My Past Experience in Mathematics Shing-Tung Yau The Chinese University of Hong Kong Sep. 19, 2003
I grew up in the (farming) countryside of Hong Kong: Yuen Long and Shatin. There was no electricity and no pipe water. I took bath in the river when I was very young. I have eight brothers and sisters and food was scarce. When I was five years old, I took an entrance examination to a good public school. I failed mathematics because I made the wrong convention: I wrote 57 to be 75, 69 to be 96.
So I entered a very small village school. There were many rough kids from the farm. In a matter of half a year, I got serious sick because of the intimidation of the rough kids and mistreatment of the teacher. I rested at home for half a year. I started to learn how to deal with difficult situation with classmates and teachers. By the time when I entered sixth grade, I was a leader of a small group of students to wander on the street.
My father was a professor. He taught me a lot of Chinese literature at that time. However, he did not realize that I did not attend school for a period of time. (Perhaps because I did well at home as I can recite most of the essays that he asked me to do.)
The reason that I did not go to school was that the teachers did not really teach. I got bored in school. Then after a while I got bored on the street, also. There was a joint examination for all primary schools. I did poorly. However, I was exactly on the borderline case. The government allowed these borderline kids to apply for private school and gave them tuition. I got into Pui Ching Middle School.
校歌 何安東 編 李竹侯 詞 培正培正何光榮，教育生涯慘淡營，培後進兮其素志，正軌道兮樹風聲，萬千氣象方蓬勃，鼓鑄群才備請纓，愛我培正謨謀遠，永為真理之干城，永為真理之干城！ 青年向上歌 我要真誠，莫負人家信任深。我要潔淨，因為有人關心。我要剛強，人間痛苦才能當。我要膽壯，奮鬥才能得勝。我要膽壯，奮鬥才能得勝。 我要愛人，愛敵人也愛淪落人。我要施贈，心誠、義重、財輕。我要虛懷，不忘我身多弱點。我要向上，學主榜樣做人。我要向上，學主榜樣做人。
This is probably one of the best middle school. Nothing was exciting in the first year of middle school. I did not do too well that year. However, I learnt much more at home: Chinese literature, Novels (Chinese and Western), Philosophy, history. All from my father and his conversations with his students. Although I did not understand Greek Philosophy. It started to impress me after listening to many conversation of my father with his students.
I started to read the famous Chinese history books: 史記、左傳 I am especially fascinated by 史記. Not only by its beautiful writings, but also by its original and responsible way to report the ancient history. Up to present days, I read this book. The global view of history from a great master resonants with the thinking of a great scientist.
The following essay strikes me: Though indulged in reading, I do not pursuit precise meanings. Nonetheless, every time I hit on something I was so overwhelmed with joy that I forgot my meals. 晉‧陶淵明 好讀書，不求甚解， 每有會意，便欣然忘食。 I am not bitter for being poor and obscure, nor am I keen on being rich and famous. 不戚戚於貧賤，不汲汲於富貴。
In following years, this has been the guiding principle of my study of many different subjects. With my father’s teaching, I started to set the goal of my life. An important quotation:
Zuo's Book of History On Immortality “The first place is to reign benevolently, the next to gain victory, and the last to say valuable words. These achievements will stand for long and not be abandoned. Thereafter they are called immortal.” 左傳 叔孫豹論三不朽 太上有立德，其次有立功，其次有立言，雖久不廢，此之謂不朽。 One needs being humble and simple to reach these goals. A former student of mine recently recited the following lines from a Tang poem during a TV interview in China: 立德立功立言之道，必以謙讓質樸為主。 「會當凌絕頂，一覽眾山小」，輕妄浮誇之言也。 “I would rather be at the summit, So all mountains will become tiny under me.” I think he was a bit too arrogant.
Sima Qian (司馬遷) on Confucius (孔子) There are so abundantly many kings and men of virtue! Widely known by their contemporaries, their names feel into oblivion soon after they perished. And yet Confucius, a man in plain cloth, has been held in great esteem by scholars of more than ten generations. 天下君王至於賢人，眾矣！ 當時則榮，沒則已焉，孔子布衣， 傳十餘世，學者宗之。
During conversations of my father with students, many important points of history of philosophy were mentioned. • Basic principles • The root of the very existence of matter (basic axioms, etc) • General phenomena • Unification of all principles (unified field theory) • Methods of understanding truth based on logic and reasoning • How to combine different knowledge and different phenomena under a general principle • The great philosophers did not simply follow others in developing their views, not even from their teachers. They created their own thoughts (based on previous works.)
