弗拉基米尔·德林费尔德Vladimir Drinfeld

弗拉基米尔·德林费尔德出生于一个犹太数学家庭。他是数学教授格尔雄·伊赫列维奇·德林费尔德（1908年2月29日-2000年8月18日）和他的妻子、古典语言学家弗里达·约瑟福夫娜·卢茨卡娅-利特瓦克（1921-2011）的儿子。由于德林费尔德的父亲是哈尔科夫大学的数学教授，我们将在给出他儿子弗拉基米尔的传记细节之前，先给出他生活的一些细节。

德林费尔德出生于乌克兰的旧康斯坦丁诺夫，从小就展现出非凡的数学天赋。他的老师帕维尔·马克西莫维奇·谢苗诺夫在内战期间坚持教学，并以各种可能的方式鼓励格尔雄·伊赫列维奇对数学的热情。1922年从七年制学校毕业后，G I 德林费尔德在1927年进入基辅大学（当时的基辅公共教育学院）之前，曾做过鞋匠学徒和锯木厂工人。在那里，他师从米哈伊尔·皮利波维奇·米哈伊尔·克拉夫丘克（1892-1942），后者邀请他参加自己的讨论班，并引导他进行独立的科学工作。他的研究由Georgii Vasilovich Pfeiffer指导，他于1931年从基辅学院毕业。他从1944年到1962年担任哈尔科夫大学数学系主任，但在第二次世界大战期间，他为疏散到巴什基里亚的乌克兰苏维埃社会主义共和国科学院工作。到1950年，他已是哈尔科夫数学研究所的副所长，但该研究所在当年根据斯大林的命令被关闭。格尔雄·德林费尔德还在哈尔科夫数学会中发挥了重要作用。他研究微分几何，特别是测度论和积分。

德林费尔德 是个神童，斯韦特兰娜·吉托米尔斯卡亚 说他[6]：-

……六岁时就能做数学，实在让人瞠目结舌。

德林费尔德 就读于哈尔科夫物理数学第27学校，这是一所为有天赋的学生开设的专门学校，他的数学职业生涯就是从这所学校开始的（[20] 或 [21]）：-

德林费尔德 在学生时代就写出了他的第一篇发表论文。他在其中证明了 戈弗雷·哈罗德·哈代 的经典著作《不等式》风格的一个漂亮结果，并解决了一个 罗伯特·亚历山大·兰金 为之写了两篇注记的问题。这篇论文至今读来仍饶有趣味。

他于1969年11月24日将这篇论文 A cyclic inequality（俄文）提交给 Matematicheskie Zametki。该文于1971年2月发表，同年，英文译本发表在 Maths Notes 上。

1969年，他十五岁时代表苏联参加了在罗马尼亚布加勒斯特举行的国际数学奥林匹克竞赛，并以满分40分的成绩获得金牌——这是一项令人难以置信的成就。当时，他是获得最高分的最年轻参赛者，但此后已有另外三人做到了这一点。他从1969年到1974年在莫斯科国立大学学习。我们注意到，他进入大学时年仅十五岁。在 [19] 的采访中，他谈到了当时在苏联身为犹太人的种种困难：-

……多年来（大约1948—1987年），反犹主义是（未公开的）政府政策，而不是个人的主动行为。莫斯科国立大学校长伊万·彼得罗夫斯基和许多其他人抵制这一政策（这需要他们付出大量努力）。另一方面，也有一些有影响力的数学家（例如当时的弗拉基米尔·安德烈耶维奇·斯捷克洛夫研究所所长），他们把个人的反犹主义加到了国家的反犹主义之上。

他于1974年毕业，留在莫斯科国立大学在尤里·伊万诺维奇·马宁的指导下进行研究。金兹堡写道[10]：-

[德林费尔德]对数学的见解在很大程度上受到他的导师尤里·伊万诺维奇·马宁以及代数几何讨论班（尤里·伊万诺维奇·马宁的讨论班）的影响，该讨论班在莫斯科国立大学定期活动了大约二十年。

