格里戈里·佩雷尔曼Grigori Perelman

格里戈里·佩雷尔曼的父母是Yakov Perelman，一位电气工程师，以及Lubov Lvovna，她曾是一所技术学院的数学教师。他们是犹太人，这会给他们的儿子带来一些问题，因为在这个国家里，人们担心犹太裔人士会有分裂的忠诚。他们的第一个孩子佩雷尔曼 Yakovlevich，通常以Grisha这个名字为人所知。Grisha小时候，他的母亲和一位私人教师都教他拉小提琴。他的父亲对培养他解决问题的能力也有重大影响。谈到他的父亲，佩雷尔曼说（见[11]）：-

他给我逻辑和其他数学问题让我思考。他弄来很多书让我读。他教我下棋。他为我感到骄傲。

他的母亲也帮助培养他的数学技能，到他十岁时，他已经参加过地区数学竞赛，并显示出显著的天赋。Lubov就如何最好地培养Grisha的数学才能寻求建议，得到的建议是把他送到一个由一位名叫Sergei Rukshin的十九岁教练经营的数学俱乐部。该俱乐部每周两次在放学后于少年宫聚会，而Rukshin，一名列宁格勒大学的本科生，有一些新颖的方法来让来到俱乐部的男孩们发挥出最佳水平。

鲁克申很快看出了佩雷尔曼的潜力，尽管起初他与组里其他聪明的孩子几乎没有什么区别。两人之间形成了一种纽带、一种默契，佩雷尔曼成了鲁克申最喜欢的学生。1980年夏天，鲁克申辅导佩雷尔曼英语，以便他能在当年9月进入列宁格勒第239特殊数学与物理学校。为了让佩雷尔曼接受这种高强度辅导，在几周内学完四年学校教育的英语内容，佩雷尔曼一家不得不整个夏天留在列宁格勒，而不是像通常那样去乡下。课程在列宁格勒的公园里一边散步一边进行，并成功达到了目的。

佩雷尔曼进入第239学校时所进的班级很不寻常，因为鲁克申辅导的那群极具天赋的数学人才被编在了同一个班。在学校里，瓦列里·雷日克既是他们的班主任，也是他们的数学老师。雷日克是一位才华非凡的数学教师，但鲁克申聚集起来的这群数学天才所在的班级，对他来说几乎是一个不可能应对的挑战。除了数学，雷日克每周还有一个晚上开办国际象棋俱乐部，佩雷尔曼参加了，并在这项运动中展现出相当高的天赋。十五岁时，佩雷尔曼参加了鲁克申开办的夏令营。这是他第一次离开母亲在外过夜，但鲁克申与佩雷尔曼之间的纽带帮助缓解了这种可能很困难的局面。鲁克申不仅把他的俱乐部男孩们训练成最优秀的数学解题者，还努力拓宽他们的兴趣。佩雷尔曼已经对小提琴和古典音乐感兴趣，但佩雷尔曼能够拓宽他的音乐兴趣。尽管他会和鲁克申一起参加夏令营，佩雷尔曼从未参加过雷日克安排的旅行。

1982年1月，佩雷尔曼被选为1982年苏联数学奥林匹克竞赛队的潜在成员。他参加了在莫斯科以北约80公里的切尔诺戈洛夫卡举行的选拔集训，在那里除了数学训练外，他们还要在体育馆接受严格的体能训练。佩雷尔曼表现出色，下一步是4月在敖德萨为期两天的集训，当时给他们的题目比奥林匹克竞赛中预期的更难。佩雷尔曼取得了满分，正如他在7月于布达佩斯举行的国际数学奥林匹克竞赛中一样。他获得了一枚金牌和一项因取得满分而颁发的特别奖。作为苏联队的一员，佩雷尔曼自动获得了大学入学资格。

