Robert Phelan Langlands


Quick Info

Born
6 October 1936
New Westminster, British Columbia, Canada

Summary
Robert P Langlands is a Canadian mathematician who works in representation theory and number theory. He has been awarded all the major mathematical prizes including the Shaw Prize, the Wolf Prize and the Abel Prize.

Biography

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 1121\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.


References (show)

  1. J G Arthur, The work of Robert Langlands, arxiv.org (5 July 2023).
    https://arxiv.org/abs/2307.02571
  2. F Barekat, Robert Langlands Interview, University of British Columbia (December 2009).
    https://web.archive.org/web/20140407083054/http://www.math.ubc.ca/Dept/Newsletters/Robert_Langlands_interview_2010.pdf
  3. B Edixhoven, Abel prize awarded to Robert Langlands, Nieuw Arch. Wiskd. (5) 20 (1) (2019), 12-18.
  4. Farzib Barekat interviews Robert P Langlands, Institute for Advanced Study (2010).
    http://publications.ias.edu/sites/default/files/interview-ubc-2009-rpl_0.pdf
  5. A Bellos, Abel Prize 2018: Robert Langlands wins for 'unified theory of maths, The Guardian (20 March 2018).
  6. A Bellos, A biography of Robert P Langlands, International Mathematical Union (2018).
    https://www.mathunion.org/fileadmin/IMU/Prizes/Abel/2018/Abelprize_2018_Langlands_bio.pdf
  7. L Cagliero, Editorial, Revista de Educación Matemática 33 (1) (2018), 3-4.
  8. Canadian mathematician Robert Langlands wins Abel Prize for 2018, The New Indian Express (21 March 2018).
    https://www.newindianexpress.com/world/2018/mar/21/canadian-mathematician-robert-langlands-wins-abel-prize-for-2018-1790351.html
  9. K Chang, Robert P Langlands Is Awarded the Abel Prize, a Top Math Honor, The New York Times (20 March 2018).
  10. S Contento, The Canadian Who Reinvented Mathematics, Toronto Star (27 March 2015).
  11. S A de Reyna, Robert P Langlands: Abel Prize 2018, Instituto de Matemáticas, Universidad de Sevilla (15 April 2018).
    https://institucional.us.es/blogimus/en/2018/04/robert-p-langlands-abel-prize-2018/
  12. R Dijkgraaf, A Mathematical Rosetta Stone, Institute for Advanced Study (2018).
    https://www.ias.edu/ideas/2018/dijkgraaf-rosetta-stone
  13. B I Dundas and C Skau, Interview with Abel Laureate Robert P Langlands, Notices of the American Mathematical Society 66 (4) (2019), 494-503.
  14. S Durand, Robert Langlands - Un explorateur de l'abstrait, Institute for Advanced Study (May 2020).
    http://publications.ias.edu/sites/default/files/entrevue-quebec-science-rpl_2.pdf
  15. E Frenkel, Preface, in Love and Math: The Heart of Hidden Reality (Basic Books, 2013).
  16. S Gelbart, The early Langlands Program - personal reflections, London Math. Soc. Lecture Note Ser. 467 (Cambridge University Press, Cambridge, 2021), 43-56.
  17. A Goodacre, My reminiscences of Bob Langlands at the University of British Columbia, London Math. Soc. Lecture Note Ser. 467 (Cambridge University Press, Cambridge, 2021).
  18. U Görtz, Eine Fields-Medaille für Bao Châu Ngô. Der Beweis des 'fundamentalen lemmas' des Langlands-Programms, Mitt. Dtsch. Math.-Ver. 19 (4) (2011), 198-203.
  19. I Gusic, Robert Langlands dobio Abelovu nagradu za 2018. g, Matematicko-fizicki list 69 (1) (2018-2019), 3-11.
  20. T Hales, Reminiscences by a student of Langlands, London Math. Soc. Lecture Note Ser. 467 (Cambridge University Press, Cambridge, 2021), 65-69.
  21. T Holmes, Mathematician with White Rock roots named to Order of Canada, Peace Arch News (21 November 2019).
  22. T Holmes, B.C.-born professor celebrated as mathematical 'visionary', The Chilliwack Progress (28 March 2018).
  23. T Holmes, Semiahmoo mural honours famed grad Robert Langlands, Today in BC (14 January 2020).
    https://www.todayinbc.com/news/semiahmoo-mural-honours-famed-grad-robert-langlands/
