安德烈·韦伊André Weil

安德烈·韦伊出生于巴黎，是犹太父母之子。他的母亲Salomea Reinherz（1879-1965），被称为Selma，来自一个移民到奥地利的俄罗斯犹太家庭，而他的父亲Bernard Bernhard Weil（1872-1955）是一名医生，其家庭曾住在阿尔萨斯的斯特拉斯堡。

关于Bernard和Selma 安德烈·韦伊的更多细节在Weil家族中给出。

其中也给出了安德烈·韦伊的妹妹Simone Adolphine 安德烈·韦伊（1909-1943）的详细情况。我们应当提到，安德烈·韦伊家族最初使用Weill这一拼写，例如安德烈·韦伊的祖父就是马克斯·亚伯拉罕 Weill。安德烈·韦伊家族来自阿尔萨斯，有权选择法国国籍，他们这样做了并搬到了巴黎。Bernard和Selma 安德烈·韦伊于1905年在巴黎结婚。次年安德烈·韦伊出生时，除了他的父母，他的祖母和几位叔伯住在巴黎，但他的祖父亚伯拉罕 Weill已在斯特拉斯堡去世。安德烈·韦伊的外祖母Hermine Reinherz是一位出色的钢琴家，与安德烈·韦伊一家住在他的家中，直到1912年，他们住在斯特拉斯堡大道。

安德烈·韦伊的母亲在他生命的最初几年监督他的教育，他在4到5岁之间学会了阅读。1912年他六岁时[6]：-

……我的母亲选择了一位杰出的小学教师，Chaintreuil小姐，她在蒙田中学教第十班[第二班]。经过几个月的辅导，她认为我能够加入她在中学的班级，尽管我年纪还稍小。

在中学的第一年之后，全家去了瑞士的Ballaigues度暑假。在安德烈·韦伊开始中学第二年之前，全家从斯特拉斯堡大道搬到了圣米歇尔大道。由于第一年表现如此出色，他跳了一级，被安排到M Monbeig教的最高班[6]：-

他是一位杰出的教师，充满非传统的想法。

1914年暑假期间，第一次世界大战爆发。安德烈·韦伊的父亲成为一名军医，照料受伤的士兵。他在多家军队医院工作，而安德烈·韦伊、他的母亲和妹妹（通常还有他的祖母）总是搬家以便与Bernard Weil在一起。安德烈·韦伊自幼便爱上了数学，他写道，到十岁时他已狂热地沉迷于数学[6]：-

有一次我重重地摔了一跤，我妹妹Simone想不出别的办法，只能跑去拿来我的代数书来安慰我。

当Bernard被派往阿尔及利亚时，家人没有随他同去，而是去了沙特尔，安德烈·韦伊在那里上了中学。到1917年10月，他的父亲已返回法国，全家搬到了拉瓦勒。次年，他在家中接受私人辅导，之后全家返回巴黎的住所。在进入圣路易中学M Collin的班级之前，他再次接受了私人辅导。正是在这所学校里，他学习了自学时遗漏的数学内容。然而，除了数学，他的生活中还有其他重要的事情，因为他热爱旅行。到十六岁时，他已读过梵文原版的《薄伽梵歌》。他自学了古典希腊语，用希腊语阅读荷马和柏拉图，还自学了拉丁语。用餐时，家人常用德语和英语交谈。他也热爱欧洲文学、艺术和音乐。1921年，他在圣路易中学的最后一年，遇到了雅克·阿达马，从那时起雅克·阿达马给了他很好的建议。每年安德烈·韦伊都赢得数学奖，并（在雅克·阿达马的建议下）为自己挑选奖品书籍。他选择了卡米耶·若尔当的Cours d'Analyse三卷本以及William Thomson和彼得·格思里·泰特的两卷本Treatise of Natural Philosophy。当时相对论是一个令人兴奋的新话题，他读了亚瑟·爱丁顿对“阿尔伯特·爱因斯坦理论”的描述。他于1922年从圣路易中学毕业，同年晚些时候，安德烈·韦伊进入巴黎高等师范学校。安德烈·韦伊写道[6]：-

