Joint Number Theory Seminars at Beijing

Monday, April 30, 2007

讨论班上幻灯片使用需注意

现在讨论班上,一般使用幻动片比较多,但年轻学生似乎还未太注意到使用幻动片,是需要注意的,也许你讲的时候是省事些,但其实意味着,底下必须做更多的准备。

我建议几点,希望大家考虑:

1. 幻灯和发表的文章是不同的,后者避免重复,前者是一定要多重复,幻动片上,可以不断出现前面的东西,定义之类,这样,就不用跳来跳去,让人找了。

2. 但有个别时候不得不跳,建议在tex中使用

\usepackage[pdfstartview=FitH]{hyperref}
\usepackage[hyperpageref]{backref}

这样,即使要跳的时候,也只要点击highlight后的编号就立刻达到,不用翻页找来找去。

3. 给大家研讲,展示幻灯时,念的时候,不要图快,就是老老实实地一字一句的念,不要"这个、那个"的代指,甚至还要加说一些公式以外的"废话",这样大家才基本能跟得上你。

4. 建议年轻学生研讲时,如果准备时间充分,还是应该学会使用tex中的Beamer格式。它的好处,是
a。是一页里,字够大,放的东西不多,从而容易集中注意力,
b。是它有逐渐释放的功能,可以类似于黑板写字,甚至更好,做到有先有后。 我们知道思维的顺序,对数学是很重要的,普通文本的缺陷正在于此。Beamer可能对于年纪稍大的老师是比较难亲近的,但是对于年轻学生来讲,还是应该把自己的研讲幻灯,做得好一点。如果数学水平稍差一些,那就更该把研讲形式做得好一些,这并不是多此一举,因为长期做这些有形的材料整理,日久之功,就会提升自己无形的思维素养。

Saturday, April 21, 2007

Online Bookmark on delicious

Dear Colleagues,

Please also remember that, we have an online bookmark for ergodicpnt on the

http://del.icio.us/ergodicpnt


If you use "google reader", or other rss reader, It will be very convenient to read the updates of the bookmark, once if you have subcribe it in your rss reader.

I have also set up another online bookmark for more general topic: Number theory on
http://del.icio.us/network/arith
which will include any sub-bookmarks from other number theorist on del.icio.us.
So you can make an online bookmark for yourself on number theory, and tell me your address, then I will include you in the network of
http://del.icio.us/network/arith

and then anyone who rss this webpage will receive your new bookmark automatically.

Sincerely Yours,

Yonghui wsunwsun@sina.com

Friday, April 20, 2007

Terence Tao" What Is Good Mathematics?" 英语原文

Arxiv上提供pdf,ps格式.
Tao的网页上提供dvi格式
http://www.math.ucla.edu/~tao/preprints/Expository/goodmath.dvi.

Timothy Gowers的数学讨论页

只要点击标题,即可进入他的网页,目录如下。

Gowers: Mathematical discussions contents page.

These pages are of various kinds, but they are nearly all attempts to show how mathematical ideas arise naturally, in the hope that some people will find them a useful supplement to university mathematics courses. Often they contain ideas that I have come across in one way or another and wish I had been told as an undergraduate. (Probably I was told several of them, and just wasn't concentrating enough to take them in at the time.)

Algebra

Why study finite-dimensional vector spaces in the abstract when they are all isomorphic to Rn?

The relationship between theory and computation in linear algebra.

Why is multiplication commutative?

How to invent some basic ideas of Galois theory.

1. Algebraic numbers and field extensions.
2. A solution to the cubic.

Lose your fear of tensor products
Analysis

A dialogue concerning the existence of the square root of two.

The meaning of continuity.

What is wrong with thinking of real numbers as infinite decimals?

A dialogue concerning the need for the real number system.

How to solve basic analysis exercises without thinking. (Gradually being revised and expanded.)

Proving that continuous functions on the closed interval [0,1] are bounded.

Finding the basic idea of a proof of the fundamental theorem of algebra.

What is the point of the mean value theorem?

A tiny remark about the Cauchy-Schwarz inequality.
Geometry

What makes non-Euclidean geometry interesting?

What makes it hard?

What makes hyperbolic geometry particularly hard?

What can I do to make it seem less hard?

How can the arc of a circle be considered straight?

Why did people want to prove the parallel postulate?

What is the historical importance of non-Euclidean geometry?