德林费尔德于1977年完成研究生学业，并于1978年在莫斯科大学通过了其“副博士”学位论文答辩。“副博士”学位论文相当于英国或美国的博士学位。我们注意到，到1978年，德林费尔德已有十三篇论文发表，并在罗伯特·朗兰兹猜想方面证明了显著的结果。然而，尽管德林费尔德才华出众，他仍难以在莫斯科获得职位。这基本上有两个原因。当然，他的犹太出身意味着他遭受反犹主义之害，正如他自己在上述引文中所描述的那样，但官方上，苏联实行一项政策，即人们的住址写在护照上，只允许在护照上出现的城市工作。由于德林费尔德护照上的住址不是莫斯科，他无法在那里找到工作。他去了乌法，乌拉尔山脉中的一个工业中心，在那里获得了在巴什基尔大学教数学的职位，该大学是该市的几所大学之一。1981年，他搬到哈尔科夫，与父母同住。他在哈尔科夫的国家乌克兰科学院下属的B I Verkin低温物理工程研究所获得了一个职位。这需要几位同事付出大量努力[19]：-

我最终得以幸存， thanks to the fact that in 1981 I was hired at the Kharkov Institute of Low Temperatures. 这并不容易：尽管V A Marchenko和该研究所的其他数学家想录用我，研究所所长B I Verkin也不反对，但还需要一封来自尼古拉·博戈柳博夫给所长的信，以保护Verkin免受全能的苏共地区委员会的干预（并且为了组织尼古拉·博戈柳博夫的信，需要我的科学导师尤里·伊万诺维奇·马宁和其他人的努力）。

然而，这一任命也有其问题[19]：-

在低温研究所工作期间，夏天我们得去集体农庄劳动（1984年必须干满40天）。这对我来说很困难，原因有二。首先，我体力不强；其次，这纯粹是烦人：我有自己的数学课程计划，而地区委员会书记却打电话给研究所，嗯，我想实验科学家们和我一样对此都感到愤怒。

德林费尔德于1986年在乔治·伯克利举行的国际数学家大会上作了重要报告。报告题为Quantum groups，回顾了德林费尔德和M Jimbo在海因茨·霍普夫代数（量子群）方面取得的成果。他讨论了量子群和量子化的概念，还谈到了西莫恩·德尼·泊松群、索菲斯·李双代数和经典Yang-Baxter方程。报告是这样开始的：-

这是一份关于海因茨·霍普夫代数（或量子群，两者大致相同）近期工作的报告，其动机来自量子逆散射方法（QISM），这是一种构造和研究可积量子系统的方法，主要由L D Faddeev及其合作者发展起来。本文中的大多数定义、构造、例子和定理都受到QISM的启发。尽管如此，我将从这些定义、构造等开始，然后解释它们与QISM的关系。这样我就颠倒了该主题的历史，希望能使其逻辑更清晰。

什么是量子群？回想一下，在经典力学和量子力学中都有两个基本概念：态和可观测量。在经典力学中，态是流形M的点，可观测量是M上的函数。在量子情形中，态是大卫·希尔伯特空间H的l维子空间，可观测量是H中的算子（我们忘记自伴条件）。从可观测量的角度更容易理解经典力学和量子力学之间的关系。在经典力学和量子力学中，可观测量都构成一个结合代数，在经典情形中是交换的，在量子情形中是非交换的。所以量子化有点像用非交换代数代替交换代数。

1988年，德林费尔德在莫斯科的弗拉基米尔·安德烈耶维奇·斯捷克洛夫研究所答辩了他的“博士”学位论文。“博士”学位论文相当于德国的特许任教资格。1990年8月21日，德林费尔德在日本京都的国际数学家大会上被授予约翰·查尔斯·菲尔兹奖章[35]：-

... 因其在量子群方面的工作以及在数论方面的工作。

A Jaffe 和 巴里·马聚尔 在 [3] 中写到 德林费尔德 的工作，该工作导致了 约翰·查尔斯·菲尔兹 奖章的授予：-

德林费尔德 的兴趣只能被描述为“广泛”。它们不仅涵盖代数几何和数论方面的工作，而且他最近的想法采取了截然不同的方向：他一直在做由物理学激发的数学问题的重大工作，包括相对较新的量子群理论。

德林费尔德 无法被轻易归类……他的突破具有人们期望革命性数学发现所具有的魔力：它们似乎具有取之不尽的推论。另一方面，它们似乎是极具个人色彩的数学作品：“只有 德林费尔德 才能想到它们！”但矛盾的是，它们似乎显然是自然的；一旦被理解，“每个人都应该想到它们！”

