課程名稱 |
微分幾何二 Differential Geometry (Ⅱ) |
開課學期 |
101-2 |
授課對象 |
理學院 數學研究所 |
授課教師 |
王金龍 |
課號 |
MATH7302 |
課程識別碼 |
221 U2940 |
班次 |
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學分 |
3 |
全/半年 |
半年 |
必/選修 |
選修 |
上課時間 |
星期三8(15:30~16:20)星期五3,4(10:20~12:10) |
上課地點 |
天數102天數102 |
備註 |
研究所數學組基礎課。 總人數上限:30人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1012dg2 |
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 |
課程大綱
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課程概述 |
An introduction to the modern theory of differential geometry, including both the linear and non-linear methods. The linear theory will include the theory of vector bundles and characteristic classes, as well as the Atiyah--Singer index theorem. The non-linear theory will include an introduction to geometric analysis, the Plateau problem, Donaldson theory and perhaps the Ricci flow if time is allowed. |
課程目標 |
Provide an essential foundation in modern geometry, with emphasizes on developments after 1950, and open a way to pursue work or research in differential topology and differential geometry. |
課程要求 |
Differential Geometry (I) |
預期每週課後學習時數 |
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Office Hours |
另約時間 |
指定閱讀 |
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參考書目 |
I-1: Lawson: Lectures on minimal submanifolds
I-2: Osserman: A survey of minimal surfaces
II-1: Berline, Getzler and Vergne: Heat kernels and Dirac operators
II-2: Gilkey: Invariance theory, heat equation, and the Atiyah--Singer index theorem
II-3: Lawson and Michelson: Spin geometry
III-1: Schoen--Yau: Lectures on differential geometry
III-2: Li: Lectures on geometric analysis
IV-1: Freed--Uhlenbeck: Instantons and four manifolds |
評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
期中考 |
40% |
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2. |
期末報告 |
40% |
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3. |
作業 |
20% |
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週次 |
日期 |
單元主題 |
第1週 |
2/20,2/22 |
Kaehler geometry and calibration. |
第2週 |
2/27,3/01 |
Weierstrass representations, minimizing sub-manifolds |
第3週 |
3/06,3/08 |
Douglas-Rado's solution to Plateau problem |
第4週 |
3/13,3/15 |
Vector bundles and characteristic classes |
第5週 |
3/20,3/22 |
Heat kernels of elliptic operators |
第6週 |
3/27,3/29 |
Clifford modules and Dirac operators |
第7週 |
4/03,4/05 |
Reading break |
第8週 |
4/10,4/12 |
Atiyah-Singer Index theorem |
第9週 |
4/17,4/19 |
Midterm Exam |
第10週 |
4/24,4/26 |
Gauss-Bonnet-Chern and Hirzebruch's signature theorem |
第11週 |
5/01,5/03 |
Milnor's exotic S^7 |
第12週 |
5/08,5/10 |
Donaldson--Freidman's exotic R^4 |
第13週 |
5/15,5/17 |
ASD moduli and its local structures |
第14週 |
5/22,5/24 |
Intro to Taubes' theorem and Seiberg-Witten |
第15週 |
5/29,5/31 |
Perspectives on modern geometry |
第16週 |
6/05,6/07 |
Final Reports I, II |
第17週 |
6/12,6/14 |
Final Reports III, IV |
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