1

• Mathematics and its role
• in Science and Technology
• Professor Shing Tung Yau
• Beijing
• March 24, 2004
•  
• It is a great honor for me to be
  recognized by the Chinese government on my
  efforts to promote scientific interchange between
  China and other counties.
•   I was born in south China and grew up in
  Hong Kong. I was educated under the colonial
  system set up by the British. Fortunately, my
  father was proud of Chinese culture and sent me
  to a Chinese middle school.

• ?????
•  
• ?????
•  
• ??
• ???????????
•  
• ??????????
• ????????????
• ????,???????
• ?????,????,
• ????????????
• ????????????,
• ???????????

2

• ?????????????????,??????????????????,??????
  ????????????? 
• ????????,????????????????,??????,??????????
  ?????????????,?????????????,?????????,?????
  

• I was also educated in The Chinese
  University of Hong Kong. While my training on
  mathematics and science in Hong Kong could not be
  said to be great, I was fortunate to learn
  Chinese culture in a reasonable thorough manner.
•   The elegance of Chinese literature had
  deep influence on my thinking. I am proud of the
  deep and continuous developing civilization of my
  mother country. While I devoted my whole life to
  study fundamental science, I also consider
  promotion of science, especially for my mother
  country, to be a responsibility of my life.

3

• While I left Hong Kong in 1969, I did
  not have passport of any country. At that time,
  it was not clear at all that the Chinese
  government would build a good relation with the
  United States. I did watch with great joy when
  Nixon visited China. In 1979, I was invited by
  Professor Hua to visit the Academic Sinica. I
  learnt a lot from Professor Huas book when I was
  in high school. I admired him and was flattered
  to be invited.

• 1969??????,????????????????????????,????????
  ???,????????,???????1979??????????????????????,???
  ??????????,????,?????????????,??????????

4

• When I stepped out from the airplane in
  the Beijing airport, I touched the soil of
  Beijing and felt great joy to return to my mother
  country. I am proud to say that when I was
  awarded the Fields Medal in mathematics, I held
  no passport of any country and should certainly
  be considered as Chinese.
• I did put in a lot of efforts to help
  Chinese mathematics. On the other hand, I
  regretted very much that I still could not settle
  down in China. Whatever I have done for Chinese
  mathematics can in no way be compared with those
  Chinese who have grown up and stayed here or
  those who returned to China on a permanent basis.

• ????,????????,??????????,?????,????????????
  ??????,????????????,????????????????????
•   ?????????????????????,?????????????,????
  ?????,????????,??????,?????????,???????

5

• It will be an exaggeration to attach
  too much importance for those who hold a
  permanent position in foreign soil to talk about
  development of this great country.
• I will therefore only make some humble
  comments based on my observations.
• As we know, China has developed
  education system since Confucius time. Knowledge
  is accessible even to non-nobleman. It was a
  break through.

• ?????????????,????????,???????????,?????
  
• ??,????????,????,??????,??????,??????
• ????????,???????????,?????????????????????,
  ????????????

6

• By Han dynasty, the government started
  to build a system to select educated man and man
  with virtue to serve the country. Even kids in
  villages have potential to be in the central
  government. The system was reasonably fair and
  China is united partially based on such
  outreached system. It may be interesting to note
  that even foreigners were able to hold high
  position in the court at that time.
• The system was polished and became
  system of examinations. The scope of examinations
  was rather large during early dynasties, like
  Tang dynasty. Even mathematics was included in
  the test. However, the examination has been much
  more focused in the past four hundred years. It
  was thought that a thorough study of Confucius
  thinking is enough to run the government. Hence
  the test has been focused on a very narrow band
  of knowledge. The damage to the creative thinking
  in China is almost fatal.

• ????,?????????????,??????,?????????????????
  ?,????,??????????,?????????????????,??????????????
  ?
•   ????????,???????????????????--????--???????
  ?????,????????????????,???????????????????????,???
  ????????,????????????,???????????????,?????????