What is geometry?

Why isn't it just obvious that a regular dodecahedron exists?
Logic, set theory etc.

What is naive about naive set theory?

Paradoxes concerning definability.

A beginner's guide to countable ordinals.

Eight metamathematical statements with proofs that can be understood by non-logicians.
Number theory

How to discover a proof of the fundamental theorem of arithmetic.

How to discover the statement and two proofs of Fermat's little theorem.

How to think of the Riemann zeta function and discover the product formula.
Probability

New: Is Cambridge biased against state-school applicants?
Topology

Watch this space.
Miscellaneous

Just-do-it proofs.

The definition of `definition'.

Is the phrase `well-defined' well-defined?

What is `solved' when one solves an equation?

The implication of implication.

Does mathematics need a philosophy? (This is a talk I gave to the new Cambridge University Society for the Philosophy of Mathematics and Mathematical Sciences.)

Doron Zeilberger's attitude to computer mathematics.

什么是好数学?- 上 -

什么是好数学?

- 上 -

- 模糊翻译作品 -

- 作者:Terence Tao 译者:卢昌海 -

译者序: 本文译自澳大利亚数学家 Terence Tao 的近作 “What is Good Mathematics?”。 Tao 是调和分析、 微分方程、 组合数学、 解析数论等领域的大师级的年轻高手。 2006 年, 31 岁的 Tao 获得了数学界的最高奖 Fields 奖, 成为该奖项七十年来最年轻的获奖者之一。 美国数学学会 (AMS) 对 Tao 的评价是: “他将精纯的技巧、 超凡入圣的独创及令人惊讶的自然观点融为一体”。 著名数学家 Charles Fefferman (1978 年的 Fields 奖得主) 的评价则是: “如果你有解决不了的问题, 那么找到出路的办法之一就是引起 Terence Tao 的兴趣”。 Tao 虽然已经具有了世界性的声誉, 但由于他的年轻, 多数人 (尤其是数学界以外的人) 对他的了解仍很有限。 Tao 的这篇短文在一定程度上阐述了他的数学观, 类似于英国数学家 Godfrey Hardy 的名著《A Mathematician's Apology》, 相信会让许多读者感兴趣 (如果哪位读者想接受 Fefferman 的忠告, 让自己的问题有朝一日引起 Tao 的兴趣, 那么读一读这篇文章可能会有所助益:-)。 不过 Tao 的这篇文章远比《A Mathematician's Apology》难读得多。 从表面上看, 它不带任何数学公式, 这点甚至比《A Mathematician's Apology》做得更为彻底 (后者还带有一些 12+12=2 之类的数学公式), 但实际上, 文章的主要部分 - 即第二节 (对应于译文 中篇 的全部及 下篇 的大部分) - 所涉及的数学概念相当密集, 足以给非数学专业的读者造成很大的困难, 因此译文对译者知识所及且能用简短方式加以说明的部分概念进行了注释。 本译文略去了原文的摘要、 文献及正文中单纯与文献有关的个别文句 (即诸如 “感兴趣的读者请参阅某某文献” 之类的文句)。

1. 数学品质的诸多方面

我们都认为数学家应该努力创造好数学。 但 “好数学” 该如何定义? 甚至是否该斗胆试图加以定义呢? 让我们先考虑前一个问题。 我们几乎立刻能够意识到有许多不同种类的数学都可以被称为是 “好” 的。 比方说, “好数学” 可以指 (不分先后顺序):

