"junlee,这么聪明的脑袋不来读数学真是可惜，真是辛苦我们这些笨笨的人了；说起最近又成为媒体中心的目前还是唯一的华人菲尔兹奖获得者丘成桐（S.T.Yau），因为本人跟这个领域也沾点边，顺便跟大家介绍一下-----
在媒体面前Yau说是与曹怀德、朱熹平完成了七大猜想之一的庞开莱猜想的封顶工作，有些言过其实，在81/2年汉密尔顿（Hamilton）开创性的引入Ricci flow来研究庞开莱猜想，2002年底俄国“马克思型”数学家Perelman在网上贴出的证明文章，照现在看来，已经把这个猜想7788了。其余的工作很大程度上是在修修补补（在Yau Claimed the complete proof of Poincare conjecture 的同时，另外两个美国数学家也贴出了"the notes on perelman's paper" to comlete the detail for Perelman's proof），或是其中一些证明的改进，或者他的一些想法在其他方向的应用。Perelman 之所以说是马克西型人物，他于80年代底90年代初证明了一个著名的几何（？）问题之后，开始接触Poincare conjecture,来美国做了几年的postdoc之后回到俄国，隐居了好些年，2002年终于又现身，大花脸的胡子，前不久菲尔兹奖委员会说是发信给他（今年夏天四年一次的菲尔兹奖又将有新人来摘取），后来说是他居然没有回信，他真是连菲尔兹奖都不想要了。哎，数学的最为尊贵的奖项阿！
说到即将诞生的菲尔兹奖，有个人不得不提到，也是个华裔，Terrence Tao,中文名字叫陶哲轩,born in Australia,刚从 Princeton 博士毕业就成为UCLA的教授，成为全美最年轻的正教授（24岁），这届的菲尔兹奖Perelman不拿的话，那就很有可能是他了，是继老Yau之后的第二个带华裔血统的人，而且即使他这次不拿，他还有两次机会，对他来说，菲尔兹奖不是拿不拿的问题，而是拿几次的问题，赞!
前不久开车去听了他的一次课, 是一门高等偏微分课程,竟然什么都不带,只拿了只粉笔, 速度真是快啊! 不愧当年12岁开始连续三次国际数学奥赛,分别获得铜银金牌!将来我要是有机会有个儿子的话, 他的名字就是Terrence Kuang,以此纪念这位千年难得的数学天才!!! "
今天之所以发文，就是因为上面提到的fields medal今天刚刚揭晓，Terrence Tao拿到这个奖项，有意思的是，the reclusive russian mathematician G.Perelman truned down the fields medal, and didn't attend the ceremony for the medal.
2006 list:
• The winners of the Fields Medals awarded today, 22nd of August, in Madrid, during the opening ceremony of the International Congress of Mathematicians ICM2006, are: Andrei Okounkov; Grigori Perelman; Terence Tao; and Wendelin Werner.
• The Nevanlinna Prize is awarded to Jon Kleinberg. The Gauss Prize goes to Kiyoshi Itô (Please see separate press release for the Gauss Prize.)
The Fields Medals are the most important international prize in the world of mathematics. They are awarded by the International Mathematical Union (IMU) every four years at the ICM (International Congress of Mathematicians). They are accompanied by strict conditions. For example, the identity of the winners must remain confidential until the presentation ceremony. The awardees are chosen by a committee whose members must also remain anonymous, and who are sworn to secrecy until the official public announcement of their decisions. The award-winners are notified separately a few weeks before the official presentation, although they too are unaware of which of their colleagues have been chosen to receive the prizes.
Between two and four Fields Medals can be awarded at each ICM, and only those mathematicians below the age of 40 (on January 1st of the year in which the Congress is held) are eligible to receive them. This is because they are meant to encourage future endeavour. They are typically awarded for a body of work, rather than a single isolated achievement.
The medals, gold-minted, are named after the Canadian mathematician John Charles Fields (1863-1932) and were first awarded at the International Congress held in Oslo in 1936.
