皮埃尔·德利涅Pierre Deligne

皮埃尔·德利涅是Albert Deligne的儿子，Albert Deligne是一家公司的管理人员，妻子是Renée Bodart。他出生在埃特尔贝克，这是与布鲁塞尔市中心一起构成大布鲁塞尔的十九个郊区之一。他是父母三个孩子中最小的，有一个比他大七岁的哥哥，还有一个姐姐。甚至在他开始接受初等教育之前，德利涅就在跟他的哥哥学习数学[33]：-

我很幸运，我的哥哥比我大七岁。当我看着温度计，意识到有正数和负数时，他会试着向我解释负一乘以负一等于正一。那是一个很大的惊喜。

德利涅于1950年9月至1956年6月在斯哈尔贝克上小学，这是布鲁塞尔市中心东北的十九个郊区中的另一个。这所学校的一位优秀教师教他阅读、写作和算术。但不仅如此，他记得自己被老师用绳子覆盖半球面和同样半径的圆盘来比较它们的表面积所吸引。用长度来测量面积让德利涅深思。他继续跟他的哥哥学习数学，所以当他在小学时，他就被教了求解二次方程的公式。

1956年9月，他在布鲁塞尔的阿道夫·马克斯中学开始接受中等教育。这所男子学校成立于1909年，后来以布鲁塞尔著名市长阿道夫·马克斯命名。德利涅继续跟他的哥哥学习，当他在高中时，他的哥哥给了他一些关于求解三次和四次方程的笔记，包括著名的复杂三次方程公式，德利涅觉得非常有趣。他在高中最喜欢的课程是几何课；学习以欧几里得的风格写证明，他发现这是一个极好的练习。他数学发展中最重要的时刻，不是来自他的哥哥，而是来自Jeff Nijs。德利涅在童子军里，他的一个朋友是高中教师Jeff Nijs的儿子。听说德利涅对数学的热情后，Jeff Nijs给了他一本尼古拉·布尔巴基的Set Theory。这是一本很难的文本，几乎会让任何高中生对数学望而却步，但对德利涅来说并非如此，他很高兴拥有他的“第一本真正的数学书”。他说[33]：-

我已经在别处读到过如何从整数开始定义有理数，然后是实数。但我记得我曾在尼古拉·布尔巴基中稍微往前看，想知道如何从集合论定义整数，并钦佩如何首先定义两个集合具有“相同数量的元素”意味着什么，并由此推导出整数的概念。

他家的一位朋友给了年轻的德利涅一本关于复变量的书，这给了他“巨大的快乐”。Jeff Nijs意识到德利涅是一个多么出色的高中生，并想出了另一个绝妙的主意；他把德利涅介绍给雅克·蒂茨，解释了他非凡的才能，并请雅克·蒂茨好好照顾他。尽管还在高中，德利涅参加了雅克·蒂茨在大学里的课程和讨论班。有一天他缺席了一次讲座，雅克·蒂茨问：“德利涅在哪里？”当被告知他不能来是因为他必须参加高中旅行时，雅克·蒂茨立即将讲座推迟到下一周。德利涅于1962年6月从中学毕业，并于同年9月进入布鲁塞尔自由大学。

德利涅 于 1962 年至 1966 年在布鲁塞尔自由大学读本科，雅克·蒂茨 认为，德利涅 最好向在巴黎高等师范学院授课的 亚历山大·格罗滕迪克 学习。1964 年 11 月，雅克·蒂茨 带 德利涅 去巴黎参加一个 尼古拉·布尔巴基 讨论班，并把他介绍给 亚历山大·格罗滕迪克。德利涅 说 [34]：-

我确实吃了一惊。他有点奇怪，剃着光头，个子很高。我们握了手，但没再做别的，直到几个月后我去巴黎参加他的讨论班。

德利涅 在 1965-66 学年于巴黎高等师范学院度过。他在 [34] 中回忆起他参加的 亚历山大·格罗滕迪克 第一堂课上的一个插曲：-

……他多次使用“上同调对象”这个说法。我知道阿贝尔群的上同调是什么，但不知道“上同调对象”是什么意思。课后我问他这个说法是什么意思。我想许多其他数学家会认为，如果你不知道答案，那就没有跟你说话的必要。他的反应完全不是这样。他非常耐心地告诉我，如果你在阿贝尔范畴中有一个长正合序列，看其中一个映射的核，再除以前一个映射的像，如此等等……我很快意识到，我在一个不那么一般的语境中知道这些。他对无知的人非常开放。我想，同一个愚蠢的问题你不该问他三次，但问两次没问题。

德利涅 于 1966 年 11 月获得 Licence en mathématiques，相当于学士学位。他继续在布鲁塞尔自由大学攻读博士学位，并于 1967 年 9 月在布鲁塞尔的国家科学研究基金会担任初级研究员，同时还是法国比尔斯-伊韦特高等科学研究所的访问成员，在那里与 亚历山大·格罗滕迪克 合作。他于 1968 年 11 月由布鲁塞尔自由大学授予数学博士学位。