7

• It is important to realize that
  Confucius considered knowledge to be virtue
• ????,????,
• ???,??????
• But Confucius did train students on
  practical matter. Some of his students became
  diplomats. Some of them became businessman.
  Some of them became general. The Greek
  philosophers Socrates also consider knowledge to
  be virtue. But pursuing of truth and beauty is
  the essence of Greek education. In this daunting
  new world, knowledge is the key to human
  well-being. Any great country will have to
  substain long term investment in learning,
  without which social advancement is an empty
  dream. Knowledge has to be based on the
  following
• 1. Virtue, Ethics
• 2. Knowledge of humanity
• 3. Basic science
• 4. Applied Science

• ????,????????????
• ????,????,
• ???,?????? 
• ????,?????????????????,???????,?????,??????
  ?????????????????????????????????????? (daunting
  new world) ?,????????????????????????????,????,???
  ???????????????
• 1. ????
• 2. ????
• 3. ????
• 4. ????

8

• Since the defeat of Opium war in the
  early nineteenth century, China realized its own
  weakness in modern technology and tried several
  times to modernize. The major development was on
  building ships, railways, mining and building
  weapons. After two centuries of trial and error,
  we finally see a true opportunity to develop
  China into a great country. While economic growth
  is unpreceding strong in modern Chinese history,
  we should remember that the true engine of
  modernization is based on acquisition and
  application of knowledge.
• However, for a developing country,
  knowledge is usually associated to applied
  science only. One tends to overemphasize
  short-term payoffs and ignores long-term
  strength. One should note that Fundamental
  science is the base of all modern technology.
  Chinas modernization should realize the
  importance of basic science.

• ?????????????,???????????????,?????????????
  ???????????????????????????????????????????,????,?
  ??????????????????????????,???????????,????????,??
  ???????,??????????? 
• ??,???????????????????????????????,????????
  ???????,??????,???????????????,???????????????

9

• Michael Atiyah who is president of
  Royal Society of England, told me the following
• For a country like China which has
  aspirations to become a world economic power, it
  cannot be lack of ambition. Certainly the
  Japanese who started off with the plan of just
  copying the west soon changed to invest in
  fundamental research.
•   Also the US is the most advanced economically
  and is supporting fundamental research in a big
  way. I would think that China world be receptive
  to the challenge of competing in all respects
  with Japan and the USA.

• ????? (Michael Atiyah) ???????????,????????
   
• ????????????,???????,???????????,??????,???
  ????,?????????
•   ???????????,????????????????????,??????,????
  ???????????

10

• ?????,?????????????,?????
• ????
• ????
• ????
• ????
• ????
• ?????
• ????
• ?????????,??????????????,???????????
  ????????,????????,??????????????,?????????????,???
  ????????

• There are several important
  branches of
• sciences and technology that will play
• fundamental role in this century
• Information technology
• Life science
• Energy science
• Material science
• Environment science
• Economics and finance
• Social science
•  While these branches do interact with
  each other, they depend heavily on the
  development of fundamental science who provide
  the principle why and how things actually work.
  Interdependence of scientific disciplines recurs
  frequently throughout history. A successful
  merge of ideas of two apparently independent
  fields always brings unexpected glory to both
  subjects.

11

• ?????,??????????????????,????????????????,??
  ???????????????????,??????????????????????????????
  ?????????????,????????????
• ???????,???????????????,???????????????????
  ??????,????????,???????

• In the nineteenth century, we saw the
  great success of unification of electricity and
  magnetism. In the twentieth century, we saw the
  great success of application of quantum mechanics
  to chemistry. We saw application of mathematics
  and physics to the building of modern computer
  which in turn become the fundamental tool of all
  science and technology. Nowadays, we are
  witnessing the glory of application of physical
  science to life science.
• Each unification started from basic
  science. By the time when it is successful,
  great breakthrough in technology was rewarded.
  It brought great economic growth for those
  countries who initiated and enhanced such
  development.

