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I grew up in the (farming) countryside of Hong Kong: I grew up in the (farming) countryside of Hong Kong: Yuen Long and Shatin.Yuen Long and Shatin.

There was no electricity and no pipe water.There was no electricity and no pipe water.

I took bath in the river when I was very young. I have I took bath in the river when I was very young. I have eight brothers and sisters and food was scarce.eight brothers and sisters and food was scarce.

When I was five years old, I took an entrance When I was five years old, I took an entrance examination to a good public school. I failed examination to a good public school. I failed mathematics because I made the wrong convention:mathematics because I made the wrong convention:

I wrote I wrote 5757 to be to be 7575,,6969 to be to be 9696..

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So I entered a very small village school.So I entered a very small village school.

There were many rough kids from the farm.There were many rough kids from the farm.

In a matter of half a year, I got serious sick because of In a matter of half a year, I got serious sick because of the intimidation of the rough kids and mistreatment of the intimidation of the rough kids and mistreatment of the teacher.the teacher.

I rested at home for half a year. I started to learn how I rested at home for half a year. I started to learn how to deal with difficult situation with classmates and to deal with difficult situation with classmates and teachers.teachers.

By the time when I entered sixth grade, I was a leader By the time when I entered sixth grade, I was a leader of a small group of students to wander on the street.of a small group of students to wander on the street.

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My father was a professor. He taught me a lot of My father was a professor. He taught me a lot of Chinese literature at that time.Chinese literature at that time.

  However, he did not realize that I did not attend However, he did not realize that I did not attend

school for a period of time.school for a period of time.  (Perhaps because I did well at home as I can recite (Perhaps because I did well at home as I can recite

most of the essays that he asked me to do.)most of the essays that he asked me to do.)

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The reason that I did not go to school was that the The reason that I did not go to school was that the teachers did not really teach. I got bored in school. teachers did not really teach. I got bored in school. Then after a while I got bored on the street, also.Then after a while I got bored on the street, also.

  There was a joint examination for all primary There was a joint examination for all primary

schools. I did poorly. However, I was exactly on the schools. I did poorly. However, I was exactly on the borderline case.borderline case.

  The government allowed these borderline kids to The government allowed these borderline kids to

apply for private school and gave them tuition.apply for private school and gave them tuition.  I got into Pui Ching Middle School.I got into Pui Ching Middle School.

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校歌校歌何安東　編　　李竹侯　詞何安東　編　　李竹侯　詞培正培正何光榮，教育生涯慘淡營，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，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，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！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。

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This is probably one of the best middle school.This is probably one of the best middle school.  Nothing was exciting in the first year of middle Nothing was exciting in the first year of middle

school. I did not do too well that year.school. I did not do too well that year.  However, I learnt much more at home: Chinese However, I learnt much more at home: Chinese

literature, Novels (Chinese and Western), Philosophy, literature, Novels (Chinese and Western), Philosophy, history. All from my father and his conversations with history. All from my father and his conversations with his students.his students.

  Although I did not understand Greek Philosophy. It Although I did not understand Greek Philosophy. It

started to impress me after listening to many started to impress me after listening to many conversation of my father with his students.conversation of my father with his students.

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I started to read the famous Chinese history books:I started to read the famous Chinese history books:史記、左傳史記、左傳

  

I am especially fascinated by I am especially fascinated by 史記史記 ..  

Not only by its beautiful writings, but also by its original Not only by its beautiful writings, but also by its original and responsible way to report the ancient history.and responsible way to report the ancient history.  

Up to present days, I read this book.Up to present days, I read this book.  

The global view of history from a great master resonants The global view of history from a great master resonants with the thinking of a great scientist.with the thinking of a great scientist.

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The following essay strikes me:The following essay strikes me:

Though indulged in reading, I do not pursuit Though indulged in reading, I do not pursuit precise meanings. Nonetheless, every time I hit precise meanings. Nonetheless, every time I hit

on something I was so overwhelmed with joy that on something I was so overwhelmed with joy that I forgot my meals.I forgot my meals.

晉晉‧‧陶淵明陶淵明好讀書，不求甚解，好讀書，不求甚解，每有會意，便欣然忘食。每有會意，便欣然忘食。I am not bitter for being poor and obscure, nor I am not bitter for being poor and obscure, nor

am I keen on being rich and famous.am I keen on being rich and famous.不戚戚於貧賤不戚戚於貧賤，，不汲汲於富貴。不汲汲於富貴。

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In following years, this has been the guiding principle In following years, this has been the guiding principle of my study of many different subjects.of my study of many different subjects.

