Riemann Surfaces

Paperback | April 24, 2011

bySimon Donaldson

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The 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 otherfields. 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 inparticular 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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The 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 otherfields. It provides a model for a large num...

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...
Format:PaperbackDimensions:304 pages, 9.21 × 6.14 × 0.68 inPublished:April 24, 2011Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0199606749

ISBN - 13:9780199606740

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Table of Contents

I. Preliminaries1. Holomorphic Functions2. Surface TopologyII. Basic Theory3. Basic Definitions4. Maps between Riemann Surfaces5. Calculus on Surfaces6. Elliptic functions and integrals7. Applications of the Euler characteristicIII. Deeper Theory8. Meromorphic Functions and the Main Theorem for Compact Riemann Surfaces9. Proof of the Main Theorem10. The Uniformisation TheoremIV. Further Developments11. Contrasts in Riemann Surface Theory12. Divisors, Line Bundles and Jacobians13. Moduli and Deformations14. Mappings and Moduli15. Ordinary Differential EquationsBibliographyIndex