12

• In the past two centuries, we saw the
  accumulation in wealth of European countries
  based on scientific and technological advances.
  The first and second world wars brought a large
  group of international scientists and engineers
  to the United States. We saw the American
  dominance of the world. The unparalleled
  prosperity of United States is driven by
  technology growing out of the investment in basic
  scientific research over the past fifty years.
  Large portions of the patent rights held by
  American companies and universities came from the
  strength of basic science.

• ????????,???????????????????????????????????
  ????????????,????????????????????,???????????,????
  ?????????????????????????????,???????????

13

• The strength as a world power relies on
  the ability to educate the population in basic
  science and to recruit the best mend in the world
  to work for them. In the case of United States,
  a lot of foreigner talents may not even speak
  English fluently. I believe China should recruit
  many non-Chinese talents even if they may not
  understand about China at the beginning. The
  reason is that science, especially pure science,
  has no border. We can be benefited much more by
  foreigners who are not biased.
• In the twenty first century, mathematics
  will be the most basic subject to be studied.
  Mathematics gives the underpinning structure for
  all science. It is not only the language for all
  science. It has a life by itself.

• ???????????,?????????????,??????????????
  ?????,????????????,?????????????????????????,?????
  ?????,?????????????????,????,?????????
• ??????,????????????????????????,?????????,?
  ????????

14

• I. Mathematics as a basic language
• Geometry is the language of spacetime.
  Calculus is the language of astronomy. Operator
  theory is the language of quantum mechanics.
  Fourier analysis is the language for wave motion.
  All these are subjects that are considered
  exceedingly beautiful by mathematicians. They
  were developed in its own right. The common
  feature of these subjects that were understood by
  mathematicians. They then give rise to the most
  fundamental reason why we have interdisciplary
  interactions between seemingly unrelated subjects.

• ?.???????
• ????????,??????????,??????????????,????????
  ?????????????????,?????????????,????????????,?????
  ?????????????????????,???????????????

15

• Let us look at common language. While
  language may be a symbol, it does convey our
  emotion and our feeling of the subject that we
  describe. Chinese poems are not the same as
  Western poems because we do emphasize the use of
  each Chinese character which carry different
  meanings in general. Even among Chinese poems,
  the number of characters that are allowed in
  composing the poem do change the feeling that we
  want to convey. Ancient Poems were more
  flexible. Han poems used five characters per
  sentence. Tang poems use seven characters. Sung
  developed Qi which had different number of
  characters for each sentences. Each style of
  poems reflects the feeling of schlors of these
  dynasties.

• ??????????????????,??????????????????,?????
  ????????,?????????????????,???????,???????????????
  ??????,???????,??????,????,????????-????????,?????
  ???????????

16

• ??,????????????????????,????????????,??????
  ??????????????,?????????????????????????,?????????
  ??????????????????????????

• Hence mathematics research does change
  the scientific development that we are working
  with. For example, a deep understanding of
  Fourier analysis or wavelet can give much
  different understanding of wave motion or
  technique of imaging. Conversely, the practical
  world did influence the development of
  mathematics. The beauty of wave motions and
  their spectrum was the most basic driving force
  to develop the subject. Their important
  application to modern technology and theoretical
  science
• cannot be exaggerated.

17

• It is difficult to imagine that Newton
  can develop classical mechanics without the great
  language of calculus whose development can be
  traced back to Archimedes.
• There is no doubt that Farady understood
  electricity and magnetism. But the final
  completion of the theory was due to the Maxwell
  equations. Its implications for understanding
  light, radio wave and modern science is
  tremendous.
• II. Mathematics as a science of order
• Besides being a language, and a subject
  of great beauty on its own right, mathematics is
  a science of order. Let us read a citation of a
  Harvard professor, Andrew Gleason

• ???????????????????,??????????????
• ????,????????????????????????????????????????
  ????????????????
• ?.????????
• ????????,??????????,????????? (a science
  of order)?????????????????????? (Andrew Gleason)
  ???

