做基礎數學研究的人是怎樣的研究模式呢？
做基礎數學研究的人是怎樣的研究模式呢？比如做實驗，編程啊，和普通的理工科有什麼區別呢？
6月30日更新：
後記----我的研究方式是非主流的，現在大部分人的做法大概就是原答案第一段。嫌我啰嗦的直接看原答案的第一段，以及這篇文章的倒數第二句。如果真的碰到困境，也可以參考這個答案，以及我的第三個答案和第二個答案，或許能找到破困的辦法。前不久參加了一個summer school，果然沒人像我這樣做數學。。。坦誠地說，我自己的方式也有優點，只是回報很慢，看人有沒有耐心了。最重要的是我的方向是PDE和幾何測度論，這個雖然也有很多有意思的問題和困難的問題，但畢竟不像代數幾何、數論這種學科，要積累好多本專著才能做問題，所以感覺好羞愧啊，所以這個答案就聽著玩玩吧。
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更新：有同學私信問到怎麼從baby習題中找前沿問題，以及如何知道自己做的別人有沒有做過。我在私信里分享了我的觀點，不一定對。這裡貼出來供大家參照：我發送給（隱去其名字）：您好，
我不敢夸夸其談，舉一舉我在研究中碰到的例子吧。因為我做分析，和您做的方向不一樣，所以可能我能這麼干。
我說一個大家都能聽得懂得baby習題：如果g是L^p函數，那麼由Holder 不等式， 對任何集合E， 有int_E g le C |E|^{1/q}. 我當時就想，這是不是一個充要條件？於是我想了很久，終於證明充分必要條件是g是weak L^p的。我當時很高興，結果發現這個結果其實是一本調和分析上面的習題。可是我的研究沒有止步，因為我學過Morrey space， 它的定義是int_Q g le C |Q|^{1/q}，對所有cube Q成立（而不是對一般集合E)。我於是想，這個空間是不是和weak L^p空間等價呢？我想了好久，從一維開始，後來到高維，我覺得這是個凸幾何問題，就自學了很多凸幾何，通過谷歌搜索，看了很多paper，終於我證明了不僅它們不等價，而且Morrey space不能嵌入到任何L^r空間中，即使r遠小於p。這個結果我當時很高興，結果後來又發現在2010年被人研究了，並用另一種方法證明。然而我當時不知道，我一直在研究，比如臨界情況，p=n，這時注意|Q|^{(n-1)/n}等價於Q的surface area，我用P(Q)來表示。於是我研究這樣frac{int_E g}{P(E)}的Minimizer存在性問題，我引用了Morrey space，它啟發我研究凸集合，我就發現了一個covering theorem, 聯繫到了boxing inequality，然後證明了存在性。再到後來這種感覺使我發現了minimizer的結構：它一定是pesudoconvex（在二維一定是convex的），並進一步證明了正則性：如果g是M^p，p&>n, (比L^p還弱很多），那麼minimizer是C^{1, alpha}。這個問題實際上和variational mean curvature問題是差不多的，但是多了很好的結構比如pseudoconvex。這個叫generalized Cheever set problem.
您可以看到，整個這個問題是由一個Baby的習題引發的。雖然我在自己研究過程中，重複發現了很多別人的結果（我當時不知道），但我有自己的東西，就是和凸幾何和Morrey space掛鉤，從而發現了minimizer的結構和正則性，這是任何人沒有研究過的。2010年那篇文章給出的證明不是用凸幾何，而我的證明是用凸幾何。而且因為自己封閉的研究，使我對這個領域有和別人不一樣的看法。
您試想：假如我一開始知道那道習題，假如我知道2010年的文章，我可能就不會做下去了，就不會發現後面的結果。所以，一個教授跟我說過：sometimes it"s better you don"t know what others are doing。您想想是不是？數學充滿了無窮的可能性，有無窮的道路。
我有足夠的耐心，所以我不怕和別人做重複，因為我覺得只要我認真做了，一定能做出別人看不見的東西。我在我的第二篇答案里寫過我對ODE那個問題的看法，儘管大數學家做了很多工作，但是我也有我的看法，我認為PDE和ODE結合可能可以證明Bressan猜想，並且我從Ambrosio的文章中得到了啟發。這種啟發式模糊的，我不能夸夸其談。 我會繼續做下去，並且我相信我可能做到他們可能做不到的東西。
其實我覺得做數學就是積累的過程，每個人只要在自己感興趣的方向付出足夠的努力，不要浮躁，那麼總能發現自己的東西。
至於如何知道自己做的別人做過沒有，我一般不關心。但是如果要關心，我會利用google搜索，或者問一問這個領域的大人物。
抱歉回復得比較長。我的能力並沒有我表現的這麼強，因為很多東西是時間磨出來的。比如我下功夫一個星期，可能並沒有走幾步，但是經年累月，是可以得到一些結果的，哪怕是廢結果，都為將來研究有幫助。我從小到大，老師都說我很笨的，他們都說我腦子不夠用，我總是他們眼裡的笨學生。聰明人很多，比我專註的也大有人在，他們沒時間寫答案，我只是恰巧過來而已，有點羞恥呢。
我剛剛跟您說的例子，已經寫好論文，但是還沒有投。我繼續壓著呢，因為我可以繼續做下去。到畢業前，我才會發出去。
我對拓撲不太懂了。我本科畢業論文做的是Koszul complex的上同調。經典的方法是Poincare duality，我當時覺得那個問題既然是僅用群論語言提出的，那麼理當能那麼做。所以我也是花了很久用類似證明Jordan canonical form的方法證明的，做了很多商空間。這是我對拓撲僅有的記憶，其他都忘了。拓撲給我的感覺，是它是一門演繹太強的學科，我後來不做拓撲，就是因為我更喜歡一些理論聯繫實際的問題，更喜歡用簡單的語言去挑戰思維極限做出的東西，這是我自己的偏執吧。所以我學了PDE，學了分析。
閑話不敘。
共勉。
昨天 23:37共 4 條對話 | 回復 | 刪除------------------------------------以下原答案-------------------------------------------
可以參考下面兩篇文章：http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf 這個是菲爾茲獎得主Tim Gowers整理的：Advice to a young mathematician。http://www.ams.org/notices/201009/rtx100901132p.pdf 這個是Schulz教授整理的，在當今數學研究已經變成一個職業後，如何抱團生存的策略。
更多的內容可以在陶哲軒的博客上找到，也可以多逛逛mathoverflow，上面還是有很多軟問題的。不過私以為看這些必須得變成自己的才行，否則就夸夸其談了，反而搞得自己不能專註，用力過猛。一切順其自然。
學術畢竟還是學術，玩其他的都是虛的。
----------------------------------------------------------------------------------------------------------下面我分享我自己的研究模式，因為我覺得每個人只要認真做了，都會形成自己的習慣。拋磚引玉吧。第一步，夯實根基，系統讀一些經典的研究生教材，要做習題，哪怕不做，也要想一想。要反覆讀的，不太可能一次就理解到位。
第二步：認真讀一些十分經典的文章，進一步提高基礎、品味、感覺。這個過程很慢很痛苦，但很值。厚積薄發。第三步：多找教授討論，因為第一他們可以把抽象的數學寫作語言轉化成人可以理解的語言；第二他們隨便比劃幾分鐘，可能可以省下你自己讀paper幾個月。第四步：海量閱讀其他文獻（我一般從monograph後面的reference挨個讀），記筆記，分門別類，學會管理學術資源。粗細結合讀。（有了做數學的經驗後，會更快切中要害。有了讀經典文章的底子，後來的灌水文章可以很快讀完。如果只是讀摘要，讀感覺，那麼好的文章也可以很快讀完。記下感覺，以後備用。）註： 在這四步當中，並不一定有絕對的先後順序，有時要停下來重新走。剛開始系統地讀研究生教材不容易。後來哪怕有了底子，讀經典文章始終是不容易的，因為那畢竟是天才提煉後的東西。這一年我基本封口不說閑話，也就沉澱下出下面一點點體會：
學習吸收速率要和消化速率要差不多（主動思考，安於所學），才能專註，才能不消耗自己的意志力，進行持久的閱讀。 （然而要忘掉自己在專註，理解東西是一種享受。）如果讀太困難的東西，寧可走神玩手機，也不能強迫自己坐下來。勞逸集合。
熟能生巧。不要怕動手浪費時間。博士五年是一個很長的時間，只要不浪費，足夠做許多許多事。一年就可以做很多很多事。有了前四步，會自然產生許多研究的問題。甚至在第一步做習題，只要好奇，不斷發問，baby的問題也可以引出十分重要的數學問題。而在讀文章過程中，更加會自然而然問出更多看似「有價值」的問題，然後坐下來推一推想一想，大多數情況自己推出來的東西，前人都做過，甚至遠遠比自己做的系統、深入，但總有一些是自己獨特的，任何人都沒有做過的。這些獨特的所得可能才是決定自己將來能不能做出別人做不出來的東西。（像我這種笨蛋，也能發現一些有意思的東西呢。）
第五步：開始進入研究了，要進行大量的嘗試。常常有意想不到的情況發生。譬如我往往預料不到自己做一個東西會做成另一個東西，也往往想不到居然會在一個看似平凡的地方卡個一兩年，而看似不可能證明的東西可能睡個好覺就證出來了。一言難盡。
上面的套路適合自己獨立讀文章、找問題。如果是adviser給的問題，那就是另外一回事了。基本上老闆讓你如何，你如何，就這樣。有的老闆控制欲太強，博士畢業好多年都逃離不了老闆的方向，我只能說這是過度依賴了。
-------------------------------------------------------------------------------------------------------我主要想分享第六步。就是在文章已經寫出來後，切磋磨光的過程。這個過程對我來說，很痛苦，是道義上的痛苦。我是一個常常較真的人，比如我覺得文章里某個東西我不滿意，雖然整體上文章一定可以投出去了，但是我會壓著，因為有些東西可以繼續做下去，或者有些證明可以簡化。 這個過程之所以痛苦，是當你發現簡化的證明可能使得你前面寫了好幾頁的東西1頁就夠了，給審稿人一種「平凡」的感覺，可能文章就投不出去了。舉個簡單的栗子吧。我以前在考慮給set of finite perimeter上定義trace的問題。十分不光滑的集合很難定義trace，所以第一步要做的工作就是，從外部或者內部用光滑集合逼近不光滑集合。我當時考慮的是用磨光後的distance function。在高維很複雜，迄今為止我也不知道能不能做。但在二維，我經過了大量的考慮後，分成了好多種情況，最後歸結到了證明下面的微積分問題：
在就證這麼個小東西，我也花了數天時間的，因為算起來很麻煩，一不小心就算錯了，要用到很多三角函數的技巧，求了三次導才證明出來的。（不信大家可以試試，這算一個比較難的微積分問題吧，至少對我而言。）我那時還不知道wolfram alpha，就算知道，最多幫助我確定我的猜測對不對，證明還是要手算的。
當時證明集合的逼近，用了10頁，再加上以此為基礎定義trace，又寫了5頁。於是這篇15頁的文章是可以投的。平心而論，這文章沒質量，但是很有意思，不算灌水，因為有足夠原創的內容。 我只是壓了壓，沒投，因為我對這種證明不滿意，覺得應該可以簡化。直到這個假期回國，因為被牆了，我什麼都做不了，於是就坐下來重新考慮以前的問題以打發時間。 在一個清晨，我突然想明白那其實是一個平面幾何問題，作幾條輔助線就把引入的變數都消掉了，用初中平面幾何的語言，1頁紙就把以前10頁的東西寫下了。 於是，一個看起來還有意思的問題，經過平面幾何的初等技巧，使得這個問題瞬間變成初中數學問題，於是這篇文章大概作廢了。 我還有一篇文章，也是本來不錯，壓得久了，突然發現簡化的辦法後，本來可以投到好的雜誌，後來只能投到中等雜誌了。這都沒啥，魚與熊掌不可兼得，為了學術誠實，只能犧牲灌水了。我不知道將來會不會因為找教職的壓力，開始降低標準去灌水。最近真的很抑鬱，寫的答案多了點。
上面的最好都忘了，自己正心誠意做下去就是了。窮則變，變則通。
端午節快樂！Benson Farb：晨興通俗報告How to do Mathematics文稿(z)
晨興通俗報告How to do Mathematics文稿
（任金波整理，歡迎糾錯）以下是我整理並翻譯成漢語的，本人才疏學淺，有些地方實在沒聽懂，其餘部分難免也有很多錯誤，翻譯的漢語對演講者的意思的傳達也可能有不準確的地方，懇請大家糾錯並不吝賜教！謝謝！
特別鳴謝：Asa，Ray，諸子越同學，胡曉文師兄，張漢雄師兄，我萬分感謝他們對我的幫助！！！
How &
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簡介
· 他是芝加哥大學最受歡迎的教授之一
· 他對什麼是好的數學、什麼是不好的數學以及怎麼做數學都有很多獨到的見解· 他是Thurston（幾何化猜想的提出者）比較傑出和特別的學生
· 他已經指導了29個博士生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!
