罗伯特·朗兰兹Robert Langlands

罗伯特·朗兰兹的父亲是朗兰兹，母亲是Kathleen Johanna Phelan。他有两个妹妹，其中一个叫Mary Fran 朗兰兹，现在是Mary McArthur。虽然出生在温哥华稍南的新威斯敏斯特，但他在更北边的兰湾和默特尔角之间海岸上的一个小村庄度过了大约五年。他的父亲在伐木场工作，因工作原因搬到了那个小村庄。当朗兰兹该上小学时，全家回到了新威斯敏斯特，朗兰兹在那里开始了在圣安学院的学业。这所宗教学校由圣安修女会于1865年建立，教师是修女；学校于1968年关闭。在这所学校读了三年，期间他完成了四年的课程，之后他转到圣彼得学校完成小学教育。这次转学是因为当时圣安学院变成了女子学校，而圣彼得学校则成为相应的男子学校。朗兰兹在圣安学院过得很愉快，但在圣彼得学校却不开心。1946年，全家再次搬家，这次搬到了海岸边的白石镇，位于新威斯敏斯特以南。

在White Rock，朗兰兹的家族经营着朗兰兹木工与建材供应公司，这使他们比镇上许多人富裕一些。他就读于南素里的Semiahmoo高中。这所学校成立于1940年，有许多老师[13]：-

……只是二战退伍军人，他们被给予教师职位更多是感谢他们在军队中的服务。

朗兰兹上学期间做过各种工作来赚钱，例如他收集空啤酒瓶，每打能得到一点小费；他有一份送报纸的路线，每天花$1\large\frac{1}{2}\normalsize$小时收集和投递，每周六天；周末和暑假他还在朗兰兹木工与建材供应公司工作。当然，在他上学的大部分时间里，他甚至没有考虑过上大学，但在最后一年，即12年级，他受到了英语文学老师Crawford Vogler的极大关注。Vogler告诉他：“如果你不上大学，那将是对上帝赐予的才能的背叛。”他还遇到了Charlotte Lorraine Cheverie，她的父亲Lorenzo Francis Cheverie给了他一本书，讲述著名科学家的故事。这本书使他对科学产生了兴趣，而Charlotte和Vogler一样，鼓励他申请不列颠哥伦比亚大学。他说[2]：-

我对这些评论感到受宠若惊，我的雄心被唤起，我当场决定参加入学考试。我努力学习并成功了，甚至赢得了大学的一小笔奖学金。

关于朗兰兹的家庭和教育的更多细节，见THIS LINK。

朗兰兹于1953年从塞米亚穆高中毕业，同年晚些时候开始在不列颠哥伦比亚大学学习。在这个阶段，他不知道自己应该学习哪些科目。他与一位大学顾问讨论此事，顾问让他参加了能力倾向测试。他在所有科目上的背景都相对薄弱，但当然，在数学方面，有能力的人即使没有深厚的背景知识也能做得非常好。能力倾向测试的结果使顾问建议他可能想成为一名会计师，但他不喜欢这个想法。下一个建议是数学或物理，这更合朗兰兹的意，所以他决定数学将是专攻的科目。第一年的课程他选了法语、英语、物理、化学和数学。在第一年快结束时，他与数学教授Stephen Arthur Jennings（1915-1979）交谈，教授告诉他，要成为数学家必须学习法语、德语和俄语。朗兰兹懂基础法语，在暑假期间通过书籍自学了德语。他在大学第二年选修了俄语课程，以及英国文学、数学、物理和逻辑课程。

朗兰兹在第三年修读的荣誉课程有微分学、积分学、线性代数和代数。他的讲师包括朗兰兹 Christian、Eugene Leimanis和Frederick Goodspeed。在开始第四年之前，他于1956年8月13日与Charlotte Lorraine Cheverie结婚；他们有四个孩子：William、Sarah、朗兰兹和Thomasin。

在本科课程的最后一年，朗兰兹修读了埃瓦里斯特·伽罗瓦理论课程，由李林学讲授；凸性课程，由杰西·道格拉斯 Derry讲授；函数论课程，由朗兰兹 Christian讲授；以及应用数学课程，由Thomas Hull讲授。他还参加了两个讨论班，一个由Harry Davis主讲，关于现代泛函分析；另一个由弗洛伦斯·南丁格尔·大卫 Murdoch主讲，关于诺特环。朗兰兹于1957年获得学士学位，并继续在不列颠哥伦比亚大学攻读硕士学位，于1958年获得。他申请了哈佛、威斯康星和耶鲁的博士项目。三所都录取了他，但只有耶鲁提供经济支持，所以他毫不犹豫地决定接受耶鲁。

