12月22日出版的《科学》杂志评选出了2006年度十大科学进展，其中证明“庞加莱猜想”名列榜首。科学专栏作者Dana Mackenzie撰文《庞加莱猜想的证明》加以介绍。
对数学家们来说，格里高利•佩雷尔曼（Perelman）的庞加莱猜想证明至少称得上是十年来的最大突破。但这是经历了十多年时间之后,他们才相信那是真的。在2006年，也就是在俄国数学家描绘出猜想证明的概要的三篇论文中第一篇发表之后近四年之时，研究者们终于达成了一个共识；佩雷尔曼已经解答了该分支最古老的一个问题。但是这个解答却触发了一场争议的风暴与一连串紧张的事件，使得这项灿烂夺目的工作有可能减色。
佩雷尔曼的证明深刻变革了两大数学分支。首先，他解决了拓扑学（研究抽象形状的数学）核心领域中一百多年来最为困难的问题。大多数数学家期望这项工作将会导致更加深远的结果，给出几何化猜想的证明：如同门捷列夫元素周期表那样，给出三维空间的清除分类。
在带给拓扑学新结果的同时，佩雷尔曼的工作为几何学引入了新的技巧。加强了几何发展方程在几何学中的核心地位，这一强大的工具可以把难以捉摸的空间转化为容易处理的对象。早期的研究发现，这类方程总会遇到“奇点”，然后就无能为力了。佩雷尔曼把这一障碍彻底清除了。
“这是数学家第一次能够理解奇点的结构，以及这一复杂系统的发展。”哈佛大学教授丘成桐在今年夏天北京的一次演讲中说，“这种方法将会给许多自然的系统的求解带来启发，比如（流体力学）Navier-Stokes方程与（广义相对论）爱因斯坦方程。”
难以驾驭的空间
亨利•庞加莱（Henri Poincare）在1904年提出了他的著名猜想。他被公认为拓扑学的奠基者，是将拓扑学（从微积分发展而来的数学分支）同分析学与几何学区分开来的第一位数学家。拓扑学经常被描述为“橡皮几何学”，因为它研究曲面在任意拉伸下保持不变的性质。但是不允许撕裂与粘合。
我们的身体，以及我们所接触的大多数对象都是三维的。但是它们的表面是二维的。在拓扑学意义下，所有不带边的二维曲面（如我们的皮肤那样自闭合）只要一个特征就可以区分：曲面上的洞的个数。没有洞的曲面就是球面；一个洞的曲面是环面。球面与环面之间不能相互转变
具有二维边界的三维物体的研究还只是开始。比如我们可以把三维空间定义成四维空间的边界。人类只能隐约感知这种空间，但是数学家可以用符号来描述并研究它们的性质。庞加莱发展了一种巧妙的工具，称为“基本群”，用来探测任意维空间中的洞，扭曲和其它性质。他猜测三维空间不能越过基本群而隐含有趣的拓扑。也就是说，具有平凡基本群的三维空间必是超球面：即四维球的边界。
谜：佩雷尔曼解决庞加莱问题
为了取得进展，拓扑学家们提出了一种以前被忽视的工具：引入距离的办法。他们将几何学与拓扑学结合起来。1982年，威廉•瑟斯顿（William Thurston，现在康奈尔大学任教）提出每个三维空间都可以切割成具有给定的唯一几何性质的片，这些几何有八种类型。这些假设后来成为著名的几何化猜想。
如果成立，那么瑟斯顿的洞察力将能够解决庞加莱猜想，因为球面不过是八种几何之一，其具有平凡基本群。在1982年，理查•汉密尔顿（Richard Hamilton，现在哥伦比亚大学任教）提出一种证明策略：从任意粗糙的空间开始，引导它流动到一个均匀的空间，也就是瑟斯顿的均称“几何”空间。为了引导流动过程，汉密尔顿提出了一类几何发展方程，以物理学中热方程为模型，并将它命名为“里奇流”（Ricci flow），向一位早期的微分几何学家Gregorio Ricci-Curbastro表示敬意。在里奇流中，高曲率的区域会向较低曲率的区域扩散，直到空间的曲率处处相等。
汉密尔顿的策略应用在二维曲面上非常成功。纤细的“瓶颈”总是膨胀。可是在三维空间中，里奇流就遇到了障碍。瓶颈有时会被剪开，将空间分为具有匀称几何的区域。尽管Hamilton作了Ricci流的大部分先驱工作，但他却没能驯服奇异性。
这样一来, 整个研究计划在九十年代中期似乎搁浅了。到了2000年, 当 Clay 数学所把“庞加莱猜想”列为悬赏百万美元征答的一个问题时，大多数学家看不出突破的可能。
突破
事实上，佩雷尔曼已经在求解庞加莱猜测的道路上顺利前进着。1995年，这位29岁的圣彼得堡公民在结束了长达三年的旅美生涯后回到俄罗斯，他遇到了汉密尔顿，学到了里奇流。
在接下来7年时间里，他几乎不与外界交流。到了2002年，佩雷尔曼在互联网上贴出了三篇预印本中的第一篇文章，给出了几何化猜想的证明构思。
在专家看来，佩雷尔曼很显然已经做出了重要的突破。第一篇文章第一节的标题中“作为梯度流的里奇流”，佩雷尔曼已经发现了汉密尔顿所忽略的一个重要细节：一个随着里奇流方向总是增加的量。鉴于与统计力学（研究热动力学的数学）的相似性，佩雷尔曼称之为“熵”（entropy）.