季理真：歡迎來到「現代數學概觀」這一國際會議的特別講座《如何從事數學研究》。我相信你們都是熱愛數學的，這也是為什麼你們來這裡的原因。我也相信你們都想成功。首先，你要選擇一個方向。在生活中，你需要找很可能共度一生的伴侶，我相信大家都會很慎重地選擇。在數學中也是這樣，你需要找一個好方向。但是，即使選擇了方向，你還是要做數學呀，那麼，問題在於「怎樣做數學」？Farb教授思考這樣的問題已經很長時間了，他有足夠的資格給出評論。就像你在海報上看到的，他是芝加哥最受歡迎的教授之一，他已經培養了29個博士生，此外，他還是Thurston最特別和最出色的的學生之一。在他的講座中，也許你可以感受到Thurston的風格。今晚，他將告訴我們，怎樣研究數學。他還會講講什麼是好的數學，什麼是壞的數學。研究中你需要知道什麼，而不必知道什麼。讓我們歡迎！
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: 非常感謝，感謝大家的到來。理真在48小時前說服我要做這個報告，所以這些幻燈片是我在48小時內做出的。理真鼓勵我能夠在這裡面對所有的學生。對理真的話先做個簡短的評註：配偶是丈夫或妻子，他們可以來，也可以離開，但是數學卻總是在這裡的。不論如何，表述觀點的時候很難表現地不傻，所以，我想，標題最好改為「一個帶有偏見的人的一些建議」，這個人當然是我，這才是我主要想講述的。
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: 現在我要開始講了，不過，首先，第一點是很明顯的
斷言1：每個數學家要做的第一件事就是找一條她自己的道路去走。
不過，聽別人走的路也可以是有用的，並且事實上，我們在學生階段總是忘記了自己到底在做什麼，忘了去想數學本身的過程，忘了去想今晚我們會談到的一些話題：數學很難，花大量時間在數學上好不好？一個人，應該花一點時間想想自己在做什麼，自己要選擇什麼樣的方向，這對於你一生都很重要，想想這個過程吧，去想一想。這是很有用的。
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.
首先我要講主題是的是
作為一個博士生
在數學中，我要簡單提一下數學研究，不是數學。當然，二者區別並不大。我只有48小時時間（去準備），所以可能不是很有條理。好吧，讓我開始吧，第一件事要說的是
A 聽取頂級人士的建議
有很多已經被寫成文的關於怎樣成為好的研究生的建議。在一些頂級數學家，比如Ravil Vakil和Jordan Ellenberg的主頁上，你可以找到很好的建議。我特別喜歡Ravil Vakil給出的如何參加報告的建議：大多數人，聽報告聽了五分鐘後，發現自己不感興趣，然後就干坐在那裡，於是就浪費了55分鐘。Vakil告訴你，其實你不必這樣，有一個「遊戲」：在每個報告中都嘗試找出三個你能學到的東西。然後，你和你的朋友… 總之，這是很好的建議，這些材料你們可以自己去閱讀，我非常鼓勵你們去看看他們是怎麼說的，他們都很能言善辯。
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，我要說的第二點是
B 如何選導師
這是極其重要的，因為它講決定你那持續四年，五年乃至六年的博士生涯，可能還會繼續影響十年，乃至五十年。所以，你得仔細想想這件事。其中，我認為最重要的是
1. 選擇一個真正能激勵你幹活的主題（使得你離不開它們）
也許你並不完全知道你身邊的教授在做什麼，你可能僅僅知道，這個教授做數論，那個教授做偏微（這是不夠的），你要嘗試了解他們的工作，讀讀他們文章的介紹部分。選擇並不容易，因為你要基於並不全面的信息做出決定，而且是很大的決定。對此我的能說的就是：那很不幸，，就是這樣，沒有其他的辦法。但是從根本上講，你將會花大量時間在這種數學上面。舉例來說，我開始讀研究生的時候，我跟的是Wu-Chung Hsiang學K理論，但後來我完全轉換了方向，因為我一直在思考「我的數學哲學是什麼」，然後很多個人原因導致我去做Thurston做的東西。舉例來說，雖然這個例子可能有點傻，我愛作為李群G的離散子群的符號」Γ」，我喜歡這種作為離散群的符號。你可能會覺得傻，但是，你要知道，從那時到現在，我寫下Γ已經不下三十萬次了，對於了解更多Γ的信息，我充滿了熱情。我是有些傻，但是我是認真的，它應該是發自內心的，換句話說，這純粹是感情的作用。我總是跟我的學生說「一定要找一個能在你心中停留足夠長時間的東西」，那麼你將會每晚都會被這個東西包圍，你睡不著的，因為你需要知道答案。如果你不這樣做，你博士畢業的時候一定是失敗的。比起 J.P Serre，我想沒有人是足夠聰明的，也沒有人聰明到不需要整個晚上熬夜（因為你需要知道答案）就能做出偉大的數學工作。總之，這個問題很難一概而論，但是，你一定要嘗試。
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. 你要選擇你導師是其專家的領域
這個相當關鍵。這個問題很多人都很感興趣，事實上，每天在mathoverflow上都有人討論，對此我的朋友Matthew Emerton給出了很好的回答。你應該去看看這些回答。這是我侄子想出來的，他馬上要去數學研究生院了。如果你對某個方向感興趣，比如說數論中的二次型吧，那麼不管你在哪個大學，你不能僅僅學二次型，你需要找個導師帶你，你需要你們學校的教授。原因是，如果一個數學分支很活躍並且有好的數學家的話，它會前進的很快，這個團隊知道很多大家不知道的東西。你的導師應該在這個團隊中。選導師要注意很多，我不打算現在全部展開。你要找一個（學術上）活躍的人，這一點你可以從他的文章和其他很多東西看出來，但是今天我講集中將：導師需要是這個領域的專家，因為不管他們做二次型還是谷山志村，或者微分拓撲，他們知道研究中正在發生什麼。如果你在你的大學中獨身一人，想去跟著（這個領域），你說我會給專家寫郵件，你會讀arvix，你會讀書，你很努力，這些都不夠。你可能會悲劇，因為你可能在做別人已經做過的工作或在研究大家都熟悉的東西。比如，Mark Kisin做他的東西的時候，你（也是這個方向）僅僅還去讀Hartshorne...所以，這個確實是很重要的。這些都會把你的興趣範圍縮小。但願你能找到與上述兩點都符合的的教授，如果沒有，那麼恐怕你得離開你的學校。但是對於大多數人，你不能，這兩點是相容的，你可以找到。
If you have a question, please feel free to stop me at any time.
如果你什麼問題，請隨時打斷。
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.你必須信任你的導師
除非你是百萬中挑一的那種人，你甚至意識不到你是否是那種人。否則，如果你的導師讓你學某種特別的東西，那麼我想說，你可以問為什麼，說出你的見解，但是，如果最終導師堅持讓你學，那麼你必須去學。否則，你無法擁有這個（師生）關係。我可以從導師的角度來講：導師會放棄你的，我們都會放棄你，如果你不去聽我的話，那麼我無法幫助你，我不想浪費我的時間。如果你有什麼爭議，那麼你可以去理論，我也會聽，但是，當一天結束的時候，你還是得完成你的任務，因為我是老闆。有時，你不明白為什麼這個問題很有趣，你有權問導師，也應該問導師，但有時回答將是…你要有更廣闊的的視野，手上的活不能停，最終你會看到它們的聯繫。你要相信這一點
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.
最後一點
4.導師=父母
導師是和你相伴一生的人。就在講座開始五分鐘前，確切的講，在我跨入這扇門和坐在這裡之間，我收到我之前一個學生的郵件，他是十二年前畢業的，現在已經是教授了。這層師生關係會伴隨你眾生，所以，（選導師）是個非常重大的決定。你要慎重。
Here is my advisor. Who I had I a rocky relationship with, but that is a different story.
How &
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.
