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

Speaker: Prof. Beson Farb (Chicago University)

Time: 1 July, 19:00-20:30

Place: 110 Hall of Morningside Center of Mathematics


Beson Farb简介

· 他是芝加哥大学最受欢迎的教授之一
· 他对什么是好的数学、什么是不好的数学以及怎么做数学都有很多独到的见解
· 他是Thurston(几何化猜想的提出者)比较杰出和特别的学生
· 他已经指导了29个博士生


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!



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. 

 评:husband and wife一句,有两种解释。第一,老婆不喜欢可以换,数学不能换,因此数学方向选择更重要。但做数学可以quit,不合理。第二种解释,老婆会跑掉,数学却不会。更符合逻辑。

  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.

 评:张五常所说的,有时要“停一停,想一想”。何谓Process? 大概是”知所先后“的意思。

  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.



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. 

 张五常的老师,最近去世的艾智仁(Armen Alchina)说过:价格决定什么,比价格由什么决定更重要。对\Gamma的执迷,原因有点无厘头,但它有效果。


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.



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



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.



 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.




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

 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.


 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.



读数博的,智力不会是问题;在美国的制度下,你不干活3年也没事,即没有迫切的压力,从而亦不会生厌;往往开始工作的时候,你会喜欢的。但总体而言,要每天坚持工作很难,你会run away,为什么呢?因为数学很难。



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.  

 这段说了很多招法,其实就是一条:用acceptance对抗”逃避“。你连everyday working都做不到,反而要"live, sleep, breath in math", 怎么可能?就是因为困难是emotional 的,所以才有希望可行。


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. ^

评: 最后一个词是强迫症的意思。重点是self motivated. 



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…


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.


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.




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,(手指低位) I think minimum to be a reasonable mathematician you need to be here(手指高位), 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. 





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.


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. 


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.


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:

  1. 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.



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.



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.


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?



首先涉及一个做事的方法:梅花桩。如果我不懂曲线的曲率,搞其他东西有什么意思?如果我懂了曲线的曲率,对其他东西也是有帮助的。所谓it is just you and that paper,是说所有理论到最后都没有用,关键是你做得动。张五常评价中国经济:一个人,跳高,没学过,没参加过专业训练,跳得很难看,但一跳居然跳了八丈!你总是要解释这个现象的。至于你不懂深奥的东西显得很cheap,伤害到ego,这不重要,做出问题才是王道。

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.


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.



  1. 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

  1. 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,

  1. 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.


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. 



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

  1. 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?

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.


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.



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.


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,


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 

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

  1. Both sides are computable.

You can use it, it’s interesting in every example.

  1. 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

  1. 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

  1. Leads to deep and important refinements (e.g Homology theory)


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




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!





WordPress.com Logo

您的留言將使用 WordPress.com 帳號。 登出 / 變更 )

Twitter picture

您的留言將使用 Twitter 帳號。 登出 / 變更 )


您的留言將使用 Facebook 帳號。 登出 / 變更 )

Google+ photo

您的留言將使用 Google+ 帳號。 登出 / 變更 )

連結到 %s

%d 位部落客按了讚: