Riemann SurfacesⅡ

  • 申立勇
  • Created: 2014-12-08
Riemann SurfacesⅡ



Course No.S070101ZJ004    

Course Category Professional Basic Course


PrerequisitesRiemann 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.

Primary Coverage

Chapter 1 Compact Riemann Surfaces

The Serre Duality TheoremThe Riemann-Hurwitz formulaHolomorphic 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 LemmaThe Theorems of Mittag-Leffler and WeierstrassThe Riemann Mapping Theorem; Fundamentals of Klein groups and Fuchs groupsLine and Vector Bundles.


1.Otto ForsterLectures on Riemann Surfaces, GTM 81Springe-Verlag 1981.

2.H.M.FarkasI.KraRiemann SurfacesGTM Vol.71Springe-Verlag,1980.

3.L.V.AhlforsL.Sario,Riemann SurfacesPrinceton,1960.                                      


Author: Wu YingyiMathematical School of GUCAS

Date June 2009