朗兰兹随后在耶鲁大学攻读博士学位。他的正式学位论文导师是Cassius Ionescu Tulcea（1923-2021），一位出生于罗马尼亚的数学家，专攻概率论、统计学和数学分析。然而，Tulcea并未建议他学位论文的题目，朗兰兹自己找到了题目。尽管他在耶鲁待了两年，但他在第一年就完成了学位论文的工作。在这两年里，他参加了各种课程，包括：Nelson Dunford的函数分析；Einar Hille的函数分析和半群；Cassius Ionescu Tulcea的索菲斯·李半群及其表示；菲利克斯·白劳德的偏微分方程；以及Stephen Gaal的解析数论。

他于1960年向耶鲁提交了学位论文Semi-groups and representations of Lie groups，并获得了博士学位。朗兰兹写道[57]：-

这篇学位论文有两个相关的部分：一部分关于索菲斯·李半群的表示，另一部分关于与索菲斯·李群的表示相关联的算子。第一部分发表在《加拿大数学杂志》上，但第二部分仅作为一则通告发表在美国国家科学院的会议录中。尽管如此，它有幸被Derek Robinson认真对待，后者将其中一些结果纳入了他关于椭圆算子与李群的书中。

在谈到他的博士论文时，朗兰兹也遗憾地表示它仍然是[57]：-

……我与偏微分方程的唯一一次积极接触，我一直希望回到这个主题，但要以不同的方式。

他现在必须找到一份学术职位。他在采访中说[2]：-

我完成博士学位时真正希望的是留在耶鲁。我爱上了那里的氛围：我在那里拥有从未在别处有过的学习和思考的自由。几位教员鼓励我留下，但我的任命被阻止了，可能是被角谷静夫阻止的。于是我接受了普林斯顿的聘约，在那里我有幸遇到了萨洛蒙·博赫纳，他的鼓励产生了决定性的、具体的后果。我不确定萨洛蒙·博赫纳是否曾理解他为我做了多少。我是一个胆怯的年轻人，而他是一位真正胆怯的老人，所以有些感情从未表达出来。

完成博士研究后，朗兰兹被任命为普林斯顿的讲师，在那里教了七年。1961年，他被提升为讲师，然后成为助理教授——1962年成为副教授。由于此前教授这门课程的埃米尔·阿廷于1958年离开普林斯顿返回德国，他被要求在一周的通知时间内讲授类域论课程。他觉得自己对类域论了解不够，无法讲授这门课程，于是去找萨洛蒙·博赫纳，说他在一周内无法学到足够的知识来讲授这门课程[13]：-

但他坚持，所以我根据谢瓦莱的论文讲授了类域论课程，那是更现代的观点，我讲完了。有三四个学生，他们说从中学到了一些东西。因此，有了这些，我开始思考一个事实：还没有非阿贝尔类域论。有些人，比如埃米尔·阿廷，并不期望会有。所以，我只是意识到了这一点，仅此而已。

从1962年8月到1963年6月，他是普林斯顿高等爱德华·斯图迪数学学院的成员。1964-65年，他在乔治·伯克利加利福尼亚大学度过，作为米勒基金会研究员和阿尔弗雷德·P·斯隆研究员。然后在1967年，他回到耶鲁大学担任正教授。然而，朗兰兹在1967-68年访问土耳其安卡拉，办公室紧邻卡希特·阿尔夫的办公室。在耶鲁五年后，他于1972年7月再次回到普林斯顿，这次是作为高等斯图迪的数学教授。他一直留在高等斯图迪，直到2007年6月退休。2007年7月，他被授予高等斯图迪的荣休称号。

朗兰兹从1960年发表的第一批论文是：On Lie semi-groups（1960）；Some holomorphic semi-groups（1960）；The dimension of spaces of automorphic forms（1963）；Dimension of spaces of automorphic forms（1966）；The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups（1966）；和Eisenstein series （1966）。

1988年，朗兰兹获得了国家科学院数学奖。他是该奖项的第一位获得者，该奖项由国家科学院设立；2012年更名为玛丽亚姆·米尔扎哈尼数学奖。授予朗兰兹的奖项引文认可了他的：-