The goal of writing history of philosophy • (求因) The origin of a philosophical thinking must come from different sources. It is our goal to find out such sources. • (明變) There are many complicated philosophical thoughts in the history. It is important to figure out the treads of their thoughts. • (評論) A cortical comments of the occurrence of all philosophies and their consequences.
I also learnt the way to do research with lasting importance. Wang Guowei (王國維) The Three Levels To Achieve Breakthrough in Research (borrowing some lines from Song Ci or lyric poems from the Sung dynasty) Level One ... Last night the west wind withered the greenery of the trees In loneliness, I mounted the tall building Casting my eyesight along all roads to the edge of the earth 晏殊 … 昨夜西風凋碧樹，獨上高樓，望盡天涯路。
Level two ... For my loosening waist belt I feel no regret, For her it is worth being haggard and thin. 柳永 … 衣帶漸寬終不悔，為伊消得人憔悴。 Level Three Looking for her a thousand times In a crowd All of a sudden As I turned my head There she was Standing in the shades of fading lights 辛棄疾 … 眾裏尋他千百度，驀然回首，那人卻在，燈火闌珊處。
In my second year of Pui Ching, I got into a problem with my teacher. The teacher was a very devoted head master of my class and clearly meant well for me. She was shocked to find out that my father was a professor and poorly paid. Her passion for my future changed my behavior in classroom. I studied plane geometry in second year of high school.
My classmates were not used to reason abstractly. The mere fact that I listened to my father’s philo-sophical discussion at home made me feel at home with axiomatic approach. In fact, I felt I can understand my father’s conversation better after I learnt geometry. The charm to prove elegant theorems based on simple axioms excites me.
With the passion for geometry, I started to develop my taste for mathematics, which included algebra. Everything became easy after I found the subject exciting. I also found the subject of history interesting. It taught me a global view of everything that I learnt. How events happened? Why they happened? What may happen in the future?
At this time, my father just finished writing his book on history of Western philosophy. His conversations with students taught me the way that we should see history in a global manner. This kind of practice deeply influence my way of looking at my research projects in the later time.
Around 400 A.D. Liu (who wrote the first book on comparative literature on Chinese writings up to that period) 文心雕龍 諸子 My body may perish along with time. My goal and my ambition will extend with my theory. My heart is in resonance with those great men in ancient days. My feeling and my theory will go forward for the next thousand years. 身與時桀，志共道申。標心於萬古之上，而送懷於千載之下。
When I was fourteen, my father passed away. This was perhaps the greatest shock to my life. For a long time, I could not believe that my father left me and the family. The financial situation of the family was really bad. It was not clear at all that we could still go to school. The tremendous will of my mother and the helps of my father’s friends and students made it possible.
This disastrous change of family situation made me much more mature. The extreme hard-ship showed difficulty of human relationship, what I learnt from my father became practical. The poems and the classical essays that I learnt became much more meaningful. I spent half a year to read classical literature and history of China. It became a way for me to relax during very tense situation in life. The beautiful poems guided me to appreciate the beauty of nature.
英國大詩人拜倫 〝希臘啊！你本是平和時代的愛嬌，你本是戰爭時代的天驕。撒芷波，歌聲高，女詩人，熱情好。更有那德羅士、菲波士榮光常照。此地是藝文舊壘，技術中潮。如今在否？算除卻太陽光線，萬般沒了。〞 〝馬拉頓前啊！山容縹緲。馬拉頓後啊！海門環繞。如此好山河，也應有自由回照。我向那波斯軍墓門憑眺。難道我為奴為隸，今生便了？不信我為奴為隸，今生便了。〞 梁啟超翻譯
From Byron’s Don Juan II -- A variation edition by Steffan Pratt Canto III, 86 1. The isles of Greece, the isles of Greece! Where burning Sappho loved and sung, Where grew the arts of war and peace, … Where Delos rose, and Phoebus sprung! Eternal summer gilds then yet, But all, except their sun, is set. 2. The mountains look on Marathon … And Marathon looks on the sea; And musing there an hour alone, I dream’d that Greece might still be free; For standing on the Persian’s grave, I could not deem myself a slave.
