亚历山大·格罗滕迪克Alexander Grothendieck

亚历山大·格罗滕迪克的名字常写作Alexandre，这是他在法国生活时采用的形式。他的父母是格罗滕迪克 Schapiro（1890-1942）和Johanna Grothendieck（1900-1957）。他的父亲以标准的俄语名字Sascha（即格罗滕迪克）为人所知，而他的母亲则被称为Hanka。为了对格罗滕迪克的成长和个性有所了解，有必要提供一些关于他父母的细节，特别是解释为什么他随母姓格罗滕迪克。格罗滕迪克 Schapiro是一名俄罗斯犹太人，出生于Novozybkov，一个靠近俄罗斯、白俄罗斯（今白俄罗斯）和乌克兰交界处的城镇。十五岁时，他在1905年开始的战争中为革命者对抗俄国沙皇。1907年，他和他的同志们被捕并被判处死刑。他的同志们都被行刑队枪决，但Schapiro虽然连续20天被带去处决，最终因年幼而被判终身监禁。他在监狱中度过了十年，直到1917年革命期间逃脱。他继续为无政府主义者战斗，多次被捕和逃脱，有一次失去了左臂。他与一名犹太妇女Rachil结婚，育有一子名叫Dodek。最终，在1921年，他逃到柏林生活了一段时间，然后以格罗滕迪克 Tanaroff的名字在巴黎生活。他以街头摄影师为生。格罗滕迪克的母亲Hanka 格罗滕迪克出生于德国汉堡，加入了几个左翼团体。她与Alfred Raddatz（人称Johannes）结婚，育有一女名叫Frode，人称Maidi，生于1924年。在一次访问柏林时，格罗滕迪克 Schapiro遇到了Hanka，他们有了一个儿子格罗滕迪克（人称Schurik），即本传记的主人公。他的出生登记名为格罗滕迪克 Raddatz，因为在他出生时，他的母亲与Alf Raddatz已婚。Alf和Hanka于1929年离婚。格罗滕迪克 Schapiro（他仍自称格罗滕迪克 Tanaroff）和Hanka从1928年到1933年与他们的儿子Schurik以及他母亲的女儩Maidi一起住在柏林。在那里他们有一家摄影工作室，为家庭提供收入。

阿道夫·希特勒于1933年1月30日成为德意志帝国总理。1933年4月1日发生了所谓的“抵制日”，犹太商店和企业遭到抵制。1933年4月7日，纳粹通过了一项法律，根据第三条款，命令非雅利安血统的公务员退休。尽管格罗滕迪克 Schapiro通过使用Tanaroff这个名字隐藏了他的犹太出身，他仍然认为柏林对犹太人来说太危险，并于1933年5月返回巴黎。Hanka和她的儿子格罗滕迪克留在柏林直到1933年12月，当时她安排五岁的格罗滕迪克由住在汉堡的牧师Wilhelm Heydorn和他的妻子Dagmar抚养。她把女儿Maidi送进了柏林的一家机构。Schurik与其他寄养儿童一起，从1934年初到1939年4月与Heydorn一家生活。他上了小学，然后开始在文理中学学习。与此同时，Hanka已在巴黎与格罗滕迪克的父亲团聚，在西班牙内战（1936-39）爆发后，他们两人都去了西班牙，支持共和派。1939年初共和派失败后，格罗滕迪克和Hanka返回法国，Hanka开始在尼姆工作。到1939年4月底，Heydorn一家作为抵抗希特勒的一部分，现在处于严重危险之中，认为他们的家对年轻的Schurik来说太危险，因此在法国找到他的父母后，把他送上火车与父亲团聚。1939年夏天，他与母亲在尼姆度过。

第二次世界大战的爆发意味着Schurik和他的父母在法国现在处于相当危险的境地。关于“不受欢迎者”的法律已于1938年11月12日通过，该法律要求所有居住在法国的德国人都被送往专门的拘留营。Hanka和Schurik被拘留在芒德附近的Rieucros营。然而，Schurik被允许继续在距离营地约5公里的村庄学校接受教育。他还得到了一些私人辅导。Schurik的父亲格罗滕迪克被拘留在Camp du Vernet。1942年，Rieucros营被关闭，因此Schurik和他的母亲被送往波城附近的Gurs集中营。Schurik设法到达了Le Chambon sur Lignon，在那里他就读于著名的Collège Cévénol。每当当局来搜查犹太人时，他都会躲在树林里。他于1945年在该学院获得了业士学位。与此同时，1942年8月，他的父亲被法国维希政府移交给纳粹，纳粹将他从Camp du Vernet带到奥斯维辛灭绝营，他在那里丧生。此时让我们注意到，Maidi在战争中幸存下来，并最终移民到美国。

