Schedule
An Introduction to Green-Tao's Theorem on Linear Prime Equations(II)
Speaker: Wang Yonghui (Capital Normal University)
Time: 9:30-11:30 am, June 14, 2008
Place: Room 703, Si Yuan Building
Abstract: In this talk, we introduce Green-Tao'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) Mobius-Nilsequences conjecture.
2) Unconditionally, the asympototic formula was obtained for that primes has 4-AP, i.e. four elements in arithmetic progression.
New Opinions on the Sieve Method with Some Applications (IV)
Speaker: Jia Chaohua (Academia Sinica)
Time: 2:00-4:00 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 Green-Tao's Theorem on Linear Prime Equations(I)
Speaker: Wang Yonghui (Capital Normal University)
Time: 9:30-11:30 am, June 7, 2008
Place: Room 703, Si Yuan Building
Abstract: In this talk, we introduce Green-Tao'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) Mobius-Nilsequences conjecture.
2) Unconditionally, the asympototic formula was obtained for that primes has 4-AP, i.e. four elements in arithmetic progression.
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.
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: Green and Tao-II.pdf
Another Proof of Li Hongze and Pan Hao's Theorem (II)
Time: 2:00-4:00 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.
New Opinions on the Sieve Method with Some Applications (I)
Speaker: Jia Chaohua (Academia Sinica)
Time: 9:30-11:30 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 Tao-I.pdf
Another Proof of Li Hongze and Pan Hao's Theorem (I)
Time: 2:00-4:00 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 I-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.
Mathematical Reviews on Ergodic Theory
Time: 2:00-4:00 pm, July 29, 2007
Place: Room 712, Si Yuan Building
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
Ergodic Theorems (IV)
Time: 2:00-4:00 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 theorems-including von NeuMann mean ergodic theorem, maximal ergodic theorem, and Birkhoff pointwise ergodic theorem-and ergodic measure-preserving transformations.
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
Ergodic Theorems (III)
Time: 2:00-4:00 pm, 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
Ergodic Theorems (II)
Time: 2:00-4:00 pm, 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
Ergodic Theorems(I)
Time: 2:00-4:00 pm, 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.
Uniform Distribution and Roth's Theorem (III)
Time: 2:00-4:00 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 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.
Vinogradov's Three-primes Theorem (III)
Place: Room 712, Si Yuan Building
Abstract: We shall introduce Vinogradov's three-primes Theorem according to a lecture of W. T. Gowers.
Uniform Distribution and Roth's Theorem (II)
Time: 2:00-4:00 pm, June 17, 2007
Place: Room 712, Si Yuan Building
Vinogradov's Three-primes Theorem (II)
Place: Room 712, Si Yuan Building
Replying for MS Dissertation
Place: Room 712, Si Yuan Building
Uniform Distribution and Roth's Theorem (I)
Time: 2:00-4:00 pm, June 10, 2007
Place: Room 712, Si Yuan Building
Vinogradov's Three-primes Theorem (I)
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.
Ternary Goldbach Problem for the Subsets of Primes with Positive Relative Densities
Speaker: Li Hongze (Shanghai Jiao Tong University)
Time: 9:30-11:30 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
$$ \underline{d}_{\P}(\P_1)+\underline{d}_{\P}(\P_2)+\underline{d}_{\P}(\P_3)>2, $$ where
$\underline{d}_{\P}(\P_i)$ is the lower density of $\P_i$ relative
to $\P$. We prove that for sufficiently large odd integer $n$,
there exist $p_i\in\P_i$ such that $n=p_1+p_2+p_3$.
Gowers Norm and Pseudorandom Measures of the Pseudorandom Binary Sequences
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
Ternary Goldbach Problem for the Subsets of Primes with Positive Relative Densities (III)
Speaker: Pan Hao (Shanghai Jiao Tong University)
Time: 9:30-11:30 am, May 20, 2007
Place: Room 712, Si Yuan Building
Small Gaps between Primes (V)
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.
Sieve Method, the Exceptional Zero and Distribution of Primes
Speaker: Lv Guangshi (Shandong University)
Place: Room 712, Si Yuan Building
Abstract: The speaker will report Friedlander and Iwaniec's lectures at ICTP, Italy, which include
1) the principle, the limitation of classical sieve method;
2) the key point, the achievement and the future of modern sieve method;
3) exceptional zero and the distribution of primes.
Reference: Iwaniec's lecture
Modular Forms and Automorphic L-functions
Place: Room 509, Si Yuan Building
Abstract: The speaker will report P. Michel's lecture at ICTP, Italy, which include
1) holomorphic modular forms; 2) Maass wave forms; 3) automorphic L-functions; 4) Eisenstein series; 5) estimate of Fourier coefficients; 6) some applications of modular foms, ect.
Reference: Kowalski's lecture
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)
Place: Room 712, Si Yuan Building
Abstract: These consecutive talks will introduce the important work on small gaps between primes by D. A. Goldston, J. Pintz and C. Y. Yildirim. We first give some history of the problem, then demonstrate their methods of tuple approximations and the way of applying the approximations. Also we will show the connection between small prime gaps and the distribution of primes in arithmetic progressions. Of course the "gaps" between my talks will be, bounded, seven!
Lecture notes: GY.pdf
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)
Place: Room 712, Si Yuan Building
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)
Place: Room 712, Si Yuan Building
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)
Place: Room 509, Si Yuan Building
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
Introduction to Selberg's Sieve Method
Two talks of "Problems and Results on Restricted Sumsets" 孙智伟
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
Three talks of "素数的多尺度分析" 贾朝华
更多计划请看 More details....
Notices
Why we hold this workshop ?
1. Understanding for non-specialist；