2010-07-09 23:58:27frank
[Russia] 數學家佩雷曼（Grigory Perelman）
在沒讀過這篇報導前，我並不知道佩雷曼（Grigory Perelman）這號人物，但是文中所提到的澳洲出生的華裔知名數學家陶哲軒（Terrence Tao）幾年前，就讀過他的報導。Terrence 那時是在研究最大的質數。
Terrence 出生在一個香港移民到澳洲的醫生家庭，從小就是天才兒童。他的兄弟也是，可見得陶氏夫婦教育的成功。在談及對子女的教育（尤其是天才兒童的教育）時，Terrence的父親問了一個令我印象深刻的問題：「羅傑(Jay Luo)現在在什麼地方？」羅傑是我小時候台灣最著名的天才兒童，12歲就從美國著名大學畢業，也是那一年（1982）全美最年輕的大學畢業生。Terrence父親的金字塔理論，令我印象深刻－－學問就像一座金字塔，要有高的學問，就像要蓋一座高的金字塔；底要廣，才蓋的高。
Terrence 出生在一個香港移民到澳洲的醫生家庭，從小就是天才兒童。他的兄弟也是，可見得陶氏夫婦教育的成功。在談及對子女的教育（尤其是天才兒童的教育）時，Terrence的父親問了一個令我印象深刻的問題：「羅傑(Jay Luo)現在在什麼地方？」羅傑是我小時候台灣最著名的天才兒童，12歲就從美國著名大學畢業，也是那一年（1982）全美最年輕的大學畢業生。Terrence父親的金字塔理論，令我印象深刻－－學問就像一座金字塔，要有高的學問，就像要蓋一座高的金字塔；底要廣，才蓋的高。
Remark: 羅傑成為美國人，住在亞特蘭大，從事電腦軟體工作。這是他的網頁。
This is Jay Lou's LinedIn page. http://www.linkedin.com/in/jayluo
上網找到了幾年前讀到介紹Terrence Tao的文章 "Journeys to the Distant Fields of Prime"
http://www.nytimes.com/2007/03/13/science/13prof.html
佩雷曼（Grigory Perelman）果然是個特立獨行之人，或許想要成就什麼事就要如此專心一意，這也是「無欲則剛」的體現嗎？欲望到底是獲致成就的驅力，還是成功的阻力呢？
這篇文章"Elusive Proof, Elusive Prover: A New Mathematical Mystery" 有對佩雷曼（Grigory Perelman）解出百年難解的「龐加萊猜想」（Poincare Conjecture）的經過說明。另外文章裡所提到的「與丘成桐相比，佩雷曼可算是另一個極端，名譽、富貴對他如浮雲。」第一印象很重要，也因此我對這位也是台灣中研院院士的丘成桐，沒甚麼好印象。在Wikipedia 有一篇 田剛丘成桐事件，也講述著與這破解百年難解的「龐加萊猜想」（Poincare Conjecture）有些關係的學術界的口水戰。田剛就是報導裡的Gang Tian of Princeton.
比較令我好奇的是為什麼中時的傅先生要在2010年寫這一篇2006年的舊聞，發表在新聞報紙上，而非個人的部落格。
2010.7.10 補記：
中央社的即時新聞：第二屆丘成桐中學數學獎頒獎－－這大概是傅先生寫這篇舊文的原因。
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August 15, 2006
Elusive Proof, Elusive Prover: A New Mathematical Mystery
By DENNIS OVERBYE
Correction Appended
Grisha Perelman, where are you?
Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.
After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.
Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.
As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.
“It’s really a great moment in mathematics,” said Bruce Kleiner of Yale, who has spent the last three years helping to explicate Dr. Perelman’s work. “It could have happened 100 years from now, or never.”
In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century.
Quoting Poincaré himself, Dr.Yau said, “Thought is only a flash in the middle of a long night, but the flash that means everything.”
But at the moment of his putative triumph, Dr. Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math’s version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up.