With my father’s teaching, I started to set the goal of With my father’s teaching, I started to set the goal of my life. An important quotation:my life. An important quotation:

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Zuo's Book of HistoryZuo's Book of History

On ImmortalityOn Immortality““The first place is to reign benevolently, the next to gain victory, The first place is to reign benevolently, the next to gain victory,

and the last to say valuable words. These achievements will stand and the last to say valuable words. These achievements will stand for long and not be abandoned. Thereafter they are called for long and not be abandoned. Thereafter they are called

immortal.”immortal.”左傳 左傳

叔孫豹論三不朽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 One needs being humble and simple to reach these goals. A former student of mine recently recited the following lines from former student of mine recently recited the following lines from a Tang poem during a TV interview in China: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。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。「會當凌絕頂，一覽眾山小」，輕妄浮誇之言也。

““I would rather be at the summit,I would rather be at the summit,So all mountains will become tiny under me.”So all mountains will become tiny under me.”

I think he was a bit too arrogant.I think he was a bit too arrogant.

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Sima Qian (Sima Qian ( 司馬遷司馬遷 ) on Confucius () on Confucius ( 孔子孔子 ))

There are so abundantly many kings and There are so abundantly many kings and men of virtue! Widely known by their men of virtue! Widely known by their contemporaries, their names feel into contemporaries, their names feel into

oblivion soon after they perished. And yet oblivion soon after they perished. And yet Confucius, a man in plain cloth, has been Confucius, a man in plain cloth, has been held in great esteem by scholars of more held in great esteem by scholars of more

than ten generations.than ten generations.

天下君王至於賢人，眾矣！天下君王至於賢人，眾矣！當時則榮，沒則已焉，孔子布衣，當時則榮，沒則已焉，孔子布衣，

傳十餘世，學者宗之。傳十餘世，學者宗之。

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During conversations of my father with students, many During conversations of my father with students, many important points of history of philosophy were important points of history of philosophy were mentioned.mentioned.

1.1. Basic principlesBasic principles The root of the very existence of matter (basic axioms, etc)The root of the very existence of matter (basic axioms, etc) General phenomenaGeneral phenomena Unification of all principles (unified field theory)Unification of all principles (unified field theory)

2.2. Methods of understanding truth based on logic and Methods of understanding truth based on logic and reasoningreasoning

3.3. How to combine different knowledge and different How to combine different knowledge and different phenomena under a general principlephenomena under a general principle

4.4. The great philosophers did not simply follow others The great philosophers did not simply follow others in developing their views, not even from their in developing their views, not even from their teachers. They created their own thoughts (based on teachers. They created their own thoughts (based on previous works.)previous works.)

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The goal of writing history of philosophyThe goal of writing history of philosophy

1.1. (( 求因求因 ) ) The origin of a philosophical thinking must come from The origin of a philosophical thinking must come from different sources. It is our goal to find out such sources.different sources. It is our goal to find out such sources.

2.2. (( 明變明變 ) ) There are many complicated philosophical thoughts in There are many complicated philosophical thoughts in the history. It is important to figure out the treads of their the history. It is important to figure out the treads of their thoughts.thoughts.

3.3. (( 評 論評 論 ) ) A cortical comments of the occurrence of all A cortical comments of the occurrence of all philosophies and their consequences.philosophies and their consequences.

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I also learnt the way to do research with lasting importance.I also learnt the way to do research with lasting importance.

Wang Guowei (Wang Guowei ( 王國維王國維 ))

The Three Levels To Achieve Breakthrough in Research The Three Levels To Achieve Breakthrough in Research (borrowing some lines from Song Ci or lyric poems from the (borrowing some lines from Song Ci or lyric poems from the Sung dynasty)Sung dynasty)  Level OneLevel One

... Last night the west wind withered the greenery of the ... Last night the west wind withered the greenery of the treestrees

In loneliness, I mounted the tall buildingIn loneliness, I mounted the tall buildingCasting my eyesight along all roads to the edge of the Casting my eyesight along all roads to the edge of the

earthearth晏殊晏殊… … 昨夜西風凋碧樹，獨上高樓，望盡天涯路。昨夜西風凋碧樹，獨上高樓，望盡天涯路。

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Level twoLevel two

  ... For my loosening waist belt I feel no regret,... For my loosening waist belt I feel no regret,For her it is worth being haggard and thin.For her it is worth being haggard and thin.柳永柳永… … 衣帶漸寬終不悔，為伊消得人憔悴。衣帶漸寬終不悔，為伊消得人憔悴。

