課程名稱 |
摩斯理論 Morse Theory |
開課學期 |
100-1 |
授課對象 |
理學院 數學研究所 |
授課教師 |
王文才 |
課號 |
MATH5335 |
課程識別碼 |
221 U5860 |
班次 |
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學分 |
3 |
全/半年 |
半年 |
必/選修 |
選修 |
上課時間 |
星期二5(12:20~13:10)星期四5,6(12:20~14:10) |
上課地點 |
天數304天數304 |
備註 |
總人數上限:30人 |
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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 |
課程大綱
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課程概述 |
Morse theory, conceived as the study of functions on manifolds, has a number of applications to geometry and topology. Many natural objects, such as geodesics, for example, arise as critical points (in particular, as minima) of a functional. Topics to be discussed include Morse functions, index, Morse lemma, Sard’s theorem, Morse inequalities, attaching handles, loop space of a manifold, energy functional. Primary applications involve conjugate points (leading to Myers’ theorem for example), determining cell decompositions (for computing Betti numbers, homology and homotopy groups), and mini-max constructions (leading to existence of certain geodesics). Depending on time and the interests of the class, additional topics might include Morse theory on singular spaces, further results on geodesics, Bott periodicity and/or outline of the proof of the Poincaré conjecture in dimensions ≥ 5. |
課程目標 |
Morse theory, conceived as the study of functions on manifolds, has a number of applications to geometry and topology. Many natural objects, such as geodesics, for example, arise as critical points (in particular, as minima) of a functional. Topics to be discussed include Morse functions, index, Morse lemma, Sard’s theorem, Morse inequalities, attaching handles, loop space of a manifold, energy functional. Primary applications involve conjugate points (leading to Myers’ theorem for example), determining cell decompositions (for computing Betti numbers, homology and homotopy groups), and mini-max constructions (leading to existence of certain geodesics). Depending on time and the interests of the class, additional topics might include Morse theory on singular spaces, further results on geodesics, Bott periodicity and/or outline of the proof of the Poincaré conjecture in dimensions ≥ 5. |
課程要求 |
Some introductory topology (e.g., CW complexes, homology and homotopy groups) would be helpful. The differential topology which is needed will be introduced or reviewed in the course. In addition, a background familiarity with geodesics and definition of curvature would be helpful, although not strictly necessary. |
預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Textbook: Morse theory, J. Milnor, Annals of Mathematics Studies 51, Princeton University Press 1963 |
參考書目 |
Textbook: Morse theory, J. Milnor, Annals of Mathematics Studies 51, Princeton University Press 1963 |
評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
作業 |
80% |
Homework problems will be assigned roughly every two weeks. |
2. |
報告 |
20% |
giving a presentation of a special topic |
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