Дисертації з теми "Riemannsk geometri"

Nous proposons dans cette thèse de nouvelles méthodes de réduction de dimension fondées sur la géométrie différentielle. Il s'agit de trouver une représentation d'un ensemble d'observations dans un espace de dimension inférieure à l'espace d'origine des données. Les méthodes de réduction de dimension constituent la pierre angulaire des statistiques et ont donc un très large éventail d'applications. Dans les statistiques euclidiennes ordinaires, les données appartiennent à un espace vectoriel et l'espace de dimension inférieure peut être un sous-espace linéaire ou une sous-variété non linéaire approximant les observations. L'étude de telles variétés lisses, la géométrie différentielle, joue naturellement un rôle important dans ce dernier cas. Lorsque l'espace des données est lui-même une variété, l'espace approximant de dimension réduite est naturellement une sous-variété de la variété initiale. Les méthodes d'analyse de ce type de données relèvent du domaine des statistiques géométriques. Les statistiques géométriques pour des observations appartenant à une variété riemannienne sont le point de départ de cette thèse, mais une partie de notre travail apporte une contribution même dans le cas de données appartenant à l'espace euclidien, mathbb{R}^d.Les formes, dans notre cas des courbes ou des surfaces discrètes ou continues, sont un exemple important de données à valeurs dans les variétés. En biologie évolutive, les chercheurs s'intéressent aux raisons et aux implications des différences morphologiques entre les espèces. Cette application motive la première contribution principale de la thèse. Nous généralisons une méthode de réduction de dimension utilisée en biologie évolutive, l'analyse en composantes principales phylogénétiques (P-PCA), pour travailler sur des données à valeur dans une variété riemannienne - afin qu'elle puisse être appliquée à des données de forme. P-PCA est une version de PCA pour des observations qui sont les feuilles d'un arbre phylogénétique. D'un point de vue statistique, la propriété importante de ces données est que les observations ne sont pas indépendantes. Nous définissons et estimons des moyennes et des covariances intrinsèquement pondérées sur une variété qui prennent en compte cette dépendance des observations. Nous définissons ensuite l'ACP phylogénétique sur une variété comme la décomposition propre de la covariance pondérée dans l'espace tangent de la moyenne pondérée. Nous montrons que l'estimateur de moyenne actuellement utilisé en biologie évolutive pour étudier la morphologie correspond à ne prendre qu'une seule étape de notre algorithme de descente de gradient riemannien pour la moyenne intrinsèque, lorsque les observations sont représentées dans l'espace des formes de Kendall.Notre deuxième contribution principale est une méthode non paramétrique de réduction de dimension fondée sur une classe très flexible de sous-variétés qui est novatrice même dans le cas de données euclidiennes. Grâce à une PCA locale, nous construisons tout d'abord un sous-fibré du fibré tangent sur la variété des données que nous appelons le sous-fibré principal. Cette distribution (au sens géométrique) induit une structure sous riemannienne. Nous montrons que les géodésiques sous-riemanniennes correspondantes restent proches de l'ensemble des observations et que l'ensemble des géodésiques partant d'un point donné génèrent localement une sous-variété qui est radialement alignée avec le sous-fibré principal, même lorsqu'il est non intégrables, ce qui apparait lorsque les données sont bruitées. Notre méthode démontre que la géométrie sous-riemannienne est le cadre naturel pour traiter de tels problèmes. Des expériences numériques illustrent la puissance de notre cadre en montrant que nous pouvons réaliser des reconstructions d'une extension importante, même en présence de niveaux de bruit assez élevés<br>In this thesis, we propose new methods for dimension reduction based on differential geometry, that is, finding a representation of a set of observations in a space of lower dimension than the original data space. Methods for dimension reduction form a cornerstone of statistics, and thus have a very wide range of applications. For instance, a lower dimensional representation of a data set allows visualization and is often necessary for subsequent statistical analyses. In ordinary Euclidean statistics, the data belong to a vector space and the lower dimensional space might be a linear subspace or a non-linear submanifold approximating the observations. The study of such smooth manifolds, differential geometry, naturally plays an important role in this last case, or when the data space is itself a known manifold. Methods for analysing this type of data form the field of geometric statistics. In this setting, the approximating space found by dimension reduction is naturally a submanifold of the given manifold. The starting point of this thesis is geometric statistics for observations belonging to a known Riemannian manifold, but parts of our work form a contribution even in the case of data belonging to Euclidean space, mathbb{R}^d.An important example of manifold valued data is shapes, in our case discrete or continuous curves or surfaces. In evolutionary biology, researchers are interested in studying reasons for and implications of morphological differences between species. Shape is one way to formalize morphology. This application motivates the first main contribution of the thesis. We generalize a dimension reduction method used in evolutionary biology, phylogenetic principal component analysis (P-PCA), to work for data on a Riemannian manifold - so that it can be applied to shape data. P-PCA is a version of PCA for observations that are assumed to be leaf nodes of a phylogenetic tree. From a statistical point of view, the important property of such data is that the observations (leaf node values) are not necessarily independent. We define and estimate intrinsic weighted means and covariances on a manifold which takes the dependency of the observations into account. We then define phylogenetic PCA on a manifold to be the eigendecomposition of the weighted covariance in the tangent space of the weighted mean. We show that the mean estimator that is currently used in evolutionary biology for studying morphology corresponds to taking only a single step of our Riemannian gradient descent algorithm for the intrinsic mean, when the observations are represented in Kendall's shape space. Our second main contribution is a non-parametric method for dimension reduction that can be used for approximating a set of observations based on a very flexible class of submanifolds. This method is novel even in the case of Euclidean data. The method works by constructing a subbundle of the tangent bundle on the data manifold via local PCA. We call this subbundle the principal subbundle. We then observe that this subbundle induces a sub-Riemannian structure and we show that the resulting sub-Riemannian geodesics with respect to this structure stay close to the set of observations. Moreover, we show that sub-Riemannian geodesics starting from a given point locally generate a submanifold which is radially aligned with the estimated subbundle, even for non-integrable subbundles. Non-integrability is likely to occur when the subbundle is estimated from noisy data, and our method demonstrates that sub-Riemannian geometry is a natural framework for dealing which such problems. Numerical experiments illustrate the power of our framework by showing that we can achieve impressively large range reconstructions even in the presence of quite high levels of noise<br>I denne afhandling præsenteres nye metoder til dimensionsreduktion, baseret p˚adifferential geometri. Det vil sige metoder til at finde en repræsentation af et datasæti et rum af lavere dimension end det opringelige rum. S˚adanne metoder spiller enhelt central rolle i statistik, og har et meget bredt anvendelsesomr˚ade. En laveredimensionalrepræsentation af et datasæt tillader visualisering og er ofte nødvendigtfor efterfølgende statistisk analyse. I traditionel, Euklidisk statistik ligger observationernei et vektor rum, og det lavere-dimensionale rum kan være et lineært underrumeller en ikke-lineær undermangfoldighed som approksimerer observationerne.Studiet af s˚adanne glatte mangfoldigheder, differential geometri, spiller en vigtig rollei sidstnævnte tilfælde, eller hvis rummet hvori observationerne ligger i sig selv er enmangfoldighed. Metoder til at analysere observationer p˚a en mangfoldighed udgørfeltet geometrisk statistik. I denne kontekst er det approksimerende rum, fundetvia dimensionsreduktion, naturligt en submangfoldighed af den givne mangfoldighed.Udgangspunktet for denne afhandling er geometrisk statistik for observationer p˚a ena priori kendt Riemannsk mangfoldighed, men dele af vores arbejde udgør et bidragselv i tilfældet med observationer i Euklidisk rum, Rd.Et vigtigt eksempel p˚a data p˚a en mangfoldighed er former, i vores tilfældediskrete kurver eller overflader. I evolutionsbiologi er forskere interesseret i at studeregrunde til og implikationer af morfologiske forskelle mellem arter. Former er ´en m˚adeat formalisere morfologi p˚a. Denne anvendelse motiverer det første hovedbidrag idenne afhandling. We generaliserer en metode til dimensionsreduktion brugt i evolutionsbiologi,phylogenetisk principal component analysis (P-PCA), til at virke for datap˚a en Riemannsk mangfoldighed - s˚a den kan anvendes til observationer af former. PPCAer en version af PCA for observationer som antages at være de yderste knuder iet phylogenetisk træ. Fra et statistisk synspunkt er den vigtige egenskab ved s˚adanneobservationer at de ikke nødvendigvis er uafhængige. We definerer og estimerer intrinsiskevægtede middelværdier og kovarianser p˚a en mangfoldighed, som tager højde fors˚adanne observationers afhængighed. Vi definerer derefter phylogenetisk PCA p˚a enmangfoldighed som egendekomposition af den vægtede kovarians i tanget-rummet tilden vægtede middelværdi. Vi viser at estimatoren af middelværdien som pt. bruges ievolutionsbiologi til at studere morfologi svarer til at tage kun et enkelt skridt af voresRiemannske gradient descent algoritme for den intrinsiske middelværdi, n˚ar formernerepræsenteres i Kendall´s form-mangfoldighed.Vores andet hovedbidrag er en ikke-parametrisk metode til dimensionsreduktionsom kan bruges til at approksimere et data sæt baseret p˚a en meget flexibel klasse afsubmangfoldigheder. Denne metode er ny ogs˚a i tilfældet med Euklidisk data. Metodenvirker ved at konstruere et under-bundt af tangentbundet p˚a datamangfoldighedenM via lokale PCA´er. Vi kalder dette underbundt principal underbundtet. Viobserverer at dette underbundt inducerer en sub-Riemannsk struktur p˚a M og vi viserat sub-Riemannske geodæter fra et givent punkt lokalt genererer en submangfoldighedsom radialt flugter med det estimerede subbundt, selv for ikke-integrable subbundter.Ved støjfyldt data forekommer ikke-integrabilitet med stor sandsynlighed, og voresmetode demonstrerer at sub-Riemannsk geometri er en naturlig tilgang til at h˚andteredette. Numeriske eksperimenter illustrerer styrkerne ved metoden ved at vise at denopn˚ar rekonstruktioner over store afstande, selv under høje niveauer af støj

