Ergodic Prime Number Theorems Seminar
（Tested Version）
Supported by the Morningside Center
Organizer:Members:Liu Chunlei Xu Fei Yao Jiayan Memberlist to be added: Advisory committee
Wang Yuan Math CoffeeMaterials
Lecture Notes for this Seminar
 ScheduleAn Introduction to GreenTao's Theorem on Linear Prime Equations(II)Speaker: Wang Yonghui (Capital Normal University) Time: 9:3011:30 am, June 14, 2008 Place: Room 703, Si Yuan Building Abstract: In this talk, we introduce GreenTao's theorem on linear prime equations: 1) Dickson's conjecture states that the system of linear prime equations has solutions in convex body. Green and Tao proved this conjecture under the hypothesis of a) Gowers inverse conjecture, b) MobiusNilsequences conjecture. 2) Unconditionally, the asympototic formula was obtained for that primes has 4AP, i.e. four elements in arithmetic progression.
New Opinions on the Sieve Method with Some Applications (IV)Speaker: Jia Chaohua (Academia Sinica) Time: 2:004:00 pm, June 14, 2008 Place: Room 703, 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.
Past Talks:
An Introduction to GreenTao's Theorem on Linear Prime Equations(I)Speaker: Wang Yonghui (Capital Normal University) Time: 9:3011:30 am, June 7, 2008 Place: Room 703, Si Yuan Building Abstract: In this talk, we introduce GreenTao's theorem on linear prime equations: 1) Dickson's conjecture states that the system of linear prime equations has solutions in convex body. Green and Tao proved this conjecture under the hypothesis of a) Gowers inverse conjecture, b) MobiusNilsequences conjecture. 2) Unconditionally, the asympototic formula was obtained for that primes has 4AP, i.e. four elements in arithmetic progression. Pseudorandom Subsets and Szemeredi Theorem (III)Speaker: Liu Huaning (Northwest University) Time: 2:004:00 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 TalkIII.pdf An Introduction to padic Dynamical Systems(II)Speaker: Yao Jiayan (Tsinghua University) Time: 9:3011:30 am, May 31, 2008 Place: Room 712, Si Yuan Building Abstract: In this talk, we shall review the recent results about padic compatible dynamical systems, including in particular, the characterization of the minimality of affine systems, and the structure of all minimal padic compatible dynamical systems.
On the Error Term in Weyl's Law for the Heisenberg ManifoldsSpeaker: Zhai Wenguang (Shandong Normal University) Time: 2:004:00 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 $t^{(l1/4)}R(t)$ has a distribution function. An Introduction to padic Dynamical Systems(I)Speaker: Yao Jiayan (Tsinghua University) Time: 9:3011:30 am, May 24, 2008 Place: Room 712, Si Yuan Building Abstract: In this talk, we shall review the recent results about padic compatible dynamical systems, including in particular, the characterization of the minimality of affine systems, and the structure of all minimal padic compatible dynamical systems.
Pseudorandom Subsets and Szemeredi Theorem (II)Speaker: Liu Huaning (Northwest University) Time: 2:004:00 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 TalkII.pdf New Opinions on the Sieve Method with Some Applications (III)Speaker: Jia Chaohua (Academia Sinica) Time: 9:3011:30 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 TaoIII.pdf
Pseudorandom Subsets and Szemeredi Theorem (I)Speaker: Liu Huaning (Northwest University) Time: 2:004:00 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 TalkI.pdf
New Opinions on the Sieve Method with Some Applications (II)Speaker: Jia Chaohua (Academia Sinica) Time: 9:3011:30 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: Green and TaoII.pdf
Speaker: Wang Yingnan (Shandong University)
Time: 2:004:00 pm, May 10, 2008 Place: Room 712, Si Yuan Building Abstract: Recently, using Green's ideas, Li Hongze and Pan Hao extended the Vinogradov's theorem. In this series of talks, we shall give another proof of Li Hongze and Pan Hao's theorem, which is on the basis of Green and Tao's transference principle. Lecture notes: Wang_Yingnan_LectureBeamer.pdf
New Opinions on the Sieve Method with Some Applications (I)
Speaker: Jia Chaohua (Academia Sinica) Time: 9:3011:30 am, May 3, 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 TaoI.pdf
Speaker: Wang Yingnan (Shandong University)
Time: 2:004:00 pm, May 3, 2008 Place: Room 712, Si Yuan Building Abstract: Recently, using Green's ideas, Li Hongze and Pan Hao extended the Vinogradov's theorem. In this series of talks, we shall give another proof of Li Hongze and Pan Hao's theorem, which is on the basis of Green and Tao's transference principle. Lecture notes: Wang Yingnan IBeamer.pdf
Notes on Szemeredi's Theorem for Length 4 (V)Speaker: Niu Chuanze (Beijing Normal University) Time: 9:3011:30 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: 9:3011:30 am, July 22, 2007 Place: Room 712, Si Yuan Building
Speaker: Wang Yingnan (Shandong University)
Time: 2:004:00 pm, July 22, 2007 Place: Room 712, Si Yuan Building Abstract: This talk is a review of Professor Yao Jiayan's lectures on ergodic prime number theory in the last year. We will mainly talk about some ergodic theoremsincluding von NeuMann mean ergodic theorem, maximal ergodic theorem, and Birkhoff pointwise ergodic theoremand ergodic measurepreserving transformations.