佩雷尔曼于1982年秋进入列宁格勒国立大学。在那里，他尤其受到Viktor Zalgaller和亚历山大·亚历山德罗夫的影响。在本科期间，他作为Rukshin的数学辅导助手，参加夏令营，但他那令人难以置信的高标准甚至让最优秀的学生都几乎无法承受。最终，Rukshin不得不停止让佩雷尔曼在夏令营中协助。然而，他的大学学业非常出色，于1987年毕业。他已经发表了许多论文：Realization of abstract k-skeletons as k-skeletons of intersections of convex polyhedra in $\mathbb{R}^{2k-1}$（俄文）（1985年）；与I V Polikanova合作A remark on Helly's theorem（俄文）（1986年）；为亚历山大·亚历山德罗夫的On the foundations of geometry（俄文）（1987年）所作的补充，其中佩雷尔曼讨论了Pasch风格的亚历山大·亚历山德罗夫公理的等价性及其一些推论；以及On the k-radii of a convex body（俄文）（1987年）。

人们可能会想象，他的成就意味着他会被弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所列宁格勒分所张开双臂欢迎为研究生。然而，在伊万·维诺格拉多夫的领导下，弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所没有接收任何犹太人，尽管现在有了新所长，旧政策依然存在。亚历山大·亚历山德罗夫写信给所长，请求允许佩雷尔曼在弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所列宁格勒分所由他指导进行研究生工作。这一请求出自亚历山大·亚历山德罗夫这样地位崇高的人，极不寻常，但获得了批准，然而，尽管亚历山大·亚历山德罗夫将是他的正式导师，实际上是由Yuri Burago承担了这一角色。佩雷尔曼于1990年通过了学位论文Saddle Surfaces in Euclidean Spaces的答辩。他已经在An example of a complete saddle surface in $\mathbb{R}^{4}$ with Gaussian curvature bounded away from zero（俄文）（1989年）中发表了论文的主要结果之一。

Burago联系了米哈伊尔·格罗莫夫，此人曾是列宁格勒国立大学的教授，但当时是巴黎郊外高等科学研究所的永久成员。他向米哈伊尔·格罗莫夫解释说，他有一个出色的学生，并询问是否可以发出邀请，让他到IHES度过一段时间。邀请允许佩雷尔曼在IHES与米哈伊尔·格罗莫夫一起研究亚历山大·亚历山德罗夫空间几个月。佩雷尔曼的第一篇重要论文，与Burago和米哈伊尔·格罗莫夫合著，是A D Aleksandrov spaces with curvatures bounded below（1992）。Tadeusz Januszkiewicz开始评论如下：-

这是一篇在许多方面都很重要的论文。它包含了对该理论基本事实的仔细且相当详细的讨论，包括定义的各种等价形式。它认识到黎曼几何的各种重要定理的归宿是亚历山大·亚历山德罗夫空间理论，在这个背景下，陈述和证明都变得更加令人满意（但不一定更容易），其他定理自然出现以完善图景。它发展了用于研究具有下方曲率界的亚历山大·亚历山德罗夫空间的一般性的有用工具。最后，它包含了对进一步结果和开放问题的充分讨论。

在访问巴黎附近的IHES后，佩雷尔曼回到列宁格勒的弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所，但多亏了米哈伊尔·格罗莫夫，佩雷尔曼被邀请到美国，在1991年于北卡罗来纳州达勒姆的杜克大学举行的几何节上演讲。他讲授了他与Burago和米哈伊尔·格罗莫夫在亚历山大·亚历山德罗夫空间上所做的工作（当时尚未发表）。1992年，佩雷尔曼被邀请在纽约大学理查·科朗特研究所度过秋季学期，作为博士后研究员，并在1993年春季学期在石溪，纽约州立大学的一个校区，再次由奖学金资助。Masha Gessen描述了此时的佩雷尔曼[1]：-

当佩雷尔曼到达美国时，他二十六岁，不再胖乎乎的，而是高大且显然健康。他的胡须已经过了其长长的尴尬簇状阶段，变得浓密、黑色且蓬松。他的头发很长。他不相信剪头发或指甲……[他穿着]每天同样的衣服——最显眼的是一件棕色灯芯绒夹克……[他吃]一种特殊的黑面包，只能从布鲁克林海滩的一家俄罗斯商店买到，佩雷尔曼从曼哈顿步行到那里。

