Yau Survey of Geometric Analysis

There’s a remarkable new paper out from Shing-Tung Yau, entitled Perspectives on Geometric Analysis. Yau is probably the dominant figure in the field of Geometric Analysis in recent years, and in this paper he gives his personal perspective on the field, including many comments on its recent history, where it is now, and where he thinks it is going.

The paper begins with a dedication to Chern, and some personal history of Yau’s interactions with him. It includes an outline of the distant and recent history of geometric analysis, mainly by giving names of the mathematicians involved. There are 755 references in the reference section, listing pretty much all the papers that Yau sees as important for one reason or another. This has to be some kind of record for number of references in a paper, especially a paper whose main text is only about 50 pages long.

Yau covers an immense amount of ground, commenting on a very wide variety of topics. This is a paper aimed at those who already know quite a bit about the subject, or who are beginning to learn it and would appreciate recommendations of what they should be reading. It includes very little in the way of expository material aimed at the beginner. There is a long section on “Ricci flow” techniques, which are the topic of a lot of current research and that Yau considers to be “the most spectacular development in the last thirty years.” He also has quite a bit to say about “Calabi-Yau” manifolds and their use in physics, commenting that they provide “a good testing ground for analysis, geometry, physics, algebraic geometry, automorphic forms and number theory.”

Another expository paper also appeared on the arXiv last night, but one of a very different nature. It’s by Ravi Vakil, a young algebraic geometer at Stanford, and it is aimed at explaining how Gromov-Witten theory has been used in recent years to study the moduli space of curves. It includes a lot of expository material about the moduli space of curves, and is designed to be understandable by the non-expert.

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22 Responses to Yau Survey of Geometric Analysis

  1. Quantoken says:

    Wow, I did not know that Yau was actually a fan of Chiung Yau‘s romantic novels. The poetry he quoted on page 3 was from one of the most popular romantic novels written by Chiung Yau, which was also made into movie.
    Curiously it was supposed to be a “she” in the middle of water but Yau made it a “he”. Clearly he is dedicating this poetry to his teacher, Chern. Very touching. I would also say that this paper is very readable.


  2. Joe Zhou says:

    Quantoken is a truly confused person. He’s spreading misinformation again. Every sentence above is misinformed.

    The poem Yau quoted is ~3000 years old, appearing in the Book of Poem edited by Confucius under the chapter of Winds of Tang.

    The person being sought in this poem is referred to with a gender-neutral third-person pronoun, impossible to translate into English without destroying the poem.

    It’s Chiung Yao, not Chiung Yau; no relation here — it’s a pen name. In fact the Yao in Chiung Yao is not even a last name. I’ll lose quite a bit of esteem for Yau if he quotes from Chiung Yao (eck!)in his paper.

  3. D R Lunsford says:

    That is really a wonderful paper! If not for the missing equations of any kind, I might be reading Pauli’s encyclopedia article on relativity. However this is exactly what is most needed, a verbal outpouring of mathematical ideas like Klein used to make.

    Readers who may not be familiar should consult “Vorlesung uber die Entwicklung der Mathematik im 19’ten Jahrhundert” for a similar example of mathematical perspective-taking on this advanced level.


  4. MathPhys says:

    “Vorlesung uber die Entwicklung der Mathematik im 19′ten Jahrhundert”

    is available in English as

    Development of Mathematics in the Nineteenth Century, and published as volume 9 of Robert Hermann’s Lie Groups Series.

    It’s a great book.

  5. mathjunkie says:

    Joe Zhou,

    Hi, just curious about what the Book of Poem edited by Confucius under the chapter of Winds of Tang is in Chinese. (Hope that Peter’s weblog can properly display Chinese characters) Thanks.

  6. Quantoken says:

    Get real!
    There are different ways of translating Chinese names into English. For example Yau himself, it could have been called “Qiu” or “Chiu” using standard mandarin pronunciation. But he prefered his native Cantonese “Yaou”, or “Yau”. I deliberately used the translation “Chiung Yau” for a little bit humor. Too bad you have no sense of humor. It’s true it’s a 3000 year old poetry, but if it were not for that romantic novel making it widely known, few Chinese even know it, lest along quote it.

