Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-29T13:28:57Z No. of bitstreams: 1 douglasferreirabrum.pdf: 632780 bytes, checksum: 1da883a558292ba219387c3fdf6f98af (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-29T19:37:42Z (GMT) No. of bitstreams: 1 douglasferreirabrum.pdf: 632780 bytes, checksum: 1da883a558292ba219387c3fdf6f98af (MD5) / Made available in DSpace on 2017-05-29T19:37:42Z (GMT). No. of bitstreams: 1 douglasferreirabrum.pdf: 632780 bytes, checksum: 1da883a558292ba219387c3fdf6f98af (MD5) Previous issue date: 2013-08-30 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Esse trabalho visa descrever o espaço de Módulos de m-uplas geodésicas complexas distintas em H2c nos casos regular, especial e degenerado. Para tal fim faremos uso da matriz de Gram e dos invariantes (d-invariantes, δ-invariantes, invariante angular e invariantes parabólicos) que descrevem unicamente a classe de congruência de PU(2, 1) de m-uplas ordenadas de geodésicas complexas distintas nos diferentes casos supracitados. / This work aims to describe the Modules space of m-tuples distinct complex geodesics in H2c in the cases regular, special and degenerate. To this end we use the Gram matrix and the invariant (d-invariant, δ-invariants, angular invariant and parabolic invariants) that define uniquely the PU(2,1)-congruence class of ordered m-uplas of distinct complex geodesics in the different cases above.

Les espaces de modules de Riemann répondent au problème de la classification des surfaces de Riemann compactes d'un genre donné. Le sujet de cette thèse est la cohomologie de l'espace des modules des courbes d'un genre donné avec un certain nombre de points marqués. La description de cet anneau a été initiée par D. Mumford puis C. Faber avait proposé une description de l'anneau tautologique des espaces de modules sans points marqués. Une première source de relations provient des relations A. Pixton démontrées par A. Pixton, R. Pandharipande et D. Zvonkine mais on ne sait pas si elles sont complètes. Une autre source de relations utilisée dans ce travail sont les relations de A. Buryak, S. Shadrin et D. Zvonkine. Avant cette thèse, il y avait peu de résultats sur l'anneau tautologique d'espaces de modules de courbes avec un nombre quelconque de points marqués. Cette thèse donne une description complète des l'anneaux tautologiques des espaces de modules de courbes de genres 0, 1, 2, 3 et 4. Un des résultats ayant demandé beaucoup de travail est le groupe de degré 2 de l'anneau tautologique des espaces de modules de courbes lisses de genre 4. Ce groupe demande un travail sur l'annulation de certaines classes tautologiques sur le bord de la compactification de Deligne-Mumford de l'espace des modules en plus d'un astucieux travail numérique. L'espace des modules des courbes réelles de genre 0 et sa théorie de l'intersection sont également étudiés. On peut alors démontrer plusieurs résultats analogues à ceux obtenus dans le cas complexe comme l'équation de la corde. On démontre une formule donnant les nombres d'intersection. / The problem of the moduli spaces of compact Riemann surfaces is the problem of the classification of compact Riemann surfaces of a certain genus. The topic of this thesis is the cohomology of the moduli spaces of curves of a certain genus with marked points and more precisely its subbring called tautological ring. The description of the tautological ring has been initiated by D. Mumford, then C. Faber conjectured a description of the moduli space of curves without marked points. A source of tautological relations are Pixton's relations proven by A. Pixton, R. Pabndharipande and D. Zvonkine. Another source of relations are relations of A. Buryak, S. Shadrin and D. Zvonkine. Before this thesis, there were only few results on the tautological ring of curves with any number of marked points. This thesis gives a complete description of the tautological rings of moduli curves of genera 0, 1, 2, 3 and 4 with any number of marked points. A result which needed a lot of work is the group of degree 2 of the tautological ring of the moudli space of smooth curves of genus 4. We need to work on the vanishing of some tautological classes on the boundary of the Deligne-Mumford compactification of the moduli space of curves and a clever numerical work.The moduli space of real curves of genus 0 and its intersection theory are also studied. Then we can show several results which are analogous to results in the complex case like the string equation. One result of this thesis is a formula giving intersection numbers of products of xi classes.x

No description available.

Mirror symmetry is a phenomenon from physics that has inspired a lot of interesting mathematics. In the Landau-Ginzburg setting, we have two constructions, the A and B models, which are created based on a choice of an affine singularity with a group of symmetries. Both models are vector spaces equipped with multiplication and a pairing (making them Frobenius algebras), and they are also Frobenius manifolds. We give a result relating stabilization of singularities in classical singularity to its counterpart in the Landau-Ginzburg setting. The A model comes from so-called FJRW theory and can be de fined up to a full cohomological field theory. The structure of this model is determined by a generating function which requires the calculation of certain numbers, which we call correlators. In some cases the their values can be computed using known techniques. Often, there is no known method for finding their values. We give new computational methods for computing concave correlators, including a formula for concave genus-zero, four-point correlators and show how to extend these results to find other correlator values. In many cases these new methods give enough information to compute the A model structure up to the level of Frobenius manifold. We give the FJRW Frobenius manifold structure for various choices of singularities and groups.

