Title：How to do mathematics?（A personal viewpoint）

Speaker：Prof. Beson Farb（Chicago University）

Time：1 July, 2013  19:00-20:30

Place：110 Hall of Morningside Center of Mathematics

Lizhen Ji：Welcome to this exceptional talk—— The special talk of the international conference "Surveys of Modern Mathematics ".I guess that we all love mathematics, that's why you are here. I am sure you all want to be successful. So, this depends, I think you have to choose a direction. In the life, It will take a lifetime for you to share with someone, so I suppose you will choose very carefully. Same thing in mathematics, you have to choose a good direction. But once you have chosen a direction, you still have to do mathematics, the question is "how to do mathematics"? And professor Farb has thought about such problem for a long time, I think he is well qualified to make a comment. As you have seen in the poster, he is one of the most popular professors in Chicago, and he has civilized 29 PhD students so far. Another thing is that he was one of the special and outstanding students of Thurston. If you have been to his lecture, probably I think that is in the trace of Thurston's style. This evening, he will tell you how to do mathematics, and he will also comment what is good mathematics, and what is bad mathematics and what you should not know and what you need not know. OK, let's welcome!

 

季理真：欢迎来到“现代数学概观”这一国际会议的特别讲座《如何从事数学研究》。我相信你们都是热爱数学的，这也是为什么你们来这里的原因。我也相信你们都想成功。首先，你要选择一个方向。在生活中，你需要找很可能共度一生的伴侣，我相信大家都会很慎重地选择。在数学中也是这样，你需要找一个好方向。但是，即使选择了方向，你还是要做数学呀，那么，问题在于“怎样做数学”？Farb教授思考这样的问题已经很长时间了，他有足够的资格给出评论。就像你在海报上看到的，他是芝加哥最受欢迎的教授之一，他已经培养了29个博士生，此外，他还是Thurston最特别和最出色的的学生之一。在他的讲座中，也许你可以感受到Thurston的风格。今晚，他将告诉我们，怎样研究数学。他还会讲讲什么是好的数学，什么是坏的数学。研究中你需要知道什么，而不必知道什么。让我们欢迎！

Farb：Thank you! Thank you all for coming! Lizhen, somehow, convinced me to do this 48 hours ago, I agreed to do this, and I made these slides in 48 hours. Lizhen is very persuasive, encouraged me to do this for all students. A quick comment, of Lizhen’s comment, spouse is husband and wife they can come and go but mathematics is always here, choosing mathematics direction might be more important but don’t tell my wife I said that. Anyway, It is scary because it’s hard to not to look foolish to say opinions so I think the title may be changed to "some advice made by a specific biased person", that is me, and that is basically I'd like to offer.

 

Farb：非常感谢，感谢大家的到来。理真在48小时前说服我要做这个报告，所以这些幻灯片是我在48小时内做出的。理真鼓励我能够在这里面对所有的学生。对理真的话先做个简短的评注：配偶是丈夫或妻子，他们可以来，也可以离开，但是数学却总是在这里的。不论如何，表述观点的时候很难表现地不傻，所以，我想，标题最好改为“一个带有偏见的人的一些建议”，这个人当然是我，这才是我主要想讲述的。

I am going to say things, but of course, the first thing which is self-evident

Claim 1 :Each mathematician must find her path

But, having said that, it can be useful to hear about the path of others, and in fact, one thing I think we forget to do as students is to actually remember to think about what we are doing, to think about the process of mathematics, to think about the issues that we going to talk about tonight. Math is hard enough, and so we spend all the time with math which is great? But one should spend a little time think about what one is doing, and what direction you want to go. These are enormously important things that stay with you for a life, think about the process, so just thinking about it. Itself is a really useful thing to do.

 

Farb：现在我要开始讲了，不过，首先，第一点是很明显的

断言1：每个数学家要做的第一件事就是找一条她自己的道路去走。

不过，听别人走的路也可以是有用的，并且事实上，我们在学生阶段总是忘记了自己到底在做什么，忘了去想数学本身的过程，忘了去想今晚我们会谈到的一些话题：数学很难，花大量时间在数学上好不好？一个人，应该花一点时间想想自己在做什么，自己要选择什么样的方向，这对于你一生都很重要，想想这个过程吧，去想一想。这是很有用的。

 

The first thing I want to talk about is

On being a Ph.D student

In mathematics, I’ll briefly mention something about research mathematics, not mathematics, but of course they are not so different. I only have 48 hours, so it is not perfectly organized. So let me just start, the first thing is

A Take advice from top people

You can find written advice on how to be a good graduate student. At the webpage of ...such as Ravil Vakil and Jordan Ellenberg, and each of these people is top mathematician, and they offer fantastic advice. One particular what I like is Ravil Vakil, which offers very good advice on how to attend a talk, since for most of us, you know, you listen for 5 minutes, and then you feel you don't like it, you sort of just sitting there, you've wasted 55 minutes of your life. But you don't have to, he discusses this, it is really great——there is a game, try to find three things you'll learn from every talk! And afterwards, you and your friends…Anyway, it’s great advice, You can read these by yourself, and I encourage you to read what these people have to say, and they are eloquent.

 

 

首先我要讲主题是的是

作为一个博士生

在数学中，我要简单提一下数学研究，不是数学。当然，二者区别并不大。我只有48小时时间（去准备），所以可能不是很有条理。好吧，让我开始吧，第一件事要说的是

A 听取顶级人士的建议

有很多已经被写成文的关于怎样成为好的研究生的建议。在一些顶级数学家，比如Ravil Vakil和Jordan Ellenberg的主页上，你可以找到很好的建议。我特别喜欢Ravil Vakil给出的如何参加报告的建议：大多数人，听报告听了五分钟后，发现自己不感兴趣，然后就干坐在那里，于是就浪费了55分钟。Vakil告诉你，其实你不必这样，有一个“游戏”：在每个报告中都尝试找出三个你能学到的东西。然后，你和你的朋友… 总之，这是很好的建议，这些材料你们可以自己去阅读，我非常鼓励你们去看看他们是怎么说的，他们都很能言善辩。

 

Ok, the second thing is

B How to choose an advisor

This is enormously important because it determines you entire graduate career which could be four, five or six years, and it probably determines next ten years, and perhaps next fifty. So one should actually think about it. And what I believe is that the most important thing is

1. Choose a topic that compels you（that you can't live without）

Probably you do not exactly know what the professors are actually doing, you sort of know that this guy does number theory, this professor she does PDE. Try to figure out what they do, try to read the introduction of their papers, this is very difficult, because you make a decision based on incomplete information, but it is a huge decision, and my solution to that is just to say tough luck, that's the way it is, there is no other method. But primarily you will be spending so much time with that mathematics. You better, it had better move you emotionally. For example, I went to graduate school to work with Wu-Chung Hsiang, a good mathematician, do K-theory, and I completed switched. I began to work with Thurston, because I was thinking all the time "what is my philosophy of mathematics" and there is a lot personal reasons for why I decided to do what Thurston is doing. But for example, I know it sounds silly, but I love the symbol "Γ" being a discrete group for a Lie group G, when I saw the Γ, the word of the discrete group, something I like it. You might think that be silly, well, I have written "Γ" for about three hundred thousand times since then, and I am excited to learn more about "Γ”. And so, I am sort of being silly, but I am actually being serious, it should be a visceral, or in other words, this is purely emotional reaction. I always said to my students "let's find something that you'll keep long enough', at the end of your Ph.D, if you have not stayed up all night——you couldn't sleep because you need to know the answer, then you wouldn't done the work. No one is smart enough except J.P Serre, no one is smart enough to do great math without staying up the whole night because you need to know, it is hard to identify that, but you have to try.

 

好的，我要说的第二点是

B 如何选导师

这是极其重要的，因为它讲决定你那持续四年，五年乃至六年的博士生涯，可能还会继续影响十年，乃至五十年。所以，你得仔细想想这件事。其中，我认为最重要的是

1. 选择一个真正能激励你干活的主题（使得你离不开它们）

也许你并不完全知道你身边的教授在做什么，你可能仅仅知道，这个教授做数论，那个教授做偏微（这是不够的），你要尝试了解他们的工作，读读他们文章的介绍部分。选择并不容易，因为你要基于并不全面的信息做出决定，而且是很大的决定。对此我的能说的就是：那很不幸，，就是这样，没有其他的办法。但是从根本上讲，你将会花大量时间在这种数学上面。举例来说，我开始读研究生的时候，我跟的是Wu-Chung Hsiang学K理论，但后来我完全转换了方向，因为我一直在思考“我的数学哲学是什么”，然后很多个人原因导致我去做Thurston做的东西。举例来说，虽然这个例子可能有点傻，我爱作为李群G的离散子群的符号”Γ”，我喜欢这种作为离散群的符号。你可能会觉得傻，但是，你要知道，从那时到现在，我写下Γ已经不下三十万次了，对于了解更多Γ的信息，我充满了热情。我是有些傻，但是我是认真的，它应该是发自内心的，换句话说，这纯粹是感情的作用。我总是跟我的学生说“一定要找一个能在你心中停留足够长时间的东西”，那么你将会每晚都会被这个东西包围，你睡不着的，因为你需要知道答案。如果你不这样做，你博士毕业的时候一定是失败的。比起 J.P Serre，我想没有人是足够聪明的，也没有人聪明到不需要整个晚上熬夜（因为你需要知道答案）就能做出伟大的数学工作。总之，这个问题很难一概而论，但是，你一定要尝试。

I think second

2. Work on an area which your advisor is an expert in

This is absolutely crucial. Some people are interested, let's say, this comes up a lot in fact people read Mathoverflow and came up with Mathoverflow and my friends Matthew Emerton had very fantastic response. You should look for that response. This came up with my nephew, who is going to mathematic graduate school, If you are interested in some specific area in Number, says, especially interested in the study of quadratic forms, well, whatever university you are at, you can't just work on quadratic forms, you need a professor to guide you, the professor at your university you will need him. The reason is if this area in mathematics is active and has excellent people, it moves quickly and the community knows things that are not available to the public, your advisor should be part of the community. There are a lot of things to choosing an advisor, I will not get into all of them right now. You should try to choose someone who is active, you can determine this by looking at what papers they have written, there are a lot of things, but today I am just going to concentrate on, The advisor needs be an expert because people who working on quadratic forms or Taniyama-Shimura or some other related things, that is another area, and differential topology, they all know what is going on. If you are some lone person in your university, and you try to follow, you say oh I’ll email the experts, I will read the Arvix, I will read books, I am a hard worker, that is not good enough. You are dead. You come up with something and it will already be done, you’ll be working on something that everybody knows. You know, Mark Kisin is working on this, but you are still just reading Hartshorne. So that is really important. This would constrain what you are really interested in. And hopefully, you can find one that both satisfies both one and two. If you don’t, the you need to leave your university. But for most of us, you can’t, these two are not incompatible, you can find both.