1945年，Schurik——我们现在开始称他为格罗滕迪克——和他的母亲搬到了蒙彼利埃附近的Maisargues村，在那里格罗滕迪克在葡萄园里干活，同时靠一小笔奖学金的资助，在蒙彼利埃大学学习数学。上学时，他对呈现给他的一些数学内容感到不满（例如见[16]）：-

在我们的高中数学课本中，最令我不满的是，对于曲线的长度、曲面的面积、立体的体积这些概念，没有任何严肃的定义。我向自己承诺，一旦有机会，我就要填补这一空白。

他发现蒙彼利埃的教授们在填补这些空白方面帮不上什么忙，所以他不得不独自钻研[8]：-

正是在蒙彼利埃，在他的本科岁月里，他经历了第一次真正的数学体验。他对所接受的教学非常不满。有人告诉过他如何计算球体或棱锥的体积，但没有人解释过体积的定义。想要用“为什么”取代“如何”，这是数学精神的一个确凿标志。格罗滕迪克的一位教授向他保证，某位昂利·勒贝格已经解决了数学中最后一些悬而未决的问题，但他的工作太难，无法讲授。格罗滕迪克独自一人，几乎没有任何提示，重新发现了昂利·勒贝格积分的一个非常一般的版本。这第一项数学工作的起源[是]在完全孤立的状态下完成的……

让我们注意，那位认为昂利·勒贝格已经解决了所有悬而未决的数学问题的蒙彼利埃教授，是某位M Soula。他曾受教于埃利·嘉当，他建议格罗滕迪克去巴黎，与埃利·嘉当一起工作。格罗滕迪克听从了这一建议，在从蒙彼利埃获得学士学位后，他在1948-49学年在巴黎高等师范学校度过。在那里，他参加了亨利·嘉当的讨论班，讨论班的内容是代数拓扑学和层论。格罗滕迪克此时与当时一流的数学家们交往，这些数学家也参加亨利·嘉当的讨论班，包括谢瓦莱、让·戴尔萨特、让·迪厄多内、罗杰·戈德芒、洛朗·施瓦茨和安德烈·韦伊。格罗滕迪克的同学之一是让-皮埃尔·塞尔。由于格罗滕迪克此时对拓扑向量空间的兴趣大于对代数拓扑的兴趣，安德烈·韦伊和亨利·嘉当都建议他去南锡，那里有一支强大的团队，包括让·迪厄多内、让·戴尔萨特、罗杰·戈德芒和洛朗·施瓦茨。

1949年，格罗滕迪克搬到了南锡大学，在那里他与母亲同住，母亲因在拘留营中感染了肺结核而偶尔卧床不起。此时，格罗滕迪克与一位女士育有一子，名叫Serge，他们从这位女士那里租房。Serge主要由他的母亲抚养。格罗滕迪克从事泛函分析的研究，与让·迪厄多内[16]合作：-

格罗滕迪克的书很少；他不是通过阅读来学习，而是尝试自己重新构建知识。他工作非常努力。

在南锡，每周六都有一个活跃的研讨会，所有教授和一些学生都参加，研究各种不同的主题。这为年轻的格罗滕迪克提供了一个极好的环境。让·迪厄多内写道[9]：-

必须建立局部凸空间的一般对偶理论：洛朗·施瓦茨和我已经开始研究莫里斯·弗雷歇空间及其直接极限，但我们遇到了一系列无法解决的问题。因此，我们将这些问题提交给格罗滕迪克，结果超出了我们最乐观的预期。在不到一年的时间里，他通过非常巧妙的新构造解决了我们所有的问题；然后，利用他开发的技术，他开始研究泛函分析中的许多其他问题。

他于1953年提交了博士论文Produits tensoriels topologiques et espaces nucléairesⓉ（拓扑张量积与核空间），1953年2月28日他的论文答辩会上，在场的人之一是Bernard Malgrange，他[16]：-

...回忆说，在格罗滕迪克写完论文后，他声称自己不再对拓扑向量空间感兴趣。“他告诉我，‘没有什么可做的了，这个主题已经死了，’”Malgrange回忆道。当时，学生被要求准备一篇“第二论文”，这不包含原创工作，但旨在展示对另一个与论文主题相去甚远的数学领域的深刻理解。格罗滕迪克的第二论文是关于层论的，这项工作可能为他后来对代数几何的兴趣埋下了种子，他将在那里做出他最伟大的工作。在格罗滕迪克的论文答辩之后，答辩在巴黎举行，Malgrange回忆说，他、格罗滕迪克和亨利·嘉当挤进一辆出租车，去洛朗·施瓦茨家吃午饭。他们打车是因为Malgrange滑雪时摔断了腿。“在出租车里，亨利·嘉当向格罗滕迪克解释了格罗滕迪克关于层论的一些错误说法，”Malgrange回忆道。

