Course Category： Professional Basic Course
Prerequisites：Riemann Surfaces I
Aims & Requirements：
This course is one of the special basic courses for the doctoral candidates and the master degree candidates of some specialities in mathematics. Riemann surfaces is the cross point of many fields in mathematics, such as complex geometry, Lie Groups, Algebraic number theory, Harmonic analysis and Topology etc. This course is divided into two parts: Riemann Surfaces I and Riemann Surfaces II. Riemann Surfaces I focuses on the basic concepts in Riemann Surfaces, such as holomorphic map, meromorphic function, branched covering, sheaf, Riemann-Roch Theorem and its simple applicattions. Riemann Surfaces II focuses on some profound theorems in Riemann Surfaces, such as Serre Duality Theorem, Abel's Theorem, Jacobi Inversion Problem, Riemann Mapping Theorem etc. By the study of Riemann Surfaces II, we wish the students can understand the ideas and the methods of Riemann Surfaces well and can apply some results to some other fields.
Chapter 1 Compact Riemann Surfaces
The Serre Duality Theorem，The Riemann-Hurwitz formula，Holomorphic vector fields on a compact Riemann surface; The existence of meromorphic functions and 1-forms with prescribed principal parts; Harmonic Differential Forms; Abel’s Theorem; Intersection Number, The Jacobi Inversion Problem; The topology of a compact Riemann Surface.
Chapter 2 Non-compact Riemann Surfaces
Harmonic functions and the Dirichlet Boundary Value Problem; Countable Topology; Weyl’s Lemma；The Theorems of Mittag-Leffler and Weierstrass；The Riemann Mapping Theorem; Fundamentals of Klein groups and Fuchs groups；Line and Vector Bundles.
1.Otto Forster，Lectures on Riemann Surfaces, GTM 81，Springe-Verlag 1981.
2.H.M.Farkas，I.Kra，Riemann Surfaces，GTM Vol.71，Springe-Verlag,1980.
Author: Wu Yingyi（Mathematical School of GUCAS）
Date： June 2009