约翰·福布斯·纳什John Forbes Nash

纳什的父亲也叫约翰·福布斯·纳什，因此我们称他为老纳什。老纳什生于1892年，童年不幸，他通过在得克萨斯农业与机械学院学习电气工程而摆脱了这种不幸。第一次世界大战期间在法国服兵役后，老纳什在得克萨斯大学讲授电气工程一年，然后加入了西弗吉尼亚州蓝田的阿巴拉契亚电力公司。纳什的母亲玛格丽特·弗吉尼亚·马丁，人称弗吉尼亚。她受过大学教育，先后在玛莎·华盛顿学院和西弗吉尼亚大学学习语言。在遇到老纳什之前，她当了十年学校教师，两人于1924年9月6日结婚。

纳什，家人这样称呼他，出生在蓝田疗养院，并在圣公会受洗。他是2：-

……一个奇特的小男孩，孤独而内向……

但他在一个充满爱的家庭中长大，周围都是亲近的亲戚，他们对他关爱有加。几年后，约翰尼有了一个妹妹，玛莎出生了。他小时候似乎对书籍表现出很大兴趣，但对和其他孩子玩耍却兴趣寥寥。约翰尼这样并非因为缺少玩伴，因为玛莎和她的表兄弟姐妹们玩着寻常的童年游戏：从书上剪下图案、在阁楼里捉迷藏、踢足球。然而，当其他人一起玩耍时，约翰尼却独自玩着玩具飞机和火柴盒汽车。

他的母亲对此的回应是热情地鼓励约翰尼接受教育，既确保他上好学校，也亲自教他。约翰尼的父亲的回应是把他当作成年人对待，在其他父母可能给孩子涂色书的时候，给他科学书籍。

约翰尼在学校的老师们肯定没有认识到他的天才，而且看来他也没有给他们什么理由去意识到他拥有非凡的才能。他们更注意到他缺乏社交技能，因此把他归类为迟钝。虽然事后聪明是容易的，但现在看来他在学校极其无聊。大约十二岁时，他就表现出对在家里的房间进行科学实验的极大兴趣。相当清楚的是，他在家里学到的比在学校学到的更多。

玛莎似乎是一个极其正常的孩子，而约翰尼则显得与其他孩子不同。她后来在人生中写道（见2）：-

约翰尼总是与众不同。［我的父母］知道他与众不同。他们也知道他聪明。他总是想按自己的方式做事。母亲坚持要我为他做事，要我把他也纳入我的朋友圈……但我并不太热衷于炫耀我这个有些古怪的哥哥。

他的父母鼓励他参加社交活动，他也没有拒绝，但体育运动、舞会、探亲以及类似活动，他视之为对读书和实验的乏味干扰。

纳什大约14岁时第一次对数学产生了兴趣。他是如何读到埃里克·坦普尔·贝尔的Men of Mathematics的，已不清楚，但可以肯定的是，这本书激励了他。他尝试并成功地为皮埃尔·德·费马的结果给出了自己的证明，这些结果埃里克·坦普尔·贝尔曾在其书中陈述过。纳什在此发现的兴奋感，与他在学校学习的数学形成了对比，后者未能引起他的兴趣。

他于1941年进入蓝田学院，在那里他既修了数学课程，也修了科学课程，尤其是化学，这是他最喜欢的科目。他开始在数学上显露才能，尤其是在解题方面，但他仍然几乎没有朋友，行为也有些古怪，这只会加深同学们对他怪异的看法。然而，此时他并未考虑以数学为职业，这并不奇怪，因为那是一个不寻常的职业。相反，他以为自己会学习电气工程并追随父亲，但他继续自己做化学实验，并参与制造炸药，这导致了他一名同学的死亡。[2]：-

无聊和青春期涌动的攻击性使他搞恶作剧，偶尔还带有恶意的成分。

他用怪异的漫画讽刺他不喜欢的同学，喜欢折磨动物，还曾试图让他妹妹坐上一把他用电池接了线的椅子。

纳什在乔治·威斯汀豪斯竞赛中赢得奖学金，并被卡内基理工学院（现卡内基-梅隆大学）录取，他于1945年6月入学，打算攻读化学工程学位。然而，很快他对数学日益增长的兴趣使他选修了张量分析和相对论课程。在那里，他接触到了约翰·L·辛格，后者最近被任命为数学系主任并教授相对论课程。约翰·L·辛格和其他数学教授很快认识到纳什非凡的数学才能，并说服他成为数学专业的学生。他们意识到他有成为职业数学家的才能，并大力鼓励他。