The obverse of the medals shows Archimedes facing right and the motto “Transire Suum Pectus Mundoque Potir”: “To transcend one's spirit and to take hold of (to master) the world”. On the reverse side, also in Latin, the inxxxxion "The mathematicians having congregated from the whole world awarded (this medal) because of outstanding writings”. The name of the Medallist is engraved on the rim of the medal.
Fields Medals ICM2006
Andrei Okounkov: "for his contributions bridging probability, representation
theory and algebraic geometry"
The work of Andrei Okounkov has revealed profound new connections between different areas of mathematics and has brought new insights into problems arising in physics. Although his work is difficult to classify because it touches on such a variety of areas, two clear themes are the use of notions of randomness and of classical ideas from representation theory. This combination has proven powerful in attacking problems from algebraic geometry and statistical mechanics. (Further inxxxxation at www.icm2006.org)
Andrei Okounkov was born in 1969 in Moscow. He received his doctorate in mathematics from Moscow State University in 1995. He is a professor of mathematics at Princeton University. He has also held positions at the Russian Academy of Sciences, the Institute for Advanced Study in Princeton, the University of Chicago, and the University of California, Berkeley. His distinctions include a Sloan Research Fellowship (2000), a Packard Fellowship (2001), and the European Mathematical Society Prize (2004).
Contact: Enrico Arbarello, ea@mat.uniroma1.it, +39 3386397112
Grigori Perelman: "for his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow"
The name of Grigori Perelman is practically a household word among the scientifically interested public. His work from 2002-2003 brought ground-breaking insights into the study of evolution equations and their singularities. Most significantly, his results provide a way of resolving two outstanding problems in topology: the Poincare Conjecture and the Thurston Geometrization Conjecture. As of the summer of 2006, the mathematical community is still in the process of checking his work to ensure that it is entirely correct and that the conjectures have been proved. After more than three years of intense scrutiny, top experts have encountered no serious problems in the work (Further inxxxxation at www.icm2006.org)
Grigori Perelman was born in 1966 in what was then the Soviet Union. He received his doctorate from St. Petersburg State University. During the 1990s he spent time in the United States, including as a Miller Fellow at the University of California, Berkeley. He was for some years a researcher in the St. Petersburg Department of the Steklov Institute of Mathematics. In 1994, he was an invited speaker at the International Congress of Mathematicians in Zurich.
Terence Tao: "for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory".
Terence Tao is a supreme problem-solver whose spectacular work has had an impact across several mathematical areas. He combines sheer technical power, an other-worldly ingenuity for hitting upon new ideas, and a startlingly natural point of view that leaves other mathematicians wondering, "Why didn't anyone see that before?". His interests range over a wide swath of mathematics, including harmonic analysis, nonlinear partial differential equations, and combinatorics. (Further inxxxxation at www.icm2006.org)
Terence Tao was born in Adelaide, Australia, in 1975. He received his PhD in mathematics in 1996 from Princeton University. He is a professor of mathematics at the University of California, Los Angeles. Among his distinctions are a Sloan Foundation Fellowship, a Packard Foundation Fellowship, and a Clay Mathematics Institute Prize Fellowship. He was awarded the Salem Prize (2000), the American Mathematical Society (AMS) Bocher Prize (2002), and the AMS Conant Prize (2005, jointly with Allen Knutson). At 31 years of age, Tao has written over 80 research papers, with over 30 collaborators.
Contact: Charles Fefferman, cf@Math.Princeton.EDU
Wendelin Werner: “for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conxxxxal field theory".