尤里·伊万诺维奇·马宁 在日本京都举行的国际数学家大会上的演讲（他本人无法亲自发表，而是由 Michio Jimbo 宣读）以这些话结束 [20]：-

我希望我向你们传达了 德林费尔德 工作的某种广泛性、概念丰富性、技术力量和美感，我们现在正以 约翰·查尔斯·菲尔兹 奖章来表彰他。对我来说，近距离观察这个才华横溢的头脑的快速发展是一种乐趣和荣幸，它教会了我很多。

有关 尤里·伊万诺维奇·马宁 演讲的更多摘录，请参见 THIS LINK。

德林费尔德的主要成就是证明了函数域上$GL(2)$的罗伯特·朗兰兹猜想，以及他在量子群理论方面的工作。虽然他仅证明了罗伯特·朗兰兹猜想的一个特例，但德林费尔德在解答中引入了重要的新思想，并取得了真正的突破。他在证明中引入了椭圆模的概念，这一概念正引领数论中一个全新的课题。迈克尔·阿蒂亚所研究的数学与数学物理之间的相互作用导致了瞬子的引入——即某些非线性偏微分方程系统，自对偶杨-米尔斯方程的解，这些方程最初由物理学家在量子场论的背景下引入。德林费尔德和尤里·伊万诺维奇·马宁利用代数几何的思想致力于瞬子的构造。德林费尔德说[19]：-

我的一项工作（与迈克尔·阿蒂亚、尤里·伊万诺维奇·马宁和N J Hitchin合作）致力于所谓的瞬子。这是代数几何中为理论物理做出有用贡献的最早例子之一，给物理学家留下了深刻印象。物理学家是聪明人；他们计数（并不总是使用严格方法）比数学家好得多，很难用任何事情让他们惊讶。但在我们的工作之后，他们意识到他们需要代数几何，10年后大多数物理学家都学会了它。现在从事弦理论工作的人都知道代数几何。20世纪80年代，我研究称为量子群的数学对象。这些对象是在量子可积系统理论的影响下发明的，该理论由物理学家开创，然后由Ludwig Dmitrievich Faddeev的列宁格勒学派的数学家发展。我试图理解他们的工作，这并不容易。然后我意识到其中起关键作用的是海因茨·霍普夫代数，可以称为量子群。这种方法有助于理解许多先前获得的结果，也获得了许多新结果。这个领域的许多人都喜欢这种方法并开始使用它。量子群已在数学物理学家中得到应用。

Chari和Thakur写道[7]：-

德林费尔德引入了德林费尔德模，并在年仅20岁时解决了罗伯特·朗兰兹纲领的很大一部分，24岁时完成了$GL(2)$情形。德林费尔德在罗伯特·朗兰兹猜想、量子群、p进统一化等方面的工作说明了他对强大而复杂技术的掌握。另一方面，他（与Vladut合作）的一页证明给出了定义在阶为$p^{2^n}$的有限域上的曲线点数的尖锐渐近上界，仅使用了高中数学巧妙地应用于众所周知的结果。他还给出了一页证明，证明二维或三维球面上任何旋转不变的有限可加测度与昂利·勒贝格测度成比例，通过巧妙地组合已知结果。

1992年，德林费尔德当选为乌克兰科学院成员。他继续住在哈尔科夫，直到1998年移居美国。1998年12月，他被任命到芝加哥大学。他在采访[19]中谈到了移居西方：-

1990年，我已经在哈尔科夫工作（National Academy of Sciences of Ukraine低温物理技术研究所），然后我有机会在西方找到工作，但我拒绝了。1998年，我收到了几所美国大学的工作邀请，我和妻子决定应该接受其中一个。我们住在乌克兰，经济正在下滑（例如，人们提前六个月领取工资）。我们的儿子九岁，我们想知道他长大后会在什么样的世界里生活。很明显，这是一个野蛮资本主义的世界。我们决定搬到一个更文明的资本主义国家。这并不容易，因为我有年迈的父母，我不能离开他们。芝加哥大学能够雇用我的母亲并为她提供健康保险。

德林费尔德被任命到芝加哥大学，同事们表达了他们的喜悦[17]：-

哈佛大学数学教授巴里·马聚尔在得知德林费尔德接受了芝加哥大学的职位后说：“这是一个极好的任命。”巴里·马聚尔说，他认为德林费尔德和Beilinson是俄罗斯最有影响力的两位数学家。“毫无疑问，芝加哥大学在这方面取得了巨大的成功。这些都是伟大的数学家，”巴里·马聚尔说。