佩雷尔曼在1992年在美国，他的母亲住在纽约的朋友家，他的父亲早些时候移民到以色列，而佩雷尔曼的妹妹Lena仍在圣彼得堡接受教育（列宁格勒在1991年恢复其原名圣彼得堡）。他结识了Jeff Cheeger和Gang Tian，他们三人定期前往普林斯顿参加高等爱德华·斯图迪研究所的研讨会。佩雷尔曼在1993年参加了在以色列的一次会议，然后接受了加州大学乔治·伯克利分校的两年Miller研究奖学金。在这些年里，他发表了一些卓越的论文。Elements of Morse theory on Aleksandrov spaces（俄语）（1993）研究了帕维尔·亚历山德罗夫空间的局部拓扑结构。Manifolds of positive Ricci curvature with almost maximal volume（1994）解决了一个关于完备黎曼流形$M^{n}$的猜想。如果这样的流形具有格雷戈里奥·里奇-库尔巴斯托罗曲率≥$n - 1$且体积接近球体的体积，那么佩雷尔曼证明它与球体同胚。然而，最大的突破是他的论文Proof of the soul conjecture of Cheeger and Gromoll（1994），它回答了Cheeger和Gromoll二十年前提出的一个问题。佩雷尔曼被邀请在1994年于苏黎世举行的国际数学家大会上发表演讲，他做了讲座Spaces with curvature bounded below。

为了理解佩雷尔曼此时开始思考的问题，我们给出[16]中对儒勒·昂利·庞加莱猜想和威廉·瑟斯顿几何化猜想的描述：-

一个具有正曲率的2-流形可以形变为一个2-球面；一个具有零曲率的2-流形可以形变为一个环面；而一个具有负曲率的2-流形可以形变为一个带有多于一个洞的环面。儒勒·昂利·庞加莱猜想，起源于法国数学家儒勒·昂利·庞加莱在1904年的工作，涉及3维流形，即3-流形。……是否每一个单连通的3-流形都可以形变为3-球面？儒勒·昂利·庞加莱猜想断言这个问题的答案是肯定的。正如2-流形一样，人们也可以希望对于3-流形进行分类。在20世纪70年代，Fields Medalist 威廉·瑟斯顿提出了一个新的猜想，后来被称为威廉·瑟斯顿几何化猜想，它给出了一种分类所有3-流形的方法。威廉·瑟斯顿几何化猜想提供了关于3-流形的一个宏阔的视野，并且实际上将儒勒·昂利·庞加莱猜想作为一个特例包含在内。威廉·瑟斯顿提出，以类似于2-流形情形的方式，3-流形可以使用几何来分类。但是这种类比并不能延伸得很远：3-流形比2-流形要多样和复杂得多。

一种可能用来攻克儒勒·昂利·庞加莱猜想的方法曾由Richard Hamilton发展出来，他在1982年开始研究一个他称之为格雷戈里奥·里奇-库尔巴斯托罗流的特定方程时引入了一个重要的想法。当佩雷尔曼在高等斯图迪研究院听讲座时，他参加了Hamilton在那里的一次讲座，并在讲座后与他交谈。佩雷尔曼回忆[11]：-

我真的很想问他一些事情。他面带微笑，而且相当耐心。他实际上告诉了我几件他几年后发表的事情。他毫不犹豫地告诉了我。Hamilton的开放和慷慨——这真的吸引了我。我不能说大多数数学家都那样行事。我当时在做不同的事情，不过偶尔我会思考格雷戈里奥·里奇-库尔巴斯托罗流。你不需要是一个伟大的数学家就能看出这对几何化会有用。我觉得我知道的并不多。我一直在问问题。

当他在伯克利做米勒研究员时，佩雷尔曼参加了Hamilton的一些进一步讲座，他开始理解为什么Hamilton无法使用格雷戈里奥·里奇-库尔巴斯托罗流在证明儒勒·昂利·庞加莱猜想方面取得任何进一步的进展。

在美国期间，佩雷尔曼收到了几份请他申请教授职位的请求。这些请求来自斯坦福和普林斯顿等顶尖机构。以色列特拉维夫大学在没有要求他提出任何申请的情况下向他提供了一个正教授职位，但他拒绝了所有邀请，并在1995年夏天他的米勒 fellowship 结束后回到了弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所圣彼得堡分所。基本上，他能够靠在美国所得报酬中积攒下来的储蓄生活，由于他生活得异常节俭，这笔储蓄相当可观。他在1996年拒绝接受一个欧洲数学会奖。佩雷尔曼在读到Hamilton于1995年发表的一篇论文时意识到Hamilton在儒勒·昂利·庞加莱猜想上没有取得进展，并在次年写信给Hamilton，解释说他可能有办法绕过这个问题，并提出与他合作。当他没有收到回复时，佩雷尔曼似乎决定独自研究解决儒勒·昂利·庞加莱猜想。