Also left hanging, for now, is $1 million offered by the Clay Mathematics Institute in Cambridge, Mass., for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom back at the beginning of the millennium.
“It’s very unusual in math that somebody announces a result this big and leaves it hanging,” said John Morgan of Columbia, one of the scholars who has also been filling in the details of Dr. Perelman’s work.
Mathematicians have been waiting for this result for more than 100 years, ever since the French polymath Henri Poincaré posed the problem in 1904. And they acknowledge that it may be another 100 years before its full implications for math and physics are understood. For now, they say, it is just beautiful, like art or a challenging new opera.
Dr. Morgan said the excitement came not from the final proof of the conjecture, which everybody felt was true, but the method, “finding deep connections between what were unrelated fields of mathematics.”
William Thurston of Cornell, the author of a deeper conjecture that includes Poincaré’s and that is now apparently proved, said, “Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition.
“You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.”
Depending on who is talking, Poincaré’s conjecture can sound either daunting or deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.
The conjecture is fundamental to topology, the branch of math that deals with shapes, sometimes described as geometry without the details. To a topologist, a sphere, a cigar and a rabbit’s head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere.
In effect, what Poincaré suggested was that anything without holes has to be a sphere. The one qualification was that this “anything” had to be what mathematicians call compact, or closed, meaning that it has a finite extent: no matter how far you strike out in one direction or another, you can get only so far away before you start coming back, the way you can never get more than 12,500 miles from home on the Earth.
In the case of two dimensions, like the surface of a sphere or a doughnut, it is easy to see what Poincaré was talking about: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.
With three dimensions, it is harder to discern the overall shape of something; we cannot see where the holes might be. “We can’t draw pictures of 3-D spaces,” Dr. Morgan said, explaining that when we envision the surface of a sphere or an apple, we are really seeing a two-dimensional object embedded in three dimensions. Indeed, astronomers are still arguing about the overall shape of the universe, wondering if its topology resembles a sphere, a bagel or something even more complicated.
Poincaré’s conjecture was subsequently generalized to any number of dimensions, but in fact the three-dimensional version has turned out to be the most difficult of all cases to prove. In 1960 Stephen Smale, now at the Toyota Technological Institute at Chicago, proved that it is true in five or more dimensions and was awarded a Fields Medal. In 1983, Michael Freedman, now at Microsoft, proved that it is true in four dimensions and also won a Fields.
“You get a Fields Medal for just getting close to this conjecture,” Dr. Morgan said.
In the late 1970’s, Dr. Thurston extended Poincaré’s conjecture, showing that it was only a special case of a more powerful and general conjecture about three-dimensional geometry, namely that any space can be decomposed into a few basic shapes.
Mathematicians had known since the time of Georg Friedrich Bernhard Riemann, in the 19th century, that in two dimensions there are only three possible shapes: flat like a sheet of paper, closed like a sphere, or curved uniformly in two opposite directions like a saddle or the flare of a trumpet. Dr. Thurston suggested that eight different shapes could be used to make up any three-dimensional space.
“Thurston’s conjecture almost leads to a list,” Dr. Morgan said. “If it is true,” he added, “Poincaré’s conjecture falls out immediately.” Dr. Thurston won a Fields in 1982.
Topologists have developed an elaborate set of tools to study and dissect shapes, including imaginary cutting and pasting, which they refer to as “surgery,” but they were not getting anywhere for a long time.
In the early 1980’s Richard Hamilton of Columbia suggested a new technique, called the Ricci flow, borrowed from the kind of mathematics that underlies Einstein’s general theory of relativity and string theory, to investigate the shapes of spaces.
Dr. Hamilton’s technique makes use of the fact that for any kind of geometric space there is a formula called the metric, which determines the distance between any pair of nearby points. Applied mathematically to this metric, the Ricci flow acts like heat, flowing through the space in question, smoothing and straightening all its bumps and curves to reveal its essential shape, the way a hair dryer shrink-wraps plastic.