据数学系主任兼Louis Block数学教授Robert Fefferman称，约翰·查尔斯·菲尔兹奖章相当于数学界的诺贝尔奖。该奖章每四年在国际数学家大会上颁发给不少于两位且不超过四位40岁以下的数学家。查尔斯·费夫曼称德林费尔德为“世界上最伟大的代数学家之一”。

德国波恩马克斯·普朗克数学研究所所长尤里·伊万诺维奇·马宁也给出了同样强烈的评价。“德林费尔德的工作深刻影响了过去二十年的数学界，”尤里·伊万诺维奇·马宁说，他曾在20世纪80年代在莫斯科大学担任德林费尔德和Beilinson的博士论文导师，并且是1998年柏林国际数学家大会约翰·查尔斯·菲尔兹奖委员会主席。“几部研究专著、讨论班笔记和数百篇论文都致力于他创建的两个数学新篇章——所谓的德林费尔德模和量子群。”

Alexander A Beilinson，也是尤里·伊万诺维奇·马宁的学生，于1998年被任命到芝加哥大学，就在德林费尔德之前不久。Beilinson和德林费尔德相识多年，在成为芝加哥的同事之前已经合作过两篇论文：Affine Kac-Moody algebras and polydifferentials（1994）和Quantization of Hitchin's fibration and Langlands' program（1996）。他们在芝加哥的合作促成了一本合著书籍Chiral algebras的出版，该书由美国数学学会于2004年出版。Francisco J Plaza Martin在评论[42]中写道：-

本书从代数几何的角度全面介绍了手性代数理论。毫无疑问，它将成为该主题的标准参考书。……手性代数起源于数学物理中关于共形场论的研究。在数学方面，手性代数的局部理论与顶点代数[理查·博赫兹]理论重叠，后者通常用表示论技术来研究。在这两种方法中，“算子乘积展开”形式体系起着至关重要的作用。正如作者所说，他们研究手性代数的动机是在D-模框架下理解几何自守形式，以及描述仿射马克·卡茨-Moody代数表示范畴的谱分解。

德林费尔德较晚的文章之一是Infinite-dimensional vector bundles in algebraic geometry: an introduction。德林费尔德在论文的引言中写道：-

这项工作的目标是表明，存在一个合理的代数几何意义下的向量丛概念，其纤维是无限维局部线性紧的，并且这些对象会“自然地”出现。我们的方法基于1958年至1972年间由H Bass、L Gruson、厄文·卡普兰斯基、M Karoubi和M Raynaud在代数中发现的一些结果和思想。

德林费尔德于2001年3月1日被任命为芝加哥大学Harry Pratt Judson杰出服务教授。2008年，他当选为美国艺术与科学院。2016年，他当选为国家科学院。

2018年，德林费尔德和Beilinson共同获得了沃尔夫奖[37]：-

……因他们在代数几何（一个将抽象代数与几何相结合的领域）、数学物理和表示论——一个有助于理解复杂代数结构的领域——方面的开创性工作。

引文继续写道[37]：-

德林费尔德对纯数学的各个分支做出了巨大贡献，主要是代数几何、算术几何和表示论——以及数学物理。以他命名的数学对象——“德林费尔德模”、“德林费尔德 Shtukas”、“德林费尔德上半平面”、“德林费尔德结合子”，以及许多其他对象，以至于他的一位推荐人开玩笑说：“人们可能会认为‘德林费尔德’是一个形容词，而不是一个人的名字”。

关于与德林费尔德相关的引文的所有部分，见THIS LINK。

2023年，德林费尔德和丘成桐共同获得邵逸夫奖。引文写道[44]：-

2023年邵逸夫数学科学奖由美国芝加哥大学Harry Pratt Judson杰出服务数学教授德林费尔德和中国清华大学讲席教授丘成桐平分，以表彰他们在数学物理、算术几何、微分几何和埃里希·凯勒几何方面的贡献。

关于该引文的更多摘录，见THIS LINK。

在[11]中，Victor Ginzburg概述了德林费尔德截至2005年的贡献。他以以下例子结尾，展示了德林费尔德非凡的洞察力：-

我想以几个例子来结束，我相信这些例子表明，德林费尔德的许多洞见仍在等待“发现”。其中一个例子与辛反射代数有关，这是P Etingof和我在2002年引入的一个概念。在这个课题上工作数年后，我们发现（2005年1月），辛反射代数的定义本质上包含在德林费尔德15年前写的论文“退化仿射埃里希·赫克代数和杨子”的两行中！尽管这篇论文本身非常有名，但似乎没有人足够仔细地阅读过德林费尔德那两行写得非常密集的文字。