熵的提出排除了一直困扰汉密尔顿的某些特定奇点。为了顺利达到目标，佩雷尔曼仍然需要识别出剩下可能产生问题的奇点。他必须证明，这些奇点在某一时刻只出现一次，而不是无穷汇聚。然后，对每个奇点，他需要给出如何修剪与光滑化，以免对里奇流造成破坏。这些步骤足够证明庞加莱猜想。为了完成几何化猜想，佩雷尔曼还需要证明，“手术化的里奇流”可以延续到无穷时间。
2003年，佩雷尔曼重游美国宣讲他的工作的时候，很多数学家怀疑他是否真的完成了
壮举。然而，到了2006年，数学界最终跟上了。三篇独立的手稿，每份都有超过300页长，填补了佩雷尔曼证明里面缺少的关键细节三篇解释工作中的两篇，一篇出自安娜堡-密歇根大学的Bruce Kleiner与John Lott, 另一篇出自哥伦比亚大学的John Morgan和剑桥麻省理工学院的田刚。
第三篇论文，作者是宾夕法尼亚Bethlehem里海大学的曹怀东与中国广州中山大学的朱熹平，则没有太多顾忌。他们宣称完成了“第一个写下来的庞加莱与瑟斯顿几何化猜想的完整证明。”今年夏天，国际数学联盟决定颁给佩雷尔曼菲尔兹数学奖，传统上数学界的最高荣誉。
不幸的转变
“从那时起，凯旋的玫瑰晕开始呈现深色的阴影。8月22日，国际数学联盟主席John Ball宣布佩雷尔曼谢绝接受菲尔兹奖章。在《纽约客》的一篇访谈中，这位隐居的数学家说他正从数学界退出，对未指明的同事“道德标准”的跌落放弃幻想。《纽约客》文章还画了一幅对影射丘成桐的漫画像，隐喻他曾为门徒曹朱索取过多的功劳。
难过的感觉充斥了随后的月份。几位数学家说他们的引言被《纽约客》扭曲了，丘成桐威胁说要起诉。Kleiner和Lott抗议说曹朱抄袭了他们的一个证明作为自己的原创，后者发表了一个更正来承认Kleiner与Lott的优先权。
今秋，美国数学会试图在2007年1月召开的路州新奥尔良会议上组织一个关于庞加莱暨几何化猜想的全明星讨论组。按照执行主任John Ewing的说法，由于 Lott拒绝和朱熹平同台，这个努力搁浅了。Ewing仍旧希望组织那样一次活动“在将来的某个时间”。直到如今，敌意令数学家们难以庆祝他们在新千年最伟大的突破。
*作者Dana Mackenzie于1983年从普林斯顿大学数学系取得博士学位，曾在杜克大学任教，现在是科学专栏作者。
原文：
Science 22 December 2006:
Vol. 314. no. 5807, pp. 1848 - 1849
DOI: 10.1126/science.314.5807.1848
NEWS
BREAKTHROUGH OF THE YEAR:
The Poincaré Conjencture--Proved
Dana Mackenzie*
The solution of a century-old mathematics problem turns out to be a bittersweet prize
CREDIT: CAMERON SLAYDEN/SCIENCE
To mathematicians, Grigori Perelman's proof of the Poincaré conjecture qualifies at least as the Breakthrough of the Decade. But it has taken them a good part of that decade to convince themselves that it was for real. In 2006, nearly 4 years after the Russian mathematician released the first of three papers outlining the proof, researchers finally reached a consensus that Perelman had solved one of the subject's most venerable problems. But the solution touched off a storm of controversy and drama that threatened to overshadow the brilliant work.
