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)》之铭言： : 今天在西班牙的马德里, 四年一度的国际数学家大会, 颁发了 Fields 奖. Fields 奖是 : 数学界等同於诺贝尔的超级大奖, 只颁给四十岁以下的数学家. : 今年一共四个人得奖. 包括 Andrei Okounkov, Wendelin Werner, Grigori Perelman, : Terence Tao. 底下来写写後两位. : 首先是 Perelman. 由於数学论文出版是复杂的过程, 通常一篇论文从投稿到出版平均要 : 一年半, 因此数学界自己有一个机制, 如果你写了定版, 可以直接丢在预印本网站上. : Perelman 在 2002 和 2003 年, 丢了一共三篇论文, 像天书一样, 只有架构的 : Poincare 猜想的证明在预印本网站上. 然後人就失去踪影. 这是一个奇人, 史丹佛哈佛 : 的聘任他都拒绝了, 最近因为盛传要得奖了被找到, 在家里陪母亲, 靠着救济金生活. 听 : 说他也拒绝了领取 Fields 奖. : Poincare 猜想已经一百年, 也是 Clay 数学研究院的世纪七大难题之一, 每一个悬赏一 : 百万美金. 想像一下橡皮筋箍在一个球面上, 橡皮筋可以因为弹性慢慢缩成一个点. 但是 : 想像一下橡皮筋如果套住游泳圈, 则橡皮筋拿不出来, 无法缩成一个点. 如果橡皮筋不管 : 如何套在某个东西上, 然後你可以把橡皮筋慢慢缩小成一个点, 则这个东西本质上和球就 : 是一样的. (所以游泳圈和球在数学上完全不一样, 一个有洞一个没洞). 用数学的语言就 : 是说, 一个单连通的闭 2-流形必和一个2-球面同胚. : Poincare 猜想就是上面这句话三维的情形, "一个单连通的闭 3-流形必和一个 3-球面同 : 胚". 说也奇怪, 四维以上的推广已经证出来(其间也好几个数学家因此拿到 Fields 奖 : ), 单单留下三维的情况. : 这个猜想在数学和物理上都太重要, 因此 Perelman 的论文在数学圈中引起非常大的讨论 : . 但是 Perelman 的证明只有架构, 行家一看却知道是真有东西. 他用 Ricci 流, 像火 : 山熔岩一样慢慢把雀斑填平, 露出物体真正的形状. 於是接下来两年这个领域的数学家陷 : 入昏天黑地, 努力理解他的架构. 三个团队独立工作, 分别在前阵子各写了 "注解", 把 : Perelman 的想法弄得更清楚. 这里是 Perelman 的那三篇论文: : http://arxiv.org/abs/math.DG/0211159 : http://arxiv.org/abs/math.DG/0303109 : http://arxiv.org/abs/math.DG/0307245 : 你以为三个团队的数学家 "注解" 是一两页的说明或勘误表吗, 别闹了, 这三个团队中每 : 一个数学家都是国际级顶尖的. 光是要看懂这些注解, 至少要刻苦读十年书. 这三个注解 : 如下: : 曹怀东 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, : 这个一共 328 页. : John Morgan 和 田刚, Ricci Flow and the Poincare Conjecture, : http://arxiv.org/abs/math.DG/0607607, : 这个 474 页. : Bruce Kleiner 和 John Lott, Notes on Perelman's Papers, : http://arxiv.org/abs/math.DG/0605667, : 这个 192 页. : Terence Tao 又是另外一个故事, 他和 Green 一起证明了质数中有一个无限多项的等差 : 数列. Tao 是一个天才, 他在八岁时去考 SAT, 然後满分 800 分他考了 760. 他在 : 11,12,13 岁 (1986,87,88) 参加国际数学奥林匹亚 (IMO, 今年在斯洛维尼亚, 我是台 : 湾队领队 ^^) 三次, 拿到铜牌, 银牌, 金牌. 13 岁拿金牌! 仍然是目前的纪录保 : 持人. 他 19 岁拿到 Princeton 博士学位, 25 岁成为 UCLA 的正教授. 他现在只有 30 : 岁, 已经得到数学界好几个大奖: 2000 年 Salem Prize, 2002 年 Bocher Prize, 2003 : 年 Clay Research Award, 现在再加上今年的 Fields 奖. : 说到 IMO, Perelman 也是 1982 年的满分金牌. 这也是为什麽每年我带队出国, 看到 : 这麽多优秀的各国学生, 我心中总是战栗不已的缘故. 不写了, 该念书了. : 2006, 森棚教官 --

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