Made available in DSpace on 2015-04-09T12:28:21Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-03-28Bitstream added on 2015-04-09T12:48:18Z : No. of bitstreams: 1 000809982.pdf: 1238279 bytes, checksum: 51811e33aad5834491b25013aa77ba4b (MD5)<br>Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)<br>O objetivo desde trabalho e estimar um cota para o n umero de pontos racionais de uma curva. Observando as várias semelhanças entre o anel dos inteiros e o anel dos polinômios em uma variável, iremos usar ferramentas da teoria dos números para resolver um problema da geometria algébrica. Desta fusão nasce uma das mais nobres areas da matemática: a geometria aritmética. Fazendo uso do célebre teorema de Riemann-Roch e das ferramentas da teoria dos números demonstraremos a hipótese de Riemann para a funço-zeta de uma curva não singular e qual consequência tal hipótese tem para a contagem de pontos racionais de uma curva<br>The aim of this work is to estimate a bound for the number of rational points of a curve. Observing the various similarities between the ring of integers and the ring of polynomials in one variable, we use tools from number theory to solve a problem of algebraic geometry. From this merger is born one of the noblest areas of mathematics: arithmetic geometry. Making use of the famous Riemann-Roch's theorem and tools of number theory we demonstrate the Riemann hypothesis for the zeta-function of a nonsingular curve and which consequence this hypothesis has to count rational points on a curve

Orientador: Parham Salehyan<br>Banca: Eduardo Tengan<br>Banca: Trajano Pires da Nóbrega Neto<br>Resumo: O objetivo desde trabalho e estimar um cota para o n umero de pontos racionais de uma curva. Observando as várias semelhanças entre o anel dos inteiros e o anel dos polinômios em uma variável, iremos usar ferramentas da teoria dos números para resolver um problema da geometria algébrica. Desta fusão nasce uma das mais nobres areas da matemática: a geometria aritmética. Fazendo uso do célebre teorema de Riemann-Roch e das ferramentas da teoria dos números demonstraremos a hipótese de Riemann para a funço-zeta de uma curva não singular e qual consequência tal hipótese tem para a contagem de pontos racionais de uma curva<br>Abstract: The aim of this work is to estimate a bound for the number of rational points of a curve. Observing the various similarities between the ring of integers and the ring of polynomials in one variable, we use tools from number theory to solve a problem of algebraic geometry. From this merger is born one of the noblest areas of mathematics: arithmetic geometry. Making use of the famous Riemann-Roch's theorem and tools of number theory we demonstrate the Riemann hypothesis for the zeta-function of a nonsingular curve and which consequence this hypothesis has to count rational points on a curve<br>Mestre

Made available in DSpace on 2015-05-14T12:14:07Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1123884 bytes, checksum: 0b9e6a7e26799ac54ef0dcb5fb98d15b (MD5) Previous issue date: 2012-02-28<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>We consider the geometry of affine connections and take, as particular examples, Weyl and Riemann-Cartan geometies. In a modern geometrical approach, we take up the problem of local embedding of manifolds in Weyl spaces and in spaces endowed with semi-symmetric torsion. We then obtain the extrinsic curvature, Weingarten operator and Gauss-Codazzi equations in the mentioned non-riemannian spaces. We investigate some important properties of a Weyl structure in the case of a warped product and carry out an analysis of the geodesics in a foliation de…ned in such a space. We consider the particular case when the embedding space is a warped product manifold and has a Riemann-Cartan geometry. As an application, we show that the torsion …eld of de bulk may provide a mechanism of geometrical con…nement. In this way, we exhibit a classical analogue of the quantum con…nement induced by scalar …elds.<br>Consideramos a geometria de uma conexão a…m e abordamos como exemplos, as geometrias de Weyl e Riemann-Cartan, esta ultima considerando o caso em que a torção é semi-simétrica. Após uma exposição moderna das propriedades destas geometrias, abordamos o problema de imersões isométricas em espaços de Weyl e de torção semi-simétrica. Introduzimos um roteiro para a obtenção da curvatura extrínseca, operador de Weingarten e das equações de Gauss-Codazzi para tais espaços. Em seguida, analisamos as propriedades de uma estrutura de Weyl em um espaço produto distorcido (EPD) e analisamos as geodésicas das folhas em tal espaço. Consideramos, também, o caso particular quando o espaço ambiente para um (EPD) com uma geometria de Riemann-Cartan. Mostramos como o confi…namento e as propriedades de estabilidade de geodésicas próximas ao mundo-brana podem ser afetadas pela torção do bulk. Deste modo, construímos um análogo clássico do confi…namento quântico inspirado em modelos de teoria de campo, substituindo um campo escalar por um campo de torção.

Thesis (Ph. D.)--University of Oregon, 2003.<br>Typescript. Includes vita and abstract. Includes bibliographical references (leaves 236-241). Also available for download via the World Wide Web; free to University of Oregon users.

Orientador: Valerio Ramos Batista<br>Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-04T03:34:02Z (GMT). No. of bitstreams: 1 Lopes_LauriclecioFigueiredo_M.pdf: 1161319 bytes, checksum: c34f319b4252610a06e72d9b93740a89 (MD5) Previous issue date: 2005<br>Resumo: Entende-se por superfícies mínimas aquelas cuja curvatura média é nula. Têm-se como exemplos clássicos o catenóide, o helicóide e a superfície de Scherk. Historicamente, elas estão relacionadas com minimização de área, porém quando realiza-se uma variação normal incluindo os bordos, a superfície original com curvatura média nula pode representar uma área localmente máxima. Em certos casos de variação com bordo fixo, tem-se realmente a minimização do funcional área. No espaço euclidiano tridimensional, o Teorema da Representação de Weierstrass expressa uma superfície mínima em termos de integrais envolvendo uma função holomorfa e uma meromorfa. A partir desta meromorfa pode-se deduzir a aplicação normal de Gauss. Conceitos como curvatura Gaussiana, curvatura total, superfícies completas e regularidade também são utilizados para deduzir propriedades das superfícies mínimas. Quando estudamos as superfícies mínimas para as quais o bordo consiste de duas circunferências disjuntas, os Teoremas de Enneper e Shiffman, o Princípio de Reflexão de Schwarz e a unicidade do Problema de Bjõrling são ferramentas importantes para a dedução das soluções, a saber, o catenóide e as superfícies de Riemann. Estas apresentam simetrias por reflexão a um plano e invariância por rotação de 180 graus em torno de uma reta. A função "P de Weierstrass" simétrica é de grande utilidade no estudo destas propriedades<br>Abstract: Minimal surfaces are known to be the ones with mean curvature zero. Classical exampIes are the catenoid, helicoid and the Scherk surface. Historically, they were associated with the property of minimizing area. However, they can even maximize it localIy for cases of normal variation which include the boundary. For fixed boundary, we shalI analyse when they realIy minimize the area functional. In the three-dimensional Euclidean space, the Weierstrass Representation Theorem expresses any minimal surface S by means of integraIs with a holomorphic and a meromorphic functions, usualIy denoted by f and g, respectively. The unitary normal N of S is fulIy determined by g. Concepts like "Gaussian curvature", "total curvature", "com pleteness" and "regularity" are also employed in order to read off some properties of minimal surfaces. Concerning the case for which the boundary of S consists of two disjoint circumferences, Enneper's and Shiffman's Theorems, The Schwarz's Reflection PrincipIe and the B6rling's Problem are fundamental tools to characterize the solutions, namely the catenoid and the Riemann's examples. AlI these are invariant by a reflectional symmetry in a plane, and also by a rotation of 180-degree around a straight line. The symmetric Weierstrass-Pfunction is very useful to deduce these properties<br>Mestrado<br>Matematica<br>Mestre em Matemática

Orientador: Valerio Ramos Batista<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-10T08:53:28Z (GMT). No. of bitstreams: 1 Lubeck_KellyRobertaMazzutti_D.pdf: 1896664 bytes, checksum: 506a84f4e0c03b2fff585bb45b6a9b1f (MD5) Previous issue date: 2007<br>Resumo: Neste trabalho apresentamos o estudo e a construção de superfícies minimas atraves de um metodo exclusivo. Em 1762, Lagrange introduziu a Equacao Diferencial das Superfícies Mnimas atraves do Calculo de Variações, e hoje a teoria de tais superfícies e umaarea de pesquisa ativa e abrangente. A elaboração de novas famílias de superfícies minimas esta baseada no metodo da Construção Reversa, desenvolvido por Hermann Karcher nos meados da década de 80. Salientamos no presente trabalho a maneira diferenciada com que os problemas de periodos foram resolvidos. Para isso, utilizaram-se as equações de uma superfície mínima limite, para a qual ja era conhecido que o problema de períodos tinha solução transversal. Tal método, que neste trabalho sera denominado "método limite", simplica de maneira consideravel o esforco em solucionar os problemas de período da família original<br>Abstract: In this work we present the study and construction of minimal surfaces through an exclusive method. In 1762, Lagrange introduced the Minimal Surfaces Diferential Equation through the Calculus of Variations, and today the theory of such surfaces builds up an active and broad research area. We obtain new families of minimal surfaces based upon the Reverse Construction Method, developed by Hermann Karcher during the eighties. In our work we stress the original fashion with which period problems are solved: One makes use of a limit minimal surface, of which the periods are known to have transversal solution. Because of that we named our technique as "limit-method", which simplies considerably the effort of solving period problems for the sought after family of minimal surfaces<br>Doutorado<br>Geometria Diferencial<br>Mestre em Matemática