1. 好的数学题解 (比如在一个重要数学问题上的重大突破);
2. 好的数学技巧 (比如对现有方法的精湛运用, 或发展新的工具);
3. 好的数学理论 (比如系统性地统一或推广一系列现有结果的概念框架或符号选择);
4. 好的数学洞察 (比如一个重要的概念简化, 或对一个统一的原理、 启示、 类比或主题的实现);
5. 好的数学发现 (比如对一个出人意料、 引人入胜的新的数学现象、 关联或反例的揭示);
6. 好的数学应用 (比如应用于物理、 工程、 计算机科学、 统计等领域的重要问题, 或将一个数学领域的结果应用于另一个数学领域);
7. 好的数学展示 (比如对新近数学课题的详尽而广博的概览, 或一个清晰而动机合理的论证);
8. 好的数学教学 (比如能让他人更有效地学习及研究数学的讲义或写作风格, 或对数学教育的贡献);
9. 好的数学远见 (比如富有成效的长远计划或猜想);
10. 好的数学品味 (比如自身有趣且对重要课题、 主题或问题有影响的研究目标);
11. 好的数学公关 (比如向非数学家或另一个领域的数学家有效地展示数学成就);
12. 好的元数学 (比如数学基础、 哲学、 历史、 学识或实践方面的进展); [译者注: 此处 “元数学” 译自 “meta-mathematics”, 不过这里所举的有些内容, 如历史、 实践等, 通常并不属于元数学的范畴。]
13. 严密的数学 (所有细节都正确、 细致而完整地给出);
14. 美丽的数学 (比如 Ramanujan 的令人惊奇的恒等式; 陈述简单漂亮, 证明却很困难的结果);
15. 优美的数学 (比如 Paul Erdős 的 “来自天书的证明” 观念; 通过最少的努力得到困难的结果); [译者注: “来自天书的证明” 译自 “proofs from the Book”。 Paul Erdős 喜欢将最优美的数学证明说成是来自 “The Book” (我将之译为 “天书”), 他有这样一句名言: 你不一定要相信上帝, 但应该相信 “The Book”。 Erdős 去世后的第三年, 即 1998 年, Martin Aigner 和 Günter M. Ziegler 以《来自天书的证明》为书名出版了一本书, 收录了几十个优美的数学证明, 以纪念 Erdős。]
16. 创造性的数学 (比如本质上新颖的原创技巧、 观点或各类结果);
17. 有用的数学 (比如会在某个领域的未来工作中被反复用到的引理或方法);
18. 强有力的数学 (比如与一个已知反例相匹配的敏锐的结果, 或从一个看起来很弱的假设推出一个强得出乎意料的结论);
19. 深刻的数学 (比如一个明显非平凡的结果, 比如理解一个无法用更初等的方法接近的微妙现象);
20. 直观的数学 (比如一个自然的、 容易形象化的论证);
21. 明确的数学 (比如对某一类型的所有客体的分类; 对一个数学课题的结论);
22. 其它[注一]。

如上所述, 数学品质这一概念是一个高维的 (high-dimensional) 概念, 并且不存在显而易见的标准排序[注二]。 我相信这是由于数学本身就是复杂和高维的, 并且会以一种自我调整及难以预料的方式而演化; 上述每种品质都代表了我们作为一个群体增进对数学的理解及运用的不同方式。 至于上述品质的相对重要性或权重, 看来并无普遍的共识。 这部分地是由于技术上的考虑: 一个特定时期的某个数学领域的发展也许更易于接纳一种特殊的方法; 部分地也是由于文化上的考虑: 任何一个特定的数学领域或学派都倾向于吸引具有相似思维、 喜爱相似方法的数学家。 它同时也反映了数学能力的多样性: 不同的数学家往往擅长不同的风格, 因而适应不同类型的数学挑战。

我相信 “好数学” 的这种多样性和差异性对于整个数学来说是非常健康的, 因为它允许我们在追求更多的数学进展及更好的理解数学这一共同目标上采取许多不同的方法, 并开发许多不同的数学天赋。 虽然上述每种品质都被普遍接受为是数学所需要的品质, 但牺牲其它所有品质为代价来单独追求其中一两种却有可能变成对一个领域的危害。 考虑下列假想的 (有点夸张的) 情形:

* 一个领域变得越来越华丽怪异, 在其中各种单独的结果为推广而推广, 为精致而精致, 而整个领域却在毫无明确目标和前进感地随意漂流。
* 一个领域变得被令人惊骇的猜想所充斥, 却毫无希望在其中任何一个猜想上取得严格进展。
* 一个领域变得主要通过特殊方法来解决一群互不关联的问题, 却没有统一的主题、 联系或目的。
* 一个领域变得过于枯燥和理论化, 不断用技术上越来越形式化的框架来重铸和统一以前的结果, 后果却是不产生任何令人激动的新突破。
* 一个领域崇尚经典结果, 不断给出这些结果的更短、 更简单及更优美的证明, 但却不产生任何经典著作以外的真正原创的新结果。