I read a lot of books in mathematics. I thought about the problems in those books. When I exhausted all those problems, I started to create my own problems as I thought that they may be challenging. The practice of creating my own problem had been the most important part of my research in the future. The textbooks in school did not satisfy me. I went to library and I went to bookstores to read books. I spent hours and hours in bookstore to read books that I could not afford to buy.
When I was fifteen, I started to teach lower grade level students to earn money. I was proud that by teaching in a novel way, I was able to transform some very poor student to become the best student in class. It was an experience to train young people. I also learnt that it is beneficial to myself to teach.
My high school teachers in mathematics were excellent. We studied rather advanced topics in mathematics. I had no difficulty with them. However, I was rather disappointed that my physics teacher was not good enough. The fundamental intuition in physics was not established during my high school year. I regretted it up to now. I had an excellent teacher in Chinese. He was my father’s friend: he taught us to think in a non-traditional manner.
We were asked to think creatively. He said that we should read good books but also bad books as possible comparisons. So I read everything. This is true even for my scientific career. A typical topic for our essay in our Chinese writing: the philosophy of a pig. So we start to dream about anything interesting. I was not the best in my high school. I did not have the highest grade in mathematics. But I think deeper than my classmates and I read much more books.
I entered The Chinese University of Hong Kong in 1966. I chose mathematics as my career although I was also very interested in the subject of history. By this time, I started to digest those advanced level mathematics books that I read in high school. I did not quite understand those books at the beginning. Suddenly I understand them and I was much better than the contemporary students.
崇基門前對聯 崇高惟博愛本天地立心無間東西溝通學術 基礎在育才當海山勝境有懷胞與陶鑄人群
Chung Chi College Anthem Men from four seas founded Chung Chi so that here might youth Honour Christ, eternal teacher, who Himself is truth. Through the long night keep the torch bright and the work begun Till the lights of faith and knowledge show the world made one. China's still evolving culture, grateful, we retain; East and West, through freely sharing, further strength obtain, By the Church upheld and nurtured, minds to duty drawn, Chung Chi, toward the very highest, lead us on, and on!
College mathematics opened my eyes. The fact that one can derive every statement in mathematics from simple axioms really excited me. After I understood how mathematics was built, I got so excited that I wrote a letter to my professor showing my great pleasure. It was a cornerstone for my appreciation of mathematics. A new Ph.D came from Berkeley to Hong Kong. His name is Stephen Salaff. He was so impressed by my performance that he wrote a book with me together. Its topic is on ordinary differential equations.
Another teacher Dr. Brody came from Princeton. He had a rather unique way of teaching. He picked an advanced book in mathematics. He assigned a chapter for the students to find mistakes in the book and corrected the mistakes. It is a good method to train us not to depend on textbook. At the same time, I trained myself to be critical about the established theorems in the book. Sometimes I generalized the theorems. Brody was very pleased by my performance when I showed what I could do in class.
The importance of such trainings is that: • I learnt how to think independently. • I found out how important it is to express mathematics in front of an audience. These points have been important for me and for my teaching.
With the help of Dr. Salaff, I was able to enter the graduate school of Berkeley despite that I did not finish my college in Hong Kong. Of course, Berkeley has the leading mathematics department in the world. I arrived in August. I met Prof. S. S. Chern who become my thesis advisor later.
When I was in Hong Kong, I was too much fascinated by very abstract mathematics. (Although I was trained quite solidly in analysis.) I thought mathematics covering a very general area is best mathematics. I thought I would study functional analysis. I learnt a lot in that subject. I read a big fat book of Dunford-Schwatz on functional analysis. I also read a lot of books on operator algebra. When I arrived in Berkeley, I met some best minds in mathematics. I changed my view.
When I met those first rated mathematicians, I was rather thirsty in learning different subjects from them. I attended many classes from 8am to 5pm. (Sometimes I ate lunch in class.) These are subjects ranging from topology, geometry, differential equations, Lie groups, number theory, combinatorials, probability theory to dynamical system. I did not understand all of them. But I focus my efforts on several of them.