Submitted by Renata Lopes (renatasil82@gmail.com) on 2018-03-28T11:17:02Z No. of bitstreams: 1 andersoncorreaporto.pdf: 567494 bytes, checksum: e685a947374868ceaa838290c83bc61a (MD5)<br>Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-04-09T19:23:34Z (GMT) No. of bitstreams: 1 andersoncorreaporto.pdf: 567494 bytes, checksum: e685a947374868ceaa838290c83bc61a (MD5)<br>Made available in DSpace on 2018-04-09T19:23:34Z (GMT). No. of bitstreams: 1 andersoncorreaporto.pdf: 567494 bytes, checksum: e685a947374868ceaa838290c83bc61a (MD5) Previous issue date: 2018-02-08<br>O objetivo desse trabalho é o estudo de conceitos básicos da Geometria Algébrica sob o ponto de vista clássico. O foco central do trabalho é o estudo do Teorema de Riemann- Roch e algumas de suas aplicações. Esse teorema constitui uma importante ferramenta no estudo da Geometria Algébrica clássica uma vez que possibilita, por exemplo, o cáculo do gênero de uma curva projetiva não singular no espaço projetivo de dimensão dois. Para o desenvolvimento do estudo do Teorema de Riemann-Roch e suas aplicações serão estudados conceitos tais como: variedades, dimensão, diferenciais de Weil, divisores, divisores sobre curvas e o anel topológico Adèle.<br>The goal of this work is the study of basic concepts of Algebraic Geometry from the classical point of view. The central focus of the paper is the study of Riemann-Roch Theorem and some of its applications. This theorem constitutes an important tool in the study of classical Algebraic Geometry since it allows, for example, the calculation of the genus of a non-singular projective curve in the projective space of dimension two. For the development of the study of the Riemann-Roch Theorem and its applications we will study concepts such as: varieties, dimension, Weil differentials, divisors, divisors on curves and the Adèle topological ring.

The Riemann Zeta Function has a deep connection with the distribution of primes. This expository thesis will explain the techniques used in proving the properties of the Rieman Zeta Function, its analytic continuation to the complex plane, and the functional equation that the the Riemann Zeta Function satisfies.

Ce travail est consacre a l'etude des proprietes h-projectives des espaces symetriques complexes. Dans le chapitre i (preliminaire) on rappelle quelques definitions et resultats sur les courbes holomorphiquement planaites, les transformations h-projectives, le tenseur de weyl h-projectif et les espaces symetriques complexes. On construit et on etudie dans le chapitre ii, deux familles de domaines symetriques de l'espace projectif complexe et on demontre que ce sont les seuls domaines symetriques. La construction effectuee au chapitre ii est utilisee, dans le chapitre iii pour obtenir une classification algebrique des espaces symetriques complexes h-projectivement plats. Dans le cas simplement connexe, on ramene la recherche des classes d'isomorphisme a la determination effectuee au chapitre ii d'orbites du groupe projectif complexe. Le cas non simplement connexe est egalement etudie. Le chapitre iv est consacre a la recherche des espaces symetriques complexes admettant des transformations h-projectives non affines. On demontre, tout d'abord, que ces espaces sont necessairement h-projectivement plats, et, en exploitant la classification du chapitre iii, on determine, a isomorphisme pres, tous les espaces symetriques complexes dont le groupe des transformations h-projectives ne se reduit pas a un groupe de transformations affines

The object of this thesis is the automated computation of the rational (co)homology of the moduli spaces of smooth marked Riemann surfaces Mg;n. This is achieved by using a computer to generate a chain complex, known in advance to have the same homology as Mg;n, and explicitly spell out the boundary operators in matrix form. As an application, we compute the Betti numbers of some moduli spaces Mg;n. Our original contribution is twofold. In Chapter 3, we develop algorithms for the enumeration of fatgraphs and their automorphisms, and the computation of the homology of the chain complex formed by fatgraphs of a given genus g and number of boundary components n. In Chapter 4, we describe a new practical parallel algorithm for performing Gaussian elimination on arbitrary matrices with exact computations: projections indicate that the size of the matrices involved in the Betti number computation can easily exceed the computational power of a single computer, so it is necessary to distribute the work over several processing units. Experimental results prove that our algorithm is in practice faster than freely available exact linear algebra codes. An effective implementation of the fatgraph algorithms presented here is available at http://code.google.com/p/fatghol. It has so far been used to compute the Betti numbers of Mg;n for (2g + n) 6 6. The Gaussian elimination code is likewise publicly available as open-source software from http://code.google.com/p/rheinfall.

Orientador: Luiz Mariano Paes de Carvalho Filho<br>Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação<br>Made available in DSpace on 2018-08-11T14:58:23Z (GMT). No. of bitstreams: 1 Jacyntho_LuizAntonio_M.pdf: 8765048 bytes, checksum: d1d39ba51eca5f10f2b9eb3fd48e367c (MD5) Previous issue date: 2008<br>Resumo: Este trabalho tem como objetivos estudar algumas realizações de Arquimedes (287 a.C. - 212 a.C., Grécia) e de Isaac Barrow (1630-1677, Inglaterra), e, também, desenvolver atividades no Geogebra para auxiliar no ensino do Cálculo Diferencial e Integral. Apresentamos a construção do conjunto dos números reais, definições e teoremas atuais que antecedem, logicamente, o Teorema Fundamental do Cálculo. Tratamos de algumas das realizações de Arquimedes: a demonstração da medida da área do círculo, utilizando o Método de Eudoxo, o "método mecânico", pelo qual ele descobriu a medida da área do segmento parabólico e a demonstração rigorosa desta medida. São discutidas algumas realizações de Isaac Barrow: o método por ele utilizado para encontrar retas tangentes a uma curva, um estudo sobre o conteúdo da Conferência I e sobre algumas proposições da Conferência X. Nesta última, será dada atenção especial à Proposição 11, que demonstra casos particulares do Teorema Fundamental do Cálculo. O trabalho termina com um conjunto de atividades baseadas no programa Geogebra. Cada atividade tem a sua função numa seqüência didática e aborda os seguintes temas: a representação do conjunto dos números reais, a proposição de Arquimedes sobre a medida da área do círculo, o cálculo de áreas, a construção da função área, o cálculo de primitivas, a interpretação de Barrow para casos particulares do Teorema fundamental do Cálculo e algumas aplicações do Teorema Fundamental do Cálculo<br>Abstract: This work has as objectives study some realizations of Archimedes (287 BC - 212 BC, Greece) and of Isaac Barrow (1630-1677, UK), and, also, develop activities in Geogebra to aid in the teaching of Differential and Integral Calculus. We present the construction of the set of the real numbers, definitions and actual theorems that precede, logically, the Fundamental Theorem of Calculus. We deal with some of Archimedes' realizations: the demonstration of the measure of the circle's area, using the Eudoxus' Method, the "mechanical method", by which he discovered the measure of the area of the parabolic segment and the rigorous demonstration of it. There are discussed some realizations of Isaac Barrow: the method used by him to find tangent straights to a curve, a study about the content of the Lecture I and about some prepositions of the Lecture X. In this last one, main attention will be given to Proposition 11, which demonstrates particular cases of the Fundamental Theorem of Calculus. The word ends with a group of activities based in the Geogebra. Each activity has its function in a didactic sequence and they are about the following themes: the representation of the set of the real numbers, the proposition of Archimedes about the measure of the area of the circle, the calculation of areas, the construction of the area function, the calculation of primitives, the interpretation of Barrow to particular cases of the Fundamental Theorem of Calculus and some applications of the Fundamental Theorem of Calculus<br>Mestrado<br>Geometria<br>Mestre em Matemática

Nessa dissertação exploramos os efeitos geométricos das Transformações de Möbius em C utilizando projeções na Esfera de Riemann. Como aplicação, apresentamos a ação de algumas transformações aplicadas em cônicas no plano. Uma atividade didática voltada aos alunos do Ensino Médio sobre Transformações de Möbius utilizando o Geogebra é apresentada.<br>In the course of this dissertation we explore the geometric effects of the Möbius Transforms in C using projections in the Riemann sphere. As an application, we present the action of some transformations applied on conics in the plane. A didactic activity aimed at high school students about Möbius Transformations using Geogebra is presented.