获得博士学位后，德利涅 前往法国比叙尔-伊韦特的高等科学研究所（IHES），在那里担任访问成员直到 1970 年 2 月，之后成为该研究所的永久成员。在 IHES，他最初与 亚历山大·格罗滕迪克 合作，研究 奥斯卡·扎里斯基 主要定理的推广。他还与 让-皮埃尔·塞尔 密切合作，在模形式附带的 $l$-进表示以及 $L$-函数的猜想函数方程方面取得了重要成果。

与 亚历山大·格罗滕迪克 和 让-皮埃尔·塞尔 合作对 德利涅 很重要，特别是因为他们的数学方法非常不同。亚历山大·格罗滕迪克 想以最大的 generality 理解一切。他对相关文献了解不多，宁愿自己证明一切。另一方面，让-皮埃尔·塞尔 对文献有极好的了解，能给出精确的参考文献。他不是总去寻找最一般的情形，而是专注于漂亮的特例。德利涅 认为这两人需要彼此，他们的合作对双方都有价值。他还发现，平衡他们的两种方法对他自己是一种有用的学习经历。他说 亚历山大·格罗滕迪克 的讲座很精彩，但他需要去听 让-皮埃尔·塞尔 的讲座才能脚踏实地。在 IHES 期间，德利涅 还与 戴维·芒福德 合作，对曲线的模空间作了新的描述：这项工作在后来由弦理论引发的发展中被大量使用。

他卓越的数学贡献很快得到认可，获得了重要奖项：1974 年，他获得 比利时皇家科学院 颁发的 François Deruyts 奖和 法国科学院 颁发的 儒勒·昂利·庞加莱 奖章。1975 年，他获得比利时国家科学基金会颁发的 A De Leeuw-Damry-Bourlart 奖。

1980 年 9 月 9 日，德利涅 与 Elena Vladimirovna Alexeeva 结婚，他是在一次访问俄罗斯时认识她的；他们有两个孩子，Natalia 和 Alexis。

德利涅 一直留在高等科学研究所，直到 1984 年他前往美国普林斯顿的高等 爱德华·斯图迪 研究所，并被任命为教授。他离开 IHES 并不是因为在那里不开心，恰恰相反，而是他觉得整个职业生涯都在同一机构度过并不好，换一个地方会让他获益良多。在 IHES 期间，他每年就一个不同主题开设讨论班，而去了普林斯顿后，他不再觉得有必要这样做。他在 [34] 中谈到了这两个机构的相似与不同之处：-

我会说，高等研究院斯图迪更古老、更大、也更稳定。两者非常相似的地方在于，都有许多年轻访问者来到这里。所以它们不是那种你可以打瞌睡的地方，因为你总会接触到年轻人，他们会告诉你，你并不像自己以为的那么优秀。两个地方都有物理学家，但我认为在普林斯顿与他们的接触对我来说比在布雷斯更有成效。在普林斯顿，有共同的讨论班。

作为数学家和物理学家之间互动的一个例子，让我们引用该研究院1997-98年度报告[48]中的话：-

这是由数学学院的德利涅教授和自然科学学院的爱德华·威滕教授领导的为期三年的数学与物理学跨学科项目的第二年，项目名称为“量子场论与规范理论的代数与几何方面”。该项目证明了参与其中的数学家和物理学家对持续而高深的互动的非凡投入。在过去几年中，理论物理学界提出了若干引人注目的数学猜想。该项目的目的是让数学家了解物理学家得出这些猜想的思维过程。……这些讲义初稿已在研究院网站上提供，并正被全世界的数学家用作这一材料的独特来源。

安德烈·韦伊在其Foundations of Algebraic Geometry（1946年）中首次给出了由系数在任意域中的方程所定义的簇的理论。这运用了奥斯卡·扎里斯基的思想，也很好地利用了几何概念。安德烈·韦伊关于多项式方程的工作引出了这样的问题：一个几何对象的哪些性质可以纯粹用代数方式确定。安德烈·韦伊的工作将关于多项式方程整数解的问题与代数几何中的问题联系起来。他利用关于代数拓扑应如何应用于这一新情境的直觉，猜想了关于多项式方程在整数上解的个数的结果。他的第三个猜想是关于zeta函数的波恩哈德·黎曼假设的推广。这些问题很快成为数学家的重大研究挑战。

安德烈·韦伊三个猜想的一个解答由德利涅于1974年给出。这项工作将代数几何与代数数论结合在一起，并使德利涅于1978年在赫尔辛基举行的国际数学家大会上被授予约翰·查尔斯·菲尔兹奖章。解决这些问题需要发展一种新的代数拓扑。雅克·蒂茨说[29]：-

这些猜想既极其难以解决（包括亚历山大·格罗滕迪克在内的最优秀的专家都曾研究过它们），又因其解答具有深远影响而极其引人关注。

德利涅还研究过许多其他重要问题。除代数几何外，他研究过的领域有大卫·希尔伯特第21问题、威廉·瓦兰斯·道格拉斯·霍奇理论、模理论、模形式、埃瓦里斯特·伽罗瓦表示、L级数以及罗伯特·朗兰兹猜想，还有代数群的表示。