Dr. Hamilton succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process, but the fates of more complicated objects were problematic. As the Ricci flow progressed, kinks and neck pinches, places of infinite density known as singularities, could appear, pinch off and even shrink away. Topologists could cut them away, but there was no guarantee that new ones would not keep popping up forever.
“All sorts of things can potentially happen in the Ricci flow,” said Robert Greene, a mathematician at the University of California, Los Angeles. Nobody knew what to do with these things, so the result was a logjam.
It was Dr. Perelman who broke the logjam. He was able to show that the singularities were all friendly. They turned into spheres or tubes. Moreover, they did it in a finite time once the Ricci flow started. That meant topologists could, in their fashion, cut them off, and allow the Ricci process to continue to its end, revealing the topologically spherical essence of the space in question, and thus proving the conjectures of both Poincaré and Thurston.
Dr. Perelman’s first paper, promising “a sketch of an eclectic proof,” came as a bolt from the blue when it was posted on the Internet in November 2002. “Nobody knew he was working on the Poincaré conjecture,” said Michael T. Anderson of the State University of New York in Stony Brook.
Dr. Perelman had already established himself as a master of differential geometry, the study of curves and surfaces, which is essential to, among other things, relativity and string theory Born in St. Petersburg in 1966, he distinguished himself as a high school student by winning a gold medal with a perfect score in the International Mathematical Olympiad in 1982. After getting a Ph.D. from St. Petersburg State, he joined the Steklov Institute of Mathematics at St. Petersburg.
In a series of postdoctoral fellowships in the United States in the early 1990’s, Dr. Perelman impressed his colleagues as “a kind of unworldly person,” in the words of Dr. Greene of U.C.L.A. — friendly, but shy and not interested in material wealth.
“He looked like Rasputin, with long hair and fingernails,” Dr. Greene said.
Asked about Dr. Perelman’s pleasures, Dr. Anderson said that he talked a lot about hiking in the woods near St. Petersburg looking for mushrooms.
Dr. Perelman returned to those woods, and the Steklov Institute, in 1995, spurning offers from Stanford and Princeton, among others. In 1996 he added to his legend by turning down a prize for young mathematicians from the European Mathematics Society.
Until his papers on Poincaré started appearing, some friends thought Dr. Perelman had left mathematics. Although they were so technical and abbreviated that few mathematicians could read them, they quickly attracted interest among experts. In the spring of 2003, Dr. Perelman came back to the United States to give a series of lectures at Stony Brook and the Massachusetts Institute of Technology, and also spoke at Columbia, New York University and Princeton.
But once he was back in St. Petersburg, he did not respond to further invitations. The e-mail gradually ceased.
“He came once, he explained things, and that was it,” Dr. Anderson said. “Anything else was superfluous.”
Recently, Dr. Perelman is said to have resigned from Steklov. E-mail messages addressed to him and to the Steklov Institute went unanswered.
In his absence, others have taken the lead in trying to verify and disseminate his work. Dr. Kleiner of Yale and John Lott of the University of Michigan have assembled a monograph annotating and explicating Dr. Perelman’s proof of the two conjectures.
Dr. Morgan of Columbia and Gang Tian of Princeton have followed Dr. Perelman’s prescription to produce a more detailed 473-page step-by-step proof only of Poincaré’s Conjecture. “Perelman did all the work,” Dr. Morgan said. “This is just explaining it.”
Both works were supported by the Clay institute, which has posted them on its Web site, claymath.org. Meanwhile, Huai-Dong Cao of Lehigh University and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, have published their own 318-page proof of both conjectures in The Asian Journal of Mathematics (www.ims.cuhk.edu.hk/).
Although these works were all hammered out in the midst of discussion and argument by experts, in workshops and lectures, they are about to receive even stricter scrutiny and perhaps crossfire. “Caution is appropriate,” said Dr. Kleiner, because the Poincaré conjecture is not just famous, but important.