La géométrie rienmannienne des sous-variétés a connu ces cinquante dernières années un essor considérable, essentiellement dans le cas compact. Cette thèse a pour but de développer des outils consacrés à l'étude des sous-variétés riemanniennes complètes. Ces outils sont proches de ceux développés par Bochner et Lichnérowicz. Ils sont particulièrement adaptés aux problèmes de rigidité de certains types de sous-variétés complètes : celles qui sont à courbure moyenne constante dans un espace hyperbolique. Il est ainsi possible d'obtenir un théorème de classification de ces sous-variétés. D'autres applications sont données pour des sous-variétés totalement réelles des espaces projectifs complexes

Orientador: Reginaldo Palazzo Junior<br>Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação<br>Made available in DSpace on 2018-08-19T09:31:01Z (GMT). No. of bitstreams: 1 Alves_AlessandroFerreira_D.pdf: 1080224 bytes, checksum: 0748952c3176e9548151bec7e6d9c71d (MD5) Previous issue date: 2011<br>Resumo: Para projetarmos um sistema de comunicação digital em espaços hiperbólicos é necessário estabelecer um procedimento sistemático de construção de reticulados como elemento base para a construção de constelações de sinais. De outra forma, em codificação de canal é de fundamental importância a caracterização das estruturas algébrica e geométrica associadas a canais discretos sem memória. Neste trabalho, apresentamos a caracterização geométrica de superfícies a partir dos possíveis emparelhamentos das arestas do polígono fundamental hiperbólico com 3 ? n ? 8 lados associado 'a superfície. Esse tratamento geométrico apresenta propriedades importantes na determinação dos reticulados hiperbólicos a serem utilizados no processo de construção de constelações de sinais, a partir de grupos fuchsianos aritméticos e da superfície de Riemann associada. Além disso, apresentamos como exemplo o desenvolvimento algébrico para a determinação dos geradores do grupo fuchsiano 'gama'8 associado ao polígono hiperbólico 'P IND. 8'<br>Abstract: In order to design a digital communication system in hyperbolic spaces is necessary to establish a systematic procedure of constructing lattices as the basic element for the construction of the signal constellations. On the other hand, in channel coding is of fundamental importance to characterize the geometric and algebraic structures associated with discrete memoryless channels. In this work, we present a geometric characterization of surfaces from the edges of the possible pairings of fundamental hyperbolic polygon with 3 ? n ? 8 sides associated with the surface. This treatment has geometric properties important in determining the hyperbolic lattices to be used in the construction of sets of signals derived from arithmetic Fuchsian groups and the associated Riemann surface<br>Doutorado<br>Telecomunicações e Telemática<br>Doutor em Engenharia Elétrica

Neste trabalho apresentamos três métodos distintos provando que S(n) = +1 X k=−1 (4k + 1)−n é um múltiplo racional de n para todos os inteiros n = 1, 2, 3, . . . O primeiro utiliza a teoria das função analíticas e funções geradoras. No segundo reduzimos o problema, via mudança de variável devida a E. Calabi, ao cálculo do volume de certos politopos em Rn enquanto que no terceiro usamos a teoria dos operadores integrais compactos. Cada um dos métodos tem um interesse intrínsico e está sujeito a generalizações para aplicações em novas situações.

The following presents a brief introduction to tropical geometry, especially tropical curves, and explains a connection to graph theory. We also give a brief summary of the Riemann-Roch property for graphs, established by Baker and Norine (2007), as well as the tools used in their proof. Various generalizations are described, including a more thorough description of the extension to strongly connected directed graphs by Asadi and Backman (2011). Building from their constructions, an algorithm to determine if a directed graph has Row Riemann-Roch Property is given and thoroughly explained.

Ingeniero Civil Matemático<br>El trabajo presentado en esta memoria se sitúa en la interfaz entre el análisis y la geometría. El interés recae en el estudio de fenómenos de concentración para dos problemas "geométricos" no lineales: la existencia de hipersuperficies con r-curvatura constante en variedades Riemannianas, y una ecuación de Schrödinger no lineal. Esta memoria se puede dividir en dos partes principales. La primera está dedicada a explorar algunos resultados sobre concentración de familias de hipersuperficies de curvatura media constante (o en general curvatura r-media constante) con topología no trivial en variedades Riemannianas compactas. Se recuerda que la curvatura r-media de una hipersuperficie se define como la r-ésima función simétrica elemental de las curvaturas principales de la hipersuperficie. Se prueba que las técnicas desarrolladas en el trabajo de Mahmoudi, Mazzeo y Pacard se pueden extender para manejar el caso de curvatura r-media con r>=1. Este fenómeno de concentración se relaciona en general con un fenómeno de resonancia, que hace el análisis particularmente delicado y que también se encuentra en el estudio de una clase de ecuaciones elípticas no lineales que presentan concentración sobre conjuntos de dimensión mayor. En la segunda parte, correspondiente al paper presentado, se prueba un nuevo resultado sobre concentración en subvariedades para una ecuación de Schrödinger no lineal con potencial definido en una variedad Riemanniana suave y compacta M o el espacio Euclídeo R^n, resolviendo en completa generalidad una conjetura planteada por Ambrosetti, Malchiodi y Ni. Precisamente, se estudian soluciones positivas de la siguiente ecuación semilineal: $$\e^2\Delta_{\bar g} u - V(z)u + u^{p} =0 en M,$$ donde (M,g) es una variedad Riemanniana n-dimensional suave, compacta y sin borde o el espacio Euclídeo R^n, e es un parámetro positivo pequeño, p>1 y V es un potencial uniformemente positivo. Se prueba que dado k=1,...,n-1 y 1<p<(n+2-k)/(n-2-k), y suponiendo que K es una subvariedad k-dimensional suave y encajada de M, que es estacionaria y no degenerada con respecto al funcional $\int_K V^{\frac{p+1}{p-1}-\frac{n-k}{2}}dvol$, entonces existe una secuencia $e=\e_j \to 0$ y soluciones positivas asociadas $u=u_\e$ que concentran sobre K en el sentido de que decaen exponencialmente a cualquier distancia positiva a K. En particular este enfoque explora una conexión entre soluciones de esta ecuación de Schrödinger no lineal y subvariedades f-minimales en variedades con densidad.

Nesta dissertação, definimos a geometria hiperbólica usando o disco de Poincaré (D2) e o semiplano superior (H2) com as respectivas propriedades. Além disso, apresentamos algumas funções e relações importantes da geometria hiperbólica; conceituamos as superfícies de Riemann, analisando suas propriedades e representações; estudamos o espaço de Teichmüller com a devida decomposição em calças. Esses temas são ferramentas necessárias para atingir o objetivo da dissertação: definir as coordenadas de Fenchel Nielsen como um sistema de coordenadas locais do espaço de Teichmüller Tg.<br>In this dissertation, we defined the hyperbolic geometry using the Poincares disk (D2) and upper half-plane (H2) with its properties. Besides, we presented some functions and important relations of the hyperbolic geometry; we conceptualize the Riemann surfaces, analyzing its properties and representations; we studied the Teichmüller Space with proper decomposition pants. These themes are essential tools to reach the goal of the work: The definition of the Fenchel Nielsen coordenates as local coordinate system of the Teichmüller space Tg.

If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the construction for Lorenzini's Riemann-Roch structure and how it relates to the Riemann-Roch theorem for finite graphs shown by Matthew Baker and Serguei Norine. We then focus on the connection between the Frobenius number and arithmetical graphs. Using the Laplacian of an arithmetical graph and a formulation of chip-firing on the vertices of an arithmetical graph, we show results that can be used to find arithmetical graphs whose g-numbers correspond to the Frobenius number of R. We describe how this can be used to quickly calculate upper bounds for the Frobenius number of R.

Orientador: Reginaldo Palazzo Junior<br>Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação<br>Made available in DSpace on 2018-08-09T06:02:14Z (GMT). No. of bitstreams: 1 Bastos_JeffersonLuizRocha_D.pdf: 792249 bytes, checksum: 7a08577655f1a651a653df4d98e29e62 (MD5) Previous issue date: 2007<br>Resumo: Tendo como base trabalhos recentes que associam o desempenho de sistemas de comunicação digital ao gênero de uma superfície compacta de Riemann, este trabalho tem como objetivo propor uma integração entre modulação e codificação de canal, tendo como base o gênero da superfície. Para atingir tais objetivos, nossa proposta é a seguinte: fixado um gênero g (g = 0,1,2,3), encontrar curvas com este gênero e fazer uma análise dos parâmetros dos códigos associados a esta curva, a fim de se obter uma modulação e um sub-código desta modulação para ser utilizado na codificação de canal<br>Abstract: Based on recent research showing that the performance of bandwidth efficent communication systems also depends on the genus of a. compact Riemann surface in which the communication channel is embedded, this study aims at proposing a combined form of modulation and coding technique when only the genus of a surface is given to the communication system designeI. To achieve this goal, the following procedure is proposed. Knowing that the channel is embedded in a surface of genus g, find algebraic curves with the given genus which will give rise to the modulation system, an (n, n, 1) type of code, and from this find the best (n, k, d) subcode, to be employed in the aforementioned combined formo Keywords: Riemann surface, algebraic curves, Goppa codes, modulation<br>Doutorado<br>Engenharia de Computação<br>Doutor em Engenharia Elétrica

Descrevemos como a operação de somar um unitão surge através do método de Dorfmeister, Pedit e Wu (DPW) que permite obter aplicações harmónicas em espaços simétricos Riemannianos compactos a partir de certas 1-formas holomorfas. Exploramos este ponto de vista para investigar quais os unitões que preservam a propriedade do tipo nito das aplicações harmónicas. Em particular, provamos que o brado de Gauss de uma aplicação harmónica do tipo nito numa Grassmanniana também é do tipo nito. Provamos que qualquer aplicação harmónica da esfera de dimensão 2 num grupo de Lie compacto semi-simples de matrizes pode ser reduzida a uma constante usando as acções de revestimento singular, isto é, as singular dressing actions introduzidas por Bergvelt e Guest. Encontramos também geradores para o grupo dos lacetes racionais das representações fundamentais de Sp(n)C e SU(n)C: em ambos os casos a classe dos geradores é um pouco maior do que a classe de factores simples (lacetes racionais com um número mínimo de singularidades, cuja acção de revestimento pode ser calculada explicitamente). Estabelecemos fórmulas explícitas para as factorizações canónicas de soluções estendidas que correspondem a aplicações harmónicas com número de unitão nito no grupo de Lie excepcional G2 em termos do modelo Grassmanniano. É dada uma descrição dos geradores do referencial de Frenet para estas aplicações harmónicas. Em particular, mostramos que aplicações harmónicas da esfera de dimensão 2 em G2 correspondem a soluções de certos sistemas algébricos de equações quadráticas e cúbicas.