除了约翰·查尔斯·菲尔兹奖章外，德利涅还在1988年获得了瑞典皇家科学院的克拉福德奖：-

……因他在代数几何方面的基础性研究。

德利涅因其杰出贡献还获得了许多其他荣誉。例如，他于1974年6月获得比利时皇家科学院颁发的Francois Deruyts奖，1974年12月获得巴黎科学院颁发的儒勒·昂利·庞加莱奖章，并于1975年获得国家科学研究基金会颁发的Doctor A De Leeuw-Damry-Bourlart奖。他于1989年获得布鲁塞尔弗拉芒大学的名誉博士学位，1995年获得巴黎高等师范学院的名誉博士学位。他于1978年当选为巴黎科学院成员，并于同年当选为美国艺术与科学院成员。

2004年，德利涅当选为伦敦数学会[15]的名誉成员：-

……以表彰他对代数几何的巨大贡献。

从整体上看，德利涅的工作涉及代数簇上同调的许多不同方面。它把亚历山大·格罗滕迪克的动机哲学从一个猜想性的纲领变成了当前代数几何与算术中许多最微妙领域的驱动力。通过一种无与伦比的洞察力、无畏的技术掌握和令人眼花缭乱的独创性的结合，德利涅单枪匹马地带来了对簇上同调的新理解，既包括经典的也包括有限特征的情形，并在几何与数论的深刻问题上有着众多应用。

同年，德利涅获得了国际巴尔赞基金会颁发的2004年巴尔赞数学奖[42]：-

……因对数学若干重要领域（如代数几何、代数与解析数论、群论、拓扑学、亚历山大·格罗滕迪克的动机理论）的重大贡献，用新的有力工具和辉煌成果丰富了这些领域，例如他对“有限域上的波恩哈德·黎曼假设”（安德烈·韦伊猜想）的惊人证明。

雅克·蒂茨作为巴尔赞奖委员会成员，于2004年9月7日在米兰宣布了该奖项。他描述了德利涅的工作，最后作了以下评论[42]：-

德利涅思维的一个显著特点是，当面对一个新问题或新理论时，他能以极快的速度理解并可以说将其基本原理化为己有，并立即能够讨论该问题或像使用一个完全熟悉的对象一样使用该理论。因此，在讨论中，他能轻松采用与他交谈的人的语言。这种灵活性是他数学工作具有普遍性的原因之一。

德利涅独自或合作撰写了约一百篇论文，其中大多数篇幅相当长。由于他文风简洁，并且习惯于从不重复写同样的东西（事实上，他相当多最好的想法从未被写下来！），他出版物的数量真实地衡量了他科学产出的丰富程度。

作为巴尔赞奖的获得者，德利涅获得了100万瑞士法郎（约合80万美元），其中一半将用于他所在领域涉及年轻研究者的研究项目。颁奖仪式于2004年11月18日在罗马的Accademia dei Lincei举行。德利涅在回复中说道[43]：-

我自己的研究主要围绕代数几何展开。这门学科如此得益于意大利学派，使我今天身在罗马倍感喜悦。代数几何诞生于几个世纪以来的一种认识：几何与代数在许多情况下是同一底层“实在”的两种不同语言的表达。作为一个经典例子，我将引用“半径为$r$的圆”与“方程$x^{2} + y^{2} = r^{2}$”之间的重要等同。构造这类类比——在其完成形式中可以成为词典——以及它们的推论：用乍看之下与所提问题毫无关系的方法来解决问题，对我来说是数学为职业数学家提供的巨大乐趣之一。这些意想不到的类比、词典和比较，也确保了数学不会分散为各自独立的子学科，并且当数学作为其他科学的工具时，是其有效性的一个来源——即使这种有效性并非数学家的首要目标。

2005年10月17日至20日，在斯图迪高等研究院举行了一次庆祝德利涅六十一岁生日的会议。2007年5月，他当选为国家科学院数学部的国际成员。他的研究兴趣在科学院网站上给出如下[44]：-

我的本行是代数几何，一门与许多其他学科相联系的学科；凡有多项式出现之处，它都可以在那里提供一种几何理解。例如：丢番图方程（方程在整数或有理数中的解）、代数被积函数的积分之间的恒等式、代数群。我着迷于代数簇所产生的上同调理论的多样性及其相互关系。我构造了一种：混合威廉·瓦兰斯·道格拉斯·霍奇理论。由于它们的性质，来自代数几何的空间和映射非常特殊。一个宏大的亚历山大·格罗滕迪克（“动机”）理论使之得到理解，但需模去一些仍然难以企及的猜想。我的一些工作给出了对某些应用而言足够的无条件变体。我还研究过自守形式（通过罗伯特·朗兰兹的哲学与代数簇的算术和上同调相关）、超平面构型、张量范畴、多重zeta值（一个始于莱昂哈德·欧拉的故事）。