2002年11月11日，佩雷尔曼将他的论文The Entropy Formula for the Ricci Flow and Its Geometric Applications发布到网上。尽管他在论文中并未声称能够解决儒勒·昂利·庞加莱猜想，但当该领域的专家读到它时，他们意识到他已经取得了解决该猜想所必需的突破。他很快收到了访问纽约州立大学石溪分校和麻省理工学院的邀请。他开始为这些访问做计划，并在出发前在网上发布了第二篇论文Ricci flow with surgery on three-manifolds，继续他的证明。他于2003年4月抵达美国，首先去了麻省理工学院，在那里他待的两周中大多数日子都在就自己的工作发表演讲。他在石溪分校度过了类似的两周，随后访问了哥伦比亚大学和普林斯顿大学并在那里讲学。他拒绝了向他提出的所有教授职位邀请，并对一些人向他施加的接受压力感到恼火。

他于2002年4月底回到圣彼得堡，并在7月将Finite extinction time for the solutions to the Ricci flow on certain three-manifolds，他工作的第三部分，放到了网上。该领域的专家们花了一些时间才使自己确信佩雷尔曼解决了儒勒·昂利·庞加莱猜想，又花了稍长一点的时间来仔细研究细节，以看出他也解决了威廉·瑟斯顿几何化猜想。他继续在圣彼得堡的弗拉基米尔·安德烈耶维奇·斯捷克洛夫数学研究所工作，在那里他被提升为高级研究员。然而在2005年12月他辞职了，说他对于数学感到失望，想尝试别的事情。2006年8月他被授予约翰·查尔斯·菲尔兹奖章：-

由于他对几何学的贡献以及他对格雷戈里奥·里奇-库尔巴斯托罗流的解析和几何结构的革命性洞见。

弗瑞兹·约翰 Lott在2006年8月于马德里举行的国际数学家大会上所作的演讲中，描述了佩雷尔曼导致获得约翰·查尔斯·菲尔兹奖章的工作[8]。

关于Lott演讲的摘录，其中给出了一些技术细节，见THIS LINK。（注意Lott措辞的谨慎选择。他说佩雷尔曼“证明了所谓的灵魂猜想”，但只是说他“给出了儒勒·昂利·庞加莱猜想和几何化猜想的证明。”）

佩雷尔曼拒绝了2006年国际数学家大会全体会议演讲者的邀请。他也拒绝了约翰·查尔斯·菲尔兹奖章，是第一个这样做的人。如果他希望避免公众关注，那他非常不成功，因为引起了巨大的公众兴趣，他受到媒体的追逐。2010年3月，克莱数学研究所宣布佩雷尔曼已满足他们为解决儒勒·昂利·庞加莱猜想而提供的100万美元奖金的条件。2010年7月，佩雷尔曼拒绝接受这100万美元，说：-

我不喜欢他们的决定，我认为这不公平。我认为美国数学家Hamilton对解决这个问题的贡献不亚于我。

让我们引用米哈伊尔·格罗莫夫的话来结束这篇传记（见[1]）：-

[佩雷尔曼]有他坚持的道德原则。这让人惊讶。他们常说他的行为奇怪，因为他诚实行事，以一种不墨守成规的方式，这在这个社区不受欢迎——尽管这应该是常态。

Grigori Yakovlevich Perelman's parents are Yakov Perelman, an electrical engineer, and Lubov Lvovna, who was a teacher of mathematics at a technical college. They were Jewish, which would present their son with some problems in a country where it was feared that those of Jewish descent had divided loyalty. Grigori Yakovlevich, their first child, is often known by the name Grisha. As a young child Grisha was taught to play the violin both by his mother and by a private tutor. His father also had a major influence in developing his son's problem solving skills. Speaking about his father, Perelman said (see [11]):-

He gave me logical and other maths problems to think about. He got a lot of books for me to read. He taught me how to play chess. He was proud of me.