<p>A compact Riemann surface can be realized as a quotient space $\mathcal{U}/\Gamma$, where $\mathcal{U}$ is the sphere $\Sigma$, the euclidian plane $\mathbb{C}$ or the hyperbolic plane $\mathcal{H}$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal{U}\rightarrow\mathcal{U}/\Gamma$.</p><p>For each $\Gamma$ acting on $\mathcal{H}$ we have a polygon $P$ such that $\mathcal{H}$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal{H}$ under $\Gamma$.</p>

Orientador: Caio Jose Colletti Negreiros<br>Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-07-31T17:46:49Z (GMT). No. of bitstreams: 1 Silva_CleusianeVieirada_M.pdf: 1600289 bytes, checksum: f73aa8034db2cd0fb651e4378b464ee6 (MD5) Previous issue date: 2002<br>Resumo: Neste trabalho, estudamos a geometria das estruturas-f invariantes e curvas fholomorfas em variedades bandeira, a construção de toros equiharmônicos em variedades bandeira complexas não-degeneradas que não são f-holomorfos para qualquer estrutura-f invariante. Calculamos a segunda variação da energia para superfícies harmônicas riemannianas fechadas em variedades bandeira munidas com métricas do tipo Borel daídiscutimos a estabilidade para o referencial de Frenet de aplicações holomorfas com respeito a uma grande classe de métricas invariantes em F(N) obtidas via perturbação de métricas Kãhler. Além disso relacionamos a teoria de torneios com as estruturas quase complexas de uma variedade bandeira. Finalmente mostramos que a métrica Killing em F(N) é (1,2)-simplética se e somente se N :S 3<br>Abstract: In this work we study the geometry of invariant f-structures and f-holomorphic curves on flag manifolds, and the construction of the equiharmonic tori on full complex flag manifolds which are not f-holomophic for any invariant f-estructure. Moreover we relate the tournament theory with the almost-complex on a flag manifolds. We compute the second variation of energy for harmonic closed Riemann surfaces into flag manifolds equipped with the Borel type metrics then we discuss stability for Frenet frames of holomorphics maps with respect to a very large class de invariants metrics F(N) obtained via perturbation of the Kãhler ones. Finally we proof that the metric Killing on F(N) is (1,2)-simplétic if and only if N :S 3<br>Mestrado<br>Mestre em Matemática

Il teorema Egregium di Gauss afferma che se due superfici hanno diverse curvature gaussiane non possono essere localmente isometriche, per questo motivo non può esistere una mappa tra la sfera e il piano che conservi le distanze. La sfera come approssimazione del nostro pianeta è stata a lungo studiata nell'ambito della cartografia nel tentativo di realizzare planisferi sempre più fedeli alla realtà. La sfera viene poi vista come modello di geometria non euclidea, nel quale non è ammesso il postulato delle parallele, ma viene assunto l'assioma di Riemann.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior<br>Em 1951, H. Hopf publicou em um prestigiado artigo um famoso resultado: Seja M uma superfície compacta de gênero zero imersa no espaço Euclidiano de dimensão três com curvatura média constante. Então M é isométrica à esfera redonda. Neste trabalho descreveremos detalhadamente do ponto de vista matemático uma generalização do resultado obtido por H. Hopf, a qual será publicada na revista Communication in Analysis and Geometry em 2007, cujos autores são Hilário Alencar, Manfredo Perdigão do Carmo e Renato Tribuzy. Neste artigo, os pesquisadores classificaram as superfí&#305;cies compactas de gênero zero imersas na variedade produto: superfícies com curvatura Gaussiana constante cartesiano o espaço Euclidiano de dimensão um e cuja diferencial da curvatura média satisfaz uma certa desigualdade envolvendo uma forma quadrátrica. Além disso, estudaremos uma extensão da classificação anterior no caso em que as superfícies estão imersas numa variedade Riemanniana simplesmente conexa, homogênea com um grupo de isometrias de dimensão quatro. Tais resultados foram obtidos recentemente por Hilário Alencar, Isabel Fernández, Manfredo Perdigão do Carmo e Renato Tribuzy. Nas demonstrações destes teoremas foram usadas técnicas de Análise Complexa, fatos de Topologia e uma generalização do Teorema de H. Hopf obtida por Abresch e Rosenberg, publicado em Acta Mathematica em 2004.

On compact Riemann surfaces, the Laplacian $\Delta$ has a discrete, non-negative spectrum of eigenvalues $\{\lambda_{i}\}$ of finite multiplicity. The spectrum is intrinsically linked to the geometry of the surface. In this work, we consider surfaces of constant negative curvature with a large symmetry group. It is not possible to explicitly calculate the eigenvalues for surfaces in this class, so we combine group theoretic and analytical methods to derive results about the spectrum. In particular, we focus on the Bolza surface and the Klein quartic. These have the highest order symmetry groups among compact Riemann surfaces of genera 2 and 3 respectively. The full automorphism group of the Bolza surface is isomorphic to $\mathrm{GL}_{2}(\mathbb{Z}_{3})\rtimes\mathbb{Z}_{2}. We analyze the irreducible representations of this group and prove that the multiplicity of $\lambda_{1}$ is 3, building on the work of Jenni, and identify the irreducible representation that corresponds to this eigenspace. This proof relies on a certain conjecture, for which we give substantial numerical evidence and a hopeful method for proving. We go on to show that $\lambda_{2}$ has multiplicity 4.

Made available in DSpace on 2014-06-11T19:22:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-02-25Bitstream added on 2014-06-13T20:28:17Z : No. of bitstreams: 1 arruda_rl_me_sjrp.pdf: 624072 bytes, checksum: 23ddd00e27d1ad781e2d1cec2cb65dee (MD5)<br>Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)<br>O objetivo principal deste trabalho é estudar o Teorema de Riemann-Roch, um dos resultados fundamentais na teoria de curvas algébricas, e apresentar algumas de suas aplicações. Este teorema é uma importante ferramenta para a classificação das curvas algébricas, pois relaciona propriedades algébricas e topológicas. Daremos uma descrição das curvas algébricas de gênero g, 1≤ g ≤ 5, e faremos um breve estudo dos pontos de inflexão de um sistema linear sobre uma curva algébrica<br>The main purpose of this work is to discuss The Riemann-Roch Theorem, wich is one of the most important results of the theory algebraic curves, and to present some applications. This theorem is an important tool of the classification of algebraic curves, sinces relates algebraic and topological properties. We will describle the algebraic curves of genus g, 1≤ g ≤ 5, and also study inflection points of a linear system on an algebraic curve

Nous étudions des problèmes de géométrie sous-Riemannienne et la diffusion subelliptique correspondante. Ces études sont motivées par un modèle de perception visuelle dû à Petitot-Citti-Sarti. Nous étudions les métriques invariantes de Carnot-Caratheodory sur SU(2)= S^3, SO(3), et SL(2). Nous calculons les cut loci globalement et la distance. Ceci est le premier calcul explicite global en géométrie sous-Riemannienne, exception faite pour le groupe d'Heisenberg. Nous présentons une définition du Laplacien hypoelliptique généralisant l'opérateur de Laplace-Beltrami. On présent une méthode de calcul du noyau de la chaleur hypoelliptique sur les groupes de Lie utilisant la transformée de Fourier non-commutative. On étudie les cas de SU(2), SO(3), SL(2), et du groupe des déplacements dans le plan. Nous étudions le modèle de perception visuelle dû à PCS, selon lequel la reconstruction d'une courbe \gamma corrompue est la minimisation d'un coût qui dépend autant de la longueur que de la courbure K. On fixe les points initial et final, ainsi que les directions initiale et finale. On prouve la non-existence des minimisants pour \int \sqrt{1+K_\gamma^2} ds. On prouve l'existence des minimisants pour J=\int \|\dot\gamma(t) \|\sqrt{1+K_\gamma^2} dt si les directions sont considérées sans orientation. Nous résolvons globalement le problème de minimisation de J sur la sphère S^2. Des minimisants présentent des points de rebroussement. Nous présentons un algorithme de reconstruction d'images basé sur le modèle, où on substitue la minimisation de J par la diffusion hypoelliptique correspondante, que l'on ressoude explicitement. Des exemples de reconstruction sont présentés<br>This thesis focuses on problems of sub-Riemannian geometry and on the corresponding subelliptic diffusion equation. These studies are motivated by a model of visual perception due to Petitot-Citti-Sarti. We study the invariant Carnot--Caratheodory metrics on SU(2)=S^3, SO(3), and SL(2). We compute the cut loci globally and the distance. This is the first explicit computation of the whole cut locus in sub-Riemannian geometry, except for the Heisenberg group. We present a definition of the hypoelliptic Laplacian that generalizes the Laplace-Beltrami operator. We present a method to compute explicitly the hypoelliptic heat kernels on Lie groups. The main tool is the noncommutative Fourier transform. We study some relevant cases: SU(2), SO(3), SL(2) and the group of rototranslations of the plane. We study the model of visual perception by PCS, for which the reconstruction of a corrupted curve \gamma is the minimization of a cost depending on length and curvature K. We fix starting and ending points as well as initial and final directions. We prove the non-existence of minimizers for \int \sqrt{1+K_\gamma^2} ds. We instead prove existence of minimizers for J=\int \| \dot\gamma(t) \|\sqrt{1+K_\gamma^2} dt if initial and final directions are considered regardless to orientation. We solve globally the problem of minimization of J on the sphere S^2. Some optimal geodesics present cusps. We present an algorithm of image reconstruction based on the model, where the minimization process is replaced by an hypoelliptic heat diffusion, that we solve explicitly. Examples of image reconstruction are provided