 

第二

2. 你要选择你导师是其专家的领域

这个相当关键。这个问题很多人都很感兴趣，事实上，每天在mathoverflow上都有人讨论，对此我的朋友Matthew Emerton给出了很好的回答。你应该去看看这些回答。这是我侄子想出来的，他马上要去数学研究生院了。如果你对某个方向感兴趣，比如说数论中的二次型吧，那么不管你在哪个大学，你不能仅仅学二次型，你需要找个导师带你，你需要你们学校的教授。原因是，如果一个数学分支很活跃并且有好的数学家的话，它会前进的很快，这个团队知道很多大家不知道的东西。你的导师应该在这个团队中。选导师要注意很多，我不打算现在全部展开。你要找一个（学术上）活跃的人，这一点你可以从他的文章和其他很多东西看出来，但是今天我讲集中将：导师需要是这个领域的专家，因为不管他们做二次型还是谷山志村，或者微分拓扑，他们知道研究中正在发生什么。如果你在你的大学中独身一人，想去跟着（这个领域），你说我会给专家写邮件，你会读arvix，你会读书，你很努力，这些都不够。你可能会悲剧，因为你可能在做别人已经做过的工作或在研究大家都熟悉的东西。比如，Mark Kisin做他的东西的时候，你（也是这个方向）仅仅还去读Hartshorne...所以，这个确实是很重要的。这些都会把你的兴趣范围缩小。但愿你能找到与上述两点都符合的的教授，如果没有，那么恐怕你得离开你的学校。但是对于大多数人，你不能，这两点是相容的，你可以找到。

If you have a question, please feel free to stop me at any time.

如果你什么问题，请随时打断。

 

Then

3. You do have to trust you advisor

Unless you are one in the million, which you probably won’t know even if you are. You need to trust your advisor, you cannot go on your own path, if your advisor tells you to learn some specific things, I would say, you are allowed to question why, but if the advisor sticks to it, if you disagree, voice your opinion, but if your advisor finally says you really need to do it, then you really need to do it. You can't have a relationship otherwise. I can see this from a advisor's view point, the advisor give up on you immediately, we will give up immediately, there is nothing I can do for you if you don't listen to me, I won’t waste my time. If you come up with a logical argument, you can debate, I will listen, but in the end of day, I am the boss, you know what can I say, you do need to have that done. Sometimes you don't know why your problem is interesting, you are allowed to ask your advisor, and you should ask your advisor, but sometime the response might be, you have to have a broader viewpoint, you are still learning keep working, finally you will see how it connects later .Trust that.

 

再次

3.你必须信任你的导师

除非你是百万中挑一的那种人，你甚至意识不到你是否是那种人。否则，如果你的导师让你学某种特别的东西，那么我想说，你可以问为什么，说出你的见解，但是，如果最终导师坚持让你学，那么你必须去学。否则，你无法拥有这个（师生）关系。我可以从导师的角度来讲：导师会放弃你的，我们都会放弃你，如果你不去听我的话，那么我无法帮助你，我不想浪费我的时间。如果你有什么争议，那么你可以去理论，我也会听，但是，当一天结束的时候，你还是得完成你的任务，因为我是老板。有时，你不明白为什么这个问题很有趣，你有权问导师，也应该问导师，但有时回答将是…你要有更广阔的的视野，手上的活不能停，最终你会看到它们的联系。你要相信这一点

At the end, you are really choosing a parent.

4. Advisor=parent

It is the people you are stuck with your whole life. Just five minutes before I give this talk, literally between the time I open the door and sat here, I got an email from one of my past students who graduated 12 years ago, he is now a professor, you never sever that relationship, it is a huge, huge decision. You have to take it seriously.

 

最后一点

4.导师=父母

导师是和你相伴一生的人。就在讲座开始五分钟前，确切的讲，在我跨入这扇门和坐在这里之间，我收到我之前一个学生的邮件，他是十二年前毕业的，现在已经是教授了。这层师生关系会伴随你众生，所以，（选导师）是个非常重大的决定。你要慎重。

 

Here is my advisor. Who I had I a rocky relationship with, but that is a different story.

How <wbr>to <wbr>do <wbr>Mathematics <wbr>by <wbr>Benson <wbr>Farb

It is a long story but I switched my direction in mathematics, and I asked Thurston to be my advisor, he agreed, he really didn’t give problems to the students, he actually gave me a problem, after few weeks of struggling, someone came, he looked at my problem and said "you know that is a very famous open question", so I stopped doing this. He wasn't very good at giving problems, but anyway, I began to work on my own problem which is the one have something to do with hyperbolic groups, and I went to my advisor and said”I would like to do my own problem”, he said this is OK, that is a good choice. He squinted his eyes and looked in the air. Here is what he said, I mean, I wanted to understand the lattice of the Lie group in complex hyperbolic geometry, he said 'Oh, I see, it's like a froth of bubbles, and the bubbles have a bounded interaction", I was stupid, and I always wrote everything he has said, I wrote down "froth, bubbles"and then I went to the library after the meeting, Ok, I am ready to go. I start my problem, pencil,”froth, bubbles”, go I didn’t know what to do with bubbles, I didn’t know what math equation to write down. After three years of horrible pains and suffering ,on my part ,I solved my problem. And if you ask me for a five words of summary of my thesis, I would say "froth of bubbles bounded interaction", my thesis is basically that. He told me just enough so that at the end I would get out what he got.

 

这是我的导师，我和他的关系很不稳定，当然这是另一个故事了。当时，一言难尽，但我改变的当初的研究方向，我希望Thurston成为我的导师，他答应了。但是他是不怎么给学生问题的，当年他就给我了一个问题，但是，在很多周的挣扎（无进展）之后，他看了我的问题，告诉我“这其实是个很有名的开放问题。。。”，于是我就不再做他的问题了。他不是很擅长给学生问题，但是，不管怎么说，我开始做我自己的问题了，这个问题是关于双曲群的，我告诉他我在做这个问题，他说没问题，这是个不错的选择。他斜着眼，望着天空。当时我想弄懂双曲几何中李群的格，他说“哦，这就像一充满气泡的泡沫，气泡间有有界的相互作用”，我当时很单纯，我把他说的话都用笔记下来，我写着“泡沫，气泡。。。”见完面之后就去图书馆查文献。是的，我已经准备好了，我开始做我的问题，铅笔 “泡沫，气泡”。最初，我根本不知道怎么处理气泡，甚至连方程怎么列都不知道，但是，经过三年的痛苦和煎熬，我最终解决了这个问题。如果你要我给出我的论文的五个词作为概括，我可以说“泡沫，气泡，有界作用”，我的论文大体就是基于这些。他告诉我这足够了，我终于明白他说的了。

 

Now, I will talk about

C. How to work

I thought for this talk it would be useful to tell you about my opinion of these things, because I had a lot of Ph.D students. I really believe the battle in graduate school is the hardest time, hardest time for research for me, I don’t know if it was like this for you. Because I knew the least of mathematics, but I am supposed to prove the theorem, now you know more mathematics, it is easier to prove theorems. So being a graduate student is the hardest time. And usually, there are so many issues that come up that don't have to do with mathematics, but I believe you guys are smart, you are getting a Ph.D in mathematics, you can do it. Some of you will write better Ph.D than others, but I think the battle is emotional, you are going to get depression, you are going be get stuck, you are going to think why I am doing math, I want to help the world. Although I think math is helping the world. That is a different story. You have to get through all this in the battle, and the battle is like work, and the working is hard, no one is forcing you to work, you have a thesis, in United States, it’s definitely now you can do nothing for three years basically. It’s hard, to everyday work, and once you are working, you usually love it, but you want to run from math, because it is hard. I really think the entire battle is emotional battle, it’s just as important as the mathematical. I wish somebody had told me these things, it would have made my life a lot easier.

 

现在讲

C. 如何工作

我想，如果在报告中讲这些东西，那么将会很有用，因为我有很多博士生。我相信研究生时期的工作是科研中最艰难的阶段，至少对我是这样，我不知道你们是不是这样。因为我这时对数学了解的最少，但我还得证明定理，现在我懂得更多数学，证明容易多了。所以做研究生是最难的。一般来说，很多问题迎面而来让你不做数学。但是，我相信你们都是聪明的，否则不会读博士。诚然，有的博士写的论文比另外的博士优秀，但是，真正的挑战是心理上的。你可能会遇到失望，可能会遇到瓶颈，甚至你会想我干嘛要做数学，我要服务于这个世界。虽然数学也是服务于世界的，当然这是另一个故事。这些困难你都要克服，当然，工作是很难的，没有人强迫你去工作，事实上，在美国，理论上你从现在开始三年啥也不干也没人管。每天工作很难，但一般你开始工作后你会喜欢它，但是你想逃离数学，因为它很难。所以，我觉得困难主要是心理上的。当初我就希望有人告诉我一些这方面的东西，让我生活容易些。

 

So, there is no doubt about it you should

1. Live, breath and sleep mathematics

You shouldn’t do this if you are not willing to live, breath and sleep mathematics. We even have some students, in Chicago, one of the top five schools in the world. ..There were a few student, years and years ago, they said" I am going home for the summer, and I need to take a break for the summer"——you can't take a break for the summer, what are you talking about, you can take a break for a week or two. but if you do not basically willing to completely immerged in mathematics, then you are silly——you shouldn’t be doing this, it’s way too hard and painful, go be an investment banker to earn money? I am just saying, like if it is going to be painful, then you may as well make money. It’s also the case though, if you do not working, I always say this to my students "look, we are paying you for the first year of undergraduate teaching, I say look , we are paying you the salary, this is the job, most people work from nine to five, that is forty hours a week, that is the minimum, under forty hours a week, you are ripping off your university, you are taking money, you are not doing your job." Of course，forty hours a week is not enough, there are several number of hours you should have pencil and papers, and it’s very hard to do this. I would say forty hours a week if you want to be an incredibly crappy mathematician. Maybe you can get away with forty hours a week but eighty hours may be more realistic. Believe, I know about the Delays, I know the about video games, I know all about these things, any excuses for not doing mathematics, but when you are doing this, you can't do that. I think I have done this before myself, it’s just a test of yourself, very hard to look in the mirror, and see the true picture of yourself, and that is also the battle. One week, I decided, every time I have a pencil and a paper, I am doing mathematics, not having mathematics in front of my face——I am writing, not just reading, without writing things down is basically worthless, I think you have to read all the time, but you have to solve problems, and try mathematics examples. I have been working on how much time I have been doing that. And I am not going to tell you the answer. It’ s a shock, Oh, my god, I thought I worked five times just now, when I was a student. It a method to scare yourself back into reality.

 

毫无疑问地，你应该

1. 让你的生活，呼吸，睡眠都被数学包围

你不应该做这个（数学）如果你不打算让你的生活，呼吸和睡眠都被数学包围。在芝加哥大学这种世界前五名的学校，也甚至有这样的学生，多年前，有个学生说“哦，夏天到了，我要回家，我准备好去过暑假生活了”——“你不能放假！你在说什么？你不能这样”，休息一周或两周是可以的，但是如果你不愿让自己泡在数学中，那你太傻了——学数学这么难，这么艰辛，为什么不去投资银行赚更多钱呢？我只是说，既然这么难，那你还是去挣钱好了。如果你不干活——我总是对我学生讲“我们给你付工资（研究生助教工作），你得工作，大多数人都从早上九点工作到下午五点，这样每周40个小时，这是底线，如果每周不足40小时，那么你是在坑你的学校——你拿了钱，但是不干事。” 当然，每周40小时肯定不够，你得额外花很多小时陪伴你的笔和纸。我想说，如果你只想成为没什么价值的数学家，那就每周工作40小时好了。当然，每周40小时也许会让你侥幸成功，但每周80个小时也许更现实些。相信我，我也知道拖延的毛病，我也知道那些电脑游戏，这些都可以作为你不做数学的借口，但是，一旦你开始做数学，你就不该这样。^这是对自己的一个测验吧，照照镜子，看看自己究竟在干什么并不容易。有一天，我决定了：每次我拿着笔，我（才算）在做数学，不要仅仅把书本放在面前——我在读，不只是在写。作为你，只读书不写下东西，那几乎就没用。我想，你应该一直去读，但是你也应该去解一些题目，试一些数学中的例子。

Thomas Edison has said

Genius is 1% of inspiration, 99% perspiration.