Notes on Szemeredi's Theorem for Length 4 (III)
Speaker: Niu Chuanze (Beijing Normal University) Time: 9:3011:30 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: 9:3011:30 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: 9:3011:30 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: 9:3011:30 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.
Speaker: Wang Yingnan (Shandong University)
Time: 2:004:00 pm, July 1, 2007 Place: Room 712, Si Yuan Building Abstract: In this lecture we talk about a proof of Roth's Theorem given by Andrew Granville. Granville first gives an introduction to Weyl's famous criterion for recognizing uniform distribution mod one. When he considers uniform distribution mod N, he formulates an analogy to Weyl's criterion along the lines: The Fourier transforms of A are all small if and only if A and all of its dilates are uniform distributed. This idea is essential to his proof of Roth's Theorem. Lecture notes: roth.pdf Bourgain's Refinement of Roth's TheoremSpeaker: Pan Hao (Shanghai Jiao Tong University) Time: 9:3011:30 am, 2:004:00 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 written by Wang Yuan (Academia Sinica)
Spoken by Jia Chaohua (Academia Sinica)
Time: 9:3011:30 am, June 17, 2007
Place: Room 712, Si Yuan Building Abstract: We shall introduce Vinogradov's threeprimes Theorem according to a lecture of W. T. Gowers. Lecture notes: Threeprimes Theorem
Speaker: Wang Yingnan (Shandong University)
Time: 2:004:00 pm, June 17, 2007 Place: Room 712, Si Yuan Building Lecture notes written by Wang Yuan (Academia Sinica)
Spoken by Jia Chaohua (Academia Sinica)
Time: 9:3010:30 am, June 10, 2007
Place: Room 712, Si Yuan Building MS student: Liu Wenxin (Beijing Normal University)
Advisor: Liu Chunlei (Beijing Normal University)
Time: 10:4012:00 am, June 10, 2007
Place: Room 712, Si Yuan Building Dissertation title: The strong asymptotic orthogonality of the Mobius function and the polynomial phases
Speaker: Wang Yingnan (Shandong University)
Time: 2:004:00 pm, June 10, 2007 Place: Room 712, Si Yuan Building Lecture notes written by Wang Yuan (Academia Sinica)
Spoken by Jia Chaohua (Academia Sinica)
Time: 9:3011:30 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: 2:004:00 pm, June 3, 2007 Place: Room 712, Si Yuan Building Abstract: Let $P$ denote the set of all primes. Suppose that $P_1$, Reference: [1] B. Green, Roth's theorem in the primes, Ann. Math., 161(2005), 16091636. [2] H. Li and H. Pan, Ternary Goldbach problem for the subsets of primes with positive relative densities, arXiv:math/0701240. Ternary Goldbach Problem for the Subsets of Primes with Positive Relative DensitiesSpeaker: Li Hongze (Shanghai Jiao Tong University) Time: 9:3011:30 am, May 26, 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
Gowers Norm and Pseudorandom Measures of the Pseudorandom Binary SequencesSpeaker: Liu Huaning (Northwest University, Xi'an)
Time: 2:004:00 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 Lecture Notes: talk_liuhuaning200705.pdf
Ternary Goldbach Problem for the Subsets of Primes with Positive Relative Densities (III)Speaker: Pan Hao (Shanghai Jiao Tong University) Time: 9:3011:30 am, May 20, 2007 Place: Room 712, Si Yuan Building
Small Gaps between Primes (V)Speaker: Gong Ke (Tong Ji University)
Time: 2:004:00 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 Lecture Notes: GPY.pdf