Level ThreeLevel Three

Looking for her a thousand timesLooking for her a thousand timesIn a crowdIn a crowd

All of a suddenAll of a suddenAs I turned my headAs I turned my head

There she wasThere she wasStanding in the shades of fading lightsStanding in the shades of fading lights辛棄疾辛棄疾… … 眾裏尋他千百度，驀然回首，那人卻在，燈火闌珊處。眾裏尋他千百度，驀然回首，那人卻在，燈火闌珊處。

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In my second year of Pui Ching, I got into a problem In my second year of Pui Ching, I got into a problem with my teacher. The teacher was a very devoted head with my teacher. The teacher was a very devoted head master of my class and clearly meant well for me. She master of my class and clearly meant well for me. She was shocked to find out that my father was a professor was shocked to find out that my father was a professor and poorly paid.and poorly paid.

  Her passion for my future changed my behavior in Her passion for my future changed my behavior in

classroom.classroom.  I studied plane geometry in second year of high I studied plane geometry in second year of high

school.school.

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My classmates were not used to reason abstractly.My classmates were not used to reason abstractly.

The mere fact that I listened to my father’s philo-The mere fact that I listened to my father’s philo-sophical discussion at home made me feel at home with sophical discussion at home made me feel at home with axiomatic approach.axiomatic approach.

  In fact, I felt I can understand my father’s In fact, I felt I can understand my father’s

conversation better after I learnt geometry.conversation better after I learnt geometry.  The charm to prove elegant theorems based on simple The charm to prove elegant theorems based on simple

axioms excites me.axioms excites me.

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With the passion for geometry, I started to develop my With the passion for geometry, I started to develop my taste for mathematics, which included algebra. taste for mathematics, which included algebra. Everything became easy after I found the subject Everything became easy after I found the subject exciting.exciting.  I also found the subject of history interesting.I also found the subject of history interesting.  It taught me a global view of everything that I learnt.It taught me a global view of everything that I learnt.

  How events happened?How events happened?Why they happened?Why they happened?

What may happen in the future?What may happen in the future?

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At this time, my father just finished writing his book At this time, my father just finished writing his book on history of Western philosophy. His conversations on history of Western philosophy. His conversations with students taught me the way that we should see with students taught me the way that we should see history in a global manner.history in a global manner.

  This kind of practice deeply influence my way of This kind of practice deeply influence my way of

looking at my research projects in the later time.looking at my research projects in the later time.

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Around 400 A.D.Around 400 A.D.

Liu (who wrote the first book on comparative literature on Liu (who wrote the first book on comparative literature on Chinese writings up to that period) Chinese writings up to that period) 文心雕龍 諸子文心雕龍 諸子  

My body may perish along with time.My body may perish along with time.My goal and my ambition will extend with my theory.My goal and my ambition will extend with my theory.

My heart is in resonance with those great men in My heart is in resonance with those great men in ancient days.ancient days.

My feeling and my theory will go forward for the next My feeling and my theory will go forward for the next thousand years.thousand years.

身與時桀身與時桀，，志共道申。標心於萬古之上志共道申。標心於萬古之上，，而送懷於千載之下。而送懷於千載之下。

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When I was fourteen, my father passed away. This When I was fourteen, my father passed away. This was perhaps the greatest shock to my life. For a long was perhaps the greatest shock to my life. For a long time, I could not believe that my father left me and the time, I could not believe that my father left me and the family.family.

  The financial situation of the family was really bad. It The financial situation of the family was really bad. It

was not clear at all that we could still go to school. The was not clear at all that we could still go to school. The tremendous will of my mother and the helps of my tremendous will of my mother and the helps of my father’s friends and students made it possible.father’s friends and students made it possible.

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This disastrous change of family situation made me This disastrous change of family situation made me much more mature. The extreme hard-ship showed much more mature. The extreme hard-ship showed difficulty of human relationship, what I learnt from my difficulty of human relationship, what I learnt from my father became practical.father became practical.

  The poems and the classical essays that I learnt The poems and the classical essays that I learnt

became much more meaningful. I spent half a year to became much more meaningful. I spent half a year to read classical literature and history of China. It became read classical literature and history of China. It became a way for me to relax during very tense situation in life. a way for me to relax during very tense situation in life. The beautiful poems guided me to appreciate the beauty The beautiful poems guided me to appreciate the beauty of nature.of nature.