I think most people have heard this, basically the greatest inventor in history. I think this is a fantastic…perspiration is sweating, it is hard work. He was trying to say the obvious thing is all about hard work. I have to say, in mathematics, I have a different quote, with all apologies to Edison, here is my own, maybe is coming from my own personal life:

Genius is 1% inspiration, 35% perspiration and 64% obsessive-compulsive disorder.

Anyway, this is a very useful mental disorder to have, obsessive-compulsive disorder, if you don’t, you can look that up,（do you have some detail or examples?）Obsessive-compulsive disorder is just when you drink, you have to touch things three time, the inability to let things go, to just hold on and never let go. The point is, you can not have vacations, you can’t do anything, it’s like mental illness, you are constantly obsessing, you are constantly thinking about that, you are looking at someone, nodding, but you are really thinking about math. I see all the good mathematicians I know have this obsessive-compulsive disorder. ^

 

托马斯爱迪生说过：天才是1%的灵感和99%的汗水。

我想大多数人都听说过这个，爱迪生是历史上最伟大的发明家之一。汗水，那就是努力工作了。这句话在于阐述世上最明显的真理——努力工作。我想说，在数学中，我有另一个注记，这是我自己说的，来源于我的体会：

天才是1%的灵感，35%的汗水和64%的“强迫症”

当然，这“强迫症”是有用的，它让你离不开你从事的东西，让你专注于此。要点是，你不能休（太久）假，你不能做任何（其他）事情，就像精神病一样，你完全集中精力，你完全在想着它。也许你在看着某个人，点着头，但是其实你在想数学。我见到的好的数学家都有这种“强迫症”。

 

next

2. Don't just read-write things down! ——Solve problems

I am very lucky I discovered this, I remember in college, when I graduated from Cornell, and I was going to go Princeton, and it was a great place. In 1989, I was going to read Milnor's book Characteristic Class, so I would go, sit by rivers and waterfalls, and read Milnor's book, it’s so beautiful and clean, by the end you are doing fancy work related to Chern, constructing Chern’s Classes from curvature tensor of manifolds, it’s very fancy stuff, then I thought I knew a very fancy mathematics, I’ve learned incredibly beautiful and sophisticated mathematics, and then, somehow, it came to me, oh, what is characteristic class, what is the second stiefel -whitney class of ... the torus? Then I realized I couldn't do anything, I knew nothing. Milnor is so clean, and his books are beautiful, but I haven’t written anything down, I thought I understood everything, but I understood nothing, I understood the words, I knew how to say them. I looked good in team, I didn’t do it for this reason, I could talk the talk, but I can do nothing, you can ask me to do something on the board, this is varying now that I see the students. And I always say, at the end of day, you, and a piece of paper and a pad, that’s it. All the BS is worthless, it’s what you can do, it’s what you can prove. If don’t know Characteristic Class and go the board to do something, then it’s completely worthless. It’s good for a while but of course you cannot do anything, it’s very compelling to just read because it’s fun to learn, you know about the higher K- theory, infinite categories, topological quantum field theory, it goes on and on. but what can you compute? If you cannot get up to the board, you know…

下面一条

2. 不要仅仅去读，要写下东西——做习题

我很庆幸自己意识到了这一点。当我从Cornell本科毕业后，我去了Princeton，这真是个好地方。那是在1989年，我开始读Milnor的书《示性类》，于是我坐在河边，坐在瀑布边，去读Milnor的书，这里很美，很干净，到了最后，我学到了很多奇妙的东西——比如从流形的曲率张量构造陈类，“我学到了不可思议的美丽和精妙的数学”，但是，突然，我想到，最简单的例子，环面吧，它的示性类，它的stiefel–whitney类是什么？（“我不知道！”），我才意识到我做不了任何事，我啥也不懂。Milnor的书很干净，很漂亮，但是我啥也没写下来，我以为我什么都知道了，但我什么都不知道。我只知道名词，我只能把它们说出来。我会吹牛，在讲台上大讲一通，但是，我什么也做不了。我总是说，一天结束的时候，只有纸和笔，一个便签本，是的，只有这些，剩下的bullshit都是没有用的——只有这些是你能用的，只有这些是你能用来证明问题用的。如果我不能用我掌握的陈类在黑板上做什么，那就完全没有用。仅仅去读，那么你什么也做不了。仅仅去读的坏习惯也是很难抗拒的，因为这样学起来有趣（很轻松），你可以学什么高阶K理论，无穷范畴或者拓扑量子场论，但是，你能计算哪些呢？如果你不能在黑板上验算，那么…

I remember giving one of my students “yes, I want to pass out in every class about differential topology”, I was teaching differential topology, I said come to my office I will give you an oral to see if you have known the basics”, so I said,” OK, what is Morse function, define Morse function”. Minor wrote a beautiful book on Morse theory, on page one of Milnor, before Morse function, it draws a beautiful picture that everybody sees, there is a torus and a height function, gradient flow, four critical points, and this is a beautiful picture, I said OK, great, but just define Morse function, it's in the page 2 of Milnor's, and he said "yes, it is very imaginary...", I said, “I don't want to know it’s imaginary, I don’t want to hear the philosophy, write down on the board the definition”, after all, it is about the derivative of the functions, and he couldn't do it. So it’s worth zero, and probably worth negative. It is very easy to deceive yourself. Probably we can cheat ourselves, it’s a nice person they weren’t try to be vain. That’s what I am trying to say, we all deceive ourselves, all the time, it’s much easier, just be aware of it, it’s so much easier, , and it is human's nature to walk on an easier road. Part of the way we check ourselves is to solve problems, just pick a book, I see some of you have Hartshorne, I encourage you work on all the problem of Hartshorne, I am glad to see it was beaten up, it did not look new, it looks like it is used, and I hope you do all the problem of Hartshorne, there is nothing like going through a book, doing all the problems, at the end, it is the greatest feeling in the world. Your books are all battered and they have coffee stains on them. There is nothing like it, that is a good easy way to check yourself, to force yourself.

How <wbr>to <wbr>do <wbr>Mathematics <wbr>by <wbr>Benson <wbr>Farb

我记得我有一个学生，他说“你的微分拓扑的课我想在课上睡觉（不想听了）”，我说“来我的办公室吧，我给你一个口试，看看最基本的内容你掌握了没有。”于是我问他“什么是Morse函数，请定义Morse函数”，Milnor写过一本很好的关于Morse理论的书，在这本书的第1页，讲Morse函数之前，它画了一个很漂亮的插图，这图中有个环面，有高度函数，梯度流，和四个临界点，这是一张很漂亮的图。我说“好的，告诉我什么是Morse 函数”，这大概在Milnor的书的第二页，然后他说：“是的，这个东西很玄妙。。。”我打断了他“我不需要知道它玄妙不玄妙，我也不想听它的哲学，请你在黑板上写出它的定义”，其实，这也就是关于函数的微分的，但是他写不出。那么它的价值就是零了，甚至价值是负的。欺骗自己是很容易的。我们欺骗自己，因为我们不希望自己一无是处。所以我才说，我们总是欺骗自己。因为欺骗自己很容易，请注意，它很容易，并且人的天性总是喜欢走容易的路。部分检验自己的方式就是做习题，选出一本书吧。我看到你们在座的有人拿着Hartshorne，我鼓励你们把它的习题全做了，我很高兴看到这书已经有些破了，它不像是新的，它像是用过的，我希望你们能够把Hartshorne的习题都做了。没有什么像拿起一本书，做完所有习题一样，可以让我们对这个世界感觉这么棒。这时你的书已经烂了，沾满了咖啡渍。从来没有其它感觉和这一样，这（做习题）是一个简单易行的检验，激励自己的方法。

3. Each week, take 10 minutes to look at yourself in the mirror

Ok， this is just an expression. Does anyone in the class know this expression. Of course, literally, I don’t need you look yourself in the mirror. It’s try to see the way the things really are, for me I am not really learning Milnor's Characteristic Class, I didn’t see that at the end. I was wasting time, it was wasting my life, it was really hard——try to do it, try really hard to do it, can I really know, can I really parametrize the surface using the tangent plane? I think this is when I was undergraduate, I am too lazy, Confucius was a great man, here is something useful to say, "what did I learn in this week",^ I have my student email me once a week, I see them every two weeks for a few hours, but every week, you are allowed to say, I did nothing, but I not going to yell at you, but it embarrasses them, so we never do it two weeks in a row, they usually never do it, but forces them realize “I haven’t learned anything” just try to learn one thing in a week. I like this exercise, because it just makes it easier for you. And "what new tool can I now use", you need to learn tools, you can’t just do work from scratch, I always say, at least once a month you should learn a big new hard technique. It’s a little overly ambitious, And "How many hours did I work", and again, people make fun of me and my student, we have some make fun of professor and then always make fun of working, working, working, I am not meaning to be hard, but literally just saying, you are here, at your current rate of your learning, you can get here, I need you to be here, you have to do something, something has to change. Your rate of learning will increase. But literally, there are no short cuts, literally, there are not short cuts, Again, I think the battle is "to be honest with yourself ". OK, to be honest with yourself.

 

 

3. 每周，都花十分钟照照镜子看看自己

当然，这只是字面意思，在座各位应该都明白。确切的说，我不需要你真的对着镜子照自己。我只是希望你认清真实的情况，比如我当初没有真正在读Milnor的《示性类》这本书。我只是在浪费时间，浪费自己的生命。这很难，但是你一定要去尝试——比如问问自己“我能通过切空间将曲面参数化吗？”下面这句话你可以问问自己“我这周学到了什么？”我要求我的学生每周给我写邮件，并且每两周我会和他们见几个小时的面。当然，每周，我都允许你说“我什么也没做”，我不会朝你喊叫的，我只会让你有点小不安，如果他两周都什么也没做到，也许他们永远也不会做到了。我会让他们意识到“我这一周啥也没学到”…总之，每周都要去学一个东西。我喜欢这个练习，因为它简易可行。紧接着，要问问自己“我学到了什么新工具？”你需要学工具，你不能光是乱写乱画，我总是说，你一个月应该学会一个新的，大的，困难的工具，当然，这目标可能有点太高了。最后，还要问问自己“我学了多少个小时？”，再一次的，人们拿我和我的学生开玩笑，关于教授的工作，我们也开玩笑，我不是说一定得怎么样，但是，你现在在这里（用手指着黑板上较低的位置），以你现在进步的速度，你可以达到这里（用手指指着稍微靠上的位置），但是我需要你达到这个位置（用手指着远比刚才高的位置），你得做点什么，你得改变！你要提升你的进步的速度。当然，确切的讲，没有捷径，没有捷径的。再一次，我要说“真正的挑战在于：要对自己诚实。”要对自己诚实。

4. How much you can learn.

If you are proving theorems, so many times I really want the theorem to be true, and I almost have it, but I realized I ducked the issues, I didn’t do encounter the key thing, this is hard. I am not doing it to laugh myself. And now here is a rule about how much you can learn, You are here, I think minimum to be a reasonable mathematician you need to be here, at the current rate of your learning, there is no way you can make it. So probably you look up something in the talks, you are reading the books, you say there is no way to learn all of that. You are correct, there is no way to learn all of that at your current rate of learning. One thing I have to say is I have definitely seen people who are not ambitious enough. And there is an expression if you shoot for mediocrity, you will succeed. So you need to change the way you think so that you can learn, here is the proposition. You have to trust me, this is absolutely true, I am absolutely not exaggerating here.