James Carlson, president of the Clay Institute, said the appearance of these papers had started the clock ticking on a two-year waiting period mandated by the rules of the Clay Millennium Prize. After two years, he said, a committee will be appointed to recommend a winner or winners if it decides the proof has stood the test of time.
“There is nothing in the rules to prevent Perelman from receiving all or part of the prize,” Dr. Carlson said, saying that Dr. Perelman and Dr. Hamilton had obviously made the main contributions to the proof.
In a lecture at M.I.T. in 2003, Dr. Perelman described himself “in a way” as Dr. Hamilton’s disciple, although they had never worked together. Dr. Hamilton, who got his Ph.D. from Princeton in 1966, is too old to win the Fields medal, which is given only up to the age of 40, but he is slated to give the major address about the Poincaré conjecture in Madrid next week. He did not respond to requests for an interview.
Allowing that Dr. Perelman, should he win the Clay Prize, might refuse the honor, Dr. Carlson said the institute could decide instead to use award money to support Russian mathematicians, the Steklov Institute or even the Math Olympiad.
Dr. Anderson said that to some extent the new round of papers already represented a kind of peer review of Dr. Perelman’s work. “All these together make the case pretty clear,” he said. “The community accepts the validity of his work. It’s commendable that the community has gotten together.”
Correction: Aug. 21, 2006
An article in Science Times on Tuesday about a mathematical problem called the Poincaré conjecture misstated the year William Thurston, the author of a deeper conjecture that includes Poincaré’s, was awarded a Fields Medal. It was 1982, not 1986.
http://www.nytimes.com/2006/08/15/science/15math.html
上網找到了幾年前讀到介紹Terrence Tao的文章 "Journeys to the Distant Fields of Prime"
http://www.nytimes.com/2007/03/13/science/13prof.html
佩雷曼（Grigory Perelman）果然是個特立獨行之人，或許想要成就什麼事就要如此專心一意，這也是「無欲則剛」的體現嗎？欲望到底是獲致成就的驅力，還是成功的阻力呢？
這篇文章"Elusive Proof, Elusive Prover: A New Mathematical Mystery" 有對佩雷曼（Grigory Perelman）解出百年難解的「龐加萊猜想」（Poincare Conjecture）的經過說明。另外文章裡所提到的「與丘成桐相比，佩雷曼可算是另一個極端，名譽、富貴對他如浮雲。」第一印象很重要，也因此我對這位也是台灣中研院院士的丘成桐，沒甚麼好印象。在Wikipedia 有一篇 田剛丘成桐事件，也講述著與這破解百年難解的「龐加萊猜想」（Poincare Conjecture）有些關係的學術界的口水戰。田剛就是報導裡的Gang Tian of Princeton.