……非凡的远见，使群表示论与自守形式论和数论建立了革命性的新关系。

让我们稍微解释一下朗兰兹的工作，正是这些工作使他获得了这一奖项。他刚完成学位论文工作，朗兰兹就开始研究自守形式。在一篇出色的论文中，他应用哈里什-钱德拉的近期结果，得到了某些自守形式空间的维数公式。随后，在接下来的几年里，他在费迪南·艾森斯坦级数上做出了深刻的结果，并进一步应用费迪南·艾森斯坦级数证明了安德烈·韦伊提出的一个数论猜想。

1967年，他给安德烈·韦伊写了一封信，信中包含深刻的数学思想，这些思想至今仍在推动整个数学研究领域。这封信是手写的，共17页，于1967年1月寄出。信中勾勒了后来被称为“朗兰兹猜想”的内容。安德烈·韦伊让人把信打字，这份打字稿在对此主题感兴趣的数学家中广泛流传。Casselman在[44]中写道，这封信包含：-

……一系列影响深远且异常准确的猜想，涉及数论、自守形式和表示论。这些猜想构成了一个仍在进行中的纲领的核心，并在三个学科中都发挥了核心作用。

朗兰兹的其他信件也被证明极为重要。1967至1968年他在安卡拉时，曾写信给让-皮埃尔·塞尔，信中的想法最终被表述为皮埃尔·德利涅-朗兰兹猜想；这一猜想最终由大卫·卡日丹和乔治·卢斯蒂格证明。

在[38]中，Judy Mendaglio试图用初等概念来介绍朗兰兹纲领：-

数学中的“万有理论”，即朗兰兹纲领，是一组猜想，旨在统一数学不同分支的知识。其思想是，数学某一领域的问题，用该领域可用的工具来分析可能非常困难。然而，如果问题中的结构能与另一个领域中类似的结构相关联，而那个领域有更好的分析工具，那么分析可能会更容易进行，结果也能关联回原问题。这样，原数学领域中更深层的结构就被揭示出来。

例如，考虑一个相对初等的问题，如寻找四维或更高维空间中固体的性质。这是一个几何问题。然而，由于我们难以想象四维或更高维的物体，我们发现分析它们的性质很困难；但是，我们可以将它们的几何结构与已被充分理解的代数结构相关联。我们在代数世界里摸索，获得一两个洞见，然后将这些洞见关联回几何问题。
...
当然，朗兰兹纲领的数学比大多数数学家所理解的还要高深，但根本上，思想是一样的。在朗兰兹纲领中，被关联的数学分支是那些连接我们对数的理解和我们对变化的理解的分支。想想算术遇上几何，它们遇上微积分，然后提高几十个档次。涉及的主要数学分支是数论（研究数的性质和关系）、代数几何（解析几何的表亲）、表示论（涉及集合和映射）以及数学物理。数学家和《爱与数学》的作者Edward Frenkel称朗兰兹的理论为“所有数学的源代码”。

我们上面提到的国家科学院数学奖当然不是朗兰兹因其工作获得的第一个奖项。1975年，他获得了耶鲁大学颁发的威尔伯·克罗斯奖章。1980年，他因对数学研究的杰出贡献获得了加拿大数学会颁发的杰弗里-Williams奖，1982年，他因在自守形式、费迪南·艾森斯坦级数和乘积公式方面的开创性工作获得了美国数学会颁发的法兰克·尼尔森·寇尔数论奖。他继续赢得重大奖项，例如，他与安德鲁·怀尔斯分享了1995-96年沃尔夫数学奖。该奖授予朗兰兹，以表彰他的：-

...在数论、自守形式和群表示领域的开拓性工作和非凡洞见。

他于1972年当选为加拿大皇家学会会士，1981年当选为伦敦皇家学会会士，1993年成为国家科学院成员，2004年成为美国哲学学会成员，2012年成为美国数学会会士。他获得了三十多个荣誉博士学位，来自包括不列颠哥伦比亚大学、麦克马斯特大学、纽约市立大学、滑铁卢大学、巴黎第七大学、麦吉尔大学、多伦多大学和芝加哥大学等机构。

朗兰兹于2000年被法国科学院授予Grande Médaille d'Or。2005年，他因论文Problems in the theory of automorphic forms（1970年）被美国数学会授予Leroy P 凯瑟琳·斯蒂尔研究开创性贡献奖。正是这篇论文引入了现在被称为朗兰兹猜想的内容。2006年，他因[67]获得Frederic Esser Nemmers数学奖：-

...连接表示论、自守形式和数论的基本愿景。

引文包含西北大学数学教授 Kari Vilonen 的引述 [67]：-

朗兰兹 纲领通过研究数论与自守形式这两个不同数学领域的对称性，假设了它们之间的深刻联系。自约 40 年前提出以来，朗兰兹 纲领一直是数学中的统一原则，并指导了数论、自守形式和表示论的研究。最近，它还进入了数学物理领域。在所有这些领域中，它仍然是一个面向未来的研究纲领。

