Classification Theory Of Riemann Surfaces
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Classification Theory of Riemann Surfaces
Author | : Leo Sario,Mitsuru Nakai |
Publsiher | : Springer Science & Business Media |
Total Pages | : 469 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 9783642482694 |
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The purpose of the present monograph is to systematically develop a classification theory of Riemann surfaces. Some first steps will also be taken toward a classification of Riemannian spaces. Four phases can be distinguished in the chronological background: the type problem; general classification; compactifications; and extension to higher dimensions. The type problem evolved in the following somewhat overlapping steps: the Riemann mapping theorem, the classical type problem, and the existence of Green's functions. The Riemann mapping theorem laid the foundation to classification theory: there are only two conformal equivalence classes of (noncompact) simply connected regions. Over half a century of efforts by leading mathematicians went into giving a rigorous proof of the theorem: RIEMANN, WEIERSTRASS, SCHWARZ, NEUMANN, POINCARE, HILBERT, WEYL, COURANT, OSGOOD, KOEBE, CARATHEODORY, MONTEL. The classical type problem was to determine whether a given simply connected covering surface of the plane is conformally equivalent to the plane or the disko The problem was in the center of interest in the thirties and early forties, with AHLFORS, KAKUTANI, KOBAYASHI, P. MYRBERG, NEVANLINNA, SPEISER, TEICHMÜLLER and others obtaining incisive specific results. The main problem of finding necessary and sufficient conditions remains, however, unsolved.
An Introduction to Riemann Surfaces
Author | : Terrence Napier,Mohan Ramachandran |
Publsiher | : Springer Science & Business Media |
Total Pages | : 563 |
Release | : 2011-09-08 |
Genre | : Mathematics |
ISBN | : 9780817646936 |
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This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the so-called L2 $\bar{\delta}$-method. This method is a powerful technique from the theory of several complex variables, and provides for a unique approach to the fundamentally different characteristics of compact and noncompact Riemann surfaces. The inclusion of continuing exercises running throughout the book, which lead to generalizations of the main theorems, as well as the exercises included in each chapter make this text ideal for a one- or two-semester graduate course.
Classification Theory of Riemannian Manifolds
Author | : S. R. Sario,M. Nakai,C. Wang,L. O. Chung |
Publsiher | : Springer |
Total Pages | : 518 |
Release | : 2006-11-15 |
Genre | : Mathematics |
ISBN | : 9783540372615 |
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Galois Theory Coverings and Riemann Surfaces
Author | : Askold Khovanskii |
Publsiher | : Springer Science & Business Media |
Total Pages | : 86 |
Release | : 2013-09-11 |
Genre | : Mathematics |
ISBN | : 9783642388415 |
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The first part of this book provides an elementary and self-contained exposition of classical Galois theory and its applications to questions of solvability of algebraic equations in explicit form. The second part describes a surprising analogy between the fundamental theorem of Galois theory and the classification of coverings over a topological space. The third part contains a geometric description of finite algebraic extensions of the field of meromorphic functions on a Riemann surface and provides an introduction to the topological Galois theory developed by the author. All results are presented in the same elementary and self-contained manner as classical Galois theory, making this book both useful and interesting to readers with a variety of backgrounds in mathematics, from advanced undergraduate students to researchers.
Contributions to the Theory of Riemann Surfaces AM 30 Volume 30
Author | : Lars Valerian Ahlfors,E. Calabi,Marston Morse,Leo Sario,Donald Clayton Spencer |
Publsiher | : Princeton University Press |
Total Pages | : 264 |
Release | : 1953-08-01 |
Genre | : Mathematics |
ISBN | : 9781400828371 |
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The description for this book, Contributions to the Theory of Riemann Surfaces. (AM-30), Volume 30, will be forthcoming.
Dessins d Enfants on Riemann Surfaces
Author | : Gareth A. Jones,Jürgen Wolfart |
Publsiher | : Springer |
Total Pages | : 259 |
Release | : 2016-03-23 |
Genre | : Mathematics |
ISBN | : 9783319247113 |
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This volume provides an introduction to dessins d'enfants and embeddings of bipartite graphs in compact Riemann surfaces. The first part of the book presents basic material, guiding the reader through the current field of research. A key point of the second part is the interplay between the automorphism groups of dessins and their Riemann surfaces, and the action of the absolute Galois group on dessins and their algebraic curves. It concludes by showing the links between the theory of dessins and other areas of arithmetic and geometry, such as the abc conjecture, complex multiplication and Beauville surfaces. Dessins d'Enfants on Riemann Surfaces will appeal to graduate students and all mathematicians interested in maps, hypermaps, Riemann surfaces, geometric group actions, and arithmetic.
Algebraic Curves and Riemann Surfaces
Author | : Rick Miranda |
Publsiher | : American Mathematical Soc. |
Total Pages | : 390 |
Release | : 1995 |
Genre | : Mathematics |
ISBN | : 9780821802687 |
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The book was easy to understand, with many examples. The exercises were well chosen, and served to give further examples and developments of the theory. --William Goldman, University of Maryland In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Duality Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one semester of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry.
Contributions to the Theory of Riemann Surfaces
Author | : Lars Valerian Ahlfors,E. Calabi,Marston Morse,Leo Sario,Donald Clayton Spencer |
Publsiher | : Princeton University Press |
Total Pages | : 280 |
Release | : 1953-08-21 |
Genre | : Mathematics |
ISBN | : 0691079390 |
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The description for this book, Contributions to the Theory of Riemann Surfaces. (AM-30), Volume 30, will be forthcoming.