2008年2月，德利涅成为沃尔夫奖的获得者，该奖此次还授予了菲利普·格里菲思和戴维·芒福德。颁奖词称，该奖授予德利涅[45]：-

……因其在混合威廉·瓦兰斯·道格拉斯·霍奇理论、安德烈·韦伊猜想、波恩哈德·黎曼-大卫·希尔伯特对应方面的工作，以及他对算术的贡献。

2008年晚些时候，他成为普林斯顿高等研究院的荣休教授斯图迪。他继续获得重大荣誉和奖项：2009年2月当选为瑞典皇家科学院外籍会士，同年4月当选为美国哲学学会会员。也许他获得的最大荣誉是2013年5月的尼尔斯·阿贝尔奖。该奖项的颁奖词指出，该奖授予德利涅（[1]和[24]）：-

……表彰其对代数几何的开创性贡献，以及这些贡献对数论、表示论及相关领域的变革性影响。……德利涅的强大概念、思想、结果和方法继续影响着代数几何乃至整个数学的发展。

上面未提及的其他荣誉包括：1978年当选Académie des Sciences, Paris外籍通讯院士；1989年获布鲁塞尔自由大学荣誉博士学位；1994年当选Académie Royale de Belgique通讯院士；1995年获巴黎高等师范学院荣誉博士学位；2003年当选Accademia nazionale dei Lincei外籍院士；2006年被比利时国王亚伯拉罕·阿德里安·艾伯特二世封为贵族。这最后一项荣誉使他成为德利涅子爵，他设计了自己的纹章，其灵感来自以下童谣[39]：-
插图：deligne_arms.jpg ↗

当三只母鸡走向田野，
第一只走在前面，
第二只跟着第一只，
第三只是最后一只。
当三只母鸡走向田野，
第一只走在前面。

这翻译为：

当三只母鸡走向田野，
第一只领头，
第二只跟着第一只，
第三只是最后一只。
当三只母鸡走向田野，
第一只领头。

德利涅的推理见[39]：-

这首韵诗旨在教给孩子们几个简单词的含义。因此，在成年人看来，它不过是一连串的同义反复。德利涅认为，数学论述的进行方式也大致如此……

2007年，比利时邮政局发行了一枚纪念他的邮票。
见THIS LINK。

2016年10月，他当选为俄罗斯科学院外籍院士。

在访谈[33]中，德利涅谈到了教学：-

我从来不必教书。我非常喜欢与人交谈。在我工作过的两个机构里，年轻人会来找我交谈。有时我回答他们的问题，但更多时候我会反问他们一些问题，这些问题有时也很有趣。所以，这种一对一接触的教学方式，试图提供有用的信息并在此过程中学习，对我来说很重要。我猜想，教那些不感兴趣、只是为了拿学分去做别的事而被迫学数学的人，一定非常痛苦。我会觉得那很令人反感。

也许记录一下这位数学天才除了数学之外还做些什么会很有趣[47]：-

他没有汽车，到哪里都骑自行车，右裤腿常年塞在袜子里。他很可能从未被人见过穿西装，而是喜欢穿土色调的旧毛衣。在普林斯顿的骤雨中，有人见过他脱到只剩腰部以上，以减少损失，不过他并没有把这一策略推向逻辑极端。

德利涅热爱自然，并设法买下了紧邻研究所树林的房子，尽管那房子曾被用作行政空间。冬天，他习惯在后院建一个雪屋，并在外面睡上几晚。夏天，他打理一个花园——土豆、西红柿、覆盆子、醋栗、韭葱、罗勒、欧芹、龙蒿、细香葱。“有些东西我非常喜欢，我尝试了，但并不成功，”他说。“我非常喜欢洋蓟，倒不是那么喜欢吃，而是当它们过熟时，会有一个花蕾，像花一样开放，非常美丽。但气候似乎对它们不利。每年我都尝试。我觉得这非常放松。”

让我们以两段关于德利涅贡献的引文结束。Peter Sarnak写道[2]：-

很少有数学家对现代数学的影响能接近德利涅。他在代数几何和算术几何方面的研究塑造了这些领域，并使他解决了许多长期存在的问题，包括安德烈·韦伊猜想（这是著名的波恩哈德·黎曼假设在有限域上簇的类似物）以及模形式理论中著名的拉马努金猜想。德利涅的基础性贡献涵盖上述领域到群的表示论、微分方程与单值化、拓扑学……他在这些论文中或为回应向他提出的问题（他非常平易近人且慷慨）而开发的许多技术和工具，是当今这些领域许多激动人心的研究的基础。德利涅的风格是希望用简单而一般的术语理解那些基本但显得非常复杂的事物。他对新见解和方法的发展，以及长期存在问题的解决，自然源于这一观点。当然，他惊人的成功很大程度上归功于他非凡的数学才能，尤其是他的抽象思维能力。

