"junlee,這麽聰明的腦袋不來讀數學真是可惜，真是辛苦我們這些笨笨的人了；說起最近又成為媒體中心的目前還是唯一的華人菲爾茲獎獲得者丘成桐（S.T.Yau），因為本人跟這個領域也沾點邊，順便跟大家介紹一下-----
在媒體麵前Yau說是與曹懷德、朱熹平完成了七大猜想之一的龐開萊猜想的封頂工作，有些言過其實，在81/2年漢密爾頓（Hamilton）開創性的引入Ricci flow來研究龐開萊猜想，2002年底俄國“馬克思型”數學家Perelman在網上貼出的證明文章，照現在看來，已經把這個猜想7788了。其餘的工作很大程度上是在修修補補（在Yau Claimed the complete proof of Poincare conjecture 的同時，另外兩個美國數學家也貼出了"the notes on perelman's paper" to comlete the detail for Perelman's proof），或是其中一些證明的改進，或者他的一些想法在其他方向的應用。Perelman 之所以說是馬克西型人物，他於80年代底90年代初證明了一個著名的幾何（？）問題之後，開始接觸Poincare conjecture,來美國做了幾年的postdoc之後回到俄國，隱居了好些年，2002年終於又現身，大花臉的胡子，前不久菲爾茲獎委員會說是發信給他（今年夏天四年一次的菲爾茲獎又將有新人來摘取），後來說是他居然沒有回信，他真是連菲爾茲獎都不想要了。哎，數學的最為尊貴的獎項阿！
說到即將誕生的菲爾茲獎，有個人不得不提到，也是個華裔，Terrence Tao,中文名字叫陶哲軒,born in Australia,剛從 Princeton 博士畢業就成為UCLA的教授，成為全美最年輕的正教授（24歲），這屆的菲爾茲獎Perelman不拿的話，那就很有可能是他了，是繼老Yau之後的第二個帶華裔血統的人，而且即使他這次不拿，他還有兩次機會，對他來說，菲爾茲獎不是拿不拿的問題，而是拿幾次的問題，讚!
前不久開車去聽了他的一次課, 是一門高等偏微分課程,竟然什麽都不帶,隻拿了隻粉筆, 速度真是快啊! 不愧當年12歲開始連續三次國際數學奧賽,分別獲得銅銀金牌!將來我要是有機會有個兒子的話, 他的名字就是Terrence Kuang,以此紀念這位千年難得的數學天才!!! "
今天之所以發文，就是因為上麵提到的fields medal今天剛剛揭曉，Terrence Tao拿到這個獎項，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
摘錄一些文章的評論：
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.”
"與Perelman的搞笑對話（仿鹿鼎記）》
前日在群與圖討論班徐老談到自己一個弟子已博士畢業卻想著出家當和尚。我聽說此事之後，頓覺數學和佛學倒是有共同之處，於是模擬《鹿鼎記》中韋小寶和澄觀的對話寫些無聊的文字。請看：topowu與Perelman對話（Perelman號稱解決三維Poincare猜想）。
topowu說道：“你剛才隨便寫寫，Poincare猜想就順利解決，這是什麽功夫？”
Perelman道：“這是‘Ricci 流’功夫，你不會嗎？”
topowu道：“我不會。不如你教了我罷。”
Perelman道：“師叔有命，自當遵從。這‘Ricci 流’功夫，也不難學，隻要問題看得準，用點時間仔細算算，也就成了。”
topowu大喜，忙道：“那好極了，你快快教我。”心想學會了這門功夫，就隨便算算，那難題便輕鬆搞定，那時要得Fields獎，還不容易？而“也不難學”四字，更是關鍵所在。天下功夫之妙，無過於此，霎時間眉花眼笑，心癢難搔。
Perelman道：“師叔的偏微分內功，不知練到了第幾層，請你解這個橢圓方程試試。”topowu道：“怎樣解法？”Perelman屈指一算，大吼一聲，拿起粉筆就寫，瞬間題目搞定。
topowu笑道：“那倒好玩。”學著他樣，也是大吼一聲，拿起粉筆就寫，但半天也未見動靜。
Perelman道：“原來師叔沒練過偏微分內功，要練這門內勁，須得先練調和分析。待我跟你聊聊調和分析，看了師叔功力深淺，再傳授偏微分。”topowu道：“調和分析我也不會。”Perelman道：“那也不妨，咱們來拆複分析。”topowu道：“什麽複分析，可沒聽見過。”
Perelman臉上微有難色，道：“那麽咱們試拆再淺一些的，試同調論好了。這個也不會？就從抽象代數II試起好了。也不會？那要試線性偏微分方程。是了，師叔年紀小，還沒學到這路功夫，抽象代數I？微分幾何？？點集拓撲？”他說一路功夫，topowu便搖一搖頭。
