哪里有人哪里就有江湖

   昂利•庞加莱是法国数学家，1854年4月29日生于南锡，1912年7月17日卒于巴黎。

    庞加莱的父母亲都出身于法国的显赫世家，几代人都居住在法国东部的洛林。庞加莱从小就显出超常的智力，他智力的重要来源之一是遗传。他的双亲智力都很高，他的双亲又可追溯到他的祖父。他的祖父曾在拿破仑政权下的圣康坦部队医院供职，1817年在鲁昂定居，先后生下两个儿子，大儿子莱昂·庞加莱即为庞加莱的父亲。绿色建筑博客~u|\^8sj

    庞加莱的父亲是当地一位著名医生，并任南锡大学医学院教授。他的母亲是一位善良、才华出众、很有教养的女性，一生的心血全部倾注到教育和照料孩子身上。庞加莱叔叔的两个儿子是法国政界的著名人物：雷蒙·庞加莱于1913至1920年间任法国总统；吕西·庞加莱曾任法国民众教育与美术部长，负责中等教育工作。

    庞加莱的童年主要接受母亲的教育。他的超常智力使他成为早熟的儿童，不仅接受知识极为迅速，而且口才也很流利。但不幸的事发生了：五岁时患了一场白喉病、九个月后喉头坏了，致使他的思想不能顺利用口头表达出来，并成为一位体弱多病的入。尽管如此，庞加莱还是乐意玩耍游戏，喜欢跳舞。当然，剧烈的运动他是无法进行。

    庞加莱特别爱好读书，读书的速度快得惊人，而且能对读过的内容迅速、准确、持久地记住。他甚至能讲出书中某件事是在第几页第几行中讲述的！庞加莱还对博物学发生过特殊的兴趣，《大洪水前的地球》一书据说给他留下了终身不忘的印象。他对自然史的兴趣也很浓，历史、地理的成绩也很优异。他在儿童时代还显露了文学才华，有的作文被老师誉为“杰作”。绿色建筑博客eUr{0Gj(y+^R

    庞加莱l862年进入南锡中学读书。初进校时虽然他的各科学习成绩十分优异，但并没有对数学产生特殊的兴趣。对数学的特殊兴趣大约开始于15岁，并很快就显露了非凡才能。从此，他习惯于一边散步，一边解数学难题。这种习惯一直保持终身。

    1870年7月19日爆发的普法战争使得庞加莱不得不中断学业。法国被战败了，法国的许多城乡被德军洗劫一空并被德军占领。为了了解时局，他很快学会了德文。他通过亲眼看到的德军的暴行，使他成了一个炽热的爱国者。绿色建筑博客w)I6za$c~

    1871年3月18日，巴黎无产者举行了武装起义，普法的反动派又很快联合起来扑灭了革命烈火，庞加莱又继续上学了。1872年庞加莱两次荣获法国公立中学生数学竞赛头等奖，从而使他于1873年被高等二科学校作第一名录取。据说，在南锡中学读书时，他的老师就誉称他为“数学巨人”。高等工科学校为了测试他的数学才能还特意设计了一套“漂亮的问题”，一方面要考出他的数学天才；另一方面也为了避免40年前伽罗瓦的教训重演。绿色建筑博客z+W {wTbF(L

    1875年～1878年，庞加莱在高等工科学校毕业后，又在国立高等矿业学校学习工程，准备当一名工程师。但他却缺少这方面的勇气，且与他的兴趣不符。

    1879年8月1日，庞加莱撰写了关于微分方程方面的博士论文，获得了博士学位。然后到卡昂大学理学院任讲师，1881年任巴黎大学教授，直到去世。这样，庞加莱一生的科学事业就和巴黎大学紧紧地联在一起了。绿色建筑博客 B^s5q9v6t B8i

    庞加莱的研究涉及数论、代数学、几何学、拓扑学等许多领域，最重要的工作是在分析学方面。他早期的主要工作是创立自守函数理论(1878)。他引进了富克斯群和克莱因群，构造了更一般的基本域。他利用后来以他的名字命名的级数构造了自守函数，并发现这种函数作为代数函数的单值化函数的效用。