比較令我好奇的是為什麼中時的傅先生要在2010年寫這一篇2006年的舊聞，發表在新聞報紙上，而非個人的部落格。
2010.7.10 補記：
中央社的即時新聞：第二屆丘成桐中學數學獎頒獎－－這大概是傅先生寫這篇舊文的原因。
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華府看天下－拒領百萬獎金的數學家
猶太後裔的俄國數學家佩雷曼（Grigory Perelman），可說是全球學界絕無僅有的奇人和怪人，現年四十四歲的佩雷曼因破解百年難解的「龐加萊猜想」（PoincareConjecture）而譽滿全球，因此獲得有數學諾貝爾獎之稱的費爾茲獎（Fields Medal）（只給給四十歲以下的數學家）和數學「千年大獎」（Millennium Prize），後者獎金高達一百萬美元，但他都拒絕接受，理由是他不配得獎，而且他所做的也不值得如此大張旗鼓的表揚。
「龐加萊猜想」是法國數學家龐加萊（Jules Henri Poincare，1854-1912）在一九○四年提出的命題，其基本理論是：「在一個封閉的三維（度）空間，假如每條封閉的曲線都能收縮成一點，這個空間一定是一個圓球」。過去一百年來無數數學家試圖證明「龐加萊猜想」，但都沒能成功，一直到二○○二年十一月佩雷曼在網上公布他證明「龐加萊猜想」的三篇文章，此一一世紀來無解的難題才終於有了正確的答案，可是還得經過行內專家的審核與鑑定，這個證明才告定案，「國際數學聯盟」在二○○六年把費爾茲獎頒給佩雷曼，但佩雷曼不接受，理由是他的證明對了，就已經夠了，得不得獎與他的證明無關。「國際數學聯盟」的主席包爾爵士（Sir John M.Ball）為了說服佩雷曼改變心意，親自出席馬德里的大會並領獎，專程去了趟聖彼得堡面見佩雷曼懇談，雖經十小時的長談，佩雷曼還是堅持初衷，包爾只好鎩羽而歸。不過二○○六年費爾茲獎得獎人有一位澳洲出生的華裔知名數學家陶哲軒（TerrenceTao，現任教於洛杉磯加州大學），陶對佩雷曼的成就極為推崇。
今年三月設於波斯頓的克雷數學研究所宣布頒給佩雷曼「千年大獎」，獎金一百萬美元，以表彰他對數學的貢獻，一直折騰到這個月佩雷曼終於決定不接受這個獎及獎金，理由是他不配，並具體指出哥倫比亞大學的數學教授韓默頓（RichardHamilton）比他更有資格得獎，因為韓默頓的「瑞奇流」（Ricci flow）理論為破解龐加萊猜想奠了基。
九○年代佩雷曼應邀到美國做研究和講學，見過韓默頓，也請教過韓一些數學上的問題，韓對佩知無不言，包括他尚未發表的論文在內，這等坦盪無私非常難得，因為學術界勾心鬥角、爭名奪利決不亞於政商界，只是外界把它看得清高罷了，佩雷曼對此知之甚詳，且極端厭惡，這是為何一九九五年他拒絕史丹福、普林斯頓等名校的重金禮聘，寧可回俄國和母親住在破公寓內，安貧樂道，和外界絕少接觸，過著隱士般的生活。最新的消息是佩雷曼已辭去俄國科學院數學研究所的工作，斷絕了和數學界一切的關係。
當二○○六年「國際數學聯盟」已經確認佩雷曼證明了「龐加萊猜想」、並決定頒發費爾茲獎給他時，中國大陸中山大學的數學家朱熹平和美國李海大學的講座教授曹懷東迫不及待聯名發表了他們對「龐加萊猜想」證明的論文，這篇長達數百頁的論文登在數學大師丘成桐主編的《亞洲數學學報》上，丘對朱、曹二人的論文，給予極高的評價，說是「龐加萊猜想」的「封頂之作」，壓低了佩雷曼的證明，儘管數學界公認佩的證明是原創性的，其他各家的論文都只是在解釋或補充佩雷曼的證明，這使佩雷曼不滿，並結下心結。
丘成桐是當今世界上頂尖的數學家不容置疑，甚至有人封他為數學皇帝或凱撒大帝，可惜他的人情世故很差，加以心高氣傲，難免人緣欠佳，以致二○○六年八月份一期的《紐約客》雜誌刊出一篇長文，對丘頗多批評，指控他突出朱熹平和曹懷東、壓抑佩雷曼的作法，「違反了學術界的基本倫理」，由於此文作者寫過《美麗境界》（A Beautiful Mind，曾拍成電影）一書，是普林斯頓數學家諾貝爾獎得主納許（John ForbesNash）的真實故事，頗負盛名，文章一經刊出，流傳甚廣，對丘的傷害亦大。
多年前我和丘成桐有過一面之緣，並曾長談，基本上他認為台灣在國際學術上沒有地位，也看不太起，儘管他有位台灣太太。他說提升學術地位不一定非有錢莫辦，並以他專精的數學為例，指出發展數學不需要花大錢，而收效卻可能很大，說得也是。
與丘成桐相比，佩雷曼可算是另一個極端，名譽、富貴對他如浮雲。套句孔老夫子讚美顏回的話：「…居陋巷，人不堪其憂，佩也不改其樂。賢栽！佩雷曼」。
Elusive Proof, Elusive Prover: A New Mathematical Mystery
By DENNIS OVERBYE
Correction Appended
Grisha Perelman, where are you?
