Wen, Chengfeng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 77-83). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

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) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / 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 / 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

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. / Tesis

Tse, Wang Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 75-77). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- On bounded interval with n jumps in inhibitor diffusivity --- p.3 / Chapter 2.1 --- Introduction --- p.3 / Chapter 2.2 --- Preliminaries --- p.5 / Chapter 2.3 --- Review of previous results in the two segment case: interior spike and spike near the jump discontinuity of the diffusion coefficient --- p.7 / Chapter 2.4 --- The construction and analysis of spiky steady-state solutions --- p.9 / Chapter 2.5 --- Stability Analysis --- p.10 / Chapter 2.6 --- Spikes near the jump discontinuity xb of the inhibitor diffusivity --- p.11 / Chapter 2.7 --- Stability Analysis II: Small Eigenvalues of the Spike near the Jump --- p.16 / Chapter 2.8 --- Existence of interior spikes for N segments --- p.20 / Chapter 2.9 --- Existence of a spike near a jump for N segments --- p.24 / Chapter 2.10 --- Appendix: The Green´ةs function for three segments --- p.25 / Chapter 3 --- On a compact Riemann surface without boundary --- p.30 / Chapter 3.1 --- Introduction --- p.30 / Chapter 3.2 --- Some Preliminaries --- p.35 / Chapter 3.3 --- Existence --- p.43 / Chapter 3.4 --- Refinement of Approximate Solution --- p.50 / Chapter 3.5 --- Stability --- p.52 / Chapter 3.6 --- Appendix I: Expansion of the Laplace-Beltrami Operator --- p.67 / Chapter 3.7 --- Appendix II: Some Technical Calculations --- p.73

Chan, Kin Wai. / "September 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 103-106). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.8 / Chapter 2 --- Background Knowledge --- p.13 / Chapter 2.1 --- Results from Riemann Surface Theory and Quasicon- formal Mappings --- p.13 / Chapter 2.1.1 --- Riemann Surfaces and the Uniformization The- orem --- p.13 / Chapter 2.1.2 --- Fuchsian Groups --- p.15 / Chapter 2.1.3 --- Quasiconformal Mappings and the Beltrami Equation --- p.17 / Chapter 2.1.4 --- Holomorphic Quadratic Differentials --- p.20 / Chapter 2.1.5 --- Nodal Riemann Surfaces --- p.21 / Chapter 2.2 --- Teichmuller Theory --- p.24 / Chapter 2.2.1 --- Teichmiiller Spaces --- p.24 / Chapter 2.2.2 --- Teichmuller's Distance --- p.26 / Chapter 2.2.3 --- The Bers Embedding --- p.26 / Chapter 2.2.4 --- Teichmuller Modular Groups and Moduli Spaces of Riemann Surfaces --- p.27 / Chapter 2.2.5 --- Infinitesimal Theory of Teichmiiller Spaces --- p.28 / Chapter 2.2.6 --- Boundary of Moduli Spaces of Riemann Sur- faces --- p.29 / Chapter 2.3 --- Schwarz-Yau Lemma --- p.30 / Chapter 3 --- Classical Canonical Metrics on the Teichnmuller Spaces and the Moduli Spaces of Riemann Surfaces --- p.31 / Chapter 3.1 --- Finsler Metrics and Bergman Metric --- p.31 / Chapter 3.1.1 --- Definitions and Properties of the Metrics --- p.32 / Chapter 3.1.2 --- Equivalences of the Metrics --- p.33 / Chapter 3.2 --- Weil-Petersson Metric --- p.36 / Chapter 3.2.1 --- Definition and Properties of the Weil-Petersson Metric --- p.36 / Chapter 3.2.2 --- Results about Harmonic Lifts --- p.37 / Chapter 3.2.3 --- Curvature Formula for the Weil-Petersson Met- ric --- p.41 / Chapter 4 --- Kahler Metrics on the Teichmiiller Spaces and the Moduli Spaces of Riemann Surfaces --- p.42 / Chapter 4.1 --- McMullen Metric --- p.42 / Chapter 4.1.1 --- Definition of the McMullen Metric --- p.42 / Chapter 4.1.2 --- Properties of the McMullen Metric --- p.43 / Chapter 4.1.3 --- Equivalence of the McMullen Metric and the Teichmuller Metric --- p.45 / Chapter 4.2 --- Kahler-Einstein Metric --- p.50 / Chapter 4.2.1 --- Existence of the Kahler-Einstein Metric --- p.50 / Chapter 4.2.2 --- A Conjecture of Yau --- p.50 / Chapter 4.3 --- Ricci Metric --- p.51 / Chapter 4.3.1 --- Definition of the Ricci Metric --- p.51 / Chapter 4.3.2 --- Curvature Formula of the Ricci Metric --- p.53 / Chapter 4.4 --- The Asymptotic Behavior of the Ricci Metric --- p.61 / Chapter 4.4.1 --- Estimates on the Asymptotics of the Ricci Metric --- p.61 / Chapter 4.4.2 --- Estimates on the Curvature of the Ricci Metric --- p.83 / Chapter 4.5 --- Perturbed Ricci Metric --- p.92 / Chapter 4.5.1 --- Definition and the Curvature Formula of the Perturbed Ricci Metric --- p.92 / Chapter 4.5.2 --- Estimates on the Curvature of the Perturbed Ricci Metric --- p.93 / Chapter 4.5.3 --- Equivalence of the Perturbed Ricci Metric and the Ricci Metric --- p.96 / Chapter 5 --- Equivalence of the Kahler Metrics on the Teichmuller Spaces and the Moduli Spaces of Riemann Surfaces --- p.98 / Chapter 5.1 --- Equivalence of the Ricci Metric and the Kahler-Einstein Metric --- p.98 / Chapter 5.2 --- Equivalence of the Ricci Metric and the McMullen Metric --- p.99 / Bibliography --- p.103