The work of Wendelin Werner and his collaborators represents one of the most exciting and fruitful interactions between mathematics and physics in recent times. Werner's research has developed a new conceptual framework for understanding critical phenomena arising in physical systems and has brought new geometric insights that were missing before. The theoretical ideas arising in this work, which combines probability theory and ideas from classical complex analysis, have had an important impact in both mathematics and physics and have potential connections to a wide variety of applications. (Further inxxxxation at www.icm2006.org)
Born in 1968 in Germany, Wendelin Werner is of French nationality. He received his PhD at the University of Paris VI in 1993. He has been professor of mathematics at the University of Paris-Sud in Orsay since 1997. From 2001 to 2006, he was also a member of the Institut Universitaire de France, and since 2005 he has been seconded part-time to the Ecole Normale Supèrieure in Paris. Among his distinctions are the Rollo Davidson Prize (1998), the European Mathematical Society Prize (2000), the Fermat Prize (2001), the Jacques Herbrand Prize (2003), the Loève Prize (2005) and the Pólya Prize (2006).
Contact: Charles Newman, newman@courant.nyu.edu
Nevanlinna Prize to Jon Kleinberg
The Nevanlinna Prize has been awarded every four years since 1982 in recognition of the most notable advances made in mathematics in the Inxxxxation Society (e.g. computational science, programming languages, algorithm analysis, etc.). This prize consists of a gold medal bearing the profile of Rolf Nevanlinna (1895-1980), rector of the University of Helsinki and president of the IMU (International Mathematical Union). Nevanlinna was the first mathematician to introduce computation into Finnish universities in 1950.
Jon Kleinberg's work has brought theoretical insights to bear on important practical questions that have become central to understanding and managing our increasingly networked world. He has worked in a wide range of areas, from network analysis and routing, to data mining, to comparative genomics and protein structure analysis. In addition to making fundamental contributions to research, Kleinberg has thought deeply about the impact of technology, in social, economic, and political spheres. (Further inxxxxation at www.icm2006.org)
Jon Kleinberg was born in 1971 in Boston, Massachusetts, USA. He received his Ph.D. in 1996 from the Massachusetts Institute of Technology. He is a professor of computer science at Cornell University. Among his distinctions are a Sloan Foundation Fellowship (1997), a Packard Foundation Fellowship (1999), and the Initiatives in Research Award of the U.S. National Academy of Sciences (2001). In 2005, Kleinberg received a MacArthur "genius" Fellowship from the John D. and Catherine T. MacArthur Foundation.
Contact: John Hopcroft, jeh@cs.cornell.edu
摘录一些文章的评论：
Terence Tao is UCLA’s first mathematician to receive the prestigious Fields Medal, often described as the “Nobel Prize in Mathematics.”
Tao, 31, was presented the prize today (August 22) at the International Congress of Mathematicians in Madrid. The Fields Medal is awarded by the International Mathematical Union every fourth year.
Tao's capture of the Fields Medal surprised few at UCLA.
“Terry is like Mozart; mathematics just flows out of him, except without Mozart’s personality problems,” said John Garnett, professor and former chair of the mathematics department.
“People all over the world say, ‘UCLA’s so lucky to have Terry Tao,'” said Tony Chan, dean of physical sciences and professor of mathematics. “The way he crosses areas would be like the best heart surgeon also being exceptional in brain surgery.”
A math prodigy from Adelaide, Australia, Tao started learning calculus as a 7-year-old high school student. By 9, he had progressed to university-level calculus; by 11, he was already burnishing his reputation at international math competitions. Tao was 20 when he earned his Ph.D. from Princeton University and joined UCLA’s faculty. By 24, he had become a full professor.
“The best students in the world in number theory all want to study with Terry,” Chan said. Graduate students have come to UCLA from as far as Romania and China.
One area in which Tao specializes is harmonic analysis, an advanced form of calculus that uses equations from physics. Some of his work involves “geometrical constructions that almost no one understands,” Garnett said. Tao is also regarded as the world’s expert on the “Kakeya conjecture,” a perplexing set of five problems in harmonic analysis. And his work with Ben Green of the University of Bristol, England - proving that prime numbers contain infinitely many progressions of all finite lengths - was lauded by Discover magazine as one of the 100 most important scientific discoveries in 2004.
“I don’t have any magical ability,” Tao said. “I look at a problem, and it looks something like one I’ve done before. I think maybe the idea that worked before will work here. . . . After awhile, I figure out what’s going on.”