這是我的導師，我和他的關係很不穩定，當然這是另一個故事了。當時，一言難盡，但我改變的當初的研究方向，我希望Thurston成為我的導師，他答應了。但是他是不怎麼給學生問題的，當年他就給我了一個問題，但是，在很多周的掙扎（無進展）之後，他看了我的問題，告訴我「這其實是個很有名的開放問題。。。」，於是我就不再做他的問題了。他不是很擅長給學生問題，但是，不管怎麼說，我開始做我自己的問題了，這個問題是關於雙曲群的，我告訴他我在做這個問題，他說沒問題，這是個不錯的選擇。他斜著眼，望著天空。當時我想弄懂雙曲幾何中李群的格，他說「哦，這就像一充滿氣泡的泡沫，氣泡間有有界的相互作用」，我當時很單純，我把他說的話都用筆記下來，我寫著「泡沫，氣泡。。。」見完面之後就去圖書館查文獻。是的，我已經準備好了，我開始做我的問題，鉛筆 「泡沫，氣泡」。最初，我根本不知道怎麼處理氣泡，甚至連方程怎麼列都不知道，但是，經過三年的痛苦和煎熬，我最終解決了這個問題。如果你要我給出我的論文的五個詞作為概括，我可以說「泡沫，氣泡，有界作用」，我的論文大體就是基於這些。他告訴我這足夠了，我終於明白他說的了。
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.
現在講
C. 如何工作
我想，如果在報告中講這些東西，那麼將會很有用，因為我有很多博士生。我相信研究生時期的工作是科研中最艱難的階段，至少對我是這樣，我不知道你們是不是這樣。因為我這時對數學了解的最少，但我還得證明定理，現在我懂得更多數學，證明容易多了。所以做研究生是最難的。一般來說，很多問題迎面而來讓你不做數學。但是，我相信你們都是聰明的，否則不會讀博士。誠然，有的博士寫的論文比另外的博士優秀，但是，真正的挑戰是心理上的。你可能會遇到失望，可能會遇到瓶頸，甚至你會想我幹嘛要做數學，我要服務於這個世界。雖然數學也是服務於世界的，當然這是另一個故事。這些困難你都要克服，當然，工作是很難的，沒有人強迫你去工作，事實上，在美國，理論上你從現在開始三年啥也不幹也沒人管。每天工作很難，但一般你開始工作後你會喜歡它，但是你想逃離數學，因為它很難。所以，我覺得困難主要是心理上的。當初我就希望有人告訴我一些這方面的東西，讓我生活容易些。
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.
毫無疑問地，你應該
1. 讓你的生活，呼吸，睡眠都被數學包圍
你不應該做這個（數學）如果你不打算讓你的生活，呼吸和睡眠都被數學包圍。在芝加哥大學這種世界前五名的學校，也甚至有這樣的學生，多年前，有個學生說「哦，夏天到了，我要回家，我準備好去過暑假生活了」——「你不能放假！你在說什麼？你不能這樣」，休息一周或兩周是可以的，但是如果你不願讓自己泡在數學中，那你太傻了——學數學這麼難，這麼艱辛，為什麼不去投資銀行賺更多錢呢？我只是說，既然這麼難，那你還是去掙錢好了。如果你不幹活——我總是對我學生講「我們給你付工資（研究生助教工作），你得工作，大多數人都從早上九點工作到下午五點，這樣每周40個小時，這是底線，如果每周不足40小時，那麼你是在坑你的學校——你拿了錢，但是不干事。」 當然，每周40小時肯定不夠，你得額外花很多小時陪伴你的筆和紙。我想說，如果你只想成為沒什麼價值的數學家，那就每周工作40小時好了。當然，每周40小時也許會讓你僥倖成功，但每周80個小時也許更現實些。相信我，我也知道拖延的毛病，我也知道那些電腦遊戲，這些都可以作為你不做數學的借口，但是，一旦你開始做數學，你就不該這樣。^這是對自己的一個測驗吧，照照鏡子，看看自己究竟在幹什麼並不容易。有一天，我決定了：每次我拿著筆，我（才算）在做數學，不要僅僅把書本放在面前——我在讀，不只是在寫。作為你，只讀書不寫下東西，那幾乎就沒用。我想，你應該一直去讀，但是你也應該去解一些題目，試一些數學中的例子。
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. ^
托馬斯愛迪生說過：天才是1%的靈感和99%的汗水。
我想大多數人都聽說過這個，愛迪生是歷史上最偉大的發明家之一。汗水，那就是努力工作了。這句話在於闡述世上最明顯的真理——努力工作。我想說，在數學中，我有另一個註記，這是我自己說的，來源於我的體會：
天才是1%的靈感，35%的汗水和64%的「強迫症」
當然，這「強迫症」是有用的，它讓你離不開你從事的東西，讓你專註於此。要點是，你不能休（太久）假，你不能做任何（其他）事情，就像精神病一樣，你完全集中精力，你完全在想著它。也許你在看著某個人，點著頭，但是其實你在想數學。我見到的好的數學家都有這種「強迫症」。
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…
下面一條
2. 不要僅僅去讀，要寫下東西——做習題
我很慶幸自己意識到了這一點。當我從Cornell本科畢業後，我去了Princeton，這真是個好地方。那是在1989年，我開始讀Milnor的書《示性類》，於是我坐在河邊，坐在瀑布邊，去讀Milnor的書，這裡很美，很乾凈，到了最後，我學到了很多奇妙的東西——比如從流形的曲率張量構造陳類，「我學到了不可思議的美麗和精妙的數學」，但是，突然，我想到，最簡單的例子，環面吧，它的示性類，它的stiefel–whitney類是什麼？（「我不知道！」），我才意識到我做不了任何事，我啥也不懂。Milnor的書很乾凈，很漂亮，但是我啥也沒寫下來，我以為我什麼都知道了，但我什麼都不知道。我只知道名詞，我只能把它們說出來。我會吹牛，在講台上大講一通，但是，我什麼也做不了。我總是說，一天結束的時候，只有紙和筆，一個便簽本，是的，只有這些，剩下的bullshit都是沒有用的——只有這些是你能用的，只有這些是你能用來證明問題用的。如果我不能用我掌握的陳類在黑板上做什麼，那就完全沒有用。僅僅去讀，那麼你什麼也做不了。僅僅去讀的壞習慣也是很難抗拒的，因為這樣學起來有趣（很輕鬆），你可以學什麼高階K理論，無窮範疇或者拓撲量子場論，但是，你能計算哪些呢？如果你不能在黑板上驗算，那麼…
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 &
我記得我有一個學生，他說「你的微分拓撲的課我想在課上睡覺（不想聽了）」，我說「來我的辦公室吧，我給你一個口試，看看最基本的內容你掌握了沒有。」於是我問他「什麼是Morse函數，請定義Morse函數」，Milnor寫過一本很好的關於Morse理論的書，在這本書的第1頁，講Morse函數之前，它畫了一個很漂亮的插圖，這圖中有個環面，有高度函數，梯度流，和四個臨界點，這是一張很漂亮的圖。我說「好的，告訴我什麼是Morse 函數」，這大概在Milnor的書的第二頁，然後他說：「是的，這個東西很玄妙。。。」我打斷了他「我不需要知道它玄妙不玄妙，我也不想聽它的哲學，請你在黑板上寫出它的定義」，其實，這也就是關於函數的微分的，但是他寫不出。那麼它的價值就是零了，甚至價值是負的。欺騙自己是很容易的。我們欺騙自己，因為我們不希望自己一無是處。所以我才說，我們總是欺騙自己。因為欺騙自己很容易，請注意，它很容易，並且人的天性總是喜歡走容易的路。部分檢驗自己的方式就是做習題，選出一本書吧。我看到你們在座的有人拿著Hartshorne，我鼓勵你們把它的習題全做了，我很高興看到這書已經有些破了，它不像是新的，它像是用過的，我希望你們能夠把Hartshorne的習題都做了。沒有什麼像拿起一本書，做完所有習題一樣，可以讓我們對這個世界感覺這麼棒。這時你的書已經爛了，沾滿了咖啡漬。從來沒有其它感覺和這一樣，這（做習題）是一個簡單易行的檢驗，激勵自己的方法。
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，都花十分鐘照照鏡子看看自己
當然，這只是字面意思，在座各位應該都明白。確切的說，我不需要你真的對著鏡子照自己。我只是希望你認清真實的情況，比如我當初沒有真正在讀Milnor的《示性類》這本書。我只是在浪費時間，浪費自己的生命。這很難，但是你一定要去嘗試——比如問問自己「我能通過切空間將曲面參數化嗎？」下面這句話你可以問問自己「我這周學到了什麼？」我要求我的學生每周給我寫郵件，並且每兩周我會和他們見幾個小時的面。當然，每周，我都允許你說「我什麼也沒做」，我不會朝你喊叫的，我只會讓你有點小不安，如果他兩周都什麼也沒做到，也許他們永遠也不會做到了。我會讓他們意識到「我這一周啥也沒學到」…總之，每周都要去學一個東西。我喜歡這個練習，因為它簡易可行。緊接著，要問問自己「我學到了什麼新工具？」你需要學工具，你不能光是亂寫亂畫，我總是說，你一個月應該學會一個新的，大的，困難的工具，當然，這目標可能有點太高了。最後，還要問問自己「我學了多少個小時？」，再一次的，人們拿我和我的學生開玩笑，關於教授的工作，我們也開玩笑，我不是說一定得怎麼樣，但是，你現在在這裡（用手指著黑板上較低的位置），以你現在進步的速度，你可以達到這裡（用手指指著稍微靠上的位置），但是我需要你達到這個位置（用手指著遠比剛才高的位置），你得做點什麼，你得改變！你要提升你的進步的速度。當然，確切的講，沒有捷徑，沒有捷徑的。再一次，我要說「真正的挑戰在於：要對自己誠實。」要對自己誠實。
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.你能學多少？
在證明定理的時候，有無數次，我都希望定理是對的，但是我發現我迴避了問題，我沒有發掘出最關鍵的東西，這很難。在這裡，我不是自嘲。現在，我要給出關於你能學多少的一個規律。你現在在這裡（手指較低位置），我想，要想成為一個像樣的數學家，你至少得到這裡（手指較高位置），以你現在的學習速度，你不可能達到這個位置。可能你在講座中聽到什麼時，或者讀書時，你會說「我無法把它們都學會。」你說對了，如果你僅僅維持現在的速度，你不可能把它們全掌握。有一件事我得說：我確實見過一些不怎麼有志向的人，有一句話叫「如果你朝著中等水平努力，你就能成功。」所以你需要對你學習能力的觀念進行改變，以下是我給出的性質。你要相信我，這是真的，這絕對是真的，我沒有誇張。
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.