Perelman's proof has fundamentally altered two distinct branches of mathematics. First, it solved a problem that for more than a century was the indigestible seed at the core of topology, the mathematical study of abstract shape. Most mathematicians expect that the work will lead to a much broader result, a proof of the geometrization conjecture: essentially, a "periodic table" that brings clarity to the study of three-dimensional spaces, much as Mendeleev's table did for chemistry.
While bringing new results to topology, Perelman's work brought new techniques to geometry. It cemented the central role of geometric evolution equations, powerful machinery for transforming hard-to-work-with spaces into more-manageable ones. Earlier studies of such equations always ran into "singularities" at which the equations break down. Perelman dynamited that roadblock.
"This is the first time that mathematicians have been able to understand the structure of singularities and the development of such a complicated system," said Shing-Tung Yau of Harvard University at a lecture in Beijing this summer. "The methods developed … should shed light on many natural systems, such as the Navier-Stokes equation [of fluid dynamics] and the Einstein equation [ of general relativity]."
Unruly spaces
Henri Poincaré, who posed his problem in 1904, is generally regarded as the founder of topology, the first mathematician to clearly distinguish it from analysis (the branch of mathematics that evolved from calculus) and geometry. Topology is often described as "rubbersheet geometry," because it deals with properties of surfaces that can undergo arbitrary amounts of stretching. Tear
ing and its opposite, sewing, are not allowed.
Our bodies, and most of the familiar objects they interact with, have three dimensions. Their surfaces, however, have only two. As far as topology is concerned, two-dimensional surfaces with no boundary (those that wrap around and close in on themselves, as our skin does) have essentially only one distinguishing feature: the number of holes in the surface. A surface with no holes is a sphere; a surface with one hole is a torus; and so on. A sphere can never be turned into a torus, or vice versa.
Three-dimensional objects with 2D surfaces, however, are just the beginning.
For example, it is possible to define curved 3D spaces as boundaries of 4D objects. Human beings can only dimly visualize such spaces, but mathematicians can use symbolic notation to describe them and explore their properties. Poincaré developed an ingenious tool, called the "fundamental group," for detecting holes, twists, and other features in spaces of any dimension. He conjectured that a 3D space cannot hide any interesting topology from the fundamental group. That is, a 3D space with a "trivial" fundamental group must be a hypersphere: the boundary of a ball in 4D space.
Although simple to state, Poincaré's conjecture proved maddeningly difficult to prove. By the early 1980s, mathematicians had proved analogous statements for spaces of every dimension higher than three--but not for the original one that Poincaré had pondered.
Enigma. Perelman (top) solved Poincaré's problem.
To make progress, topologists reached for a tool they had neglected: a way to specify distance. They set about recombining topology with geometry. In 1982, William Thurston (now of Cornell University) theorized that every 3D space can be carved up so that each piece has a unique uniform geometry, and that those geometries come in only eight possible types. This hypothesis became known as the geometrization conjecture.
If true, Thurston's insight would solve the Poincaré conjecture, because a sphere is the only one of the eight geometries that admits a trivial fundamental group. In 1982, Richard Hamilton (now of Columbia University) proposed a possible strategy for proving it: Start with any lumpy space, and then let it flow toward a uniform one. The result would be a tidy "geometrized" space à la Thurston. To guide the flow, Hamilton proposed a geometric evolution equation modeled after the heat equation of physics and named it "Ricci flow" in honor of Gregorio Ricci-Curbastro, an early differential geometer. In Ricci flow, regions of high curvature tend to diffuse out into the regions of lower curvature, until the space has equal curvature throughout.