第二个例子同样令人惊叹。我正在准备一门表示论课程，我定期在芝加哥教授这门课。Volodya向我提到，他有一些旧的笔记，里面有表示论的习题，是20世纪80年代为他在哈尔科夫的学生写的。像往常一样，Volodya的笔记非常系统；它们既包含习题也包含解答。在笔记的中间某处，我发现了一段关于“q-模拟”的题外话，其中包含的计算本质上等价于Beilinson、乔治·卢斯蒂格和MacPherson在10年后发现的量子群的重要几何构造！

Vladimir Drinfeld was born into a Jewish mathematical family. He was the son of the mathematics professor Gershon Ikhelevich Drinfeld (29 February 1908-18 August 2000) and his wife, the classical philologist Frida Iosifovna Lutskaya-Litvak (1921-2011). Since Vladimir's father was a professor of mathematics at Kharkov University, we shall give some details of his life before giving biographical details of his son Vldimir.

Gershon Ikhelevich Drinfeld was born in Starokostiantyniv, Ukraine and showed remarkable mathematical talents from a young age. His teacher, Pavel Maksimovich Semenov, taught through the civil war and encourage Gershon Ikhelevich's passion for mathematics in every possible way. After graduating from seven-year school in 1922, G I Drinfeld was a shoemaker's apprentice and a sawmill worker before he entered Kiev University (then the Kiev Institute of Public Education) in 1927. There he was taught by Mikhail Pylypovych Kravchuk (1892-1942) who invited him to take part in his seminar and introduced him to independent scientific work. His research was supervised by Georgii Vasilovich Pfeiffer and he graduated from the Kiev Institute in 1931. He became head of the Mathematics Department at Kharkov University from 1944 to 1962 but during World War II he worked for the Academy of Sciences of the Ukrainian SSR which was evacuated to Bashkiria. By 1950 he was deputy director of the Kharkov Institute of Mathematics but it was closed in that year on the orders of Stalin. Gershon Drinfeld also played a major role in the Kharkov Mathematical Society. He worked on differential geometry, particularly on measure theory and integration.

Vladimir Drinfeld was a child prodigy and Svetlana Jitomirskaya said he [6]:-

... could do maths at age six that really made people's jaws drop.

Drinfeld studied at Kharkov Physics and Mathematics School no. 27, a specialised school for talented pupils, and his mathematical career started while he was at this school ([20] or [21]):-

Drinfeld has written his first published paper when he was a schoolboy. He proved there a nice result in the style of Hardy's classic treatise "Inequalities" and solved a problem to which R A Rankin devoted two notes. This paper still makes interesting reading.

He submitted this paper, A cyclic inequality (Russian), to Matematicheskie Zametki on 24 November 1969. It was published in February 1971 and, in the same year, an English translation was published in Maths Notes.

In 1969, at the age of fifteen, he represented the Soviet Union at the International Mathematical Olympiad in Bucharest, Romania, and was awarded a gold medal after obtaining full marks, namely 40 points - an incredible achievement. At that time, he was the youngest competitor to achieve the highest score but three others have achieved this since. He studied at Moscow State University from 1969 until 1974. We note that he was only fifteen years of age when he entered the university. In the interview [19] he spoke about the difficulties of being a Jew in the USSR at this time:-

... for many years (roughly 1948-1987) anti-Semitism was the (unpublicised) government policy, and not the initiative of individuals. The rector of Moscow State University I G Petrovsky and many others resisted this policy (this required a lot of effort from them). On the other hand, there were influential mathematicians (for example, the then director of the Steklov Institute) who added their personal anti-Semitism to the state one.

He graduated in 1974 and remained at Moscow State University to undertake research under Yuri Ivanovich Manin's supervision. Ginzburg writes [10]:-

[Drinfeld's] vision of mathematics was, to a great extent, influenced by Yu I Manin, his advisor, and by the Algebraic Geometry Seminar (Manin's Seminar) that functioned with regularity at Moscow State University for about two decades.