"与Perelman的搞笑对话（仿鹿鼎记）》
前日在群与图讨论班徐老谈到自己一个弟子已博士毕业却想着出家当和尚。我听说此事之后，顿觉数学和佛学倒是有共同之处，于是模拟《鹿鼎记》中韦小宝和澄观的对话写些无聊的文字。请看：topowu与Perelman对话（Perelman号称解决三维Poincare猜想）。
topowu说道：“你刚才随便写写，Poincare猜想就顺利解决，这是什么功夫？”
Perelman道：“这是‘Ricci 流’功夫，你不会吗？”
topowu道：“我不会。不如你教了我罢。”
Perelman道：“师叔有命，自当遵从。这‘Ricci 流’功夫，也不难学，只要问题看得准，用点时间仔细算算，也就成了。”
topowu大喜，忙道：“那好极了，你快快教我。”心想学会了这门功夫，就随便算算，那难题便轻松搞定，那时要得Fields奖，还不容易？而“也不难学”四字，更是关键所在。天下功夫之妙，无过于此，霎时间眉花眼笑，心痒难搔。
Perelman道：“师叔的偏微分内功，不知练到了第几层，请你解这个椭圆方程试试。”topowu道：“怎样解法？”Perelman屈指一算，大吼一声，拿起粉笔就写，瞬间题目搞定。
topowu笑道：“那倒好玩。”学着他样，也是大吼一声，拿起粉笔就写，但半天也未见动静。
Perelman道：“原来师叔没练过偏微分内功，要练这门内劲，须得先练调和分析。待我跟你聊聊调和分析，看了师叔功力深浅，再传授偏微分。”topowu道：“调和分析我也不会。”Perelman道：“那也不妨，咱们来拆复分析。”topowu道：“什么复分析，可没听见过。”
Perelman脸上微有难色，道：“那么咱们试拆再浅一些的，试同调论好了。这个也不会？就从抽象代数II试起好了。也不会？那要试线性偏微分方程。是了，师叔年纪小，还没学到这路功夫，抽象代数I？微分几何？？点集拓扑？”他说一路功夫，topowu便摇一摇头。
Perelman见topowu什么科目都不会，也不生气，说道：“咱们低维拓扑武功循序渐进，入门之后先学点集拓扑，熟习之后，再学微分几何，然后学抽象代数I，内功外功有相当根柢了，可以学线性偏微分方程。如果不学线性偏微分方程，那么学现代分析基础也可以……”topowu口唇一动，便想说：“这现代分析基础我倒会。”随即忍住，知道XXX所教的这些什么现代分析基础，十条定理中只怕有九条半没说清楚，这个“会”字，无论如何说不上。只听Perelman续道：“不论学线性偏微分方程或现代分析基础，聪明勤力的，学三四年也差不多了。如果悟性高，可以跟着学复分析。学到复分析，别的大学的一般弟子，就不大能比你强了。是否能学调和分析，要看各人性子是否适合学数学。”
topowu倒抽了口凉气，说道：“你说那Ricci流并不难学，可是从点集拓扑练起，一门门科目学将下来，练成这Ricci流内功，要几年功夫？”
Perelman微笑道：“师侄从大二开始学点集拓扑，总算运气极好，能入名校学习，学得比一般人扎实，到40岁，于这内功已略窥门径。”
topowu道：“你从大二练起，到了40岁时略跪什么门闩，那么总共练了二十二年才练成？”Perelman甚是得意，道：“以二十二年而练成Ricci流内功，近一个世纪，我名列第三。”顿了一顿，又道：“不过老衲的内力修为平平，若以功力而论，恐怕排名在七十名以下。”说到这里，又不禁沮丧。
topowu心想：“管你排第三也好，第七十三也好，老子前世不修，似乎没从娘胎里带来什么武功，要花二十二年时光来练这指法，我都四五十岁老头子啦。老子还得个屁的Fields！”说道：“人家伽罗华年纪轻轻，你要练二三十年才比得过他，实在差劲之至。”
Perelman早想到了此节，一直在心下盘算，说道：“是，是！咱们武功如此给人家比了下去，实在……实在不……不大好。”
topowu道：“什么不大好，简直糟糕之极。咱们低维拓扑这一下子，可就抓不到武林中的牛耳朵，马耳朵了。你是牛校教授，不想个法子，怎对得起一个世纪来这个方向的高人？你死了以后，见到庞什么莱、布什么尔，大家责问你，说你只是吃饭拉屎，却不管事，不想法子保全低维拓扑的威名，岂不羞也羞死了？”
Perelman老脸通红，十分惶恐，连连点头，道：“师叔指点得是，待师侄回去，翻查图书馆中的Paper，看有什么妙法，可以速成。”topowu喜道：“是啊，你倘若查不出来，咱们也不用再在数学界中混了。不如让他们搞代数的来当我们的老板。”
Perelman一怔，问道：“他们搞代数的，怎么能做我们搞低维拓扑的老板？”
topowu道：“谁教你想不出速成的法子？你自己丢脸，那也不用说了，低维拓扑从此在数学圈中没了立足之地，本方向几千名教授，都要去改拜他们搞代数的为师了。大家都说，花了几十年时光来学低维拓扑，又有什么用？人家伽罗华脑袋灵光一闪就解决几百年的难题。不如大家都搞代数去算了。”