纳什很快在数学上志存高远。他两次参加威廉·洛厄尔·普特南数学竞赛，但尽管表现不错，却未能进入前五名。在纳什看来这是一次失败，而他对此反应很糟。对纳什来说，普特南数学竞赛并不是唯一不顺心的事。尽管他的数学教授们对他赞誉有加，同学们却觉得他是个非常奇怪的人。他身体强壮，这使他免于被欺负，但同学们却乐于取笑纳什，他们把他看作一个笨拙不成熟、会像孩子一样发脾气的人。他的一位同学写道：-

他是个乡下男孩，即使按我们的标准也算不上世故。他行为古怪，会在钢琴上反复弹奏同一个和弦，把融化的冰淇淋蛋筒放在他脱下的衣服上让它融化，踩在室友睡觉的身体上去关灯。

另一位写道：-

他极其孤独。

另一位同学写道：——

我们折磨可怜的弗瑞兹·约翰。我们非常不友善。我们令人讨厌。我们感觉到他有心理问题。

他表现出同性恋倾向，爬到其他男孩的床上，他们的反应是取笑他被男孩吸引并羞辱他。他们对他搞残酷的恶作剧，他的回应是请同学们用数学问题挑战他。最后他替许多学生做了作业。

纳什于1948年获得数学学士和硕士学位。此时，他已被哈佛、普林斯顿、芝加哥和密歇根大学的数学项目录取。他认为哈佛是顶尖大学，因此想去那里，但另一方面，他们给他的奖学金不如普林斯顿慷慨。纳什觉得普林斯顿热切希望他去，而他认为自己在普特南数学竞赛中缺乏成功意味着哈佛不那么热情。他花了一段时间做决定，同时约翰·L·辛格和其他教授鼓励他接受普林斯顿。当所罗门·莱夫谢茨向他提供普林斯顿最负盛名的奖学金时，纳什决定去那里学习。

1948年9月，纳什进入普林斯顿，在那里他对广泛的纯数学表现出兴趣：拓扑学、代数几何、博弈论和逻辑是他的兴趣所在，但他似乎避免听课。通常那些决定不通过听课学习的人会转向书籍，但这似乎不适用于纳什，他决定不“二手”学习数学，而是自己发展主题。在许多方面，这种方法很成功，因为它确实促成了他发展成为最原创的数学家之一，会以全新的方式攻击问题。

1949年，在攻读博士学位期间，他写了一篇论文，45年后这篇论文赢得了诺贝尔经济学奖。在此期间，纳什建立了博弈论的数学原理。P Ordeshook写道：——

纳什均衡$n$元组的概念也许是非合作博弈论中最重要的思想。……无论我们是在分析候选人的竞选策略、战争的起因、立法机构中的议程操纵，还是利益集团的行为，对事件的预测都归结为对均衡的寻找和描述。简言之，均衡策略就是我们对人们所预测的东西。

约翰·米尔诺，一位同学，描述了纳什在普林斯顿的岁月[6]：-

他总是充满数学想法，不仅在博弈论上，在几何和拓扑学上也是如此。然而，我对这段时间最生动的记忆是在公共休息室里玩过的许多游戏。我被介绍给了围棋和战争棋，还有一种巧妙的拓扑游戏，我们以发明者的名字命名为纳什。

事实上，游戏“纳什”几乎与丹麦的Piet Hein独立发明的Hex相同。

以下是三位同学的评论：-

纳什与众不同。如果他和一个二十人的房间在一起，他们在谈话，如果你问一个观察者谁让你觉得奇怪，那会是纳什。这不是他有意为之。这是他的举止。他的冷漠。

纳什完全令人毛骨悚然。他不会看你。他会花很多时间回答问题。如果他认为问题愚蠢，他根本不会回答。他没有情感。这是骄傲和其他东西的混合。他如此孤立，但在这一切之下，确实有温暖和对人的欣赏。

我们很多人会轻视纳什说的话。……我不想听。和这个人在一起你不觉得舒服。

他有想法，并且非常确信这些想法很重要。到达普林斯顿后不久，他就去见了阿尔伯特·爱因斯坦，并告诉他自己关于引力的一个想法。在向阿尔伯特·爱因斯坦解释了大约一个小时的复杂数学之后，阿尔伯特·爱因斯坦建议他去多学一些物理。显然，几年后确实有一位物理学家发表了类似的想法。

1950年，纳什在普林斯顿获得博士学位，学位论文题为Non-cooperative Games。那年夏天，他在兰德公司工作，他在博弈论方面的工作使他成为冷战冲突方面的顶尖专家，而冷战冲突主导了兰德公司的工作。在接下来的几年里，他时不时在那里工作，因为该公司试图将博弈论应用于军事和外交战略。1950年秋天回到普林斯顿后，他开始认真研究纯数学问题。看起来，一个刚刚提出了有朝一日会被认为值得获得诺贝尔奖的想法的人，找一份学术职位应该没有问题。然而，纳什的工作在当时并未被视为具有突出的重要性，他看出自己需要通过其他方式来确立地位。我们还应该注意到，这其实并不是转向纯数学，因为他一直认为自己是一个纯数学家。在撰写关于博弈论的学位论文之前，他已经获得了关于流形和代数簇的结果。他那个著名的定理——任何紧实流形都微分同胚于某个实代数簇的一个分支——被纳什视为一个可能的备用结果，以防他在博弈论方面的工作被认为不适合作为博士论文。他在最近的一次采访中说：-