Cette thèse est consacrée au développement d'outils de traitement pour l'Imagerie par Résonance Magnétique du Tenseur de Diffusion (IRM-TD). Cette technique d'IRM récente est d'une grande importance pour comprendre le fonctionnement du cerveau ou pour améliorer le diagnostic de pathologies neurologiques. Nous proposons des méthodes de traitement basées sur la géométrie Riemannienne, les équations aux dérivées partielles et les techniques de propagation de front. La première partie de ce travail est théorique. Après des rappels sur le système nerveux humain, l'IRM et la géométrie différentielle, nous étudions l'espace des lois normales multivariées. L'introduction d'une structure Riemannienne sur cet espace nous permet de définir des statistiques et des schémas numériques intrinsèques qui sont à la base des algorithmes proposés dans la seconde partie. Les propriétés de cet espace sont importantes pour l'IRM-TD car les tenseurs de diffusion sont les matrices de covariance de lois normales modélisant la diffusion des molécules d'eau en chaque voxel du milieu imagé. La seconde partie est méthodologique. Nous y introduisons des approches originales pour l'estimation et la régularisation d'IRM-TD. Puis nous montrons comment évaluer le degré de connectivité entre aires corticales et introduisons un modèle statistique d'évolution de surface permettant de segmenter ces images. Finalement, nous proposons une méthode de recalage non-rigide. La dernière partie de cette thèse est consacrée à l'analyse des connexions entre le cortex cérébral et les noyaux gris centraux, impliquées dans des tâches motrices, et à l'étude du réseau anatomo-fonctionnel du cortex visuel humain<br>This thesis deals with the development of new processing tools for Diffusion Tensor Magnetic Resonance Imaging (DT-MRI). This recent MRI technique is of utmost importance to acquire a better understanding of the brain mechanisms and to improve the diagnosis of neurological disorders. We introduce new algorithms relying on Riemannian geometry, partial differential equations and front propagation techniques. The first part of this work is theoretical. After a few reminders about the human nervous system, MRI and differential geometry, we study the space of multivariate normal distributions. The introduction of a Riemannian structure on that space allows us to define statistics and intrinsic numerical schemes that will constitute the core of the algorithms proposed in the second part. The properties of that space are important for DT-MRI since diffusion tensors are the covariance matrices of normal laws modeling the diffusion of water molecules at each voxel of the acquired volume. The second part of this thesis is methodological. We start with the introduction of original approaches for the estimation and regularization of DT-MRI. We then show how to evaluate the degree of connectivity between cortical areas. Next, we introduce a statistical surface evolution framework for the segmentation of those images. Finally, we propose a non-rigid registration method. The last part of this thesis is dedicated to the application of our tools to two important neuroscience problems: the analysis of the connections between the cerebral cortex and the basal ganglia, implicated in motor tasks, and the study of the anatomo-functional network of the human visual cortex

Le flot de Ricci est une équation aux dérivées partielles qui régit l’évolution d’une métrique riemannienne dépendant d’un paramètre de temps sur une variété différentielle. D’abord introduit et étudié par R. Hamilton, il est à l’origine de la solution de la conjecture de géométrisation des variétés compactes de dimension 3 par G. Perelman en 2001. La théorie classique concernant l’existence en temps court des solutions, due à Hamilton et à Shi, garantit (en dimension quelconque) l’existence d’un flot soit sur une variété compacte, soit lorsque la métrique initiale est complète avec une borne sur la norme du tenseur de courbure. En l’absence de cette borne, on conjecture qu’on peut trouver, à partir de la dimension 3, des données initiales pour lesquelles il n’existe pas de solution. Dans cette thèse, on démontre des théorèmes d’existence en temps court du flot sous des hypothèses plus faibles qu’une borne sur la norme du tenseur de courbure. Pour cela, on introduit une construction générale qui, pour une métrique riemannienne g quelconque sur une variété M, pas nécessairement complète, permet de produire une solution de l’équation du flot sur un domaine ouvert D de l’espace-temps M * [0,T] qui contient la tranche de temps initiale, avec g pour donnée initiale. On montre ensuite que sous des hypothèses adaptées sur la métrique g, on contrôle la forme du domaine D. En particulier, lorsque la métrique g est complète, D contient un ensemble de la forme M * [0,t], avec t&gt;0, ce qui revient à dire qu’il existe un flot au sens classique dont la donnée initiale est g. Les « hypothèses adaptées » qui conduisent à des théorèmes d’existence sont de trois types. Dans tout les cas, on suppose une minoration uniforme du volume des boules de rayon au plus 1, à quoi on ajoute : a) en dimension 3, une minoration du tenseur de Ricci, b) en dimension n, une minoration d’une notion de courbure dite « courbure isotrope I » ou bien c) en dimension n, une borne sur la norme du tenseur de Ricci et une hypothèse qui garantit la proximité au sens métrique des boules de rayon au plus 1 avec une boule de même rayon dans un espace métrique obtenu comme le produit cartésien d’un espace de dimension 3 et d’un facteur euclidien de dimension n-3. De plus, avec ces résultats d’existence viennent des estimations sur les propriétés de régularisation du flot quantifiées en fonction des hypothèses sur la donnée initiale. La possibilité ainsi offerte de régulariser, globalement ou localement, pour un temps et avec des estimations quantifiés, une métrique initiale a des conséquence sur les espaces métriques singuliers obtenus comme limites, pour la distance de Gromov-Hausdorff, de suites de variétés satisfaisant uniformément aux conditions a), b) ou c). En effet, des théorèmes de compacité classiques pour le flot de Ricci permettent d’extraire un flot limite, étant donnée une suite de métriques initiales satisfaisant uniformément à ces hypothèses, et possédant donc toutes un flot pour un temps contrôlé. Lorsque les métriques en question approchent, pour la topologie de Gromov-Hausdorff, un espace singulier, cette solution limite s’interprète comme un flot régularisant l’espace singulier en question, et son existence contraint la topologie de cet espace singulier<br>The Ricci Flow is a partial differential equation governing the evolution of a Riemannian metric depending on a time parameter t on a differential manifold. It was first introduced and studied by R. Hamilton, and eventually led to the solution of the Geometrization conjecture for closed three-dimensional manifolds by G. Perelman in 2001. The classical short-time existence theory for the Ricci Flow, due to Hamilton and Shi, asserts, in any dimension, the existence of a flow starting from any initial metric when the underlying manifold in compact, or for any complete initial metric with a bound on the norm of the curvature tensor otherwise. In the absence of such a bound, though, the conjecture is that starting from dimension 3 one can find such initial data for which there is no solution. In this thesis, we prove short-time existence theorems under hypotheses weaker than a bound on the norm of the curvature tensor. To do this, we introduce a general construction which, for any Riemannian metric g (not necessarily complete) on a manifold M, allows us to produce a solution to the equation of the flow on an open domain D of the space-time M * [0,T] which contains the initial time slice, with g as an initial datum. We proceed to show that under suitable hypotheses on g, one can control the shape of the domain D, so that in particular, D contains a subset of the form M * [0,t] with t&gt;0 if g is complete. By « suitable hypothesis », we mean one of the following. In any case, we assume a lower bound on the volume of balls of radius at most 1, plus a) in dimension 3, a lower bound on the Ricci tensor, b) in dimension n, a lower bound on the so-called « isotropic curvature I » or c) in dimension n, a bound on the norm of the Ricci tensor, as well as a hypothesis which garanties the metric proximity of every ball of radius at most $1$ with a ball of the same radius in a metric product between a three-dimensional metric space and a $n-3$ dimensional Euclidian factor. Moreover, with these existence results come estimates on the existence time and regularization properties of the flow, quantified in term of the hypotheses on the initial data. The possibility to regularize metrics, locally or globally, with such estimates has consequences in terms of the metric spaces obtained as limits, in the Gromov-Hausdorff topology, of sequences of manifolds uniformly satisfying a), b) or c). Indeed, the classical compactness theorems for the Ricci Flow allow for the extraction of a limit flow for any sequence of initial metrics uniformly satisfying the hypotheses and thus possessing a flow for a controlled amount of time. In the case when these metrics approach a singular space in the Gromov-Hausdorff topology, such a limit solution can be interpreted as a flow regularizing the singular limit space, the existence of which puts constraints on the topology of this space

