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課程名稱 |
複分析II
Functions of A Complex Variable II |
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開課學期 |
106-2 |
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授課對象 |
理學院 數學系 |
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授課教師 |
莊武諺 |
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課號 |
MATH4005 |
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課程識別碼 |
201 49800 |
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班次 |
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學分 |
3.0 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期二6,7(13:20~15:10)星期四6,7(13:20~15:10) |
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上課地點 |
天數304天數304 |
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備註 |
總人數上限：40人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1062complexanalysis |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
complex dynamics, Julia set, linear ODE, monodromy groups, analytic continuations, elliptic functions, modular forms, holomorphic/meromorphic functions/1-forms on Riemann surfaces, Riemann-Roch theorem, uniformization, Abel-Jacobi theorem, Torelli theorem, Abelian varieties. |
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課程目標 |
This course is a continuation of ”Complex Analysis I” from the previous semester. We will cover more advanced topics, including modular forms, Riemann surfaces, and Abelian varieties. |
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課程要求 |
Course prerequisite: general topology and Complex Analysis I. |
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預期每週課後學習時數 |
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Office Hours |
每週四 14:40~15:30
每週二 14:40~15:30 |
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指定閱讀 |
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參考書目 |
1. Ahlfors, Complex Analysis.
2. Stein and Shakarchi, Complex Analysis.
3. Gamelin, Complex analysis.
4. Serre, A Course in Arithmetic.
5. Farkas and Kra, Riemann surfaces.
6. Some references will be supplemented along the way. |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Homework |
50% |
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2. |
Presentation |
50% |
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週次 |
日期 |
單元主題 |
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第1週 |
2/27,3/01 |
2/27: analytic continuation, linear differential equations, hypergeometric differential equations. [Ahlfors, chap.8]
3/01: hypergeometric differential equations. |
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第2週 |
3/06,3/08 |
3/06: modular forms. [Serre, chap.6]
3/08: modular forms. [Serre, chap.6] |
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第3週 |
3/13,3/15 |
3/13: modular forms. [Serre, chap.6]
3/15: Hecke operators. [Serre, chap.6] |
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第4週 |
3/20,3/22 |
3/20: Hecke operators. [Serre, chap.6]
3/22: Riemann surfaces. [FK, chap.1] |
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第5週 |
3/27,3/29 |
3/27: holomorphic maps, differential forms. [FK, chap.1]
3/29: differential forms, Weyl's lemma. [FK, chap.1, chap.2] |
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第6週 |
4/03,4/05 |
no class |
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第7週 |
4/10,4/12 |
4/10: Weyl's lemma, harmonic differentials. [FK, chap.2.2, 2.3]
4/12: harmonic differentials, meromorphic functions and differentials. [FK, chap.2.4, 2.5] |
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第8週 |
4/17,4/19 |
4/17: harmonic differentials, meromorphic functions and differentials. [FK, chap.2.4, 2.5]
4/19: topology of compact Riemann surfaces. [FK, chap.1.2] |
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第9週 |
4/24,4/26 |
4/24: harmonic and holomorphic differentials, bilinear relation. [FK, chap.3.1, 3.2]
4/26: bilinear relation, periods of meromorphi differentials, simplest case of Riemann-Roch. [FK, chap.3.3, 3.4] |
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第10週 |
5/01,5/03 |
5/01: divisors and the Riemann-Roch theorem. [FK, chap.3.4]
5/03: Weierstrass points. [FK, chap.3.5] |
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第11週 |
5/08,5/10 |
5/08: Weierstrass points. [FK, chap.3.5]
5/10: Abel-Jacobi theorem. [FK, chap.3.6] |
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第12週 |
5/15,5/17 |
5/15: Abel-Jacobi theorem, Jacobi inversion theorem. [FK, chap.3.6]
5/17: more on Jacobian varieties. [FK, chap.3.11] |
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第13週 |
5/22,5/24 |
5/22: Universal property of Jacobian varieties, symmetric products and integral divisors. [FK, chap.3.11]
5/24: symmetric products and integral divisors. [FK, chap.3.11] |
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第14週 |
5/29,5/31 |
5/29: Torelli theorem. [FK, chap.3.12]
5/31: Torelli theorem. [FK, chap.3.12] |
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第15週 |
6/05,6/07 |
6/05: Uniformization. [Gamelin, chap.16]
6/07: Uniformization. [Gamelin, chap.16] |
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第16週 |
6/12,6/14 |
6/12: Uniformization. [Gamelin, chap.16]
6/14: no class. |
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第17週 |
6/19,6/21 |
no class |
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