格罗滕迪克在1953至1955年间在圣保罗大学度过，随后一年在堪萨斯大学。然而正是在此期间，他的研究兴趣发生了变化，转向了拓扑学和几何学。事实上，在此期间，格罗滕迪克一直得到国家科学研究中心的支持，支持始于1950年。因此，在1956年离开堪萨斯后，他回到了国家科学研究中心。大约在这个时候，他成为了尼古拉·布尔巴基数学家群体的一员，该群体包括安德烈·韦伊、亨利·嘉当和让·迪厄多内。然而在1959年，他获得了新成立的高等科学研究所的研究职位，并接受了。在[2]中，格罗滕迪克职业生涯的下一阶段描述如下：-

毫不夸张地说，格罗滕迪克在IHES的1959至1970年是‘黄金时代’，在此期间，一个全新的数学学派在格罗滕迪克富有魅力的领导下蓬勃发展。格罗滕迪克的代数几何讨论班Ⓣ确立了IHES作为世界代数几何中心的地位，并使他成为其推动力。

在此期间，格罗滕迪克的工作为几何学、数论、拓扑学和复分析提供了统一的主题。他在1960年代引入了概型理论，这使得安德烈·韦伊的某些数论猜想得以解决。他研究了与数学逻辑高度相关的拓扑斯理论。他给出了波恩哈德·黎曼-Roch定理的代数证明。他提供了曲线基本群的代数定义。

再次引用[2]：-

仅仅列举格罗滕迪克最著名的贡献就令人震撼：拓扑张量积与核空间、作为导出函子的层上同调、概型、K-理论以及格罗滕迪克-波恩哈德·黎曼-Roch、强调相对于基的工作、通过函子定义和构造几何对象、纤维范畴与下降、栈、格罗滕迪克拓扑（site）与拓扑斯、导出范畴、局部与整体对偶的形式体系（‘六种运算’）、étale上同调与L-函数的上同调解释、晶体上同调、‘标准猜想’、动机与‘权重的瑜伽’、张量范畴与动机埃瓦里斯特·伽罗瓦群。很难想象所有这些都出自一个人的头脑。

在1950年代末，格罗滕迪克与Mireille同居，他们有三个孩子：Johanna、Mathieu和Alexandre。几年后他与Mireille结婚。Luc Illusie于1964年开始参加格罗滕迪克在高等科学研究所的讨论班。他写道[13]：-

他在董事会发言时精力充沛，但注意回顾所有必要的材料。他非常精确。陈述如此简洁，以至于连我这个对主题一无所知的人也能理解形式结构。进展很快，但如此清晰，以至于我能做笔记。

Illusie接着描述了拜访格罗滕迪克讨论他的工作[13]：-

当时他住在Bures-sur-Yvette的rue de Moulon，一座小白亭子里，有底层和一层。他在那里的办公室简朴，冬天寒冷。他有一幅他父亲的铅笔画肖像，桌子上还有他母亲的遗容面具。他的桌子后面有文件柜。当他想要某份文件时，他只需转过身，就能立刻找到。他很有条理。

Valentin Poénaru在这些年也认识格罗滕迪克。他写道[21]：-

格罗滕迪克我在这时认识的是一个非常令人印象深刻的人，当我说这话时，我不仅仅想到数学。Shourik，我这样称呼他，是我见过的最坚强、最有魅力的人之一。我认为他是一个直接从陀思妥耶夫斯基作品中走出来的人物。他也是一个非常善良和慷慨的人。他似乎总是心情愉快，精神非常平衡，而且以他自己的方式，有一种生活的乐趣。当时，他有能力在想睡的时候睡觉，想睡多少小时就睡多少小时，以便之后更好地投入工作。事实上，他的工作能力对我来说是奇迹般的。

[2]的作者写道：-

他于1966年获得约翰·查尔斯·菲尔兹奖章。回顾这一时期，人们惊叹于格罗滕迪克与同事和学生分享想法的慷慨，他和合作者致力于细致编辑的精力，以及他们出发探索新领域时的兴奋。