 

 

4.你能学多少？

在证明定理的时候，有无数次，我都希望定理是对的，但是我发现我回避了问题，我没有发掘出最关键的东西，这很难。在这里，我不是自嘲。现在，我要给出关于你能学多少的一个规律。你现在在这里（手指较低位置），我想，要想成为一个像样的数学家，你至少得到这里（手指较高位置），以你现在的学习速度，你不可能达到这个位置。可能你在讲座中听到什么时，或者读书时，你会说“我无法把它们都学会。”你说对了，如果你仅仅维持现在的速度，你不可能把它们全掌握。有一件事我得说：我确实见过一些不怎么有志向的人，有一句话叫“如果你朝着中等水平努力，你就能成功。”所以你需要对你学习能力的观念进行改变，以下是我给出的性质。你要相信我，这是真的，这绝对是真的，我没有夸张。

Proposition:

Let X =amount you think you can learn in this month

Let Y=amount you can learn in one month.

Then.1. Y=2X

2. Statement 1 holds replacing X by Y.

And this is different for each person. Maybe you know, you are not meant for graduate school, some people, rate of leaning doesn’t change, they have to leave math, I think for a lot of people, why they can’t learn. First Y=2X,

You absolutely learn twice as much as you think. And if you don’t try, so when I say learn, one paper, have note books, write a hundred pages a day, can’t? I have a student, he said ”look I can’t work with you anymore” I said “you are just sort of doing minimum, and I am yelling at you” Something has to change, I said, “ok, here's the book, do all the problems” he gave me a 75 pages document——he solved all the problems in the book, oh, it was 75 pages, he did in a week, “my god, I didn't thought you do that, you just did it,” he originally said he could never do that. I said, I am sure you can do this, and he did, you never look back, you did a fantastic job of fifteen papers. He is really a good mathematician.

And here is the second thing, once you do it, you say hey I can learn 2X, so then we have X, right? But now I said to them you are wrong again, you can do double again. I am always aware of infinite re-occursion. But unless there are certain people, then if it is probably that they’ll learn something anyways, OK,

Just remember this, again, when I was at graduate school, undergraduate school was in Cornell. It’s a good school that have smart graduate student. I would say, difference from Cornell and Princeton, the graduate students are try to learn tons of stuff at the same time, and again it doesn't mean they whipping through the pages like I read Milnor’s book, it means try to do every problem of Hartshorne. I will say Some of students in Cornell at that time they were very smart, but I remember saying they say” let's go through such and such book, and do the problems,” “Oh, man, we can't do that”, and I look back now, you should be able to do five times of that math, but the things it they never tried, if you never try, you will never reach that higher level.

性质：令

X=你认为你能在一个月能学的东西

Y=你实际上一个月能学的东西

则

Y=2X
断言1在把X换成Y后仍然成立
当然，我想说，这个确实因人而异。有人学东西的速度确实是一成不变得，当然，这些人就不能再继续做数学了。我想对于很多人，为什么他们学不到呢..首先，Y=2X

你完全可以学你认为你能学到的两倍的东西，如果你不去试的话…我说“学”，我是指一张纸，一个笔记本，每天用完一百张草稿纸。我有个学生，他说“我没法再跟你读了”我说“你只是达到了最低标准，你得改变”，于是我跟他说“这本书，拿去，把所有题目做了”，于是，他后来给了我一个75页的文档——他完成了书中的所有习题，是的，他一周就完成了。“哦，天哪，真没想到，居然做到了！”他开始认为他一定做不到的。我说我确定你能做到。现在，他已经写了15篇论文了，他是一个很好的数学家。

下面说第二件事，当你完成的时候，你说“我能做到2X了，但也就2X了”，你又错了，你能学的还可以再加倍。这样可以一直进行下去。你总能学得更多。

你就把它记牢了。当我再读研究生的时候——我本科在Cornell读的——它是个很好的学校，有很多聪明的研究生。聪明的研究生。但Cornell和Princeton的学生区别在于，后者的学生总是尝试着同时学海量的东西，当然他们不是像我读Milnor的书那样仅仅快速翻着书页，他们是要尝试着去做Hartshorne的所有习题的，我想说那时很多Cornell的学生是很聪明的，但是当有人提到“我们学这本书吧，并且做它的习题”时，回应总是“天哪，我们不可能做到”，现在回想起来，我们有能力学相当于那时学的五倍的内容，但问题在于我们总是不去尝试。如果不去尝试，就不可能达到这个更高的水平。

Now I am going to talk about

D. What to learn, and how to learn it.

1. Take your advisor advice, she knows best

I have basically discussed this. You do have to trust someone’s guidance, but if you don’t trust them, choose someone else. It’s trust, there is no way around it, you have to trust. You can’t make this on your own, you just can’t.

2. Learn big pictures as well as details

This is true for the scientific reasons as well. I have definitely talk to people even in postdocs, I said ”oh, what did you do in your thesis” “I refuted the cohomology of this level five of the subgroup, of every coefficients,” ”why did you do that?” "I don't know" I mean why you are even interest then? And you have to try to learn a big picture. And even you are not doing the major thing for your thesis, in the history of math, Serre's thesis and Tate's thesis, that’s better, everybody else did normal thesis. You are in the picture and and just a piece of the puzzle, it’s not going to be the biggest piece, it’s just going to be the first step of your research, you should know what the whole picture looks like, otherwise I don’t even know how your student should be interested, I talked to so many students and this is a big one. And I know the details could be complicated, if your advisor is doing the minimum model program, this is really big one, oh, my god, it’s a lot mathematics, that is really difficult, you might do some very specific things, you should definitely try to learn the big picture, this is not very interesting otherwise you’ll never prove big theorems, you are living in fear of the big picture. You’ll never be able to communicate what you are doing to others, but you do need to that, especially when you are young, you need to communicate, you need to get a job somewhere. Communication is part of them. And actually you can’t only get the big picture if you actually work through every detail. One without the others is not worth very much. Maybe it’s the good time to say two. Another common mistake with students, I remember having a student, I can remember his name, I said Justin” is it OK, try to do this computation, I think you should go this way” and the student say ”I’ll see you in two weeks” and the student comes back two weeks later, he says “well, I tried it your way, here is why it didn't work,” and you can explain it in 30 seconds, and I said ” this must takes you under five minutes to discover, did you tried anything else? “"No",” oh what did you do for the two weeks? “I mean, math is very hard, so the students were sort of clinging on the details, and the other students refuse to look, if it doesn’t work, there is no other avenue. You know I totally stumped on this They will spend a week thinking about the non-orientable case, the paper doesn't say orientable, you know probably it doesn't matter, the guy probably forgot it, how important is that detail? So this is really a big thing, we are proving theorems, if you are proving a theorem that takes more than ten pages, a lot of people prove in thirty, forty, fifty or sixty pages, you need to paint the big picture, you need to say that this is going to be these five main steps. OK, step one, you need to break it down.

下面我要讲

D.学什么，怎么学？

1.信任你的导师，她最懂行

刚才已经基本上谈过这个了，你总需要某个人对你的指导，如果你不相信某人，你应该另选他人。别无他法，你一定要信任你的导师。你不能靠你一个人完成学业，你不能。

2. 整体框架和细节都要学

从科学的角度讲，这也是对的。我和很多人，甚至是博士后谈，我问他“你的论文在做什么”，他说一堆抽象的东西，“为什么你要做这个？”“~~我不知道！”那你为什么对它感兴趣呢？你要去学整体框架，即使这不是你的论文的主要任务。在历史上，Serre和Tate都写出了很好的博士论文，其他人的论文水平就相对来说比较一般。你处在这个大框架中，这是个巨大的拼图，当然你做的不必是这个拼图最大的一块——这只是你科研的第一步。你要明白大框架是什么，否则你都无法给你的学生解释为什么你喜欢它，我经常和我的学生说起这个，这很重要。我当然知道细节可以使很复杂的，如果你的导师在做minimum model program，天哪，这其中有太多的数学，这非常难，你在做非常专的工作，但你还要学习大的框架，这可能不会很有趣，但是不这样的话，如果你害怕大框架的话，你就无法证明大定理，并且，你永远无法和他人交流你在做什么，而这正是你需要做的，尤其是在年轻的时候，更是需要交流。因为你需要找工作，而交流就是最重要的一部分。当然了，仅仅研究每个细节是不足以让你了解整个大框架的。一个人不和别人合作的话，是不会有大出息的。再讲另一个学生常犯的错误，我以前有个学生，他的名字我还记得，我说“Justin，你试试算下这个，我想可以这样算”，然后学生说“我两周后来见您”，两周后，学生回来了，说我的方法行不通，并解释了一下。但是，解释我的方法行不通最多只需30秒。我于是说“发现这个最多花了你5分钟时间吧，你有没有尝试其它方法？”“没有。”“那你这两周都干什么去了？！”…我想说的是，数学很难，我有的学生过于迷恋细节，而有的人如果一次不行，就不愿再试。我有一次被一个问题挂住了，我花了一周时间去想不可定向的情形，因为文章没有说可定向，可能这根本无所谓的，写文章的家伙估计是忘了，你认为这个细节会有多要紧呢？这一点很重要，如果你要证一个篇幅十页，甚至三十页，四十页，五十页，六十页的定理，你一定要有一个整体框架。你要能说出：“这其中有五个主要步骤，第一步是…”，你需要把它们拆分。

 

And this represent, you need to work through every detail.

3. It is too hard to remember all the details, remember key principles

You need to work through details, there is no question, but it is too hard to remember them all, it’s really a good way to learn mathematics to go through the details, write outlines. When I was a graduate, I sort of wrote an outline of all the mathematics, no, a big chunk of mathematics. But just find the key principles. And I remember one example：change coordinate techniques.

Example:

1. Change coordinates

We’ve all see this, we take a really hard problem, and you either change basis or change coordinates or applying Mobius transformation, one of the famous ones is, I actually forget the theorem, and I only remember the principle, but that is fine, because principle is more important, I just remember it is a problem we take a circle, and what do you do? Take another circle tangent, maybe one tangent to both, and there is some theorem about this, I forget the theorem, I don't care, but I just remember the trick, OK, maybe, no, no, no, this is correct. I just remember the trick, maybe, the outer circle is l, the inner circle is l', you can, here, with a Mobius transformation these two circles meeting at a point, oh, on the Riemann Sphere, I can handle this by the Mobius transformation, and make this l_1…Sorry, this is l, and this l'. And then, any theorem, I forget the theorem, but now it’s all trivial. Another one is like：you just remember:

Holomorphic maps tend to be distance non-increasing
^When I say “tend to be”, it depends on the domain and the target, this is not always true, it’s that you should know. This is just one that came to my head and I use recently, even though I am not an expert of holomorphic maps ^, I remember this principle, I need something like which is going to be distance non-increasing, Oh, wait, if I can make it holomorphic, that could be true, and it is very useful. You see what I mean? Of course you need to know the details of what exactly this means, but I can reconstruct that now. new generalization. So I think learning key principles is a very useful way to learn mathematics because now if you ask me to prove Thurston’s hyperbolization theorem, I can list the outline, maybe even the next level. I think it’s good to learn like webpages, if the hyperbolic structure, that’s how I like to learn it, ant then, you know, you can click on any one of the five steps, you know Mostow’s Rigidity there is another one, five steps, and then, if you are interested in one of them, you can just click on it, and I can tell you the five steps of that, and maybe the next level, at some point, I don’t remember the details well on the top of my head. I think that’s really important, cause there are so much detail, you can get overwhelmed.