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英國大詩人拜倫英國大詩人拜倫  〝希臘啊！你本是平和時代的愛嬌，你本是戰爭時代的〝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。〞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〝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。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，今生便難道我為奴為隸，今生便了？不信我為奴為隸，今生便了。〞了。〞  

梁啟超翻譯梁啟超翻譯

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From Byron’s Don Juan II -- A variation edition by Steffan PrattFrom Byron’s Don Juan II -- A variation edition by Steffan PrattCanto III, 86Canto III, 86

1. The isles of Greece, the isles of Greece!1. The isles of Greece, the isles of Greece! Where burning Sappho loved and sung,Where burning Sappho loved and sung, Where grew the arts of war and peace,Where grew the arts of war and peace, … … Where Delos rose, and Phoebus sprung!Where Delos rose, and Phoebus sprung! Eternal summer gilds then yet,Eternal summer gilds then yet, But all, except their sun, is set.But all, except their sun, is set.

2. The mountains look on Marathon …2. The mountains look on Marathon … And Marathon looks on the sea;And Marathon looks on the sea; And musing there an hour alone,And musing there an hour alone, I dream’d that Greece might still be free;I dream’d that Greece might still be free; For standing on the Persian’s grave, I could not deem myself a For standing on the Persian’s grave, I could not deem myself a

slave.slave.

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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 thoughtthought 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.

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When I was fifteen, I started to teach lower grade When I was fifteen, I started to teach lower grade level students to earn money. I was proud that by level students to earn money. I was proud that by teaching in a novel way, I was able to transform some teaching in a novel way, I was able to transform some very poor student to become the best student in class.very poor student to become the best student in class.

  It was an experience to train young people. I also It was an experience to train young people. I also

learnt that it is beneficial to myself to teach.learnt that it is beneficial to myself to teach.

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My high school teachers in mathematics were My high school teachers in mathematics were excellent. We studied rather advanced topics in excellent. We studied rather advanced topics in mathematics. I had no difficulty with them.mathematics. I had no difficulty with them.

  However, I was rather disappointed that my physics However, I was rather disappointed that my physics

teacher was not good enough. The fundamental teacher was not good enough. The fundamental intuition in physics was not established during my high intuition in physics was not established during my high school year. I regretted it up to now.school year. I regretted it up to now.

  I had an excellent teacher in Chinese. He was my I had an excellent teacher in Chinese. He was my

father’s friend: he taught us to think in a non-traditional father’s friend: he taught us to think in a non-traditional manner.manner.

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We were asked to think creatively. He said that we We were asked to think creatively. He said that we should read good books but also bad books as possible should read good books but also bad books as possible comparisons. So I read everything. This is true even comparisons. So I read everything. This is true even for my scientific career.for my scientific career.

  

A typical topic for our essay in our Chinese writing: A typical topic for our essay in our Chinese writing: the philosophy of a pig.the philosophy of a pig.

  

So we start to dream about anything interesting.So we start to dream about anything interesting.  

I was not the best in my high school. I did not have I was not the best in my high school. I did not have the highest grade in mathematics. But I think deeper the highest grade in mathematics. But I think deeper than my classmates and I read much more books.than my classmates and I read much more books.

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I entered The Chinese University of Hong I entered The Chinese University of Hong Kong in 1966. I chose mathematics as my Kong in 1966. I chose mathematics as my career although I was also very interested in career although I was also very interested in the subject of history.the subject of history.

  By this time, I started to digest those By this time, I started to digest those advanced level mathematics books that I read advanced level mathematics books that I read in high school. I did not quite understand in high school. I did not quite understand those books at the beginning. Suddenly I those books at the beginning. Suddenly I understand them and I was much better than understand them and I was much better than the contemporary students.the contemporary students.

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  崇基門前對聯崇基門前對聯

  崇高惟博愛本天地立心無間東西溝通學術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基礎在育才當海山勝境有懷胞與陶鑄人群群    

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Chung Chi College AnthemChung Chi College Anthem

Men from four seas founded Chung Chi so that here might youthMen from four seas founded Chung Chi so that here might youthHonour Christ, eternal teacher, who Himself is truth.Honour Christ, eternal teacher, who Himself is truth.Through the long night keep the torch bright and the work begunThrough the long night keep the torch bright and the work begunTill the lights of faith and knowledge show the world made one.Till the lights of faith and knowledge show the world made one.

China's still evolving culture, grateful, we retain;China's still evolving culture, grateful, we retain;East and West, through freely sharing, further strength obtain,East and West, through freely sharing, further strength obtain,By the Church upheld and nurtured, minds to duty drawn,By the Church upheld and nurtured, minds to duty drawn,Chung Chi, toward the very highest, lead us on, and on!Chung Chi, toward the very highest, lead us on, and on!

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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.

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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.