我在纯数学方面发展出了一个非常好的想法。我得到了后来成为实代数流形的东西。我本可以更早发表它，但并没有急于发表。我花了一些时间来撰写它。有人暗示我是个神童。另一次，有人建议我应该被称为“虫子脑瓜”，因为我有想法，但这些想法有点像是虫子般不完善或并非完全可靠。所以那可能是对精神问题的一种预兆。我是说，就表面来看。

1952年，纳什在Annals of Mathematics上发表了Real Algebraic Manifolds。这篇论文最重要的结果是，两个实代数流形等价当且仅当它们解析同胚。尽管这篇关于流形的论文的发表确立了他作为一流数学家的地位，但普林斯顿并非所有人都愿意看到他加入那里的教职。这与他的数学能力无关——所有人都承认他的数学能力出众——而是因为一些数学家，比如埃米尔·阿廷，觉得由于纳什好斗的个性，他们无法接受他作为同事。

保罗·哈尔莫斯在1953年初收到了Warren Ambrose关于纳什的以下信件（例如参见[2]）：-

这里一如既往，没有什么重要消息。Martin正在任命纳什为助理教授（不是伊利诺伊州的那个纳什，而是由诺曼·斯廷罗德从普林斯顿出来的那个），我对此相当恼火。纳什是个幼稚的聪明家伙，想要“基本上原创”，我想这对那些自身有一些基本原创性的人来说倒也无妨。他还以各种方式做出与这一哲学相悖的愚蠢行为。他最近听说了关于将黎曼流形等距嵌入欧几里得空间的未解决问题，觉得这是他的专长，前提是这个问题足够有价值，值得他付出努力；于是他着手写信给数学学会的每个人去核实这一点，被告知这很可能值得，然后他就宣布自己已经解决了它，只差细节，并告诉Mackey他想在哈佛讨论会上谈论它。与此同时，他去找诺曼·莱文森询问一个介入其中的微分方程，而诺曼·莱文森说它是一个偏微分方程系统，如果他能够[达到]单个常微分方程的那种本质上更简单的类似物，那将会是一篇非常好的论文——而纳什对整个事情只有最模糊的概念。所以大家普遍承认他毫无进展，并且比那些洞察力不如我的人之前所认为的更加出丑。但我们已经得到了他，从而避免了可能得到一个真正数学家的可能性。他是个聪明人，但自负得要命，幼稚得像诺伯特·维纳，急躁得像X，喧闹得像Y，其中X和Y是任意的。

这封信的作者Ambrose和纳什有一段时间一直互相看不顺眼。他们互相开过愚蠢的玩笑，而Ambrose似乎无法像其他人学会的那样无视纳什的挖苦。正是Ambrose对纳什说：-

如果你这么厉害，为什么不解决流形的嵌入定理。

从1952年起，纳什在麻省理工学院任教，但他的教学方式不寻常（且不受学生欢迎），他的考试方法也极不正统。然而，他在实代数簇理论、黎曼几何、抛物型和椭圆型方程方面的研究，在这些课题的发展中极为深刻且意义重大。他的论文$C$1 isometric imbeddings发表于1954年，陈省身在一篇评论中指出它：-

……包含了一些令人惊讶的结果，关于具有正定$C^{0}$度量的黎曼流形到欧几里得空间的$C^{1}$等距嵌入。

纳什在1956年发表的论文The imbedding problem for Riemannian manifolds中继续发展这项工作。这篇论文包含了他著名的深刻隐函数定理。此后，纳什研究了一些想法，这些想法后来出现在他于1958年在American Journal of Mathematics上发表的论文Continuity of solutions of parabolic and elliptic equations中。然而，当纳什发现E De 恩尼奥·德乔吉用完全不同的方法证明了类似结果时，他非常失望。

纳什在几年内取得的杰出成果使他成为1958年菲尔兹奖的有力竞争者，但由于他在委员会做出决定时关于抛物线和椭圆方程的工作仍未发表，他没有获得。人们想象委员会会期望他成为1962年约翰·查尔斯·菲尔兹奖章的主要竞争者，甚至可能是几乎确定的人选，但精神疾病在这些决定做出之前很久就毁掉了他的职业生涯。