朗兰兹 和 理查德.泰勒 于 2007 年共同获得邵逸夫奖 [59]：-

朗兰兹 开创了一种统一的数学愿景，极大地扩展了前几个世纪数学的遗产，将素数数与对称性联系起来。这种统一源于 卡尔·弗里德里希·高斯 和 大卫·希尔伯特 的互反律理论，现在被称为 朗兰兹 纲领。它提供了一个研究方向，指导了数学家过去四十年，并将在未来许多年继续如此。

获奖后，朗兰兹 作了 Reflections on receiving the Shaw prize 讲座（见 [31]）。Balasubramanian Sury 在评论这篇论文时写道 [63]：-

这是作者在香港获得邵逸夫奖时所作讲座的文本。它读起来绝对引人入胜。这篇精湛阐述的内容如此扣人心弦，几乎不可能不读完就放下文章。因此，评论者不详细描述内容，而是鼓励感兴趣的读者自己阅读文本，只需引用文章中的以下文字：“许多数学家对过去四十年自守形式理论发展的看法与我不同，如果不是根本性的，也肯定是本质上的。一些差异是由于误解，这是该理论与具有不同气质和训练的数学家所从事领域多种关系的自然结果。稍加解释，这些误解就可以消除。这个奖项是一个机会。其他差异是方法论立场冲突的结果，大多未被认识到，当然也未解决。它们的解决肯定需要对主题有比目前更深入的理解。在这次讲座中，我试图描述当前未解决的情况。我的重点将放在我自己的立场上，尽管我在这里的目的不是倡导而是解释它。”对于任何对函子性猜想的历史发展和背后动机感兴趣的人来说，这是必读之作。

2015年，朗兰兹在其150周年之际当选为伦敦数学会荣誉会员。简短引文如下：-

朗兰兹作为同名研究纲领的提出者（1967年）和首位开发者，在数学史上确立了自己的地位。朗兰兹纲领的深刻结果和富有远见的猜想将数论和表示论的核心主题联系起来。

完整引文开头[55]：-

朗兰兹是现代数学的巨人之一。通过将强大的技术能力与非凡的想象力和远见相结合，他展示了如何统一以前被认为相当不同的数学主要领域。更确切地说，朗兰兹将自守形式的传统领域——最初是复变量理论的一部分——转变为一个非常不同的理论，其经典根源现在几乎无法辨认。在朗兰兹手中，自守形式理论已成为统一的宏大力量，代表着数学对称性的基本定律。这些定律支配着数学许多不同部分的内部结构，最显著的是来自数论和算术代数几何。

尼尔斯·阿贝尔奖被公认为授予数学家的最高奖项。它于2018年颁发给朗兰兹[51]：-

... 因为他将表示论与数论联系起来的富有远见的纲领。

以下是新闻稿50中的一段引文：-

朗兰兹因其可追溯到1967年1月的工作而获得了尼尔斯·阿贝尔奖。当时他30岁，是普林斯顿大学的副教授，在圣诞假期期间工作。他给60岁的伟大法国数学家安德烈·韦伊写了一封17页的信，概述了他的一些新的数学见解。

“如果你愿意把它当作纯粹的推测来读，我会很感激，”他写道。“如果不愿意——我相信你手边有废纸篓。”

幸运的是，这封信没有落入废纸篓。他的信引入了一种理论，开创了一种全新的数学思考方式：它暗示了两个此前被认为无关的领域——数论与调和分析——之间的深刻联系。
...
朗兰兹的见解如此激进、如此丰富，以至于他提出的连接这些数学领域的机制催生了一个名为朗兰兹纲领的项目。在过去五十年里，该纲领汇聚了数百位世界上最优秀的数学家。现代数学中没有任何其他项目具有如此广阔的范围、产生了如此多深刻的成果，并有如此多的人致力于其中。其深度和广度不断增长，朗兰兹纲领现在常被描述为数学的大统一理论。