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The importance of such trainings is that:The importance of such trainings is that: I learnt how to think independently.I learnt how to think independently. I found out how important it is to express I found out how important it is to express

mathematics in front of an audience.mathematics in front of an audience.

These points have been important for me and for my These points have been important for me and for my teaching.teaching.

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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.

37

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.

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When I met those first rated mathematicians, I was When I met those first rated mathematicians, I was rather thirsty in learning different subjects from them. I rather thirsty in learning different subjects from them. I attended many classes from 8am to 5pm. (Sometimes I attended many classes from 8am to 5pm. (Sometimes I ate lunch in class.) These are subjects ranging from ate lunch in class.) These are subjects ranging from topology, geometry, differential equations, Lie groups, topology, geometry, differential equations, Lie groups, number theory, combinatorials, probability theory to number theory, combinatorials, probability theory to dynamical system. I did not understand all of them. dynamical system. I did not understand all of them. But I focus my efforts on several of them.But I focus my efforts on several of them.

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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.

40

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.

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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.

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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.

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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.

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In that summer, I asked Prof. Chern to be my advisor. In that summer, I asked Prof. Chern to be my advisor. He agreed and after one month, he said that my papers He agreed and after one month, he said that my papers in the first year should be enough for my thesis. I was in the first year should be enough for my thesis. I was surprised because I thought they are not good enough surprised because I thought they are not good enough and I wanted to learn more.and I wanted to learn more.

  In any case, in the second year, I learnt more in the In any case, in the second year, I learnt more in the

field of complex geometry and topology. Prof. Chern field of complex geometry and topology. Prof. Chern had a great expectation on me. He suggested me to had a great expectation on me. He suggested me to work on Rieman hypothesis. Unfortunately up to now, work on Rieman hypothesis. Unfortunately up to now, I never thought about it.I never thought about it.

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Instead I pursued the general understanding of Instead I pursued the general understanding of curvature of space. I decided a key to understand such curvature of space. I decided a key to understand such a concept was a proposal made by Calabi in early a concept was a proposal made by Calabi in early fifties. Nobody believed what Calabi’s thought is true. fifties. Nobody believed what Calabi’s thought is true. I started to think about it. It is not the standard thing I started to think about it. It is not the standard thing that a geometer would do in those days. It is clearly a that a geometer would do in those days. It is clearly a hard question of analysis. Nobody would touch such hard question of analysis. Nobody would touch such problem.problem.

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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.

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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.

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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).

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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.

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Tao Yuan-Ming (Tao Yuan-Ming ( 晉晉‧‧陶淵明陶淵明 ))Long I lived checked by the bars of a cage. Now I Long I lived checked by the bars of a cage. Now I

have turned again to Nature and Freedom.have turned again to Nature and Freedom.久在樊籠裡，復得返自然。久在樊籠裡，復得返自然。

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When I arrived at Stanford, there was a big When I arrived at Stanford, there was a big conference in geometry. A physicist was involved to conference in geometry. A physicist was involved to give a talk on general relativity.give a talk on general relativity.

Although I did not understand Physics well, I Although I did not understand Physics well, I immediately fell in love of the geometry problem immediately fell in love of the geometry problem associated to general relativity. It is fascinating to give associated to general relativity. It is fascinating to give physical meaning of space that we saw and vice versa. physical meaning of space that we saw and vice versa.

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The problem was too difficult to solve at that time. The problem was too difficult to solve at that time. But I kept that in mind.But I kept that in mind.

During the conference, I thought I found a way to During the conference, I thought I found a way to disprove the proposal of Prof. Calabi. I was asked to disprove the proposal of Prof. Calabi. I was asked to give a presentation of my thoughts. It all sounded good. give a presentation of my thoughts. It all sounded good. Nobody objected. So every people was happy that the Nobody objected. So every people was happy that the general expection was true: the Calabi Conjecture is general expection was true: the Calabi Conjecture is wrong after all.wrong after all.

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After two months, Prof. Calabi wrote to me for After two months, Prof. Calabi wrote to me for clarification of my thoughts.clarification of my thoughts.

I found a serious gap on my reasoning. It was the I found a serious gap on my reasoning. It was the most painful period of my research life. I could not most painful period of my research life. I could not sleep.sleep.