在麻省理工学院期间，纳什开始出现个人生活问题，这些问题是在他一直遭受的社交困难之外的。同事们说：-

纳什总是与一些带有浪漫气质的男性结成深厚的友谊。他非常青春期，总是和男孩子们在一起。他非常爱尝试——大多只是接吻。

他遇到了 Eleanor Stier，他们有一个儿子，John David Stier，出生于1953年6月19日。Eleanor 是一个害羞的女孩，缺乏自信，有点害怕男人，不想卷入其中。她在纳什身上找到了一个比她更没有经验的人，并觉得这很有吸引力。[2]：-

纳什在寻找更愿意付出而非索取的情感伴侣，而Eleanor正是那种人。

纳什不想娶Eleanor，尽管她努力劝说。1954年夏天，在为兰德公司工作期间，纳什在一次诱捕同性恋者的警方行动中被捕。他被兰德公司解雇了。

纳什在麻省理工学院的一名学生Alicia Larde与他变得友好，到1955年夏天他们经常见面。此时他还与一名男性研究生Jack Bricker有着特殊的友谊。Eleanor在1956年春天发现了Alicia的事，当时她来到纳什的家，发现他和Alicia在床上。纳什对一位朋友说：-

我完美的小世界毁了，我完美的小世界毁了。

Alicia 在发现纳什与 Eleanor 有一个孩子时似乎并没有太难过，并推断既然这段婚外情已经持续了三年，纳什可能对她并不认真。1956年，纳什的父母发现了他与 Eleanor 持续不断的婚外情以及他的儿子John David Stier。这一打击可能促成了纳什的父亲不久后的去世，但即使不是这样，纳什也可能自责。1957年2月，纳什与 Alicia 结婚；到1958年秋天，她怀孕了，但几个月后，即1958年底，纳什的精神状态变得非常紊乱。

在一次新年聚会上，纳什在午夜出现，只穿着尿布，身上系着一条写有“1959”的腰带。他整晚大部分时间都蜷缩在妻子的腿上，就像他打扮成的婴儿一样。有些人形容他的行为比平时更古怪。1月4日，他回到大学，开始教授他的博弈论课程。他对全班同学的开场白是：-

我想到一个问题。你们为什么在这里？

一名学生立即退选了这门课！纳什请一名研究生接管他的课程，然后消失了几个星期。当他回来时，他走进公共休息室，拿着一份《纽约时报》，说里面包含来自外太空的加密信息，这些信息只给他一个人。有几天，人们以为他在开一个精心设计的私人玩笑。

诺伯特·维纳是最早认识到纳什的极端怪癖和人格问题实际上是某种医学障碍症状的人之一。在数月古怪行为之后，Alicia将丈夫非自愿地送进麦克莱恩医院，这是波士顿郊外的一家私立精神病医院。纳什一出院就突然从麻省理工学院辞职，取走养老金，前往欧洲，打算放弃美国公民身份。Alicia把刚出生的儿子留给母亲，跟随生病的纳什。她随后使纳什被驱逐出境——回到美国。

他们回来后，两人在普林斯顿定居，Alicia在那里找了一份工作。纳什的病情持续，使他变成一个令人害怕的人物。他大部分时间都在普林斯顿校园里闲逛，以第三人称称自己为Johann von 杰森·约翰·拿骚，写毫无意义的明信片，给以前的同事打电话。他们坚忍地听着他无休止地讨论数字命理学和世界政治事务。丈夫日益恶化的状况使Alicia越来越抑郁。

1961年1月，沮丧的Alicia、约翰的母亲和他的妹妹Martha做出了艰难的决定，将他送进新泽西州特伦顿州立医院，在那里他每周五天接受胰岛素昏迷疗法，这是一种激进且有风险的治疗，持续了一个半月。随后是一段漫长而悲伤的经历，包括住院治疗、暂时康复，然后进一步治疗。Alicia于1962年与纳什离婚。纳什与Eleanor和John David住了一段时间。1970年，Alicia试图帮助他，让他作为寄宿者住进来，但他似乎已与世隔绝，脱离普通社会，尽管他大部分时间都在普林斯顿数学系度过。强烈推荐[2]这本书，因为它对纳什的精神痛苦作了感人的叙述。

多年来，纳什慢慢康复了。他在1996年第十届世界精神病学大会上发表了一篇论文，描述了他的疾病；该论文报道于[3]。1958年，他被描述为：-

...世界上最 promising 的年轻数学家...