这番言语只把Perelman听得额头汗水涔涔而下，双手不住发抖，颤声道：“是，是！那……那太丢人了。”topowu道：“可不是吗？那时候咱们也不叫低维拓扑了。”Perelman问道：“那……那叫什么？”topowu道：“不如干脆叫低维代数好啦，低维拓扑改成低维代数。只消将山门上的牌匾取下来，刮掉那个‘拓扑’字，换上一个‘代数’字，那也容易得紧。”Perelman脸如土色，忙道：“不成，不成！我……我这就去想法子。师叔，恕师侄不陪了。”合十行礼，转身便走。
topowu道：“且慢！这件事须得严守秘密。倘若有人知道了，可大大的不妥。”Perelman问道：“为什么？”topowu道：“大家信不过你，也不知你想不想得出法子。而大家都想一举成名，在现实考虑之下，都去改学代数，咱们偌大低维拓扑，岂不就此散了？”
Perelman道：“师叔指点的是。此事有关本派兴衰存亡，那是万万说不得的。”心中好生感激，心想这位师叔年纪虽小，却眼光远大，前辈师尊，果然了得，若非他灵台明澈，具卓识高见，低维拓扑不免变了低维代数，百年主流，万劫不复。
topowu见他匆匆而去，袍袖颤动，显是十分惊惧，心想：“他拚了老命去想法子，总会有些门道想出来。我这番话人人都知破绽百出，但只要他不和旁人商量，谅这他也不知我在骗他。”想起得了Fields后的荣誉，一阵心猿意马，拿起本书看了看，却发现身边没了Perelman指导，单身一人，什么也学不动，只得叹了口气，回到自已宿舍休息。"
"Grisha Perelman, where are you?
Three years ago, a Russian mathematician by the name of Grigory Perelman 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, Perelman, known as Grisha, disappeared back into the Russian woods in the spring of 2003, leaving the world's mathematicians to pick up the pieces and decide whether 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 University, who has spent the last three years helping to explicate 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 the Poincaré conjecture could be one of the major pillars of math in the 21st century.
But at the moment of his putative triumph, 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 the $1 million offered by the Clay Mathematics Institute in Cambridge, Massachusetts, for the first published proof of the conjecture, one of seven outstanding questions for which they offered a ransom 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 University in New York, one of the scholars who has also been filling in the details of 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.
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 University, the author of a deeper conjecture, which includes Poincaré's and which 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," saying that curiosity is tied in some way with intuition.
"You don't see what you're seeing until you see it," Thurston said, "but when you do see it, it lets you see many other things."