18

• Mathematics is the science of order
  its object is to find, describe and understand
  the order that underlies apparently complex
  situations. The principal tools of mathematics
  are concepts which enable us to describe this
  order. Precisely because mathematicians have been
  searching for centuries for the most efficient
  concepts for describing obscure instances of
  order. Their tools are applicable to the outside
  world for the real world is the very epitome of
  a complex situation in which there is a great
  deal of order.
• Therefore it is not surprising to see
  that mathematics has strong applications in
  economics where several Nobel Laurents were
  awarded based on their works related to
  mathematics. We expect its application to Social
  science and history through Data mining and
  statistic.

• ?????????,?????????????????????????????,???
  ???????????????????????????????????????????????,??
  ??????????????,??????????
•   ????,?????????,???????????????????,????????
  ??

19

• ?.???????
• ???????,???????????????,?? (N. Wiener)
  ????,?????????????????????????????????,???????????
  ????Bucy-Kalman ???????????????,??????????????????
  

• III. Mathematics as a tool
• A lot of important mathematics were
  developed with sole motivation to study problems
  in engineer. For example, N. Wiener and his
  students pioneer the study of information
  science. However, the theory of stochastic
  differential equations, the theory of Wiener
  measure, and the theory of entropy have gone far
  beyond the original purpose of their study. The
  theory of Bucy-Kalman filter is absolutely
  fundamental for modern Control theory. The
  theory of Shock wave is important for design of
  airplanes.

20

• IV. Mathematics as a subject of beauty
• The purest branch of mathematics is
  perhaps number theory. It goes back to ancient
  days in Babylon, in Greece and other countries.
  It is a subject of great beauty. No great
  mathematicians can resist the appreciation of it.
  Yet in the past twenty years, we see its
  importance in application to question of
  security. Crytography depends in a large scale on
  questions related to factorization of integers to
  prime numbers, development of Self-corrected
  Codes depends on algebraic geometry.
• Geometry grew out from the desire to
  survey land and for navigation. While it still
  serves the same purpose, its power has gone far
  beyond the original motivation. It is the basic
  building block of fundamental physics of
  spacetime. Its applications include computer
  graphics, Crystallography and tomography.

• ?.?????????
• ??????,??????????????????????????????????,?
  ???????????????????,??????????????????????????????
  ?????????????????????????
• ??????????????????????????,????????????,?????
  ????????

21

• Practically all branches of mathematics
  that were driven by desire to pursue beauty have
  found great use in practical world.
• V. Mathematics in industry
• In 1995, the Society for industrial and
  applied mathematics published a report.
• They survey 75 managers in industry through
  telephone calls. Nearly half (49) of them
  characterized mathematics as an underlying
  requirement or tool for them. Among those
  managers, their education background

• ????????????????????,???????????????
• ?.?????
• 1995????????????????
• ???????????????????????????49?????????????????
  ????????????

Area of managers degrees Ph.D Masters
Mathematics 16 11
Engineering 13 6
Physics 13 3
Statistics / Biostatistics 9 5
Business / Management 0 11
Computer science 0 6
Chemistry / Biology 0 3
?? ?? ??
?? 16 11
?? 13 6
?? 13 3
?? / ???? 9 5
?? / ?? 0 11
??? 0 6
?? / ?? 0 3
22

• The report found the following
• Application of mathematics

• ????????
•  ?????

Algebra and number theory Cryptography
Computational fluid dynamics Air craft and automobile design
Differential Equations Aerodynamics, porous media, finance
Discrete mathematics Communication and information security
Formal systems and logic Computer security, verification
Geometry Computer-aided engineering and design
Optimization Asset allocation, shape and system design
Parallel Algorithms Weather modeling and prediction crash simulation
Statistics Design of experiments, analysis of large data sets
Stochastic processes Signal analysis
????? ???
?????? ???????
???? ???????????
???? ???????
??????? ???????
?? ????????
??? ????????????
???? ?????????
?? ????????????
???? ????
23

• VI. Overview of policy on mathematics
  
• in China
• Chinese government certainly realizes
  the importance of modern technology. There is no
  question that Chinese science and technology has
  experienced tremendous growth in the past ten
  years in terms of outputs of papers. To look at
  mathematics alone, we have the following table
  which shows that the number of Chinese
  mathematics papers grew from 6 to 10 of the
  product of the whole world in a matter of ten
  years. (Although one should note that this
  figure include Chinese mathematicians all over
  the world, a lot of them reside in America).