Hélène Esnault写道[2]：-

德利涅的工作构建了几代代数几何学家和算术几何学家思考和写作的语言。许多代数几何学家多年来床头柜上放着德利涅思想的几行文字，以便冥想和进一步思考。所有代数几何学家都对德利涅的工作怀有最深切的钦佩。

德利涅本人在论述亚历山大·格罗滕迪克时写道[46]：-

我感到极其幸运，亚历山大·格罗滕迪克是我的导师。我从他那里学到的东西，尤其是动机的哲学，一直是我最喜欢的那些著作中的指导线索，例如混合威廉·瓦兰斯·道格拉斯·霍奇结构的形式体系。从他以及他的榜样中，我还学会了不以证明的困难为荣：困难意味着我们尚未理解。理想是能够描绘出一幅证明显而易见的图景。

Pierre Deligne was the son of Albert Deligne, a company administrator, and his wife Renée Bodart. He was born in Etterbeek, one of the nineteen suburban districts that, together with central Brussels, make up Greater Brussels. He was the youngest of his parents three children, having a brother who was seven years older than he was, and also an elder sister. Even before he began his primary education, Pierre was learning mathematics from his brother [33]:-

I was lucky that my brother was seven years older than me. When I looked at the thermometer and realised that there were positive and negative numbers, he would try to explain to me that minus one times minus one is plus one. That was a big surprise.

Pierre attended primary school at Schaerbeek, another of the nineteen suburban districts northeast of central Brussels, from September 1950 to June 1956. An excellent teacher at this school taught him reading, writing and arithmetic. But more than this, he remembered being fascinated by the teacher comparing the surface area of a half-sphere with a disc of the same radius by covering both with rope. Using length to measure area made Pierre think deeply. He continued to learn mathematics from his brother and so when in primary school he was taught the formula for solving quadratic equations.

In September of 1956 he began his secondary schooling at the Athénée Adolphe Max in Brussels. This boys' school was founded in 1909 and later named for Adolphe Max, a famous mayor of Brussels. Pierre continued to learn from his brother and, when in the high school, his brother gave him some notes about solving cubic and quartic equations, including the famous complicated formula for cubics which Pierre found very interesting. His favourite lessons at high school were geometry lessons; learning to write proofs in the style of Euclid he found an excellent exercise. The most important event in his mathematical development came about, not from his brother, but from Jeff Nijs. Pierre was in the Boy Scouts where one of his friends was the son of the high school teacher Jeff Nijs. Hearing about Pierre's passion for mathematics, Jeff Nijs gave him a copy of Bourbaki's Set Theory. This is a difficult text and one which should put off mathematics almost any high school pupil, but not so for Pierre Deligne who was delighted to have his "first real mathematical book." He said [33]:-

I had already read elsewhere how rational numbers, then real numbers, could be defined starting from the integers. But I remember wondering how integers could be defined from set theory, looking a little ahead in Bourbaki, and admiring how one could first define what it means for two sets to have the "same number of elements", and derive from this the notion of integers.

A friend of his family gave the young Deligne a book on complex variable which gave him "tremendous joy." Jeff Nijs realised what a remarkable high school student Deligne was and came up with another excellent idea; he introduced Deligne to Jacques Tits, explained his extraordinary talent, and asked Tits to take good care of him. Although still at high school, Deligne attended Tits's courses and seminars at the university. One day he was absent from a lecture and Tits asked, "Where is Deligne?" When told he could not be there because he had to be on a high school trip, Tits promptly postponed the lecture until the following week. Deligne graduated from secondary school in June 1962 and entered the Free University of Brussels in September of that year.

Although Deligne was an undergraduate at the Free University of Brussels from 1962 to 1966, Tits decided that it would be in Deligne's best interests to learn from Alexander Grothendieck who was lecturing at the École Normale Supérieure in Paris. In November 1964 Tits took Deligne to Paris to attend a Bourbaki seminar and introduce him to Grothendieck. Deligne said [34]:-

I was really taken aback. He was a little strange, with his shaved head, a very tall man. We shook hands but did nothing more until I went to Paris a few months later to attend his seminar

Deligne spent the academic year 1965-66 at the École Normale Supérieure in Paris. He recalls in [34] an incident from the first lecture by Grothendieck that he attended:-

... he used the expression "cohomology object" many times. I knew what cohomology was for abelian groups, but I did not know the meaning of "cohomology object". After the lecture I asked him what he meant by this expression. I think that many other mathematicians would have thought that if you didn't know the answer, there wouldn't be any point to speak to you. This was not his reaction at all. Very patiently he told me that if you have a long exact sequence in an abelian category and you look at the kernel of one map, you divide by the image of the previous one and so on… I recognised quickly that I knew about this in a less general context. He was very open to people who were ignorant. I think that you should not ask him the same stupid question three times, but twice was all right.