Récemment, le développement de systèmes intelligents dédiés pour la compréhension du comportement humain est devenu un axe de recherche très important. En effet, il est très important de comprendre le comportement humain pour rendre les machines capables d'aider et interagir avec les humains. Pour cela, plusieurs approches de l'état de l'art commencent par détecter automatiquement un ensemble de points 2D ou 3D, appelés marqueurs, sur le corps et/ou le visage humain à partir de données visuelles. L’analyse des séquences temporelles de ces marqueurs pose plusieurs défis dus aux erreurs de suivi et aux variabilités temporelles et de pose. Dans cette thèse, nous proposons deux nouvelles représentations spatio-temporelles avec des outils de calcul appropriés pour la compréhension du comportement humain. La première consiste à représenter une séquence temporelle de marqueurs par une trajectoire de matrices de Gram. Les matrices de Gram sont des matrices semi-définies positives de rang fixe et vivent dans un espace non-linéaire dans lequel les outils d’apprentissage automatique conventionnels ne peuvent pas être appliqués directement. Nous évaluons l’efficacité de notre approche dans plusieurs applications, impliquant des marqueurs 2D et 3D de visages et de corps humain, tels que la reconnaissance des émotions à partir des expressions faciales la reconnaissance d’actions et des émotions à partir des données de profondeur 3D. La deuxième représentation proposée dans cette thèse est basée sur les coordonnées barycentriques des marqueurs de visages 2D. Cette représentation permet d’utiliser les outils de calcul et d’apprentissage automatique tels que les techniques d’apprentissage de métrique. Les résultats obtenus en reconnaissance des expressions faciales et en mesure automatique de la sévérité de la dépression à partir du visage montrent tout l’intérêt de la représentation barycentrique combinée à des techniques d’apprentissage automatique. Les résultats obtenus avec les deux méthodes proposées sur des bases de données réelles montrent la compétitivité de nos approches avec les méthodes récentes de l’état de l’art<br>Developing intelligent systems dedicated to human behavior understanding has been a very hot research topic in the few recent decades. Indeed, it is crucial to understand the human behavior in order to make machines able to interact with, assist, and help humans in their daily life.. Recent breakthroughs in computer vision and machine learning have made this possible. For instance, human-related computer vision problems can be approached by first detecting and tracking 2D or 3D landmark points from visual data. Two relevant examples of this are given by the facial landmarks detected on the human face and the skeletons tracked along videos of human bodies. These techniques generate temporal sequences of landmark configurations, which exhibit several distortions in their analysis, especially in uncontrolled environments, due to view variations, inaccurate detection and tracking, missing data, etc. In this thesis, we propose two novel space-time representations of human landmark sequences along with suitable computational tools for human behavior understanding. Firstly, we propose a representation based on trajectories of Gram matrices of human landmarks. Gram matrices are positive semi-definite matrices of fixed rank and lie on a nonlinear manifold where standard computational and machine learning techniques could not be applied in a straightforward way. To overcome this issue, we make use of some notions of the Riemannian geometry and derive suitable computational tools for analyzing Gram trajectories. We evaluate the proposed approach in several human related applications involving 2D and 3D landmarks of human faces and bodies such us emotion recognition from facial expression and body movements and also action recognition from skeletons. Secondly, we propose another representation based on the barycentric coordinates of 2D facial landmarks. While being related to the Gram trajectory representation and robust to view variations, the barycentric representation allows to directly work with standard computational tools. The evaluation of this second approach is conducted on two face analysis tasks namely, facial expression recognition and depression severity level assessment. The obtained results with the two proposed approaches on real benchmarks are competitive with respect to recent state-of-the-art methods

Cette thèse s'intéresse à des aspects divers des structures affines complexes branchées sur les surfaces de Riemann.Dans une première partie, nous étudions un invariant algébrique de ces structures appelé holonomie, qui est une représentation du groupe fondamental de la surface sous-jacente dans le groupe affine. Nous démontrons un théorème caractérisant les représentations se réalisant comme l'holonomie d'une structure affine.Nous nous intéressons ensuite à la géométrie de certains espaces de modules de telles structures qui viennent naturellement avec une structure hyperbolique complexe. Nous décrivons cette géométrie en terme de dégénérescences de structures affines.Enfin, nous regardons une sous-classe de structures affines dont chaque élément induit une famille de feuilletages sur la surface sous-jacente. Nous relions ces feuilletages à des systèmes dynamiques unidimensionnels appelés échanges d'intervalles affines et nous étudions un cas particulier en détails<br>This thesis deals with several aspects of branched, complex affine structures on Riemann surfaces.In a first chapter, we study an algebraic invariant of these structures called holonomy, which is a representation of the fundamental group of the underlying surface into the affine group. We prove a theorem characterising such representations that arise as the holonomy of an affine structure.In a second part, we study certain moduli spaces of affine tori which happen to have an additional complex hyperbolic structure. We analyse the geometry of this structures in terms of degenerations of the underlying affine tori.Finally, we narrow our interest to a subclass of affine structures each element of which inducing a family of foliations on the underlying topological surface. We link these foliations to 1-dimensional dynamical systems called affine interval exchange transformations and study a particular case in details

Cette thèse s'appuie sur et reflète l'interaction entre la théorie des modèles et la géométrie de Berkovich. En utilisant les méthodes de Hrushovski et Loeser, nous montrerons que plusieurs phénomènes topologiques concernant des analytifications de variétés sont contrôlés par certains complexes simpliciaux contenus dans les analytifications. Ce travail comporte les résultats suivants. Soit $k$ un corps algébriquement clos et complet pour une valuation non-archimédienne non-triviale à valeurs réelles. 1) Soit $\phi : C' \to C$ un morphisme fini entre deux courbes projectives, lisses et irréductibles. Le morphisme $\phi$ induit un morphisme $\phi^{an} : C'^{an} \to C^{an}$ entre les deux analytifications. Nous construisons une paire de rétractions par déformations qui sont compatible pour le morphisme $\phi^{an}$. Les images des déformations $\Upsilon_{C'^{an}}$, $\Upsilon_{C^{an}$ sont des sous-espaces fermés de $C'^{an}$ and $C^{an}$ et homéomorphes à des graphes finis. Ce type de sous-espace est appelé \emph{squelette}. En outre, les espaces analytiques $C'^{an} \smallsetminus \Upsilon_{C'^{an}}$ et $C^{an} \smallsetminus \Upsilon_{C^{an}}$ se décomposent en une union disjointe de copies de disques unités de Berkovich. Un squelette $\Upsilon \subset C^{an}$ peut-être décomposé en un ensemble des sommets et un ensemble d'arêtes et on peut définir son genre $g(\Upsilon)$.Nous montrons que $g(\Upsilon)$ est un invariant bien défini de la courbe $C$. On appelle cet invariant $g^{an}(C)$. Le morphisme $\phi^{an}$ induira un morphisme $\Upsilon_{C'^{an}} \to \Upsilon_{C^{an}}$ entre les deux squelettes. Nous montrons que le genre du squelette $\Upsilon_{C'^{an}}$ peut être calculé en utilisant certains invariants associés aux points de $\Upsilon_{C^{an}}$. 2) Soit $\phi$ un endomorphisme fini de $\mathbb{P}^1_k$. Soit $x \in \mathbb{P}^1_k(k)$ et $f(x)$ le rayon de la plus grande boule de Berkovich de centre $x$, sur laquelle le morphisme $\phi^{an}$ est une fibration topologique. Nous voyons que la fonction $f : \mathbb{P}_k^1(k) \to \mathbb{R}_{\geq 0}$ est contrôlée par un graphe fini et non-vide contenu dans $\mathbb{P}^{1,an}_k$. Nous montrons que ce résultat peut être généralisé au cas d'un morphisme fini $\phi : V' \to V$ entre deux variétés intégrales, projectives avec $V$ normale<br>This thesis is a reflection of the interaction between Berkovich geometry and model theory. Using the results of Hrushovski and Loeser, we show that several interesting topological phenomena that concern the analytifications of varieties are governed by certain finite simplicial complexes embedded in them. Our work consists of the following two sets of results. Let k be an algebraically closed non-Archimedean non trivially real valued field which is complete with respect to its valuation. 1) Let $\phi : C' \to C$ be a finite morphism between smooth projective irreducible $k$-curves.The morphism $\phi$ induces a morphism $\phi^{an} : C'^{an} \to C^{an}$ between the Berkovich analytifications of the curves. We construct a pair of deformation retractions of $C'^{an}$ and $C^{an}$ which are compatible with the morphism $\phi^{\mathrm{an}}$ andwhose images $\Upsilon_{C'^{an}}$, $\Upsilon_{C^{an}}$ are closed subspaces of $C'^{an}$, $C^{an}$ that are homeomorphic to finite metric graphs. We refer to such closed subspaces as skeleta.In addition, the subspaces $\Upsilon_{C'^{an}}$ and $\Upsilon_{C^{an}}$ are such that their complements in their respective analytifications decompose into the disjoint union of isomorphic copies of Berkovich open balls. The skeleta can be seen as the union of vertices and edges, thus allowing us to define their genus. The genus of a skeleton in a curve $C$ is in fact an invariant of the curve which we call $g^{an}(C)$. The pair of compatible deformation retractions forces the morphism $\phi^{an}$ to restrict to a map $\Upsilon_{C'^{an}} \to \Upsilon_{C^{an}}$. We study how the genus of $\Upsilon_{C'^{an}}$ can be calculated using the morphism $\phi^{an}_{|\Upsilon_{C'^{an}}$ and invariants defined on $\Upsilon_{C^{an}}$. 2) Let $\phi$ be a finite endomorphism of $\mathbb{P}^1_k$. Given a closed point $x \in \mathbb{P}^1_k$, we are interested in the radius $f(x)$ of the largest Berkovich open ball centered at $x$ over which the morphism $\phi^{\mathrm{an}}$ is a topological fibration. Interestingly, the function $f : \mathbb{P}_k^1(k) \to \mathbb{R}_{\geq 0}$ admits a strong tameness property in that it is controlled by a non-empty finite graph contained in $\mathbb{P}^{1,an}_k$. We show that this result can be generalized to the case of finite morphisms $\phi : V' \to V$ between integral projective $k$-varieties where $V$ is normal