菲尔兹奖的引文如下：-

格罗滕迪克在安德烈·韦伊和奥斯卡·扎里斯基的工作基础上，在代数拓扑学中取得了根本性进展。他引入了K理论（格罗滕迪克群和环）的思想。他在著名的“Tohoko论文”中彻底改变了同调代数。

本引文中所指的“Tohoko论文”是关于阿贝尔范畴、模层、消解、导出函子和格罗滕迪克谱序列的。然而，颁发约翰·查尔斯·菲尔兹奖章存在某些困难，因为原定颁发该奖的1966年国际数学家大会于1966年8月在莫斯科举行。格罗滕迪克在观点上始终是坚定的和平主义者，他曾反对20世纪60年代的军事集结。作为政治抗议，他拒绝前往莫斯科领取约翰·查尔斯·菲尔兹奖章。在大会上，IHES所长Léon Motchane代表格罗滕迪克领取了约翰·查尔斯·菲尔兹奖章。格罗滕迪克没有公开声明不去莫斯科的原因，但他宣称自己是世界公民，并申请联合国公民身份。1967年11月和12月，他访问了北越，当时那里正遭到美国人的轰炸[18]：-

格罗滕迪克的第一批讲座——他称之为“一般性导引报告”——在河内举行。但由于首都遭到更猛烈的轰炸，高层决定将所有人转移到河内大学数学系的秘密地点。格罗滕迪克写道：“随后我在河内大学位于城外（距首都约100公里）的疏散地点度过了一周半；这段时间主要用于一个更专门的范畴论与同调代数讨论班，有三十到四十名听众，其中大多数人从河内跟随我而来，此前参加了那些一般性导引讲座。”这是数学史上的一件非凡事件：20世纪数学的巨人之一，在一个极度贫困、正被世界上前所未有的最强大军事力量“炸回石器时代”（美国空军将军Curtis Le May的说法）的国家的偏远森林藏身处，讲授了一门同调代数短期课程。

他在发现IHES的部分资金来自军方后，于1970年离开了IHES。他在1969年发现了这一点，并与IHES的其他教授一起说服所长Léon Motchane不再接受法国军方的资助。然而，几个月后，当IHES预算非常紧张时，所长背弃了诺言。格罗滕迪克试图说服所有教授辞职以示抗议，但其他人拒绝效仿他。格罗滕迪克的辞职信日期为1970年5月25日。然而，格罗滕迪克还有其他问题，因为他写道，此时他正遭受“精神停滞”。他放弃了将数学作为主要精力焦点，转向政治抗议，特别是反对核扩散。然而，与他数学工作的惊人影响相比，他的政治运动相当无效。1970-72年，他担任法兰西学院访问教授，随后于1972-73年在奥赛担任类似职务。1973年，他接受了蒙彼利埃大学教授职位。他在蒙彼利埃授课并有一些研究生。1973年至1980年，他住在Lodeve附近的Villecun，然后搬到Carpentras附近的Mormoiron。1984-88年，他请假在国家科学研究中心指导研究。他于1988年60岁时退休，出版物[2]的出版就是为了纪念他的60岁生日。与接受1966年约翰·查尔斯·菲尔兹奖章形成对比的是，格罗滕迪克于1988年拒绝了Crafoord奖。

1980年至1990年间，格罗滕迪克写下了数千页的文字，其中一些包含他的数学思考，另一些则包含非数学的沉思。他的数学手稿有La longue marche à travers la théorie de Galois Ⓣ（穿越埃瓦里斯特·伽罗瓦理论的漫长征程）（1981年）、A la poursuite des champs Ⓣ（追寻域）（1983年）、Esquisse d'un programme Ⓣ（一个纲要方案）（1983年）和Les dérivateurs Ⓣ（推导）（1987年），而他的非数学著作有Eloge Ⓣ（赞美）（1981年，但显然后来丢失了）、Récoltes et Semailles Ⓣ（作物与种子）（1983-85年）和La clef des songes Ⓣ（梦之钥）（1986年）。1991年8月，他突然离家，没有通知任何人，去了一个未知的地方。在那里，他花时间写了一部极其庞大的物理学著作，以及关于自由选择、决定论和恶的存在等主题的哲学沉思。他几乎拒绝了一切人际接触。

关于他这一时期生活的更多信息见THIS LINK。

让我们以格罗滕迪克与在班戈的Ronnie Brown的通信中的一些引文作为结束。Ronnie写道：-

他可能完全沉浸在数学思想中。一封信中提到他正在思考一种新的形式理论，并且不得不提醒自己要吃饭和睡觉。他对细节和小事物有强烈的兴趣。他反对‘le snobisme’，因此对Henry Whitehead的评论感到高兴：‘年轻人的势利在于认为一个定理是平凡的，因为证明是平凡的。’他写道，数学不仅在于做困难的事情，还在于提供框架使困难的事情变得容易（从而为困难任务提供新机会！）。