    1883年，庞加莱提出了一般的单值化定理(1907年，他和克贝相互独立地给出完全的证明)。同年，他进而研究一般解析函数论，研究了整函数的亏格及其与泰勒展开的系数或函数绝对值的增长率之间的关系，它同皮卡定理构成后来的整函数及亚纯函数理论发展的基础。他又是多复变函数论的先驱者之一。

    庞加莱为了研究行星轨道和卫星轨道的稳定性问题，在1881～1886年发表的四篇关于微分方程所确定的积分曲线的论文中，创立了微分方程的定性理论。他研究了微分方程的解在四种类型的奇点(焦点、鞍点、结点、中心)附近的性态。他提出根据解对极限环(他求出的一种特殊的封闭曲线)的关系，可以判定解的稳定性。绿色建筑博客)d2w?gc?\_Ec

    1885年，瑞典国王奥斯卡二世设立“n体问题”奖，引起庞加莱研究天体力学问题的兴趣。他以关于当三体中的两个的质量比另一个小得多时的三体问题的周期解的论文获奖，还证明了这种限制性三体问题的周期解的数目同连续统的势一样大。这以后，他又进行了大量天体力学研究，引进了渐进展开的方法，得出严格的天体力学计算技术。绿色建筑博客D:Z%\ [*]2vrz g)i

    庞加莱还开创了动力系统理论，1895年证明了“庞加莱回归定理”。他在天体力学方面的另一重要结果是，在引力作用下，转动流体的形状除了已知的旋转椭球体、不等轴椭球体和环状体外，还有三种庞加莱梨形体存在。

    庞加莱对数学物理和偏微分方程也有贡献。他用括去法证明了狄利克雷问题解的存在性，这一方法后来促使位势论有新发展。他还研究拉普拉斯算子的特征值问题，给出了特征值和特征函数存在性的严格证明。他在积分方程中引进复参数方法，促进了弗雷德霍姆理论的发展。

    庞加莱对现代数学最重要的影响是创立组合拓扑学。1892年他发表勒第一篇论文，1895～1904年，他在六篇论文中建立了组合拓扑学。他还引进贝蒂数、挠系数和基本群等重要概念，创造流形的三角剖分、单纯复合形、重心重分、对偶复合形、复合形的关连系数矩阵等工具，借助它们推广欧拉多面体定理成为欧拉—庞加莱公式，并证明流形的同调对偶定理。

    庞加莱的思想预示了德·拉姆定理和霍奇理论。他还提出庞加莱猜想，在“庞加莱的最后定理”中，他把限制性三体问题的周期解的存在问题，归结为满足某种条件的平面连续变换不动点的存在问题。绿色建筑博客+m/p]yf#hD

    庞加莱在数论和代数学方面的工作不多，但很有影响。他的《有理数域上的代数几何学》一书开创了丢番图方程的有理解的研究。他定义了曲线的秩数，成为丢番图几何的重要研究对象。他在代数学中引进群代数并证明其分解定理。第一次引进代数中的左理想和右理想的概念。证明了李代数第三基本定理及坎贝尔—豪斯多夫公式。还引进李代数的包络代数，并对其基加以描述，证明了庞加莱—伯克霍夫—维特定理。绿色建筑博客-`'dL-PBJn3p

    庞加莱对经典物理学有深入而广泛的研究，对狭义相对论的创立有贡献。他从1899年开始研究电子理论，首先认识到洛伦茨变换构成群。绿色建筑博客X7|:HI'T6Z Un

    庞加莱的哲学著作《科学与假设》、《科学的价值》、《科学与方法》也有着重大的影响。他是约定主义的代表人物，认为科学公理是方便的定义或约定，可以在一切可能的约定中进行选择，但需以实验事实为依据，避开一切矛盾。在数学上，他不同意罗素、希尔伯特的观点，反对无穷集合的概念，赞成潜在的无穷，认为数学最基本的直观概念是自然数，反对把自然数归结为集合论。这使他成为直觉主义的先驱者之一。绿色建筑博客|
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    1905年，匈牙利科学院颁发一项奖金为l0000金克朗的鲍尔约奖。这个奖是要奖给在过去25年为数学发展作出过最大贡献的数学家。由于庞加莱从1879年就开始从事数学研究，并在数学的几乎整个领域都作出了杰出贡献，因而此项奖又非他莫属。

