Pseudorandom Subsets and Szemeredi Theorem (III)

Speaker:    Liu Huaning (Northwest University)

Time:        2:00-4:00 pm,  June 7,  2008

Place:        Room 703,  Si Yuan Building

Abstract:  Roth proved Szemeredi theorem for progressions of length three. His proof could be organized as follows.                     

        1) Define an appropriate notion of pseudorandomness.

        2) Prove that every pseudorandom subset of $\{1,2,\cdots,N\}$ contains roughly the number of arithmetic progressions of length $k$ that you would expect.

        3) Prove that if $A\subset \{1,2,\cdots,N\}$ has size $\delta N$ and is not pseudorandom, then there exists an arithmetic progression $P\subset \{1,2,\cdots,N\}$ with length tending to infinite $N$, such that $|A\cap P|\geq (\delta+\epsilon)|P|$, for some $\epsilon>0$ that depends on $\delta$ and $k$ only.

        In this series of talks, we shall introduce how Gowers proved Semeredi theorem following the above scheme.

Lecture notes: Liu Huaning's Talk-III.pdf

 

  

An Introduction to p-adic Dynamical Systems(II)

Speaker:    Yao Jiayan (Tsinghua University)

Time:        9:30-11:30 am,  May 31,  2008

Place:        Room 712,  Si Yuan Building

Abstract:  In this talk, we shall review the recent results about p-adic compatible dynamical systems, including in particular, the characterization of the minimality of affine systems, and the structure of all minimal p-adic compatible dynamical systems.

 

On the Error Term in Weyl's Law for the Heisenberg Manifolds

Speaker:    Zhai Wenguang (Shandong Normal University)

Time:        2:00-4:00 pm,  May 31,  2008

Place:        Room 712,  Si Yuan Building

Abstract:  For a fixed integer $l\geq 1$  , let $R(t)$ denote the error term in the Weyl's law of  a $(2l+1)$-dimensional Heisenberg manifold with the  metric $g_l.$ We shall prove the asymptotic formula of the $k$-th power moment for any integers $3\leq k\leq 9.$ We shall also prove that the function $t^{-(l-1/4)}R(t)$ has a distribution function.

 

 

An Introduction to p-adic Dynamical Systems(I)

Speaker:    Yao Jiayan (Tsinghua University)

Time:        9:30-11:30 am,  May 24,  2008

Place:        Room 712,  Si Yuan Building

Abstract:  In this talk, we shall review the recent results about p-adic compatible dynamical systems, including in particular, the characterization of the minimality of affine systems, and the structure of all minimal p-adic compatible dynamical systems.

 

Pseudorandom Subsets and Szemeredi Theorem (II)

Speaker:    Liu Huaning (Northwest University)

Time:        2:00-4:00 pm,  May 24,  2008

Place:        Room 712,  Si Yuan Building

Abstract:  Roth proved Szemeredi theorem for progressions of length three. His proof could be organized as follows.                     

        1) Define an appropriate notion of pseudorandomness.

        2) Prove that every pseudorandom subset of $\{1,2,\cdots,N\}$ contains roughly the number of arithmetic progressions of length $k$ that you would expect.

        3) Prove that if $A\subset \{1,2,\cdots,N\}$ has size $\delta N$ and is not pseudorandom, then there exists an arithmetic progression $P\subset \{1,2,\cdots,N\}$ with length tending to infinite $N$, such that $|A\cap P|\geq (\delta+\epsilon)|P|$, for some $\epsilon>0$ that depends on $\delta$ and $k$ only.

        In this series of talks, we shall introduce how Gowers proved Semeredi theorem following the above scheme.

Lecture notes:  Liu Huaning's Talk-II.pdf

 

 

New Opinions on the Sieve Method with Some Applications (III) 

Speaker:    Jia Chaohua (Academia Sinica)

Time:        9:30-11:30 am,  May 17,  2008

Place:        Room 712,  Si Yuan Building

Abstract:  In this series of talks, we shall introduce the paper  "Restriction theory of the Selberg sieve, with applications" written by B. Green and T. Tao, in which there are some new opinions on the sieve method with some interesting applications.