• ?.  ??????
• ????????????,???????????????????????,??????
  ??,???????????,?????,???????????,??6???10?????,?
  ??????????????????,??????????

24

• Mathematics Review
• (by American Mathematics Society)
• The output of papers by Chinese mathematics

• ?????????
• ????????

Number of Papers Percentage within all published papers
1990 3472 6.1
1991 3944 6.9
1992 4158 7.1
1993 4458 7.9
1994 4654 8.1
1995 5201 8.5
1996 5369 8.6
1997 5800 8.8
1998 6399 9.6
1999 6587 9.5
2000 6677 9.6
2001 6845 9.9
2002 7239 10.4
Number of Papers Percentage within all published papers
1990 3472 6.1
1991 3944 6.9
1992 4158 7.1
1993 4458 7.9
1994 4654 8.1
1995 5201 8.5
1996 5369 8.6
1997 5800 8.8
1998 6399 9.6
1999 6587 9.5
2000 6677 9.6
2001 6845 9.9
2002 7239 10.4
25

• While quantity of publication does give
  an indication of research activities, more detail
  study of published papers show that only a very
  limited amount of these Chinese papers appeared
  in first class journals. The major problem that
  Chinese scientists are facing is how to lift
  quality of research papers. There are not much
  Chinese papers that may pioneer a subject or
  direction of importance. The overemphasize of
  quantity of papers does have negative effect on
  research.

• ??,?????,?????????????????????,????????????
  ??,????????????????????????????????,?????????,????
  ??????????,??????????

26

• Chinese mathematicians are actually
  brilliant and creative. In the late fifties and
  early sixties, Prof. Hua and Prof. Feng had
  pioneered several subjects of mathematics that
  lead the world. The fact that they were more
  isolated in those days actually helps them to
  look for directions of their own.
• The governments overemphasize of the
  leadership of scientists that reside in foreign
  countries create a psycological burden on the
  local scientists who are very capable. Let me
  give an example.
• In the last year, a major news
  appeared in mathematics is the possible
  resolution of a problem of Poincare in topology.
  It depends on the understanding of an equation
  developed by Richard Hamilton. It was a Russian
  called Perelman who claims to be successful in
  carrying out Hamiltons program. He worked on
  this problem very hard in the past seven years.
  Become his claim of success, very few people paid
  attention to this line of research.

• ?????????,?????????????,??????????????????,
  ?????????????????,?????????,????????????
• ???????????,???????,?????????????????????
• ??????????,????????????????????????????????
  ? (R. Hamilton) ????????????????? (Perelman)
  ???,?????????,????????????

27

• However, I am a good friend with
  Richard Hamilton and had also contributed to this
  program. I recognized the importance of his work
  in 1996. So I went to China and told all the
  major leaders in mathematics the importance of
  Hamiltons equation. I urged all the experts in
  the country to study this equation and I told
  them that the reward of research in this equation
  would be tremendous. I sent two of my Ph.D
  students from Hong Kong to run seminars on this
  work in Beijing.
• To my great surprise, experts in
  Beijing, taking advice from some Chinese
  mathematician in American (including my former
  student who was a famous professor in MIT),
  consider the paper of Hamilton too difficult and
  not rewarding. They ordered the seminar to go
  into different directions, against the wishes of
  many young brilliant Chinese participants.

• ???????????????1996?,????????????????????,???
  ????????????,???????,????,????????????????????????
  ?,????????????????
• ??????,??????,???????????????????,?????????
  ???????,???????????,?????,?????????????,??????????
  ???

