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Aryasomayajula, Naga Venkata Anilatmaja. "Bounds for Green's functions on hyperbolic Riemann surfaces of finite volume". Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://dx.doi.org/10.18452/16828.
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Im Jahr 2006, in einem Papier in Compositio Titel "Bounds auf kanonische Green-Funktionen" J. Jorgenson und J. Kramer, haben optimale Schranken für den hyperbolischen und kanonischen Green-Funktionen auf einem kompakten hyperbolischen Riemannschen Fläche definiert abgeleitet. Diese Schätzungen wurden im Hinblick auf abgeleitete Invarianten aus hyperbolischen Geometrie der Riemannschen Fläche. Als Anwendung abgeleitet sie Schranken für die kanonische Green-Funktionen durch Abdeckungen und für Familien von Modulkurven. In dieser Arbeit erweitern wir ihre Methoden nichtkompakten hyperbolischen Riemann Oberflächen und leiten ähnliche Schranken für den hyperbolischen und kanonischen Green-Funktionen auf einem nichtkompakten hyperbolischen Riemannschen Fläche definiert.In 2006, in a paper in Compositio titled "Bounds on canonical Green''s functions", J. Jorgenson and J. Kramer have derived optimal bounds for the hyperbolic and canonical Green''s functions defined on a compact hyperbolic Riemann surface. These estimates were derived in terms of invariants coming from hyperbolic geometry of the Riemann surface. As an application, they deduced bounds for the canonical Green''s functions through covers and for families of modular curves. In this thesis, we extend their methods to noncompact hyperbolic Riemann surfaces and derive similar bounds for the hyperbolic and canonical Green''s functions defined on a noncompact hyperbolic Riemann surface.
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Cook, Joseph. "Properties of eigenvalues on Riemann surfaces with large symmetry groups". Thesis, Loughborough University, 2018. https://dspace.lboro.ac.uk/2134/36294.
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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.
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Ghazouani, Selim. "Structures affines complexes sur les surfaces de Riemann". Thesis, Paris Sciences et Lettres (ComUE), 2017. http://www.theses.fr/2017PSLEE022/document.
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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 détailsThis 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
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Nualart, Riera Joan. "On the hyperbolic uniformization of Shimura curves with an Atkin-Lehner quotient of genus 0". Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/396134.
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The main goal of this thesis is to contribute to the explicit hyperbolic uniformization of Shimura curves. We will restrict to the case of curves attached to Eichler orders in rational quaternion algebras whose maximal Atkin-Lehner quotient has genus 0, which despite multiple differences bears some resemblance to the classical modular case. We will provide an approach to obtain an explicit uniformization of these curves and some of their covers, together with several applications. We will illustrate all the applications with plenty of examples.L’objectiu principal d’aquesta tesi és contribuir a la uniformització hiperbòlica explícita de les corbes de Shimura. Ens restringim a les corbes associades a ordres d’Eichler dins d’àlgebres de quaternions racionals tals que el seu quocient pel grup d’involucions d’Atkin-Lehner és de gènere 0. Aquest cas,tot I que presenta nombroses diferències amb el cas modular clàssic, també hi té certes similituds. Utilitzem aquest fet per a discutir una aproximació al problema de l’obtenció d’uniformitzacions hiperbòliques explícites d’aquestes corbes i d’alguns recobriments, així com també algunes aplicacions, que il·lustrem amb abundants exemples. Per a entendre millor el problema, començarem introduint breument el seu rerefons històric. Després explicarem en detall les nostres contribucions i el contingut de la memòria.
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Aryasomayajula, Naga Venkata Anilatmaja [Verfasser], Jürg [Akademischer Betreuer] Kramer, Robin de [Akademischer Betreuer] Jong i Jay [Akademischer Betreuer] Jorgenson. "Bounds for Green's functions on hyperbolic Riemann surfaces of finite volume / Naga Venkata Anilatmaja Aryasomayajula. Gutachter: Jürg Kramer ; Robin de Jong ; Jay Jorgenson". Berlin : Humboldt Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://d-nb.info/1043593225/34.
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Alves, Alessandro Ferreira. "Análise dos emparelhamentos de arestas de polígonos hiperbólicos para a construção de constelações de sinais geometricamente uniformes". [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/261080.