His mother also helped develop his mathematical skills and, by the time he was ten, he had taken part in district mathematics competitions and shown a marked talent. Lubov sought advice about how best to develop Grisha's mathematical talents and was advised to send him to a mathematics club run by a nineteen year old coach named Sergei Rukshin. The club met twice a week at the Palace of Pioneers at the end of the school day and Rukshin, an undergraduate student at Leningrad University, had some novel ways of getting the best out of the boys who came to the club.

Rukshin quickly saw Perelman's potential even though at first there was little to distinguish him from other bright children in the group. There developed a bond, an understanding, between the two with Perelman becoming Rukshin's favourite pupil. In the summer of 1980 Rukshin tutored Perelman in English so that he could enter Leningrad's Special Mathematics and Physics School Number 239 in September of that year. To allow Perelman to get this intense tuition, learning the English covered in four years of schooling in a few weeks, the Perelman family had to remain in Leningrad over the summer rather than going to the country which would have been the norm. Lessons were conducted walking round the parks of Leningrad and successfully achieved their aim.

The class that Perelman entered in School 239 was unusual in that the group of highly talented mathematicians tutored by Rukshin were put into the same class. At the school Valery Ryzhik became both their class teacher and their mathematics teacher. Ryzhik was an extraordinarily talented mathematics teacher but the class containing Rukshin's collection of mathematical geniuses proved almost an impossible challenge for him. As well as mathematics, Ryzhik ran a chess club on one evening a week which Perelman attended, showing considerable talents at the game. When he was fifteen, Perelman attended the summer camp run by Rukshin. This was the first time that he had spent a night away from his mother but the bond between Rukshin and Perelman helped the potentially difficult situation. Rukshin not only trained his club boys to be the best solvers of mathematics problems but also tried to broaden their interests. Perelman was already interested in the violin and classical music but Perelman was able to broaden his musical interests. Although he would attend camps with Rukshin, Perelman never took part in the trips arranged by Ryzhik.

In January 1982 Perelman was chosen as a potential member of the 1982 Soviet Mathematical Olympiad team. He attended a selection session in Chernogolovka, about 80 km north of Moscow, where in addition to the mathematical training they were subjected to stiff physical exercises in the gym. Perelman excelled and the next step was a two-day session in Odessa in April when they were given harder problems than those expected at the Olympiad competition. Perelman achieved full marks as he did at the International Mathematical Olympiad competition in Budapest in July. He received a gold medal and a special prize for achieving a perfect score. Being a member of the Soviet team gave Perelman automatic entry to university.

Perelman entered Leningrad State University in autumn 1982. There he was particularly influenced by Viktor Zalgaller and Aleksandr Danilovic Aleksandrov. During his undergraduate years he assisted Rukshin as a mathematics tutor, going to summer camps, but his incredibly high standards gave even outstanding students an almost impossible time. Eventually Rukshin had to stop Perelman assisting at the summer camps. His university work, however, was exceptional and he graduated in 1987. He had already published a number of papers: Realization of abstract k-skeletons as k-skeletons of intersections of convex polyhedra in $\mathbb{R}^{2k-1}$ (Russian) (1985); (with I V Polikanova) A remark on Helly's theorem (Russian) (1986); a supplement to A D Aleksandrov's, On the foundations of geometry (Russian) (1987) in which Perelman discussed the equivalence of a Pasch-style axiom of Aleksandrov and some of its consequences; and On the k-radii of a convex body (Russian) (1987).

One might imagine that his achievements would mean that he would be welcomed as a graduate student at the Leningrad branch of the Steklov Mathematics Institute with open arms. However, under Ivan Vinogradov's leadership the Steklov Mathematics Institute had accepted no Jews and, although it now had a new director, the old policies persisted. Aleksandr Danilovic Aleksandrov wrote to the director requesting that Perelman be allowed to undertake graduate work under his supervision at the Leningrad branch of the Steklov Mathematics Institute. The request, highly unusual coming from someone of Aleksandrov's high standing, was granted but, although Aleksandrov would be his official advisor, in practice it was Yuri Burago who took on the role. Perelman defended his thesis Saddle Surfaces in Euclidean Spaces in 1990. He had already published one of the main results of the thesis in An example of a complete saddle surface in $\mathbb{R}^{4}$ with Gaussian curvature bounded away from zero (Russian) (1989).