Drinfeld completed his postgraduate studies in 1977 and he defended his "candidate" thesis in 1978 at Moscow University. The "candidate" thesis is the Russian equivalent of the British or American Ph.D. We note that by 1978, Drinfeld had thirteen papers in print and had proved remarkable results concerning the Langlands conjectures. Despite being extraordinarily talented, however, it was difficult for Drinfeld to obtain a position in Moscow. There were basically two reasons for this. Certainly his Jewish origins meant that he suffered from anti-Semitism, as he himself described in the above quote, but officially the Soviet Union operated a policy that people had their addresses in their passports and were only allowed to work in the town which appeared in this address. Since the address which was in Drinfeld's passport was not Moscow, he could not get a job there. He went to Ufa, an industrial centre in the Ural mountains, where he obtained a position teaching mathematics at Bashkir University, one of several universities in the city. In 1981 he moved to Kharkov and lived with his parents. He obtained a position working at the B I Verkin Physical Engineering Institute of Low Temperatures, part of the National Ukrainian Academy of Sciences, in Kharkov. This required much effort from several colleagues [19]:-

I ultimately survived thanks to the fact that in 1981 I was hired at the Kharkov Institute of Low Temperatures. It was not easy: although V A Marchenko and other mathematicians from this institute wanted to take me, and the director of the institute B I Verkin was not against it, but a letter from N N Bogolyubov to the director was also needed in order to protect Verkin from the all-powerful regional committee of the CPSU (and to organise Bogolyubov's letter, it took the efforts of my scientific supervisor Yu I Manin and other people).

The appointment, however, had its problems [19]:-

While working at the Institute of Low Temperatures, we had to work on a collective farm in the summer (in 1984 this had to be done for 40 days). This was difficult for me for two reasons. Firstly, I'm not physically strong, and secondly, it was just annoying: I have my own mathematics lesson plan, and the regional committee secretary calls the institute and, well, I think the experimental scientists were as infuriated by all this as I was.

Drinfeld gave an important lecture at the International Congress of Mathematicians in Berkeley in 1986. Entitled Quantum groups, the talk reviewed the results obtained by Drinfeld and M Jimbo on Hopf algebras (quantum groups). He discussed the concepts of quantum groups and quantisation, and also talked about Poisson groups, Lie bi-algebras and the classical Yang-Baxter equation. The talk began as follows:-

This is a report on recent works on Hopf algebras (or quantum groups, which is more or less the same) motivated by the quantum inverse scattering method (QISM), a method for constructing and studying integrable quantum systems, which was developed mostly by L D Faddeev and his collaborators. Most of the definitions, constructions, examples, and theorems in this paper are inspired by the QISM. Nevertheless I will begin with these definitions, constructions, etc. and then explain their relation to the QISM. Thus I reverse the history of the subject, hoping to make its logic clearer.

What is a quantum group? Recall that both in classical and in quantum mechanics there are two basic concepts: state and observable. In classical mechanics states are points of a manifold M and observables are functions on M. In the quantum case states are l-dimensional subspaces of a Hilbert space H and observables are operators in H (we forget the self-adjointness condition). The relation between classical and quantum mechanics is easier to understand in terms of observables. Both in classical and in quantum mechanics observables form an associative algebra which is commutative in the classical case and non-commutative in the quantum case. So quantisation is something like replacing commutative algebras by non-commutative ones.

In 1988 Drinfeld defended his "doctor" thesis at Steklov Institute, Moscow. The "doctor" thesis is the Russian equivalent of the German habilitation. On 21 August 1990 Drinfeld was awarded a Fields Medal at the International Congress of Mathematicians in Kyoto, Japan [35]:-

... for his work on quantum groups and for his work in number theory.

A Jaffe and B Mazur write in [3] about Drinfeld's work which led to the award of the Fields Medal:-

Drinfeld's interests can only be described as "broad". Not only do they span work in algebraic geometry and number theory, but his most recent ideas have taken a strikingly different direction: he has been doing significant work on mathematical questions motivated by physics, including the relatively new theory of quantum groups.

Drinfeld defies any easy classification ... His breakthroughs have the magic that one would expect of a revolutionary mathematical discovery: they have seemingly inexhaustible consequences. On the other hand, they seem deeply personal pieces of mathematics: "only Drinfeld could have thought of them!" But contradictorily they seem transparently natural; once understood, "everyone should have thought of them!"

Manin ends his address to the International Congress of Mathematicians in Kyoto, Japan (which he could not give in person but was read by Michio Jimbo) with these words [20]:-

I hope that I conveyed to you some sense of broadness, conceptual richness, technical strength and beauty of Drinfeld's work for which we are now honouring him with the Fields Medal. For me, it was a pleasure and a privilege to observe at a close distance the rapid development of this brilliant mind which taught me so much.

For more extracts from Manin's address, see THIS LINK.