    1906年，庞加莱当选为巴黎科学院主席；1908年，他被选为法国科学院院士，这是一位法国科学家所能达到的最高地位。1908年庞加莱因前列腺增大而未能前往罗马，虽经意大利外科医生作了手术，使他能继续如前一样精力充沛地工作，但好景不长。绿色建筑博客6l'Q5D2B5{
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    1912年春天，庞加莱再次病倒了，7月9日作了第二次手术；7月l7日在穿衣服时，突然因血栓梗塞，在巴黎逝世，终年仅58岁!

    庞加莱被公认是19世纪后四分之一和二十世纪初的领袖数学家，是对于数学和它的应用具有全面知识的最后一个人。绿色建筑博客%m.P:k
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    罗素认为，本世纪初法兰西最伟大的人物就是昂利·庞加莱。“当我最近在盖•吕萨街庞加莱通风的休息处拜访他时，……我的舌头一下子失去了功能，直到我用了一些时间(可能有两、三分钟)仔细端详和承受了可谓他思想的外部形式的年轻面貌时，我才发现自己能够开始说话了。”

    这位“如此美貌，如此年轻”的孩子，竟然是那些洪水般涌来、预示了柯西的一个后继者的到来的论文作者，这是创办《美国数学杂志》的英国数学家西尔维斯待于1885年见到庞加莱的心情写照。

    阿达马这位曾在函数论、数论、微分方程、泛函分析、微分几何、集合论、数学基础等领域作出过杰出贡献的法国数学家认为，庞加莱“整个地改变了数学科学的状况，在一切方向上打开了新的道路。”绿色建筑博客q|/Rm*S

    庞加莱逝世80年来的历史告诉我们，罗素、西尔维斯特、阿达马等的论断是多么正确！庞加莱一生发表的科学论文约500篇、科学著作约30部，几乎涉及到数学的所有领域以及理论物理、天体物理等的许多重要领域。

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围绕庞加勒猜想获得证明的恨爱情仇。。。

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MANIFOLD DESTINY
Se7}l(Vq@T'Il0A legendary problem and the battle over who solved it.

by SYLVIA NASAR AND DAVID GRUBER绿色建筑博客%c c$lEoJ
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Issue of 2006-08-28
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On the evening of June 20th, several hundred physicists, including a Nobel laureate, assembled in an auditorium at the Friendship Hotel inBeijingfor a lecture by the Chinese mathematician Shing-Tung Yau. In the late nineteen-seventies, when Yau was in his twenties, he had made a series of breakthroughs that helped launch the string-theory revolution in physics and earned him, in addition to a Fields Medal—the most coveted award in mathematics—a reputation in both disciplines as a thinker of unrivalled technical power.

Yau had since become a professor of mathematics at Harvard and the director of mathematics institutes inBeijingandHong Kong, dividing his time between theUnited StatesandChina. His lecture at the Friendship Hotel was part of an international conference on string theory, which he had organized with the support of the Chinese government, in part to promote the country’s recent advances in theoretical physics. (More than six thousand students attended the keynote address, which was delivered by Yau’s close friend Stephen Hawking, in the Great Hall of the People.) The subject of Yau’s talk was something that few in his audience knew much about: the Poincaré conjecture, a century-old conundrum about the characteristics of three-dimensional spheres, which, because it has important implications for mathematics and cosmology and because it has eluded all attempts at solution, is regarded by mathematicians as a holy grail.绿色建筑博客#e5M\4q"S
Yau, a stocky man of fifty-seven, stood at a lectern in shirtsleeves and black-rimmed glasses and, with his hands in his pockets, described how two of his students, Xi-Ping Zhu and Huai-Dong Cao, had completed a proof of the Poincaré conjecture a few weeks earlier. “I’m very positive about Zhu and Cao’s work,” Yau said. “Chinese mathematicians should have every reason to be proud of such a big success in completely solving the puzzle.” He said that Zhu and Cao were indebted to his longtime American collaborator Richard Hamilton, who deserved most of the credit for solving the Poincaré. He also mentioned Grigory Perelman, a Russian mathematician who, he acknowledged, had made an important contribution. Nevertheless, Yau said, “in Perelman’s work, spectacular as it is, many key ideas of the proofs are sketched or outlined, and complete details are often missing.” He added, “We would like to get Perelman to make comments. But Perelman resides inSt. Petersburgand refuses to communicate with other people.”绿色建筑博客d|j8F^(F7vy0j