  24. A W Knapp, Group representations and harmonic analysis from Euler to Langlands I, Notices Amer. Math. Soc. 43 (4) (1996), 410-415. http://www.ams.org/notices/199604/knapp.pdf
    25, A W Knapp, Group representations and harmonic analysis II, Notices Amer. Math. Soc. 43 (5) (1996), 537-549. http://www.ams.org/notices/199605/knapp-2.pdf
  25. A W Knapp, Group representations and harmonic analysis II, Notices Amer. Math. Soc. 43 (5) (1996), 537-549. http://www.ams.org/notices/199605/knapp-2.pdf
  26. Langlands and Taylor awarded Shaw Prize, Notices Amer. Math. Soc. 54 (8) (2007), 1001.
  27. R P Langlands, Commencement address at the University of Toronto, June 1993, University of Toronto (June 1993).
    http://publications.ias.edu/sites/default/files/UT-commencement-1993-rpl_0.pdf
  28. R P Langlands, Response upon receiving the Grande Médaille d'Or of the Académie des sciences, Institute for Advanced Study (2000).
    http://publications.ias.edu/rpl/paper/110
  29. R P Langlands, Recollections of a year in Turkey with Cahit Arf, Institute for Advanced Study (24 January 2020).
    http://publications.ias.edu/sites/default/files/Cahit-Arf-rpl.pdf
  30. R P Langlands, Mathematical retrospectives, Institute for Advanced Study (22 May 2018).
    http://publications.ias.edu/sites/default/files/mathematical-retrospections-rpl_0.pdf
  31. R P Langlands, Reflections on receiving the Shaw prize, Institute for Advanced Study (4 May 2023).
    http://publications.ias.edu/sites/default/files/shaw-reflexions-rpl_0.pdf
  32. R P Langlands, A succinct biography, Institute for Advanced Study (31 January 2020).
  33. R P Langlands, Is there beauty in mathematical theories?, University of Notre Dame (January 2010).
    http://publications.ias.edu/rpl/section/29
  34. R P Langlands, Autobiography of Robert Langlands, The Shaw Prize (2007).
    https://www.shawprize.org/prizes-and-laureates/mathematical-sciences/2007/autobiography-of-robert-langlands
  35. C Levesque, Robert P Langlands: l'homme derrière le mathématicien, London Math. Soc. Lecture Note Ser. 467 (Cambridge University Press, Cambridge, 2021), 88-99.
  36. D Mackenzie, Fermat's Last Theorem's First Cousin, Science 287 (5454) (2000), 792-793.
  37. G Marasingha, Abel prize for Langlands' unified theory, Math. Today (Southend-on-Sea) 54 (3) (2018), 80-81.
  38. J Mendaglio, An honour for Canadian mathematician Robert Langlands, Gazette - Ontario Association for Mathematics 56 (4) (2018), 10-11.
  39. J Mueller, On the genesis of Robert P Langlands' Conjectures and his letter to André Weil, Bulletin of the American Mathematical Society (25 January 2018).
    https://www.ams.org/journals/bull/2018-55-04/S0273-0979-2018-01609-1/supplementary-information/S0273-0979-2018-01609-1-original-version.pdf
  40. J Mueller, A glimpse at the genesis of the Langlands program, London Math. Soc. Lecture Note Ser. 467 (Cambridge University Press, Cambridge, 2021), 3-42.
  41. C Pichet, Un homme de culture et de nature, London Math. Soc. Lecture Note Ser. 467 (Cambridge University Press, Cambridge, 2021), 100-106.
  42. Robert Langlands awarded Abel Prize, Department of Mathematics, Yale University (27 March 2018).
    https://math.yale.edu/news/robert-p-langlands-awarded-abel-prize
  43. Robert Langlands wins Abel Prize 2018 for 'unified theory of maths', Middle East Technical University (28 May 2018).
    https://math.metu.edu.tr/en/announcement/robert-langlands-wins-abel-prize-2018-unified-theory-maths
  44. Robert P Langlands receives first NAS Award in Mathematics, Notices Amer. Math. Soc. 35 (4) (1988), 509-510.
  45. Robert P Langlands, National Academy of Sciences.
    https://www.nasonline.org/member-directory/members/47401.html
  46. Robert P Langlands, Maryam Mirzakhani Prize in Mathematics, National Academy of Sciences.