关于2018年尼尔斯·阿贝尔奖更技术性的引文，见THIS LINK。

他继续获得荣誉，例如2019年被任命为加拿大勋章同伴，2020年1月10日Semiahmoo高中安装了一幅壁画，庆祝他对数学的贡献。
见THIS LINK。

Casselman在[44]结尾给出了以下总结：-

[朗兰兹']惊人的洞察力为整整一代从事自守形式与表示论研究的数学家提供了看似无限的、深刻、有趣且最重要的是可接近的问题，供他们钻研。

在访谈13中，最后一个问题是朗兰兹是否有某种非数学的爱好或兴趣。他回答说：-

爱好？我没有任何爱好。但是，你知道，确实你会想看看其他东西，你知道。历史是迷人的：现代史、古代史、地球的历史、宇宙的历史——这些东西都令人着迷。一生中不曾花些时间去思考这些，实在可惜——当然不是全部，只是稍微想一想。

Robert P Langlands' father was Robert Langlands and his mother was Kathleen Johanna Phelan. He has two younger sisters, one of whom, Mary Fran Langlands is now Mary McArthur. Although born in New Westminster, a little south of Vancouver, he spent about five years further north in a small hamlet on the coast between Lang Bay and Myrtle Point. His father worked at lumberyards and had moved to the hamlet because of his work. When it was time for Robert to begin elementary school, the family returned to New Westminster where Robert began his schooling at St Ann's Academy. This religious school had been established by the Sisters of Saint Ann in 1865 and the teachers were nuns; the school closed in 1968. After three years at this school, during which time he covered the work of four years, he went to St Peter's School to complete his elementary schooling. The move took place since at this time St Ann's became a girls' school and St Peter's became the corresponding boys' school. Robert had enjoyed St Ann's, but was not happy at St Peter's. In 1946 the family moved again, this time to White Rock on the coast, this time south of New Westminster.

In White Rock, Robert's family ran the Langlands Millwork and Builders' Supplies and this made them somewhat better off than many in the town. He attended the Semiahmoo High School in South Surrey. This school, founded in 1940, had many teachers who [13]:-

... were just former members of the army in World War II, who were given positions as teachers more as a gratitude for their service in the army.

Langlands did various jobs to earn money while at school, for example he collected empty beer bottles for which he was paid a small fee per dozen, he had a newspaper round spending $1\large\frac{1}{2}\normalsize$ hours a day collecting and delivering them six days a week, and he worked at weekends and in the summer vacations at Langlands Millwork and Builders' Supplies. Certainly through much of his time at school, he did not even consider going to a university, but in his final year, the 12th grade, he received much attention from his English literature teacher Crawford Vogler. Vogler told him it "would be a betrayal of God-given talents for you not to attend university." He had also met Charlotte Lorraine Cheverie whose father, Lorenzo Francis Cheverie, gave him a book about famous scientists. The book gave him an interest in science and Charlotte, like Vogler, encouraged him to apply to the University of British Columbia. He said [2]:-

I was flattered by the comments, my ambition was aroused, and I decided then and there to write the entrance examinations. I worked hard and was successful, even winning a small fellowship from the University.

For more details of Langlands' family and education, see THIS LINK.

Langlands graduated from Semiahmoo High School in 1953 and later that year began his studies at the University of British Columbia. At this stage he did not have any idea which subjects he should study. He discussed this with a university counsellor who had him take aptitude tests. His background in all subjects was relatively poor but, of course, in mathematics someone with ability can do extremely well without great background knowledge. The results of the aptitude tests made the counsellor suggest that he might want to become an accountant, but he did not like this idea. The next suggestion of mathematics or physics was more to Langlands liking so he decided that mathematics would be the topic to specialise in. For his first year courses he took French, English, physics, chemistry and mathematics. Towards the end of his first year he spoke with his mathematics professor, Stephen Arthur Jennings (1915-1979) who told him that to become a mathematician one had to learn French, German and Russian. Langlands, who knew basic French, taught himself German from books over the summer vacation. He took a course in Russian in his second year at university as well as courses in English literature, mathematics, physics and logic.

The honours courses Langlands took in his third year were differential calculus, integral calculus, linear algebra and algebra. His lecturers included Robert Christian, Eugene Leimanis and Frederick Goodspeed. Before beginning his fourth year, he married Charlotte Lorraine Cheverie on 13 August 1956; they have four children, William, Sarah, Robert and Thomasin.

In the final year of his undergraduate course, Langlands took a course on Galois theory given by Rimhak Ree, a course on convexity given by Douglas Derry, a course on function theory given by Robert Christian and a course on applied mathematics given by Thomas Hull. He also participated in two seminars, one by Harry Davis was on modern functional analysis and the other by David Murdoch on Noetherian rings. Langlands was awarded his B.A. in 1957 and continued to study at the University of British Columbia for his master's degree which was awarded in 1958. He applied to study for a Ph.D. at Harvard, Wisconsin and Yale. All three accepted him but only Yale offered financial support so he had no difficulty in deciding to accept Yale.