For ninety minutes, Yau discussed some of the technical details of his students’ proof. When he was finished, no one asked any questions. That night, however, a Brazilian physicist posted a report of the lecture on his blog. “Looks likeChinasoon will take the lead also in mathematics,” he wrote.

Grigory Perelman is indeed reclusive. He left his job as a researcher at the Steklov Institute of Mathematics, inSt. Petersburg, last December; he has few friends; and he lives with his mother in an apartment on the outskirts of the city. Although he had never granted an interview before, he was cordial and frank when we visited him, in late June, shortly after Yau’s conference inBeijing, taking us on a long walking tour of the city. “I’m looking for some friends, and they don’t have to be mathematicians,” he said. The week before the conference, Perelman had spent hours discussing the Poincaré conjecture with Sir John M. Ball, the fifty-eight-year-old president of the International Mathematical Union, the discipline’s influential professional association. The meeting, which took place at a conference center in a stately mansion overlooking theNevaRiver, was highly unusual. At the end of May, a committee of nine prominent mathematicians had voted to award Perelman a Fields Medal for his work on the Poincaré, and Ball had gone toSt. Petersburgto persuade him to accept the prize in a public ceremony at the I.M.U.’s quadrennial congress, inMadrid, on August 22nd.

The Fields Medal, like the Nobel Prize, grew, in part, out of a desire to elevate science above national animosities. German mathematicians were excluded from the first I.M.U. congress, in 1924, and, though the ban was lifted before the next one, the trauma it caused led, in 1936, to the establishment of the Fields, a prize intended to be “as purely international and impersonal as possible.”绿色建筑博客Xf5y2sbk

However, the Fields Medal, which is awarded every four years, to between two and four mathematicians, is supposed not only to reward past achievements but also to stimulate future research; for this reason, it is given only to mathematicians aged forty and younger. In recent decades, as the number of professional mathematicians has grown, the Fields Medal has become increasingly prestigious. Only forty-four medals have been awarded in nearly seventy years—including three for work closely related to the Poincaré conjecture—and no mathematician has ever refused the prize. Nevertheless, Perelman told Ball that he had no intention of accepting it. “I refuse,” he said simply.

Over a period of eight months, beginning in November, 2002, Perelman posted a proof of the Poincaré on the Internet in three installments. Like a sonnet or an aria, a mathematical proof has a distinct form and set of conventions. It begins with axioms, or accepted truths, and employs a series of logical statements to arrive at a conclusion. If the logic is deemed to be watertight, then the result is a theorem. Unlike proof in law or science, which is based on evidence and therefore subject to qualification and revision, a proof of a theorem is definitive. Judgments about the accuracy of a proof are mediated by peer-reviewed journals; to insure fairness, reviewers are supposed to be carefully chosen by journal editors, and the identity of a scholar whose paper is under consideration is kept secret. Publication implies that a proof is complete, correct, and original.绿色建筑博客H3x5L6`%k[wV

By these standards, Perelman’s proof was unorthodox. It was astonishingly brief for such an ambitious piece of work; logic sequences that could have been elaborated over many pages were often severely compressed. Moreover, the proof made no direct mention of the Poincaré and included many elegant results that were irrelevant to the central argument. But, four years later, at least two teams of experts had vetted the proof and had found no significant gaps or errors in it. A consensus was emerging in the math community: Perelman had solved the Poincaré. Even so, the proof’s complexity—and Perelman’s use of shorthand in making some of his most important claims—made it vulnerable to challenge. Few mathematicians had the expertise necessary to evaluate and defend it.