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For about two weeks, I could not sleep as I For about two weeks, I could not sleep as I considered that my reputation was badly damaged by considered that my reputation was badly damaged by not able to reproduce what I claimed. (Although I never not able to reproduce what I claimed. (Although I never wrote any announcement of it.) However, the pain of wrote any announcement of it.) However, the pain of going through of each single details of the problem going through of each single details of the problem convinced me that the opposite direction should be convinced me that the opposite direction should be right. The argument to give counterexample to the right. The argument to give counterexample to the Calabi Conjecture was that if it were true, something Calabi Conjecture was that if it were true, something must happen. Hence a few years later, when I settled must happen. Hence a few years later, when I settled this problem, I knew many natural consequences of it.this problem, I knew many natural consequences of it.

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After I decided that it must be true, I worked towards After I decided that it must be true, I worked towards the right direction. Many preparatory works were done the right direction. Many preparatory works were done to prepare for the final proof. I worked with Cheng on to prepare for the final proof. I worked with Cheng on understanding my questions related to Monge-Ampere understanding my questions related to Monge-Ampere equations, affine geometry, maximal surfaces and many equations, affine geometry, maximal surfaces and many related problems. Richard Schoen worked with me on related problems. Richard Schoen worked with me on harmonic maps. Schoen, Simon and I worked on harmonic maps. Schoen, Simon and I worked on minimal surfaces. In a matter of two years, we minimal surfaces. In a matter of two years, we understood a great deal of nonlinear analysis related to understood a great deal of nonlinear analysis related to geometry. It was an exciting period of time in geometry. It was an exciting period of time in geometry.geometry.

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Qu Yuan (Qu Yuan ( 屈原屈原 ))

This is what I am after.This is what I am after.I will not turn back even nine I will not turn back even nine

deaths are ahead.deaths are ahead.

固余心之所善兮，雖九死而猶未悔。固余心之所善兮，雖九死而猶未悔。

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Right after I got married, I got the right idea to finish Right after I got married, I got the right idea to finish the proof of the Calabi Conjecture. I felt I finally the proof of the Calabi Conjecture. I felt I finally understood curvature of Kahler geometry.understood curvature of Kahler geometry.

Many important applications were found to solve Many important applications were found to solve some old problems in algebric geometry.some old problems in algebric geometry.

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““One problem is that string theory requires 10 One problem is that string theory requires 10 dimensions of space-time, whereas we appear to live in dimensions of space-time, whereas we appear to live in four. Dr. Strominger remembered being excited when he four. Dr. Strominger remembered being excited when he found a paper by the mathematician Dr. Shing-Tung Yau, found a paper by the mathematician Dr. Shing-Tung Yau, now of Harvard and the Chinese University of Hong now of Harvard and the Chinese University of Hong Kong. It proved a conjecture by Dr. Eugenio Calabi, now Kong. It proved a conjecture by Dr. Eugenio Calabi, now retired from the University of Pennsylvania, that the extra retired from the University of Pennsylvania, that the extra dimensions could be curled up in microscopically dimensions could be curled up in microscopically invisible ways like the loops in a carpet.”invisible ways like the loops in a carpet.”

One Cosmic Question, Too Many AnswersOne Cosmic Question, Too Many Answers

September 2, 2003September 2, 2003

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When I finished the proof of the Calabi Conjecture, I When I finished the proof of the Calabi Conjecture, I felt I have set up a framework for mathematicians to felt I have set up a framework for mathematicians to merge two important fields together: nonlinear partial merge two important fields together: nonlinear partial differential equation and geometry.differential equation and geometry.

In 1976, I was in UCLA and I met my friend Meeks In 1976, I was in UCLA and I met my friend Meeks who was my classmate in graduate school. He was not who was my classmate in graduate school. He was not in good shape. But he was a very original in good shape. But he was a very original mathematician. So I proposed to work with him on mathematician. So I proposed to work with him on relating ideas from minimal surfaces and topology of relating ideas from minimal surfaces and topology of three dimensional manifold together.three dimensional manifold together.

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We had a great success. We solved two classical We had a great success. We solved two classical problems in both fields:problems in both fields:

1.1. If the boundary of a soap film is convex, the soap film If the boundary of a soap film is convex, the soap film cannot cross itself.cannot cross itself.

2.2. Together with works with Thurston, the famous problem of Together with works with Thurston, the famous problem of Smith Conjecture is solved.Smith Conjecture is solved.

Once the direction was set in the right way, many Once the direction was set in the right way, many classical questions can be answered.classical questions can be answered.

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Next year, I visited Berkeley where I gave seminars Next year, I visited Berkeley where I gave seminars on nonlinear problems in geometry. Both Richard on nonlinear problems in geometry. Both Richard Schoen and S. Y. Cheng were there. With Schoen, we Schoen and S. Y. Cheng were there. With Schoen, we finally solved the problem that I was excited about in finally solved the problem that I was excited about in general relativity.general relativity.