Lecture notes: Green and Tao-III.pdf

 

Pseudorandom Subsets and Szemeredi Theorem (I)

Speaker:    Liu Huaning (Northwest University)

Time:        2:00-4:00 pm,  May 17,  2008

Place:        Room 712,  Si Yuan Building

Abstract:  Roth proved Szemeredi theorem for progressions of length three. His proof could be organized as follows.                     

        1) Define an appropriate notion of pseudorandomness.

        2) Prove that every pseudorandom subset of $\{1,2,\cdots,N\}$ contains roughly the number of arithmetic progressions of length $k$ that you would expect.

        3) Prove that if $A\subset \{1,2,\cdots,N\}$ has size $\delta N$ and is not pseudorandom, then there exists an arithmetic progression $P\subset \{1,2,\cdots,N\}$ with length tending to infinite $N$, such that $|A\cap P|\geq (\delta+\epsilon)|P|$, for some $\epsilon>0$ that depends on $\delta$ and $k$ only.

        In this series of talks, we shall introduce how Gowers proved Semeredi theorem following the above scheme.

Lecture notes: Liu Huaning's Talk-I.pdf

 

 

 

  

New Opinions on the Sieve Method with Some Applications (II) 

Speaker:    Jia Chaohua (Academia Sinica)

Time:        9:30-11:30 am,  May 10,  2008

Place:        Room 712,  Si Yuan Building

Abstract:  In this series of talks, we shall introduce the paper  "Restriction theory of the Selberg sieve, with applications" written by B. Green and T. Tao, in which there are some new opinions on the sieve method with some interesting applications.

Lecture notes: Wang_Yingnan_Lecture-Beamer.pdf

 

 

 

 

 

 Previous Talks:

 

 

Notes on Szemeredi's Theorem for Length 4 (V)

Speaker:   Niu Chuanze (Beijing Normal University)

Time:       9:30-11:30 am,  July 29,  2007

Place:       Room 712,  Si Yuan Building

Abstract: In 1998 Gowers gave a new proof of Szemeredi's theorem for length 4. In this talk, we shall discuss notes of Terence Tao on Gowers' paper.

 

 

 

 

 

Notes on Szemeredi's Theorem for Length 4 (IV)

Speaker:   Niu Chuanze (Beijing Normal University)

Time:       9:30-11:30 am,  July 22,  2007

Place:       Room 712,  Si Yuan Building

 

Notes on Szemeredi's Theorem for Length 4 (III)

Speaker:   Niu Chuanze (Beijing Normal University)

Time:       9:30-11:30 am,  July 20,  2007

Place:       Room 610,  Morningside Center

 

Notes on Szemeredi's Theorem for Length 4 (II)

Speaker:   Niu Chuanze (Beijing Normal University)

Time:       9:30-11:30 am,  July 15,  2007

Place:       Room 712,  Si Yuan Building
 

Notes on Szemeredi's Theorem for Length 4 (I)

Speaker:   Niu Chuanze (Beijing Normal University)

Time:       9:30-11:30 am,  July 8,  2007

Place:       Room 712,  Si Yuan Building

 

Distribution of Primes and its Application to Dynamics of the Omega Function

Speaker:   Chen Yonggao (Nanjing Normal University)

Time:       9:30-11:30 am,  July 1,  2007

Place:       Room 712,  Si Yuan Building

Abstract:  In this talk, firstly we prove that for any subset A of the prime numbers of positive relative upper density and any nonzero integer a, the set a+A contains arbitrary long sequences which have the same largest prime factor. An application to dynamics of the omega function is given. Secondly we talk about congruent covering systems and its application to dynamics of the omega function is given.

 

 

 

 

Bourgain's Refinement of Roth's Theorem

Speaker:   Pan Hao  (Shanghai Jiao Tong University)

Time:       9:30-11:30 am,  2:00-4:00 pm,  June 23,  2007

Place:       Room 712,  Si Yuan Building

Abstract:  In this lecture, we shall introduce Bourgain's refinement of Roth's theorem.