 

当然，目前来说，你还是需要检验每个细节，但是

4. 记住每个细节太难，要掌握主要思想

你要检验细节，但很难全记住它们。写下证明大意是学数学的好方法。我读研究生的时候，写下了海量数学定理的证明大意，当然了，只是写主要思想。

例子：

坐标变换
我们都见过，当我们面对一个困难的问题的时候，我们要么变换基，要么变换坐标，或者用Mobius变换，其中最有名的是…我忘了定理了，但我记得思想，但是没关系了，因为思想更重要。我只记得是先画一个圆，然后呢？再画一个圆和它相切，然后（再一个）…也许和两个都相切。然后关于它有个定理，我忘了定理了，我不在乎，我只记得它玩的什么把戏。外面的圆记为l，里面的圆记为l’，用一个Mobius变换…这两个圆切于一点。在黎曼球面上…我可以用一个Mobius变换，把l变成这个，把l’变成这个（直线）。然后…我忘了定理了，但这已经显然了。另一件事是，你要记住：

全纯映射趋于让距离不增
当我说“趋于”，我是指它取决于映射从哪里映到哪里，这并不总是真的，你要了解这一点。这东西我最近才意识到，并且开始使用它，虽然我不是这方面的专家。我知道这个思想，并且我需要一些东西它是让距离不增的。哦，等等，如果我能让它是全纯的，那就对了，这个很有用。你知道我要说什么对吧。当然，你需要知道每个细节在讲什么，但是，我现在就可以给你重新叙述出来。

所以，我认为记住核心思想是学习数学的好方法，比如，你现在要让我证明Thurston的hyperbolization theorem，我可以列出提纲，甚至下一级。就像网页一样，比如我想了解双曲结构，你可以点开五个步骤中的任何一个，你想知道Mostow’s Rigidity, 那就是另外五个步骤了。如果你对其中任何一个感兴趣，你可以点开它，我可以给你讲这五个步骤，甚至下一级。当然，现在，我记不得所有细节。这（写提纲）很重要，因为细节太多了，（全记住）你会被压垮的。

Number four

4. Work on basic examples! I think you should

a. Know basic examples better than you know your girlfriend/boyfriend

I guess if you are married, you shouldn’t have a boyfriend or girlfriend. This is something again that I would say, now spend a lot of time basic examples, and I really mean basic, so my thesis is on manifolds with pinched negative curvature, low dimension, includes hyperbolic manifolds the secret is...don't tell anyone, I understood the hyperbolic structure on punctured torus incredibly well, and when I understood this, right away all the general things , I did not exactly follow, there are some new ideas here, but this is incredibly useful, I spend a lot of time drawing pictures, having to do with this surface. The easier the example the better, and I would say, when I got to university in Chicago, I am interested in studying about transferring Thurston's ideas to understand a lot of things in semi-simple Lie groups, in Chicago, which is the world center for that. Super-rigidity and very fancy things, and I talk to post docs and students, and then we talk about very sophisticated things and I was trying to learn these sophisticated things, like semi-simple Lie-groups and all of that machinery, and Lizhen to lecture on, there is a lot of stuff. But I founded they have no feel, you have to discover properties that I really really knew, three dimensional Lie groups incredibly well, this is a small elementary thing, but I really really really want to do them, and my whole career a based on that, I can learn machinery later, and I did learn it.

 

第四点

要研究基本的例子。你需要：
了解基本的例子胜过你了解你的女朋友/男朋友
当然，在座的可能有人已经结婚了，那么你不该有男朋友或女朋友了~我想说的是，现在就开始花时间研究基本的例子吧，我是指非常基本的。我的博士论文是关于负曲率流形的，低维…双曲流形，秘密是什么呢？别告诉别人啊~我对去掉一个点的环面的双曲结构超级了解！当我弄懂这个的时候，一般情形也就迎刃而解了。当然（其他情形）会有一些不同，但是这仍然很有用。我曾花了大量时间去画和这个曲面相关的图。例子越简单越好，我想说的是…当我到Chicago的时候，我就致力于把Thurston的一些思想用于了解半单李群中的很多东西，Chicago是世界上这方面的中心，还有super-rigidity和其他玄妙的东西。我和哪里的学生和博后去交流，然后我们讨论各种深奥的数学，这些深奥的数学都是我想去学的。比如与半单李群相关的，这里面包含着大量的东西。但是我发现他们（对一些东西）没感觉，三维李群，这是很初等的东西，但是我非常致力于这些。我的研究均基于此。至于结构，我可以以后再学，但是我终究学会了。

b. Have a huge list at your fingertips, check every statement against this.

Richard Feynman, I got this from Richard Feynman, is one of the great minds in the last hundred years. And he said he always just learn examples and examples, and he always had huge list of examples at his fingertips and every statement against this. and he said, he could always impress people because they would say, take a Kahler manifold with holomorphic vector field, he would always give incredibly simple example. the torus, and he would say, no, there is a counter example, you are amazing, but it’s not, he just in hand, and this is just absolutely number one, and this is, I would say a huge trap, and again, I really feel the whole thesis thing. You know, I am trying to get from here to there. And there is just landmines everywhere, and if I step on one, I am dead, if I don’t know enough examples and I get to enamored with them, big theories, I am dead, if I don’t work enough, I am dead, it’s all these little issues, but it’s a wonderful thing, yeah.

b．手中要有一大堆东西（解决问题的基本元素），验证每一项不满足它的叙述

Richard Feynman，我从他这里学到的，他是前一个世纪最伟大的科学家。他说，他总是学例子，他手边总有一大堆例子，然后检验与之对立的叙述。他说…他总是让人印象深刻，因为别人总是说：来，我们来取一个有全纯向量场的Kahler流形…但是Feynman总是找简单的惊人的例子：环面。他也总说“不，这里有个反例的”“哇塞，太不可思议了”——其实不是的，这就在他手边。这完全是第一位的（检查反例！），这也是个很大的陷阱。我完全感受到了这一点。你知道吗，我尝试着从这里走到那里，并且到处都是地雷，如果我踩中了一个，那么我死定了，另外，如果我工作地不够，我也死定了。就是这么回事儿，这也是很棒的一件事。

c. No example is too easy, don't worry about your ego.

Definitely when I was a student, I switched areas, I said oh my god, I don't know anything about geometry, I did my thesis in geometry , differential geometry, that’s why I said, I have to learn the curves in the plane, curvature of the curve, that is called the second derivative, and I was dealing this, and my classmate in Princeton, were making fun of me, "you know, we are doing the moduli spaces, you are still doing... are you kidding me?" And I somehow, I don't know why. I didn’t let it get to me. I had a big ego. And it knocked my ego. They are doing fancy things, but at the end of the day, you know there are great people at that time but I was the first one who prove the theorem. Because ego is one thing, but I really understood curvature, if you don’t understand the curvature of the curve in the plane, and you are working on thesis on the curvature, you know biholomorphic, sectional curvature of the Kahler Manifold, if you don’t know the curvature of the curve in the plane, what are you going to discover? What are you going to do? You can manipulate the symbols, is what you can do. You are not going to prove a theorem.

At the end of day, it is just you and a piece of paper,（you are living in your office and unfortunately I wish it was）there is no place to hide.

Your understanding is now, it is just you and that paper, and if you don't understand the curve in the plane, you are screwed, you are not going to prove anything. So ,you know, you can put if off for a while but you have to prove the theorem. So your ego at that point, what's it worth?

C．没有例子太容易了。不要对自己的自负过于担心。

当我是学生的时候，我换了方向，哦，天哪，我完全不懂几何。我的论文是关于几何的，微分几何的。这就是我为啥这样说——我得从平面曲线开始学起。曲线的曲率，叫做二阶导（做出说悄悄话的样子，意思是当时自己很弱，连这都不懂），然后我就一直和它打交道，然后我的Princeton的同学就取笑我：“我们都在搞模空间，你居然（还在搞这么初等的东西）…你在搞笑吗？”但是，不知为什么，我没有让这些话影响我，我很自负。那些同学的话触碰了我的自尊。他们都在做时髦的东西，但是到了最后，要知道，虽然当时有很多优秀的对手，但我还是第一个证明了这个定理。自负是一回事，但是我真的对曲率非常了解，如果不懂平面上曲线的曲率…你在写关于曲率的论文，关于Kahler流形的双全纯截面曲率，如果你不知道平面曲线的曲率，你能发现什么呢？你能做什么呢？你除了玩符号，啥也干不了。你无法证明定理的。

在一天结束的时候，只有你，一张纸，没有隐藏之处（很不幸，虽然我希望有）。（意思是，你要是没弄懂，那谁也骗不了）

你的理解…只有你和纸…如果你不理解平面的曲线，你在自欺欺人，你什么也证明不了。当然，你可以暂时把它敷衍过去，但是，你还是得证明定理啊。那么这时候，你的自负有什么意义？（意思是，你想自负，可以，但基本的东西得弄懂）

 

D. Other people

You are human being, math is pure, you want to feel that the math you are doing is absolutely purely in mathematics, you know what, we all love math, but you shouldn't feel bad, you pay attention to other people, it’s human nature, the first is you should learn is:

1. Learn as much as you can from other students

The most I ever learned was from a fellow graduate student who was one year older than me. I just rose up mathematics, rose up levels, just talking to him all the time, and I taught him things, I don’t remember teaching like…But your professor, you are not going to be with them all the time, if you are fellow students, you can be with them all the time, every day, six hours, that’s you can learn from, you might feel somewhat competitive, there is an advisor and has two students, maybe it’s clear that the advisor thinks more highly of the other student, too bad, you can still learn, it's an amazing resource. So, learn from other students.

 

D．其他人

你是人，数学很干净，你希望你做的数学是完全干净的。但是，你要知道，我们都热爱数学，但是你不该感到糟糕，你应该关注一下别人，这是人的天性（？？）。首先你要学会的是：

尽量多地向其他同学学习
所有人中，我向一个比我大一届的研究生学地东西最多。我不停地提高，我总是和他在讨论，并且我也教给他一些东西，不过我不记得教了哪些…但是你的教授们，你不可能总是和他们在一起。但是你的同学就不一样了，你可以和他们每天都讨论甚至六小时，你可以向他们学习。当然，有的情况很棘手，导师有两个学生，导师认为另一个学生好（比你可以请教的）——太糟了，其实你还是可以学的，这是非常好的资源。总之，向其他同学学习。

 

2. Mimic the style of great mathematician, but develop your own style too.

Nothing is wrong if you have no style. If you are first year student you have no style, you can’t develop style, you don’t know mathematics, you don’t know research method. you can’t start from nothing, the easiest way, to, I think to, sort of pick up a really good mathematician and mimic. It’s some great people, obviously, you shouldn’t just inherit them, and be tiny copies of them, I think mimicking their style is a really good way to get really good, really fast, in terms of the style, math what they are doing.