After giving a series of lectures on the proof in theUnited Statesin 2003, Perelman returned toSt. Petersburg. Since then, although he had continued to answer queries about it by e-mail, he had had minimal contact with colleagues and, for reasons no one understood, had not tried to publish it. Still, there was little doubt that Perelman, who turned forty on June 13th, deserved a Fields Medal. As Ball planned the I.M.U.’s 2006 congress, he began to conceive of it as a historic event. More than three thousand mathematicians would be attending, and King Juan Carlos ofSpainhad agreed to preside over the awards ceremony. The I.M.U.’s newsletter predicted that the congress would be remembered as “the occasion when this conjecture became a theorem.” Ball, determined to make sure that Perelman would be there, decided to go toSt. Petersburg.

Ball wanted to keep his visit a secret—the names of Fields Medal recipients are announced officially at the awards ceremony—and the conference center where he met with Perelman was deserted. For ten hours over two days, he tried to persuade Perelman to agree to accept the prize. Perelman, a slender, balding man with a curly beard, bushy eyebrows, and blue-green eyes, listened politely. He had not spoken English for three years, but he fluently parried Ball’s entreaties, at one point taking Ball on a long walk—one of Perelman’s favorite activities. As he summed up the conversation two weeks later: “He proposed to me three alternatives: accept and come; accept and don’t come, and we will send you the medal later; third, I don’t accept the prize. From the very beginning, I told him I have chosen the third one.” The Fields Medal held no interest for him, Perelman explained. “It was completely irrelevant for me,” he said. “Everybody understood that if the proof is correct then no other recognition is needed.”绿色建筑博客K7D.j~/m(V"T2E(B9O

Proofs of the Poincaré have been announced nearly every year since the conjecture was formulated, by Henri Poincaré, more than a hundred years ago. Poincaré was a cousin of Raymond Poincaré, the President of France during the First World War, and one of the most creative mathematicians of the nineteenth century. Slight, myopic, and notoriously absent-minded, he conceived his famous problem in 1904, eight years before he died, and tucked it as an offhand question into the end of a sixty-five-page paper.绿色建筑博客F/Hj:i2VW6L6ku

Poincaré didn’t make much progress on proving the conjecture. “Cette question nous entraînerait trop loin” (“This question would take us too far”), he wrote. He was a founder of topology, also known as “rubber-sheet geometry,” for its focus on the intrinsic properties of spaces. From a topologist’s perspective, there is no difference between a bagel and a coffee cup with a handle. Each has a single hole and can be manipulated to resemble the other without being torn or cut. Poincaré used the term “manifold” to describe such an abstract topological space. The simplest possible two-dimensional manifold is the surface of a soccer ball, which, to a topologist, is a sphere—even when it is stomped on, stretched, or crumpled. The proof that an object is a so-called two-sphere, since it can take on any number of shapes, is that it is “simply connected,” meaning that no holes puncture it. Unlike a soccer ball, a bagel is not a true sphere. If you tie a slipknot around a soccer ball, you can easily pull the slipknot closed by sliding it along the surface of the ball. But if you tie a slipknot around a bagel through the hole in its middle you cannot pull the slipknot closed without tearing the bagel.绿色建筑博客Uy2E)DP8Ah7j
Two-dimensional manifolds were well understood by the mid-nineteenth century. But it remained unclear whether what was true for two dimensions was also true for three. Poincaré proposed that all closed, simply connected, three-dimensional manifolds—those which lack holes and are of finite extent—were spheres. The conjecture was potentially important for scientists studying the largest known three-dimensional manifold: the universe. Proving it mathematically, however, was far from easy. Most attempts were merely embarrassing, but some led to important mathematical discoveries, including proofs of Dehn’s Lemma, the Sphere Theorem, and the Loop Theorem, which are now fundamental concepts in topology.

By the nineteen-sixties, topology had become one of the most productive areas of mathematics, and young topologists were launching regular attacks on the Poincaré. To the astonishment of most mathematicians, it turned out that manifolds of the fourth, fifth, and higher dimensions were more tractable than those of the third dimension. By 1982, Poincaré’s conjecture had been proved in all dimensions except the third. In 2000, the Clay Mathematics Institute, a private foundation that promotes mathematical research, named the Poincaré one of the seven most important outstanding problems in mathematics and offered a million dollars to anyone who could prove it.绿色建筑博客6L K-Dws

“My whole life as a mathematician has been dominated by the Poincaré conjecture,” John Morgan, the head of the mathematics department atColumbiaUniversity, said. “I never thought I’d see a solution. I thought nobody could touch it.”