Lecture notes: bourgain.dvi

 

 

Lecture notes: Three-primes Theorem

 

 

 

 

Replying for MS Dissertation

 

MS student: Liu Wenxin (Beijing Normal University)

 

Advisor:      Liu Chunlei (Beijing Normal University)

 

Time:          10:40-12:00 am,  June 10,  2007

Place:          Room 712,  Si Yuan Building

Dissertation title: The strong asymptotic orthogonality of the Mobius function and the polynomial phases

 

  

Uniform Distribution and Roth's Theorem (I)

 

Speaker:   Wang Yingnan  (Shandong University)

Time:       2:00-4:00 pm,  June 10,  2007

Place:       Room 712,  Si Yuan Building

 

Vinogradov's Three-primes Theorem (I)

 

Lecture notes written by Wang Yuan (Academia Sinica)

 

Spoken by Jia Chaohua (Academia Sinica)

 

Time:       9:30-11:30 am,  June 3,  2007

Place:       Room 712,  Si Yuan Building

 

 

Ternary Goldbach Problem for the Subsets of Primes with Positive Relative Densities (IV)

Speaker:   Pan Hao  (Shanghai Jiao Tong University)

Time:       2:00-4:00 pm,  June 3,  2007

Place:       Room 712,  Si Yuan Building

Abstract: Let $P$ denote the set of all primes. Suppose that $P_1$,
$P_2$, $P_3$ are three subsets of $P$ with                                        $$                                                                                             \underline{d}_{P}(P_1)+\underline{d}_{P}(P_2)+\underline{d}_{P}(P_3)>2,                                                                                                    $$                                                                                                     where $\underline{d}_{P}(P_i)$ is the lower density of $P_i$ relative to $P$. Using the method of Green in [2],  we [2] shall prove that for sufficiently large odd integer $n$, there exist $p_i\in P_i$ such that                                                                  $$                                                                                   n=p_1+p_2+p_3                                                                                $$.

Reference: [1]  B. Green,  Roth's theorem in the primes, Ann. Math., 161(2005), 1609-1636.

[2]  H. Li and H. Pan,  Ternary Goldbach problem for the subsets of primes with positive relative densities,  arXiv:math/0701240.

 

 

 

Speaker:  Liu Huaning (Northwest University,  Xi'an)

 

Time:      2:00-4:00 pm,  May 26,  2007

Place:      Room 712,  Si Yuan Building

Abstract: Recently there has been much progress in the study of arithmetic progressions. An important tool in these developments is the Gowers uniformity norm. In the past decade in a series of papers, C. Mauduit, J. Rivat and A. Sarkozy studied the pseudorandomness of the pseudorandom binary sequences. In this talk we introduce the developments of the pseudorandom binary sequences, and study the Gowers norm for the pseudorandom binary sequences. Some examples are given to show that the
``good'' pseudorandom sequences have small Gowers norm.

Lecture Notes: talk_liuhuaning200705.pdf

 

 

Speaker:  Gong Ke  (Tong Ji University)

 

Time:       2:00-4:00 pm,  May 20,  2007

Place:       Room 712,  Si Yuan Building

Abstract:  In this talk, following Goldston, Pintz and Y{\i}ld{\i}r{\i}m's work, we will introduce their proof of
$$                                                                                                      \liminf_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=0                        $$
and show the connection between small prime gaps and the level of distribution of primes in APs.

 

Lecture Notes: GPY.pdf

 

 

Sieve Method, the Exceptional Zero and Distribution of Primes

Speaker:  Lv Guangshi (Shandong University)

Reference: Iwaniec's lecture

 

Modular Forms and Automorphic L-functions

Ternary Goldbach Problem for the Subsets of Primes with Positive Relative Densities (II)

Speaker:   Pan Hao  (Shanghai Jiao Tong University)

Time:       9:30-11:30 am,  May 13,  2007

Place:       Room 712,  Si Yuan Building

 

Small Gaps between Primes (IV)

Ternary Goldbach Problem for the Subsets of Primes with Positive Relative Densities (I)

Speaker:   Pan Hao  (Shanghai Jiao Tong University)

Time:       9:30-11:30 am,  April 29,  2007

Place:       Room 712,  Si Yuan Building

 

Small Gaps between Primes (III)

Roth's Theorem in the Primes

Speaker:   Pan Hao  (Shanghai Jiao Tong University)

Time:       9:30-11:30 am,  April 22,  2007

Place:       Room 712,  Si Yuan Building

Abstract:  B. Green proved that if a subset A of primes satisfies
        \limsup_{x\to\infty}\frac{|A\cap[1,x]|}{x/\log x}>0,
then A contains a non-trivial 3-term arithmetic progression. In
this lecture, we shall discuss this result.

