### In Schedule

#### Analytic part of Green-Tao Theorem

Wang Yonghui, Capital Normal University

Time: 2007 April 1, April 8, April 15

Room: To be annouced in Institute of Math, since MCM building has just been decorated.

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].

### Past Schedule

#### Relative Szemer\'edi theorem for Pseudorandom measures.

Speaker: Liang Zhibin, Peking University.

Time: Four talks， Jan 16 and Jan 30, 2007 9:30am-11:30am and 2:00pm-4:00pm

Abstracts: My talks is based on section 6-8 of Green-Tao's paper. I will introduce the proof of Szemer\'edi theorem relative to a pseudoramdon measure. It will take about three speeches for me to complete all the proof. In the first speech I will give the definition of the dual norm of Gowers norm and the Gowers anti-uniform functions. In the second speech I will introduce the $\sigma$-algebra and generalised Koopman-von Neumann structure theorem. In the last time I will complete the left of the proof of Szemer\'edi theorem relative to a pseudoramdon measure, using Furstenberg tower.

#### Brauer-Manin obstruction for integral points.

Time&Place: Jan. 24 2:00am-4:00am (Beijing Normal University, 北师大教八楼（数学楼）214室)

Slides: Xufei2006-vietnam.pdf

Abstract: Most classical methods, for example circle method, are purely local. Namely the local-global principle holds. However one can not expect this principle is true in general. In this talk, I'll explain how to refine the local-principle by using the reciprocity law. This is simply analogue to study the rational points case developed by Manin, Swinnerton-Dyer, Sansuc, Colliot-Thelene, Borovoi, Skorobogatov, Harari ... and so on. Our main result is that the integral Brauer-Manin obstruction is the only obstruction from local to global for the integral points of schemes of finite type over ring of integers of number fields, whose generic fiber are the homogenous spaces of semi-simple, simply connected algebraic groups of non-compact type. Some interesting examples will be provided. As application, the sum of three integral squares over imaginary quadratic fields and cyclotomic fields are determined. This is a joint work with Colliot-Thelene and Dasheng Wei.

#### 关于受限和集基数下界的估计(I)(II)

Zhiwei Sun, Nanjin University

First Talk: Problems and Results on Restricted Sumsets

Time&Place: Jan. 24 9:30am-11:30am (Beijing Normal University, 北师大教八楼（数学楼）214室)

Second Talk： Combinatorial aspects of Szemeredi's theorem

Time: Jan. 30 (Institute of Mathematics)

#### Prime points in the intersection of two sublattices

Abstract: A famous and difficult open conjecture of Dickson predicts the

distribution of prime points in a given affine sublattice. Assuming

the Gowers Inverse conjecture and the Mobius conjecture of

finite parameter s, Green-Tao verified Dickson's conjecture for

affine sublattices which are ranges of affine linear maps of

complexity $s$. In this paper, as an application of Green-Tao's

theorem on Dickson's conjecture, we study the distribution of prime

points in affine sublattices of finite index in a given affine

sublattice.

#### 素数包含任意长的算术级数(I)：Gowers范数和广义von Neumann定理

刘文新（北京师范大学）

时间：2006年12月19日，上午9:30-11:30

Second talk：Jan-2-2007 2:00pm-4:00pm

#### Primes on Orbits

刘建亚 (山东大学) Liu Jianya (ShanDong Univ.)

Dec 25, 9:30-11:30am，中科院数学院大楼509室

Dec 26, 2:00-4:00pm， 中科院数学院大楼511室

Abstract: Firstly, I will introduce a recent conjecture of Peter Sarnak on the distribution of primes on orbits, which may be viewed as a non-abelian extension of Dirichlet's theorem, the Hardy-Littlewood prime k-tuple conjecture, Schinzel's hypothesis, etc.

After that, I will report a joint result with Sarnak concerning prime solutions to ternary quadratic equations.

Reference on Sarnak's Conjecture:

1. http://www.math.princeton.edu/sarnak/rademacher1.pdf by Sarnak, handwriting-version，53Pages，6 hours talk at Fall 2006 - Hans Rademacher Lectures in Mathematics.

And there is a typed-version SarnakConj.pdf by some student in Shandong University.

Abstract: In this talk we shall give a short introduction to Szemeredi's theorem and related results in combinatorial number theory and in ergodic theory, and then show Szemeredi's theorem, by using the multiple Poincare recurrence theorem in ergodic theory.

Talk2: Birkhoff ergodic theorem and its applications

Abstract: In this talk we shall give a proof of Birkhoff pointwise ergodic theorem, and then present some applications.

Talk3: Multiple Poincare recurrence theorem and its proof

Abstract: In this talk we shall apply the results presented in the last talk to prove the multiple Poincare recurrence theorem.

#### 素数的多尺度分析

贾朝华 10月31日9:30-11:30am，11月7日9:30-11:30am，11月14日9:30-11:30am。

地点：晨兴中心510

(Based on Tao's slides),

Lecture Notes by Jia Chaohua Notes

Study Note by Wang Yonghui Note-Mscale.pdf

### In Preparation