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New York Times on 2006 Fields Medalists
Four Are Given Highest Honor in Mathematics
By KENNETH CHANG
Published: August 22, 2006
Grigory Perelman, a reclusive Russian mathematician who solved a key piece in
a century-old puzzle known as the Poincare conjecture, was one of four
mathematicians awarded the Fields Medal today.
The Fields Medal, often described as mathematics? equivalent to the Nobel
Prize, is given every four years. The other Fields medalists, announced at
the International Congress for Mathematicians in Madrid, are Andrei Okounkov,
a professor of mathematics at Princeton; Terence Tao, 31, a professor of
mathematics at the University of California, Los Angeles; and Wendelin Werner,
a professor of mathematics at the University of Paris-Sud in Orsay.
Dr. Perelman, 40, is known not only for his work on the Poincare conjecture,
among the most heralded unsolved math problems, but also because he has
declined previous mathematical prizes and has turned down job offers from
Princeton, Stanford and other universities. He has said he wants no part of
$1 million that the Clay Mathematics Institute in Cambridge, Mass. has offered
for the first published proof of the conjecture.
According to an article in the Aug. 28 issue of The New Yorker, Sir John M.
Ball, president of the International Mathematical Union, the organization
that chooses the Fields medalists, visited Dr. Perelman in St. Petersburg,
where he lives with his mother, to persuade him to accept a Fields Honor,
but Dr. Perelman said, "I refuse."
The union decided to bestow a medal on Dr. Perelman anyway.
Beginning in 2002, Dr. Perelman, who was then at the Steklov Institute of
Mathematics of the Russian Academy of Sciences in St. Petersburg, published a
series of papers on the Internet and gave lectures at several American
universities describing how he had overcome a roadblock in the proof of the
Poincare conjecture.
The conjecture, devised by Henri Poincare in 1904, essentially says that the
only shape that does not have any holes and fits within a finite space is a
sphere. That is certainly true looking at two-dimensional surfaces in the
everyday three-dimensional world, but the conjecture says the same is true
for three-dimensional surfaces embedded in four dimensions.
Dr. Perelman solved a difficult problem that other mathematicians had
encountered when trying to prove the conjecture using a technique called
Ricci flow that smoothes out bumps in a surface and transforms the surfaces
into simpler forms.
Dr. Okounkov, born in 1969 in Moscow, was recognized for work that tied
together different fields of mathematics that had seemed unrelated.
"This is the striking feature of Okounkov's work, finding unexpected links,"
said Enrico Arbarello, a professor of geometry at the University of Rome in
Italy.
Dr. Okounkov's work has found use in describing the changing surfaces of
melting crystals. The boundary between melted and non-melted is created
randomly, but the random process inevitably produces a border in the shape
of a heart.
Dr. Tao, a native of Australia and one of the youngest Fields Medal winners
ever at age 31, has worked in several different fields, producing significant
advances in the understanding of prime numbers, techniques that might lead to
simplifying the equations of Einstein's theory of general relativity and the
equations of quantum mechanics that describe how light bounces around in a
fiber optic cable.
Dr. Werner, born in Germany in 1968, has also worked at the intersection of
mathematics and physics, describing phenomena like percolation and shapes
produced by the random paths of Brownian motion.
The medal was conceived by John Charles Fields, a Canadian mathematician,
"in recognition of work already done and as an encouragement for further
achievements on the part of the recipient."
Since 1936, when the medal was first awarded, judges have interpreted the
terms of Dr. Fields' trust fund to mean that the award should usually be
limited to mathematicians 40 years old or younger.
※ 引述《Giawgwan (教官)》之銘言：
: 今天在西班牙的馬德里, 四年一度的國際數學家大會, 頒發了 Fields 獎. Fields 獎是
: 數學界等同於諾貝爾的超級大獎, 只頒給四十歲以下的數學家.
: 今年一共四個人得獎. 包括 Andrei Okounkov, Wendelin Werner, Grigori Perelman,
: Terence Tao. 底下來寫寫後兩位.
: 首先是 Perelman. 由於數學論文出版是複雜的過程, 通常一篇論文從投稿到出版平均要
: 一年半, 因此數學界自己有一個機制, 如果你寫了定版, 可以直接丟在預印本網站上.
