Riemann Surfaces

Simon Donaldson (Royal Society Research Professor, Imperial College, London)

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English

24 March 2011

Mathematics & Sciences; Complex analysis, complex variables; Differential & Riemannian geometry; Algebraic geometry; Topology

Series: Oxford Graduate Texts in Mathematics

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 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.

By: Simon Donaldson (Royal Society Research Professor Imperial College London)
Country of Publication: United Kingdom
Dimensions: Height: 240mm, Width: 160mm, Spine: 21mm
Weight: 562g
ISBN: 9780198526391
ISBN 10: 0198526393
Series: Oxford Graduate Texts in Mathematics
Publication Date: 24 March 2011
Audience: College/higher education , Further / Higher Education
Format: Hardback
Publisher's Status: Active

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

Reviews for Riemann Surfaces

This book may be recommended to readers of all levels, from beginners to specialists. -- Mathematical Reviews This compact book is a fabulous contribution to the literature on the gorgeous and important subject of Riemann surfaces... The book is simply wonderful. --MAA Reviews