格罗滕迪克写道（1982年）：-

密码0或群概念的引入也是普遍的胡说，数学几千年来或多或少停滞不前，因为周围没有人采取这种幼稚的步骤……

格罗滕迪克关于数学中思辨的想法见THIS LINK。

以下是格罗滕迪克在这段通信中的另外两段引文（1983年）：-

你提出的“这样的表述如何能导致计算”这个问题一点也不困扰我！在我作为数学家的整个生涯中，做出明确、优雅的计算的可能性总是自然而然地出现，作为对所发生之事彻底概念理解的副产品。因此，我从不费心考虑所得结果是否适合这个或那个，而只是试图去理解——结果总是表明，理解才是最重要的。

[通常解决问题的途径是]将新概念从黑暗中带出来。

Alexander Grothendieck's first name is often written as Alexandre, the form he adopted when living in France. His parents were Alexander Schapiro (1890-1942) and Johanna Grothendieck (1900-1957). His father was known by the standard Russian name of Sascha (for Alexander) while his mother was called Hanka. In order to get some understanding of Grothendieck's upbringing and personality, it is necessary to give some details about his parents, and in particular explain why he has his mother's name of Grothendieck. Alexander Schapiro was a Russian Jew, born in Novozybkov, a town near the point where Russia, White Russia (now Belarus), and Ukraine met. At the age of fifteen he fought for the revolutionaries against the Russian tsars in the war beginning in 1905. In 1907 he and his comrades were captured and sentenced to death. His comrades were all shot by firing squad but Schapiro, although led to be executed for 20 consecutive days, was eventually given life imprisonment because of his youth. He was ten years in prison before escaping during the Revolution of 1917. He continued to fight for the anarchists and was captured and escaped several times, on one occasion losing his left arm. He married a Jewish woman Rachil and they had a son named Dodek. Eventually, in 1921, he escaped to live for a while in Berlin and then in Paris under the name Alexander Tanaroff. He made a living as a street photographer. Grothendieck's mother, Hanka Grothendieck, was born in Hamburg, in Germany, and joined several left-wing groups. She married Alfred Raddatz (known as Johannes) and they had a daughter named Frode, known as Maidi, born in 1924. On a visit to Berlin, Alexander Schapiro met Hanka and they had a son, Alexander (known as Schurik), who is the subject of this biography. His birth was recorded under the name Alexander Raddatz since, at the time of his birth, his mother was marred to Alf Raddatz. Alf and Hanka were divorced in 1929. Alexander Schapiro (who still called himself Alexander Tanaroff) and Hanka lived with their son Schurik and his mother's daughter Maidi in Berlin from 1928 to 1933. There they had a photographic studio which provided the family income.

Adolf Hitler became Chancellor of the German Reich on 30 January 1933. On 1 April 1933 there was the so-called "boycott day" when Jewish shops and businesses were boycotted. On 7 April 1933 the Nazis passed a law which, under clause three, ordered the retirement of civil servants who were not of Aryan descent. Although Alexander Schapiro was hiding his Jewish origins by using the name Tanaroff, he still considered that Berlin was too dangerous a place for a Jew and he returned to Paris in May 1933. Hanka and her son Alexander remained in Berlin until December 1933 when she arranged for five year old Alexander to be fostered by the pastor Wilhelm Heydorn and his wife Dagmar who lived in Hamburg. She put her daughter Maidi into an institution in Berlin. Schurik, together with other foster children, lived with the Heydorns from the beginning of 1934 until April 1939. He attended elementary school and then began his studies at the Gymnasium. Meanwhile Hanka had joined Alexander's father in Paris and, after the outbreak of the Spanish Civil War (1936-39), they both went to Spain where they supported the Republicans. After the defeat of the Republicans in early 1939, Alexander and Hanka returned to France where Hanka began working in Nîmes. By the end of April 1939 the Heydorns, who were now in serious danger as part of the resistance against Hitler, considered that their home was too dangerous a place for young Schurik so, having located his parents in France, put him on a train to join his father. He spent the summer of 1939 with his mother in Nîmes.