模仿大数学家的风格，但也要发展自己的风格
（暂时）没有风格不是错。如果你只是一年级的学生，当然你没有自己的风格了，因为你尚且不懂数学，尚且不懂科研方法，你是从零开始的。我想，最快的方法是选定一个大数学家，模仿他。当然对于这些大数学家，你不应该仅仅继承他们，成为他们的小小的复制品。我想，模仿他们（大数学家）的风格是非常好的，学他们的风格，他们做的数学.

 

Don't worry about how much worse you are than others
This is very difficult. I had a student, who became my student as a teenager, who was like phenomenal, I had seven other students at the same time, and they couldn’t help but tell that this guy is probably way better than them.

And I can see, it is sort of shook their confidence, there is nothing I can say, except

You are not a good judge of yourself（or, probably, you are not a good judge of others）
Either, you might not actually be as bad as you think, you may not be as good as you think. As also a conflict I think some people seem really good. I mean, you are human being, so you are going to care, it’s not going to be... You have to deal with it. You have… I would say,

There is nothing you can do about it.
So you better get over it and better learn to live with it. The best thing about going to Princeton for me, I came out as an undergraduate, I won the prize for best undergraduate, and I thought “Hey, I am really great”, and I went to Princeton, and within one day, I realized that, oh, my god, I stink, I am nothing to them, these people are level…and I became a student of Thurston, and I just had everything ripped. It is a really tough time , at the very end, I got over it, you know what?

c. There will always be someone many levels better than you. Get used to it. Right now!

Right now. In fact, it is always funny, there are Field’s medalists, you know that there is always the next cut, so even fields’ medalists going to be top five in the world, there are , compare themselves to the people from previous generations ,it just never ends. So I have to tell you a story. When I was younger, do you guys know about Mozart and Salieri. You know Salieri? Salieri was a … There is a play called Amadeus , about Mozart. Mozart has a great genius, and Salieri, it is all about how he is mediocre, but he is good enough, only he see how amazing Mozart is, and Mozart…and every time he sees Mozart, it pains him, because he realizes he is mediocre, he knows what it is to be great. And so my parents…when I was in the beginning of my career, I was doing well, my parents are excited, "Wow, Beson, are you going to win a fields’ medal?" “no, no, no, there is a series of people, so I am not one of the great people, I am just a schmuck.” I said” But, I am Salieri, at least I can appreciate the work of the great people, I am not one of them but I can appreciate. “So I walked around for years, said” I am Salieri”, which I thought was pretty good then. And then, I went in Thurston's birthday conference, and Dennis Sullivan was one of the great mathematicians, he gets up and he says “Well, I was sort of thought of as one of the top topologists in the world, I was a professor in Berkeley, I thought I was Mozart, I was in a talk, some graduate student raise their hands says ‘ I have a counter-example’,” He brought him to the board, he started making a diagram and putting dots and…”and so Sullivan says ”my jaw dropped, and I knew two things about them, the firstly the first purely geometric argument I saw, and number two I realized is I am not Mozart, I am Salieri, and that was Mozart, and that’s Thurston.” and the second Dennis Sullivan said that, I knew I was not Salieri, I told this story to a group of people, just as I told you right now. And someone was on the table, and he is laughing, I said ”what are you laughing about?” He said “You know Beson, I used to think I was Salieri, and then I heard Dennis say that, Sullivan say this” and I realized I wasn’t Salieri” ,and I realized that I wasn’t even the guy that knows about Salieri. So I am not telling the story anymore. There is infinite layer of things. Get used to it.

Good news about this is：You still have a lot to discover and contribute.

Those people are human, they can’t do everything, So that is a good news：you still have a lot to discover and contribute, just because there are fantastic people. Again this is something I hear people who say “If I am not the best, I am not going to do this, need I leave mathematics?” “You know what, you are not the best, I am telling you now, Nobody here is the best. Nobody here is” I don’t know anybody who is.

3. 不管你比别人差多少，也不要担心（这是本文最出彩的地方）

这一点很难做到。我有个学生，他很小的时候就成了我的学生，很明显地我能觉察到…在同一时间，我还有七个学生，他们都情不自禁地觉得这个小家伙比他们强太多了。我能看到他们的自信心受打击了。对此，我能说的不多，除了：

a．你无资格评价你自己（也许也无资格评价他人）

也许，你不像你想象中那样坏，也不像想象中那么好。当然，有的人确实相当好。当然，你是人，你要注意…这并不是…你得接受它。你得…我想说：

b． 对此，你无能为力

所以，你最好适应它，学会以这样的方式生存。我去Princeton最大的收获就是…我本科的时候，我获得了“最佳本科生”的荣誉，我暗暗觉得“我真了不起呢！”。但是，当我到了Princeton，一天之内，我就发现我简直糟透了，我跟他们比起来啥也不是，这些人的级别实在是…后来成为了Thurston的学生，我感到之前一切的光环都被剥夺了。这段时间确实非常艰难，但我还是克服了这些困难。你知道事实如何吗？

c． 世上总会有人比你强好几个档次。现在就适应它吧。

对，就是现在！事实上，很有趣的是，即使是菲尔兹奖得主，也能决出个世界前五，他们还可以和他们的前人进行比较，这种比较是无休止的。所以，我现在要给你们讲个故事，你们知道Mozart和Salieri吗？你知道Salieri是谁吗？有个电影叫《莫扎特传》，是关于Mozart的：Mozart是天才，Salieri相比要平庸的多，但是他也足够优秀，因为只有他能明白Mozart有多么地天才。每次他见到Mozart，他就很痛苦，因为相比之下他只是庸才。所以，我的父母…当我的事业开始起步的时候，我做地还不错，我的父母就很兴奋地对我说：“Benson，你准备好拿菲尔兹奖了吗？”“不不不…这个圈子里面有一票人呢，我不是最好的之一，跟最好的人比起来我只是个笨蛋，但是…”我说道“我是Salieri，至少我能欣赏最优秀的人的工作，我不是他们的一份子，但我至少能欣赏。”然后，有好几年，我都这样告诉自己“我是Salieri”，感觉也还不错呢。

但是后来，我参加了Thurston的生日晚会，（在场的人中）最出色的数学家之一是Dennis Sullivan，他站起来说：“我之前觉得自己是世上最出色的的拓扑学家，当时我是Berkeley的教授，我以为我是Mozart，然后有一次，我做一个报告，一个研究生举起手来，说他有个反例”，于是他把这个研究生带上讲台，然后他开始画一些图…“我的下巴快掉了（惊呆了），我知道了两件事，第一件事是这是我接触的第一个纯几何的论述，第二，我意识到自己不是Mozart，他才是，那个人就是…Thurston！”，当然，Dennis说的第二件事暗示我，我不是Salieri…我把这件事跟一群人讲过，就跟今天这样，有人就笑了，我问他为什么笑，他说“Benson，你知道吗？我一直以为我是Salieri，我现在听了Dennis说了这些，我意识到我不是Salieri”，然后，我（Benson）进一步意识到我甚至连能欣赏Salieri的人都算不上。这个故事今天就不继续讲了。总之，人的档次是分了太多层的。你要习惯它。

好消息是：你仍然可以探索出许多东西，做出贡献

那些大牛们也是人，他们不可能把所有问题都做了。所以这是个好消息：你仍然可以探索出许多东西，做出贡献。我听到有些人说：“我不是最优秀的，我做不到这个，我需要离开数学吗？”我的回答是：“你知道吗，你确实不是最优秀的，我可以跟你明说，但是，在这里没有人是最优秀的。”我不知道谁才是。

E. Taste

I think taste in mathematics is incredibly important, you need to develop your own taste, meaning like what is good, what is bad mathematics. That is for you to say. There is no book , I don’t think there is right answer, I was going to say” there is no right answer. “But there is, I can’t define it, it’s like what Thurgood Marshall, who was a famous court justice of the United States, there was a case on pornography. Have you heard this word? Pornography? Dirty pictures! Thurgood Marshall and the lawyer said：Define it! If you are going to make a law about it, define it! And he said “I cannot define it but I know it when see it. This is same with good mathematics. But you can be influenced by those with great taste.

1. Develop your own taste, but also be influenced by those with great taste.

This is the easiest way to get good taste, so at least you be influenced and then you construct your style. So, here is the list of personal favorites：Serre, Milnor, Thurston, McMullen...just they are fantastic writers, their mathematics is everything I think is good about mathematics, this is biased, of course, my interest in Geometry and topology, it is also as Serre, they all amazing writers, their math is everything that’s the definition of what I think is good mathematics.

2. Once you start working on a problem, you should find an answer to the questions.

"Why is this interesting?" "Where does it fit into big pictures?"

Anyway the summary about the graduate school, that’s all I can say about being a graduate student:

The Summary ：this path is hard. This path is definitely very very hard. But definitely I believe that theorem. The truth of this theorem. That is:

Theorem 3. Everything worth doing is hard.

Anything worth doing is hard. There is nothing like it, it’s nothing like the journey, of course, actually getting there is actually a little, deflating, I like the journey. Even though, I don’t like mountain climbing.

At least I am physically lazy, not mental. Everything worth doing is hard, so it is worth doing, math is something like that.

E． 品味

我觉得数学品味是特别重要的，你要发展出自己的品味，也就是什么是好的数学，什么是坏的数学。当然，这是由你决定的。没有书上告诉你什么是正确答案，我想说“没有正确答案的”。但是，它（好的数学）就在那里啊，我不能定义（什么是好的数学），但是就像Thurgood Marshall那样，这人是美国著名法官，当时他正在处理关于色情图片的案子。你知道这个词吧？色情图片？…Thurgood Marshall和律师说：“请定义它（色情图片）”，然后他又说：“我无法定义它，但是它（色情图片）就在眼前啊！”数学也是这样。当然，你可以和伟大的人学习他们的品味。

1. 发展自己的品味，但也应当受到伟大的人的影响

发展好的品味最简单的方法就是接受伟大数学家的影响，然后发展自己的风格。以下列出一些伟大的人：Serre, Milnor, Thurston, McMullen…他们写的东西都很好，他们的数学我认为那就都是好的数学。当然，这绝对是带有偏见的，因为我的兴趣在几何跟拓扑上…Serre也是，他们都特别善于写东西。他们的数学我想我都可以定义为好的数学。

2. 一旦你开始做一个问题，你就应该问问自己：

“这个问题是否有意思？”“这个问题在整体大框架中扮演什么角色？”

最后，关于我想说的关于如果做研究生的总结是

总结：（数学）这条路很难，这条路毫无疑问非常非常难。但是，我也完全相信下面的定理：

定理3：任何值得去做的事情都很难

对，每件值得去做的事情都很难。当然，这不像是旅程，旅程结束了，也就显得不重要了，我喜欢这个过程。当然，我不喜欢爬山…我很懒的，我是指在身体上，不是精神上… 总之，每件值得去做的事情都很难，所以它才值得去做，数学就是这样的。

II On doing research, one opinion

So, any questions about “being a graduate student”?