性質：令
X=你認為你能在一個月能學的東西
Y=你實際上一個月能學的東西
則
Y=2X
斷言1在把X換成Y後仍然成立 當然，我想說，這個確實因人而異。有人學東西的速度確實是一成不變得，當然，這些人就不能再繼續做數學了。我想對於很多人，為什麼他們學不到呢..首先，Y=2X你完全可以學你認為你能學到的兩倍的東西，如果你不去試的話…我說「學」，我是指一張紙，一個筆記本，每天用完一百張草稿紙。我有個學生，他說「我沒法再跟你讀了」我說「你只是達到了最低標準，你得改變」，於是我跟他說「這本書，拿去，把所有題目做了」，於是，他後來給了我一個75頁的文檔——他完成了書中的所有習題，是的，他一周就完成了。「哦，天哪，真沒想到，居然做到了！」他開始認為他一定做不到的。我說我確定你能做到。現在，他已經寫了15篇論文了，他是一個很好的數學家。
下面說第二件事，當你完成的時候，你說「我能做到2X了，但也就2X了」，你又錯了，你能學的還可以再加倍。這樣可以一直進行下去。你總能學得更多。
你就把它記牢了。當我再讀研究生的時候——我本科在Cornell讀的——它是個很好的學校，有很多聰明的研究生。聰明的研究生。但Cornell和Princeton的學生區別在於，後者的學生總是嘗試著同時學海量的東西，當然他們不是像我讀Milnor的書那樣僅僅快速翻著書頁，他們是要嘗試著去做Hartshorne的所有習題的，我想說那時很多Cornell的學生是很聰明的，但是當有人提到「我們學這本書吧，並且做它的習題」時，回應總是「天哪，我們不可能做到」，現在回想起來，我們有能力學相當於那時學的五倍的內容，但問題在於我們總是不去嘗試。如果不去嘗試，就不可能達到這個更高的水平。
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.
下面我要講
D.學什麼，怎麼學？
1.信任你的導師，她最懂行
剛才已經基本上談過這個了，你總需要某個人對你的指導，如果你不相信某人，你應該另選他人。別無他法，你一定要信任你的導師。你不能靠你一個人完成學業，你不能。
2. 整體框架和細節都要學
從科學的角度講，這也是對的。我和很多人，甚至是博士後談，我問他「你的論文在做什麼」，他說一堆抽象的東西，「為什麼你要做這個？」「~~我不知道！」那你為什麼對它感興趣呢？你要去學整體框架，即使這不是你的論文的主要任務。在歷史上，Serre和Tate都寫出了很好的博士論文，其他人的論文水平就相對來說比較一般。你處在這個大框架中，這是個巨大的拼圖，當然你做的不必是這個拼圖最大的一塊——這只是你科研的第一步。你要明白大框架是什麼，否則你都無法給你的學生解釋為什麼你喜歡它，我經常和我的學生說起這個，這很重要。我當然知道細節可以使很複雜的，如果你的導師在做minimum model program，天哪，這其中有太多的數學，這非常難，你在做非常專的工作，但你還要學習大的框架，這可能不會很有趣，但是不這樣的話，如果你害怕大框架的話，你就無法證明大定理，並且，你永遠無法和他人交流你在做什麼，而這正是你需要做的，尤其是在年輕的時候，更是需要交流。因為你需要找工作，而交流就是最重要的一部分。當然了，僅僅研究每個細節是不足以讓你了解整個大框架的。一個人不和別人合作的話，是不會有大出息的。再講另一個學生常犯的錯誤，我以前有個學生，他的名字我還記得，我說「Justin，你試試算下這個，我想可以這樣算」，然後學生說「我兩周後來見您」，兩周後，學生回來了，說我的方法行不通，並解釋了一下。但是，解釋我的方法行不通最多只需30秒。我於是說「發現這個最多花了你5分鐘時間吧，你有沒有嘗試其它方法？」「沒有。」「那你這兩周都幹什麼去了？！」…我想說的是，數學很難，我有的學生過於迷戀細節，而有的人如果一次不行，就不願再試。我有一次被一個問題掛住了，我花了一周時間去想不可定向的情形，因為文章沒有說可定向，可能這根本無所謂的，寫文章的傢伙估計是忘了，你認為這個細節會有多要緊呢？這一點很重要，如果你要證一個篇幅十頁，甚至三十頁，四十頁，五十頁，六十頁的定理，你一定要有一個整體框架。你要能說出：「這其中有五個主要步驟，第一步是…」，你需要把它們拆分。
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.當然，目前來說，你還是需要檢驗每個細節，但是
4. 記住每個細節太難，要掌握主要思想
你要檢驗細節，但很難全記住它們。寫下證明大意是學數學的好方法。我讀研究生的時候，寫下了海量數學定理的證明大意，當然了，只是寫主要思想。
例子：
坐標變換
我們都見過，當我們面對一個困難的問題的時候，我們要麼變換基，要麼變換坐標，或者用Mobius變換，其中最有名的是…我忘了定理了，但我記得思想，但是沒關係了，因為思想更重要。我只記得是先畫一個圓，然後呢？再畫一個圓和它相切，然後（再一個）…也許和兩個都相切。然後關於它有個定理，我忘了定理了，我不在乎，我只記得它玩的什麼把戲。外面的圓記為l，裡面的圓記為l』，用一個Mobius變換…這兩個圓切於一點。在黎曼球面上…我可以用一個Mobius變換，把l變成這個，把l』變成這個（直線）。然後…我忘了定理了，但這已經顯然了。另一件事是，你要記住：全純映射趨於讓距離不增
當我說「趨於」，我是指它取決於映射從哪裡映到哪裡，這並不總是真的，你要了解這一點。這東西我最近才意識到，並且開始使用它，雖然我不是這方面的專家。我知道這個思想，並且我需要一些東西它是讓距離不增的。哦，等等，如果我能讓它是全純的，那就對了，這個很有用。你知道我要說什麼對吧。當然，你需要知道每個細節在講什麼，但是，我現在就可以給你重新敘述出來。所以，我認為記住核心思想是學習數學的好方法，比如，你現在要讓我證明Thurston的hyperbolization theorem，我可以列出提綱，甚至下一級。就像網頁一樣，比如我想了解雙曲結構，你可以點開五個步驟中的任何一個，你想知道Mostow』s Rigidity, 那就是另外五個步驟了。如果你對其中任何一個感興趣，你可以點開它，我可以給你講這五個步驟，甚至下一級。當然，現在，我記不得所有細節。這（寫提綱）很重要，因為細節太多了，（全記住）你會被壓垮的。
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.
第四點
要研究基本的例子。你需要：
了解基本的例子勝過你了解你的女朋友/男朋友 當然，在座的可能有人已經結婚了，那麼你不該有男朋友或女朋友了~我想說的是，現在就開始花時間研究基本的例子吧，我是指非常基本的。我的博士論文是關於負曲率流形的，低維…雙曲流形，秘密是什麼呢？別告訴別人啊~我對去掉一個點的環面的雙曲結構超級了解！當我弄懂這個的時候，一般情形也就迎刃而解了。當然（其他情形）會有一些不同，但是這仍然很有用。我曾花了大量時間去畫和這個曲面相關的圖。例子越簡單越好，我想說的是…當我到Chicago的時候，我就致力於把Thurston的一些思想用於了解半單李群中的很多東西，Chicago是世界上這方面的中心，還有super-rigidity和其他玄妙的東西。我和哪裡的學生和博後去交流，然後我們討論各種深奧的數學，這些深奧的數學都是我想去學的。比如與半單李群相關的，這裡面包含著大量的東西。但是我發現他們（對一些東西）沒感覺，三維李群，這是很初等的東西，但是我非常致力於這些。我的研究均基於此。至於結構，我可以以後再學，但是我終究學會了。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．手中要有一大堆東西（解決問題的基本元素），驗證每一項不滿足它的敘述
Richard Feynman，我從他這裡學到的，他是前一個世紀最偉大的科學家。他說，他總是學例子，他手邊總有一大堆例子，然後檢驗與之對立的敘述。他說…他總是讓人印象深刻，因為別人總是說：來，我們來取一個有全純向量場的Kahler流形…但是Feynman總是找簡單的驚人的例子：環面。他也總說「不，這裡有個反例的」「哇塞，太不可思議了」——其實不是的，這就在他手邊。這完全是第一位的（檢查反例！），這也是個很大的陷阱。我完全感受到了這一點。你知道嗎，我嘗試著從這裡走到那裡，並且到處都是地雷，如果我踩中了一個，那麼我死定了，另外，如果我工作地不夠，我也死定了。就是這麼回事兒，這也是很棒的一件事。
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．沒有例子太容易了。不要對自己的自負過於擔心。
當我是學生的時候，我換了方向，哦，天哪，我完全不懂幾何。我的論文是關於幾何的，微分幾何的。這就是我為啥這樣說——我得從平面曲線開始學起。曲線的曲率，叫做二階導（做出說悄悄話的樣子，意思是當時自己很弱，連這都不懂），然後我就一直和它打交道，然後我的Princeton的同學就取笑我：「我們都在搞模空間，你居然（還在搞這麼初等的東西）…你在搞笑嗎？」但是，不知為什麼，我沒有讓這些話影響我，我很自負。那些同學的話觸碰了我的自尊。他們都在做時髦的東西，但是到了最後，要知道，雖然當時有很多優秀的對手，但我還是第一個證明了這個定理。自負是一回事，但是我真的對曲率非常了解，如果不懂平面上曲線的曲率…你在寫關於曲率的論文，關於Kahler流形的雙全純截面曲率，如果你不知道平面曲線的曲率，你能發現什麼呢？你能做什麼呢？你除了玩符號，啥也幹不了。你無法證明定理的。
在一天結束的時候，只有你，一張紙，沒有隱藏之處（很不幸，雖然我希望有）。（意思是，你要是沒弄懂，那誰也騙不了）
你的理解…只有你和紙…如果你不理解平面的曲線，你在自欺欺人，你什麼也證明不了。當然，你可以暫時把它敷衍過去，但是，你還是得證明定理啊。那麼這時候，你的自負有什麼意義？（意思是，你想自負，可以，但基本的東西得弄懂）
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，數學很乾凈，你希望你做的數學是完全乾凈的。但是，你要知道，我們都熱愛數學，但是你不該感到糟糕，你應該關注一下別人，這是人的天性（？？）。首先你要學會的是：
盡量多地向其他同學學習
所有人中，我向一個比我大一屆的研究生學地東西最多。我不停地提高，我總是和他在討論，並且我也教給他一些東西，不過我不記得教了哪些…但是你的教授們，你不可能總是和他們在一起。但是你的同學就不一樣了，你可以和他們每天都討論甚至六小時，你可以向他們學習。當然，有的情況很棘手，導師有兩個學生，導師認為另一個學生好（比你可以請教的）——太糟了，其實你還是可以學的，這是非常好的資源。總之，向其他同學學習。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.