Orientador: Parham Salehyan<br>Banca: Eduardo de Sequeira Esteves<br>Banca: Jéfferson Luiz Rocha Bastos<br>Resumo: O objetivo principal deste trabalho é estudar o Teorema de Riemann-Roch, um dos resultados fundamentais na teoria de curvas algébricas, e apresentar algumas de suas aplicações. Este teorema é uma importante ferramenta para a classificação das curvas algébricas, pois relaciona propriedades algébricas e topológicas. Daremos uma descrição das curvas algébricas de gênero g, 1≤ g ≤ 5, e faremos um breve estudo dos pontos de inflexão de um sistema linear sobre uma curva algébrica<br>Abstract: The main purpose of this work is to discuss The Riemann-Roch Theorem, wich is one of the most important results of the theory algebraic curves, and to present some applications. This theorem is an important tool of the classification of algebraic curves, sinces relates algebraic and topological properties. We will describle the algebraic curves of genus g, 1≤ g ≤ 5, and also study inflection points of a linear system on an algebraic curve<br>Mestre

Esta tesis presenta una visión global y prácticamente autocontenida de los avances que se llevaron a cabo en la décadas de los años 1960 y 1970 con respecto al estudio de las estructuras de contacto en variedades diferenciables. Nuestro objetivo principal sería exhibir explícitamente estructuras métricas de contacto en las denominadas variedades de Brieskorn, que surgen como el conjunto de ceros de los llamados polinomios de Brieskorn-Pham intersecado con la esfera unitaria. Para ello comenzaremos desarrollando a grandes rasgos los conceptos relacionados a la geometría simpléctica, la geometría compleja y las variedades de Kähler. Luego realizaremos un esbozo de prueba del teorema de Boothby-Wang, que constituye una generalización de la fibración de Hopf. A continuación presentaremos la construcción de estructuras métricas de contacto, en particular, las denominadas estructuras de Sasaki. El objetivo de ello es obtener estructuras de Sasaki en las variedades de Brieskorn, las cuales exhibiremos en coordenadas a fin de obtener un procedimiento para construirlas en una variedad de Brieskorn arbitraria. Por último, relacionaremos lo estudiado con la fibración de Boothby-Wang para probar que las estructuras construidas pueden ser proyectadas como hipersuperficies en el espacio proyectivo complejo. Debido a la naturaleza de las nociones presentadas, se espera que el lector tenga un conocimiento elemental de la geometría riemanniana.<br>Tesis

On s'intéresse dans ce travail au théorème de Grothendieck-Riemann-Roch. Grothendieck et son école en ont démontré une forme très générale dans les années 60 tout en conjecturant l'existence d'une forme encore plus générale. Nous posons une conjecture intermédiaire entre les résultats connus et les conjectures les plus générales de Grothendieck, puis nous la démontrons dans deux cas particuliers. Plus précisément on conjecture que le théorème de Grothendieck-Riemann-Roch est vrai pour un morphisme propre localement d'intersection complète entre deux schémas divisoriels d'égale caractéristique. On démontre des cas particuliers de cette conjecture, dans le cas de la caractéristique positive d'une part, dans le cas où les schémas sont supposés réguliers et tels que le polynôme $T^k-1$ y ait $k$ racines distinctes d'autre part. Le théorème de Grothendieck-Riemann-Roch étant équivalent au théorème d'Adams-Riemann-Roch modulo torsion, on démontre des résultats de type Adams-Riemann-Roch pour en déduire des résultats de type Grothendieck-Riemann-Roch.

Clasificamos las superficies mínimas del grupo de Heisenberg, H₃, que son invariantes con respecto a un subgrupo unidimensional de isometrías de (H₃, g), haciendo uso de las técnicas de los grupos de transformaciones.

El trabajo trata sobre encontrar una representación para superficies espaciales inmersas en L3 con curvatura media constante y con métrica de Lorentz. Basado en el paper [1], esto conlleva a estudiar la aplicación de Gauss, la ecuación de Beltrami y la fórmula de representación para la superficie espaciales inmersa en L3, en función de la aplicación de Gauss y la curvatura media de la superficie. Entre otros, se ha utilizado principalmente las bibliografías [2], [3], [7], [13], [14].<br>Tesis

La reconnaissance des biomètries douces (genre, âge, etc.)trouve ses applications dans plusieurs domaines. Les approches proposéesse basent sur l’analyse de l’apparence (images 2D), très sensiblesaux changements de la pose et à l’illumination, et surtout pauvre en descriptionsmorphologiques. Dans cette thèse, nous proposons d’exploiterla forme 3D du visage. Basée sur une approche Riemannienne d’analysede formes 3D, nous introduisons quatre descriptions denses à savoir: lasymétrie bilatérale, la moyenneté, la configuration spatiale et les variationslocales de sa forme. Les évaluations faites sur la base FRGCv2 montrentque l’approche proposée est capable de reconnaître des biomètries douces.A notre connaissance, c’est la première étude menée sur l’estimation del’âge, et c’est aussi la première étude qui propose d’explorer les corrélationsentre les attributs faciaux, à partir de formes 3D<br>Soft-Biometric (gender, age, etc.) recognition has shown growingapplications in different domains. Previous 2D face based studies aresensitive to illumination and pose changes, and insufficient to representthe facial morphology. To overcome these problems, this thesis employsthe 3D face in Soft-Biometric recognition. Based on a Riemannian shapeanalysis of facial radial curves, four types of Dense Scalar Field (DSF) featuresare proposed, which represent the Averageness, the Symmetry, theglobal Spatiality and the local Gradient of 3D face. Experiments with RandomForest on the 3D FRGCv2 dataset demonstrate the effectiveness ofthe proposed features in Soft-Biometric recognition. Furtherly, we demonstratethe correlations of Soft-Biometrics are useful in the recognition. Tothe best of our knowledge, this is the first work which studies age estimation,and the correlations of Soft-Biometrics, using 3D face

L'ensemble des opérateurs de Cauchy-Riemann réels sur un fibré vectoriel complexe N muni d'une structure réelle cN au-dessus d'une courbe réelle est un espace affine de dimension infinie. L'union des déterminants de ces opérateurs est un fibré en droites réelles au-dessus de cet espace. L'objet de cette thèse est l'étude de l'action des automorphismes du fibré (N, cN) sur les orientations de ce fibré déterminant ainsi que de ses conséquences sur l'orientabilité des espaces de modules de courbes réelles dans une variété symplectique réelle. Nous commençons par interpréter l'action des automorphismes qui induisent l'identité sur le fibré en droites complexes det(N) en termes d'action sur les structures Pin± de la partie réelle de N. Nous remarquons ensuite qu'un automorphisme au-dessus de l'identité agit sur les classes de bordisme de structures Spin réelles de la courbe et nous utilisons cette action afin d'obtenir une description en termes topologiques de l'action sur les orientations du fibré déterminant. Enfin, pour comprendre l'action des automorphismes de (N, cN) qui ne relèvent pas l'identité, nous introduisons la notion de relevé d'un difféomorphisme de la courbe associé à un diviseur compatible avec (N, cN) et nous calculons le signe de l'action d'un tel relevé sur les orientations du fibré déterminant. Dans une dernière partie, nous appliquons les résultats obtenus à l'étude de l'orientabilité des espaces de modules de courbes réelles dans des variétés symplectiques réelles. Nous calculons en particulier la première classe de Stiefel-Whitney de l'espace de modules des courbes réelles dans l'espace projectif complexe de dimension trois.

Made available in DSpace on 2016-06-02T20:28:25Z (GMT). No. of bitstreams: 1 3238.pdf: 1181256 bytes, checksum: eca19c8ec7e2ed4661569fb38346ccb7 (MD5) Previous issue date: 2010-04-24<br>Financiadora de Estudos e Projetos<br>In 1989, H.Karcher elaborated a method for the construction of minimal surfaces, denominated reverse construction method given in [10]. In that work it was rewritten the theory of elliptic functions using an approach more geometrical than analytical. This allows to better control the behavior and the image values of those functions, making it easier his application in minimal surfaces. In this master s thesis, we will present basic tools of the theory of symmetric elliptic functions, describing explicitly the symmetric &#8472;-Weierstraß and the function &#947;, that will be applied in the reverse construction method for an example of minimal surface.<br>Em 1989, H.Karcher elaborou um método para a construção de superfícies mínimas, denominada método de construção reversa dado em [10]. Nesse trabalho foi reescrita a teoria de funções elíticas utilizando uma abordagem mais geométrica do que analítica. Desse modo, ele conseguiu controlar o comportamento e os valores imagens dessas funções, facilitando sua aplicação em superfícies mínimas. Neste trabalho de mestrado, apresentamos ferramentas básicas da teoria de funções elíticas simétricas, descrevendo explicitamente a &#8472;-Weierstraß simétrica e a função &#947;, que serão aplicadas no método de construção reversa para um exemplo de superfície mínima.

The aim of this thesis is to study in detail the work of S. Kobayashi on the Riemannian geometry on principal S1-bundles. To be more precise, we explain how to obtain metrics with constant scalar curvature on these bundles. The method that we use is based in [18]. The basic idea behind Kobayashi’s construction is to slightly deform the Hopf fibration S1 ‹→ S2n+1 −→ CPn in a such a way that the corresponding sectional curvatures are not far from the produced by the standard metrics on the sphere and the complex projective space on the Hopf fibration. This deformations can be controlled applying the notions of Riemaniann and Kahlerian pinching (see Chapter 3). Furthermore, thanks to a technique developed by Hatakeyama in [14], it is possible to obtain less generic metrics but with a larger set of symmetries on the total space: Sasaki metrics. Actually, If one chooses as a base space a K¨ahler-Einstein manifold with positive scalar curvature one can obtain a Sasaki-Einstein metric.<br>Tesis