Deligne received his Licence en mathématiques in November 1966, the equivalent of a B.A. He continued to study for his doctorate at the Free University of Brussels and in September 1967 he was a junior scientist at the Fond National de la Recherche Scientifique in Brussels, at the same time being a guest at the Institut des Hautes Études Scientifiques at Bures-sur-Yvette in France where he worked with Alexandre Grothendieck. He was awarded his Doctorat en mathématiques by the Free University of Brussels in November 1968.

After the award of his doctorate, Deligne went to the Institut des Hautes Études Scientifiques (IHES) at Bures-sur-Yvette in France where he was a visiting member until February 1970 after which he became a permanent member of the Institute. At the IHES he worked with Grothendieck initially on the generalisation of Zariski's main theorem. He also worked closely with Jean-Pierre Serre, leading to important results on the $l$-adic representations attached to modular forms, and the conjectural functional equations of $L$-functions.

Working with both Grothendieck and Serre was important for Deligne, particularly since their approach to mathematics was very different. Grothendieck wanted to understand everything in the utmost generality. He had not much knowledge of the relevant literature, preferring to prove everything for himself. Serre, on the other hand, had an excellent knowledge of the literature and could give precise references. Rather than always looking for the most general situation, he concentrated on beautiful special cases. Deligne thought that these two needed each other and their collaboration was valuable to them both. He also found that balancing their two approaches was a useful learning experience for him. He said Grothendieck's lectures were wonderful but he needed to go to Serre's lectures in order to keep his feet on the ground. During this period at the IHES, Deligne also collaborated with David Mumford on a new description of the moduli spaces for curves: this work has been much used in later developments arising from string theory.

His remarkable mathematical contributions were quickly recognised with the award of major prizes: in 1974 he received the François Deruyts Prize, awarded by the Belgium Royal Academy of Sciences, and the Henri Poincaré Medal, awarded by the French Academy of Sciences. He received the A De Leeuw-Damry-Bourlart Prize in 1975 from the Belgian National Science Foundation.

On 9 September 1980, Deligne married Elena Vladimirovna Alexeeva whom he met on one of his visits to Russia; they have two children, Natalia and Alexis.

Deligne remained based at the Institut des Hautes Études Scientifiques until 1984 when he went to the Institute for Advanced Study at Princeton in the United States, where he was appointed a professor. He did not leave the IHES because he was unhappy there, quite the contrary, but he felt that it was not good to spend the whole of one's career at the same institution and that he would gain much from a move. While at the IHES he gave a seminar each year on a different topic and by going to Princeton he no longer felt it necessary to do this. He talked about the similarities and differences in the two institutions in [34]:-

I would say that the Institute for Advanced Study is older, bigger, and more stable. Both are very similar in the way that there are many young visitors who come there. So they are not places where you can fall asleep since you will always be in contact with young people who will tell you that you are not as good as you think you are. In both places there are physicists, but I think the contact with them was more fruitful for me in Princeton than it was in Bures. In Princeton, there have been common seminars.

As an example of the interaction between mathematicians and physicists, let us quote from the Institute's Report for 1997-98 [48]:-

This was the second year of a three-year interdisciplinary programme in mathematics and physics, led by Professor Pierre Deligne, School of Mathematics, and Professor Edward Witten, School of Natural Sciences, and titled "Algebraic and Geometric Aspects of Quantum Field Theory and Gauge Theory." The programme is evidence of a remarkable commitment to sustained and sophisticated interaction on the part of participating mathematicians and physicists. Over the past few years the theoretical physics community has produced a number of remarkable mathematical conjectures. The aim of this program is for mathematicians to learn the physicists' thought processes that lead to these conjectures. ... Preliminary versions of these lecture notes have been available on the Institute's web site and are being used by mathematicians worldwide as a unique source for this material.

André Weil gave for the first time a theory of varieties defined by equations with coefficients in an arbitrary field, in his Foundations of Algebraic Geometry (1946). This used Zariski's ideas and also made good use of geometric concepts. Weil's work on polynomial equations led to questions on what properties of a geometric object can be determined purely algebraically. Weil's work related questions about integer solutions of polynomial equations to questions in algebraic geometry. He conjectured results about the number of solutions of polynomial equations over the integers using intuition on how algebraic topology should apply in this novel situation. The third of his conjectures was a generalisation of the Riemann hypothesis on the zeta function. These problems quickly became major research challenges to mathematicians.

A solution of the three Weil conjectures was given by Deligne in 1974. This work brought together algebraic geometry and algebraic number theory and it led to Deligne being awarded a Fields Medal at the International Congress of Mathematicians in Helsinki in 1978. A solution to these problems required the development of a new kind of algebraic topology. Jacques Tits said [29]:-

These conjectures were both exceptionally hard to settle (the best specialists, including A Grothendieck, had worked on them) and most interesting in view of the far-reaching consequences of their solution.

Deligne has worked on many other important problems. The areas on which he has worked, in addition to algebraic geometry, are Hilbert's 21st problem, Hodge theory, theory of moduli, modular forms, Galois representations, L-series and the Langlands conjectures, and representations of algebraic groups.

