Made available in DSpace on 2015-03-26T13:45:31Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1613593 bytes, checksum: 9f102a91f9dec62a3656d30b4f7a490c (MD5) Previous issue date: 2010-02-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This study deals with the surface of the compact quotient M2=G where the surface M2 is either the Euclidean plane or the plane spherical or the hyperbolic plane, G is a group of isometries of their surfaces, and this group is generated by matching of edges of polygons. The Poincaré theorem that provides a method of finding the group of isometries G the functions that the pair of edges of the polygons involved. By using this theorem we construct two new pairings of generalized edges (Chapter 4) associated with the tessellations {12η 8,4} e {12μ 12,4}, respectively. These tessellations provide packing of spheres whose packing density is very close to the maximum 3/π. Such pairings are the starting point for finding codes with optimal transmission rates for Multiple-Input Multiple-Output (MIMO). / Este estudo aborda a construção de superfícies compactas pelo quociente M2/G onde a superfície M2 ou é o plano euclidiano, ou é o plano esférico, ou é o plano hiperbólico, G é um grupo de isometrias das respectivas superfícies e esse grupo é gerado pelos emparelhamentos de arestas dos polígonos. O Teorema de Poincaré fornece um método de encontrar o grupo de isometrias G que consiste das funções de emparelhamento de arestas dos polígonos associados. Mediante o uso deste teorema nós construímos dois novos emparelhamentos de arestas generalizados (Capítulo 4), associados as tesselações {12η 8,4} e {12μ 12,4}, respectivamente. Estas tesselações fornecem empacotamento de esferas cuja densidade de empacotamento é bem próxima do valor máximo 3/π. Tais emparelhamentos são o ponto de partida para a busca de códigos com ótimas taxas de transmissão para canais de múltiplas entradas e múltiplas e saídas (MIMO).

Made available in DSpace on 2015-03-26T13:45:33Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1007963 bytes, checksum: 8fb51039076c92104d50598359cf19d8 (MD5) Previous issue date: 2011-02-24 / This work has as main objective the study of side-pairing patterns for hyperbolic polygons with 12g−6 edges and angles 2π/3 generated by trivalent graphs, in the case when the quotient of the hyperbolic plane by a Fuchsian group Γ (generated by the side-pairing of the polygon), H2/Γ , is a closed surface of genus g, g ≥ 2. So we did a study in case of g = 2, based on [10] and for the case of g = 3, based on [17]. In this work, we deduce two ways to get closed paths in the trivalent graphs cited in [10] and [17] and we contribute with exemples and results for cases of g > 3. Moreover, we find generalizations for some of these side-pairing patterns. / Este trabalho tem como objetivo principal o estudo de emparelhamentos de arestas para polígonos hiperbólicos com 12g − 6 arestas e ângulos iguais a 2π/3 gerados por meio de grafos trivalentes, no caso em que o quociente do plano hiperbólico por um grupo Fuchsiano Γ (gerado pelo emparelhamento do polígono), H2/Γ , é uma superfície fechada de gênero g, g ≥ 2. Assim, fizemos um estudo para o caso de g = 2 baseado em [10] e para o caso de g = 3, baseado em [17]. Neste trabalho, nós deduzimos duas formas de obter os caminhos fechados nos grafos trivalentes citados em [10] e [17] e contribuímos com exemplos e resultados para casos em que g > 3. Além disso, encontramos generalizações para alguns desses emparelhamentos de arestas.

In this dissertation we study lattices in quotient spaces. The basic quotient spaces are: (1) n-dimensional euclidean tori, obtained from quotient of Rn by discrete groups of isometries ge- nerated by linearly independent translations and (2) hyperbolic °at g-tori (tori of genus g ¸ 2), obtained from quotient of hyperbolic plane by fuchsian groups. In the euclidean environment, the considered lattices are provided of the additive group Z2; while in the hyperbolic case the studied lattices are the geometrically uniform and the cyclic ones. / Neste trabalho estudamos reticulados em espaços quocientes. Os espaços quocientes considerados foram: (1) toros euclidianos n-dimensionais, obtidos pelo quociente de Rn por grupos discretos de isometrias gerados por translações linearmente independentes e (2) g-toros planos hiperbólicos (g ¸ 2) ; obtidos pelo quociente do plano hiperbólico por grupos fuchsianos. No caso euclidiano, os reticulados considerados foram provenientes de Z2; enquanto que no caso hiperbólico os reticulados estudados foram os geometricamente uniformes e os cíclicos. / Mestre em Matemática