Depending on who is talking, the Poincaré 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, or 20,100 kilometers, from home on 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.
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 Anderson of the State University of New York in Stony Brook.
Perelman had already established himself as a master of differential geometry, the study of curves and surfaces. 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 doctorate 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 1990s, Perelman impressed his colleagues as "a kind of unworldly person," in the words of Robert Greene, a mathematician at the University of California, Los Angeles - friendly, but shy and not interested in material wealth.
"He looked like Rasputin, with long hair and fingernails," Greene said.
Asked about Perelman's pleasures, Anderson said that he talked a lot about hiking in the woods near St. Petersburg looking for mushrooms.
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 the Poincaré conjecture started appearing, some friends thought 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, Perelman came back to the United States to give a series of lectures.
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," Anderson said. "Anything else was superfluous."
Recently, 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"
Terence Tao has been years ahead of everyone else his entire life. Tao started taking high school classes at age eight; by 11, he was learning calculus and thriving in international mathematics competitions. He was only 21 when he earned his Ph.D. from Princeton University, and joined UCLA's faculty that year. The UCLA College promoted Tao to full professor of mathematics at 24.
"Terry is like Mozart -- except without Mozart's personality problems," said John Garnett, professor and former chair of mathematics in the UCLA College. "Mathematics just flows out of him." "Mathematicians with Terry's abilities appear only once in a generation," said Garnett. "He's probably the best mathematician in the world right now. Terry can unravel an enormously complicated mathematical problem and reduce it to something very simple. We're amazingly lucky to have him at UCLA."
The Fields Medal is considered the Nobel Prize for mathematics, said Tony Chan, Dean of Physical Sciences in the College, and professor and former chair of mathematics. The medal is given every fourth year by the International Mathematical Union, and will be given next summer in Madrid.
No one from UCLA has ever won the Fields Medal, but will that change in 2006?
"Terry is very creative and one of the most talented mathematicians I have seen in the last two decades," Chan said. "He has solved problems that have stumped others. I think the breadth and depth of his work, taken together, should make him a worthy candidate for the Fields Medal. What is even more amazing is that Terry is still so young. If he were a company, he would be Microsoft right before it sent public. If you could invest in him, you would certainly want to, because the payoff will be enormous."
"Terry will be a leading candidate for the 2006 Fields Medal," Garnett said.
While most mathematicians focus on just one branch of mathematics, Tao works in several areas, some completely unrelated to the others, and is a leading figure in four distinct areas, Garnett said.
Tao's branches of mathematics include a theoretical field called harmonic analysis, an advanced form of calculus that uses equations from physics. Some of this work involves, in Garnett's words, "geometrical constructions that almost no one understands."
Tao, 29, also works in a related field, non-linear partial differential equations, and in the entirely distinct fields of algebraic geometry, number theory, and combinatorics.
Tao and colleagues have taken on complex mathematical problems, in one case expanding on work begun by the one of the founders of formal mathematical studies more than 2,200 years ago. Work on prime numbers by Tao and University of Bristol mathematician Ben Green was acknowledged by Discover magazine as one of the 100 most important discoveries in science for 2004.
Green and Tao expanded on theories that originated with the Greek mathematician Euclid. Euclid proved there is an infinite quantity of prime numbers (a number divisible only by itself and one). Green and Tao proved that the prime numbers contain infinitely many progressions of all finite lengths. An example of an equally spaced progression of primes, of length three, is 3, 7, 11; the largest known progression of prime numbers is length 24, with each of the numbers containing more than two dozen digits. Green and Tao's discovery reveals that somewhere in the prime numbers, there is a progression of length 100, and length 1,000, and every other finite length. They also demonstrated that there are an infinite number of such progressions in the primes.
To prove this, Tao and Green spent two years analyzing all four proofs of a theorem named for Hungarian mathematician Endre Szemeredi. Very few mathematicians understand all four proofs, and Szemeredi's theorem does not apply to prime numbers.
