Riemann Surfaces

Simon Donaldson - Collection Oxford Graduate Texts in Mathematics

256 pages, parution le 01/03/2011

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Résumé

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.

L'auteur - Simon Donaldson

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

Sommaire

Preliminaries
- Holomorphic Functions
- Surface Topology
Basic Theory
- Basic Definitions
- Maps between Riemann Surfaces
- Calculus on Surfaces
- Elliptic functions and integrals
- Applications of the Euler characteristic
Deeper Theory
- Meromorphic Functions and the Main Theorem for Compact Riemann Surfaces
- Proof of the Main Theorem
- The Uniformisation Theorem
Further Developments
- Contrasts in Riemann Surface Theory
- Divisors, Line Bundles and Jacobians
- Moduli and Deformations
- Mappings and Moduli
- Ordinary Differential Equations

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Caractéristiques techniques

	PAPIER
Éditeur(s)	Oxford University Press
Auteur(s)	Simon Donaldson
Collection	Oxford Graduate Texts in Mathematics
Parution	01/03/2011
Nb. de pages	256
Couverture	Broché
Intérieur	Noir et Blanc
EAN13	9780199606740
ISBN13	978-0-19-960674-0