Submitted by (lucia.rodrigues@ufrpe.br) on 2017-03-28T14:00:56Z No. of bitstreams: 1 Carlos Bino de Souza.pdf: 2371603 bytes, checksum: d2f0bb2e430fc899161fe573fbae4e50 (MD5) / Made available in DSpace on 2017-03-28T14:00:56Z (GMT). No. of bitstreams: 1 Carlos Bino de Souza.pdf: 2371603 bytes, checksum: d2f0bb2e430fc899161fe573fbae4e50 (MD5) Previous issue date: 2015-08-26 / Euclid wrote a book in 13 volumes called Elements where systematized all the mathematical knowledge of his time. In this work, the 5 postulates of Euclidean geometry were presented. For several years, the 5th Postulate was frequently asked, this inquiries it was discovered that there are several other possible geometries, including hyperbolic geometry. Beltrimi proved that hyperbolic geometry is consistent if Euclidean geometry is consistent. Hilbert showed that Euclidean geometry is consistent if the arithmetic is consistent and presented an axiomatic system that capped the gaps in Euclid’s axiomatic system. Poincaré created a model, called the Poincaré disk, to represent the plan of hyperbolic geometry. The objective of this work is to show that the Poincaré disk model is consistent with reference Axioms Hilbert, replacing only the Axioms of Parallel to "On a point outside a line passes through the two parallel straight lines given", by constructions of Euclidean geometry. / Euclides escreveu uma obra em 13 volumes chamada de Elementos onde sistematizava todo o conhecimento matemático do seu tempo. Nesta obra, foram apresentados os 5 postulados da Geometria Euclidiana. Durante vários anos, o 5o Postulado foi muito questionado, desses questionamentos descobriu-se a existência de várias outras Geometrias possíveis, entre elas a Geometria Hiperbólica. Beltrimi provou que a Geometria Hiperbólica é consistente se a Geometria Euclidiana é consistente. Hilbert mostrou que a Geometria Euclidiana é consistente se a Aritmética é consistente e apresentou um sistema axiomático que preencheu as lacunas do sistema axiomático de Euclides. Poincaré criou um Modelo, chamado de Disco de Poincaré, para representar o plano da Geometria Hiperbólica. O objetivo deste trabalho é mostrar que o Modelo de Disco de poincaré é consistente, tomando como referência os Axiomas de Hilbert, substituindo apenas os Axiomas das Paralelas para "Por um ponto fora de uma reta passam duas retas paralelas à reta dada", através de construções da Geometria Euclidiana.

Orientadores: Reginaldo Palazzo Júnior, Cátia Regina de Oliveira Quilles Queiroz / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-25T14:53:45Z (GMT). No. of bitstreams: 1 Benedito_CintyaWinkdeOliveira_D.pdf: 1485856 bytes, checksum: 50adbb3cffa1343c4a0cd9b3d7586173 (MD5) Previous issue date: 2014 / Resumo: Na busca por novos sistemas de comunicações muitos trabalhos têm sido realizados com o objetivo de obter constelações de sinais e códigos geometricamente uniformes no plano hiperbólico. Neste contexto, nossa proposta é identificar uma estrutura algébrica e geométrica para que códigos e reticulados possam ser construídos neste espaço. O problema central deste trabalho consiste em construir grupos fuchsianos provenientes de tesselações hiperbólicas regulares {p,q} utilizando diversos tipos de emparelhamentos e identificá-los com álgebras e ordens dos quatérnios, definindo-os assim como aritmético. Desta forma, propomos um algoritmo para construir grupos fuchsianos aritméticos provenientes de tesselações hiperbólicas regulares {p,q} cujo polígono hiperbólico regular gera uma superfície orientada de gênero maior ou igual a dois. Para isso, fornecemos uma condição necessária para que estes grupos possam ser obtidos, esta condição será denominada condição de Fermat devido a sua identificação com os números de Fermat. Através da construção destes grupos, mostramos que existe um isomorfismo entre dois grupos fuchsianos aritméticos provenientes de uma tesselação {p,q} a partir de emparelhamentos diferentes. Além disso, descrevemos alguns dos corpos de números que utilizamos para construir grupos fuchsianos aritméticos, como subcorpos maximais reais de corpos ciclotômicos, a fim de propor uma relação entre os reticulados hiperbólicos e os reticulados euclidianos. Reticulados hiperbólicos completos obtidos através da identificação de grupos fuchsianos com ordens maximais dos quatérnios também são apresentados. Desta forma, obtemos um rotulamento completo dos pontos da constelação de sinal associada / Abstract: In the search for new communications systems many studies have been conducted with the goal of obtaining signal constellations and geometrically uniform codes in the hyperbolic plane. In this context, our proposal is to identify an algebraic and geometric structures for constructing codes and lattices in this space. The central problem of this work is to construct fuchsian groups derived from hyperbolic tessellations {p,q} using different edge-pairings sets and identify them with quaternion algebras and quaternion orders, by setting it as arithmetic. We also propose an algorithm to construct arithmetic fuchsian groups from a tessellation {p,q} whose regular hyperbolic polygon generates an oriented and compact surface with genus greater or equal than 2. For that we provide a necessary condition for these groups to be obtained, this necessary condition is called Fermat condition due to its identification with the Fermat numbers. By the construction of these groups, it is also shown an isomorphism between two arithmetic fuchsian groups derived from a tessellation {p,q} via different edge-pairings sets. Furthermore, we will describe some of the number fields that we use to construct arithmetic fuchsian groups as maximal real subfields of cyclotomic fields in order to propose a relationship between hyperbolic lattices and euclidean lattices. Complete hyperbolic lattices obtained by identifying fuchsian groups with maximal quaternion orders will also be presented. In this way we have a complete labeling of the points of the corresponding signal constellation / Doutorado / Telecomunicações e Telemática / Doutora em Engenharia Elétrica