Perelman見topowu什麽科目都不會，也不生氣，說道：“咱們低維拓撲武功循序漸進，入門之後先學點集拓撲，熟習之後，再學微分幾何，然後學抽象代數I，內功外功有相當根柢了，可以學線性偏微分方程。如果不學線性偏微分方程，那麽學現代分析基礎也可以……”topowu口唇一動，便想說：“這現代分析基礎我倒會。”隨即忍住，知道XXX所教的這些什麽現代分析基礎，十條定理中隻怕有九條半沒說清楚，這個“會”字，無論如何說不上。隻聽Perelman續道：“不論學線性偏微分方程或現代分析基礎，聰明勤力的，學三四年也差不多了。如果悟性高，可以跟著學複分析。學到複分析，別的大學的一般弟子，就不大能比你強了。是否能學調和分析，要看各人性子是否適合學數學。”
topowu倒抽了口涼氣，說道：“你說那Ricci流並不難學，可是從點集拓撲練起，一門門科目學將下來，練成這Ricci流內功，要幾年功夫？”
Perelman微笑道：“師侄從大二開始學點集拓撲，總算運氣極好，能入名校學習，學得比一般人紮實，到40歲，於這內功已略窺門徑。”
topowu道：“你從大二練起，到了40歲時略跪什麽門閂，那麽總共練了二十二年才練成？”Perelman甚是得意，道：“以二十二年而練成Ricci流內功，近一個世紀，我名列第三。”頓了一頓，又道：“不過老衲的內力修為平平，若以功力而論，恐怕排名在七十名以下。”說到這裏，又不禁沮喪。
topowu心想：“管你排第三也好，第七十三也好，老子前世不修，似乎沒從娘胎裏帶來什麽武功，要花二十二年時光來練這指法，我都四五十歲老頭子啦。老子還得個屁的Fields！”說道：“人家伽羅華年紀輕輕，你要練二三十年才比得過他，實在差勁之至。”
Perelman早想到了此節，一直在心下盤算，說道：“是，是！咱們武功如此給人家比了下去，實在……實在不……不大好。”
topowu道：“什麽不大好，簡直糟糕之極。咱們低維拓撲這一下子，可就抓不到武林中的牛耳朵，馬耳朵了。你是牛校教授，不想個法子，怎對得起一個世紀來這個方向的高人？你死了以後，見到龐什麽萊、布什麽爾，大家責問你，說你隻是吃飯拉屎，卻不管事，不想法子保全低維拓撲的威名，豈不羞也羞死了？”
Perelman老臉通紅，十分惶恐，連連點頭，道：“師叔指點得是，待師侄回去，翻查圖書館中的Paper，看有什麽妙法，可以速成。”topowu喜道：“是啊，你倘若查不出來，咱們也不用再在數學界中混了。不如讓他們搞代數的來當我們的老板。”
Perelman一怔，問道：“他們搞代數的，怎麽能做我們搞低維拓撲的老板？”
topowu道：“誰教你想不出速成的法子？你自己丟臉，那也不用說了，低維拓撲從此在數學圈中沒了立足之地，本方向幾千名教授，都要去改拜他們搞代數的為師了。大家都說，花了幾十年時光來學低維拓撲，又有什麽用？人家伽羅華腦袋靈光一閃就解決幾百年的難題。不如大家都搞代數去算了。”
這番言語隻把Perelman聽得額頭汗水涔涔而下，雙手不住發抖，顫聲道：“是，是！那……那太丟人了。”topowu道：“可不是嗎？那時候咱們也不叫低維拓撲了。”Perelman問道：“那……那叫什麽？”topowu道：“不如幹脆叫低維代數好啦，低維拓撲改成低維代數。隻消將山門上的牌匾取下來，刮掉那個‘拓撲’字，換上一個‘代數’字，那也容易得緊。”Perelman臉如土色，忙道：“不成，不成！我……我這就去想法子。師叔，恕師侄不陪了。”合十行禮，轉身便走。
topowu道：“且慢！這件事須得嚴守秘密。倘若有人知道了，可大大的不妥。”Perelman問道：“為什麽？”topowu道：“大家信不過你，也不知你想不想得出法子。而大家都想一舉成名，在現實考慮之下，都去改學代數，咱們偌大低維拓撲，豈不就此散了？”
Perelman道：“師叔指點的是。此事有關本派興衰存亡，那是萬萬說不得的。”心中好生感激，心想這位師叔年紀雖小，卻眼光遠大，前輩師尊，果然了得，若非他靈台明澈，具卓識高見，低維拓撲不免變了低維代數，百年主流，萬劫不複。
topowu見他匆匆而去，袍袖顫動，顯是十分驚懼，心想：“他拚了老命去想法子，總會有些門道想出來。我這番話人人都知破綻百出，但隻要他不和旁人商量，諒這他也不知我在騙他。”想起得了Fields後的榮譽，一陣心猿意馬，拿起本書看了看，卻發現身邊沒了Perelman指導，單身一人，什麽也學不動，隻得歎了口氣，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."