[pt] Espaços de moduli de polígonos em R(3) com comprimento dos lados fixados é um exemplo amplamente estudado de variedade simplética. Esses espaços podem ser descritos como quociente simplético de um número finito de órbitas coadjuntas pelo grupo SU(2). Nesta tese esses espaços de moduli são identificados como folhas simpléticas de uma variedade de Poisson que pode ser construída como quociente. Essa construção é a seguir generalizada ao caso de um produto de um número finito de órbitas coadjuntas pelo grupo SU(n), e o resultado principal desse trabalho de tese descreve como esses espaços de moduli de polígonos generalizados formam uma folheação em folhas simpléticas de uma variedade de Poisson. / [en] Moduli spaces of polygons in R(3)with fixed sides length are a widely studied example of symplectic manifold that can be described as the symplectic quotient of a finite number of SU(2)−coadjoint orbits by the diagonal action of the group SU(2). In this thesis these spaces are identified as the symplectic leaves of a Poisson manifold, that can itself be obtained by a quotient procedure. The construction is then generalized to the case of the quotient of a product of finitely many SU(n)−coadjoint orbits by the diagonal action of SU(n), and the main result of this thesis describes how these moduli spaces of generalized polygons fit together as the symplectic leaves of a quotient Poisson manifold.

Es sei A_2 der toroidal kompaktifizierte Modulraum prinzipal polarisierter komplexer abelscher Flächen, und M_k(Sp_4(Z)) das Geradenbündel Siegel'scher Modulformen von Gewicht k auf A_2, versehen mit der Petersson-Metrik. Betrachtet man A_2 als komplexe Faser einer arithmetischen Varietät über Spec(Z), und M_k(Sp_4(Z)) als das von einem Geradenbündel auf dieser arithmetischen Varietät induzierte Geradenbündel, so kann man die Frage nach dem arithmetischen Grad dieses Geradenbündels stellen. Wir stellen nachfolgend den Grad als Ausdruck in speziellen Werten der logarithmischen Ableitung der Riemann'schen Zeta-Funktion dar. Der arithmetische Grad setzt sich aus einem Beitrag vom Schnitt über den endlichen Fasern und einem Integral von Green'schen Formen über die komplexe Faser zusammen. Die Berechnung des von der komplexen Faser A_2 induzierten Anteils am arithmetischen Grad erfolgt durch eine spezifische Wahl von Schnitten von M_k(Sp_4(Z)), deren Eigenschaften bekannt oder durch ihre Darstellung als Polynome in Theta-Funktionen ableitbar sind. Mittels eines induktiven Arguments werden wir das Integral über das Stern-Produkt der zugehörigen Green'schen Formen auf eine Summe von Integralen über spezielle Zykel zurückführen, die beim sukzessiven Schneiden der zu den Schnitten gehörigen Divisoren auftauchen. Bei diesem Prozess entstehen Randterme in Form von Integralen um den toroidalen Rand. Wir werden zeigen, dass diese verschwinden, indem wir Minkowski-Theorie anwenden und eine bestimmte Wahl der Teilung der Eins treffen, die in der arithmetischen Schnitttheorie für logarithmisch singuläre Metriken auftaucht. Die Integrale über die speziellen Zykel berechnen wir durch Zurückführen auf ein Resultat von Kudla sowie auf eine modulare Version der Jensen-Formel. / Let A_2 be the toroidally compactified moduli stack of principally polarized complex abelian surfaces, and let M_k(Sp_4(Z)) be the line bundle of Siegel modular forms of weight k on A_2, equipped with the Petersson metric. Viewing A_2 as the complex fibre of an arithmetic variety over Spec(Z), and M_k(Sp_4(Z)) as the complex line bundle induced by a line bundle on this arithmetic variety, we can ask for the arithmetic degree of this line bundle. We will state a formula for the arithmetic degree in terms of special values of the logarithmic derivative of the Riemann zeta-function. The arithmetic degree consists of a contribution from intersection over Spec(Z), and from an integral of Green forms over the complex fibre. The computation of the summand of the arithmetic degree coming from the complex fibre A_2 will be approached by making a specific choice of sections of M_k(Sp_4(Z)), whose behaviour is well-known or can be worked out by their representation via theta-functions. With an induction argument, we will trace back the integral over the star-product of the corresponding Green forms to a sum of integrals over particular cycles on A_2 coming from the successive intersection of the divisors of these sections, as well as some boundary terms in the form of integrals around the toroidal boundary. We will prove that the boundary terms vanish, using Minkowski theory and a specific choice of the partition of unity that appears in arithmetic intersection theory for logarithmically singular metrics. The integrals over the special cycles will be traced back to results of Kudla and an application of a modular version of Jensen's formula.