Burago contacted Mikhael Leonidovich Gromov who had been a professor at Leningrad State University, but was at this time a permanent member of the Institut des Hautes Études Scientifiques outside Paris. He explained to Gromov that he had an outstanding student and asked if an invitation could be issued for him to spend time at IHES. The invitation allowed Perelman to spend several months at IHES working with Gromov on Aleksandrov spaces. Perelman's first major paper, written jointly with Burago and Gromov, was A D Aleksandrov spaces with curvatures bounded below (1992). Tadeusz Januszkiewicz begins a review as follows:-

This is an important paper in many respects. It contains a careful and fairly detailed discussion of basic facts of the theory, including various equivalent forms of definitions. It recognizes that the home of various important theorems of Riemannian geometry is the theory of Aleksandrov spaces, that both statements and proofs become more satisfactory (but not necessarily easier) in this context, and other theorems emerge naturally to complete the picture. It develops useful tools for studying Aleksandrov spaces with curvature bounded below in full generality. Finally, it contains an ample discussion of further results and open problems.

After visiting the IHES near Paris, Perelman returned to the Steklov Mathematics Institute in Leningrad but, thanks to Gromov, Perelman was invited to the United States to talk at the 1991 Geometry Festival held at Duke University in Durham, North Carolina. He lectured on the work which he had done on Aleksandrov spaces with Burago and Gromov (which had not been published at that time). In 1992 Perelman was invited to spend the autumn semester at the Courant Institute, New York University, on a postdoctoral fellowship, and the spring 1993 semester at Stony Brook, a campus of the State University of New York, again funded by a fellowship. Masha Gessen describes Perelman at this time [1]:-

By the time Perelman arrived in the United states, he was twenty-six, no longer pudgy but tall and apparently fit. His beard had passed out of its extended awkward-tuft stage and was thick, black and bushy. His hair was long. He did not believe in cutting hair or fingernails ... [he wore] the same clothes every day - most notably a brown corduroy jacket ... [he ate] a particular kind of black bread that could be procured only from a Russian store in Brooklyn Beach, where Perelman walked from Manhatten.

While Perelman was in the United States in 1992, his mother stayed with friends in New York, his father had earlier emigrated to Israel, and Perelman's young sister Lena was still being educated in St Petersburg (Leningrad returned to its original name of St Petersburg in 1991). He got to know Jeff Cheeger and Gang Tian, and the three of them regularly travelled to Princeton to attend seminars at the Institute for Advanced Study. Perelman attended a conference in Israel in 1993 then accepted a two-year Miller Research Fellowship at the University of California, Berkeley. He published some remarkable papers during these years. Elements of Morse theory on Aleksandrov spaces (Russian) (1993) investigates the local topological structure of Aleksandrov spaces. Manifolds of positive Ricci curvature with almost maximal volume (1994) solves a conjecture about a complete Riemannian manifold $M^{n}$. If such a manifold has Ricci curvature ≥ $n - 1$ and volume close to that of the sphere then Perelman proved it is homeomorphic to the sphere. The biggest breakthrough, however, was his paper Proof of the soul conjecture of Cheeger and Gromoll (1994) which answered a question asked by Cheeger and Gromoll twenty years earlier. Perelman was invited to address the International Congress of Mathematicians in Zürich in 1994 and he gave the lecture Spaces with curvature bounded below.

To understand the problems that Perelman was beginning to think about around this time, we give the description of the Poincaré Conjecture and the Thurston Geometrization Conjecture from [16]:-

A 2-manifold with positive curvature can be deformed into a 2-sphere; one with zero curvature can be deformed into a torus; and one with negative curvature can be deformed into a torus with more than one hole. The Poincaré Conjecture, which originated with the French mathematician Henri Poincaré in 1904, concerns 3-dimensional manifolds, or 3-manifolds. ... Can every simply connected 3-manifold be deformed into the 3-sphere? The Poincaré Conjecture asserts that the answer to this question is yes. Just as with 2- manifolds, one could also hope for a classification of 3-manifolds. In the 1970s, Fields Medalist William Thurston made a new conjecture, which came to be called the Thurston Geometrization Conjecture and which gives a way to classify all 3-manifolds. The Thurston Geometrization Conjecture provides a sweeping vision of 3-manifolds and actually includes the Poincaré Conjecture as a special case. Thurston proposed that, in a way analogous to the case of 2-manifolds, 3-manifolds can be classified using geometry. But the analogy does not extend very far: 3-manifolds are much more diverse and complex than 2-manifolds.