在“高师”，我们过去常这样称呼它，学生们被分成小组，共用宿舍。我首先关心的，甚至在学校开学之前，就是找到合得来的学习伙伴。我们共有五个人……

另外四人是伊夫·罗卡尔、让·戴尔萨特、Paul Labérenne（1902-1985）和Jean Barbotte。从进入巴黎高等师范学校起，安德烈·韦伊就参加了法兰西公学院雅克·阿达马的讨论班。他在这个讨论班上作了关于多复变幂级数收敛域的演讲。他参加的数学课程包括昂利·勒贝格和埃米尔·皮卡的课程。然而，他继续保持着数学之外的兴趣，并在索邦大学选修了梵文课程。他于1925年毕业，尽管在理性力学考试中交了一张白卷（他认为理性力学不属于数学），但仍名列班级第一。

毕业后，他利用暑假在法国阿尔卑斯山徒步旅行，总是随身带着一个笔记本，在上面进行数学计算。此时，他特别着迷于求解丢番图方程。暑假过后，他去了罗马，靠索邦大学的奖学金在那里待了六个月。他听了维多·沃尔泰拉和弗朗切斯科·塞维里的讲座，并作了一场关于路易斯·乔尔·莫德尔猜想的报告。然而，他当然没有把所有时间都花在数学上，因为他趁机研究了意大利绘画。洛克菲勒基金会奖学金资助他访问哥廷根，他在那里度过了1927年的大部分时间，并完成了他的第一项实质性数学研究，关于代数曲线理论。在哥廷根，他遇到了理查·科朗特、埃米·诺特等人，从与他们的讨论中获益。

随后，他在巴黎大学攻读博士学位，由雅克·阿达马指导。他在学位论文中发展了他在哥廷根开始研究的代数曲线理论的思想。然而，雅克·阿达马希望他这位才华横溢的学生志存高远，尝试证明路易斯·乔尔·莫德尔猜想。安德烈·韦伊选择不听从导师的建议。他后来写道：-

我的决定是明智的：证明路易斯·乔尔·莫德尔的猜想花了半个多世纪。

他于1928年在巴黎获得博士学位，学位论文为Arithmétique des courbes algébriques Ⓣ（代数曲线的算术）。当时法国实行义务兵役制，因此安德烈·韦伊在1928-29年服兵役，退役时军衔为中尉。之后他在多所大学任教，例如1930年至1932年在印度阿利加尔穆斯林大学。他曾先与海得拉巴教育部长Syed Masood商谈，获得阿利加尔大学法国文明讲席的任命，但尽管有此承诺，他却收到Syed Masood的电报：-

无法设立法国文明讲席。数学讲席空缺。

他利用一切机会充分利用这些年[11]：-

他竭尽全力抓住机会沉浸于印度的方方面面：文化、宗教、文学、人民、历史、风景、考古等等，四处旅行，常常在原始的条件下。

在印度待了两年后回到法国，他于1933年至第二次世界大战爆发期间在斯特拉斯堡大学工作。亨利·嘉当当时也在斯特拉斯堡任职，两人经常讨论教学。正是在这里，他参与了以尼古拉·布尔巴基为笔名写作的著名数学家团体。亨利·嘉当描述了这一想法是如何产生的[4]：-

安德烈·韦伊和我在1934年都在斯特拉斯堡大学。我经常和他谈论我教的微分和积分微积分课程。……我常常思考教这门课的最佳方式，因为现有的教科书并不令人满意……我多次与安德烈·韦伊讨论我的担忧。有一天，他告诉我：“我受够了，我们需要彻底解决这个问题。我们需要写一本好的分析教科书。然后你就不会再抱怨了！”

我们在下面给出更多关于尼古拉·布尔巴基合作的细节。

尼古拉·布尔巴基的创始人之一是René de Possel。勒内与伊芙琳结婚，安德烈·韦伊在尼古拉·布尔巴基成立时遇到了她。勒内和伊芙琳离婚了，在等待离婚手续办妥相当长一段时间后，安德烈·韦伊于1937年10月30日与伊芙琳·德·波塞尔结婚。他们有两个女儿，西尔维（1942年9月12日出生）和妮科莱特（1946年12月6日出生）。