Gromov’s compactness theorem for metric spaces, a compactness theorem for the space of compact metric spaces equipped with the Gromov-Hausdorff distance, is a theorem with many applications. In this thesis, we give a generalisation of this landmark result, more precisely, we give a compactness theorem for the space of distance measure spaces equipped with the generalised Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple (X, d,µ), where (X, d) forms a distance space (a generalisation of a metric space where, we allow the distance between two points to be infinity) and µ is a finite Borel measure. Using this result we prove that the Deligne-Mumford compactification is the completion of the moduli space of Riemann surfaces under the generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov compactness theorem for J-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics. While Gromov compactness theorem for J-holomorphic curves in symplectic manifolds, is an important tool in symplectic topology, its applicability is limited by the lack of general methods to construct pseudo-holomorphic curves. One hopes that considering a more general class of objects in place of pseudo-holomorphic curves will be useful. Generalising the domain of pseudo-holomorphic curves from Riemann surfaces to Riemann surface laminations is a natural choice. Theorems such as the uniformisation theorem for surface laminations by Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and topological classification of “almost all" leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations, as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation, we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this thesis is: which fat graphs are systolic graphs for some surface -we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs. A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges. We call this combinatorial admissibility. Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long enough sides, in terms of a lower bound on the angles, are close to geodesics. Using the above result, it is easy to see that a subgraph of an admissible graph is admissible. Hence it suffices to characterize minimal non-admissible fat graphs. Another major result of this thesis is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

Le problème de la limite continue de la gravitation quantique à boucle est encore ouvert. En effet, la dynamique précise n’est pas connue et nous ne disposons pas des outils nécessaires à l’étude de cette limite le cas échéant. Dans cette thèse, nous étudions quelques méthodes de coarse-graining (étude à gros grains) qui devraient contribuer à cette entreprise. Nous nous concentrons sur deux aspects du flot: la détermination d’observables naturelles à grandes échelles d’un côté et la manière de s’abstraire du problème de la dynamique à graphe variable en la projetant sur des graphes fixes de l'autre.Pour déterminer les observables aux grandes distances, nous étudions le cas des tétraèdres hyperboliques et leur description naturelle dans un langage proche de celui de la gravitation quantique à boucle. Les holonomies de surface en particulier jouent un rôle important. Cela dégage la structure des double spin networks constitués d'un graphe et de son dual, structure qui semble aussi apparaître dans les travaux de Freidel et al. Pour résoudre le problème des graphes variables, nous considérons et définissons les loopy spin networks. Ils encodent par des boucles la courbure locale d'un vertex effectif et permettent ainsi de décrire différents graphes en les masquant via le processus de coarse-graining. De plus, leur définition donne un procédé naturel systématique de coarse-graining pour passer d'une échelle à une autre.Ensemble, ces deux principaux résultats posent le fondement d'un programme de coarse-graining pour les théories invariantes sous difféomorphismes. / The continuum limit of loop quantum gravity is still an open problem. Indeed, no proper dynamics in known to start with and we still lack the mathematical tools to study its would-be continuum limit. In the present PhD dissertation, we will investigate some coarse-graining methods that should become helpful in this enterprise. We concentrate on two aspects of the theory's coarse-graining: finding natural large scale observables on one hand and studying how the dynamics of varying graphs could be cast onto fixed graphs on the other hand.To determine large scale observables, we study the case of hyperbolic tetrahedra and their natural description in a language close to loop quantum gravity. The surface holonomies in particular play an important role. This highlights the structure of double spin networks, which consist in a graph and its dual, which seems to also appear in works from Freidel et al. To solve the problem of varying graphs, we consider and define loopy spin networks. They encode the local curvature with loops around an effective vertex and allow to describe different graphs by hidding them in a coarse-graining process. Moreover, their definition gives a natural procedure for coarse-graining allowing to relate different scales.Together, these two results constitute the foundation of a coarse-graining programme for diffeomorphism invariant theories.