Langlands then studied at Yale University for his doctorate. His formal thesis advisor was Cassius Ionescu Tulcea (1923-2021), a Romanian born mathematician who specialised in probability theory, statistics and mathematical analysis. Tulcea, however, did not suggest the topic of his thesis which Langlands found for himself. Although he was at Yale for two years, he completed the work for his thesis in his first year there. During these two years he attended a variety of courses, including: Nelson Dunford on functional analysis; Einar Hille on functional analysis and semigroups; Cassius Ionescu Tulcea on Lie semi-groups and their representations; Felix Browder on partial differential equations; and Stephen Gaal on analytic number theory.

He submitted his thesis Semi-groups and representations of Lie groups to Yale in 1960 and received the degree of Ph.D. Langlands wrote [57]:-

There are two, related parts to this thesis: one on representations of Lie semi-groups and one on operators associated to representations of Lie groups. The first part was published in the 'Canadian Journal of Mathematics', but the second was published only as an announcement in the Proceedings of the National Academy of Sciences of the USA. It nevertheless had the good fortune to be taken seriously by Derek Robinson, who incorporated some of the results into his book on Elliptic Operators and Lie Groups.

Also writing about his doctoral thesis Langlands regretted that it remains [57]:-

... my only active encounter with partial differential equations, a subject to which I had always hoped to return but in a different vein.

He now had to find an academic position. He said in the interview [2]:-

What I really hoped to do when I completed my Ph.D. was to stay at Yale. I had fallen in love with the atmosphere there: I had a freedom to study and think that I had never had elsewhere. Several of the faculty encouraged me to stay, but my appointment was blocked, probably by Kakutani. So I accepted the offer from Princeton, where I had the great good fortune to meet Salomon Bochner, whose encouragement had decisive, concrete consequences. I am not sure that Bochner ever understood how much he had done for me. I was a timid young man and he was a genuinely timid old man, so that there were some feelings that were never expressed.

Appointed to Princeton as an instructor after completing his doctoral studies, Langlands taught there for seven years. In 1961 he was promoted to Lecturer then to Assistant Professor - Associate Professor in 1962. At one week's notice, he was asked to give a course on class field theory since Emil Artin, who had been teaching this course, had left Princeton in 1958 to return to Germany. Feeling he did not know enough about class field theory to give the course, he went to Bochner saying there was no way he could learn enough in a week to give the course [13]:-

But he insisted so I gave a course on class field theory from Chevalley's paper, which is the more modern view, and I got through it. There were three or four students, who said they learned something from it. So, with that, I began to think about the fact that there was no non-abelian class field theory yet. Some people, like Artin, didn't expect there to be any. So, I was just aware of it, that's all.

From August 1962 to June 1963 he was a member of the School of Mathematics of the Institute for Advanced Study at Princeton. He spent 1964-65 at the University of California, Berkeley as a Miller Foundation Fellow and an Alfred P Sloan Fellow. Then in 1967 he returned to Yale University as a full professor. However Langlands spent 1967-68 visiting in Ankara, Turkey having an office next to that of Cahit Arf. After five years at Yale he returned again to Princeton in July 1972, this time as professor of mathematics at the Institute for Advanced Study. He remained at the Institute for Advanced Study until he retired in June 2007. He was made Emeritus at the Institute for Advanced Study in July 2007.

The first papers that Langlands published from 1960 were: On Lie semi-groups (1960); Some holomorphic semi-groups (1960); The dimension of spaces of automorphic forms (1963); Dimension of spaces of automorphic forms (1966); The volume of the fundamental domain for some arithmetical subgroups of Chevalley groups (1966); and Eisenstein series (1966).

In 1988 Langlands received the National Academy of Sciences Award in Mathematics. He was the first recipient of this award which was established by the National Academy of Sciences; it was renamed the Maryam Mirzakhani Prize in Mathematics in 2012. The citation for the award to Langlands recognises his:-

... extraordinary vision that has brought the theory of group representations into a revolutionary new relationship with the theory of automorphic forms and number theory.

Let us explain a little about Langlands' work which led to this award. As soon as he had completed his doctoral work, Langlands began to work on automorphic forms. In a remarkable paper he applied recent results by Harish-Chandra to obtain a formula for the dimension of certain spaces of automorphic forms. Then, over the next couple of years, he produced deep results on Eisenstein series and went on to apply Eisenstein series to prove a number theory conjecture due to Weil.

