The Classification Theorem For Compact Surfaces
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Author |
: Jean Gallier |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 184 |
Release |
: 2013-02-05 |
ISBN-10 |
: 9783642343643 |
ISBN-13 |
: 3642343643 |
Rating |
: 4/5 (43 Downloads) |
Synopsis A Guide to the Classification Theorem for Compact Surfaces by : Jean Gallier
This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those without detailed background knowledge, or too informal to afford students a comprehensive insight into the subject. Its dedicated, student-centred approach details a near-complete proof of this theorem, widely admired for its efficacy and formal beauty. The authors present the technical tools needed to deploy the method effectively as well as demonstrating their use in a clearly structured, worked example. Ideal for students whose mastery of algebraic topology may be a work-in-progress, the text introduces key notions such as fundamental groups, homology groups, and the Euler-Poincaré characteristic. These prerequisites are the subject of detailed appendices that enable focused, discrete learning where it is required, without interrupting the carefully planned structure of the core exposition. Gently guiding readers through the principles, theory, and applications of the classification theorem, the authors aim to foster genuine confidence in its use and in so doing encourage readers to move on to a deeper exploration of the versatile and valuable techniques available in algebraic topology.
Author |
: Jean Gallier |
Publisher |
: Springer |
Total Pages |
: 0 |
Release |
: 2015-03-07 |
ISBN-10 |
: 3642437109 |
ISBN-13 |
: 9783642437106 |
Rating |
: 4/5 (09 Downloads) |
Synopsis A Guide to the Classification Theorem for Compact Surfaces by : Jean Gallier
This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those without detailed background knowledge, or too informal to afford students a comprehensive insight into the subject. Its dedicated, student-centred approach details a near-complete proof of this theorem, widely admired for its efficacy and formal beauty. The authors present the technical tools needed to deploy the method effectively as well as demonstrating their use in a clearly structured, worked example. Ideal for students whose mastery of algebraic topology may be a work-in-progress, the text introduces key notions such as fundamental groups, homology groups, and the Euler-Poincaré characteristic. These prerequisites are the subject of detailed appendices that enable focused, discrete learning where it is required, without interrupting the carefully planned structure of the core exposition. Gently guiding readers through the principles, theory, and applications of the classification theorem, the authors aim to foster genuine confidence in its use and in so doing encourage readers to move on to a deeper exploration of the versatile and valuable techniques available in algebraic topology.
Author |
: Jean Gallier |
Publisher |
: Springer |
Total Pages |
: 0 |
Release |
: 2013-02-06 |
ISBN-10 |
: 3642343635 |
ISBN-13 |
: 9783642343636 |
Rating |
: 4/5 (35 Downloads) |
Synopsis A Guide to the Classification Theorem for Compact Surfaces by : Jean Gallier
This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those without detailed background knowledge, or too informal to afford students a comprehensive insight into the subject. Its dedicated, student-centred approach details a near-complete proof of this theorem, widely admired for its efficacy and formal beauty. The authors present the technical tools needed to deploy the method effectively as well as demonstrating their use in a clearly structured, worked example. Ideal for students whose mastery of algebraic topology may be a work-in-progress, the text introduces key notions such as fundamental groups, homology groups, and the Euler-Poincaré characteristic. These prerequisites are the subject of detailed appendices that enable focused, discrete learning where it is required, without interrupting the carefully planned structure of the core exposition. Gently guiding readers through the principles, theory, and applications of the classification theorem, the authors aim to foster genuine confidence in its use and in so doing encourage readers to move on to a deeper exploration of the versatile and valuable techniques available in algebraic topology.
Author |
: Norbert A'Campo |
Publisher |
: Springer Nature |
Total Pages |
: 282 |
Release |
: 2021-10-27 |
ISBN-10 |
: 9783030890322 |
ISBN-13 |
: 3030890325 |
Rating |
: 4/5 (22 Downloads) |
Synopsis Topological, Differential and Conformal Geometry of Surfaces by : Norbert A'Campo
This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes’ Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss–Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow’s Theorem on compact holomorphic submanifolds in complex projective spaces. Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.