Hamilton's strategy works perfectly in 2D surfaces. Slender "necks," like the one seen on the cover of this issue, always expand. In 3D spaces, however, Ricci flow can run into snags. Necks sometimes pinch off, separating the space into regions with different uniform geometries. Although Hamilton did a great deal of pioneering work on Ricci flow, he could not tame the singularities.
As a result, the whole program of research seemed to run aground in the mid-1990s. In 2000, when the Clay Mathematics Institute named the Poincaré conjecture as one of its $1 million Millennium Prize problems, most mathematicians believed that no breakthrough was in sight.
Fascinating. A computer rendering of a 3D space with uniform hyperbolic geome try.
The breakthrough
In fact, Perelman was already well on his way to a solution. In 1995, the 29-year-old St. Petersburg native had returned to Russia after a 3-year sojourn in the United States, where he had met Hamilton and learned about Ricci flow.
For the next 7 years, he remained mostly incommunicado. Then, in November 2002, Perelman posted on the Internet the first of three preprints outlining a proposed proof of the geometrization conjecture.
To experts, it was immediately clear that Perelman had made a major breakthrough. It was in the title of the first section of the first paper: "Ricci Flow as a Gradient Flow." Perelman had spotted an important detail that Hamilton had missed: a quantity that always increases during the flow, giving it a direction. By analogy with statistical mechanics, the mathematics underlying the laws of thermodynamics, Perelman called the quantity "entropy."
The entropy ruled out specific singularities that had stymied Hamilton. To reach a safe harbor, however, Perelman still had to identify the remaining types of singularities that might cause problems. He had to show that they occurred one at a time instead of accumulating in an infinite pileup. Then, for each singularity, he had to show how to prune and smooth it before it could sabotage Ricci flow. Those steps would be enough to prove Poincaré. To complete the geometrization conjecture, Perelman had to show, additionally, that the " Ricci flow with surgery" procedure could be continued for an infinitely long time.
In 2003, when Perelman revisited the United States to lecture on his work, many mathematicians doubted that he could have pulled off all of these feats. By 2006, however, the mathematical community had finally caught up. Three separate manuscripts, each more than 300 pages in length, filled in key missing details of Perelman's proof.
Two of the papers--one authored by Bruce Kleiner and John Lott of the University of Michigan, Ann Arbor, the other by John Morgan of Columbia University and Gang Tian of the Massachusetts Institute of Technology in Cambridge--stopped short of the geometrization conjecture, because Perelman's explanation of the final step had been too sketchy. (Both groups are still working on it.)
They did, however, include enough math to nail down the Poincaré conjecture. The third paper, by Huai-Dong Cao of Lehigh University in Bethlehem, Pennsylvania, and Xi-Ping Zhu of Zhongshan University in Guangzhou, China, was less circumspect. Cao and Zhu claimed to have "the first written account of a complete proof of the Poincaré conjecture and the geometrization conjecture of Thurston." This summer, the International Mathematical Union (IMU) decided to award Perelman the Fields Medal, traditionally considered the highest honor in mathematics.
Anticlimax
Since then, the rosy glow of triumph has taken on darker hues. On 22 August, IMU President John Ball announced that Perelman had declined the Fields Medal. In an interview in The New Yorker, the reclusive mathematician said he was retiring from mathematics, disenchanted by unspecified lapses in "ethical standards" by colleagues. The New Yorker article also painted an unflattering portrait of Yau, intimating that he had claimed too much credit for his protégés Cao and Zhu.
In the ensuing months, hard feelings have abounded. Certain mathematicians claimed that their quotes were distorted in the New Yorker, and Yau threatened to sue. Kleiner and Lott complained that Cao and Zhu had copied a proof of theirs and claimed it as original, and the latter pair grudgingly printed an erratum acknowledging Kleiner and Lott's priority.
This fall, the American Mathematical Society attempted to organize an all-star panel on the Poincaré and geometrization conjectures at its January 2007 meeting in New Orleans, Louisiana. According to executive Director John Ewing, the effort fell apart when Lott refused to share the stage with Zhu. Ewing still hopes to organize such an event "at some time in the future." For the time being, however, the animosity continues to make it hard for mathematicians to celebrate their greatest breakthrough of the new millennium.
Dana Mackenzie is a freelance science writer in Santa Cruz, California.
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