In 1967 he wrote a letter to Weil which contains profound mathematical ideas which continue to drive a whole area of mathematical research. The letter was 17 pages hand-written and sent in January 1967. It sketched what soon became known as "the Langlands conjectures". Weil had the letter typed and this typed version circulated widely among mathematicians interested in the topics. Casselman writes in [44] that the letter contained:-

... a collection of far-reaching and uncannily accurate conjectures relating number theory, automorphic forms, and representation theory. These have formed the core of a program still being carried out, and have come to play a central role in all three subjects.

Other letters of Langlands also proved remarkably important. While he was in Ankara in 1967-68 he wrote to Serre with ideas which would eventually be formulated as the Deligne-Langlands conjecture; this was proved eventually by Kazhdan and Lusztig.

In [38], Judy Mendaglio attempted to give an idea of the Langlands' Program using elementary ideas:-

The "theory of everything" in mathematics, the Langlands Program, is a set of conjectures that seek to unify knowledge from different branches of mathematics. The idea is that a problem in one area of mathematics may be very difficult to analyse using the tools available in that area. However, if the structures within the problem can be related to similar structures in a different field, where there are better analytical tools available, then the analysis may be conducted with less difficulty and the results related back to the original problem. In this way, even deeper structures in the original area of mathematics are revealed.

For example, consider a somewhat elementary problem such as finding the properties of solids in four or more dimensions. That is a geometry problem. However, since we have difficulty imagining objects in four or more dimensions, we find analysing their properties difficult; however, we can relate their geometric structures to algebraic structures that are well understood. We muck about in the world of algebra, have an insight or two, and then relate those insights back to the geometry problem.
...
Of course, the mathematics of the Langlands Program is of a more advanced level than even most mathematicians understand, but, fundamentally, the idea is the same. In the Langlands Program, the branches of mathematics that are being related are those connecting our understanding of numbers and our understanding of change. Think arithmetic meets geometry, and they meet calculus, and then take it up a few dozen notches. The main branches of mathematics that are involved are number theory (the study of properties and relationships of numbers), algebraic geometry (a cousin of analytic geometry), representation theory (which concerns sets and mappings), and mathematical physics. Mathematician and author of 'Love and Math', Edward Frenkel, has called Langlands' theories "the source code of all mathematics."

The National Academy of Sciences Award in Mathematics which we referred to above was certainly not the first award which Langlands received for his work. In 1975 he had been awarded the Wilbur Cross Medal from Yale University. He received the Jeffery-Williams Prize from the Canadian Mathematical Society in 1980 for outstanding contributions to mathematical research, and the Cole Prize in Number Theory from the American Mathematical Society in 1982 for his pioneering work on automorphic forms, Eisenstein series and product formulae. He continued to win major awards, for example he shared the 1995-96 Wolf Prize in Mathematics with Andrew Wiles. The Prize was awarded to Langlands for his:-

... path-blazing work and extraordinary insights in the fields of number theory, automorphic forms, and group representation.

Elected a Fellow of the Royal Society of Canada in 1972, he was elected a Fellow of the Royal Society of London in 1981, a member of the National Academy of Sciences in 1993, a member of the American Philosophical Society in 2004, and a fellow of the American Mathematical Society in 2012. He has received over thirty honorary doctorates from intitutions including the University of British Columbia, McMaster University, The City University of New York, the University of Waterloo, the University of Paris VII, McGill University, the University of Toronto, and the University of Chicago.

Langlands was awarded the Grande Médaille d'Or by the French Academy of Sciences in 2000. In 2005 he was awarded the Leroy P Steele Prize for Seminal Contribution to Research by the American Mathematical Society for his paper Problems in the theory of automorphic forms (1970). This is the paper that introduced what are now known as the Langlands conjectures. In 2006 he received the Frederic Esser Nemmers Prize in Mathematics for his [67]:-

... fundamental vision connecting representation theory, automorphic forms and number theory.

The citation contains quotes from Kari Vilonen, professor of mathematics at Northwestern University [67]:-

The Langlands program postulates a deep relationship between two different areas of mathematics, number theory and automorphic forms, via a study of their symmetries. Since its initiation about 40 years ago, the Langlands program has served as a unifying principle in mathematics and has guided research in number theory, automorphic forms and representation theory. Recently, it also had entered mathematical physics. It remains a research program for the future in all these areas.

Robert Langlands and Richard Taylor jointly received the Shaw Prize in 2007 [59]:-

Robert Langlands initiated a unifying vision of mathematics that has greatly extended the legacy of the mathematics of previous centuries, connecting prime numbers with symmetry. This unification, which grew out of the Reciprocity Theory of Gauss and Hilbert, is now referred to as the Langlands program. It provides a direction of research which has guided mathematicians over the past forty years and will continue to do so for years to come.