A possible approach to attacking the Poincaré Conjecture had been developed by Richard Hamilton who had introduced a significant idea in 1982 when he began to study a particular equation he called the Ricci flow. When Perelman was going to lectures at the Institute for Advanced Study he attended a lecture there by Hamilton and talked with him after the lecture. Perelman recalled [11]:-

I really wanted to ask him something. 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. 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.

When he was a Miller fellow at Berkeley, Perelman attended some further lectures by Hamilton and he began to understand why Hamilton could not make any further progress towards proving the Poincaré Conjecture using the Ricci flow.

While he was in the United States, Perelman received several requests asking him to apply for professorships. These came from top institutions such as Stanford and Princeton. He was offered a full professorship, without making any application, by Tel Aviv University in Israel, but he turned down all the offers and returned to the St Petersburg branch of the Steklov Mathematics Institute after his Miller fellowship came to an end in the summer of 1995. Basically he was able to live on the savings he had made from the money paid to him in the United States which was quite considerable since he had lived exceptionally frugally. He refused to accept a European Mathematical Society prize in 1996. Perelman had realised that Hamilton was making no progress with the Poincaré Conjecture when he read a paper Hamilton published in 1995 and, in the following year, he wrote to Hamilton explaining that he might have a way round the problem and offering to collaborate with him. When he received no reply, Perelman seems to have decided to work on solving the Poincaré Conjecture alone.

On 11 November 2002, Perelman put his paper The Entropy Formula for the Ricci Flow and Its Geometric Applications on the web. Although he did not claim in the paper to be able to solve the Poincaré Conjecture, when experts in the subject read it they realised that he had made the breakthrough necessary to solve the Conjecture. Quickly he received invitations to visit the Stony Brook campus of the State University of New York and the Massachusetts Institute of Technology. He began making plans for the visits and, before setting off, he posted a second paper Ricci flow with surgery on three-manifolds on the web continuing his proof. He arrived in the United States in April 2003 and went first to the Massachusetts Institute of Technology where he gave talks on his work for most days in the two weeks he was there. He spent two similar weeks at Stony Brook followed by visits to Columbia University and Princeton University where he gave lectures. He turned down all offers of professorships that were made to him, becoming annoyed at the pressure some put on him to accept.

He returned to St Petersburg at the end of April 2002 and, in July, put Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, the third instalment of his work, on the web. It took some time for experts in the field to convince themselves that Perelman had solved the Poincaré Conjecture and a little longer to work through the details to see that he had also solved the Thurston Geometrization Conjecture. He continued working at the Steklov Mathematics Institute in St Petersburg where he was promoted to Senior Researcher. However in December 2005 he resigned, saying that he was disappointed in mathematics and wanted to try something else. In August 2006 he was awarded a Fields medal:-

For his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.

John Lott described Perelman's work leading to the award of a Fields Medal in a lecture he gave to the International Congress of Mathematicians in Madrid in August 2006 [8].

For an extract of Lott's talk, giving some technical details, see THIS LINK. (Note the careful choice of Lott's language. He says the Perelman "proved the so-called Soul Conjecture," but only that he "presented proofs of the Poincaré conjecture and the geometrization conjecture.")

Perelman refused the invitation to be a plenary speaker at the 2006 International Congress of Mathematicians. He also refused the award of the Fields Medal, the first person to have done so. If his hope had been to avoid publicity he was highly unsuccessful since huge public interest was generated and he was hounded by the press. In March 2010 the Clay Mathematics Institute announced that Perelman had met the conditions for the award of one million US dollars which they had offered for the solution of the Poincaré Conjecture. In July 2010 Perelman refused to accept the million dollars, saying:-

I do not like their decision, I consider it unfair. I consider that the American mathematician Hamilton's contribution to the solution of the problem is no less than mine.

Let us end this biography by quoting Mikhael Gromov (see [1]):-

[Perelman] has moral principles to which he holds. And this surprises people. They often say he acts strangely because he acts honestly, in a nonconformist manner, which is unpopular in this community - even though it should be the norm.