où a est un entier compris entre d et N-d.Ensuite, on se base sur une construction de Pearson et Bellissard qui construisent un triplet spectral sur les ensembles de Cantor ultramétriques. On suit leur construction dans le cas d'ensembles de Cantor auto-similaires. Elle s'applique en particulier aux transversales d'espaces de pavages de substitution.Enfin, on fait le lien entre la distance usuelle sur l'enveloppe d'un pavage et la complexité de ce pavage. Les liens entre complexité et métrique permettent de donner une preuve directe du fait suivant : la complexité des pavages de substitution apériodiques de dimension d croît comme n à la puissance d.La question de liens entre la complexité et la topologie (et pas seulement avec la distance) reste ouverte. Nous apportons cependant des réponses partielles dans cette direction. / Since the 1980s, the theory of aperiodic tilings developed quickly, motivated by the discovery of metallic alloys which crystallize in an aperiodic structure. This highlighted the need for new models of crystals.Two models of aperiodic tilings are specifically studied in this dissertation. First, the cut-and-project method, then the inflation and substitution method. Two point of view are developed for the study of these objects: the study of the complexity function associated to a tiling, and the metric study of the associated tiling space.In a first part, the asymptotic behaviour of the complexity function for cut-and-project tilings is studied. The results stated here generalize formerly known results in the specific case of dimension 1. It is proved that for an (N,d) canonical projection tiling without periods, the complexity grows like n to the a, with a an integer greater or equal to d but lesser or equal to N-d.A second part is based on a construction by Pearson and Bellissard of a spectral triple for ultrametric Cantor sets. Their construction is applied to self-similar Cantor sets. It applies in particular to the transversal of substitution tiling spaces.In a last part, the links between the complexity function of a tiling and the usual distance on its associated tiling space are made explicit. These links can provide a direct and complete proof of the following fact: the complexity of an aperiodic d-dimensional substitution tiling grows asymptotically as n to the d, up to constants. These links between complexity and distance raises the question of links between complexity and topology. Partial answers are given in this direction.

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The main purpose of this thesis is to calculate the integral cohomology ring of the symmetric square of quaternionic projective space, which has been an open problem since computations with symmetric squares were first proposed in the 1930's. The geometry of this particular case forms an essential part of the thesis, and unexpected results concerning two universal Pin(4) bundles are also included. The cohomological computations involve a commutative ladder of long exact sequences, which arise by decomposing the symmetric square and the corresponding Borel space in compatible ways. The geometry and the cohomology of the configuration space of unordered pairs of distinct points in quaternionic projective space, and of the Thom space MPin(4), also feature, and seem to be of independent interest.

Orientador: Alice Kimie Miwa Libardi / Banca: João Peres Vieira / Banca: Gerson Petronilho / Resumo: O objetivo desta dissertação é apresentar uma abordagem cohomológica do Teorema de Cauchy e alguns resultados equivalentes a que um subconjunto aberto e conexo de C seja simplesmente conexo. Ressaltamos que um dos objetivos desta dissertação, inserida no Mestrado Profissional, Matemática Universitária, é estabelecer uma conexão entre as diversas áreas da Matemática, dando uma visão global da mesma, necessária ao professor universitário. Desta forma, o tema escolhido "Teorema de Cauchy"é um assunto visto na graduação, porém a abordagem usando grupos de cohomologia, números de voltas, espaços de recobrimento, feixes de germes de funções holomorfas, contribuem para o enriquecimento da formação da mestranda / Abstract: In this work we present a cohomological approach of the Cauchy's Theorem and also present several characterizations of simply connected domains of C / Mestre

Introduzimos o conceito de estrutura de Poisson não comutativa em álgebras associativas e mostra como este conceito se relaciona com o caso clássico, quando a álgebra em questão é a álgebra de funções em uma variedade de Poisson. Mostramos como quocientes simpléticos, não necessariamente suaves, fornecem exemplos de estruturas de Poisson não comutativas. / We introduce the concept of noncommutative Poisson structure on associative algebras and shows how this concept is related to the classical case, that is, the algebra under study is the algebra of functions on a Poisson manifold. We also show how symplectic quotients, not necessarily smooth, provides examples of noncommutative Poisson structures.

D'une part, á partir des propriétés de la cohomologie de Floer, invariant associé á une variété symplectique, je définis et étudie une notion d'hyperbolicité symplectique et une capacité symplectique la mesurant. D'autre part, pour une variété , on dispose des notions classiques d'hyperbolicités complexes, définies à partir des courbes pseudo-holomorphes. J'étudie donc les liens entre ces deux notions d'hyperbolicités quand une variété est munie de structures pseudo-complexe et symplectique compatibles. J'explique principalement comment la non-hyperbolicité symplectique implique l'existence de courbes pseudo-holomorphes, et donc ainsi la non-hyperbolicité complexe. Cette analyse me permet à la fois de mieux comprendre la cohomologie de Floer, et d'obtenir de nouveaux résultats sur l'hyperbolicité complexe. J'établis notamment des résultats de stabilité pour la non-hyperbolicité complexe par déformation de la structure pseudo-complexe dans l'ensemble des structures pseudo-complexes compatibles à une structure symplectique non-hyperbolique fixée, généralisant ainsi un théorème de Bangert énoncant ce même résultat dans le cas particulier du tore standard. Par ailleurs, j'aborde la question de l'hyperbolicité complexe des feuilletages: en exhibant un tenseur invariant associé au feuilletage, j'étudie l'existence de cylindres holomorphes feuilletés.

Finding new examples of compact simply connected spaces admitting a Riemannian metric of positive sectional curvature is a fundamental problem in differential geometry. Likewise, studying topological properties of families of manifolds is very interesting to topologists. The Eschenburg spaces combine both of those interests: they are positively curved Riemannian manifolds whose topological classification is known. There is a second family consisting of the Witten manifolds: they are the examples of compact simply connected spaces admitting Einstein metrics of positive Ricci curvature. Thirdly, there is a notion of generalized Witten manifold as well. Topologically, all three families share the same cohomology ring. This common ring structure motivates the definition of a manifold of type r, where r is the order of the fourth cohomology group. In 1991, M. Kreck and S. Stolz classified manifolds M of type r up to homeomorphism and dieomorphism using invariants s̄[subscript i](M) and s[subscript i](M), for i = 1, 2, 3. This gave rise to many new examples of nondieomorphic but homeomorphic manifolds. In this dissertation, new versions of the homeomorphism and dieomorphism classification of manifolds of type r are proven. In particular, we can replace s̄₁ and s̄₃ by the first Pontrjagin class and the self-linking number in the homeomorphism classification of spin manifolds of type r. As the formulas of the two latter invariants are in general much easier to compute, this simplifies the classification of these manifolds up to homeomorphism significantly. / Graduation date: 2012

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