Goldman [Gol86] introduced a Lie algebra structure on the free vector space generated by the free homotopy classes of oriented closed curves in any orientable surface F . This Lie bracket is known as the Goldman bracket and the Lie algebra is known as the Goldman Lie algebra. In this dissertation, we compute the center of the Goldman Lie algebra for any hyperbolic surface of finite type. We use hyperbolic geometry and geometric group theory to prove our theorems. We show that for any hyperbolic surface of finite type, the center of the Goldman Lie algebra is generated by closed curves which are either homotopically trivial or homotopic to boundary components or punctures. We use these results to identify the quotient of the Goldman Lie algebra of a non-closed surface by its center as a sub-algebra of the first Hochschild cohomology of the fundamental group. Using hyperbolic geometry, we prove a special case of a theorem of Chas [Cha10], namely, the geometric intersection number between two simple closed geodesics is the same as the number of terms (counted with multiplicity) in the Goldman bracket between them. We also construct infinitely many pairs of length equivalent curves in any hyperbolic surface F of finite type. Our construction shows that given a self- intersecting geodesic x of F and any self-intersection point P of x, we get a sequence of such pairs.

Dans cette thèse, on s'intéresse à la construction et à la déformation de structures CR-sphériques sur des variétés de dimension 3. Pour le faire, on étudie en détail l'espace hyperbolique complexe, son groupe d'isométries et des objets géométriques liés à cet espace. On montre un théorème de chirurgie qui permet de construire des structures CR-sphériques sur des chirurgies de Dehn d'une variété à pointe portant une structure CR-sphérique : il s'applique aux structures de Deraux-Falbel sur le complémentaire du noeud de huit et à celles de Schwartz et de Parker-Will sur le complémentaire de l'entrelacs de Whitehead. On définit aussi les variétés de caractères de groupes de type fini pour les formes réelles de SL(n,C) comme des sous-ensembles de la variété des caractères SL(n,C) fixes par des involutions anti-holomorphes. Ces variétés de caractères, dont on étudie en détail l'exemple du groupe Z/3Z*Z/3Z, fournissent des espaces de déformation pour des représentations d'holonomie de structures CR-sphériques. À l'aide de ces espaces de déformations, et des outils liés aux sphères visuelles dans CP^2, on construit une déformation explicite du domaine de Ford construit par Parker et Will et qui donne une uniformisation CR-sphérique sur le complémentaire de l'entrelacs de Whitehead. Cette déformation fournit une infinité d'uniformisations CR-sphériques sur une chirurgie de Dehn particulière de cette variété, et des uniformisations CR-sphériques sur une infinité de chirurgies de Dehn sur le complémentaire de l'entrelacs de Whitehead. / In this thesis, we study the construction and deformation of spherical-CR structures on three dimensional manifolds. In order to do it, we give a detailed description of the complex hyperbolic plane, its group of isometries and some geometric objects attached to this space such as bisectors and extors. We show a surgery theorem which allows to construct spherical-CR on Dehn surgeries of a cusped spherical-CR manifold : this theorem can be applied for the Deraux-Falbel structure on the figure eight knot complement and for Schwartz's and Parker-Will structures on the Whitehead link complement. We also define the character varieties for a real form of SL(n,C) for finitely generated groups as some subsets of the SL(n,C)-character variety invariant under an anti-holomorphic involution. We study in detail the example of the group Z/3Z*Z/3Z. These character varieties give deformation spaces for the holonomy representations of spherical-CR structures. With these deformation spaces and tools related to the visual spheres of a point in CP^2, we construct an explicit deformation of the Ford domain constructed by Parker and Will, which gives a spherical-CR uniformisation of the Whitehead link complement. This deformation provides infinitely many spherical-CR uniformisations of a particular Dehn surgery of the manifold, and spherical-CR unifomisations for infinitely many Dehn surgeries of the Whitehead link complement.