在上述每种情形下, 有关领域会在短期内出现大量的工作和进展, 但从长远看却有边缘化和无法吸引更年轻的数学家的危险。 幸运的是, 当一个领域不断接受挑战, 并因其与其它数学领域 (或相关学科) 的关联而获得新生, 或受到并尊重多种 “好数学” 的文化熏陶时, 它不太可能会以这种方式而衰落。 这些自我纠错机制有助于使数学保持平衡、 统一、 多产和活跃。

现在让我们转而考虑前面提出的另一个问题, 即我们到底该不该试图对 “好数学” 下定义。 下定义有让我们变得傲慢自大的危险, 特别是, 我们有可能因为一个真正数学进展的奇异个例不满足主流定义[注三]而忽视它。 另一方面, 相反的观点 - 即在任何数学研究领域中所有方法都同样适用并该得到同样资源[注四], 或所有数学贡献都同样重要 - 也是有风险的。 那样的观点就其理想主义而言也许是令人钦佩的, 但它侵蚀了数学的方向感和目的感, 并且还可能导致数学资源的不合理分配[注五]。 真实的情形处于两者之间, 对于每个数学领域, 现存的结果、 传统、 直觉和经验 (或它们的缺失) 预示着哪种方法可能会富有成效, 从而应当得到大多数的资源; 那种方法更具试探性, 从而或许只要少数有独立头脑的数学家去进行探究以避免遗漏。 比方说, 在已经发展成熟的领域, 比较合理的做法也许是追求系统方案, 以严格的方式发展普遍理论, 稳妥地延用卓有成效的方法及业已确立的直觉; 而在较新的、 不太稳定的领域, 更应该强调的也许是提出和解决猜想, 尝试不同的方法, 以及在一定程度上依赖不严格的启示和类比。 因此, 从策略上讲比较合理的做法是, 在每个领域内就数学进展中什么品质最应该受到鼓励做一个起码是部分的 (但与时俱进的) 调查, 以便在该领域的每个发展阶段都能最有效地发展和推进该领域。 比方说, 某个领域也许急需解决一些紧迫的问题; 另一个领域也许在翘首以待一个可以理顺大量已有成果的理论框架, 或一个宏大的方案或一系列猜想来激发新的结果; 其它领域则也许会从对关键定理的新的、 更简单及更概念化的证明中获益匪浅; 而更多的领域也许需要更大的公开性, 以及关于其课题的透彻介绍, 以吸引更多的兴趣和参与。 因此, 对什么是好数学的确定会并且也应当高度依赖一个领域自身的状况。 这种确定还应当不断地更新与争论, 无论是在领域内还是从通过旁观者。 如前所述, 有关一个领域应当如何发展的调查, 若不及时检验和更正, 很有可能会导致该领域内的不平衡。

上面的讨论似乎表明评价数学品质虽然重要, 却是一件复杂得毫无希望的事情, 特别是由于许多好的数学成就在上述某些品质上或许得分很高, 在其它品质上却不然; 同时, 这些品质中有许多是主观而难以精确度量的 (除非是事后诸葛)。 然而, 一个令人瞩目的现象是[注六]: 上述一种意义上的好数学往往倾向于引致许多其它意义上的好数学, 由此产生了一个试探性的猜测, 即有关高品质数学的普遍观念也许毕竟还是存在的, 上述所有特定衡量标准都代表了发现新数学的不同途径, 或一个数学故事发展过程中的不同阶段或方面。

>> 请阅读中篇 <<

二零零七年三月十日译于纽约
http://www.changhai.org/

原文注释

[注一] 上述列举无意以完备自居。 尤其是, 它主要着眼于研究性数学文献中的数学, 而非课堂、 教材或自然科学等接近数学的学科中的数学。

[注二] 特别值得指出的是数学严格性虽然非常重要, 却只是界定高品质数学的因素之一。

[注三] 一个相关的困难是, 除了数学严格性这一引人注目的例外, 上述品质大都有点主观, 因而含有某种不精确性与不确定性。 我们感谢 Gil Kalai 强调了这一点。

[注四] 稀缺资源的例子包括钱、 时间、 注意力、 才能及顶尖刊物的版面。

[注五] 这一问题的另一个解决方法是利用数学资源也是多维这一事实。 比如人们可以为展示、 创造性等等设立奖项, 或为不同类型的成果设立不同的杂志。 我感谢 Gil Kalai 对这一点的洞察。

[注六] 这一现象与 Wigner 所发现的 “数学的不合理有效性” (unreasonable effectiveness of mathematics) 有一定的关联。 [译者注: Wigner 的这一说法见于他 1960 年发表的文章 "The Unreasonable Effectiveness of Mathematics in the Natural Sciences"。]

Ten Lessons I wish I had been Taught

Ten Lessons I wish I had been Taught



Below is a wonderful talk by the mathematician Gian-Carlo Rota on Ten Lessons. You can find this elsewhere on the web, but his advice is so good that anyone in research or academia should read it.