战争对安德烈·韦伊来说是一场灾难，他在敌对行动爆发前就决定通过去美国来避免服兵役。然而，当战争宣布时，他正在芬兰，拜访罗尔夫·内万林纳和拉尔斯·阿尔福斯。他不想回到法国以避免被迫参军，但在当时从欧洲的战争中逃脱并不是一件简单的事。安德烈·韦伊于1939年11月在芬兰被捕，当在他的房间里发现俄语信件时（它们实际上是来自列夫·庞特里亚金描述数学研究的信），情况看起来相当糟糕。安德烈·韦伊自己写道：-

他们发现的手稿显得可疑——就像索菲斯·李的手稿一样，他于1870年在巴黎因间谍罪被捕。他们还在壁橱底部发现了几卷速记纸。当我说这些是巴尔扎克小说的文本时，这个解释一定显得牵强。还有一封俄语信，我相信是来自列夫·庞特里亚金，回复我在初夏写的一封关于可能访问列宁格勒的信；以及一包属于尼古拉·布尔巴基的名片，他是波尔达维亚皇家科学院院士……

有一天，罗尔夫·内万林纳被告知他们即将以间谍罪处决安德烈·韦伊，他说服当局改为驱逐安德烈·韦伊。他于1939年12月12日从监狱获释，先被送往瑞典，然后送往英国，最后被送回法国，在那里被关进监狱。埃米尔·博雷尔写道[12]：-

他在监狱中的条件，起初有些艰苦，逐渐改善：他可以与家人通信，偶尔见面，与姐姐有活跃的通信，并能收到一些书和工作。那时，他证明了他最著名的结果之一，“有限域上曲线的波恩哈德·黎曼假设”。

安德烈·韦伊在鲁昂监狱中写的一封信现存于THIS LINK。

安德烈·韦伊当时无疑处于极大的危险之中，部分因为他是犹太人，部分因为他有一个妹妹Simone 安德烈·韦伊，是一位神秘主义哲学家和法国抵抗运动的领导人。这种困境的危险使安德烈·韦伊认定参军是更好的选择，他成功地争取到了释放，条件是他确实参军。1940年5月3日，他在鲁昂受审。埃利·嘉当前往鲁昂在审判中为他作证。他被释放出狱，成为一名陆军列兵。既然利用参军作为出狱的理由，安德烈·韦伊就无意尽可能久地服役。一有机会逃往美国，他立刻抓住，于1941年1月与妻子和父母一同前往。在美国，得到洛克菲勒基金会资助，他去了宾夕法尼亚州，从1941年起在哈弗福德学院任教，之后在利哈伊大学任教。1945年，他接受了巴西圣保罗大学的一个职位，在那里待到1947年。1947年，安德烈·韦伊回到美国，被任命为芝加哥大学教员，这一职位他一直担任到1958年。陈省身写道[14]：-

在Stone时期，我们成为芝加哥大学的同事。在Stone的领导下，芝加哥成为一个活跃的数学中心，拥有优秀的学生。我们经常接触，沿着密歇根湖南岸长时间散步，那时那里还很安全。

从1958年起，他在普林斯顿大学高等爱德华·斯图迪研究所工作。他于1976年退休，当时成为荣休教授。

安德烈·韦伊的研究方向是数论、代数几何和群论。他的工作概述于[55]：-

从20世纪40年代开始，安德烈·韦伊通过为抽象代数几何和现代阿贝尔簇理论奠定基础，开启了代数几何和数论的迅速发展。他在代数曲线方面的工作影响了广泛的领域，包括数学之外的一些领域，如基本粒子物理学和弦理论。

事实上，安德烈·韦伊在这一领域的工作是丘成桐等数学家工作的基础，丘成桐因在三维代数几何方面的工作于1982年被授予约翰·查尔斯·菲尔兹奖章，该工作对量子场论有重大应用。丘成桐并不是唯一因继续安德烈·韦伊开创的工作而获得菲尔兹奖的数学家。1978年，皮埃尔·德利涅因解决安德烈·韦伊猜想而被授予约翰·查尔斯·菲尔兹奖章。我们再次引用[55]来描述安德烈·韦伊的基本贡献：-