The outbreak of World War II meant that Schurik and his parents were now in considerable danger in France. The law concerning 'undesirables' had been passed on 12 November 1938 which required all Germans living in France to be sent to special internment camps. Hanka and Schurik were interned in the Rieucros Camp near Mende. However, Schurik was allowed to continue his education at the village school about 5 km from the Camp. He also was given some private tutoring. Schurik's father, Alexander, was interned at the Camp du Vernet. In 1942 the Rieucros Camp was closed so Schurik and his mother were sent to the Gurs concentration camp near Pau. Schurik somehow managed to get to Le Chambon sur Lignon where he attended the famous Collège Cévénol. He would hide in the woods every time the authorities came round looking for Jews. He obtained his baccalauréat at the College in 1945. Meanwhile, in August 1942 his father had been handed over by the French Vichy government to the Nazis who took him from the Camp du Vernet to the Auschwitz extermination camp where he perished. Let us note at this point that Maidi survived the war and eventually emigrated to the United States.

In 1945 Schurik, who we will now start to call Grothendieck, and his mother moved to the village of Maisargues near Montpellier where Grothendieck worked in the vineyards and also, with the aid of a small scholarship, studied mathematics at the University of Montpellier. While at school he had felt dissatisfied with some of the mathematics that had been presented to him (see, for example [16]):-

What was least satisfying to me in our high school mathematics books was the absence of any serious definition of the notion of length of a curve, of area of a surface, of volume of a solid. I promised myself I would fill this gap when I had the chance.

He didn't find his professors at Montpellier much help in filling these gaps so he had to work on his own [8]:-

It was in Montpellier, during his undergraduate days, that he underwent his first real mathematical experience. He was very dissatisfied with the teaching he was receiving. He had been told how to compute the volume of a sphere or a pyramid, but no one had explained the definition of volume. It is an unmistakable sign of a mathematical spirit to want to replace the "how" with a "why". A professor of Grothendieck's assured him that a certain Lebesgue had resolved the last outstanding problems in mathematics, but that his work would be too difficult to teach. Alone, with almost no hints, Grothendieck rediscovered a very general version of the Lebesgue integral. The genesis of this first mathematical piece of work [was] accomplished in total isolation ...

Let us note that the professor at Montpellier who believed that Lebesgue had solved all outstanding mathematics problems was a certain M Soula. He had been taught by Élie Cartan and he advised Grothendieck to go to Paris and work with Cartan. Grothendieck followed that advice and, after graduating from Montpellier with his licence, he spent the year 1948-49 at the École Normale Supérieure in Paris. There he attended Henri Cartan's seminar which was on algebraic topology and sheaf theory. Grothendieck was now rubbing shoulders with the leading mathematicians of the day who were also attending Henri Cartan's seminar including Claude Chevalley, Jean Delsarte, Jean Dieudonné, Roger Godement, Laurent Schwartz, and André Weil. One of Grothendieck's fellow students was Jean-Pierre Serre. Since Grothendieck was at this time more interested in topological vector spaces than he was in algebraic topology, André Weil and Henri Cartan both advised him to go to Nancy where there was a strong team including Jean Dieudonné, Jean Delsarte, Roger Godement and Laurent Schwartz.

In 1949 Grothendieck moved to the University of Nancy where he lived with his mother who was occasionally bedridden due to tuberculosis contracted in the internment camps. At this time Grothendieck had a son named Serge with the lady from whom they rented rooms. Serge was brought up largely by his mother. Grothendieck worked on functional analysis with Dieudonné [16]:-

Grothendieck had very few books; rather than learning things by reading, he would try to reconstruct them on his own. And he worked very hard.

At Nancy there was an active seminar every Saturday in which all the professors and some of the students participated, which studied a variety of different topics. It provided a wonderful environment for the young Grothendieck. Dieudonné writes [9]:-

A general theory of duality for locally convex spaces had to be worked out: Schwartz and I had started its study for Fréchet spaces and their direct limits, but we had met a series of problems we could not solve. We therefore proposed them to Grothendieck, and the result turned out to exceed our most sanguine expectations. In less than a year, he had solved all our problems by very ingenious new constructions; then, with the techniques he had developed, he started to work on many other questions in functional analysis.

He presented his doctoral thesis Produits tensoriels topologiques et espaces nucléaires Ⓣ in 1953 and one of those present at his thesis defence on 28 February 1953 was Bernard Malgrange who [16]:-

... recalled that after Grothendieck wrote his thesis he asserted that he was no longer interested in topological vector spaces. "He told me, 'There is nothing more to do, the subject is dead'," Malgrange recalled. At that time, students were required to prepare a "second thesis", which did not contain original work but which was intended to demonstrate depth of understanding of another area of mathematics far removed from the thesis topic. Grothendieck's second thesis was on sheaf theory, and this work may have planted the seeds for his interest in algebraic geometry, where he was to do his greatest work. After Grothendieck's thesis defense, which took place in Paris, Malgrange recalled that he, Grothendieck, and Henri Cartan piled into a taxicab to go to lunch at the home of Laurent Schwartz. They took a cab because Malgrange had broken his leg skiing. "In the taxi Cartan explained to Grothendieck some wrong things Grothendieck had said about sheaf theory," Malgrange recalled.