Drinfeld's main achievements are his proof of the Langlands conjecture for $GL(2)$ over a functional field; and his work in quantum group theory. Although he only proved a special case of the Langlands conjecture, Drinfeld has introduced important new ideas in his solution and made a real breakthrough. He introduced the idea of an elliptic module in his proof and this notion is leading to a whole new topic within number theory. The interactions between mathematics and mathematical physics studied by Atiyah led to the introduction of instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory. Drinfeld and Manin worked on the construction of instantons using ideas from algebraic geometry. Drinfeld said [19]:-

One of my works (joint with M F Atiyah, Yu I Manin and N J Hitchin) was devoted to the so-called instantons. This was one of the first examples of something useful for theoretical physics being done in algebraic geometry, which impressed physicists. Physicists are smart people; they can count (not always using strict methods) much better than mathematicians, and it is difficult to surprise them with anything. But after our work they realised that they needed algebraic geometry, and after 10 years most physicists had learned it. People working in string theory now know algebraic geometry. In the 1980s I worked on mathematical objects called quantum groups. These objects were invented under the influence of the theory of quantum integrable systems, which was started by physicists and then developed by mathematicians of the Leningrad school of Ludwig Dmitrievich Faddeev. I tried to understand their work, which was not easy. Then I realised that the key role there is played by Hopf algebras, which can be called quantum groups. This approach helped the understanding of many previously obtained results and also obtain a number of new ones. Many people working in this field liked this approach and began to use it. Quantum groups have come into use among mathematical physicists.

Chari and Thakur write [7]:-

Drinfeld introduced Drinfeld modules and solved a substantial part of the Langlands programme when he was just 20 years old and completed the $GL(2)$ case when he was 24. Drinfeld's work on Langlands conjectures, quantum groups, p-adic uniformizations etc. illustrate his mastery over powerful and involved techniques. On the other hand, his one page proof (jointly with Vladut) giving a sharp asymptotic upper bound for the number of points of a curve defined over a finite field of order $p^{2^n}$, uses only high-school algebra applied nicely to well-known results. He also gave a one page proof of the fact that any rotation invariant finitely additive measure on the two or three dimensional sphere is proportional to Lebesgue measure by using a clever combination of known results.

In 1992 Drinfeld was elected a member of the Ukrainian Academy of Sciences. He continued to live in Kharkov until 1998 when he emigrated to the United States. In December 1998, he was appointed to the University of Chicago. He spoke about the move to the West in the interview [19]:-

In 1990, I already worked in Kharkov (Physico-Technical Institute of Low Temperatures of the National Academy of Sciences of Ukraine), then I had the opportunity to get a job in the West, but I refused. In 1998, I received several job offers from American universities, and my wife and I decided that we should accept one of them. We lived in Ukraine, whose economy was going downhill (for example, people received advances on their salaries six months late). Our son was nine years old, and we wondered what kind of world he would live in when he grew up. It was clear that this was a world of wild capitalism. We decided to move to a country of more civilised capitalism. It was not easy, since I had elderly parents, I could not leave without them. The University of Chicago was able to employ my mother and provide her with health insurance.

On Drinfeld's appointment to Chicago, colleagues expressed their delight [17]:-

Barry Mazur, professor of mathematics at Harvard University, upon learning that Drinfeld had accepted a position at Chicago, said "It's a wonderful appointment." Mazur said he regards Drinfeld and Beilinson as Russia's two most influential mathematicians. "There's no question that Chicago has achieved a great coup there. These are great mathematicians," Mazur said.

A Fields Medal is the equivalent of a Nobel Prize in mathematics, according to Robert Fefferman, Chairman of the Mathematics Department and the Louis Block Professor in Mathematics. The medals are awarded to no fewer than two and no more than four mathematicians under the age of 40 every four years at the International Congress of Mathematicians. Fefferman called Drinfeld "one of the greatest algebraists in the world."

Yuri Manin, director of the Max Planck Institute for Mathematics in Bonn, Germany, offered an equally strong assessment. "Drinfeld's work deeply influenced the world of mathematics of the last two decades," said Manin, who served as Drinfeld's and Beilinson's Ph.D. thesis adviser at Moscow University in the 1980s and was the chairman of the Fields Prize Committee at the Berlin ICM 1998. "Several research monographs, Seminar Notes and hundreds of papers were dedicated to the two new chapters of mathematics created by him - the so-called Drinfeld modules and quantum groups."

