Our main result is that there are infinitely many primes of the form a² + b² such that a² + 4b² has at most 5 prime factors. We prove this by first developing the theory of $L$-functions for Gaussian primes by using standard methods. We then give an exposition of the Siegel--Walfisz Theorem for Gaussian primes and a corresponding Prime Number Theorem for Gaussian Arithmetic Progressions. Finally, we prove the main result by using the developed theory together with Sieve Theory and specifically a weighted linear sieve result to bound the number of prime factors of a² + 4b². For the application of the sieve, we need to derive a specific version of the Bombieri--Vinogradov Theorem for Gaussian primes which, in turn, requires a suitable version of the Large Sieve. We are also able to get the number of prime factors of a² + 4b² as low as 3 if we assume the Generalised Riemann Hypothesis.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

In this thesis, we investigate the period mapping of Teichmuller space into the Siegel upper half space. This is constructed from integrals of a basis of holomorphic one-forms along closed curves of a basis of the Riemann surface.  We consider the Riemann, Teichmuller and Torelli moduli spaces and their representation in the Siegel upper half space, and its relation to orbits of a symplectic and a set of positive polarizations of a vector space of dimension equal to the genus of the surface. / October 2016

Master of Science / Mathematics / Nathan Albin / Pietro Poggi-Corradini / This report explores singular integration, both real and complex, focusing on the the Cauchy type integral, culminating in the proof of generalized Sokhotski-Plemelj formulae and the applications of such to a Riemann-Hilbert problem.

Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / We investigate aspects of differential K-theory. In particular, we give a direct proof that the Freed-Lott differential analytic index is well defined, and a short proof of the differential Grothendieck-Riemann-Roch theorem in the setting of Freed-Lott differential K-theory. We also construct explicit ring isomorphisms between Freed-Lott differential K-theory and Simons-Sullivan differential K-theory, define the Simons-Sullivan differential analytic index, and prove the differential Grothendieck-Riemann-Roch theorem in the setting of Simons-Sullivan differential K-theory. / 2031-01-02

Orientador: Prof. Dr. Valério Ramos Batista. / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática, 2009. / A tese de mestrado versa sobre o artigo A family of triply periodic Costa surfaces, que apresenta uma demonstração completa de unicidade e convergência para uma família contínua a um parâmetro de Superfícies Mínimas Triplamente Periódicas. No artigo, a demonstração é norteada por simulações numéricas em MatLab, que motivam as provas teóricas. Entretanto, o presente trabalho não contemplará esta parte numérica, por dar prioridade aos argumentos Geométricos do artigo. De fato, a Geometria é uma importante ferramenta para outras áreas, mesmo da própria Matemática, não apenas por facilitar demonstrações, mas também por torná-las acessíveis. Dentre as sub-áreas da Matemática, obviamente a mais visual é a Geometria, que mesmo equipada com técnicas como Variáveis Complexas, Diferenciabilidade, Homologia, etc., não perde sua concretividade: curvas, superfícies, rotação, etc. O trabalho [RamosBatista2] é inovador, pois apresenta as primeiras superfícies mínimas triplamente periódicas cuja construção explícita não pode ser realizada pelo Método de Conjugação de Plateau. Além da unicidade e convergência mencionadas acima, traz uma descrição explícita dos membros-limite. É raro encontrar um estudo tão completo como neste artigo. A família de superfícies é obtida pelo método de construção reversa introduzido por Karcher em 1989. Tal método consiste dos seguintes passos: 1) esboço da superfície; 2) compacticação; 3) hipóteses de simetria; 4) equação algébrica; 5) obtenção dos dados de Weierstraÿ; 6) vericação de involuções e hipóteses de simetria; 7) análise de períodos; e 8) mergulho. As ferramentas teóricas deste método são apresentadas no Capítulo 2 da presente Tese de Mestrado. / This present work deals with the article A family of triply periodic Costa surfaces, which brings a complete demonstration for including uniqueness and convergence of a continuous one-parameter family of Triply Periodic Minimal Surfaces. In the paper, the theoretical proofs are motivated by numerical evidences obtained through the software MatLab. However, this present work will not include the numerics, because we give preference to the geometric arguments of the paper. Indeed, Geometry is an important tool for other research areas, even inside Mathematics itself, not just for easing demonstrations a lot, but also because it makes them accessible. Among the sub-areas in Mathematics, obviously the most visually appealing is the Geometry. Even equipped with techniques like Complex Variables, Dierentiability and Homology, it never loses its concreteness: curves, surfaces, rotations, etc. The paper [RamosBatista2] is innovative because presents the rst triply periodic minimal surfaces of which the explicit construction cannot be accomplished by Plateau's Conjugate Method. Besides uniqueness and convergence mentioned above, it brings an explicit description of the limit-members. Such a complete study is rare to nd. The family of surfaces is obtained via the reverse construction method introduced by Karcher in 1989. This method consists of the following steps: 1) drafting the soughtafter surface; 2) compactication; 3) symmetry hypotheses; 4) algebraic equation; 5) Weierstraÿ data; 6) checking involutions from symmetry hypotheses; 7) period analysis; 8) embeddedness. The main theoretical tools for this method are presented in Chapter 2 of this Master Thesis.