In addition to the Fields Medal, Deligne was awarded the Crafoord Prize of the Royal Swedish Academy of Sciences in 1988:-

... for his fundamental research in algebraic geometry.

Deligne has been awarded many other honours for his outstanding contributions. For example he was awarded the Francois Deruyts prize by the Royal Belgium Academy of Sciences in June 1974, the Henri Poincaré medal by the Paris Academy of Sciences in December 1974, and the Doctor A De Leeuw-Damry-Bourlart Prize by the Fond National de la Recherche Scientifique in 1975. He has received honorary doctorates from the Flemish University of Brussels in 1989, and from the École Normale Supérieure in 1995. He has been elected a member of the Paris Academy of Sciences in 1978 and by the American Academy of Arts and Sciences in the same year.

In 2004 Deligne was elected an honorary member of the London Mathematical Society [15]:-

... in recognition of his monumental contributions to algebraic geometry.

Viewed as a whole, Deligne's work concerns many different aspects of the cohomology of algebraic varieties. It has turned Grothendieck's philosophy of motives from a conjectural program into what is the driving force behind many of the most subtle areas of current algebraic geometry and arithmetic. Through an unparalleled blend of penetrating insights, fearless technical mastery and dazzling ingenuity, Deligne has singlehandedly brought about a new understanding of the cohomology of varieties, both classical and in finite characteristic, with numerous applications to deep problems in geometry and number theory.

In the same year Deligne received the 2004 Balzan Prize in Mathematics awarded by the International Balzan Foundation [42]:-

... for major contributions to several important domains of mathematics (like algebraic geometry, algebraic and analytic number theory, group theory, topology, Grothendieck theory of motives), enriching them with new and powerful tools and with magnificent results such as his spectacular proof of the "Riemann hypothesis over finite fields" (Weil conjectures).

Jacques Tits, as a member of the Balzan Prize committee, announced the prize on 7 September 2004 in Milan. He described Deligne's work, then ended by making the following comments [42]:-

A remarkable feature of Pierre Deligne's thinking is that, when confronted with a new problem or a new theory, he understands and, so to speak, makes his own its basic principles at a tremendous speed, and is immediately able to discuss the problem or use the theory as a completely familiar object. Thus, he readily adopts the language of the people he is talking to when engaged in discussions. This flexibility is one of the reasons for the universality of his mathematical work.

Alone or in collaboration, Pierre Deligne has written about a hundred papers, most of them of sizeable length. Because of the conciseness of his style and of his habit of never writing the same thing twice (in fact, quite a few of his best ideas have never been written!), the volume of his publications is a true measure of the richness of his scientific production.

As winner of the Balzan Prize, Deligne received 1 million Swiss francs (about US$800,000), half of which would go to research projects involving young researchers in his field. The prize ceremony took place on 18 November 2004 in the Accademia dei Lincei in Rome. Deligne said, as part of his reply [43]:-

My own research has mainly revolved around algebraic geometry. That this discipline owes so much to the Italian school doubles my pleasure in being in Rome today. Algebraic geometry was born from the realisation, over several centuries, that geometry and algebra are in many cases the expression in two different languages of the same underlying "reality". As a classic example, I will cite the significant identity between "circle of radius $r$" and "equation $x^{2} + y^{2} = r^{2}$". The construction of such analogies which, in their finished form, can become dictionaries, and their corollary: the resolution of problems by methods which, at first sight, seem to have nothing to do with the problem posed, are for me one of the great joys that mathematics offers the professional mathematician. These unexpected analogies, dictionaries and comparisons also ensure that mathematics is not scattered over autonomous sub-disciplines, and is a source of its effectiveness when it serves as a tool for other sciences - even if this effectiveness is not the first goal of the mathematician.

A conference on the occasion of the sixty-first birthday of Pierre Deligne took place at the Institute for Advanced Study on 17-20 October 2005. In May 2007 he was elected as an International Member of the Mathematics Section of the National Academy of Sciences. His Research Interests are given on the Academy's website as follows [44]:-

My home is algebraic geometry, a discipline connected to many others; wherever polynomials appear, it can be there to provide a geometric understanding. For instance: Diophantine equations (solutions of equations in integers or rationals), identities between integrals with algebraic integrands, algebraic groups. I am fascinated by the multiplicity of cohomology theories algebraic varieties give rise to, and their interrelation. I constructed one: mixed Hodge theory. Thanks to their properties, spaces and maps coming from algebraic geometry are very special. A grandiose theory of Grothendieck ("motives") makes sense of it, modulo conjectures which remain inaccessible. Some of my works give unconditional variants sufficients for some applications. I have also worked with automorphic forms (related with the arithmetic and the cohomology of algebraic varieties by Langlands' philosophy), with configurations of hyperplanes, tensor categories, multizeta values (a story beginning with Euler).

In February 2008, Deligne was a recipient of the Wolf Prize, given also on this occasion to Phillip Griffiths, and David Mumford. The citation states that the prize is awarded to Deligne [45]:-

... for his work on mixed Hodge theory, the Weil conjectures, the Riemann-Hilbert correspondence, and for his contributions to arithmetic.