但他很快开始感到：-

……我所在大学麻省理工学院的工作人员，以及后来整个波士顿的人，对我的举止都很奇怪。……我开始到处看到隐蔽的共产党人……我开始认为自己是一个具有重大宗教重要性的人，并且一直听到声音。我开始在脑海中听到类似电话的声音，来自反对我思想的人。……这种谵妄就像一场我似乎永远无法醒来的梦。

尽管由于他的精神状况而多次住院，他的数学工作仍然一个成功接着一个成功。他说：-

我不敢说数学与疯狂之间有直接关系，但毫无疑问，伟大的数学家患有躁狂特征、谵妄和精神分裂症的症状。

在1990年代，纳什从他自1959年以来一直患有精神分裂症中康复。他产生最高质量数学的能力并没有完全离开他。他说：-

如果我不能在工作中产生好的东西，我就不会把自己视为康复了。

纳什因在博弈论方面的工作（与 Harsanyi 和赖因哈德·泽尔滕共同）获得了1994年诺贝尔经济学奖。1999年，他被美国数学会授予 Leroy P 凯瑟琳·斯蒂尔 奖：-

……因对研究做出了开创性贡献。

纳什和路易·尼伦伯格于2015年获得尼尔斯·阿贝尔奖，获奖理由为：-

……对非线性偏微分方程理论及其在几何分析中的应用做出了引人注目且具有开创性的贡献。

在挪威领取该奖几天后，纳什 和他的妻子 Alicia 在新泽西收费公路上因出租车事故丧生。

John F Nash's father, also called John Forbes Nash so we shall refer to him as John Nash Senior, was a native of Texas. John Nash Senior was born in 1892 and had an unhappy childhood from which he escaped when he studied electrical engineering at Texas Agricultural and Mechanical. After military service in France during World War I, John Nash Senior lectured on electrical engineering for a year at the University of Texas before joining the Appalachian Power Company in Bluefield, West Virginia. John F Nash's mother, Margaret Virginia Martin, was known as Virginia. She had a university education, studying languages at the Martha Washington College and then at West Virginia University. She was a school teacher for ten years before meeting John Nash Senior, and the two were married on 6 September 1924.

Johnny Nash, as he was called by his family, was born in Bluefield Sanitarium and baptised into the Episcopal Church. He was [2]:-

... a singular little boy, solitary and introverted ...

but he was brought up in a loving family surrounded by close relations who showed him much affection. After a couple of years Johnny had a sister when Martha was born. He seems to have shown a lot of interest in books when he was young but little interest in playing with other children. It was not because of lack of children that Johnny behaved in this way, for Martha and her cousins played the usual childhood games: cutting patterns out of books, playing hide-and-seek in the attic, playing football. However while the others played together Johnny played by himself with toy airplanes and matchbox cars.

His mother responded by enthusiastically encouraging Johnny's education, both by seeing that he got good schooling and also by teaching him herself. Johnny's father responded by treating him like an adult, giving him science books when other parents might give their children colouring books.

Johnny's teachers at school certainly did not recognise his genius, and it would appear that he gave them little reason to realise that he had extraordinary talents. They were more conscious of his lack of social skills and, because of this, labelled him as backward. Although it is easy to be wise after the event, it now would appear that he was extremely bored at school. By the time he was about twelve years old he was showing great interest in carrying out scientific experiments in his room at home. It is fairly clear that he learnt more at home than he did at school.

Martha seems to have been a remarkably normal child while Johnny seemed different from other children. She wrote later in life (see [2]):-

Johnny was always different. [My parents] knew he was different. And they knew he was bright. He always wanted to do things his way. Mother insisted I do things for him, that I include him in my friendships. ... but I wasn't too keen on showing off my somewhat odd brother.

His parents encouraged him to take part in social activities and he did not refuse, but sports, dances, visits to relatives and similar events he treated as tedious distractions from his books and experiments.

Nash first showed an interest in mathematics when he was about 14 years old. Quite how he came to read E T Bell's Men of Mathematics is unclear but certainly this book inspired him. He tried, and succeeded, in proving for himself results due to Fermat which Bell stated in his book. The excitement that Nash found here was in contrast to the mathematics that he studied at school which failed to interest him.

He entered Bluefield College in 1941 and there he took mathematics courses as well as science courses, in particular studying chemistry, which was a favourite topic. He began to show abilities in mathematics, particularly in problem solving, but still with hardly any friends and behaving in a somewhat eccentric manner, this only added to his fellow pupils view of him as peculiar. He did not consider a career in mathematics at this time, however, which is not surprising since it was an unusual profession. Rather he assumed that he would study electrical engineering and follow his father but he continued to conduct his own chemistry experiments and was involved in making explosives which led to the death of one of his fellow pupils. [2]:-

Boredom and simmering adolescent aggression led him to play pranks, occasionally ones with a nasty edge.

He caricatured classmates he disliked with weird cartoons, enjoyed torturing animals, and once tried to get his sister to sit in a chair he had wired up with batteries.

Nash won a scholarship in the George Westinghouse Competition and was accepted by the Carnegie Institute of Technology (now Carnegie-Mellon University) which he entered in June 1945 with the intention of taking a degree in chemical engineering. Soon, however, his growing interest in mathematics had him take courses on tensor calculus and relativity. There he came in contact with John Synge who had recently been appointed as Head of the Mathematics Department and taught the relativity course. Synge and the other mathematics professors quickly recognised Nash's remarkable mathematical talents and persuaded him to become a mathematics specialist. They realised that he had the talent to become a professional mathematician and strongly encouraged him.