Orientador: Marta Cilene Gadotti / Banca: Paulo Leandro Dattori da Silva / Banca: Ricardo Parreira da Silva / Resumo: Este trabalho apresenta resultados importantes sobre a Teoria de Integração. Inicialmente é desenvolvida uma parte sobre Teoria da Medida, necessária para introduzir a integral de Lebesgue e suas propriedades. Também é apresentada a integral de Riemann-Stieltjes. Em seguida, são demonstrados resultados importantes sobre converg ência envolvendo as integrais de Lebesgue, resultados estes que não são válidos para integrais de Riemann. Para apresentar tais temas, usa-se mais fortemente as referências [1], [2], [3] e [4] / Abstract: This study presents important results on Integration of Theory. The rst of all part is developed on Measure Theory which is necessary to introduce the Lebesgue integral and its properties and we introduce. It also shows the Riemann-Stieltjes integral. Important results are proved on convergence involving the integrals of Lebesgue, which are not valid for the Riemann integral. Im order to present these themes we strongly use the references [1], [2], [3] and [4] / Mestre

Made available in DSpace on 2015-04-22T22:16:15Z (GMT). No. of bitstreams: 1 Cleiton Lira.pdf: 415012 bytes, checksum: 5984795059a4a2d4ed692c1fb90fd8ec (MD5) Previous issue date: 2009-03-25 / CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico / In this work , we will give a detailed statement of congruence theorem for CP n The result obtained by J. Bolton and L. M. Woodward . We show that ψ and ψe are harmonic maps of a Riemann surface in CP n With Γ -1 = 1 and Γe - Γ0 = Γe0 where or ψ is pseudo- holomorphic or UEP , 0 = Up , 0 for p = 2 . . . N + 1, then There is an isometry g of CP n such that ψe = gψ . Moreover, if ψ is substantially g is then single. / Neste trabalho, daremos uma demonstração detalhada do Teorema da congruência para CPn, resultado obtido por J. Bolton e L.M. Woodward. Mostraremos que se e e são aplicações harmônicas de uma superfície de Riemann em CPn, com 􀀀􀀀1 = e􀀀 􀀀1 e 􀀀0 = e􀀀 0 em que ou é pseudo-holomorfa ou eUp;0 = Up;0 para p = 2; : : : ; n + 1, então existe uma isometria g de CPn tal que e = g . Além disso, se é substancial então g é unica

Submitted by Kamila Costa (kamilavasconceloscosta@gmail.com) on 2015-06-18T20:37:32Z No. of bitstreams: 1 Dissertação-Euderley de C Nunes.pdf: 25821021 bytes, checksum: 4f3d1be54be4ba8e8d8c720131d61fb6 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-07-06T17:47:02Z (GMT) No. of bitstreams: 1 Dissertação-Euderley de C Nunes.pdf: 25821021 bytes, checksum: 4f3d1be54be4ba8e8d8c720131d61fb6 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-07-06T17:54:01Z (GMT) No. of bitstreams: 1 Dissertação-Euderley de C Nunes.pdf: 25821021 bytes, checksum: 4f3d1be54be4ba8e8d8c720131d61fb6 (MD5) / Made available in DSpace on 2015-07-06T17:54:32Z (GMT). No. of bitstreams: 1 Dissertação-Euderley de C Nunes.pdf: 25821021 bytes, checksum: 4f3d1be54be4ba8e8d8c720131d61fb6 (MD5) Previous issue date: 2015-03-04 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Aiming to contribute to the teaching of basic education, this work will present a simple approach through the use of a well-known technique in antiquity, which is the stereographic projection. This paper will deal with the studies developed by Georg Friedrich Bernhard Riemann (1826-1866), which demonstrates how to design stereographically a sphere on a plane, called the complex plane. For this, we will show that the use of complex numbers has great relevance for understanding of the techniques commonly used in the development of cartography and other areas. We will present the set of complex numbers and then de ne the stereographic projection and some of its main properties, where we use the Geogebra software version 5.0, seeing that the software produces 3D animations, which will support in understanding the stereographic projection and of their properties by the high school students and teachers. Thus, this research will serve as a motivating element for students and teachers that seek to improve their knowledge because the study by Riemann is based on complex numbers which are studied in the course of primary education. / Com o objetivo de contribuir com o ensino da educação básica, este trabalho apresentará através de uma abordagem simples o uso de uma técnica muito conhecida na antiguidade, que é a projeção estereográ ca. Este trabalho abordará os estudos desenvolvidos por Georg Friedrich Bernhard Riemann (1826-1866), que demonstra como projetar estereogra camente uma esfera sobre um plano, denominado de plano complexo. Para isso, mostraremos que o uso dos números complexos terá grande relevância para compreendermos uma das técnicas mais usadas no desenvolvimento da cartogra a e outras áreas. Apresentaremos o conjunto dos números complexos e em seguida de niremos a projeção estereográ ca e algumas de suas principais propriedades, onde faremos o uso do software Geogebra versão 5.0, visto que este software produz animações em 3D, que servirão de suporte para a compreensão da projeção estereográ ca e de suas respectivas propriedades por parte dos alunos e professores do ensino médio. Com isso, esta pesquisa servirá de elemento motivador para alunos e professores que busquem aprimorar seus conhecimentos, pois o estudo desenvolvido por Riemann tem como base os números complexos que são estudados no decorrer do ensino básico.

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.