安德烈·韦伊的主要成就之一是他证明了代数函数域的同余zeta函数的波恩哈德·黎曼假设。1949年，他提出了关于有限域上代数簇的同余ζ函数的某些猜想。这些后来被称为安德烈·韦伊猜想的猜想，源于他对代数簇拓扑学的深刻洞察，并为该领域的后续发展提供了指导原则。

安德烈·韦伊将数论与代数几何结合起来的工作成果极为丰硕。当今深入研究许多课题的基础都是由安德烈·韦伊在这项工作中奠定的，例如模形式理论、自守函数和自守表示理论的基础。然而，安德烈·韦伊的工作在许多其他新的数学课题中也具有重大重要性。他对拓扑学、微分几何和复解析几何做出了实质性贡献。他不仅对这些领域做出了贡献，更重要的是，当他研究拓扑群上的调和分析和示性类时，他的工作揭示了这些领域之间的基本关系。他的关于theta函数的几何理论和埃里希·凯勒几何的工作也将这些领域结合在了一起。

安德烈·韦伊与让·迪厄多内等人一起以尼古拉·布尔巴基的名义写作，这是他们在20世纪30年代开始的一个项目，旨在对数学进行统一的描述。其目的是扭转他们不喜欢的一种趋势，即数学中缺乏严谨性。尼古拉·布尔巴基的影响多年来一直很大，但现在已不那么重要，因为它基本上成功地实现了促进严谨性和抽象性的目标。

安德烈·韦伊通过他的著作做出了重大贡献，这些著作包括Arithmétique et géométrie sur les variétés algébriques Ⓣ（代数簇的算术与几何）（1935年）、Sur les espaces à structure uniforme et sur la topologie générale Ⓣ（论一致结构空间与一般拓扑学）（1937年）、L'intégration dans les groupes topologiques et ses applications Ⓣ（拓扑群中的积分及其应用）（1940年）、Foundations of Algebraic Geometry（1946年）、 Sur les courbes algébriques et les variétés qui s'en déduisent Ⓣ（论代数曲线及由它们导出的簇）（1948年）、Variétés abéliennes et courbes algébriques Ⓣ（阿贝尔簇与代数曲线）（1948年）、Introduction à l'étude des variétés kählériennes Ⓣ（Kähler簇研究导论）（1958年）、Discontinuous subgroups of classical groups（1958年）、Adeles and algebraic groups （1961年）、Basic number theory（1967年）、Dirichlet Series and Automorphic Forms（1971年）、Essais historiques sur la théorie des nombres Ⓣ（数论史论集）（1975年）、Elliptic Functions According to Eisenstein and Kronecker（1976年）、（与Maxwell Rosenlicht合著）Number Theory for Beginners（1979年）、Adeles and Algebraic Groups（1982年）、Number Theory: An Approach Through History From Hammurapi to Legendre（1984年）以及Correspondance entre Henri Cartan et André Weil Ⓣ（亨利·嘉当与安德烈·韦伊之间的通信）（1928-1991年）（2011年）。

你可以在THIS LINK看到安德烈·韦伊的Algebraic Geometry的序言。

关于其中一些著作的书评摘录，见THIS LINK。

关于安德烈·韦伊两本数学史著作的一些摘录，见THIS LINK。

关于安德烈·韦伊对数学教学和数学未来的思考的一些摘录，见THIS LINK。

安德烈·韦伊因其杰出的数学成就获得了许多荣誉。其中包括1959年成为伦敦数学会的荣誉会员，以及1966年当选为伦敦皇家学会的会士。此外，他还当选为巴黎的科学院和美国国家科学院的成员。他拒绝接受荣誉博士学位，这就是为什么我们没有可列出的荣誉学位。

安德烈·韦伊是1950年在哈佛举行的国际数学家大会的受邀演讲者，当时他作了关于Number Theory and Algebraic Geometry的报告，并在1954年阿姆斯特丹举行的下一届国际数学家大会上再次受邀，作了Abstract versus Classical Algebraic Geometry的演讲。1979年，安德烈·韦伊被授予沃尔夫奖，次年，American Mathematical Society授予他凯瑟琳·斯蒂尔奖。1994年，他获得了日本稻盛财团颁发的京都奖：-

