Course No.：S070100XJ007
Course Category: Basic course of the subject
Period/Credits: 40/2
Prerequisites：Basic Topology, Functions of one complex variable
Aims & Requirements：
This course is one of the basic courses for the doctoral candidates and the master degree candidates of all of subjects 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 I, we wish the students can understand some basic concepts, methods and techniques.
Primary Coverage：
Chapter 1 Some basic concepts of Riemann Surfaces
The definition of Riemann surface and some classical examples of Riemann surfaces; The definition of holomorphic map and some properties of holomorphic map; Homotopy of Curves and the Fundamental Group; Branched and Unbranched Coverings; The Universal Covering and Covering Transformations.
Chapter 2 Sheaf
The definitions of presheaf , sheaf , germ and stalk; topology of the presheaf, Analytic Continuation, the construction of Riemann surfaces; Algebraic Functions.
Chapter 3 Differential Forms and the Integration of Differential Forms
and operators; The definitions of holomorphic 1-form and meromorphic 1-form; the definition of residue and the Residue Theorem; Period Homomorphism and automorphic function.
Chapter 4 The Riemann-Roch Theorem
Cohomology Groups; Dolbeault’s Lemma; Finiteness Theorem; The Exact Cohomology Sequence; The Riemann-Roch Theorem and some simple applications.
References：
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.
3.L.V.Ahlfors，L.Sario,，Riemann Surfaces，Princeton,1960.
Author: Yingyi Wu（Mathematical School of GUCAS）
Date： June 2009