    https://www.nasonline.org/programs/awards/mathematics.html
  47. Robert P Langlands, Institute for Advanced Study.
    https://www.ias.edu/scholars/langlands
  48. Robert P Langlands awarded 2018 Abel Prize, Institute for Advanced Study (20 March 2018).
    https://www.ias.edu/news/press-releases/2018/abel
  49. Robert Langlands awarded Abel Prize, Notices Amer. Math. Soc. 65 (6) (2018), 670-672.
  50. Robert P Langlands receives the Abel Prize, Press Release, Norwegian Academy of Science and Letters (2018).
    https://abelprize.no/sites/default/files/2021-04/pressrelease_English_Abel_2018%20Robert%20Langlands.pdf
  51. Robert P Langlands, Abel Prize Citation, Norwegian Academy of Science and Letters (2018).
    https://abelprize.no/sites/default/files/2021-04/citation_English_Abel_2018%20Robert%20Langlands.pdf
  52. Robert P Langlands, Wolf Prize Laureate in Mathematics 1995/6, Wolf Foundation.
    https://wolffund.org.il/2018/12/10/robert-p-langlands/
  53. Robert P Langlands, in American Men & Women of Science. A biographical dictionary of today's leaders in physical, biological and related sciences (33rd Edition) (Cengage Learning, Detroit, 2015).
  54. Robert P Langlands, in Susan Charters (ed.), Canadian Who's Who (University of Toronto Press, Toronto, 2012).
  55. Robert Langlands, Honorary Members 2015, Bulletin of the London Mathematical Society 48 (3) (2016), 557-576.
  56. D W Robinson, In the beginning: Langlands' doctoral thesis, London Math. Soc. Lecture Note Ser. 467 (Cambridge University Press, Cambridge, 2021), 130-175
  57. Semi-groups and representation theory by Robert Phelan Langlands, Mathematics Department, Yale University (1960), Institute for Advanced Study (January 2018).
    https://publications.ias.edu/rpl/paper/4
  58. C Saçlioğlu, Langlands and Turkey, London Math. Soc. Lecture Note Ser. 467 (Cambridge University Press, Cambridge, 2021), 59-64.
  59. Shaw Prize 2007 Press Release, The Shaw Prize (11 September 2007).
    https://www.shawprize.org/prizes-and-laureates/mathematical-sciences/2007/press-release
  60. K Shi, Robert P Langlands, Who's Who of Well-known Mathematicians 3 (Zhejiang Ocean University, China, 2012), 34-38.
  61. T Spencer, Robert Langlands' work in mathematical physics, London Math. Soc. Lecture Note Ser. 467 (Cambridge University Press, Cambridge, 2021), 403-413.
  62. B Sury, Robert Langlands wins the 2018 Abel Prize, Resonance (May 2018), 613-617.
  63. B Sury, Review: Reflexions on receiving the Shaw Prize, by Robert Langlands, Mathematical Reviews MR2767520 (2012d:22001).
  64. K D Thomas, Robert Langlands: Far-Reaching Mathematics, Institute for Advanced Study (2007).
    https://www.ias.edu/ideas/2007/langlands-mathematics
  65. K D Thomas, Modern Mathematics and the Langlands Program, Institute for Advanced Study (2010).
    https://www.ias.edu/ideas/modern-mathematics-and-langlands-program
  66. The work of Robert Langlands, Institute for Advanced Study.
    https://publications.ias.edu/rpl
  67. The work of Robert Langlands, The University of British Columbia.
    http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/
  68. P V Tremmel, Nemmers awards in economics and mathematics announced, Northwestern University (15 March 2006).
    https://www.northwestern.edu/newscenter/stories/2006/03/nemmers.html
  69. E Witten, Talking Points: Edward Witten on Geometric Langlands, Institute for Advanced Study (2007).
    https://www.ias.edu/ideas/talking-points-edward-witten-geometric-langlands
  70. S Yasuda, The work of Laurent Lafforgue - Establishing the Langlands correspondence for the GL_r over function fields, Sūgaku 60 (4) (2008), 415-424.

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Written by J J O'Connor and E F Robertson
Last Update December 2023