模仿大數學家的風格，但也要發展自己的風格
（暫時）沒有風格不是錯。如果你只是一年級的學生，當然你沒有自己的風格了，因為你尚且不懂數學，尚且不懂科研方法，你是從零開始的。我想，最快的方法是選定一個大數學家，模仿他。當然對於這些大數學家，你不應該僅僅繼承他們，成為他們的小小的複製品。我想，模仿他們（大數學家）的風格是非常好的，學他們的風格，他們做的數學.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. 不管你比別人差多少，也不要擔心（這是本文最出彩的地方）
這一點很難做到。我有個學生，他很小的時候就成了我的學生，很明顯地我能覺察到…在同一時間，我還有七個學生，他們都情不自禁地覺得這個小傢伙比他們強太多了。我能看到他們的自信心受打擊了。對此，我能說的不多，除了：
a．你無資格評價你自己（也許也無資格評價他人）
也許，你不像你想像中那樣壞，也不像想像中那麼好。當然，有的人確實相當好。當然，你是人，你要注意…這並不是…你得接受它。你得…我想說：
b． 對此，你無能為力
所以，你最好適應它，學會以這樣的方式生存。我去Princeton最大的收穫就是…我本科的時候，我獲得了「最佳本科生」的榮譽，我暗暗覺得「我真了不起呢！」。但是，當我到了Princeton，一天之內，我就發現我簡直糟透了，我跟他們比起來啥也不是，這些人的級別實在是…後來成為了Thurston的學生，我感到之前一切的光環都被剝奪了。這段時間確實非常艱難，但我還是克服了這些困難。你知道事實如何嗎？
c． 世上總會有人比你強好幾個檔次。現在就適應它吧。
對，就是現在！事實上，很有趣的是，即使是菲爾茲獎得主，也能決出個世界前五，他們還可以和他們的前人進行比較，這種比較是無休止的。所以，我現在要給你們講個故事，你們知道Mozart和Salieri嗎？你知道Salieri是誰嗎？有個電影叫《莫扎特傳》，是關於Mozart的：Mozart是天才，Salieri相比要平庸的多，但是他也足夠優秀，因為只有他能明白Mozart有多麼地天才。每次他見到Mozart，他就很痛苦，因為相比之下他只是庸才。所以，我的父母…當我的事業開始起步的時候，我做地還不錯，我的父母就很興奮地對我說：「Benson，你準備好拿菲爾茲獎了嗎？」「不不不…這個圈子裡面有一票人呢，我不是最好的之一，跟最好的人比起來我只是個笨蛋，但是…」我說道「我是Salieri，至少我能欣賞最優秀的人的工作，我不是他們的一份子，但我至少能欣賞。」然後，有好幾年，我都這樣告訴自己「我是Salieri」，感覺也還不錯呢。
但是後來，我參加了Thurston的生日晚會，（在場的人中）最出色的數學家之一是Dennis Sullivan，他站起來說：「我之前覺得自己是世上最出色的的拓撲學家，當時我是Berkeley的教授，我以為我是Mozart，然後有一次，我做一個報告，一個研究生舉起手來，說他有個反例」，於是他把這個研究生帶上講台，然後他開始畫一些圖…「我的下巴快掉了（驚呆了），我知道了兩件事，第一件事是這是我接觸的第一個純幾何的論述，第二，我意識到自己不是Mozart，他才是，那個人就是…Thurston！」，當然，Dennis說的第二件事暗示我，我不是Salieri…我把這件事跟一群人講過，就跟今天這樣，有人就笑了，我問他為什麼笑，他說「Benson，你知道嗎？我一直以為我是Salieri，我現在聽了Dennis說了這些，我意識到我不是Salieri」，然後，我（Benson）進一步意識到我甚至連能欣賞Salieri的人都算不上。這個故事今天就不繼續講了。總之，人的檔次是分了太多層的。你要習慣它。
好消息是：你仍然可以探索出許多東西，做出貢獻
那些大牛們也是人，他們不可能把所有問題都做了。所以這是個好消息：你仍然可以探索出許多東西，做出貢獻。我聽到有些人說：「我不是最優秀的，我做不到這個，我需要離開數學嗎？」我的回答是：「你知道嗎，你確實不是最優秀的，我可以跟你明說，但是，在這裡沒有人是最優秀的。」我不知道誰才是。
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
我覺得數學品味是特別重要的，你要發展出自己的品味，也就是什麼是好的數學，什麼是壞的數學。當然，這是由你決定的。沒有書上告訴你什麼是正確答案，我想說「沒有正確答案的」。但是，它（好的數學）就在那裡啊，我不能定義（什麼是好的數學），但是就像Thurgood Marshall那樣，這人是美國著名法官，當時他正在處理關於色情圖片的案子。你知道這個詞吧？色情圖片？…Thurgood Marshall和律師說：「請定義它（色情圖片）」，然後他又說：「我無法定義它，但是它（色情圖片）就在眼前啊！」數學也是這樣。當然，你可以和偉大的人學習他們的品味。
1. 發展自己的品味，但也應當受到偉大的人的影響
發展好的品味最簡單的方法就是接受偉大數學家的影響，然後發展自己的風格。以下列出一些偉大的人：Serre, Milnor, Thurston, McMullen…他們寫的東西都很好，他們的數學我認為那就都是好的數學。當然，這絕對是帶有偏見的，因為我的興趣在幾何跟拓撲上…Serre也是，他們都特別善於寫東西。他們的數學我想我都可以定義為好的數學。
2. 一旦你開始做一個問題，你就應該問問自己：
「這個問題是否有意思？」「這個問題在整體大框架中扮演什麼角色？」
最後，關於我想說的關於如果做研究生的總結是
總結：（數學）這條路很難，這條路毫無疑問非常非常難。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難
對，每件值得去做的事情都很難。當然，這不像是旅程，旅程結束了，也就顯得不重要了，我喜歡這個過程。當然，我不喜歡爬山…我很懶的，我是指在身體上，不是精神上… 總之，每件值得去做的事情都很難，所以它才值得去做，數學就是這樣的。
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 &
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．關於如何做研究的個人觀點
有沒有關於「作為博士生」這一塊兒的問題？
這一節要短得多，我將在五分鐘內講完（雖然他最後沒有）。不不不，應該是一個關於「什麼才是足夠好」的一個正確的見解。就什麼是好的數學而言：
A． 抽象vs具體
就什麼是好的數學而言，我想，你應該把精力花在你認為是好的和重要的數學上，如果你不這樣，你就做不了好的數學。很重要的抉擇是：抽象vs具體。具體，就是很明顯，有例子，抽象就是抽象
1. 不要被花哨的名詞牽著鼻子走
有一些Chicago的學生，學一些很花哨的東西，它很抽象。我就對學生說，你能說成其中一個定理嗎？他做不到——那你幹嘛喜歡這個東西，你連一個定理都說不出來？！
以下是一個小測驗，如果你在從事一個學科，那你至少應該能說出一個你喜歡的定理吧。比如，你要研究代數拓撲的話，那麼你很可能會說出Lefschetz不動點定理，至少它算是比較好的吧。如果你研究離散子群，可能你會說出Mostow剛性定理。在座各位，你們看到Cantor的對角線證法，有多少人不認為自己的人生煥然一新了呢？有誰的人生沒有被此征服呢？我相信你們都是這樣，否則大家就不會來這裡了。
當然，我相信如果你壓根不喜歡這種數學，那也就只能這樣了。但是，僅僅有名詞會很讓人感到無聊，哪怕它可能很有趣。一個理論，如果它不能闡明基本例子的話，它就是完全沒有用的。 完全沒用！
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，他是非常不可思議並具有影響力的數學家。但是，他也讓很多人誤會了，因為很多人認為他建立了一套完全抽象，沒什麼例子的理論——大錯特錯，這完全是誤解！他其實是在嘗試推廣非常特殊且具體的例子，一旦理論發展了，他就不再去想具體的例子了，比如發展Scheme的時候。但是，他發展etale上同調確實是為了解決Weil猜想這個非常具體的問題——代數簇的數點問題。Grothendieck不是僅僅依附於抽象的東西之上的。人們總是說：「啊，我要成為Grothendieck」…這些人，你們知道…
但是，當然了：一個隨意給出的定理就像是彈鋼琴的猴子（有時你會挺好動聽的東西，但是，整體來看，它不會很深刻）
就像四十年前那樣，有很多非常有趣的數學問題，你在八年級的時候就可以考慮它。會有無窮多的東西出現在腦海中…每個例子都應該闡明理論，每個理論也應該闡明例子。你一定要非常仔細地嘗試。記住以下這句話是有益的：數學是用來發現，解釋並理解現象的。一些有趣的事情發生了，我教了一門迷你課程，是關於Euler示性類的余不變性的，觀察到一些…一些數的出現，這和有限群論有關，還有相關的，比如模形式。
有一種現象，大家看文章，看什麼quasi, n-category（花哨的東西）…這能闡明什麼例子呢？你能告訴我什麼呢？有什麼例子呢？你能拿它做什麼呢？你對此有什麼理解？——啥也沒有！
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.