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Identified as one of the 7 Millennium Problems, the Riemann zeta hypothesis has successfully evaded mathematicians for over 100 years. Simply stated, Riemann conjectured that all of the nontrivial zeroes of his zeta function have real part equal to 1/2. This thesis attempts to explore the theory behind Riemann’s zeta function by first starting with Euler’s zeta series and building up to Riemann’s function. Along the way we will develop the math required to handle this theory in hopes that by the end the reader will have immersed themselves enough to pursue their own exploration and research into this fascinating subject.

Ce travail est consacré à la recherche de surfaces de Riemann (\it compactes) extrê\-mes (i.e. maxima locaux) pour la systole, ou tout au moins parfaites. En genre 4, on donne une nouvelle surface extrême et deux surfaces parfaites non extrêmes (ce sont les premiers exemples de telles surfaces en genre $\leq 10$). La méthode consiste à réaliser géométriquement les groupes d'automorphismes à 4 points de branchements. En effet, le lieu des points fixes dans l'espace de Teichmüller $T_g$ d'un tel groupe, dépend d'un paramètre complexe qu'on peut alors ajuster pour maximiser la systole. On étudie ensuite les propriétés variationnelles dans $T_g$ des surfaces obtenues. Par extension de cette méthode, on trouve également une nouvelle surface extrême en genre 6, ainsi qu'une suite infinie de surfaces parfaites non extrêmes de genre $g>3$. En outre, on retrouve, de manière unifiée, les surfaces déjà connues en genre $\leq 5$. La méthode employée pour la recherche de surfaces parfaites, permet de trouver parallèlement un certain nombre de surfaces eutactiques, qui sont intéressantes à classifier en elles-mêmes puisque ce sont les points critiques de la fonction systole. Enfin, le dernier chapitre, développant une toute autre approche, concerne une méthode purement algébrique qui permet de redémontrer l'extrémalité des surfaces respectivement de Bolza et de Klein.

This thesis is a study of circle packings for arbitrary combinatorial tori in the geometric setting of affine tori. Certain new tools needed for this study, such as face labels instead of the usual vertex labels, are described. It is shown that to each combinatorial torus there corresponds a two real parameter family of affine packing labels. A construction of circle packings for combinatorial fundamental domains from affine packing labels is given. It is demonstrated that such circle packings have two affine side-pairing maps, and also that these side-pairing maps depend continuously on the two real parameters.

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