Grigory Perelman did not plan to become a mathematician. “There was never a decision point,” he said when we met. We were outside the apartment building where he lives, in Kupchino, a neighborhood of drab high-rises. Perelman’s father, who was an electrical engineer, encouraged his interest in math. “He gave me logical and other math problems to think about,” Perelman said. “He got a lot of books for me to read. He taught me how to play chess. He was proud of me.” Among the books his father gave him was a copy of “Physics for Entertainment,” which had been a best-seller in theSoviet Unionin the nineteen-thirties. In the foreword, the book’s author describes the contents as “conundrums, brain-teasers, entertaining anecdotes, and unexpected comparisons,” adding, “I have quoted extensively from Jules Verne, H. G. Wells, Mark Twain and other writers, because, besides providing entertainment, the fantastic experiments these writers describe may well serve as instructive illustrations at physics classes.” The book’s topics included how to jump from a moving car, and why, “according to the law of buoyancy, we would never drown in theDead Sea.”

The notion that Russian society considered worthwhile what Perelman did for pleasure came as a surprise. By the time he was fourteen, he was the star performer of a local math club. In 1982, the year that Shing-Tung Yau won a Fields Medal, Perelman earned a perfect score and the gold medal at the International Mathematical Olympiad, inBudapest. He was friendly with his teammates but not close—“I had no close friends,” he said. He was one of two or three Jews in his grade, and he had a passion for opera, which also set him apart from his peers. His mother, a math teacher at a technical college, played the violin and began taking him to the opera when he was six. By the time Perelman was fifteen, he was spending his pocket money on records. He was thrilled to own a recording of a famous 1946 performance of “La Traviata,” featuring Licia Albanese as Violetta. “Her voice was very good,” he said.

AtLeningradUniversity, which Perelman entered in 1982, at the age of sixteen, he took advanced classes in geometry and solved a problem posed by Yuri Burago, a mathematician at the Steklov Institute, who later became his Ph.D. adviser. “There are a lot of students of high ability who speak before thinking,” Burago said. “Grisha was different. He thought deeply. His answers were always correct. He always checked very, very carefully.” Burago added, “He was not fast. Speed means nothing. Math doesn’t depend on speed. It is about deep.”

At the Steklov in the early nineties, Perelman became an expert on the geometry of Riemannian and Alexandrov spaces—extensions of traditional Euclidean geometry—and began to publish articles in the leading Russian and American mathematics journals. In 1992, Perelman was invited to spend a semester each atNew YorkUniversityandStonyBrookUniversity. By the time he left for theUnited States, that fall, the Russian economy had collapsed. Dan Stroock, a mathematician at M.I.T., recalls smuggling wads of dollars into the country to deliver to a retired mathematician at the Steklov, who, like many of his colleagues, had become destitute.
QY9Jdt,t7r$^/j+lu9H0Perelman was pleased to be in theUnited States, the capital of the international mathematics community. He wore the same brown corduroy jacket every day and told friends at N.Y.U. that he lived on a diet of bread, cheese, and milk. He liked to walk toBrooklyn, where he had relatives and could buy traditional Russian brown bread. Some of his colleagues were taken aback by his fingernails, which were several inches long. “If they grow, why wouldn’t I let them grow?” he would say when someone asked why he didn’t cut them. Once a week, he and a young Chinese mathematician named Gang Tian drove toPrinceton, to attend a seminar at the Institute for Advanced Study.