關於你是不是僅僅在玩弄符號，以下有兩個快速檢驗方法
我是向Thurston學到的這個。首先，訪問一下Mathoverflow這個網站，如果你還不知道的話。你去看看Thurston關於此說的話，當時他已經過了科研的高峰期了，但是他還在和數學打交道。他讓我感到我就是僅僅在玩弄符號。事實上，他經常遇到一些數學中的對象，他編了一個電腦程序來研究他想處理的東西，他確實把數學對象的性質發掘地很深，而不僅僅是為了證明定理。就是這麼回事。以下是這兩個檢驗方法：
你的理論是不是僅僅證明了一個舊定理？
關於一些基本例子，你的理論能不能說出一些以前沒有的東西（比如觀點，認識）？ 這是個很方便的檢驗方法。我想說的是有很多當下流行的數學都不滿足這些。幾百頁的論文，我想說…很多人都在3-流形上定義了很多東西，我不做3-流形，但我知道Thurston對此的方法，他的方法解決了幾乎所有的問題，大量的問題。而人們總是在定義花哨的不變數。去年我參加了一個報告，一個人定義了一個非常花哨的不變數，有點讓人討厭。我思考著，坐下來：這東西難道不是說…人們總是單單把花哨的東西往上貼，他說：「是的，這些不變數可以分類所有扭結…」我於是舉起手說「你能否給我一個2-knot的例子？」我給本科生上課的時候…三葉結是一個非平凡的扭結，對吧，為什麼？關於此有一套理論，二年級學生就能理解。我跟他說：「你這文章七十頁，是個很大的理論，那你的理論能分出三葉結嗎？」「不能」… 我的意思是，這不是個孤立的定理，這是個整體。這些花哨的東西，我不明白情況為什麼如此極端，如此多的人去做它們，其中還有一些很有名的數學家。我真的很想與之爭辯，這實在太極端了。很多學生也被引誘於此。我想說：這到底是什麼？我喜歡它嗎？也許本科的時候，我會喜歡這個吧，事實上，本科生們總是被誘導著去學習最糟糕的的數學了。我不知道為什麼會這樣。抱歉…對於這個問題的回答，我的要求很低，你只需要告訴我三葉結是不是平凡的就行了。對吧，二年級的學生也會做。
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,
一個好的數學的標誌
這一理論讓我們對已有的東西有新的理解
我不知道現在什麼時間，但是，我想講一些讓我非常興奮的東西。這幾年，我一直在演講模空間M_g，非常玄妙的東西，其中g是Riemann曲面的虧格。這個東西在數學中很基本，也很fancy。有個東西叫Euler Class，它屬於S^1的微分同胚群的分類空間的第二個上同調群。它給出了非常fancy的代數幾何，這個（Euler Class）完全給出了這些東西。最近吧，六個月以內，我才意識到，這個Euler Class，如果限制到…Z/10，這個群作用在圓周上，二階同調類是…映射P：Z/10× Z/10——&>Z。我不是在講什麼大定理，但是，我想跟你說的是…你可能要我為啥我要考慮Z/10對吧。這是因為…fancy的東西大家都在講，我就不說了，我就把注意力集中在Z/10，這個東西我們最早在一年級就學過了，這叫做「進位閉上鏈」，它是取了兩個數a，b，組成一個數對兒(a,b)，如果a+b大於9，那麼就映到1，,a+b小於等於9的話就映到0，Z/10的元素就是指0到9的數字，這是個閉上鏈。於是，我回過頭來，我看到了進位，我可以看到我一年級學到的東西，我也能看到Euler Class。對我來說…在這裡我不是在講數學，我只是想闡釋我們都熟悉的容易的東西。但是我喜歡這個，因為它確確實實是Euler Class，它確實告訴我一些關於進位的東西，我也真的見到了Euler Class。它真的回溯到了加法！
我們總是忘了一些東西。你是否知道你在高中學的數學，以及你甚至Princeton的教授正在從事的東西，本質上是一回事呢？我們忘了，因為我們光忙著做模空間的上同調了，但是真正大數學家做的真正好的數學，比如Sullivan的，比如Serre的，比如Milnor的，他們的數學總是回溯到初等的東西的，他們的數學能回溯到微積分，回溯到線性代數。他們想的（意思是回溯到的東西）比我的簡單的多，這也就是他們比我強的原因。不論如何，對我來說，這是好數學的一個標誌。想想這個東西回溯到了什麼？對此我非常興奮，因為我意識到Euer Class這個東西真了不起。作為流形上圓叢的不變數，你可以看到進位。
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 &
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)
最後，我打算快速地舉上兩個好的例子。考慮到這裡有一些本科生，我打算講一些大家都知道的。e.g. 廣義Stokes公式
這是個關於微分形式的非常有名的定理，為什麼它偉大呢？因為它真的非常一般。
1.兩邊都是可以計算的
你可以使用它，這在每個例子中都是很有趣的。
2． 它蘊含了三個偉大的定理，這三個定理我們之前在不同的背景中已經見過了(Gauss, Green, Stokes)。
下面的例子值得說很多。
e.g． Euler特徵的拓撲不變數
1. 這是個偉大的定理，它在每個例子中都有豐富的內容（圓盤，2維球面，Calabi-Yau流形）
2. 它聯繫了數學中很多不同的分支。
3. 它可以計算
你能計算一些東西。這是另一個讓我感到驚訝的。
人們給出定義，事實上，你可以給出無窮多定義，但是如果你不能拿它做點計算，那真的是毫無意義的。抱歉，我真的很想說，因為很多數學都是這樣。甚至很多聰明的人也這樣干。好，那麼第四點是
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
最後說說
B．和他人協作
1. 很多聰明的人已經做了大量的工作，如果你忽略這些，那麼你之後做重複前人的工作
做這種決定總是很難的，「我要自己去學」「我不需要別人」，這太愚蠢了，你什麼做不到，你知道為什麼嗎？因為你不這樣做，會有很多其他聰明的人這樣做。你需要知道他們在做什麼。
另一方面
2. 已經說過了，（對別人的工作）不必太畢恭畢敬，說不定你可以做的更好呢。你應該以自己的方式理解問題。
總之，不要太畢恭畢敬，最好以自己的方式理解問題。我最喜歡的書…
1和2矛盾嗎？我想最好的回答是下面這句話，它摘自《浮士德》
那些你的父輩們遺贈給你的，如果你想擁有它們，你還是應該重新自己獲得。
——歌德《浮士德》
我不知道這句話對於非英語母語者，這會不會很難懂？因為它是從德語翻譯來的。就是說，那些你的父輩們遺贈給我們的東西，比如數學知識，比如Gauss-Bonnet定理的證明，所有這些，我們之前的大數學家都已經遺贈給我們了。但你應該憑自己的努力去獲得，就好比它們是全新的一樣。如果你需要Gauss-Bonnet定理的證明，你從書上獲得點提示就好，如果可以的話，最好自己去證明。這本書對我來說很有意義，因為當時我第一次和Thurston做定理，那是Mostow剛性定理高階情形的另證，當然也推廣了一些。但是我們倆都不知道Mostow的猜想，我們只需要綱要。我們確實用了很多Mostow的東西，但我們並不知道他具體怎麼證的，但我還是自己完成了，我現在就可以給你證明。這很有意義，這也是上面問題（1和2矛盾嗎）的回答。
順便說下，這一點我也在《廚藝之樂》這本書的第一頁學到了，當然我沒有讀過，我「假裝」讀過吧…
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!
最後：
3. 盡量多地和他人討論，在你遇到瓶頸的時候請假專家（哪怕你沒有遇到瓶頸）
我有個非常害羞的學生，他正在做一個問題，然後這個領域的世界做的最好的人來訪問，我告訴他「把你的問題和他討論下」，兩周後，我問他「你有沒有和他談？」「沒有。」我回答道「太糟了，如果你24小時內再不和他討論，那你就別做我的學生了！」如果你很害羞，我的回答是「太糟了，你得和人討論啊，你遇到瓶頸的時候應該請教專家！」我的學生說了，「這算不算一種『欺騙』呢，你在研究一個問題，問專家就有點欺騙的成分了——很明顯，你應該自食其力搞定它」，但是，你真的遇到瓶頸了啊，你該怎麼辦？對於我，即使我遇到一個簡單的表示論的問題，我也會寫給Drinfeld請教他。
所以，我的回答是：數學很難，如果可以的話，那就「欺騙」吧~
因為它很難，所以我說你可以去「欺騙」，再說一件事：
別忘了要過得開心！
我想說，純粹為了開心有點過分了，（純粹為了開心）那不是我做數學的原因，當然，做數學確實讓我快樂。我們每個人做數學都有原因，對我來說，一切生活都是和成長相關的，並非為了快樂，雖然我願意變得快樂。你得找到自己做數學的理由。我不打算把它講得更複雜了，總之，快樂是好事啊。我就講到這裡，謝謝！
侵權立即刪。。。。側重點可能不是數學研究，可能是從博士跨入數學研究階段一些方法。值得一讀。希望對你有幫助。謝邀。
看到 @justlikemath 答案里那句「我們不要想著從網上搜一個方法，告訴你沒有的。」 很有感觸。雖然我不懂代數幾何，看不懂他的答案，不過我自己現在考慮的問題（已經在知乎寫了3篇文章講這個東西了。。6維正曲率流形。。），網上能搜到的文獻也是寥寥無幾，基本還是要自己想自己做——雖然用的方法大部分還是前人用過的，但是畢竟別人是用在別的場合別的問題下面，要搬過來還是費點腦子。
純數的研究模式，就是只需要最基本的工具就行了——紙筆，電腦（有時候還是要用一些程序／軟體的），有人或許還喜歡用黑板／白板，然後要麼就是自己想，要麼和別人討論，就是最樸素的那種工作方式，如果你不懂一個數學家正在思考的問題，那麼你看著他成天冥思苦想大概也覺得挺無聊的。
給年輕數學工作者的建議 ---Michael Atiyah 轉自地址裡面中文，侵權刪。
Advice to a Young Mathematician
Michael Atiyah
Warning
What follows is very much a personal view based on
my own experience and re?ecting my personality, the
type of mathematics that I work on, and my style
of work. However, mathematicians vary widely in all
these characteristics and you should follow your own
instinct. You may learn from others but interpret what
you learn in your own way. Originality comes by break-
ing away, in some respects, from the practice of the
past.
Motivation
A research mathematician, like a creative artist, has
to be passionately interested in the subject and fully
dedicated to it. Without strong internal motivation
you cannot succeed, but if you enjoy mathematics the
satisfaction you can get from solving hard problems is
immense.
The ?rst year or two of research is the most di?cult.
There is so much to learn. One struggles unsuccess-
fully with small problems and one has serious doubts
about one』s ability to prove anything interesting. I
went through such a period in my second year of
research, and Jean-Pierre Serre, perhaps the outstand-
ing mathematician of my generation, told me that he
too had contemplated giving up at one stage.
Only the mediocre are supremely con?dent of their
ability. The better you are, the higher the standards
you set yourself—you can see beyond your immediate
reach.
Many would-be mathematicians also have talents
and interests in other directions and they may have
a di?cult choice to make between embarking on a
mathematical career and pursuing something else. The
great Gauss is reputed to have wavered between math-
ematics and philology, Pascal deserted mathematics at
an early age for theology, while Descartes and Leib-
niz are also famous as philosophers. Some mathemati-
cians move into physics (e.g., Freeman Dyson) while
others (e.g., Harish Chandra, Raoul Bott) have moved
the other way. You should not regard mathematics as
a closed world, and the interaction between mathe-
matics and other disciplines is healthy both for the
individual and for society.
Psychology
Because of the intense mental concentration required
in mathematics, psychological pressures can be consid-
erable, even when things are going well. Depending on
your personality this may be a major or only a minor
problem, but one can take steps to reduce the ten-
sion. Interaction with fellow students—attending lec-
tures, seminars, and conferences—both widens one』s
horizons and provides important social support. Too
much isolation and introspection can be dangerous
and time spent in apparently idle conversation is not
really wasted.