……以表彰杰出的成就和创造力。

京都奖的颁奖词写道：-

安德烈·韦伊通过对整个数学科学的深刻理解和敏锐洞察所取得的成果和提出的问题，将继续对数学科学的发展产生不可估量的影响，并极大地促进科学的发展，以及人类精神的深化与提升。

他在[49]中被志村五郎描述如下：-

然而，在我看来，他主要仍是一个具有两个相互关联特征的人物：首先，他灵活且乐于接受他人的新想法和新方向，这与当今许多只能在既定框架内工作的年轻人截然不同。其次，更重要的是，与此类似，他对数学有着深刻而透彻的理解，或者更确切地说，他不知疲倦地努力理解每一个基本数学现象的真正意义，并以更清晰的形式和更好的视角呈现它。他通过赋予每个主题新概念和建立新框架来做到这一点，始终以新颖而根本的方式。换句话说，他不仅仅是一个解决问题的人。

Komaravolu Chandrasekharan在[14]中写道：-

他以脾气急躁和突然的挑衅性干预而闻名，这有时会导致激烈的对抗。那是他性格中不太讨人喜欢的一面。正是在他的著作中，他的个性真正展现出来——作为风格大师，拥有深厚的阅读、反思和自我审视储备，以及与创造性想象力的热线联系。

André Weil was born in Paris, the son of Jewish parents. His mother, Salomea Reinherz (1879-1965) known as Selma, came from a family of Russian Jews who had emigrated to Austria, while his father, Bernard Bernhard Weil (1872-1955), was a medical doctor whose family had lived in Strasbourg, Alsace.

More details of Bernard and Selma Weil are given at THIS LINK.

where details are also given of André's sister Simone Adolphine Weil (1909-1943). We should mention that the Weil family originally used the spelling Weill, for example André's grandfather was Abraham Weill. The Weil family being from Alsace had the right to opt for French nationality and they had done this and moved to Paris. Bernard and Selma Weil were married in Paris in 1905. When André was born in the following year, in addition to his parents, his paternal grandmother and several uncles were living in Paris but his paternal grandfather Abraham Weill had died in Strasbourg. André's maternal grandmother Hermine Reinherz, who was an excellent pianist, lived in his family home with the Weil family which, until 1912, was on the Boulevard de Strasbourg.

André's mother supervised his education for the first few years of his life and he had learned to read between the ages of 4 and 5. When he was six years old in 1912 [6]:-

... my mother chose an exceptional elementary teacher, Mademoiselle Chaintreuil, who taught the tenth form [the second class] at the Lycée Montaigne. After several months of tutoring, she deemed me capable, even though I was a little too young, of joining her class at the lycée.

After this first year at the lycée, the family went to Ballaigues in Switzerland for their summer holidays. The family moved from the Boulevard de Strasbourg to the Boulevard Saint-Michel before André began his second year at the lycée. Having done so well in this first year, he missed out a form and was put into the top section taught by M Monbeig [6]:-

He was an exceptional teacher, full of unconventional ideas.

During the summer vacation of 1914 World War I broke out. André's father became an army doctor tending the wounded soldiers. He worked in a variety of military hospitals and André, his mother and sister (and usually his grandmother), always moved to be with Bernard Weil. André fell in love with mathematics at an early age, and he writes that by the age of ten he was passionately addicted to it [6]:-

Once when I took a painful fall, my sister Simone could think of nothing for it but to run and fetch my algebra book, to comfort me.

When Bernard was sent to Algeria the family didn't go with him but went to Chartes where André attended the lycée. By October 1917 when his father had returned to France, the family moved to Laval. In the following year he was tutored privately before the family returned to their home in Paris. Again he received private tutoring before entering M Collin's class at the Lycée Saint-Louis. It was in this school that studied the parts of mathematics that he had missed out in the studies he had made on his own. There were other things of importance in his life as well as mathematics, however, for he loved to travel. By the age of sixteen he had read the 'Bhagavad Gita' in the original Sanskrit. He had taught himself classical Greek, read Homer and Plato in Greek, and had also taught himself Latin. At meal times the family often conversed in German and English. He also loved European literature, art and music. In 1921, his final year at the Lycée Saint-Louis, he met Jacques Hadamard who gave him good advice from that time on. Every year Weil won the mathematics prize and chose himself (with Hadamard's advice) books for his prize. He chose the three volumes of Camille Jordan's Cours d'Analyse as well as William Thomson and Peter Guthrie Tait's two-volume Treatise of Natural Philosophy. At this time relativity was an exciting new topic and he read Arthur Eddington's description of "Einstein's theory". He graduated from the Lycée Saint-Louis in 1922 and, later that year, Weil entered the École Normale Supérieure in Paris. Weil writes [6]:-

At the "École", as we used to call it, the students were divided into groups sharing quarters. My first concern, even before school started, was to find companionable study mates. There were five of us ...