28

• Incidentally, the group of Chinese
  experts did study the field of geometric analysis
  (closely related to Hamiltons work) under my
  guidance in the eighties. In fact, they took
  notes of some of my lectures with Schoen in
  America in the eighties. The book was written in
  Chinese and published ten years before the
  English version. A large group of Chinese
  mathematicians learnt from the book and they
  wrote a lot of papers in the subject. However,
  due to the drive of mass production, only the
  easier part of the book was studied. The leaders
  of this subject are still yet to find a direction
  for their research in China.

• ?????,???????????????????????????(??????????????
  )????,????????????????????????????????????????????
  ????????,???????????,?????????,??????????????,????
  ??????????????????

29

• Fortunately, mathematicians in south China
  were not influenced by this small group of
  influential mathematics in the North. Prof. Zhu
  of Zhongshan University was doing research in The
  Chinese University of Hong Kong. He accepted the
  challenge and studied this equation deeply. He
  was able to achieve first rated results.
  Unfortunately, he is still not recognized in the
  North.
• I believe that in this new century,
  many important new fields are opening up to be
  studied. Mathematics will be the major tool.
  Chinese mathematicians will play a vital role in
  such development. But they should be open minded
  and creative in their thinking. They should make
  up their mind on what is most important, not
  depending on the urge of mass production and
  fashion only.

• ????????????????????????????????????,??????
  ,?????????,??????????,??????????????,???????????
• ???????????,???????????????????????????????
  ??????,?????????????,?????,????????????

30

• The very major problem that Chinese
  mathematics faces is lack of leadership. The only
  living world-class mathematician in China is
  Prof. S. S. Chern and he is over ninety years
  old. Many universities try to solve the problem
  by offering many visiting Professors who may
  commit their time of one to two months to the
  Chinese institutes.

• ????????????,?????????????????,???????????,
  ??????????????????,?????????,????????,???????????

31

• However, scientific leadership can
  only depend on full time resident in China.
  Unfortunately, many universities are very proud
  and content to put famous names on their list of
  professors without promoting the actual goal of
  science and technology. Most of them are Chinese
  mathematicians resided in foreign soil. They
  cannot devote much time for academic development
  in China. Some of their scientific achievements
  are overexaggerated by their counter parts in
  China. The results of such collaboration have not
  been as successful as was claimed by the
  administrators. On the other hand, I do believe
  international collaboration is very important for
  China.
• I believe scientists with no prior link to
  Chinese community should be encouraged to be
  invited. A true international scientific
  atmosphere will bring Chinese scientific strength
  to a new plateau.

• ??????????????????????????????????,????????
  ,????????????,????????????????????,???????,???????
  ??????????????,???????????????,????,??????????????
  ?????
• ????????????????????????????????,??????????
  ?????

32

• Perhaps this can be achieved by modeling
  after the Institute for Advanced in Princeton
  where Einstein and many great theoretical
  scientists spent their life in leading
  theoretical research for the whole world. Most
  of these leading scientists came from different
  countries. I believe it is important to form
  such an institute with a noble goal. It should
  be open to scientists all over the world. The
  permanent faculty should have high respect from
  the government and should have world-class
  quality. It should not be limited to Chinese
  descendent only. I wish China should be
  world-class in research in a short time.

• ???????,???????????????????????????????????,
  ???????????,??????????????????,???????????????,???
  ????????,???????????????????????????????????????

33

• PoincareScience is no more a collection
  of facts than a house is a collection of bricks.
  The architecture of science is provided by
  mathematics. 
• Of course without bricks or knowledge
  of the bricks, there is no way to make a
  successful design. A close collaboration of
  mathematicians with other scientists will lay the
  foundation for science. Mathematicians should be
  encouraged to interact with other branch of
  science. It is the nature of mathematics to
  broaden itself in response to scientific needs
  and to deepen itself through integral analysis.
  Therefore it should be the center of all science
  for this new century.
• I do hope the Chinese mathematicians to
  be open minded in both national and interactions
  with other disciplines.

• ??????,??????????
• --???
•   ??,?????????????,???????????????????????
  ????,????????????????????????????????????,????????
  ???????????,????????????????,??????,?????????????
  
•   
• --?--