Enjoy,
Kevin Knuth
Albany NY

Ten Lessons I wish I had been Taught
Gian-Carlo Rota

MIT, April 20 , 1996 on the occasion of the Rotafest

Allow me to begin by allaying one of your worries. I will not spend the next half hour thanking you for participating in this conference, or for your taking time away from work to travel to Cambridge.

And to allay another of your probable worries, let me add that you are not about to be subjected to a recollection of past events similar to the ones I’ve been publishing for some years, with a straight face and an occasional embellishment of reality.

Having discarded these two choices for this talk, I was left without a title. Luckily I remembered an MIT colloquium that took place in the late fifties; it was one of the first I attended at MIT. The speaker was Eugenio Calabi. Sitting in the front row of the audience were Norbert Wiener, asleep as usual until the time came to applaud, and Dirk Struik who had been one of Calabi’s teachers when Calabi was an undergraduate at MIT in the forties. The subject of the lecture was beyond my competence. After the first five minutes I was completely lost. At the end of the lecture, an arcane dialogue took place between the speaker and some members of the audience, Ambrose and Singer if I remember correctly. There followed a period of tense silence. Professor Struik broke the ice. He raised his hand and said: “Give us something to take home!” Calabi obliged, and in the next five minutes he explained in beautiful simple terms the gist of his lecture. Everybody filed out with a feeling of satisfaction.

Dirk Struik was right: a speaker should try to give his audience something they can take home. But what? I have been collecting some random bits of advice that I keep repeating to myself, do’s and don’ts of which I have been and will always be guilty. Some of you have been exposed to one or more of these tidbits. Collecting these items and presenting them in one speech may be one of the less obnoxious among options of equal presumptuousness. The advice we give others is the advice that we ourselves need. Since it is too late for me to learn these lessons, I will discharge my unfulfilled duty by dishing them out to you. They will be stated in order of increasing controversiality.
1 Lecturing

The following four requirements of a good lecture do not seem to be altogether obvious, judging from the mathematics lectures I have been listening to for the past forty-six years.

a. Every lecture should make only one main point The German philosopher G. W. F. Hegel wrote that any philosopher who uses the word “and” too often cannot be a good philosopher. I think he was right, at least insofar as lecturing goes. Every lecture should state one main point and repeat it over and over, like a theme with variations. An audience is like a herd of cows, moving slowly in the direction they are being driven towards. If we make one point, we have a good chance that the audience will take the right direction; if we make several points, then the cows will scatter all over the field. The audience will lose interest and everyone will go back to the thoughts they interrupted in order to come to our lecture.

b. Never run overtime Running overtime is the one unforgivable error a lecturer can make. After fifty minutes (one microcentury as von Neumann used to say) everybody’s attention will turn elsewhere even if we are trying to prove the Riemann hypothesis. One minute overtime can destroy the best of lectures.

c. Relate to your audience As you enter the lecture hall, try to spot someone in the audience with whose work you have some familiarity. Quickly rearrange your presentation so as to manage to mention some of that person’s work. In this way, you will guarantee that at least one person will follow with rapt attention, and you will make a friend to boot.

Everyone in the audience has come to listen to your lecture with the secret hope of hearing their work mentioned.

d. Give them something to take home It is not easy to follow Professor Struik’s advice. It is easier to state what features of a lecture the audience will always remember, and the answer is not pretty. I often meet, in airports, in the street and occasionally in embarrassing situations, MIT alumni who have taken one or more courses from me. Most of the time they admit that they have forgotten the subject of the course, and all the mathematics I thought I had taught them. However, they will gladly recall some joke, some anecdote, some quirk, some side remark, or some mistake I made.
2 Blackboard Technique

Two points.

a. Make sure the blackboard is spotless It is particularly important to erase those distracting whirls that are left when we run the eraser over the blackboard in a non uniform fashion.