Nash quickly aspired to great things in mathematics. He took the William Lowell Putnam Mathematics Competition twice but, although he did well, he did not make the top five. It was a failure in Nash's eyes and one which he took badly. The Putnam Mathematics Competition was not the only thing going badly for Nash. Although his mathematics professors heaped praise on him, his fellow students found him a very strange person. Physically he was strong and this saved him from being bullied, but his fellow students took delight in making fun of Nash whom they saw as an awkward immature person displaying childish tantrums. One of his fellow students wrote:-

He was a country boy unsophisticated even by our standards. He behaved oddly, playing a single chord on a piano over and over, leaving a melting ice cream cone melting on top of his cast-off clothing, walking on his roommate's sleeping body to turn off the light.

Another wrote:-

He was extremely lonely.

And a third fellow student wrote:-

We tormented poor John. We were very unkind. We were obnoxious. We sensed he had a mental problem.

He showed homosexual tendencies, climbing into bed with the other boys who reacted by making fun of the fact that he was attracted to boys and humiliated him. They played cruel pranks on him and he reacted by asking his fellow students to challenge him with mathematics problems. He ended up doing the homework of many of the students.

Nash received a BA and an MA in mathematics in 1948. By this time he had been accepted into the mathematics programme at Harvard, Princeton, Chicago and Michigan. He felt that Harvard was the leading university and so he wanted to go there, but on the other hand their offer to him was less generous than that of Princeton. Nash felt that Princeton were keen that he went there while he felt that his lack of success in the Putnam Mathematics Competition meant that Harvard were less enthusiastic. He took a while to make his decision, while he was encouraged by Synge and his other professors to accept Princeton. When Lefschetz offered him the most prestigious Fellowship that Princeton had, Nash made his decision to study there.

In September 1948 Nash entered Princeton where he showed an interest in a broad range of pure mathematics: topology, algebraic geometry, game theory and logic were among his interests but he seems to have avoided attending lectures. Usually those who decide not to learn through lectures turn to books but this appears not to be so for Nash, who decided not to learn mathematics "second-hand" but rather to develop topics himself. In many ways this approach was successful for it did contribute to him developing into one of the most original of mathematicians who would attack a problem in a totally novel way.

In 1949, while studying for his doctorate, he wrote a paper which 45 years later was to win a Nobel prize for economics. During this period Nash established the mathematical principles of game theory. P Ordeshook wrote:-

The concept of a Nash equilibrium $n$-tuple is perhaps the most important idea in noncooperative game theory. ... Whether we are analysing candidates' election strategies, the causes of war, agenda manipulation in legislatures, or the actions of interest groups, predictions about events reduce to a search for and description of equilibria. Put simply, equilibrium strategies are the things that we predict about people.

Milnor, who was a fellow student, describes Nash during his years at Princeton in [6]:-

He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well. However, my most vivid memory of this time is of the many games which were played in the common room. I was introduced to Go and Kriegspiel, and also to an ingenious topological game which we called Nash in honour of the inventor.

In fact the game "Nash" was almost identical to Hex which had been invented independently by Piet Hein in Denmark.

Here are three comments from fellow students:-

Nash was out of the ordinary. If he was in a room with twenty people, and they were talking, if you asked an observer who struck you as odd it would have been Nash. It was not anything he consciously did. It was his bearing. His aloofness.

Nash was totally spooky. He wouldn't look at you. he'd take a lot of time answering a question. If he thought the question was foolish he wouldn't answer at all. He had no affect. It was a mixture of pride and something else. He was so isolated but there really was underneath it all a warmth and appreciation of people.

A lot of us would discount what Nash said. ... I wouldn't want to listen. You didn't feel comfortable with the person.

He had ideas and was very sure they were important. He went to see Einstein not long after he arrived in Princeton and told him about an idea he had regarding gravity. After explaining complicated mathematics to Einstein for about an hour, Einstein advised him to go and learn more physics. Apparently a physicist did publish a similar idea some years later.