  7. Alice says:

    actually, Miss Chiung Yau ever wrote a lyric by adapting that poem from the Book of Poem.. You might feel interested in it. LINK to speical blog (in hinese)–> http://william.cswiz.org/blog/archives/2005-04-10/poem-and-song-part1/

  8. Quantoken says:




    Yau destroyed the beauty of this ancient Chinese poetry. He should at least give it a taste of ancient-ness, using Shakespear’s language style.

  9. mathjunkie says:


    Thanks for posting the whole poem. I didn’t know that you even knew the Chinese language. The Chinese poem is very difficult for me. I can’t understand it though I am Chinese.

  10. Lubos Motl says:

    It is actually a Czech poem from 3500 B.C. originally. Let me tell you the original poem in the simplified combined Czech Chinese:

    游蒹葭伊蒼,白露為蒹霜。Cestující v tramvaji
    水所謂伊人,在水一方伊。mají úsměv na líci,
    遡洄遡從之,道阻方且長;když známou znělku zahrají
    遡游之從之,宛在水中央。v Hellichově ulici.

    蒹葭凄洄凄,白之露未晞。Jemná ruka umělcova
    所謂伊游人,之在水之湄。v duši kreslí ideál
    遡洄躋從之,道阻人且躋;krásnější než všechna slova,
    遡游洄從之,宛在水中坻。a tak je nechám opodál.

    蒹葭采阻采,白露水未已。Kde bys hledal v zemi české
    所謂中伊人,在中水之涘。krásku ve snu zjevenou?
    遡洄從阻之,道游阻且右;Na matfyzu dívky hezké
    遡游從阻之,宛在水中沚。nedostaly zelenou.

  11. robert says:

    In less enlightened times it was a commonplace to categorise the Chinese as inscrutable. Is Quantoken having a laugh? Is mathjunkie being facetious? Does ???????; ?? or whatever appear differently on other people’s screens? (??>;’?) Is the quantity in brackets a reasonable representation of Charlie Chan? This sequence of posts started out among the more sensible and informative; somehow it was subverted along the way.

  12. Luboš Motl says:

    Dear Robert, you seem to be the only person here who can’t read Chinese. Go to your Control Panel / Regional Settings / Languages, and click Install files for East Asian languages. If you don’t have Windows, then uninstall your other OS and buy a CD with XP. Otherwise your future is not too bright because the future belongs to China.

  13. Quantoken says:

    Ho ho ho, Lubos:-) You thought no one else here but you knows Czech language? You actually plagiarized your romantic poetry from this Czech web site 🙂


  14. Qui-Quien Hu says:

    and anyway the first line of that Czech poem contains the word
    “tramvaji” which means streetcar.
    So how can it be from 3500 BC?

  15. worldspace says:

    It’s a romantic poem from between 500 BC and 1100 BC in ancient China.

  16. worldspace says:

    Yau wrote a wonderful paper! It’s very useful for those who have begun their career in Geometric Analysis.

  17. Luboš Motl says:

    Dear Quantoken, as an internet searcher, you’re a superstar. 🙂

    Qui-Quien Hu: the electric streetcars were operating in Bohemia long before the Slavs came to the territory. See the history at


    2000 years later, Marco Polo actually made his journeys to Asia in order to sell some of these streetcars.

  18. Adrian H. says:

    Back to the Yau paper.

    On the first page Yau says that he set Chinese students the problem of a `Jordan curve bounding two surfaces…’ He then says that this resulted in an ugly fight at the 60th Anniversary meeting of the Chinese mathematical Society and Professor Wang was forced to resign!

    Can anyone throw some light on this extraordinary fracas? What is it about Jordan curves that would cause a professor to resign? What on earth were they fighting about?

    (Ah! Academia — such a bucolic environment for seeking the Truth!)

  19. Juan R. says:

    Otherwise your future is not too bright because the future belongs to China.

    I thought that future belongs to anyone in China or not.

    Juan R.

    Center for CANONICAL |SCIENCE)

  20. worldspace says:

    Let’s focus on “Geometric Analysis”. It’s really a long one and there are a lot of useful materials. Has anyone read the whole article of Yau’s??

  21. Passby says:

    quite funny…

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