By starting with a spotless blackboard, you will subtly convey the impression that the lecture they are about to hear is equally spotless.

b. Start writing on the top left hand corner What we write on the blackboard should correspond to what we want an attentive listener to take down in his notebook. It is preferable to write slowly and in a large handwriting, with no abbreviations. Those members of the audience who are taking notes are doing us a favor, and it is up to us to help them with their copying. When slides are used instead of the blackboard, the speaker should spend some time explaining each slide, preferably by adding sentences that are inessential, repetitive or superfluous, so as to allow any member of the audience time to copy our slide. We all fall prey to the illusion that a listener will find the time to read the copy of the slides we hand them after the lecture. This is wishful thinking.

3 Publish the same result several times

After getting my degree, I worked for a few years in functional analysis. I bought a copy of Frederick Riesz’ Collected Papers as soon as the big thick heavy oversize volume was published. However, as I began to leaf through, I could not help but notice that the pages were extra thick, almost like cardboard. Strangely, each of Riesz’ publications had been reset in exceptionally large type. I was fond of Riesz’ papers, which were invariably beautifully written and gave the reader a feeling of definitiveness.As I looked through his Collected Papers however, another picture emerged. The editors had gone out of their way to publish every little scrap Riesz had ever published. It was clear that Riesz’ publications were few. What is more surprising is that the papers had been published several times. Riesz would publish the first rough version of an idea in some obscure Hungarian journal. A few years later, he would send a series of notes to the French Academy’s Comptes Rendus in which the same material was further elaborated. A few more years would pass, and he would publish the definitive paper, either in French or in English. Adam Koranyi, who took courses with Frederick Riesz, told me that Riesz would lecture on the same subject year after year, while meditating on the definitive version to be written. No wonder the final version was perfect.

Riesz’ example is worth following. The mathematical community is split into small groups, each one with its own customs, notation and terminology. It may soon be indispensable to present the same result in several versions, each one accessible to a specific group; the price one might have to pay otherwise is to have our work rediscovered by someone who uses a different language and notation, and who will rightly claim it as his own.

4 You are more likely to be remembered by your expository work

Let us look at two examples, beginning with Hilbert. When we think of Hilbert, we think of a few of his great theorems, like his basis theorem. But Hilbert’s name is more often remembered for his work in number theory, his Zahlbericht, his book Foundations of Geometry and for his text on integral equations. The term “Hilbertspace” was introduced by Stone and von Neumann in recognition of Hilbert’s textbook on integral equations, in which the word “spectrum” was first defined at least twenty years before the discovery of quantum mechanics. Hilbert’s textbook on integral equations is in large part expository, leaning on the work of Hellinger and several other mathematicians whose names are now forgotten.Similarly, Hilbert’s Foundations of Geometry, the book that made Hilbert’s name a household word among mathematicians, contains little original work, and reaps the harvest of the work of several geometers, such as Kohn, Schur (not the Schur you have heard of), Wiener (another Wiener), Pasch, Pieri and several other Italians.

Again, Hilbert’s Zahlbericht, a fundamental contribution that revolutionized the field of number theory, was originally a survey that Hilbert was commissioned to write for publication in the Bulletin ofthe German Mathematical Society.

William Feller is another example. Feller is remembered as the author of the most successful treatise on probability ever written. Few probabilists of our day are able to cite more than a couple of Feller’s research papers; most mathematicians are not even aware that Feller had a previous life in convex geometry.

Allow me to digress with a personal reminiscence. I sometimes publish in a branch of philosophy called phenomenology. After publishing my first paper in this subject, I felt deeply hurt when, at a meeting of the Society for Phenomenology and Existential Philosophy, I was rudely told in no uncertain terms that everything I wrote in my paper was well known. This scenario occurred more than once, and I was eventually forced to reconsider my publishing standards in phenomenology.

It so happens that the fundamental treatises of phenomenology are written in thick, heavy philosophical German. Tradition demands that no examples ever be given of what one is talking about. One day I decided, not without serious misgivings, to publish a paper that was essentially an updating of some paragraphs from a book by Edmund Husserl, with a few examples added. While I was waiting for the worst at the next meeting of the Society for Phenomenology and Existential Philosophy, a prominent phenomenologist rushed towards me with a smile on his face. He was full of praise for my paper, and he strongly encouraged me to further develop the novel and original ideas presented in it.