Collaboration, initially with fellow students or one』s
supervisor, has many bene?ts and long-term collabo-
ration with coworkers can be extremely fruitful both in
mathematical terms and at the personal level. There is
always the need for hard quiet thought on one』s own,
but this can be enhanced and balanced by discussion
and exchange of ideas with friends.
Problems versus Theory
Mathematicians are sometimes categorized as either
「problem solvers」 or 「theorists.」 It is certainly true
that there are extreme cases that highlight this divi-
sion (Erd?os versus Grothendieck, for example) but
most mathematicians lie somewhere in between, with
their work involving both the solution of problems and
the development of some theory. In fact, a theory that
does not lead to the solution of concrete and inter-
esting problems is not worth having. Conversely, any
really deep problem tends to stimulate the develop-
ment of theory for its solution (Fermat』s Last Theo-
rem being a classic example).
What bearing does this have on a beginning stu-
dent? Although one has to read books and papers and
absorb general concepts and techniques (theory), real-
istically, a student has to focus on one or more speci?c
problems. This provides something to chew on and to
test one』s mettle. A de?nite problem, which one strug-
gles with and understands in detail, is also an invalu-
able benchmark against which to measure the utility
and strength of available theories.
1
2
Princeton Companion to Mathematics Proof
Depending on how the research goes, the eventual
PhD thesis may strip away most of the theory and
focus only on the essential problem, or else it may
describe a wider scenario into which the problem nat-
urally ?ts.
The Role of Curiosity
The driving force in research is curiosity. When is a
particular result true? Is that the best proof, or is
there a more natural or elegant one? What is the most
general context in which the result holds?
If you keep asking yourself such questions when
reading a paper or listening to a lecture, then sooner
or later a glimmer of an answer will emerge—some
possible route to investigate. When this happens to
me I always take time out to pursue the idea to see
where it leads or whether it will stand up to scrutiny.
Nine times out of ten it turns out to be a blind alley,
but occasionally one strikes gold. The di?culty is in
knowing when an idea that is initially promising is in
fact going nowhere. At this stage one has to cut one』s
losses and return to the main road. Often the decision
is not clear-cut and in fact I frequently return to a
previously discarded idea and give it another try.
Ironically, good ideas can emerge unexpectedly from
a bad lecture or seminar. I often ?nd myself listen-
ing to a lecture where the result is beautiful and the
proof ugly and complicated. Instead of trying to fol-
low a messy proof on the blackboard, I spend the rest
of the hour thinking about producing a more elegant
proof. Usually, but not always, without success, but
even then my time is better spent, since I have thought
hard about the problem in my own way. This is much
better than passively following another person』s rea-
soning.
Examples
If you are, like me, someone who prefers large vis-
tas and powerful theories (I was in?uenced but not
converted by Grothendieck) then it is essential to be
able to test general results by applying them to sim-
ple examples. Over the years I have built up a large
array of such examples, drawn from a variety of ?elds.
These are examples where one can do concrete cal-
culations, sometimes with elaborate formulas, that
help to make the general theory understandable. They
keep your feet on the ground. Interestingly enough,
Grothendieck eschewed examples, but fortunately he
was in close touch with Serre who was able to rec-
tify this omission. There is no clear-cut distinction
between example and theory. Many of my favourite
examples come from my early training in classical pro-
jective geometry: the twisted cubic, the quadric sur-
face, or the Klein representation of lines in 3-space.
Nothing could be more concrete or classical and all
can be looked at algebraically or geometrically, but
each illustrates and is the ?rst case in a large class of
examples which then become a theory: the theory of
rational curves, of homogeneous spaces, or of Grass-
mannians.
Another aspect of examples is that they can lead
o? in di?erent directions. One example can generalize
in several di?erent ways or illustrate several di?erent
principles. For instance, the classical conic is a rational
curve, a quadric, and a Grassmannian all in one.
But most of all a good example is a thing of beauty.
It shines and convinces. It gives insight and under-
standing. It provides the bedrock of belief.
Proof
We are all taught that 「proof」 is the central fea-
ture of mathematics, that Euclidean geometry with its
careful array of axioms and propositions has provided
the essential framework for modern thought since
the Renaissance. Mathematicians pride themselves on
absolute certainty, in comparison with the tentative
steps of natural scientists, let alone the woolly think-
ing of other areas.
It is true that, since G¨odel, absolute certainty has
been undermined, and the more mundane assault of
computer proofs of interminable length has induced
some humility. Despite all this, proof retains its car-
dinal role in mathematics and a serious gap in your
argument will lead to your paper being rejected.
However, it is a mistake to identify research in math-
ematics with the process of producing proofs. In fact,
one could say that all the really creative aspects of
mathematical research precede the proof stage. To
take the metaphor of the 「stage」 further, you have
to start with the idea, develop the plot, write the
dialogue, and provide the theatrical instructions. The
actual production can be viewed as the 「proof」: the
implementation of an idea.
In mathematics, ideas and concepts come ?rst, then
come questions and problems. At this stage the search
Princeton Companion to Mathematics Proof
3
for solutions begins, one looks for a method or strat-
egy. Once you have convinced yourself that the prob-
lem has been well-posed, and that you have the right
tools for the job, you then begin to think hard about
the technicalities of the proof.
Before long you may realize, perhaps by ?nding
counterexamples, that the problem was incorrectly
formulated. Sometimes there is a gap between the ini-
tial intuitive idea and its formalization. You left out
some hidden assumption, you overlooked some techni-
cal detail, you tried to be too general. You then have to
go back and re?ne your formalization of the problem.
It would be an unfair exaggeration to say that mathe-
maticians rig their questions so that they can answer
them, but there is undoubtedly a grain of truth in the
statement. The art in good mathematics, and math-
ematics is an art, is to identify and tackle problems
that are both interesting and solvable.
Proof is the end product of a long interaction
between creative imagination and critical reasoning.
Without proof the program remains incomplete, but
without the imaginative input it never gets started.
One can see here an analogy with the work of the cre-
ative artist in other ?elds: writer, painter, composer,
or architect. The vision comes ?rst, it develops into
an idea that gets tentatively sketched out, and ?nally
comes the long technical process of erecting the work
of art. But the technique and the vision have to remain
in touch, each modifying the other according to its
own rules.
Strategy
In the previous section I discussed the philosophy
of proof and its role in the whole creative process.
Now let me turn to the most down-to-earth question
of interest to the young practitioner. What strategy
should one adopt? How do you actually go about ?nd-
ing a proof?
This question makes little sense in the abstract.
As I explained in the previous section a good prob-
lem always has antecedents: it arises from some back-
ground, it has roots. You have to understand these
roots in order to make progress. That is why it is
always better to ?nd your own problem, asking your
own questions, rather than getting it on a plate from
your supervisor. If you know where a problem comes
from, why the question has been asked, then you are
half way towards its solution. In fact, asking the right
question is often as di?cult as solving it. Finding the
right context is an essential ?rst step.
So, in brief, you need to have a good knowledge of
the history of the problem. You should know what
sort of methods have worked with similar problems
and what their limitations are.
It is a good idea to start thinking hard about a
problem as soon as you have fully absorbed it. To get
to grips with it, there is no substitute for a hands-
on approach. You should investigate special cases and
try to identify where the essential di?culty lies. The
more you know about the background and previous
methods, the more techniques and tricks you can try.
On the other hand, ignorance is sometimes bliss. J. E.
Littlewood is reported to have set each of his research
students to work on a disguised version of the Rie-
mann hypothesis, letting them know what he had
done only after six months. He argued that the stu-
dent would not have the con?dence to attack such a
famous problem directly, but might make progress if
not told of the fame of his opponent! The policy may
not have led to a proof of the Riemann hypothesis,
but it certainly led to resilient and battle-hardened
students.
My own approach has been to try to avoid the
direct onslaught and look for indirect approaches. This
involves connecting your problem with ideas and tech-
niques from di?erent ?elds that may shed unexpected
light on it. If this strategy succeeds, it can lead to
a beautiful and simple proof, which also 「explains」
why something is true. In fact, I believe the search for
an explanation, for understanding, is what we should
really be aiming for. Proof is simply part of that pro-
cess, and sometimes its consequence.
As part of the search for new methods it is a good
idea to broaden your horizons. Talking to people will
extend your general education and will sometimes
introduce you to new ideas and techniques. Very occa-
sionally you may get a productive idea for your own
research or even for a new direction.
If you need to learn a new subject, consult the liter-
ature but, even better, ?nd a friendly expert and get
instruction 「from the horse』s mouth」—it gives more
insight more quickly.
As well as looking forward, and being alert to new
developments, you should not forget the past. Many
powerful mathematical results from earlier eras have
4
Princeton Companion to Mathematics Proof
got buried and have been forgotten, coming to light
only when they have been independently rediscovered.
These results are not easy to ?nd, partly because
terminology and style change, but they can be gold
mines. As usual with gold mines, you have to be lucky
to strike one, and the rewards go to the pioneers.
Independence
At the start of your research your relationship with
your supervisor can be crucial, so choose carefully,
bearing in mind subject matter, personality, and track
record. Few supervisors score highly on all three.
Moreover, if things do not work out well during the
?rst year or so, or if your interests diverge signi?cantly,
then do not hesitate to change supervisors or even uni-
versities. Your supervisor will not be o?ended and may
even be relieved!
Sometimes you may be part of a large group and
may interact with other members of the faculty, so
that you e?ectively have more than one supervisor.
This can be helpful in that it provides di?erent inputs
and alternative modes of work. You may also learn
much from fellow students in such large groups, which
is why choosing a department with a large graduate
school is a good idea.
Once you have successfully earned your PhD you
enter a new stage. Although you may still carry on
collaborating with your supervisor and remain part of
the same research group, it is healthy for your future
development to move elsewhere for a year or more.
This opens you up to new in?uences and opportuni-
ties. This is the time when you have the chance to
carve out a niche for yourself in the mathematical
world. In general, it is not a good idea to continue
too closely in the line of your PhD thesis for too long.
You have to show your independence by branching
out. It need not be a radical change of direction but
there should be some clear novelty and not simply a
routine continuation of your thesis.
Style
In writing up your thesis your supervisor will normally
assist you in the manner of presentation and organi-
zation. But acquiring a personal style is an important
part of your mathematical development. Although the
needs may vary, depending on the kind of mathemat-
ics, many aspects are common to all subjects. Here
are a number of hints on how to write a good paper.
(1) Think through the whole logical structure of the
paper before you start to write.
(2) Break up long complex proofs into short inter-
mediate steps (lemmas, propositions, etc.) that will
help the reader.
(3) Write clear coherent English (or the language
of your choice). Remember that mathematics is also a
form of literature.
(4) Be as succinct as it is possible to be while
remaining clear and easy to understand. This is a dif-
?cult balance to achieve.