Nesta disertação estudamos algumas ferramentas básicas relacionadas aos espaços de Teichmüller. Introduzimos o espaço de Teichmüller de gênero g ≥ 1, denotado por Tg. O objetivo principal é construir as coordenadas de Fenchel-Nielsen ωG : Tg → R3g-3+ × R3g-3 para cada grafo trivalente marcado G. / In this dissertation we study some basic tools related to Teichmüller space. We introduce the Teichmüller space of genus g ≥ 1, denoted by Tg. The main goal is to construct the Fenchel-Nielsen coordinates ωG : Tg → R3g-3+ × R3g-3 to each marked cubic graph G.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-02-25T10:58:39Z No. of bitstreams: 1 rafaeldasilvalima.pdf: 522996 bytes, checksum: 5a2e8f3b92223160a83315d7eb1cac50 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2016-03-03T13:26:29Z (GMT) No. of bitstreams: 1 rafaeldasilvalima.pdf: 522996 bytes, checksum: 5a2e8f3b92223160a83315d7eb1cac50 (MD5) / Made available in DSpace on 2016-03-03T13:26:29Z (GMT). No. of bitstreams: 1 rafaeldasilvalima.pdf: 522996 bytes, checksum: 5a2e8f3b92223160a83315d7eb1cac50 (MD5) Previous issue date: 2014-04-11 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O objetivo desse trabalho, é a construção do espaço de módulos para o conjunto de quá- druplas ordenadas de pontos na fronteira do espaço hiperbólico complexo. Para isso, utilizaremos o conceito de matriz de Gram como critério de congruência, e a parametrização do espaço de con gurações será feito pelo invariante angular de Cartan e a razão-cruzada. Exempli caremos algumas situações geométricas. / The aim of this work is the construction of a moduli space for the con guration space ordered quadruples of points on the boundary of the complex hyperbolic space. For this use the concept of Gram matrix as a criterion of congruence, and parametrization the con guration space will be done by the Cartan invariant and cross-ratio. Will be exempli ed some geometric situations.

Nesta disertação estudamos algumas ferramentas básicas relacionadas aos espaços de Teichmüller. Introduzimos o espaço de Teichmüller de gênero g ≥ 1, denotado por Tg. O objetivo principal é construir as coordenadas de Fenchel-Nielsen ωG : Tg → R3g-3+ × R3g-3 para cada grafo trivalente marcado G. / In this dissertation we study some basic tools related to Teichmüller space. We introduce the Teichmüller space of genus g ≥ 1, denoted by Tg. The main goal is to construct the Fenchel-Nielsen coordinates ωG : Tg → R3g-3+ × R3g-3 to each marked cubic graph G.

Cette thèse traite de plusieurs aspects algorithmiques des courbes algébriques. La première partie décrit et implémente en Magma un algorithme de calcul des tordues pour les courbes sur les corps finis et en étudie la complexité. Dans le cas hyperellitptique, il s’agit du premier algorithme complet pour faire cela en tout genre. La deuxième partie construit des familles représentatives pour les courbes non hyperelliptiques de genre 3 afin de permettre leur énumération efficace en lien avec le problème de l’obstruction de Serre. Cette partie a fait l’objet d’une publication dans ANTS et une annexe de la thèse est constituée d’un préprint étudiant un modèle statistique pour l’interprétation des données obtenues. La dernière partie de la thèse étudie les invariants et covariants des formes binaires en lien avec la description de l’espace de modules des courbes de genre 2. On y décrit en particulier une nouvelle opération pour engendrer des covariants en petite caractéristique. On étudie aussi l’application d’une nouvelle stratégie (dite de Geyer-Sturmfels) pour obtenir les algèbres de séparants et on l’applique au cas du degré 4 et du degré 6. Enfin, un dernier chapitre montre la validité d’un algorithme de reconstruction pour les courbes de genre 2 à partir de leurs invariants en toute caractéristique différente de 2 et l’implémente en SAGE. / This thesis addresses several algorithmic aspects of algebraic curves.The first part describe and plug in Magma a computational algorithm of twists for the curves over finite fields and study it's complexity. In the hyperelliptic case, it is the first complete algorithm to do this in all genus. The second part builts representatives family for the non hyperelliptic curves of genus 3 to enable them effective enumeration in connection with the Serre obstruction problem. This part has been published in ANTS and an annex of this thesis is made up of a preprint studing a statistic model for interpreting the data obtained.The last part of the thesis studies the invariants and covariants of binary forms in connexion with the description of the moduli space of curves of genus 2. A new operation in particular is described to generate covariants in small characteristic. We study to the implementation of a new strategy (called Geyer-Sturmfels) to get the algebras of separants and we apply it of the case of degree 4 ans 6. Finally, the last chapter shows the validity of a reconstruction algorithm for genus 2 curves from their invariants in all characteristic diferent from 2 and implements it in SAGE .