For several decades, the institute and nearbyPrincetonUniversityhad been centers of topological research. In the late seventies, William Thurston, aPrincetonmathematician who liked to test out his ideas using scissors and construction paper, proposed a taxonomy for classifying manifolds of three dimensions. He argued that, while the manifolds could be made to take on many different shapes, they nonetheless had a “preferred” geometry, just as a piece of silk draped over a dressmaker’s mannequin takes on the mannequin’s form.绿色建筑博客:u~
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Thurston proposed that every three-dimensional manifold could be broken down into one or more of eight types of component, including a spherical type. Thurston’s theory—which became known as the geometrization conjecture—describes all possible three-dimensional manifolds and is thus a powerful generalization of the Poincaré. If it was confirmed, then Poincaré’s conjecture would be, too. Proving Thurston and Poincaré “definitely swings open doors,” Barry Mazur, a mathematician at Harvard, said. The implications of the conjectures for other disciplines may not be apparent for years, but for mathematicians the problems are fundamental. “This is a kind of twentieth-century Pythagorean theorem,” Mazur added. “It changes the landscape.”绿色建筑博客/x'?UD i2l&w

In 1982, Thurston won a Fields Medal for his contributions to topology. That year, Richard Hamilton, a mathematician at Cornell, published a paper on an equation called the Ricci flow, which he suspected could be relevant for solving Thurston’s conjecture and thus the Poincaré. Like a heat equation, which describes how heat distributes itself evenly through a substance—flowing from hotter to cooler parts of a metal sheet, for example—to create a more uniform temperature, the Ricci flow, by smoothing out irregularities, gives manifolds a more uniform geometry.绿色建筑博客!^~3WE xwsN.K

Hamilton, the son of aCincinnatidoctor, defied the math profession’s nerdy stereotype. Brash and irreverent, he rode horses, windsurfed, and had a succession of girlfriends. He treated math as merely one of life’s pleasures. At forty-nine, he was considered a brilliant lecturer, but he had published relatively little beyond a series of seminal articles on the Ricci flow, and he had few graduate students. Perelman had readHamilton’s papers and went to hear him give a talk at the Institute for Advanced Study. Afterward, Perelman shyly spoke to him.
5L1f5z#O-\Wu0“I really wanted to ask him something,” Perelman recalled. “He was smiling, and he was quite patient. He actually told me a couple of things that he published a few years later. He did not hesitate to tell me.Hamilton’s openness and generosity—it really attracted me. I can’t say that most mathematicians act like that.

“I was working on different things, though occasionally I would think about the Ricci flow,” Perelman added. “You didn’t have to be a great mathematician to see that this would be useful for geometrization. I felt I didn’t know very much. I kept asking questions.”绿色建筑博客%o)l8\T3qE
Shing-Tung Yau was also askingHamiltonquestions about the Ricci flow. Yau and Hamilton had met in the seventies, and had become close, despite considerable differences in temperament and background. A mathematician at theUniversityofCaliforniaatSan Diegowho knows both men called them “the mathematical loves of each other’s lives.”绿色建筑博客 p5e4EWS*Q0O4yO

Yau’s family moved toHong Kongfrom mainlandChinain 1949, when he was five months old, along with hundreds of thousands of other refugees fleeing Mao’s armies. The previous year, his father, a relief worker for the United Nations, had lost most of the family’s savings in a series of failed ventures. InHong Kong, to support his wife and eight children, he tutored college students in classical Chinese literature and philosophy.

When Yau was fourteen, his father died of kidney cancer, leaving his mother dependent on handouts from Christian missionaries and whatever small sums she earned from selling handicrafts. Until then, Yau had been an indifferent student. But he began to devote himself to schoolwork, tutoring other students in math to make money. “Part of the thing that drives Yau is that he sees his own life as being his father’s revenge,” said Dan Stroock, the M.I.T. mathematician, who has known Yau for twenty years. “Yau’s father was like the Talmudist whose children are starving.”

Yau studied math at theChineseUniversityofHong Kong, where he attracted the attention of Shiing-Shen Chern, the preëminent Chinese mathematician, who helped him win a scholarship to theUniversityofCaliforniaatBerkeley. Chern was the author of a famous theorem combining topology and geometry. He spent most of his career in theUnited States, atBerkeley. He made frequent visits toHong Kong,Taiwan, and, later,China, where he was a revered symbol of Chinese intellectual achievement, to promote the study of math and science.