This is much shorter, and I will give it five minutes. So I am giving one opinion. No, no, no, it is correct opinion about what’s good enough. So, this is big one. In terms of what is good mathematics. I think

A. abstract VS concrete

About what is good mathematics. I think You should spend time on mathematics you think is good and you think is important, if you don't, you won’t do good mathematics, and there is nothing to find. The big decision is abstract vs concrete. Concrete, means, explicit, examples, abstract is abstract

Don't be seduced by fancy words.
There are definitely graduate students, at the university of Chicago, oh, ^ the word is not mathematics, it's very fancy, it's very abstract, so I say to the student, what’s the naming of the single theorem for? He can’t do it, why the hell are you like that if you can’t name a single theorem?

Here is a little test, if you are doing a certain topic, you should at least know one theorem you like, If you are going to algebraic topology, you know what, you should probably choose Lefschetz fixed point theorem, because it doesn’t get much better, if you are going to discrete subgroup, you better like Mostow’s rigidity, how many people here do not were not are not just their life changed, when they saw Cantor’s diagonal proof? Whose life is not, who is not overwhelmed. Right? We all were, you wouldn't be here, you mean that like a test question in mathematics?

How <wbr>to <wbr>do <wbr>Mathematics <wbr>by <wbr>Benson <wbr>Farb

I really believe if you don’t like that math basically, that’s it, that’s the height, it doesn’t get much better. So, but anyway, just the words it’s going to get boring. Even though they can be fun. Theory that sheds no light on specific examples I believe it's absolutely worthless. Absolutely worthless.

II．关于如何做研究的个人观点

有没有关于“作为博士生”这一块儿的问题？

这一节要短得多，我将在五分钟内讲完（虽然他最后没有）。不不不，应该是一个关于“什么才是足够好”的一个正确的见解。就什么是好的数学而言：

A． 抽象vs具体

就什么是好的数学而言，我想，你应该把精力花在你认为是好的和重要的数学上，如果你不这样，你就做不了好的数学。很重要的抉择是：抽象vs具体。具体，就是很明显，有例子，抽象就是抽象

1. 不要被花哨的名词牵着鼻子走

有一些Chicago的学生，学一些很花哨的东西，它很抽象。我就对学生说，你能说成其中一个定理吗？他做不到——那你干嘛喜欢这个东西，你连一个定理都说不出来？！

以下是一个小测验，如果你在从事一个学科，那你至少应该能说出一个你喜欢的定理吧。比如，你要研究代数拓扑的话，那么你很可能会说出Lefschetz不动点定理，至少它算是比较好的吧。如果你研究离散子群，可能你会说出Mostow刚性定理。在座各位，你们看到Cantor的对角线证法，有多少人不认为自己的人生焕然一新了呢？有谁的人生没有被此征服呢？我相信你们都是这样，否则大家就不会来这里了。

当然，我相信如果你压根不喜欢这种数学，那也就只能这样了。但是，仅仅有名词会很让人感到无聊，哪怕它可能很有趣。一个理论，如果它不能阐明基本例子的话，它就是完全没有用的。 完全没用！

 

Grothendieck, I think, was an amazing mathematician and one of the most influential people, obviously, but he ruined a lot of people, because they think that he build big theories with no examples, that is absolutely false, it is a complete misunderstanding. Well, he was trying to generalize incredibly specific concrete examples, he didn’t want to know about examples and think about them only he develop, like theories of schemes, it was hopeless to define, he was developing etale cohomology to solve Weil's conjecture, incredibly concrete stuff——counting points of varieties. So just clinging to this abstract stuff that is not Grothendieck, people like this, ”oh, I am going to be like Grothendieck”,this is a huge, huge thing.

But of course, Random theorm without a theory is like a monkey playing a piano（sometimes you hear something beautiful, but taken as a whole, it isn't very profound）, I think this is basically the state of like forty years ago before, you know it was a bunch of random fun math problem, that you do in eighth grade. There are infinitely many problems you can come up with. Everything should shed light…every example should shed light on theories and every theory should shed light on examples. You do have to try to thread and needle, so to speak. I think it is useful to remember, that math is about discovering, explaining and understanding phenomena, something interesting is happened, I just taught a mini course, we did coinvariance of Euler Character Class, something is happening, some list of numbers looks like its occurring, go through finite group theory, and some other field, modular functions. It is a phenomena, definitely, you look at papers, like, every quasi, n-category is…what do you shedding light on? What have you just told me? what example, what did you discover, what’s your explaining, what are your understanding with that? Nothing.

 

Grothendieck，他是非常不可思议并具有影响力的数学家。但是，他也让很多人误会了，因为很多人认为他建立了一套完全抽象，没什么例子的理论——大错特错，这完全是误解！他其实是在尝试推广非常特殊且具体的例子，一旦理论发展了，他就不再去想具体的例子了，比如发展Scheme的时候。但是，他发展etale上同调确实是为了解决Weil猜想这个非常具体的问题——代数簇的数点问题。Grothendieck不是仅仅依附于抽象的东西之上的。人们总是说：“啊，我要成为Grothendieck”…这些人，你们知道…

但是，当然了：一个随意给出的定理就像是弹钢琴的猴子（有时你会挺好动听的东西，但是，整体来看，它不会很深刻）

就像四十年前那样，有很多非常有趣的数学问题，你在八年级的时候就可以考虑它。会有无穷多的东西出现在脑海中…每个例子都应该阐明理论，每个理论也应该阐明例子。你一定要非常仔细地尝试。记住以下这句话是有益的：数学是用来发现，解释并理解现象的。一些有趣的事情发生了，我教了一门迷你课程，是关于Euler示性类的余不变性的，观察到一些…一些数的出现，这和有限群论有关，还有相关的，比如模形式。

有一种现象，大家看文章，看什么quasi, n-category（花哨的东西）…这能阐明什么例子呢？你能告诉我什么呢？有什么例子呢？你能拿它做什么呢？你对此有什么理解？——啥也没有！

Here are two quick checks that you are not pushing symbols around

This is something I learned from Thurston, firstly, go to Mathoverflow, the website, if you don’t already know, look up Thurston’s quotes. He was already long passed his prime, but Thurston posted on there, and he was encountering mathematics. He make me feel like I was crunching symbols. He really encountered the mathematical objects, he made a computer program, that would like bring up the thing he was trying to study, he got in there and he was really using all the properties, he wasn’t just trying to pump theorems. That’s what it is all about. Here are two things, here is like little checks:

1. Does your theory reprove an old theorem?

2. Does it say anything new or interesting about a fundamental, basic example?

This is an easy check. I could think there are lot of current mathematics, doesn’t even satisfy these, and there are hundreds of papers, I would say that ... a lot of People define a lot of in 3-Manifolds, I don’t do 3-Manifold theory, but I know about the Thurston’s approach, the Thurston’s approach solved almost all the problem, a huge part of the problem. People define fancy invariants , and I went to a talk once, within last year, someone defines a very fancy thing, very obnoxious. And I am thinking, sitting there, this isn’t what it means for things to add up… they are just attaching these fancy things, and he says” yes, I believe these invariants can distinguish all knots...” and I raised my hand and said, can you give me an example of two knots in this case. I know from when I taught undergraduates, the trefoil is knotted, why? There is an argument that you can understand in the second…so a second grader can distinguish the trefoil, I say you big theory, your seventy pages paper, to me, can you tell the trefoil or not? No…I mean, that’s not a isolated theorem, there are whole fields, like this, in my opinion, it’s so extreme that I have no idea why people doing it, and it’s even people who are well known inside mathematics, is that extreme. And I love to debate it, I mean it’s that extreme, it’s that extreme. And this is a thing that students are just drawn to that stuff and I go like” what is it about, was I like this, probably, I was like this too as undergraduate? Undergraduate are drawn to the crappiest mathematics, I don’t know why they do. Sorry, the answer to this question. I have a very low bar, I just want you to tell me something can you tell the trefoil is knotted? Fine, even the second grader can do it.

关于你是不是仅仅在玩弄符号，以下有两个快速检验方法

我是向Thurston学到的这个。首先，访问一下Mathoverflow这个网站，如果你还不知道的话。你去看看Thurston关于此说的话，当时他已经过了科研的高峰期了，但是他还在和数学打交道。他让我感到我就是仅仅在玩弄符号。事实上，他经常遇到一些数学中的对象，他编了一个电脑程序来研究他想处理的东西，他确实把数学对象的性质发掘地很深，而不仅仅是为了证明定理。就是这么回事。以下是这两个检验方法：

你的理论是不是仅仅证明了一个旧定理？
关于一些基本例子，你的理论能不能说出一些以前没有的东西（比如观点，认识）？
这是个很方便的检验方法。我想说的是有很多当下流行的数学都不满足这些。几百页的论文，我想说…很多人都在3-流形上定义了很多东西，我不做3-流形，但我知道Thurston对此的方法，他的方法解决了几乎所有的问题，大量的问题。而人们总是在定义花哨的不变量。去年我参加了一个报告，一个人定义了一个非常花哨的不变量，有点让人讨厌。我思考着，坐下来：这东西难道不是说…人们总是单单把花哨的东西往上贴，他说：“是的，这些不变量可以分类所有扭结…”我于是举起手说“你能否给我一个2-knot的例子？”

我给本科生上课的时候…三叶结是一个非平凡的扭结，对吧，为什么？关于此有一套理论，二年级学生就能理解。我跟他说：“你这文章七十页，是个很大的理论，那你的理论能分出三叶结吗？”“不能”… 我的意思是，这不是个孤立的定理，这是个整体。这些花哨的东西，我不明白情况为什么如此极端，如此多的人去做它们，其中还有一些很有名的数学家。我真的很想与之争辩，这实在太极端了。很多学生也被引诱于此。我想说：这到底是什么？我喜欢它吗？也许本科的时候，我会喜欢这个吧，事实上，本科生们总是被诱导着去学习最糟糕的的数学了。我不知道为什么会这样。抱歉…对于这个问题的回答，我的要求很低，你只需要告诉我三叶结是不是平凡的就行了。对吧，二年级的学生也会做。

 

One sign of good mathematics.

It teaches us something new about an idea we thought we knew about

I don’t know the time, but I want to say something I am really excited about. For the last few years I worked a lot on something very very fancy——Moduli space M_g, g is the genus of the Riemann surface, the very fundamental object in mathematics, very very fancy thing, there is something called the Euler Class, the two dimensional Euler Class, and this is really living in, like, second cohomology of the classifying space diffeomorphism of S^1，real coefficient stuff, but anyway, somehow, for this, it gives topological ring, a fanciest most beautiful algebraic geometry, this thing gives rise to all of them. I just realized, recently, in last six months, that this Euler class, is somehow living on the circle, so if you restricted all the way down, to Z/10, a group of rotation is acting on the circle, that is a group, 2-dimensional cohomology class, is just the function P：Z/10× Z/10——>Z. This is not about any great theorem, but I am going to tell you, I realized…You might wonder why I restrict to Z/10? Here is why. This fancy thing that everyone has been talking about, thinking about too. I am not going to ,we restrict to Z/10, is something we learned in the first grade, it is called the carrying cocycle, it takes two numbers a and b, a pair（a,b）, and it sends to 1, if a+b is greater than 9, otherwise maps to 0, something Z/10 is a digit from 0 to 9, that is a cocycle, so it gives 1 if a+b is greater than 9, and 0 if a+b is less than or equal to 9. Then, I look back I can see the carrying, I can see what I learned in the first grade, in the Euler class. And so to me, I am not doing mathematics here, I am just try to understand well known easy things. But I love this, I knew that was definitely Euler Class, because it told me something about carrying, really, about addition. And it reflecting all the way back to addition.

