課程資訊
課程名稱
 黎曼面導論
Introduction to Riemann Surfaces 
開課學期
 104-2 
授課對象
 理學院  數學系  
授課教師
 蔡忠潤 
課號
 MATH5344 
課程識別碼
 221 U6090 
班次
  
學分
 3 
全/半年
 半年 
必/選修
 選修 
上課時間
 星期二6,7(13:20~15:10)星期四6,7(13:20~15:10) 
上課地點
 天數302天數302 
備註
 二四(13:45~15:00)
總人數上限：40人 
Ceiba 課程網頁
 http://ceiba.ntu.edu.tw/1042RS 
課程簡介影片
 
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課程概述

http://www.math.ntu.edu.tw/courses/super_pages.php?ID=allcourses 

課程目標
  
課程要求
  
預期每週課後學習時數
  
Office Hours
  
指定閱讀
  
參考書目
  
評量方式
(僅供參考)
    
課程進度
週次
日期
單元主題
Week 1
 2/23,2/25  basic examples and properties of Riemann surfaces, maps, differential forms. Reference: [FK, §I.1] 
Week 2
 3/01,3/03  more about the differential forms on a Riemann surface, Weyl's lemma. Reference: [FK, §I.3, I.4, II.1 and II.2] 
Week 3
 3/08,3/10  harmonic differential, Hodge decomposition, harmonic function with singularities. Reference: [FK, §II.3 and II.4] 
Week 4
 3/15,3/17  meromorphic differential, and meromorphic function. topology of compact Riemann surfaces. Reference: [FK, §II.4, II.5 and I.2] 
Week 5
 3/22,3/24  topology of compact Riemann surfaces (continued), harmonic and holomorphic differentials, bilinear relation. Reference: [FK, §I.2] 
Week 6
 3/29,3/31  bilinear relation (continued), periods of meromorphi differentials, simplest case of Riemann-Roch. Reference: [FK, §III.3 and III.4] 
Week 7
 4/07  divisors and the Riemann-Roch theorem. Reference: [FK, §III.4] 
Week 8
 4/12,4/14  some applications of the Riemann-Roch theorem, Weierstrass points. Reference: [FK, §III.4 and III.5] 
Week 9
 4/19,4/21  Abel's theorem and Jacobi inversion problem. Reference: [FK, §III.6] 
Week 10
 4/26,4/28  midterm

Elliptic integral, elliptic function and Weierstrass ℘-function. 
Week 11
 5/03,5/05  Restoration: no class this week. 
Week 12
 5/10,5/12  Uniformization theorem: Perron's method, Green's function. Reference: [Gamelin, §XVI] 
Week 13
 5/17,5/19  Uniformization theorem: bipolar Green's function and the uniformization theorem. Reference: [Gamelin, §XVI] 
Week 14
 5/24,5/26  Torelli theorem: more on the Jacobian varieties. Reference: [FK, §III.11] 
Week 15
 5/31,6/02  Torelli theorem: tranlation properties of W_r, sketch of the proof of Torelli theorem. Reference: [FK, §III.11] 
Week 16
 6/07  (Jacobi) theta function. Reference: [FK, §VI] 
Week 17
 FINAL  final report (writing repoart)

Please let me know the topic you choose by June 5. The deadline of the final is June 30.