The four others were Yves Rocard, Jean Delsarte, Paul Labérenne (1902-1985), and Jean Barbotte. Right from the time he entered the École Normale Supérieure, Weil attended Hadamard's seminar at the Collège de France. He gave a talk to this seminar on domains of convergence of power series in several complex variables. Among the mathematics courses he attended were those of Henri-Léon Lebesgue and Charles-Émile Picard. However, he continued to have interests outside mathematics and took a course in Sanskrit at the Sorbonne. He graduated in 1925 being ranked first in the class despite returning a blank paper for rational mechanics which he did not consider to be part of mathematics.

After graduating he spent the summer vacation walking in the French Alps, always taking a notebook with him in which he made his mathematical calculations. At this time he was particularly fascinated by solving Diophantine equations. After the summer vacation he went to Rome where he spent six months supported by a scholarship from the Sorbonne. He attended lectures by Vito Volterra and Francesco Severi, and gave a lecture on the Mordell conjecture. However, he certainly didn't devote all his time to mathematics for he took the opportunity to study Italian painting. A Rockefeller Foundation fellowship funded a visit to Göttingen where he spent most of 1927 and produced his first substantial piece of mathematical research on the theory of algebraic curves. In Göttingen he met Richard Courant, Emmy Noether and others, profiting from discussions with them.

He then undertook research for his doctorate in the University of Paris, supervised by Jacques Hadamard. He developed for his thesis the ideas on the theory of algebraic curves which he had begun to study at Göttingen. However, Hadamard wanted his brilliant student to aim higher and try to prove the Mordell Conjecture. Weil chose not to follow his supervisor's advice. He wrote later:-

My decision was a wise one: it was to take more than half a century to prove Mordell's Conjecture.

He received his doctorate from Paris in 1928 for his thesis Arithmétique des courbes algébriques Ⓣ. At this time military service was compulsory in France, so Weil undertook these duties in the year 1928-29, leaving with the rank of lieutenant. He then taught at different universities, for example the Aligarh Muslim University in India from 1930 to 1932. He had first discussed with Syed Masood, the Minister of Education for Hyderabad, obtaining an appointment to a chair in French Civilization at Aligarh University but, despite the promise, he received a telegram from Syed Masood:-

Impossible to create chair of French civilisation. Mathematics chair open.

He took every opportunity to make the most of these years [11]:-

He used to the hilt the opportunity to immerse himself in all aspects of India: culture, religion, literature, people, history, scenery, archaeology, and so on, travelling all over, often under primitive conditions.

Returning to France after the two years in India, he worked at the University of Strasbourg from 1933 until the outbreak of World War II. Henri Cartan was on the staff at Strasbourg at this time and the two often discussed teaching. It was here that he became involved with the famous group of mathematicians writing under the name Nicolas Bourbaki. Henri Cartan described how the idea came about [4]:-

André Weil and I were both at the University of Strasbourg in 1934. I often talked with him about the course on differential and integral calculus that I was teaching. ... I often wondered about the best way to teach this course because the existing textbooks were not satisfactory ... I discussed my concerns several times with André Weil. One beautiful day he told me, "I've had it, we need to fix this for good. We need to write a good textbook an analysis. Then you'll stop complaining!"

We give more details of the Bourbaki collaboration below.

One of the founders of Bourbaki was René de Possel. René was married to Eveline and Weil met her when Bourbaki was being set up. René and Eveline were divorced and, after waiting a considerable time for the divorce to come through, Weil married Eveline de Possel on 30 October 1937. They had two daughters, Sylvie (born 12 September 1942) and Nicolette (born 6 December 1946).