Here is something that we all forget, you know that math we learned in high school, the stuff what you are doing now even the Princeton professors, did you know they are the same thing? We forget this because we do cohomology of moduli space, but really good mathematics by best people, like Sullivan, like Serre, and Milnor, their math reflects back, on calculus, on linear algebra. They think much more simply than me think which is why they are much better than us. But anyway, to me, that’s the sign of good math. What is that thing reflecting on? So I got really excited about this, and I thought, oh, I saw Euler Class is really something. That is an invariant of circle bundles over all manifolds, you can see carrying, anyway,

一个好的数学的标志

这一理论让我们对已有的东西有新的理解

我不知道现在什么时间，但是，我想讲一些让我非常兴奋的东西。这几年，我一直在演讲模空间M_g，非常玄妙的东西，其中g是Riemann曲面的亏格。这个东西在数学中很基本，也很fancy。有个东西叫Euler Class，它属于S^1的微分同胚群的分类空间的第二个上同调群。它给出了非常fancy的代数几何，这个（Euler Class）完全给出了这些东西。最近吧，六个月以内，我才意识到，这个Euler Class，如果限制到…Z/10，这个群作用在圆周上，二阶同调类是…映射P：Z/10× Z/10——>Z。我不是在讲什么大定理，但是，我想跟你说的是…你可能要我为啥我要考虑Z/10对吧。这是因为…fancy的东西大家都在讲，我就不说了，我就把注意力集中在Z/10，这个东西我们最早在一年级就学过了，这叫做“进位闭上链”，它是取了两个数a，b，组成一个数对儿（a,b），如果a+b大于9，那么就映到1，,a+b小于等于9的话就映到0，Z/10的元素就是指0到9的数字，这是个闭上链。于是，我回过头来，我看到了进位，我可以看到我一年级学到的东西，我也能看到Euler Class。对我来说…在这里我不是在讲数学，我只是想阐释我们都熟悉的容易的东西。但是我喜欢这个，因为它确确实实是Euler Class，它确实告诉我一些关于进位的东西，我也真的见到了Euler Class。它真的回溯到了加法！

 

我们总是忘了一些东西。你是否知道你在高中学的数学，以及你甚至Princeton的教授正在从事的东西，本质上是一回事呢？我们忘了，因为我们光忙着做模空间的上同调了，但是真正大数学家做的真正好的数学，比如Sullivan的，比如Serre的，比如Milnor的，他们的数学总是回溯到初等的东西的，他们的数学能回溯到微积分，回溯到线性代数。他们想的（意思是回溯到的东西）比我的简单的多，这也就是他们比我强的原因。不论如何，对我来说，这是好数学的一个标志。想想这个东西回溯到了什么？对此我非常兴奋，因为我意识到Euer Class这个东西真了不起。作为流形上圆丛的不变量，你可以看到进位。

OK. Finally, I would explain two quick examples of good theorems. Maybe we all know because I know there might be some undergraduate student.

Example：Generalized Stokes Theorem

How <wbr>to <wbr>do <wbr>Mathematics <wbr>by <wbr>Benson <wbr>Farb

This is really well known, theory of differential forms, why it is a great theorem. It’s completely general,

Both sides are computable.
You can use it, it’s interesting in every example.

Implies three great theorems that were sort of we viewed differently（Gauss, Green, Stokes）
That is a big example, there are so many things I can say about that.

Example:Topological invariance of Euler Characteristic

It is a general theorem that is rich and surprising in every single example（like disk, and for the 2-sphere, and for Calabi-Yau Manifold）.
Keep these things in mind.

If you are not doing something that’s not like them

2. Connects and relates to so many different areas of mathematics

3. Computable

And something you can compute. That is other things that just dumfounds me.

People make these definitions, you can make infinitely many definitions, if you can’t compute anything, this is really worthless. Sorry, I feel compel to say this because such a huge chunk of mathematics is like this. Smart people are doing this. This of course leads us to our fourth point which is

Leads to deep and important refinements（e.g Homology theory）
最后，我打算快速地举上两个好的例子。考虑到这里有一些本科生，我打算讲一些大家都知道的。

e.g. 广义Stokes公式

这是个关于微分形式的非常有名的定理，为什么它伟大呢？因为它真的非常一般。

1.两边都是可以计算的

你可以使用它，这在每个例子中都是很有趣的。

2． 它蕴含了三个伟大的定理，这三个定理我们之前在不同的背景中已经见过了（Gauss, Green, Stokes）。

下面的例子值得说很多。

e.g． Euler特征的拓扑不变量

1. 这是个伟大的定理，它在每个例子中都有丰富的内容（圆盘，2维球面，Calabi-Yau流形）

2. 它联系了数学中很多不同的分支。

3. 它可以计算

你能计算一些东西。这是另一个让我感到惊讶的。

人们给出定义，事实上，你可以给出无穷多定义，但是如果你不能拿它做点计算，那真的是毫无意义的。抱歉，我真的很想说，因为很多数学都是这样。甚至很多聪明的人也这样干。好，那么第四点是

 

So finally about

B. Work of others

1. Lots of smart people have figured out lots of things, If you ignore these things, you will spend all the time re-inventing the wheel

So, it’s always a hard decision, “I will learn this in my own” "I don't need others", that is stupid, you will never get anywhere, you know what? There are other smart people they can figure out these things too. You need to know what they did, I hate this, because you will be working hard or something and someone will come in with me a great, new idea. Yes, it’s fun to learn it. But sometimes it’s like, oh, I can neither do one of the two things, it’s not there, it’s not there, it’s not there. But it’s there. You have to learn what the other people did.

On the other hand.

2. Having said this, Don't be too respectful, maybe you can do better. Understanding things for yourself.

Anyway, don’t be too respectful, I think it’s good to understand things in your own way. And my favorite book…

Aren't 1 and 2 contradictory? And here is the best quote, is from Faust

"That which thy fathers have bequeathed to thee, earn it anew if thou wouldst possess it"——Goethe, Faust

I don’t know if this is hard for non-native speakers, cause this is a translation, of course, of German, that is supposed to be like that which your fathers have given to you, like the knowledge of mathematics, like the proofs of the Gauss-Bonnet theorem, all of the things we’ve been given by the great mathematicians who came before us, you have to earn it, as if it were new, so you would clearly possess the knowledge of it. so prove it in your way, if you need the proof of the Gauss-Bonnet, get a few hints from the book, then do it yourself if you can. This to me is a very meaningful book, because the first theorem I worked seriously with Thurston, it gives another proof of Mostow’s rigidity in higher rank, it generalized though, but neither of us really knew Mostow’s conjecture, we have the outline. We use many things about Mostow, but we still don’t know the proof, but I made it my own, I can give you a proof right now. This is really meaningful, and this is the answer to this.

By the way, I learned this from the first page of the book The joy of cooking, it's like recipes. Actually I didn’t, I pretend that I did, just would say that

 

最后说说

B．和他人协作

1. 很多聪明的人已经做了大量的工作，如果你忽略这些，那么你之后做重复前人的工作

做这种决定总是很难的，“我要自己去学”“我不需要别人”，这太愚蠢了，你什么做不到，你知道为什么吗？因为你不这样做，会有很多其他聪明的人这样做。你需要知道他们在做什么。

另一方面

2. 已经说过了，（对别人的工作）不必太毕恭毕敬，说不定你可以做的更好呢。你应该以自己的方式理解问题。

总之，不要太毕恭毕敬，最好以自己的方式理解问题。我最喜欢的书…

1和2矛盾吗？我想最好的回答是下面这句话，它摘自《浮士德》

那些你的父辈们遗赠给你的，如果你想拥有它们，你还是应该重新自己获得。

——歌德《浮士德》

我不知道这句话对于非英语母语者，这会不会很难懂？因为它是从德语翻译来的。就是说，那些你的父辈们遗赠给我们的东西，比如数学知识，比如Gauss-Bonnet定理的证明，所有这些，我们之前的大数学家都已经遗赠给我们了。但你应该凭自己的努力去获得，就好比它们是全新的一样。如果你需要Gauss-Bonnet定理的证明，你从书上获得点提示就好，如果可以的话，最好自己去证明。这本书对我来说很有意义，因为当时我第一次和Thurston做定理，那是Mostow刚性定理高阶情形的另证，当然也推广了一些。但是我们俩都不知道Mostow的猜想，我们只需要纲要。我们确实用了很多Mostow的东西，但我们并不知道他具体怎么证的，但我还是自己完成了，我现在就可以给你证明。这很有意义，这也是上面问题（1和2矛盾吗）的回答。

顺便说下，这一点我也在《厨艺之乐》这本书的第一页学到了，当然我没有读过，我“假装”读过吧…

 

Finally

3. Talk to other people as much as you can, ask experts when you are stuck（ and even you aren't）

I had a very shy student, and working on something and literally the world expert on it is down the hall, I said ”talk to him” Two weeks later, “oh, did you talk to Daniel?””No, I am shy” Here is my answer “Too bad! If you don’t talk to him in the next 24 hours, that’s it. Out of here” If you are shy, my answer to you is “too bad, you have to talk to people, talk to experts when you are stuck”, on a research problem, Some of my student said “Isn’t that sort of like 'cheating’”? You are working on a research problem, asking expert is like cheating, you obviously should try to solve it yourself and work on it ,but when you are stuck what you are going to do? Works great, even elementary representation theory questions , I go write to Drinfeld, anyway,

And my response to that is：Math is hard, “Cheat” if you can

It’s very hard, so you have to “cheat”, one more thing:

Don’t forget to have fun

Actually, I somehow, said that, and I realized that actually think fun is over-rated, that’s not why I do math, I have fun doing math .We all have to do it for some reasons, for me, all life is about growing, it’s not about happiness, I like to be happy, I guess, you have to find your own thing, so, actually I won’t complicate this, having fun is good. So, I stop here, thank you!

最后：

3. 尽量多地和他人讨论，在你遇到瓶颈的时候请假专家（哪怕你没有遇到瓶颈）

我有个非常害羞的学生，他正在做一个问题，然后这个领域的世界做的最好的人来访问，我告诉他“把你的问题和他讨论下”，两周后，我问他“你有没有和他谈？”“没有。”我回答道“太糟了，如果你24小时内再不和他讨论，那你就别做我的学生了！”如果你很害羞，我的回答是“太糟了，你得和人讨论啊，你遇到瓶颈的时候应该请教专家！”我的学生说了，“这算不算一种‘欺骗’呢，你在研究一个问题，问专家就有点欺骗的成分了——很明显，你应该自食其力搞定它”，但是，你真的遇到瓶颈了啊，你该怎么办？对于我，即使我遇到一个简单的表示论的问题，我也会写给Drinfeld请教他。

所以，我的回答是：数学很难，如果可以的话，那就“欺骗”吧~

因为它很难，所以我说你可以去“欺骗”，再说一件事：

别忘了要过得开心！

我想说，纯粹为了开心有点过分了，（纯粹为了开心）那不是我做数学的原因，当然，做数学确实让我快乐。我们每个人做数学都有原因，对我来说，一切生活都是和成长相关的，并非为了快乐，虽然我愿意变得快乐。你得找到自己做数学的理由。我不打算把它讲得更复杂了，总之，快乐是好事啊。我就讲到这里，谢谢！

任金波整理