In 1950 Nash received his doctorate from Princeton with a thesis entitled Non-cooperative Games. In the summer of that year he worked for the RAND Corporation where his work on game theory made him a leading expert on the Cold War conflict which dominated RAND's work. He worked there from time to time over the next few years as the Corporation tried to apply game theory to military and diplomatic strategy. Back at Princeton in the autumn of 1950 he began to work seriously on pure mathematical problems. It might seem that someone who had just introduced ideas which would, one day, be considered worthy of a Nobel Prize would have no problems finding an academic post. However, Nash's work was not seen at the time to be of outstanding importance and he saw that he needed to make his mark in other ways. We should also note that it was not really a move towards pure mathematics for he had always considered himself a pure mathematician. He had already obtained results on manifolds and algebraic varieties before writing his thesis on game theory. His famous theorem, that any compact real manifold is diffeomorphic to a component of a real-algebraic variety, was thought of by Nash as a possible result to fall back on if his work on game theory was not considered suitable for a doctoral thesis. He said in a recent interview:-

I developed a very good idea in pure mathematics. I got what became Real Algebraic Manifolds. I could have published that earlier, but it wasn't rushed to publication. I took some time in writing it up. Somebody suggested that I was a prodigy. Another time it was suggested that I should be called "bug brains", because I had ideas, but they were sort of buggy or not perfectly sound. So that might have been an anticipation of mental problems. I mean, taking it at face value.

In 1952 Nash published Real Algebraic Manifolds in the Annals of Mathematics. The most important result in this paper is that two real algebraic manifolds are equivalent if and only if they are analytically homeomorphic. Although publication of this paper on manifolds established him as a leading mathematician, not everyone at Princeton was prepared to see him join the Faculty there. This was nothing to do with his mathematical ability which everyone accepted as outstanding, but rather some mathematicians such as Artin felt that they could not have Nash as a colleague due to his aggressive personality.

Halmos received the following letter in early 1953 from Warren Ambrose relating to Nash (see for example [2]):-

There's no significant news from here, as always. Martin is appointing John Nash to an Assistant Professorship (not the Nash at Illinois, the one out of Princeton by Steenrod) and I'm pretty annoyed at that. Nash is a childish bright guy who wants to be "basically original," which I suppose is fine for those who have some basic originality in them. He also makes a damned fool of himself in various ways contrary to this philosophy. He recently heard of the unsolved problem about imbedding a Riemannian manifold isometrically in Euclidean space, felt that this was his sort of thing, provided the problem were sufficiently worthwhile to justify his efforts; so he proceeded to write to everyone in the math society to check on that, was told that it probably was, and proceeded to announce that he had solved it, modulo details, and told Mackey he would like to talk about it at the Harvard colloquium. Meanwhile he went to Levinson to inquire about a differential equation that intervened and Levinson says it is a system of partial differential equations and if he could only [get] to the essentially simpler analog of a single ordinary differential equation it would be a damned good paper - and Nash had only the vaguest notions about the whole thing. So it is generally conceded he is getting nowhere and making an even bigger ass of himself than he has been previously supposed by those with less insight than myself. But we've got him and saved ourselves the possibility of having gotten a real mathematician. He's a bright guy but conceited as Hell, childish as Wiener, hasty as X, obstreperous as Y, for arbitrary X and Y.

Ambrose, the author of this letter, and Nash had rubbed each other the wrong way for a while. They had played silly pranks on each other and Ambrose seems not to have been able to ignore Nash's digs in the way others had learned to do. It had been Ambrose who had said to Nash:-

If you're so good, why don't you solve the embedding theorem for manifolds.

From 1952 Nash had taught at the Massachusetts Institute of Technology but his teaching was unusual (and unpopular with students) and his examining methods were highly unorthodox. His research on the theory of real algebraic varieties, Riemannian geometry, parabolic and elliptic equations was, however, extremely deep and significant in the development of all these topics. His paper $C$1 isometric imbeddings was published in 1954 and Chern, in a review, noted that it:-

... contains some surprising results on the $C^{1}$-isometric imbedding into an Euclidean space of a Riemannian manifold with a positive definite $C^{0}$-metric.

Nash continued to develop this work in the paper The imbedding problem for Riemannian manifolds published in 1956. This paper contains his famous deep implicit function theorem. After this Nash worked on ideas that would appear in his paper Continuity of solutions of parabolic and elliptic equations which was published in the American Journal of Mathematics in 1958. Nash, however, was very disappointed when he discovered that E De Giorgi had proved similar results by completely different methods.

The outstanding results which Nash had obtained in the course of a few years put him into contention for a 1958 Fields Medal but since his work on parabolic and elliptic equations was still unpublished when the Committee made their decisions he did not make it. One imagines that the Committee would have expected him to be a leading contender, perhaps even a virtual certainty, for a 1962 Fields Medal but mental illness destroyed his career long before those decisions were made.

During his time at MIT Nash began to have personal problems with his life which were in addition to the social difficulties he had always suffered. Colleagues said:-

Nash was always forming intense friendships with men that had a romantic quality. He was very adolescent, always with the boys. He was very experimental - mostly he just kissed.

He met Eleanor Stier and they had a son, John David Stier, who was born on 19 June 1953. Eleanor was a shy girl, lacking confidence, a little afraid of men, did not want to be involved. She found in Nash someone who was even less experienced than she was and found that attractive. [2]:-

Nash was looking for emotional partners who were more interested in giving than receiving, and Eleanor, was very much that sort.