The problem is called positive mass conjecture and is The problem is called positive mass conjecture and is fundamental for general relativity. (Only if the mass is fundamental for general relativity. (Only if the mass is positive that spacetime can be stable.)positive that spacetime can be stable.)

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In 1978, I returned to Stanford, I applied ideas of In 1978, I returned to Stanford, I applied ideas of minimal surfaces to solve a famous problem (Frenkel minimal surfaces to solve a famous problem (Frenkel Conjecture) in complex geometry with Y.T. Siu. I also Conjecture) in complex geometry with Y.T. Siu. I also introduced ideas of harmonic map to studying discrete introduced ideas of harmonic map to studying discrete group symmetries. These ideas are still useful up to group symmetries. These ideas are still useful up to now.now.

Schoen and I developed the structure theory of Schoen and I developed the structure theory of manifolds with positive scalar curvature based on our manifolds with positive scalar curvature based on our works on general relativity.works on general relativity.

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In 1979, we had a special year in differential In 1979, we had a special year in differential geometry in the Institute for Advanced Study. geometry in the Institute for Advanced Study. Practically all geometers came. We set a good Practically all geometers came. We set a good direction for geometry. I proposed one hundred direction for geometry. I proposed one hundred interesting open problems for the field of geometry. interesting open problems for the field of geometry. Some of them were solved and some of them were not. Some of them were solved and some of them were not.

The 1970’s have been one of the most fruitful era of The 1970’s have been one of the most fruitful era of geometry.geometry.

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By late seventies, I was well-recognized by my By late seventies, I was well-recognized by my colleagues. There were many news coverage on colleagues. There were many news coverage on problems that I solved.problems that I solved.

However, it will be misleading to think that my goal However, it will be misleading to think that my goal is to earn medals and gain recognition. It has never is to earn medals and gain recognition. It has never been the priority of my study.been the priority of my study.

I am interested in Mathematics because it is so I am interested in Mathematics because it is so exciting to see how human thought can be extended to exciting to see how human thought can be extended to understand nature. The beauty of nature from the point understand nature. The beauty of nature from the point of view of geometry is everlasting.of view of geometry is everlasting.

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Together with my friends Schoen, Simon, Cheng, Together with my friends Schoen, Simon, Cheng, Meeks, Uhlenbeck, Hamilton, and later by Donaldson, Meeks, Uhlenbeck, Hamilton, and later by Donaldson, Taubes and Huisken, and others, Nonlinear Analysis in Taubes and Huisken, and others, Nonlinear Analysis in Geometry has been established as a rich subject. Its Geometry has been established as a rich subject. Its importance in understanding beauty of nature can never importance in understanding beauty of nature can never be underestimated. The most recent developments be underestimated. The most recent developments show their importance in Physics and in Applied show their importance in Physics and in Applied Science.Science.

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Once a natural merge of several great subjects: Once a natural merge of several great subjects: Geometry, Non-linear Analysis, Algebraic Geometry, Geometry, Non-linear Analysis, Algebraic Geometry, Mathematical PhysicsMathematical Physics, is done, classical and difficult , is done, classical and difficult problems were naturally solved. Problem solving can problems were naturally solved. Problem solving can be considered as lampstands on our road to understand be considered as lampstands on our road to understand nature.nature.

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Confucius Confucius (( 孔子孔子 )) Studying without thinking goes Studying without thinking goes

nowhere; thinking without nowhere; thinking without studying leads to bewilderment.studying leads to bewilderment.

學而不思則罔，思而不學則殆。學而不思則罔，思而不學則殆。

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Han (Han ( 韓愈 韓愈 600 AD) on the process of learning:600 AD) on the process of learning:

When general people praise my writing, I am not When general people praise my writing, I am not pleased.pleased.

When general people slight my writing, I am not When general people slight my writing, I am not depressed.depressed.

譽之則不以為喜，毀之則不以為憂。譽之則不以為喜，毀之則不以為憂。

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C.F. Gauss (1817)C.F. Gauss (1817)

I am becoming more and more I am becoming more and more convincing that the necessity of our convincing that the necessity of our geometry cannot be proved, at least not geometry cannot be proved, at least not by human reason nor for human reason. by human reason nor for human reason. Perhaps in another life we will be able to Perhaps in another life we will be able to obtain insight into the nature of space obtain insight into the nature of space which is now unattainable.which is now unattainable.

Until then we must place geometry Until then we must place geometry not in the same class with arithmetic not in the same class with arithmetic which is purely a priori, but with which is purely a priori, but with mechanics.mechanics.

