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    Riemann Surfaces (Oxford Graduate Texts in Mathematics) (Paperback) By (author) Simon K. Donaldson

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    DescriptionThe theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, and diverse topics in mathematical physics. This graduate text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind. Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved. Following this section, the remainder of the text illustrates various facets of the more advanced theory.


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  • Full bibliographic data for Riemann Surfaces

    Title
    Riemann Surfaces
    Authors and contributors
    By (author) Simon K. Donaldson
    Physical properties
    Format: Paperback
    Number of pages: 304
    Width: 152 mm
    Height: 230 mm
    Thickness: 18 mm
    Weight: 440 g
    Language
    English
    ISBN
    ISBN 13: 9780199606740
    ISBN 10: 0199606749
    Classifications

    BIC E4L: MAT
    Nielsen BookScan Product Class 3: S7.8
    B&T Book Type: NF
    B&T Modifier: Region of Publication: 03
    Ingram Subject Code: SE
    B&T General Subject: 710
    B&T Modifier: Academic Level: 02
    B&T Modifier: Text Format: 06
    BIC subject category V2: PBP
    Abridged Dewey: 530
    LC classification: QC
    B&T Merchandise Category: UP
    DC22: 515.93
    LC subject heading:
    BISAC V2.8: MAT034000, MAT012030, MAT038000
    BIC subject category V2: PBKD
    BISAC V2.8: SCI040000
    BIC subject category V2: PBMW, PBMP
    Warengruppen-Systematik des deutschen Buchhandels: 16260
    BISAC V2.8: MAT012010
    Thema V1.0: PBMW, PBKD, PBMP, PBP
    Illustrations note
    46 line diagrams
    Publisher
    Oxford University Press
    Imprint name
    Oxford University Press
    Publication date
    19 May 2011
    Publication City/Country
    Oxford
    Author Information
    Simon Donaldson gained a BA from Cambridge in 1979. In 1980 he began graduate work in Oxford, supervised by Nigel Hitchin and Sir Michael Atiyah. His PhD thesis studied mathematical aspects of Yang-Mills theory. In 1986, aged 29, he was awarded a Fields Medal and was elected to the Royal Society. He was Wallis Professor of Mathematics in Oxford between 1985 and 1998 when he moved to Imperial College London. Most of his work since has been on the interface between differential geometry and complex algebraic geometry. The recipient of numerous awards, including the Shaw Prize in 2009 with Clifford Taubes, he is also a Foreign Member of the US, French & Swedish academies. Donaldson has supervised more than 40 doctoral students, many of whom have gone on to become leading figures in research.
    Review quote
    This clear and insightful text, from one of the foremost researchers in the field, is suitable for graduate or advanced undergraduate course on Riemann surfaces, and will provide fascinating reading for professional mathematicians. L'Enseignement Mathematique
    Table of contents
    I PRELIMINARIES ; 1. Holomorphic Functions ; 2. Surface Topology ; II BASIC THEORY ; 3. Basic Definitions ; 4. Maps between Riemann Surfaces ; 5. Calculus on Surfaces ; 6. Elliptic functions and integrals ; 7. Applications of the Euler characteristic ; III DEEPER THEORY ; 8. Meromorphic Functions and the Main Theorem for Compact Riemann Surfaces ; 9. Proof of the Main Theorem ; 10. The Uniformisation Theorem ; IV FURTHER DEVELOPMENTS ; 11. Contrasts in Riemann Surface Theory ; 12. Divisors, Line Bundles and Jacobians ; 13. Moduli and Deformations ; 14. Mappings and Moduli ; 15. Ordinary Differential Equations ; Bibliography ; Index