Grothendieck spent the years 1953-55 at the University of São Paulo and then he spent the following year at the University of Kansas. However it was during this period that his research interests changed and they moved towards topology and geometry. In fact during this period Grothendieck had been supported by the Centre National de la Recherche Scientifique, the support beginning in 1950. After leaving Kansas in 1956 he therefore returned to the Centre National de la Recherche Scientifique. Around this time, he became one of the Bourbaki group of mathematicians which included André Weil, Henri Cartan and Jean Dieudonné. However in 1959 he was offered a research position in the newly formed Institut des Hautes Études Scientifiques which he accepted. In [2] the next period in Grothendieck's career is described as follows:-

It is no exaggeration to speak of Grothendieck's years 1959-70 at the IHES as a 'Golden Age', during which a whole new school of mathematics flourished under Grothendieck's charismatic leadership. Grothendieck's Séminaire de Géométrie Algébrique Ⓣ established the IHES as a world centre of algebraic geometry, and him as its driving force.

During this period Grothendieck's work provided unifying themes in geometry, number theory, topology and complex analysis. He introduced the theory of schemes in the 1960s which allowed certain of Weil's number theory conjectures to be solved. He worked on the theory of topoi which are highly relevant to mathematical logic. He gave an algebraic proof of the Riemann-Roch theorem. He provided an algebraic definition of the fundamental group of a curve.

Again quoting from [2]:-

The mere enumeration of Grothendieck's best known contributions is overwhelming: topological tensor products and nuclear spaces, sheaf cohomology as derived functors, schemes, K-theory and Grothendieck-Riemann-Roch, the emphasis on working relative to a base, defining and constructing geometric objects via the functors they are to represent, fibred categories and descent, stacks, Grothendieck topologies (sites) and topoi, derived categories, formalisms of local and global duality (the 'six operations'), étale cohomology and the cohomological interpretation of L-functions, crystalline cohomology, 'standard conjectures', motives and the 'yoga of weights', tensor categories and motivic Galois groups. It is difficult to imagine that they all sprang from a single mind.

In the late 1950s, Grothendieck lived with Mireille, with whom he had three children: Johanna, Mathieu, and Alexandre. He married Mireille a few years later. Luc Illusie began attending Grothendieck's seminar at the Institut des Hautes Études Scientifiques in 1964. He writes [13]:-

He spoke with great energy at the board but taking care to recall all the necessary material. He was very precise. The presentation was so neat that even I, who knew nothing of the topic, could understand the formal structure. It was going fast but so clearly that I could take notes.

Illusie goes on to describe visiting Grothendieck to discuss his work [13]:-

At the time he lived at Bures-sur-Yvette, rue de Moulon, in a little white pavilion, with a ground floor and one storey. His office there was austere, and cold in the winter. He had a portrait of his father in pencil, and also on the table there was the mortuary mask of his mother. Behind his desk he had filing cabinets. When he wanted some document, he would just turn back, and find it in no time. He was well organized.

Valentin Poénaru also knew Grothendieck during these years. He writes [21]:-

The Grothendieck I knew at this time was a very impressive person, and when I say this I am not thinking only of mathematics. Shourik, as I called him, was one of the strongest and most charismatic people I have ever met. I think of him as a character straight out of Dostoyevsky. He was also a person of great kindness and generosity. He seemed always to be in good spirits, with great mental equilibrium and also, in his own way, a certain joie de vivre. At the time, he had the capacity to be able to sleep when he wanted to, and for the number of hours he wanted to, in order to take up his work all the better afterward. In fact, his capacity for work was to me something miraculous.

The authors of [2] write:-

He received the Fields Medal in 1966. In looking back at this period, one marvels at the generosity with which Grothendieck shared his ideas with colleagues and students, the energy he and his collaborators devoted to meticulous redaction, the excitement with which they set out to explore a new land.

The citation for the Fields Medal reads:-

Grothendieck built on work of Weil and Zariski and effected fundamental advances in algebraic topology. He introduced the idea of K-theory (the Grothendieck groups and rings). He revolutionised homological algebra in his celebrated "Tohoko paper".