Three years ago, a Russian mathematician by the name of Grigory Perelman, a k a Grisha, in St. Petersburg, announced that he had solved a famous and intractable mathematical problem, known as the Poincaré conjecture, about the nature of space.
After posting a few short papers on the Internet and making a whirlwind lecture tour of the United States, Dr. Perelman disappeared back into the Russian woods in the spring of 2003, leaving the world’s mathematicians to pick up the pieces and decide if he was right.
Now they say they have finished his work, and the evidence is circulating among scholars in the form of three book-length papers with about 1,000 pages of dense mathematics and prose between them.
As a result there is a growing feeling, a cautious optimism that they have finally achieved a landmark not just of mathematics, but of human thought.
“It’s really a great moment in mathematics,” said Bruce Kleiner of Yale, who has spent the last three years helping to explicate Dr. Perelman’s work. “It could have happened 100 years from now, or never.”
In a speech at a conference in Beijing this summer, Shing-Tung Yau of Harvard said the understanding of three-dimensional space brought about by Poincaré’s conjecture could be one of the major pillars of math in the 21st century.
Quoting Poincaré himself, Dr.Yau said, “Thought is only a flash in the middle of a long night, but the flash that means everything.”
But at the moment of his putative triumph, Dr. Perelman is nowhere in sight. He is an odds-on favorite to win a Fields Medal, math’s version of the Nobel Prize, when the International Mathematics Union convenes in Madrid next Tuesday. But there is no indication whether he will show up.
Also left hanging, for now, is $1 million offered by the Clay Mathematics Institute in Cambridge, Mass., for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom back at the beginning of the millennium.
“It’s very unusual in math that somebody announces a result this big and leaves it hanging,” said John Morgan of Columbia, one of the scholars who has also been filling in the details of Dr. Perelman’s work.
Mathematicians have been waiting for this result for more than 100 years, ever since the French polymath Henri Poincaré posed the problem in 1904. And they acknowledge that it may be another 100 years before its full implications for math and physics are understood. For now, they say, it is just beautiful, like art or a challenging new opera.
Dr. Morgan said the excitement came not from the final proof of the conjecture, which everybody felt was true, but the method, “finding deep connections between what were unrelated fields of mathematics.”
William Thurston of Cornell, the author of a deeper conjecture that includes Poincaré’s and that is now apparently proved, said, “Math is really about the human mind, about how people can think effectively, and why curiosity is quite a good guide,” explaining that curiosity is tied in some way with intuition.
“You don’t see what you’re seeing until you see it,” Dr. Thurston said, “but when you do see it, it lets you see many other things.”
Depending on who is talking, Poincaré’s conjecture can sound either daunting or deceptively simple. It asserts that if any loop in a certain kind of three-dimensional space can be shrunk to a point without ripping or tearing either the loop or the space, the space is equivalent to a sphere.
The conjecture is fundamental to topology, the branch of math that deals with shapes, sometimes described as geometry without the details. To a topologist, a sphere, a cigar and a rabbit’s head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere.
In effect, what Poincaré suggested was that anything without holes has to be a sphere. The one qualification was that this “anything” had to be what mathematicians call compact, or closed, meaning that it has a finite extent: no matter how far you strike out in one direction or another, you can get only so far away before you start coming back, the way you can never get more than 12,500 miles from home on the Earth.
In the case of two dimensions, like the surface of a sphere or a doughnut, it is easy to see what Poincaré was talking about: imagine a rubber band stretched around an apple or a doughnut; on the apple, the rubber band can be shrunk without limit, but on the doughnut it is stopped by the hole.