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Orientador: Reginaldo Palazzo JuniorTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação
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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'
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
Doutorado
Telecomunicações e Telemática
Doutor em Engenharia Elétrica
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Finski, Siarhei. "On some problems of holomorphic analytic torsion". Thesis, Sorbonne Paris Cité, 2019. https://theses.md.univ-paris-diderot.fr/FINSKI_Siarhei_va.pdf.
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Le but de cette thèse est d'étudier la torsion analytique dans deux contextes différents. Dans le premier contexte, on étudie l'asymptotique de la torsion analytique, quand un fibré vectoriel holomorphe hermitien est tordué par une puissance croissant du fibré en droites positif. Dans le deuxième contexte, on généralise la théorie de la torsion analytique pour des surfaces de Riemann avec des pointes hyperboliques. Motivé par des singularités de la métrique complète de courbure scalaire constante -1 sur des surfaces de Riemann stables épointées, on demande que la métrique sur la surface de Riemann soit lisse seulement en dehors d'un nombre fini des points au voisinage auxquelles elle peut avoir des singularités comme la métrique de Poincaré sur un disque épointé. On fixe un fibré vectoriel holomorphe hermitien qui peut avoir au pire des singularités logarithmiques au voisinage des points marqués. Pour ces données, en renormalisant la trace de l'opérateur de la chaleur, on construit la torsion analytique et on étudie ces propriétésIn the first context, we study the asymptotics of the analytic torsion, when a Hermitian holomorphic vector bundle is twisted by an increasing power of a positive line bundle. In the second context, we generalize the theory of analytic torsion for surfaces with hyperbolic cusps. Motivated by singularities appearing in complete metrics of constant scalar curvature -1 on stable Riemann surfaces, we suppose that the metric on the surface is smooth outside a finite number points in the neighborhood of which it can to have singularities like Poincaré metric has on a punctured disc. We fix a Hermitian holomorphic vector bundle which has at worst logarithmic singularities in the neighborhood of the marked points. For these data, by renormalizing the trace of the heat operator, we construct the analytic torsion and study its properties. Then we study the properties of the analytic torsion in family setting: we prove the curvature theorem, we study the behavior of the analytic torsion when the cusps are created by degeneration and we give some applications to the moduli spaces of pointed curves
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Oliveira, Júnior João de Deus. "Construção de superfícies utilizando o Teorema de Poincaré". Universidade Federal de Viçosa, 2010. http://locus.ufv.br/handle/123456789/4901.
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Previous issue date: 2010-02-24Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
This study deals with the surface of the compact quotient M2=G where the surface M2 is either the Euclidean plane or the plane spherical or the hyperbolic plane, G is a group of isometries of their surfaces, and this group is generated by matching of edges of polygons. The Poincaré theorem that provides a method of finding the group of isometries G the functions that the pair of edges of the polygons involved. By using this theorem we construct two new pairings of generalized edges (Chapter 4) associated with the tessellations {12η 8,4} e {12μ 12,4}, respectively. These tessellations provide packing of spheres whose packing density is very close to the maximum 3/π. Such pairings are the starting point for finding codes with optimal transmission rates for Multiple-Input Multiple-Output (MIMO).
Este estudo aborda a construção de superfícies compactas pelo quociente M2/G onde a superfície M2 ou é o plano euclidiano, ou é o plano esférico, ou é o plano hiperbólico, G é um grupo de isometrias das respectivas superfícies e esse grupo é gerado pelos emparelhamentos de arestas dos polígonos. O Teorema de Poincaré fornece um método de encontrar o grupo de isometrias G que consiste das funções de emparelhamento de arestas dos polígonos associados. Mediante o uso deste teorema nós construímos dois novos emparelhamentos de arestas generalizados (Capítulo 4), associados as tesselações {12η 8,4} e {12μ 12,4}, respectivamente. Estas tesselações fornecem empacotamento de esferas cuja densidade de empacotamento é bem próxima do valor máximo 3/π. Tais emparelhamentos são o ponto de partida para a busca de códigos com ótimas taxas de transmissão para canais de múltiplas entradas e múltiplas e saídas (MIMO).