After winning the prize, Langlands gave the lecture Reflections on receiving the Shaw prize (see [31]). Balasubramanian Sury wrote in a review of this paper [63]:-

This is the text of a lecture delivered in Hong Kong on the occasion of the author receiving the Shaw Prize. It makes for absolutely fascinating reading. The contents of the masterly exposition are so riveting that it is scarcely possible to put the article down without finishing it. Therefore, instead of giving a detailed description of the contents, the reviewer encourages the interested reader to peruse the text himself by just quoting the following text from the article: "a number of mathematicians have a perception of the development of the theory of automorphic forms over the last four decades that differs from mine if not in a radical, certainly in an essential way. Some of the differences are a result of misapprehensions that are a natural consequence of the variety of the theory's relations to fields practiced by mathematicians with many different temperaments and training. With a little explanation these misapprehensions can be dissipated. The prize is an opportunity to do so. Others are the result of conflicting methodological stances, mostly unrecognised and certainly unresolved. Their resolution will certainly demand a deeper understanding of the subject than is yet available. In this lecture I attempt to describe the current, unresolved situation. My emphasis will be on my own stance, although my purpose here is not to advocate but to explain it." For anyone interested in the historical development of, and the motivating ideas behind, the functoriality conjecture(s), this is a must-read.

In 2015 Langlands was elected to Honorary Membership of the London Mathematical Society in its 150th Anniversary year. The short citation reads:-

Professor Langlands secured his place in history of mathematics as the proposer (in 1967) and first developer of the eponymous research programme. The deep results and visionary conjectures of the Langlands Programme relate the core themes in number theory and representation theory.

The full citation begins [55]:-

Robert Langlands is one of the giants of modern mathematics. By combining great technical power with extraordinary imagination and vision, he has shown how to unify major areas of mathematics that were previously believed to be quite distinct. More precisely, Langlands has transformed the traditional area of automorphic forms, originally a part of the theory of complex variables, into a very different theory whose classical roots are now almost unrecognisable. In Langlands' hands, the theory of automorphic forms has become a grand force for unification, representing what seem to be the fundamental laws of mathematical symmetry. These laws govern the internal structure of many diverse parts of mathematics, most notably from number theory and arithmetic algebraic geometry.

The Abel Prize is recognised as the highest possible award to a mathematician. It was presented to Langlands in 2018 [51]:-

... for his visionary program connecting representation theory to number theory.

Here is a quote from the Press Release [50]:-

Robert P Langlands has been awarded the Abel Prize for his work dating back to January 1967. He was then a 30-year-old associate professor at Princeton, working during the Christmas break. He wrote a 17-page letter to the great French mathematician André Weil, aged 60, outlining some of his new mathematical insights.

"If you are willing to read it as pure speculation I would appreciate that," he wrote. "If not - I am sure you have a waste basket handy."

Fortunately, the letter did not end up in a waste basket. His letter introduced a theory that created a completely new way of thinking about mathematics: it suggested deep links between two areas, number theory and harmonic analysis, which had previously been considered as unrelated.
...
Langlands' insights were so radical and so rich that the mechanisms he suggested to bridge these mathematical fields led to a project named the Langlands program. The program has enlisted hundreds of the world's best mathematicians over the last fifty years. No other project in modern mathematics has as wide a scope, has produced so many deep results, and has so many people working on it. Its depth and breadth have grown and the Langlands program is now frequently described as a grand unified theory of mathematics.

For the more technical Citation for the Abel Prize 2018, see THIS LINK.

He continued to receive honours, for example he was appointed Companion of the Order of Canada in 2019 and on 10 January 2020 Semiahmoo High School installed a mural celebrating his contributions to mathematics.
See THIS LINK.

Casselman, in [44], ends with the following summary:-

[Langlands'] astounding insight has provided a whole generation of mathematicians working in automorphic forms and representation theory with a seemingly unlimited expanse of deep, interesting, and above all approachable problems to work away on.

In the interview [13] the final question was whether Langlands had non-mathematical passions or interests of some sort. He replied:-

Passions? I don't have any passions. But, you know, it is true that you want to take a look at other things, you know. History is fascinating: modern history, ancient history, the Earth's history, the Universe's history - these things are all fascinating. It is a shame to go through life and not have spent some time contemplating on that - certainly not everything of course but just to think about it a little bit.