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課程名稱 |
組合學二
Combinatorics (Ⅱ) |
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開課學期 |
100-1 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
張鎮華 |
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課號 |
MATH7702 |
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課程識別碼 |
221 U3300 |
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班次 |
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學分 |
3 |
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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 |
星期二1(8:10~9:00)星期五1,2(8:10~10:00) |
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上課地點 |
天數204天數204 |
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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/1001combinatorics2 |
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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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課程概述 |
The course basicaly follows the book “Additice Combinatorics” edited by Granville et al. [1], using Tao and Vu’s book [2] as an important addition. It will start at the classical theorem of arithmetic progressions by van der Waerden, and try to end at Szemeredi’s theorem, at least for the case of k = 3. Various tools as well as related topics will also be touched. |
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課程目標 |
Additive combinatorics is currently a highly active area of resaerch for several reasons, for example its many applications to additive number theory. One remarkable feature of the field is the use of tools from many diverse fields of mathematics, including elementary combinatorics, harmonic analysis, convex geometry, incidence geometry, graph theory, probability, algebraic geometry, and ergodic theory; this wealth of perspectives makes addivitive combinatorics a rich, fascinating, and multi-faceted subject. The purpose of this course is to introduce basic tools in this subject. |
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課程要求 |
代數導論。 |
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預期每週課後學習時數 |
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Office Hours |
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參考書目 |
[1] A. Granville, M. B. Nathanson and J. Solymosi ed., Additive Combinatorics,
AMS, 2007.
[2] T. Tao and V. Vu, Additive Combinatorics, Cambridge University Press, 2006.
[3] K. Soundararajan, Additive Combinatorics, Lecture Notes, Winter 2007.
[4] R. L. Graham, Rudiments of Ramsey Theory, AMS, 1979.
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指定閱讀 |
[1] A. Granville, M. B. Nathanson and J. Solymosi ed., Additive Combinatorics, AMS, 2007.
[2] T. Tao and V. Vu, Additive Combinatorics, Cambridge University Press, 2006.
[3] K. Soundararajan, Additive Combinatorics, Lecture Notes, Winter 2007.
[4] R. L. Graham, Rudiments of Ramsey Theory, AMS, 1979.
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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. |
作業 |
30% |
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2. |
期中考試 |
35% |
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3. |
期末考試 |
35% |
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週次 |
日期 |
單元主題 |
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第1週 |
9/13,9/16 |
Introduction / van der Waerden's theorem |
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第2週 |
9/20,9/23 |
van der Waerden's theorem / Hales-Jewett's theorem |
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第3週 |
9/27,9/30 |
Dynamical proof of van der Waerden's theorem |
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第4週 |
10/04,10/07 |
The Furstenberg correspondence principle vs Szemered's theorem |
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第5週 |
10/11,10/14 |
Princeton Lecture Note [4]---Chapters 2 and 5, Semeredi's regularity lemma |
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第6週 |
10/18,10/21 |
Princeton Lecture Note [4]----Chapter 6,Roth's proof for Szemeredi's theorem with k = 3 |
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第7週 |
10/25,10/28 |
Chapter 2 in Tao and Vu [2]: Sum set estimates |
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第8週 |
11/01,11/04 |
Chapter 2 in Tao and Vu [2]: Sum set estimates |
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第9週 |
11/08,11/11 |
Chapter 2 in Tao and Vu [2]: Sum set estimates |
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第10週 |
11/15,11/18 |
Chapter 2 in Tao and Vu [2]: Sum set estimates |
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第11週 |
11/22,11/25 |
Chapter 9 in tao and Vu: Algebraic Method |
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第12週 |
11/29,12/02 |
Chapter 9 in tao and Vu: Algebraic Method |
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第13週 |
12/06,12/09 |
(P1) Scetion 5.1 up to Theorem 5.7. ----- (P2) Section 5.1 from Proposition 5.8. |
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第14週 |
12/13,12/16 |
(P2) Section 5.1 from Proposition 5.8. -----(P3) Section 5.2 up to Corollary 5.16. |
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第15週 |
12/20,12/23 |
(P4) Scetion 5.2 from Theorem 5.17. |
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第16週 |
12/27,12/30 |
(P5) Section 5.3. |
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第17週 |
1/03,1/06 |
(P6) Section 5.4. |
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第18週 |
1/10, 1/13 |
(P7) Section 5.5. |
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