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An abstract two dimensional space with complex coordinates is An abstract two dimensional space with complex coordinates is called Riemann surface.called Riemann surface.

In the following pictures, we use these concepts to draw a map In the following pictures, we use these concepts to draw a map over the surfaces of a teapot, a bunny, and a minimal surface. over the surfaces of a teapot, a bunny, and a minimal surface. Note that this map is a generalization of latitude and longitude Note that this map is a generalization of latitude and longitude of the globe. They are orthogonal to each other.of the globe. They are orthogonal to each other.

Let’s now demonstrate an old concept of geometry to Let’s now demonstrate an old concept of geometry to computer graphicscomputer graphics

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Application of Gaussian Curvature to Application of Gaussian Curvature to Face RecognitionFace Recognition

Note that the color on the face corresponds to level sets Note that the color on the face corresponds to level sets of the Gaussian curvature of the faceof the Gaussian curvature of the face

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Riemann SurfaceRiemann Surface

Holomorphic transformation between surfacesHolomorphic transformation between surfaces

demodemo

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In the last ten years, my coauthors and I are very In the last ten years, my coauthors and I are very much involved in understanding the role of much involved in understanding the role of fundamental physics in geometry. In order to gain the fundamental physics in geometry. In order to gain the intuition behind the motivation from physics. I spent a intuition behind the motivation from physics. I spent a lot of time attending their seminars in physics lot of time attending their seminars in physics department. We were able to obtain deep theorems in department. We were able to obtain deep theorems in mathematics based on this interaction. A very mathematics based on this interaction. A very important concept is duality:important concept is duality:

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Duality is a beautiful and elegant concept.Duality is a beautiful and elegant concept.

It says that some theory with strong coupling is the It says that some theory with strong coupling is the same as other theory with weak coupling.same as other theory with weak coupling.

It is very similar to Chinese Taoism or Yingyang.It is very similar to Chinese Taoism or Yingyang.

But the concept of duality is more rigorous and But the concept of duality is more rigorous and quantitive. It enables us to calculate some quantitive. It enables us to calculate some mathematical quantities that are otherwise difficult to mathematical quantities that are otherwise difficult to calculate.calculate.

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I do not hesitate to say that I do not hesitate to say that mathematics deserves to be mathematics deserves to be cultivated for its own sake, and that cultivated for its own sake, and that the theories of not admitting the theories of not admitting applications to physics deserve to be applications to physics deserve to be studied as well as others.studied as well as others.

- H. POINCARE- H. POINCARE

為數學而數學為數學而數學，，實屬顯然實屬顯然，，何須三何須三思。於無用諸物理學之種種數學理思。於無用諸物理學之種種數學理論論，，均需一視同仁均需一視同仁，，與其它理論無與其它理論無分軒輊。分軒輊。- - 彭加箂彭加箂

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… … and, believe me, if I were again and, believe me, if I were again beginning my studies I would beginning my studies I would follow the advice of Plato and start follow the advice of Plato and start with mathematics.with mathematics.

- GALILEO- GALILEO

使余復稚年，童蒙初習，則願從使余復稚年，童蒙初習，則願從柏拉圖之教晦，自數學始。柏拉圖之教晦，自數學始。- - 伽里略伽里略

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My father took part in the establishment of the Chinese My father took part in the establishment of the Chinese University in its early stages. His three sons, including the University in its early stages. His three sons, including the speaker, were educated in the same school. I am deeply speaker, were educated in the same school. I am deeply honored to be able to make contribution towards the honored to be able to make contribution towards the university.university.

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Hong Kong has played an important role in modern Hong Kong has played an important role in modern history. There are many outstanding scholars from Hong history. There are many outstanding scholars from Hong Kong who are now working all over the world. Among the Kong who are now working all over the world. Among the four Hong Kong professors in Harvard, it is interesting to four Hong Kong professors in Harvard, it is interesting to observe that three of them have been professors of the observe that three of them have been professors of the Chinese University. I firmly believe that under the support Chinese University. I firmly believe that under the support of our government and our society with vision, Hong Kong of our government and our society with vision, Hong Kong is the city with the greatest potential to nourish future Nobel is the city with the greatest potential to nourish future Nobel or Fields laureates. Our university should see this as a goal, or Fields laureates. Our university should see this as a goal, not a burden.not a burden.

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Alumni of Mathematics DepartmentAlumni of Mathematics Department

http://www.math.cuhk.edu.hk/alumnihttp://www.math.cuhk.edu.hk/alumni

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Thank you !Thank you !