The "Tohoko paper" referred to in this citation is about abelian categories, sheaves of modules, resolutions, derived functors, and the Grothendieck spectral sequence. However, there were certain difficulties in making the presentation of the Fields Medal since the 1966 International Congress of Mathematicians at which the presentation was to be made was held in Moscow in August 1966. Now Grothendieck was always strongly pacifist in his views and he had campaigned against the military built-up of the 1960s. As a political protest, he refused to travel to Moscow to receive the Fields medal. At the Congress, Léon Motchane, director of IHES, received the Fields Medal on Grothendieck's behalf. Grothendieck made no public statement about the reasons for not going to Moscow but he declared himself a citizen of the world and requested United Nations citizenship. In November and December of 1967 he visited North Vietnam which, at that time, was being bombed by the Americans [18]:-

Grothendieck's first lectures - which he describes as "general orientation talks" - were given in Hanoi. But because of intensified bombing of the capital, a high-level decision was made to move everyone to the secret location of the Faculty of Mathematics of Hanoi University. Grothendieck writes: "I then spent a week and a half at Hanoi University in evacuation outside the city (about 100 km from the capital); this time was largely devoted to a more specialized seminar on categories and homological algebra, with thirty to forty listeners, most of whom had followed me from Hanoi after attending the general orientation lectures." It was a remarkable event in the history of mathematics: one of the giants of 20th century mathematics delivering a short course on homological algebra in a remote forest hideout in a desperately poor country that was being "bombed back into the stone age" (U.S. Air Force General Curtis Le May's phrase) by the most powerful military force the world had ever known.

He left the IHES in 1970 after he discovered that some of their funding came from military sources. He discovered this in 1969 and, along with the other professors at the IHES, he persuaded the director, Léon Motchane, to take no further funding from the French military. However, when a few months later the IHES budget was very tight, the director went back on his word. Grothendieck tried to persuade all the professors to resign in protest but the others refused to follow his example. Grothendieck's letter of resignation was dated 25 May 1970. However, Grothendieck had other problems for he wrote that at this time he was suffering a "spiritual stagnation". He abandoned mathematics as the main focus of his energies and turned to political protest, particularly against nuclear proliferation. However, in contrast to the amazing impact of his mathematical work, his political campaigns were rather ineffective. In 1970-72 he held an appointment as visiting professor at the Collège de France, then a similar appointment at Orsay for 1972-73. In 1973 he accepted an appointment as professor at the University of Montpellier. He lectured and has some graduate students in Montpellier. He lived in Villecun near Lodeve from 1973 to 1980, then he moved to live in Mormoiron near Carpentras. He took leave during 1984-88 to direct research at the Centre National de la Recherche Scientifique. He retired at age 60 in 1988 and the publication [2] was produced to honour that 60th birthday. In contrast to his acceptance of the 1966 Fields Medal, Grothendieck declined the Crafoord Prize in 1988.

Between 1980 and 1990, Grothendieck wrote literally thousands of pages some containing his mathematical thoughts and others containing non-mathematical meditations. His mathematical manuscripts are La longue marche à travers la théorie de Galois Ⓣ (1981), A la poursuite des champs Ⓣ (1983), Esquisse d'un programme Ⓣ (1983), and Les dérivateurs Ⓣ (1987), while his non-mathematical works are the Eloge Ⓣ (1981, but apparently subsequently lost), Récoltes et Semailles Ⓣ (1983-85), and La clef des songes Ⓣ (1986). In August 1991 he left home suddenly, without informing anyone, for an unknown location. There he spent his time writing an extremely large work on physics as well as philosophical meditations on themes such as free choice, determinism and the existence of evil. He refused practically every human contact.

More about his life during this period is at THIS LINK.

Let us end with some quotations from Grothendieck's correspondence with Ronnie Brown at Bangor. Ronnie writes:-

He could be totally absorbed in mathematical ideas. One letter remarks that he is thinking of a new theory of form, and has to remind himself to eat and to sleep. He had a strong interest in detail and small things. He was against 'le snobisme', and so was delighted with a comment of Henry Whitehead: 'It is the snobbishness of the young to suppose that a theorem is trivial because the proof is trivial.' He wrote that mathematics is not only about doing difficult things, but also providing the framework to make difficult things easy (thus giving new opportunities for difficult tasks!).

Grothendieck writes (in 1982):-

The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps ...

For Grothendieck's ideas on speculation in mathematics see THIS LINK.

Here are two further quotes by Grothendieck from this correspondence (in 1983):-

The question you raise "how can such a formulation lead to computations" doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered.

[Often the route to solving problems is] to bring new concepts out of the dark.