The war was a disaster for Weil who had decided before hostilities broke out that he would avoid military service by going to the United States. However, he was in Finland, visiting Rolf Nevanlinna and Lars Ahlfors, when war was declared. He didn't want to return to France to avoid being forced into the army, but it was not a simple matter to escape from the war in Europe at this time. Weil was arrested in Finland in November 1939 and when letters in Russian were found in his room (they were actually from Pontryagin describing mathematical research) things looked pretty black. Weil himself wrote:-

The manuscripts they found appeared suspicious - like those of Sophus Lie, arrested on charges of spying in Paris, in 1870. They also found several rolls of stenotypewritten paper at the bottom of a closet. When I said these were the text of a Balzac novel, the explanation must have seemed far-fetched. There was also a letter in Russian, from Pontryagin, I believe, in response to a letter I had written at the beginning of the summer regarding a possible visit to Leningrad; and a packet of calling cards belonging to Nicolas Bourbaki, member of the Royal Academy of Poldavia ...

One day Nevanlinna was told that they were about to execute Weil as a spy, and he was able to persuade the authorities to deport Weil instead. He was released from prison on 12 December 1939 and he was sent first to Sweden, then to England before finally being sent back to France where he was put in prison. Borel writes [12]:-

His conditions in prison, at first somewhat hard, gradually improved: he could communicate with, and occasionally see, his family, had a lively correspondence with his sister, and could receive some books and work. At that time, he proved one of his most famous results, the "Riemann hypothesis for curves over finite fields."

A letter that Weil wrote while in prison at Rouen is at THIS LINK.

Weil was certainly in great danger at this time, partly because he was Jewish, partly because he had a sister Simone Weil who was a mystic philosopher and a leading figure in the French Resistance. The dangers of his predicament made Weil decide that being in the army was a better bet and he was able to argue successfully for his release on the condition that indeed he did join the army. On 3 May 1940 he was tried in Rouen. Élie Cartan went to Rouen to testify in his favour at his trial. He was released from prison and became an army private. Having used the army as a reason to get out of prison, Weil had no intention of serving any longer than he possibly could. As soon as the chance to escape to the United States came, he took it at once travelling there with his wife and parents in January 1941. In the United States, supported by the Rockefeller Foundation, he went to Pennsylvania where he taught from 1941 at Haverford College and then at Lehigh University. In 1945 he accepted a position in São Paulo University, Brazil, where he remained until 1947. In 1947 Weil returned to the United States and he was appointed to the faculty of the University of Chicago, a position he continued to hold until 1958. Shiing-Shen Chern writes [14]:-

We became colleagues at the University of Chicago during the Stone period. Under Stone's leadership Chicago became an active mathematical centre with excellent students. We had constant contact and took long walks along the south coast of Lake Michigan when it was still safe.

From 1958 he worked at the Institute for Advanced Study at Princeton University. He retired in 1976, becoming Professor Emeritus at that time.

Weil's research was in number theory, algebraic geometry and group theory. His work is summarised in [55]:-

Beginning in the 1940s, Weil started the rapid advance of algebraic geometry and number theory by laying the foundations for abstract algebraic geometry and the modern theory of abelian varieties. His work on algebraic curves has influenced a wide variety of areas, including some outside mathematics, such as elementary particle physics and string theory.

In fact Weil's work in this area was basic to work by mathematicians such as Shing-Tung Yau who was awarded a Fields Medal in 1982 for work in three dimensional algebraic geometry which has major applications to quantum field theory. Yau is not the only mathematician who received a Fields Medal for work which continued that begun by Weil. In 1978 Pierre Deligne was awarded a Fields Medal for solving the Weil Conjectures. Again we quote [55] for a description of Weil's fundamental contribution:-

One of Weil's major achievements was his proof of the Riemann hypothesis for the congruence zeta functions of algebraic function fields. In 1949 he raised certain conjectures about the congruence zeta function of algebraic varieties over finite fields. These Weil conjectures, as they came to be called, grew out of his deep insight into the topology of algebraic varieties and provided guiding principles for subsequent developments in the field.