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Larsson, David. "Algorithmic Construction of Fundamental Polygons for Certain Fuchsian Groups". Thesis, Linköpings universitet, Matematik och tillämpad matematik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-119916.
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The work of mathematical giants, such as Lobachevsky, Gauss, Riemann, Klein and Poincaré, to name a few, lies at the foundation of the study of the highly structured Riemann surfaces, which allow definition of holomorphic maps, corresponding to analytic maps in the theory of complex analysis. A topological result of Poincaré states that every path-connected Riemann surface can be realised by a construction of identifying congruent points in the complex plane, the Riemann sphere or the hyperbolic plane; just three simply connected surfaces that cover the underlying Riemann surface. This requires the discontinuous action of a discrete subgroup of the automorphisms of the corresponding space. In the hyperbolic plane, which is the richest source for Riemann surfaces, these groups are called Fuchsian, and there are several ways to study the action of such groups geometrically by computing fundamental domains. What is accomplished in this thesis is a combination of the methods found by Reidemeister & Schreier, Singerman and Voight, and thus provides a unified way of finding Dirichlet domains for subgroups of cofinite groups with a given index. Several examples are considered in-depth.
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Aigon, Aline. "Transformations hyperboliques et courbes algébriques en genre 2 et 3". Montpellier 2, 2001. http://www.theses.fr/2001MON20129.
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Bouzoubaa, Taoufik. "Compactification d'espaces de structures hyperboliques". Rennes 1, 1992. http://www.theses.fr/1992REN10060.
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Dans cette thèse, en premier, on étudie l'équivalence semi-algébrique de l'espace de Teichmuller d'une surface fermée de genre g2, tg. Pour cela on établit un homéomorphisme semi-algébrique de tg (étant identifié à un semi-algébrique) sur r#6#g##6. En seconde, on étudie la compactification via le spectre réel des espaces de structures hyperboliques de dimensions n>2, en passant par les représentations d'un groupe finiment engendré dans so(n,1). On aboutit à associer aux points idéaux de cette compactification une action de par des isométries d'un arbre tf#n, quotient de l'espace n-hyperbolique h#nf sur un corps réel clos non-archimédien. Finalement, on donne une vision un peu plus géométrique de la compactification via le spectre réel d'un semi-algebrique fermé (non borné)
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Turaça, Angélica. "As coordenadas de Fenchel-Nielsen". Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-27082015-073617/.
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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.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.
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Bartolini, Gabriel. "On Poicarés Uniformization Theorem". Thesis, Linköping University, Department of Mathematics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-7968.
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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 $p:\mathcal{U}\rightarrow\mathcal{U}/\Gamma$.
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 $\Gamma$.
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Divakaran, D. "Compactness Theorems for The Spaces of Distance Measure Spaces and Riemann Surface Laminations". Thesis, 2014. http://hdl.handle.net/2005/3131.
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Gromov’s compactness theorem for metric spaces, a compactness theorem for the space of compact metric spaces equipped with the Gromov-Hausdorff distance, is a theorem with many applications. In this thesis, we give a generalisation of this landmark result, more precisely, we give a compactness theorem for the space of distance measure spaces equipped with the generalised Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple (X, d,µ), where (X, d) forms a distance space (a generalisation of a metric space where, we allow the distance between two points to be infinity) and µ is a finite Borel measure.
Using this result we prove that the Deligne-Mumford compactification is the completion of the moduli space of Riemann surfaces under the generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov compactness theorem for J-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.
While Gromov compactness theorem for J-holomorphic curves in symplectic manifolds, is an important tool in symplectic topology, its applicability is limited by the lack of general methods to construct pseudo-holomorphic curves. One hopes that considering a more general class of objects in place of pseudo-holomorphic curves will be useful. Generalising the domain of pseudo-holomorphic curves from Riemann surfaces to Riemann surface laminations is a natural choice. Theorems such as the uniformisation theorem for surface laminations by Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and topological classification of “almost all" leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations, as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation, we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.
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Ebel, Tobias [Verfasser]. "Equivariant analytic torsion on hyperbolic Riemann surfaces and the arithmetic Lefschetz trace of an Atkin-Lehner involution on a compact Shimura curve / vorgelegt von Tobias Ebel". 2006. http://d-nb.info/983214786/34.
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