With three dimensions, it is harder to discern the overall shape of something; we cannot see where the holes might be. “We can’t draw pictures of 3-D spaces,” Dr. Morgan said, explaining that when we envision the surface of a sphere or an apple, we are really seeing a two-dimensional object embedded in three dimensions. Indeed, astronomers are still arguing about the overall shape of the universe, wondering if its topology resembles a sphere, a bagel or something even more complicated.
Poincaré’s conjecture was subsequently generalized to any number of dimensions, but in fact the three-dimensional version has turned out to be the most difficult of all cases to prove. In 1960 Stephen Smale, now at the Toyota Technological Institute at Chicago, proved that it is true in five or more dimensions and was awarded a Fields Medal. In 1983, Michael Freedman, now at Microsoft, proved that it is true in four dimensions and also won a Fields.
“You get a Fields Medal for just getting close to this conjecture,” Dr. Morgan said.
In the late 1970’s, Dr. Thurston extended Poincaré’s conjecture, showing that it was only a special case of a more powerful and general conjecture about three-dimensional geometry, namely that any space can be decomposed into a few basic shapes.
Mathematicians had known since the time of Georg Friedrich Bernhard Riemann, in the 19th century, that in two dimensions there are only three possible shapes: flat like a sheet of paper, closed like a sphere, or curved uniformly in two opposite directions like a saddle or the flare of a trumpet. Dr. Thurston suggested that eight different shapes could be used to make up any three-dimensional space.
“Thurston’s conjecture almost leads to a list,” Dr. Morgan said. “If it is true,” he added, “Poincaré’s conjecture falls out immediately.” Dr. Thurston won a Fields in 1982.
Topologists have developed an elaborate set of tools to study and dissect shapes, including imaginary cutting and pasting, which they refer to as “surgery,” but they were not getting anywhere for a long time.
In the early 1980’s Richard Hamilton of Columbia suggested a new technique, called the Ricci flow, borrowed from the kind of mathematics that underlies Einstein’s general theory of relativity and string theory, to investigate the shapes of spaces.
Dr. Hamilton’s technique makes use of the fact that for any kind of geometric space there is a formula called the metric, which determines the distance between any pair of nearby points. Applied mathematically to this metric, the Ricci flow acts like heat, flowing through the space in question, smoothing and straightening all its bumps and curves to reveal its essential shape, the way a hair dryer shrink-wraps plastic.
Dr. Hamilton succeeded in showing that certain generally round objects, like a head, would evolve into spheres under this process, but the fates of more complicated objects were problematic. As the Ricci flow progressed, kinks and neck pinches, places of infinite density known as singularities, could appear, pinch off and even shrink away. Topologists could cut them away, but there was no guarantee that new ones would not keep popping up forever.
“All sorts of things can potentially happen in the Ricci flow,” said Robert Greene, a mathematician at the University of California, Los Angeles. Nobody knew what to do with these things, so the result was a logjam.
It was Dr. Perelman who broke the logjam. He was able to show that the singularities were all friendly. They turned into spheres or tubes. Moreover, they did it in a finite time once the Ricci flow started. That meant topologists could, in their fashion, cut them off, and allow the Ricci process to continue to its end, revealing the topologically spherical essence of the space in question, and thus proving the conjectures of both Poincaré and Thurston.
Dr. Perelman’s first paper, promising “a sketch of an eclectic proof,” came as a bolt from the blue when it was posted on the Internet in November 2002. “Nobody knew he was working on the Poincaré conjecture,” said Michael T. Anderson of the State University of New York in Stony Brook.
Dr. Perelman had already established himself as a master of differential geometry, the study of curves and surfaces, which is essential to, among other things, relativity and string theory Born in St. Petersburg in 1966, he distinguished himself as a high school student by winning a gold medal with a perfect score in the International Mathematical Olympiad in 1982. After getting a Ph.D. from St. Petersburg State, he joined the Steklov Institute of Mathematics at St. Petersburg.