When I learnt topology, it was so differently from what I learnt before. There were fifty students in class. All of them seem to be smart and far better than me. They could perform and talked nicely. However, I did my home work well and in a short time, I found out that I was not bad, after all. The key is to work out all those tough home works and think about them thoroughly.
I read a book of John Milnor and was fascinated by the description of the concept of curvature. John Milnor is an excellent topologist. I started to think about problems that is related to questions in the book. I spent a lot of time in the library. There was no office for graduate students. There were many famous professors in Berkeley. Soon I realize that they are human beings after all. I read many journals and books in the library.
I started to be able to prove some nontrivial theorems in the second quarter. They are related to some theorems in group theory that I learnt over some casual conversation with a teacher in college. I applied it to geometry. My professors were surprised and pleased by my progress. One of the professors started to work with me. We wrote two papers. Professor Chern was on sabbatical leave. When he returned, he was very pleased.
I did not think what I did was great. I was very impressed by Prof. Morrey on his teaching of nonlinear partial differential equations. He taught nonlinear technique. It was not fashionable. The book he wrote was difficult to read. I thought those technique that he developed are very deep and must be important for the future of geometry. I learnt those technology. Despite of Prof. Morrey’s big name, very few students or faculties cared about his course. At the end, I was the only student in class and Prof. Morrey taught me in his office. This course built the foundation of my mathematics career.
After I wrote several papers, Prof. Chern told many people how brilliant I was although I did not think he knew my works well. I started to think more thoroughly about mathematics and geometry in particular. I worked on other parts of geometry. However, results did not come easy. My friend S. Y. Cheng came from Hong Kong that summer. We shared an apartment right next to the campus and I became more relax.
In that summer, I asked Prof. Chern to be my advisor. He agreed and after one month, he said that my papers in the first year should be enough for my thesis. I was surprised because I thought they are not good enough and I wanted to learn more. In any case, in the second year, I learnt more in the field of complex geometry and topology. Prof. Chern had a great expectation on me. He suggested me to work on Rieman hypothesis. Unfortunately up to now, I never thought about it.
Instead I pursued the general understanding of curvature of space. I decided a key to understand such a concept was a proposal made by Calabi in early fifties. Nobody believed what Calabi’s thought is true. I started to think about it. It is not the standard thing that a geometer would do in those days. It is clearly a hard question of analysis. Nobody would touch such problem.
I started to develop my taste into learning how to introduce analytic methods into geometry. Previous to this, there were attempts to apply nonlinear theory to surfaces in three space. I wanted to deal with an abstract space in arbitrary dimension. Because of Prof. Morrey and Chern’s interests in minimal surfaces, I also developed interest into this fascinating subject. In particular, I was interesting into harmonic maps. In general, I studied Calculus of variation.
I was interested in all analytic aspects of geometry. The basic idea is to merge the subject of nonlinear differential equation and geometry. In order to understand nonlinear equation, it is fundamental to understand linear equations. Hence I establish the first major theorem for harmonic functions on manifolds. I got my friend S. Y. Cheng to look into eigenvalue and eigenfunction problems. Together we wrote several important papers on the subject. They are still fundamental for modern research.
When I graduated, I got several offers. My teacher Chern suggested me to go to Institute for Advanced Study. The salary was less than half of what I could have gotten from Harvard. But I went to the Institute for Advanced Study. I met some other group of distinguished mathematicians. I developed some taste into topology, especially the theory of symmetries of space. I did solve some important problems in this subject based on analytical ideas I developed (group actions on manifolds).
Because of problem of visa, I went to New York State University of Stony Brook. At that point, it was supposed to be the center of metric geometry. It was indeed a good place, full of energetic young geometers. I learnt their technique. But I did not think that was the right direction for geometry. After one year, I went to Stanford where there was no geometer. It is a very peaceful environment and is very good in nonlinear partial differential equations. I met my very good friend Leon Simon and my former student Richard Schoen. Together we developed the subject of nonlinear analysis in geometry.
Tao Yuan-Ming (晉‧陶淵明) Long I lived checked by the bars of a cage. Now I have turned again to Nature and Freedom. 久在樊籠裡，復得返自然。