(5) Identify papers that you have enjoyed reading
and imitate their style.
(6) When you have ?nished writing the bulk of your
paper go back and write an introduction that explains
clearly the structure and main results as well as the
general context. Avoid unnecessary jargon and aim
at a general mathematical reader, not just a narrow
expert.
(7) Try out your ?rst draft on a colleague and take
heed of any suggestions or criticisms. If even your close
friend or collaborator has di?culty understanding it,
then you have failed and need to try harder.
(8) If you are not in a desperate hurry to publish,
put your paper aside for a few weeks and work on
something else. Then return to your paper and read
it with a fresh mind. It will read di?erently and you
may see how to improve it.
(9) Do not hesitate to rewrite the paper, perhaps
from a totally new angle, if you become convinced
that this will make it clearer and easier to read. Well-
written papers become 「classics」 and are widely read
by future mathematicians. Badly written papers are
ignored or, if they are su?ciently important, they get
rewritten by others.本文只是個人觀點, 基於我自己的個性, 經驗, 我所研究的數學以及我的工作風格. 然而不
同數學家在性格, 特點方面差異很大,所以讀者要依靠自己的直覺. 你可以從別人那裡學習,
但是你必須要以自己的方式來理解你所學到的東西. 從某個方面來講,創造力就是要掙脫以
往實踐的束縛.
科研的動機
一個研究型的數學工作者, 就像一個具有創造性的藝術家, 必須要對於自己從事的工作充
滿激情, 並全力以致. 沒有強烈的內在動力, 則很難有所成就. 如果你能夠享受數學, 那麼
你從解決難題中獲得的滿足將是巨大的.
開始研究的第一年或者前兩年是最困難的, 因為有很多的東西要學. 當一個人嘗試著去解
決小問題而不斷的失敗時, 會對與自己是否有能力證明任何有意義的東西產生嚴重的懷疑 .
我自己在開始做研究的第二年也經歷了這樣的時期. 讓-皮埃爾.賽爾(Jean-Pierre Serre)，可
能是我們這一代傑出的數學家，曾告訴我他也曾在某個時期考慮放棄。
只有平庸的人才會對於自己的能力有無限的自信。你越優秀，則給自己設定的目標越高 --------
因為你可以看到暫時無法到達的地方。
許多將要成為數學家的人對於其他的領域也有興趣，並且也具有天分。所以他們可能面對一
個很困難的抉擇：是否以數學為業？據說偉大的高斯曾在數學和哲學之間猶豫不決，帕斯卡
在很年輕的時候就為了神學而放棄了數學，而笛卡爾和萊布尼茲則即是傑出的數學家，同時
又是著名哲學家。有些數學家轉行去研究物理學（如 Freeman Dyson）, 而另一些人則反
其道而行之（如 Harish Chandra, Raoul Bott）。所以你不應將數學看成一個隔絕的王國，數
學同其他學科之間的相互影響無論對於個人還是社會都是有益的。
心理的層面
因為數學常常需要高強度集中的腦力活動，即使在事情進展順利的時候，對於心理上的壓力
也是巨大的。取決於你的個性，這會是一個輕微或者嚴重的問題。但你可以通過做某些事情
來減少這些壓力。如通過和自己的學生交流，參加講座，會議等，這些會開闊你的視野，並
且為你提供重要的社交上的支持。過度的自我封閉和反省則是危險的，那些看似花在閑談上
的時間則並不是真正的浪費。
問題和理論
數學家有時被劃分為『解決難題的人』和『理論家』這兩類。當然在數學界有這樣極端的例子來
證明這些人確實存在（如 Erodos versus Grothedieck），但絕大多數的數學家都介於這兩者
之間的某個地方，他們的工作中既有要解決問題也要發展一些理論。實際上，一個不能夠解
決具體有意義問題的理論，是一個不值得擁有的理論。反而言之，任何深刻的問題都傾向於
刺激與其解答相關理論的發展（費馬大定理就是一個經典的例子）。 這對於一個剛開始科研
的學生有什麼影響呢？雖然一個人要靠讀書，看文獻來學習一般的概念和技巧（理論），但
實際上學生最後必須把注意力集中在一個或者幾個具體的問題上。這將會給他提供一種體
驗，並讓他判斷自己的勇氣。一個具體明確的問題，當一個人通過努力解決之並且完全的理
解相關細節的時候，就會成為一個無價的標尺，可以用來去檢測已有理論的功用和實力。 取
決於研究實際上的進行，最終的博士論文可能是剝掉絕大多數的理論，只是集中到關鍵的問
題上，或者論文可能最後是描述問題所在的一個廣泛的背景。 好奇心的作用 好奇心是科研
的驅動力。在什麼情況下一個特殊的結論是正確的？這是不是最好的證明，或者另有更自然
的，更優美的的證明？結論成立的最一般的情況是什麼？ 如果你能在讀文獻或者聽報告的
時候不斷的問自己這樣的問題，那麼遲早解答的曙光—–即那些可能的研究途徑，就會在你
腦海中閃現。當這樣的情況出現在我的身上時，我總是給自己時間去追蹤這個想法，看他可
以把我帶到哪裡，是否經得住仔細的檢查。十次有九次最後發現自己走到死胡同，但是偶爾
也會發現金子。困難的是知道什麼情況下一個開始看似很有前景的理論是不實際的。這時候，
你必須儘快的結束並且回到原來的軌道上。但常常這樣的決定不是清晰明確的，實際上我自
己常常會撿起以前拋棄的理論，再給他們一次機會。 有點諷刺意味的是，好的想法常常出
其不意的在糟糕的會議或者報告之間產生。我經常發現自己去聽一些結論漂亮但證明卻醜陋
複雜的報告。不是去緊跟黑板上混亂的證明，我讓自己在餘下的時間裡去思考一個優雅的證
明。經常，但不是總是，不會成功，但即使是這樣我的時間也被很好的利用了，因為我已經
努力的用自己的方式去思考了這個問題。這比被動的去接受另一個人的推理要好的多。
策略
(略去若干)
現在讓我們來談一下青年工作者最感興趣的話題：即要採取怎樣的策略？實際中怎樣才能發
現一個證明？
這樣抽象的問是沒有意義的。我在上一節說到，每一個好的問題都會有緣起：它源於某種背
景，有一定的根基。要取得進展你必須要理解這些基礎。這就是為什麼最好是找你自己感興
趣的問題，而不只是簡單的從你的導師那裡得到題目。如果你知道一個問題的起源，為什麼
提出這個問題，那麼你離解決這個問題的距離已經縮短一半了。實際上，提出正確的問題往
往和解決問題一樣困難，了解正確的背景是最基礎的第一步。
所以，簡單來講，你需要對於問題的歷史有很好的了解。你需要知道什麼樣的方法曾經用在
類似的問題上，以及他們的局限是什麼。
當你被一個問題深深吸引的時候，最好及時的用心的去思考它。去了解一個問題，沒有比實
際的去嘗試這個問題更好的辦法。你應該去研究特例，儘力的弄清楚問題關鍵難點所在。當
你對於問題的背景和以前的方法有越多的了解的時候，你就可以嘗試更多的技術和技巧。但
另一方面，忽略有些時候也是一種好處。據說 J.E. Littlewood 讓他的每一個學生開始的時候
都去解決一個經過偽裝的黎曼猜想，在他們工作六個月之後才告訴他們真相。他解釋說如果
學生在開始的時候就知道，那麼他們就沒有足夠的信心直接去嘗試一個這麼著名的問題，但
是當他們沒有被告知問題的聲望的時候，他們則有可能取得進展！ 這樣的策略也許不能夠
解決黎曼猜想，但是它一定可以訓練出充滿活力和歷經磨練的學生。
我自己的方法是盡量的繞開正面的進攻去尋找間接的途徑。把你的問題和其他領域的技術以
及想法連接起來也許會有未預見的啟發。如果這個策略奏效，它將會產生一個簡單的美麗的
證明，會『解釋』為什麼一些事情是正確的。實際上，我相信對於解釋，理解的尋找才是我們
真正的目標。證明只這個過程中一個簡單的步驟，或則，有時候是結果。
在尋找新的方法的時候開拓你的眼界是非常有用的。和人們談論會延伸你的教育並且有時會
帶給你新的想法和技術。你常常可能會得到對於你自己的研究富有成效的想法甚至一個新的
方向。
如果你需要了解一個新的方向，去參考文獻當然是一個很好的途徑，但是更好的是能夠從這
個領域中找到一個友好的專家，讓他來指導，這樣往往會使你更快並獲得更多對於這個領域
的深刻見解。
當我們向前看的時候，對於新的發展保持警覺，但是也不能忘記過去。從前很多強大的數學
結果被埋沒和遺忘了，只有在被獨立的重新發現之後才重見陽光。這些結果並不容易發現，
部分是由於術語以及風格的變化，但是它們有可能是金礦。就像通常對於金礦一樣，你要有
足夠的運氣去發現一個，而獎賞只給那些先行者。
風格
在寫論文的時候，導師通常會在表述和組織上給你指導。但對於你自己在數學上的發展，取
得自己的風格是很重要的一部分。雖然不同的數學具體的需要可能不同，但在很多方面是一
致的。下面是關於如何寫好一篇文章的一些小建議。
（1） 在開始寫文章之前，想清楚整篇文章的邏輯結構。
（2） 把繁瑣的證明分化為簡短的步驟會幫助讀者理解。
（3） 用條理清楚的英語，要知道數學也是一種文學形式。
（4） 在保持清楚和易於理解的基礎上盡量的簡潔，這是一個很難取得的平衡。
（5） 挑選你欣賞的文章並模仿他們的風格。
（6） 當完成文章主體部分後，回頭再去寫簡介部分，使之能清楚的介紹整篇文章的結構及
內容。盡量避免困難的術語，要面向一般的數學讀者而不只是很少的專家。
（7） 把第一稿給同事讀一下，要注意所有的建議和批評。如果連你的好友或合作者都難以
理解，表明你沒有達到目標，需要更多的努力。
（8） 如果你不是非常著急要發表，那麼可以把你的文章放在一邊幾周，去做別的事情。然
後回過頭用全新的頭腦再來讀你的文章，你將會感到不同並可能發現可以改進的地方。
（9） 不要猶豫去重寫你的文章，也許從一個全新的角度，如果你相信這樣會使之更容易理
解和明了。寫的好的文章會變成『經典』並被以後的數學家廣泛的閱讀。寫的壞的文章通常被
忽略；如果文章非常重要的話，它們通常會被別人重寫。
參閱Imre Lakatos所著之《證明與反駁》
研究生專業基礎數學的，等開學來答
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