In a series of postdoctoral fellowships in the United States in the early 1990’s, Dr. Perelman impressed his colleagues as “a kind of unworldly person,” in the words of Dr. Greene of U.C.L.A. — friendly, but shy and not interested in material wealth.
“He looked like Rasputin, with long hair and fingernails,” Dr. Greene said.
Asked about Dr. Perelman’s pleasures, Dr. Anderson said that he talked a lot about hiking in the woods near St. Petersburg looking for mushrooms.
Dr. Perelman returned to those woods, and the Steklov Institute, in 1995, spurning offers from Stanford and Princeton, among others. In 1996 he added to his legend by turning down a prize for young mathematicians from the European Mathematics Society.
Until his papers on Poincaré started appearing, some friends thought Dr. Perelman had left mathematics. Although they were so technical and abbreviated that few mathematicians could read them, they quickly attracted interest among experts. In the spring of 2003, Dr. Perelman came back to the United States to give a series of lectures at Stony Brook and the Massachusetts Institute of Technology, and also spoke at Columbia, New York University and Princeton.
But once he was back in St. Petersburg, he did not respond to further invitations. The e-mail gradually ceased.
“He came once, he explained things, and that was it,” Dr. Anderson said. “Anything else was superfluous.”
Recently, Dr. Perelman is said to have resigned from Steklov. E-mail messages addressed to him and to the Steklov Institute went unanswered.
In his absence, others have taken the lead in trying to verify and disseminate his work. Dr. Kleiner of Yale and John Lott of the University of Michigan have assembled a monograph annotating and explicating Dr. Perelman’s proof of the two conjectures.
Dr. Morgan of Columbia and Gang Tian of Princeton have followed Dr. Perelman’s prescription to produce a more detailed 473-page step-by-step proof only of Poincaré’s Conjecture. “Perelman did all the work,” Dr. Morgan said. “This is just explaining it.”
Both works were supported by the Clay institute, which has posted them on its Web site, claymath.org. Meanwhile, Huai-Dong Cao of Lehigh University and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, have published their own 318-page proof of both conjectures in The Asian Journal of Mathematics (www.ims.cuhk.edu.hk/).
Although these works were all hammered out in the midst of discussion and argument by experts, in workshops and lectures, they are about to receive even stricter scrutiny and perhaps crossfire. “Caution is appropriate,” said Dr. Kleiner, because the Poincaré conjecture is not just famous, but important.
James Carlson, president of the Clay Institute, said the appearance of these papers had started the clock ticking on a two-year waiting period mandated by the rules of the Clay Millennium Prize. After two years, he said, a committee will be appointed to recommend a winner or winners if it decides the proof has stood the test of time.
“There is nothing in the rules to prevent Perelman from receiving all or part of the prize,” Dr. Carlson said, saying that Dr. Perelman and Dr. Hamilton had obviously made the main contributions to the proof.
In a lecture at M.I.T. in 2003, Dr. Perelman described himself “in a way” as Dr. Hamilton’s disciple, although they had never worked together. Dr. Hamilton, who got his Ph.D. from Princeton in 1966, is too old to win the Fields medal, which is given only up to the age of 40, but he is slated to give the major address about the Poincaré conjecture in Madrid next week. He did not respond to requests for an interview.
Allowing that Dr. Perelman, should he win the Clay Prize, might refuse the honor, Dr. Carlson said the institute could decide instead to use award money to support Russian mathematicians, the Steklov Institute or even the Math Olympiad.
Dr. Anderson said that to some extent the new round of papers already represented a kind of peer review of Dr. Perelman’s work. “All these together make the case pretty clear,” he said. “The community accepts the validity of his work. It’s commendable that the community has gotten together.”
Correction: Aug. 21, 2006
An article in Science Times on Tuesday about a mathematical problem called the Poincaré conjecture misstated the year William Thurston, the author of a deeper conjecture that includes Poincaré’s, was awarded a Fields Medal. It was 1982, not 1986.
http://www.nytimes.com/2006/08/15/science/15math.html