Later in 2008 he became Emeritus at the Institute for Advanced Study, Princeton. He continued to receive major honours and awards: he was elected a Foreign Member of the Royal Swedish Academy of Sciences in February 2009 and a member of the American Philosophical Society in April of that year. Perhaps the greatest honour he received was the Abel Prize in May 2013. The citation for the Prize states that the prize is awarded to Deligne ([1] and [24]):-

... for seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields. ... Deligne's powerful concepts, ideas, results and methods continue to influence the development of algebraic geometry, as well as mathematics as a whole.

Other honours, not mentioned above, include: elected Membre Associé Étranger, Académie des Sciences, Paris (1978); Doctor honoris causa of the Vrije Universiteit Brussel (1989); elected Membre Associé, Académie Royale de Belgique (1994); Doctor honoris causa of École Normale Supérieure (1995); elected foreign member, Accademia nazionale dei Lincei (2003); ennobled by Albert II, King of the Belgians (2006). This last mentioned honour made him Vicomte Deligne, and he designed his coat of arms which was inspired by the following nursery rhyme [39]:-
插图：deligne_arms.jpg ↗

Quand trois poules vont aux champs,
La première va devant,
La deuxième suit la première,
La troisième est la dernière.
Quand trois poules vont aux champs,
La première va devant.

This translates as:

As three hens head for the fields,
The first one leads,
The second follows the first,
The third one is last.
As three hens head for the fields,
The first one leads.

Deligne's reasoning is given in [39]:-

This rhyme is intended to teach children the meaning of a few simple words. As such, it appears to adults as a succession of tautologies. Deligne argues that mathematical discourse proceeds in much the same way ...

In 2007 the Belgian post office issued a postage stamp in his honour.
See THIS LINK.

In October 2016 he was elected a Foreign Member of the Russian Academy of Sciences.

In the interview [33] Deligne speaks about teaching:-

I never had to teach. I like very much to speak with people. In the two institutions where I have worked young people come to speak with me. Sometimes I answer their questions, but more often I ask them counter-questions that sometimes are interesting, too. So this aspect of teaching with one-to-one contact, trying to give useful information and learning in the process, is important to me. I suspect it must be very painful to teach people who are not interested, but are forced to learn mathematics because they need the grade to do something else. I would find that repulsive.

Perhaps it would be interesting to record a little about what this mathematical genius does other than mathematics [47]:-

He doesn't own a car and bicycles everywhere, keeping his right pant leg perennially tucked into his sock. He has quite possibly never been seen in a suit, favouring instead well-worn sweaters in earth tones. In Princeton's flash rainstorms he has been known to strip down to the waist to minimise the damage, though he does not take this strategy to the logical extreme.

Deligne loves nature and negotiated to buy the house right next to the Institute woods, even though it had been used for administrative space. In wintertime, he has taken to building an igloo in his backyard and spending a few nights sleeping outside. In summertime, he maintains a garden - potatoes, tomatoes, raspberries, gooseberries, leeks, basil, parsley, tarragon, chives. "Some things I like very much and I try but I am not successful," he says. "I like very much artichoke, not so much to eat them but when they get overripe there is a bud and they open like a flower and they are beautiful. But the climate does not seem to be good for them. Every year I try. I find it very relaxing."

Let us end with two quotes about Deligne's contributions. Peter Sarnak writes [2]:-

There are very few mathematicians whose impact on modern mathematics comes close to that of Deligne. His research in algebraic geometry and arithmetic geometry have shaped these fields and led him to the solution of a number of long standing problems, including the Weil Conjectures (which are the analogues of the notorious Riemann Hypothesis for varieties over finite fields) and the celebrated Ramanujan Conjecture in the theory of modular forms. Deligne's foundational contributions range from the above fields to representation theory of groups, differential equations and monodromy, topology ... Many of the techniques and tools that he developed either in these papers or in response to questions posed to him (he is very approachable and generous) are at the bottom of much of the exciting research that is going in these fields today. Deligne's style is that of wanting to understand in simple and general terms things that are fundamental but appear to be very complex. His development of new insights and methods, as well as the solution of long standing problems, follow naturally from this point of view. Of course his striking success has a lot to do with his tremendous mathematical talent and especially his ability to think abstractly.

Hélène Esnault writes [2]:-

Deligne's work has structured the language in which generations of algebraic geometers and arithmeticians think and write. Many algebraic geometers have slept for years with a few lines of Pierre Deligne's ideas on their night table, to meditate and think further. All algebraic geometers have the deepest admiration for Deligne's work.

Deligne himself, writing about Grothendieck, says [46]:-

I feel extremely fortunate that Grothendieck was my Master. What I learned from him, especially the philosophy of motives, has been a guiding thread in the works of mine I like the most, such as the formalism of mixed Hodge structures. From him and his example, I have also learned not to take glory in the difficulty of a proof: difficulty means we have not understood. The ideal is to be able to paint a landscape in which the proof is obvious.