Weil's work on bringing together number theory and algebraic geometry was highly fruitful. The foundations of many topics studied in depth today were laid by Weil in this work, such as the foundations of the theory of modular forms, automorphic functions and automorphic representations. However, Weil's work was of major importance in a number of other new mathematical topics. He contributed substantially to topology, differential geometry and complex analytic geometry. It was not just to these areas that he contributed but, even more importantly, his work brought out fundamental relationships between the areas when he studied harmonic analysis on topological groups and characteristic classes. Also bringing these areas together was his work on the geometric theory of the theta function and Kähler geometry.

Together with Dieudonné and others, Weil wrote under the name Nicolas Bourbaki, a project they began in the 1930s, in which they attempted to give a unified description of mathematics. The purpose was to reverse a trend which they disliked, namely that of a lack of rigour in mathematics. The influence of Bourbaki has been great over many years but it is now less important since it has basically succeeded in its aim of promoting rigour and abstraction.

Weil made a major contribution through his books that include Arithmétique et géométrie sur les variétés algébriques Ⓣ (1935), Sur les espaces à structure uniforme et sur la topologie générale Ⓣ (1937), L'intégration dans les groupes topologiques et ses applications Ⓣ (1940), Foundations of Algebraic Geometry (1946), Sur les courbes algébriques et les variétés qui s'en déduisent Ⓣ (1948), Variétés abéliennes et courbes algébriques Ⓣ (1948), Introduction à l'étude des variétés kählériennes Ⓣ (1958), Discontinuous subgroups of classical groups (1958), Adeles and algebraic groups (1961), Basic number theory (1967), Dirichlet Series and Automorphic Forms (1971), Essais historiques sur la théorie des nombres Ⓣ (1975), Elliptic Functions According to Eisenstein and Kronecker (1976), (with Maxwell Rosenlicht) Number Theory for Beginners (1979), Adeles and Algebraic Groups (1982), Number Theory: An Approach Through History From Hammurapi to Legendre (1984), and Correspondance entre Henri Cartan et André Weil Ⓣ (1928-1991) (2011).

You can see the preface to Weil's Algebraic Geometry at THIS LINK.

For extracts from reviews of some of these books see THIS LINK.

For some extracts from two of Weil's books on the history of mathematics see THIS LINK.

For some extracts from Weil's thoughts on the teaching of mathematics and the future of mathematics see THIS LINK.

Weil received many honours for his outstanding mathematics. Among these has been honorary membership of the London Mathematical Society in 1959 and election to a Fellowship of the Royal Society of London in 1966. In addition he has been elected to the Academy of Sciences in Paris and to the National Academy of Sciences in the United States. He refused to accept honorary doctorates which explains why there are none for us to list.

Weil was an invited speaker at the International Congress of Mathematicians in 1950 at Harvard when he gave an address on Number Theory and Algebraic Geometry and again at the following International Congress in 1954 in Amsterdam when he gave the lecture Abstract versus Classical Algebraic Geometry. In 1979 Weil was awarded the Wolf Prize and, in the following year, the American Mathematical Society awarded him their Steele Prize. In 1994 he received the Kyoto Prize from the Inamori Foundation of Japan:-

... for outstanding achievement and creativity.

The citation for the Kyoto Prize reads:-

The results achieved and problems raised by André Weil through his deep understanding of and sharp insight into mathematical sciences in general will continue to have immeasurable influence on the development of mathematical sciences, and to contribute greatly to the development of science, as well as the deepening and uplifting of the human spirit.

He is described by Goro Shimura as follows in [49]:-

In my mind, however, he will remain chiefly as the figure with two mutually related characteristics: First, he was flexible and receptive to new ideas of others and new directions, quite unlike many of the younger people these days who can work only within a well-established framework. Second, more importantly and in a similar vein, he had a deep and penetrating understanding of mathematics, or, rather, he strived tirelessly to understand the real meaning of every basic mathematical phenomenon and to present it in a clearer form and in a better perspective. He did so by endowing each subject with new concepts and setting up new frameworks, always in a fresh and fundamental way. In other words, he was not a mere problem solver.

Komaravolu Chandrasekharan writes in [14]:-

He was known for his short temper and for his sudden, provocative interventions, which sometimes resulted in abrasive confrontations. That was the less endearing side of his personality. It is in his writings that his personality really shows through - as a master of style, with deep reserves of reading, reflection, and self-scrutiny, with a hotline to the creative imagination.