: Perelman 在 2002 和 2003 年, 丟了一共三篇論文, 像天書一樣, 只有架構的
: Poincare 猜想的證明在預印本網站上. 然後人就失去蹤影. 這是一個奇人, 史丹佛哈佛
: 的聘任他都拒絕了, 最近因為盛傳要得獎了被找到, 在家裡陪母親, 靠著救濟金生活. 聽
: 說他也拒絕了領取 Fields 獎.
: Poincare 猜想已經一百年, 也是 Clay 數學研究院的世紀七大難題之一, 每一個懸賞一
: 百萬美金. 想像一下橡皮筋箍在一個球面上, 橡皮筋可以因為彈性慢慢縮成一個點. 但是
: 想像一下橡皮筋如果套住游泳圈, 則橡皮筋拿不出來, 無法縮成一個點. 如果橡皮筋不管
: 如何套在某個東西上, 然後你可以把橡皮筋慢慢縮小成一個點, 則這個東西本質上和球就
: 是一樣的. (所以游泳圈和球在數學上完全不一樣, 一個有洞一個沒洞). 用數學的語言就
: 是說, 一個單連通的閉 2-流形必和一個2-球面同胚.
: Poincare 猜想就是上面這句話三維的情形, "一個單連通的閉 3-流形必和一個 3-球面同
: 胚". 說也奇怪, 四維以上的推廣已經證出來(其間也好幾個數學家因此拿到 Fields 獎
: ), 單單留下三維的情況.
: 這個猜想在數學和物理上都太重要, 因此 Perelman 的論文在數學圈中引起非常大的討論
: . 但是 Perelman 的證明只有架構, 行家一看卻知道是真有東西. 他用 Ricci 流, 像火
: 山熔岩一樣慢慢把雀斑填平, 露出物體真正的形狀. 於是接下來兩年這個領域的數學家陷
: 入昏天黑地, 努力理解他的架構. 三個團隊獨立工作, 分別在前陣子各寫了 "註解", 把
: Perelman 的想法弄得更清楚. 這裡是 Perelman 的那三篇論文:
: http://arxiv.org/abs/math.DG/0211159
: http://arxiv.org/abs/math.DG/0303109
: http://arxiv.org/abs/math.DG/0307245
: 你以為三個團隊的數學家 "註解" 是一兩頁的說明或勘誤表嗎, 別鬧了, 這三個團隊中每
: 一個數學家都是國際級頂尖的. 光是要看懂這些註解, 至少要刻苦讀十年書. 這三個註解
: 如下:
: 曹懷東 和 朱熹平, A Complete Proof of the Poincare and Geometrization
: Conjectures - application of the Hamilton-Perelman theory of the Ricci flow,
: http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf,
: 這個一共 328 頁.
: John Morgan 和 田剛, Ricci Flow and the Poincare Conjecture,
: http://arxiv.org/abs/math.DG/0607607,
: 這個 474 頁.
: Bruce Kleiner 和 John Lott, Notes on Perelman's Papers,
: http://arxiv.org/abs/math.DG/0605667,
: 這個 192 頁.
: Terence Tao 又是另外一個故事, 他和 Green 一起證明了質數中有一個無限多項的等差
: 數列. Tao 是一個天才, 他在八歲時去考 SAT, 然後滿分 800 分他考了 760. 他在
: 11,12,13 歲 (1986,87,88) 參加國際數學奧林匹亞 (IMO, 今年在斯洛維尼亞, 我是台
: 灣隊領隊 ^^) 三次, 拿到銅牌, 銀牌, 金牌. 13 歲拿金牌! 仍然是目前的紀錄保
: 持人. 他 19 歲拿到 Princeton 博士學位, 25 歲成為 UCLA 的正教授. 他現在只有 30
: 歲, 已經得到數學界好幾個大獎: 2000 年 Salem Prize, 2002 年 Bocher Prize, 2003
: 年 Clay Research Award, 現在再加上今年的 Fields 獎.
: 說到 IMO, Perelman 也是 1982 年的滿分金牌. 這也是為什麼每年我帶隊出國, 看到
: 這麼多優秀的各國學生, 我心中總是戰慄不已的緣故. 不寫了, 該唸書了.
: 2006, 森棚教官
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