課程資訊
課程名稱
組合學二
Combinatorics (Ⅱ) 
開課學期
100-1 
授課對象
理學院  數學系  
授課教師
張鎮華 
課號
MATH7702 
課程識別碼
221 U3300 
班次
 
學分
全/半年
半年 
必/選修
選修 
上課時間
星期二1(8:10~9:00)星期五1,2(8:10~10:00) 
上課地點
天數204天數204 
備註
總人數上限:40人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1001combinatorics2 
課程簡介影片
 
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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.  

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. 
課程要求
代數導論。 
預期每週課後學習時數
 
Office Hours
 
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.
 
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.
 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
作業 
30% 
 
2. 
期中考試 
35% 
 
3. 
期末考試 
35% 
 
 
課程進度
週次
日期
單元主題
第1週
9/13,9/16  Introduction / van der Waerden's theorem 
第2週
9/20,9/23  van der Waerden's theorem / Hales-Jewett's theorem 
第3週
9/27,9/30  Dynamical proof of van der Waerden's theorem 
第4週
10/04,10/07  The Furstenberg correspondence principle vs Szemered's theorem 
第5週
10/11,10/14  Princeton Lecture Note [4]---Chapters 2 and 5, Semeredi's regularity lemma 
第6週
10/18,10/21  Princeton Lecture Note [4]----Chapter 6,Roth's proof for Szemeredi's theorem with k = 3 
第7週
10/25,10/28  Chapter 2 in Tao and Vu [2]: Sum set estimates  
第8週
11/01,11/04  Chapter 2 in Tao and Vu [2]: Sum set estimates  
第9週
11/08,11/11  Chapter 2 in Tao and Vu [2]: Sum set estimates  
第10週
11/15,11/18  Chapter 2 in Tao and Vu [2]: Sum set estimates  
第11週
11/22,11/25  Chapter 9 in tao and Vu: Algebraic Method 
第12週
11/29,12/02  Chapter 9 in tao and Vu: Algebraic Method 
第13週
12/06,12/09  (P1) Scetion 5.1 up to Theorem 5.7. ----- (P2) Section 5.1 from Proposition 5.8. 
第14週
12/13,12/16  (P2) Section 5.1 from Proposition 5.8. -----(P3) Section 5.2 up to Corollary 5.16. 
第15週
12/20,12/23  (P4) Scetion 5.2 from Theorem 5.17. 
第16週
12/27,12/30  (P5) Section 5.3. 
第17週
1/03,1/06  (P6) Section 5.4. 
第18週
1/10, 1/13  (P7) Section 5.5.