Nash did not want to marry Eleanor although she tried hard to persuade him. In the summer of 1954, while working for RAND, Nash was arrested in a police operation to trap homosexuals. He was dismissed from RAND.

One of Nash's students at MIT, Alicia Larde, became friendly with him and by the summer of 1955 they were seeing each other regularly. He also had a special friendship with a male graduate student at this time: Jack Bricker. Eleanor found out about Alicia in the spring of 1956 when she came to Nash's house and found him in bed with Alicia. Nash said to a friend:-

My perfect little world is ruined, my perfect little world is ruined.

Alicia did not seem too upset at discovering that Nash had a child with Eleanor and deduced that since the affair had been going on for three years, Nash was probably not serious about her. In 1956 Nash's parents found out about his continuing affair with Eleanor and about his son John David Stier. The shock may have contributed to the death of Nash's father soon after, but even if it did not Nash may have blamed himself. In February of 1957 Nash married Alicia; by the autumn of 1958 she was pregnant but, a couple of months later near the end of 1958, Nash's mental state became very disturbed.

At a New Year's Party Nash appeared at midnight dressed only with a nappy and a sash with "1959" written on it. He spent most of the evening curled up, like the baby he was dressed as, on his wife's lap. Some described his behaviour as stranger than usual. On 4 January he was back at the university and started to teach his game theory course. His opening comments to the class were:-

The question occurs to me. Why are you here?

One student immediately dropped the course! Nash asked a graduate student to take over his course and vanished for a couple of weeks. When he returned he walked into the common room with a copy of the New York Times saying that it contained encrypted messages from outer space that were meant only for him. For a few days people thought he was playing an elaborate private joke.

Norbert Wiener was one of the first to recognize that Nash's extreme eccentricities and personality problems were actually symptoms of a medical disorder. After months of bizarre behaviour, Alicia had her husband involuntarily hospitalised at McLean Hospital, a private psychiatric hospital outside of Boston. Upon his release, Nash abruptly resigned from MIT, withdrew his pension, and went to Europe, where he intended to renounce his US citizenship. Alicia left her newborn son with her mother, and followed the ill Nash. She then had Nash deported - back to the United States.

After their return, the two settled in Princeton where Alicia took a job. Nash's illness continued, transforming him into a frightening figure. He spent most of his time hanging around on the Princeton campus, talking about himself in the third person as Johann von Nassau, writing nonsensical postcards and making phone calls to former colleagues. They stoically listened to his endless discussions of numerology and world political affairs. Her husband's worsening condition depressed Alicia more and more.

In January 1961 the despondent Alicia, John's mother, and his sister Martha made the difficult decision to commit him to Trenton State Hospital in New Jersey where he endured insulin-coma therapy, an aggressive and risky treatment, five days a week for a month and a half. A long sad episode followed which included periods of hospital treatment, temporary recovery, then further treatment. Alicia divorced Nash in 1962. Nash spent a while with Eleanor and John David. In 1970 Alicia tried to help him taking him in as a boarder, but he appeared to be lost to the world, removed from ordinary society, although he spent much of his time in the Mathematics Department at Princeton. The book [2] is highly recommended for its moving account of Nash's mental sufferings.

Slowly over many years Nash recovered. He delivered a paper at the tenth World Congress of Psychiatry in 1996 describing his illness; it is reported in [3]. He was described in 1958 as the:-

... most promising young mathematician in the world ...

but he soon began to feel that:-

... the staff at my university, the Massachusetts Institute of Technology, and later all of Boston were behaving strangely towards me. ... I started to see crypto-communists everywhere ... I started to think I was a man of great religious importance, and to hear voices all the time. I began to hear something like telephone calls in my head, from people opposed to my ideas. ...The delirium was like a dream from which I seemed never to awake.

Despite spending periods in hospital because of his mental condition, his mathematical work continued to have success after success. He said:-

I would not dare to say that there is a direct relation between mathematics and madness, but there is no doubt that great mathematicians suffer from maniacal characteristics, delirium and symptoms of schizophrenia.

In the 1990s Nash made a recovery from the schizophrenia from which he had suffered since 1959. His ability to produce mathematics of the highest quality did not totally leave him. He said:-

I would not treat myself as recovered if I could not produce good things in my work.

Nash was awarded (jointly with Harsanyi and Selten) the 1994 Nobel Prize in Economic Science for his work on game theory. In 1999 he was awarded the Leroy P Steele Prize by the American Mathematical Society:-

... for a seminal contribution to research.

Nash and Louis Nirenberg were awarded the Abel prize in 2015 for:-

... striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.

A few days after picking up the prize in Norway, Nash and his wife Alicia were killed in an accident to their taxi on the New Jersey turnpike.