"We took Szemeredi's theorem and goosed it so that it handles primes," Tao said. "To do that, we borrowed from each of the four proofs to build an extended version of Szemeredi's theorem. Every time Ben and I got stuck, there was always an idea from one of the four proofs that we could somehow shoehorn into our argument." Tao is also known among mathematics researchers for his work on the "Kakeya conjecture," a perplexing set of five problems in harmonic analysis. One of Tao's proofs extends more than 50 pages, in which he and two colleagues obtained the most precise known estimate of the size of a particular geometric dimension in Euclidean space. The issue involves the most space-efficient way to fully rotate an object in three dimensions, a question that interests theoretical mathematicians.
Tao and colleagues Allen Knutson at UC Berkeley and Chris Woodward at Rutgers solved an old problem (proving a conjecture proposed by former UCLA professor Alfred Horn) for which they developed a method that also solved longstanding problems in algebraic geometry ?describing equations geometrically ?and representation theory.
Speaking of this work, Tao said, "Other mathematicians gave the impression that the puzzle required so much effort that it was not worth making the attempt ?that first you have to understand this 100-page paper and that 100-page paper before even starting. We used a different approach to solve a key missing gap."
Solving some problems comes out of less formal collaborations. Tao found a surprising result to an applied mathematics problem involving image processing with Caltech mathematician Emmanuel Candes in a collaboration forged while they were taking their children to UCLA's Fernald Child Care Center.
"A lot of our work came at the pre-school while we were dropping off our kids," Tao said.
How does Tao describe his success?
"I don't have any magical ability," he said. "I look at a problem, and it looks something like one I've already done; I think maybe the idea that worked before will work here. When nothing's working out; then I think of a small trick that makes it a little better, but still is not quite right. I play with the problem, and after a while, I figure out what's going on.
"Most mathematicians faced with a problem, will try to solve the problem directly. Even if they get it, they might not understand exactly what they did. Before I work out any details, I work on the strategy. Once I have a strategy, a very complicated problem can split up into a lot of mini-problems. I've never really been satisfied with just solving the problem; I want to see what happens if I make some changes.
"If I experiment enough, I get a deeper understanding," said Tao, whose work is supported by the David and Lucille Packard Foundation. "After a while, when something similar comes along, I get an idea of what works and what doesn't work.
"It's not about being smart or even fast," Tao added. "It's like climbing a cliff; if you're very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there. Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness and good tools; you still need a plan ?that's the hard part ?and you have to see the bigger picture."
His views about mathematics have changed over the years.
"When I was a kid, I had a romanticized notion of mathematics -- that hard problems were solved in Eureka moments of inspiration," he said. "With me, it's always, 憀et's try this that gets me part of the way. Or, that doesn't work, so now let's try this. Oh, there's a little shortcut here.'
"You work on it long enough and you happen to make progress towards a hard problem by a back door at some point. At the end, it's usually, 'oh, I've solved the problem.'"
Tao concentrates on one math problem at a time, but keeps a couple of dozen others in the back of his mind, "hoping one day I'll figure out a way to solve them. If there's a problem that looks like I should be able to solve it but I can't, that gnaws at me."
Does theoretical mathematics have applications beyond the theory?
"Mathematicians often work on pure problems that may not have applications for 20 years -- and then a physicist or computer scientist or engineer has a real-life problem that requires the solution of a mathematical problem, and finds that someone already solved it 20 years ago," Tao said.
"When Einstein developed his theory of relativity, he needed a theory of curved space. Einstein found that a mathematician devised exactly the theory he needed more than 30 years earlier."
Will Tao become an even better mathematician in another decade or so?
"Experience helps a lot," he said. "I may get a little slower, but I'll have access to a larger database of tricks; I'll know better what will work and what won't. I'll get d閖?vu more often, seeing a problem that reminds me of something."
What does Tao think of his success?
"I'm very happy," he said. "Maybe when I'm in my 60s, I'll look back at what I've done, but now I would rather work on the problems."