Reference:  B. Green,  Roth's theorem in the primes,  Ann. Math.,  161(2005), 1609-1636.  Green's paper

 

Small Gaps between Primes (II)

Analytic Part of Green-Tao Theorem (III)

Speaker:   Wang Yonghui  (Capital Normal University)

Time:       9:30-11:30 am,  April 15,  2007

Place:       Room 509,  Si Yuan Building

Abstract:  Green-Tao's paper can be viewed as several parts. Firstly, they separated the characteristic function of primes into  two parts, the Gowers anti-uniform (bounded part) and Gowers uniform (oscillary part but very small in Gowers norm) . Secondly, applying Szmeredi theorem to Gowers anti-uniform (main term), and applying generalized von-Neumann theorem to the remaining terms which contains at least one Gowers uniform, the Green-Tao's theorem is then obtained. In the separation of  the charactericstic function of primes,  it only suffices to assume the condition that the chosen characteristic function is bounded by a psedudorandom measure. Although the former parts is concluded with self-contained ergodic theory,  the third part on how to prove an arithmetic measure to be pseudorandom, in fact,  is totally an analytic number theory method, which is attributed to Goldston-Pintz-Yildirim's great breakthrough on the small gaps of primes [NT/0504336] [NT/0508185].

Sliders-3 for the third talk

 

Small Gaps between Primes (I)

Analytic Part of Green-Tao Theorem (II)

Speaker:   Wang Yonghui  (Capital Normal University)

Time:       9:30-11:30 am,  April 8,  2007

Place:       Room 712,  Si Yuan Building

slides-2 for the second talk

 

The Application of Sieve Method

Analytic Part of Green-Tao Theorem (I)

Speaker:   Wang Yonghui  (Capital Normal University)

Time:       9:30-11:30 am,  April 1,  2007

Place:       Room 509,  Si Yuan Building

Slides-1  for the first talk

 

 

Introduction to Selberg's Sieve Method

 

 

 

 

Four talks of "Relative Szemer\'edi theorem for Pseudorandom measures" 梁志斌

 

One talk of "Brauer-Manin obstruction for integral points"  徐飞

Two talks of "Prime Points in the Intersection of Two Sublattices" 刘春雷

Four talks of "Gowers norm and Generalized von Neumann Theorem" 刘文新

 

Two talks of "primes on orbits" 刘建亚

 

Four talks of "ergodic proof of Szemeredi's theorem" 姚家燕

 2. Understanding for specialist and our project-plan；

 

 3. More to be added.

 

研究生申请和资助。

    讨论班是开放式的，谁都能参加。但晨兴中心还会给正式学生成员一些资助名额。每月会有些资助（≤ 700元），同时外地来的同学会提供免费住宿和报销一次火车票（硬座）。但我们对学生的资格，要求是比较严的：

 

1. 一定是自己特别热爱数学、专心数学的，对其他想得比较少的；

2. 对Green-Tao的定理，非常感兴趣，愿意学习，不止了解，而是要深入去学的学生；

 

4. 如果你已经在学习相关文章，可以把学习笔记电邮给王永晖，以便我们加深考虑；

 

5. 对老师有礼貌，知道搽黑板，愿意把讨论班的一些任务以及杂事认真干的同学；

 

6. 这样的同学一定很少，所以为方便我们判断，你可以请自己的导师给我们推荐，但原则上，一名导师只能推荐一名学生，如果特别优秀，请让导师给贾朝华老师联系，推荐两名。

 

电邮联系时，请填写MemberList.xls , 报告题目若不清楚，可暂且不填。

 

建议和意见

如果您有些建议和意见，可以电邮给我们，也可以发在这个讨论班专用的博客上，建议和意见请集中发在下面帖子中：

advices-and-comments