Author |
: Richard Evan Schwartz |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 330 |
Release |
: 2011 |
ISBN-10 |
: 9780821853689 |
ISBN-13 |
: 0821853686 |
Rating |
: 4/5 (89 Downloads) |
Synopsis Mostly Surfaces by : Richard Evan Schwartz
The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.
Author |
: Alexander Jasper |
Publisher |
: |
Total Pages |
: 50 |
Release |
: 2014 |
ISBN-10 |
: OCLC:1194412898 |
ISBN-13 |
: |
Rating |
: 4/5 (98 Downloads) |
Synopsis The Classification Theorem for Compact Surfaces by : Alexander Jasper
Author |
: Clark Bray |
Publisher |
: Springer Nature |
Total Pages |
: 216 |
Release |
: 2021-06-18 |
ISBN-10 |
: 9783030706081 |
ISBN-13 |
: 3030706087 |
Rating |
: 4/5 (81 Downloads) |
Synopsis Algebraic Topology by : Clark Bray
Algebraic Topology is an introductory textbook based on a class for advanced high-school students at the Stanford University Mathematics Camp (SUMaC) that the authors have taught for many years. Each chapter, or lecture, corresponds to one day of class at SUMaC. The book begins with the preliminaries needed for the formal definition of a surface. Other topics covered in the book include the classification of surfaces, group theory, the fundamental group, and homology. This book assumes no background in abstract algebra or real analysis, and the material from those subjects is presented as needed in the text. This makes the book readable to undergraduates or high-school students who do not have the background typically assumed in an algebraic topology book or class. The book contains many examples and exercises, allowing it to be used for both self-study and for an introductory undergraduate topology course.
Author |
: Bojan Mohar |
Publisher |
: Johns Hopkins University Press |
Total Pages |
: 0 |
Release |
: 2001-08-02 |
ISBN-10 |
: 0801866898 |
ISBN-13 |
: 9780801866890 |
Rating |
: 4/5 (98 Downloads) |
Synopsis Graphs on Surfaces by : Bojan Mohar
Graph theory is one of the fastest growing branches of mathematics. Until recently, it was regarded as a branch of combinatorics and was best known by the famous four-color theorem stating that any map can be colored using only four colors such that no two bordering countries have the same color. Now graph theory is an area of its own with many deep results and beautiful open problems. Graph theory has numerous applications in almost every field of science and has attracted new interest because of its relevance to such technological problems as computer and telephone networking and, of course, the internet. In this new book in the Johns Hopkins Studies in the Mathematical Science series, Bojan Mohar and Carsten Thomassen look at a relatively new area of graph theory: that associated with curved surfaces. Graphs on surfaces form a natural link between discrete and continuous mathematics. The book provides a rigorous and concise introduction to graphs on surfaces and surveys some of the recent developments in this area. Among the basic results discussed are Kuratowski's theorem and other planarity criteria, the Jordan Curve Theorem and some of its extensions, the classification of surfaces, and the Heffter-Edmonds-Ringel rotation principle, which makes it possible to treat graphs on surfaces in a purely combinatorial way. The genus of a graph, contractability of cycles, edge-width, and face-width are treated purely combinatorially, and several results related to these concepts are included. The extension by Robertson and Seymour of Kuratowski's theorem to higher surfaces is discussed in detail, and a shorter proof is presented. The book concludes with a survey of recent developments on coloring graphs on surfaces.
Author |
: Benson Farb |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 371 |
Release |
: 2013-08-16 |
ISBN-10 |
: 9780821898871 |
ISBN-13 |
: 0821898876 |
Rating |
: 4/5 (71 Downloads) |
Synopsis Moduli Spaces of Riemann Surfaces by : Benson Farb
Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
Author |
: Leo Sario |
Publisher |
: Springer |
Total Pages |
: 480 |
Release |
: 1970 |
ISBN-10 |
: UCAL:B5008555 |
ISBN-13 |
: |
Rating |
: 4/5 (55 Downloads) |
Synopsis Classification Theory of Riemann Surfaces by : Leo Sario
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.