An authoritative but accessible text on one dimensional complex manifolds or Riemann surfaces. Dealing with the main results on Riemann surfaces from a variety of points of view; it pulls together material from global analysis, topology, and algebraic geometry, and covers the essential mathematical methods and tools.

• 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

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.