Ergodic Prime Number Theorems Seminar

（Tested Version）

Supported by the Morningside Center

Jia Chaohua

Liu Chunlei

Xu Fei

Yao Jiayan

Wang Yuan

Zhang Shouwu

### Math Coffee

BLOG For JNTBEIJING

Literature for Coffee-time

《中国数学会通讯》文章选登

### Materials

Number Theory Web Source

Lecture Notes for this Seminar

BookMarks for this Seminar

## 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)

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)

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)

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)

Speaker:    Wang Yingnan  (Shandong University)

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)

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)

Speaker:    Wang Yingnan  (Shandong University)

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

Speaker:   Wang Yingnan  (Shandong University)

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)

Speaker:   Wang Yingnan  (Shandong University)

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)

Speaker:   Wang Yingnan  (Shandong University)

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)

Speaker:   Wang Yingnan  (Shandong University)

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)

Speaker:   Wang Yingnan  (Shandong University)

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)

Speaker:   Wang Yingnan  (Shandong University)

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.

Lecture notes: bourgain.dvi

Lecture notes written by Wang Yuan (Academia Sinica)

Spoken by Jia Chaohua (Academia Sinica)

Time:       9:30-11:30 am,  June 17,  2007

Place:       Room 712,  Si Yuan Building

Abstract:  We shall introduce Vinogradov's three-primes Theorem according to a lecture of W. T. Gowers.

Lecture notes: Three-primes Theorem

#### Uniform Distribution and Roth's Theorem (II)

Speaker:   Wang Yingnan  (Shandong University)

Time:       2:00-4:00 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:       9:30-10:30 am,  June 10,  2007

Place:       Room 712,  Si Yuan Building

MS student: Liu Wenxin (Beijing Normal University)

Advisor:      Liu Chunlei (Beijing Normal University)

Time:          10:40-12:00 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

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.

#### 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

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

#### 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)

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)

Time:      9:00-11:00 am, May 14, 2007

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

Speaker:  Ji Guanghua (Shandong University)

Time:      1:30-3:30 pm, May 14, 2007

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)

Speaker:  Gong Ke  (Tong Ji University)

Time:       2:00-4:00 pm,  May 13,  2007

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)

Speaker:  Gong Ke  (Tong Ji University)

Time:       2:00-4:00 pm,  April 29,  2007

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)

Speaker:  Gong Ke  (Tong Ji University)

Time:       2:00-4:00 pm,  April 22,  2007

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)

Speaker:  Gong Ke  (Tong Ji University)

Time:       2:00-4:00 pm,  April 15,  2007

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

Gong Ke  (Tong Ji University)

Time:       2:00-4:00 pm,  April 8,  2007

Place:       Room 712,  Si Yuan Building

Lecture notes: Notes of Jia Chaohua

#### 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

Time:       2:00-4:00 pm,  April 1,  2007

Place:       Room 509,  Si Yuan Building

Lecture notes: Introduction to Selberg sieve method-Jia.pdf

Two talks of "Problems and Results on Restricted Sumsets" 孙智伟

Two talks of "primes on orbits" 刘建亚

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

Three talks of "素数的多尺度分析"  贾朝华

## Notices

### 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；

3. 所以，如果你申请后获得资助，希望能全程参加（或至少一月以上），而不要才来两个星期就走，如果仅把晨兴的资助当作一次旅游机会，那对你和你导师的声誉会造成不良影响

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。