In 1969, Yau started graduate school atBerkeley, enrolling in seven graduate courses each term and auditing several others. He sent half of his scholarship money back to his mother inChinaand impressed his professors with his tenacity. He was obliged to share credit for his first major result when he learned that two other mathematicians were working on the same problem. In 1976, he proved a twenty-year-old conjecture pertaining to a type of manifold that is now crucial to string theory. A French mathematician had formulated a proof of the problem, which is known as Calabi’s conjecture, but Yau’s, because it was more general, was more powerful. (Physicists now refer to Calabi-Yau manifolds.) “He was not so much thinking up some original way of looking at a subject but solving extremely hard technical problems that at the time only he could solve, by sheer intellect and force of will,” Phillip Griffiths, a geometer and a former director of the Institute for Advanced Study, said.绿色建筑博客!Od(s8B$xA']g

In 1980, when Yau was thirty, he became one of the youngest mathematicians ever to be appointed to the permanent faculty of the Institute for Advanced Study, and he began to attract talented students. He won a Fields Medal two years later, the first Chinese ever to do so. By this time, Chern was seventy years old and on the verge of retirement. According to a relative of Chern’s, “Yau decided that he was going to be the next famous Chinese mathematician and that it was time for Chern to step down.”绿色建筑博客+RXr*KN te*i

Harvard had been trying to recruit Yau, and when, in 1983, it was about to make him a second offer Phillip Griffiths told the dean of faculty a version of a story from “The Romance of the Three Kingdoms,” a Chinese classic. In the third century A.D., a Chinese warlord dreamed of creating an empire, but the most brilliant general in China was working for a rival. Three times, the warlord went to his enemy’s kingdom to seek out the general. Impressed, the general agreed to join him, and together they succeeded in founding a dynasty. Taking the hint, the dean flew to Philadelphia, where Yau lived at the time, to make him an offer. Even so, Yau turned down the job. Finally, in 1987, he agreed to go to Harvard.绿色建筑博客a-O&D6b2J7F)GY&h

Yau’s entrepreneurial drive extended to collaborations with colleagues and students, and, in addition to conducting his own research, he began organizing seminars. He frequently allied himself with brilliantly inventive mathematicians, including Richard Schoen and William Meeks. But Yau was especially impressed by Hamilton, as much for his swagger as for his imagination. “I can have fun with Hamilton,” Yau told us during the string-theory conference in Beijing. “I can go swimming with him. I go out with him and his girlfriends and all that.” Yau was convinced that Hamilton could use the Ricci-flow equation to solve the Poincaré and Thurston conjectures, and he urged him to focus on the problems. “Meeting Yau changed his mathematical life,” a friend of both mathematicians said of Hamilton. “This was the first time he had been on to something extremely big. Talking to Yau gave him courage and direction.”绿色建筑博客2nry*LdYL5I

Yau believed that if he could help solve the Poincaré it would be a victory not just for him but also for China. In the mid-nineties, Yau and several other Chinese scholars began meeting with President Jiang Zemin to discuss how to rebuild the country’s scientific institutions, which had been largely destroyed during the Cultural Revolution. Chinese universities were in dire condition. According to Steve Smale, who won a Fields for proving the Poincaré in higher dimensions, and who, after retiring from Berkeley, taught in Hong Kong, Peking University had “halls filled with the smell of urine, one common room, one office for all the assistant professors,” and paid its faculty wretchedly low salaries. Yau persuaded a Hong Kong real-estate mogul to help finance a mathematics institute at the Chinese Academy of Sciences, in Beijing, and to endow a Fields-style medal for Chinese mathematicians under the age of forty-five. On his trips to China, Yau touted Hamilton and their joint work on the Ricci flow and the Poincaré as a model for young Chinese mathematicians. As he put it in Beijing, “They always say that the whole country should learn from Mao or some big heroes. So I made a joke to them, but I was half serious. I said the whole country should learn from Hamilton.”

Grigory Perelman was learning from Hamilton already. In 1993, he began a two-year fellowship at Berkeley. While he was there, Hamilton gave several talks on campus, and in one he mentioned that he was working on the Poincaré. Hamilton’s Ricci-flow strategy was extremely technical and tricky to execute. After one of his talks at Berkeley, he told Perelman about his biggest obstacle. As a space is smoothed under the Ricci flow, some regions deform into what mathematicians refer to as “singularities.” S

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