5 Every mathematician has only a few tricks

A long time ago an older and well known number theorist made some disparaging remarks about Paul Erdos’ work. You admire contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdos’ work could be reduced to a few tricks which Erdos repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert’s beautiful results have been completely forgotten. But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory, it was surprising to verify that Hilbert’s proofs relied on the same few tricks. Even Hilbert had only a few tricks!
6 Do not worry about your mistakes

Once more let me begin with Hilbert. When the Germans were planning to publish Hilbert’s collected papers and to present him with a set on the occasion of one of his later birthdays, they realized that they could not publish the papers in their original versions because they were full of errors, some of them quite serious. Thereupon they hired a young unemployed mathematician, Olga Taussky-Todd, to go over Hilbert’s papers and correct all mistakes. Olga labored for three years; it turned out that all mistake scould be corrected without any major changes in the statement of the theorems. There was one exception, a paper Hilbert wrote in his old age, which could not be fixed; it was a purported proof of the continuum hypothesis, you will find it in a volume of the Mathematische Annalen of the early thirties. At last, on Hilbert’s birthday, a freshly printed set of Hilbert’s collected papers was presented to the Geheimrat. Hilbert leafed through them carefully and did not notice anything.Now let us shift to the other end of the spectrum, and allow me to relate another personal anecdote. In the summer of 1979, while attending a philosophy meeting in Pittsburgh, I was struck with a case of detached retinas. Thanks to Joni’s prompt intervention, I managed to be operated on in the nick of time and my eyesight was saved.

On the morning after the operation, while I was lying on a hospital bed with my eyes bandaged, Joni dropped in to visit. Since I was to remain in that Pittsburgh hospital for at least a week, we decided to write a paper. Joni fished a manuscript out of my suitcase, and I mentioned to her that the text had a few mistakes which she could help me fix.

There followed twenty minutes of silence while she went through the draft. “Why, it is all wrong!” she finally remarked in her youthful voice. She was right. Every statement in the manuscript had something wrong. Nevertheless, after laboring for a while, she managed to correct every mistake, and the paper was eventually published.

There are two kinds of mistakes. There are fatal mistakes that destroy a theory; but there are also contingent ones, which are useful in testing the stability of a theory.

7 Use the Feynman method

Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say: “How did he do it? He must be a genius!”
8 Give lavish acknowledgments

I have always felt miffed after reading a paper in which I felt I was not being given proper credit, and it is safe to conjecture that the same happens to everyone else. One day, I tried an experiment. After writing a rather long paper, I began to draft a thorough bibliography. On the spur of the moment, I decided to cite a few papers which had nothing whatsoever to do with the content of my paper, to see what might happen.Somewhat to my surprise, I received letters from two of the authors whose papers I believed were irrelevant to my article. Both letters were written in an emotionally charged tone. Each of the authors warmly congratulated me for being the first to acknowledge their contribution to the field.

9 Write informative introductions

Nowadays, reading a mathematics paper from top to bottom is a rare event. If we wish our paper to be read, we had better provide our prospective readers with strong motivation to do so. A lengthy introduction, summarizing the history of the subject, giving everybody his due, and perhaps enticingly outlining the content of the paper in a discursive manner, will go some of the way towards getting us a couple of readers.As the editor of the journal Advances in Mathematics, I have often sent submitted papers back to the authors with the recommendation that they lengthen their introduction. On occasion I received by return mail a message from the author, stating that the same paper had been previously rejected by Annals of Mathematics because the introduction was already too long.

10 Be prepared for old age

My late friend Stan Ulam used to remark that his life was sharply divided into two halves. In the first half, he was always the youngest person in the group; in the second half, he was always the oldest. There was no transitional period.I now realize how right he was. The etiquette of old age does not seem to have been written up, and we have to learn it the hard way. It depends on a basic realization, which takes time to adjust to. You must realize that, after reaching a certain age, you are no longer viewed as a person. You become an institution, and you are treated the way institutions are treated. You are expected to behave like a piece of period furniture, an architectural landmark, or an incunabulum.

It matters little whether you keep publishing or not. If your papers are no good, they will say, “What did you expect? He is a fixture!” and if an occasional paper of yours is found to be interesting, they will say, “What did you expect? He has been working at this all his life!” The only sensible response is to enjoy playing your newly-found role as an institution.