Alexander A Beilinson, also a student of Manin's, had been appointed to the University of Chicago in 1998, just a short time before Drinfeld. Beilinson and Drinfeld had known each other for many years and had already collaborated on two papers before becoming colleagues in Chicago: Affine Kac-Moody algebras and polydifferentials (1994) and Quantization of Hitchin's fibration and Langlands' program (1996). Their collaboration in Chicago led to the publication of a jointly authored book Chiral algebras published by the American Mathematical Society in 2004. Francisco J Plaza Martin writes in a review [42]:-

This book presents a comprehensive approach to the theory of chiral algebras from the point of view of algebraic geometry. Without a doubt, it will become a standard reference on the subject. ... Chiral algebras arose in mathematical physics in the study of conformal field theory. On the mathematical side, the local theory of chiral algebras overlaps the theory of vertex algebras [R E Borcherds], which are normally studied with representation theory techniques. In these two approaches the "operator product expansion" formalism plays an essential role. As the authors say, their motivation for studying chiral algebras was the understanding of geometric automorphic forms in the D-module setting as well as the description of a spectral decomposition of the category of representations of an affine Kac-Moody algebra.

One of Drinfeld's most later articles is Infinite-dimensional vector bundles in algebraic geometry: an introduction. Drinfeld writes in the introduction to the paper:-

The goal of this work is to show that there is a reasonable algebro-geometric notion of vector bundle with infinite-dimensional locally linearly compact fibers and that these objects appear 'in nature'. Our approach is based on some results and ideas discovered in algebra during the period 1958-1972 by H Bass, L Gruson, I Kaplansky, M Karoubi, and M Raynaud.

Drinfeld was named Harry Pratt Judson Distinguished Service Professor at the University of Chicago on 1 March 2001. In 2008 he was elected to the American Academy of Arts and Sciences. He was elected to the National Academy of Sciences in 2016.

In 2018 Drinfeld and Beilinson were jointly awarded the Wolf Prize [37]:-

... for their ground-breaking work in algebraic geometry (a field that integrates abstract algebra with geometry), in mathematical physics and in presentation theory, a field which helps to understand complex algebraic structures.

The citation continues [37]:-

Drinfeld has contributed greatly to various branches of pure mathematics, mainly algebraic geometry, arithmetic geometry and the theory of representation - as well as mathematical physics. The mathematical objects named after him - the "Drinfeld Modules", the "Drinfeld Shtukas", the "Drinfeld Upper Half Plane", the "Drinfeld Associator", and so many others that one of his endorsers jokingly said, "one could think that "Drinfeld" was an adjective, not the name of a person".

For all the parts of the citation relevant to Drinfeld, see THIS LINK.

In 2023 Drinfeld and Shing-Tung Yau were jointly awarded the Shaw Prize. The citation states [44]:-

The Shaw Prize in Mathematical Sciences 2023 is awarded in equal shares to Vladimir Drinfeld, Harry Pratt Judson Distinguished Service Professor of Mathematics at the University of Chicago, USA and Shing-Tung Yau, Chair Professor at Tsinghua University, PRC, for their contributions related to mathematical physics, to arithmetic geometry, to differential geometry and to Kähler geometry.

For further extracts from this citation, see THIS LINK.

In [11] Victor Ginzburg gives an overview of Drinfeld's contributions up to 2005. He ends with the following examples showing Drinfeld's remarkable insight:-

I would like to finish with a couple of examples that show, I believe, that many of Drinfeld's insights are still awaiting "discovery." One such example is related to symplectic reflection algebras, a notion introduced by P Etingof and myself in 2002. After having worked on the subject for several years, we discovered (in January 2005) that the definition of symplectic reflection algebras was essentially contained in two lines of Drinfeld's paper "Degenerate Affine Hecke Algebras and Yangians," written 15 years earlier! Although the paper itself is very well known, it seems nobody has read those two lines of Drinfeld's very densely written text carefully enough.

The second example is equally amazing. I was preparing for a course on representation theory, which I teach regularly in Chicago. Volodya mentioned to me that he had some old notes with exercises on representation theory, written for his students in Kharkov back in the 1980s. As usual, Volodya's notes were very systematic; they contained both the exercises and the solutions. Somewhere in the middle of the notes, I found a digression on "q-analogues" that contained computations equivalent, essentially, to the important geometric construction of the quantum group discovered by Beilinson, Lusztig, and MacPherson 10 years later!