James Greig Arthur

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18 May 1944
Hamilton, Ontario, Canada

James Arthur is a Canadian mathematician who has made outstanding contributions to the Langlands programme which relates arithmetic and algebra with analysis and spectral theory. He has been awarded many prizes including the prestigious Wolf Prize and the Leroy P Steele Prize for Lifetime Achievement.


James Arthur was the son of James Greig Arthur and Katherine Mary Patricia Scott (1918-2011). James Sr was the son of Rev A J Arthur. Katherine, daughter of Henry Duke Scott and Lillian Mary Scott, was a graduate of St Clement's School and of Trinity College, University of Toronto. James and Katherine had four children. Their first two children were boys: James Greig Arthur, the subject of this biography, and Philip Duke Arthur, born 25 January 1947, who became a public accountant. Their two girls were Elizabeth and Katherine. James was born in Hamilton, Ontario and attended Upper Canada College, a boy's school in Toronto, Canada. This school, founded in 1829, served as a high quality feeder school for the University of Toronto. James Arthur was Head Boy at the College in 1962, the year he graduated. James' father and James's younger brother Philip both attended this school. In 2020 when Arthur was awarded the Upper Canada College Old Boy of Distinction Award, he spoke about his days at the College [44]:-
I was fortunate to go to Upper Canada College. I got a really great foundation. The masters were by and large original thinkers and pretty passionate about what they were teaching. Most of my classmates were imbued with a sense of passion for what they do in life. It was infectious. You learn that attitude. We have memories that go way back. I have a lot of happy memories. There was a sense of irreverence among the boys, and some of their jokes and pranks, while not always amusing to the long-suffering masters, are still funny to us today. By my last year, I was really excited about mathematics and decided it was what I wanted to do as a career.
After graduating from the Upper Canada College, James Arthur entered Trinity College, one of the colleges of the University of Toronto. The staff of the Department of Mathematics when Arthur began his studies included Albert Fenton Pillow, Frederick Valentine Atkinson, George Francis Denton Duff, H S M Coxeter, Paul George Rooney and G de B Robinson. Two other leading mathematicians took up appointments while Arthur was an undergraduate, namely Hans Heilbronn (who arrived in 1964) and Jacob Lionel Bakst Cooper (who arrived in 1965). Arthur graduated with a B.Sc. in 1966 and continued to study at Toronto for his Master's Degree which he was awarded in 1967.

Many students from the United States and Canada who went on to win major prizes for outstanding research contributions, took part in the William Lowell Putnam Competition and were Putnam fellows, received an Honourable Mention, or were highly placed. James Arthur was certainly an excellent student who was in the Toronto team for the Putnam Competition but, at this stage, did not show the "genius" ability that some others showed. He said [5]:-
I was not a prodigy in mathematics as a child. As a matter of fact, I am quite happy that my record for the Putnam exams was not available to the Prize Committee. But I do remember being fascinated even as a child by what was said to be the magic and power of mathematics.
Arthur went to Yale University to undertake research for a Ph.D. advised by Robert Langlands. He was awarded a Ph.D. in 1970 for his thesis Harmonic Analysis of Tempered Distributions on Semisimple Lie Groups of Real Rank One. In it he writes [9]:-
I would like to express my deep gratitude to Professor Robert Langlands for his constant guidance and encouragement. He suggested the problem for this dissertation, and over the past two years has been more than generous with both his time and his ideas. I would also like to acknowledge a debt of a different sort to Professor Harish-Chandra. As will become evident to the reader, the results of this dissertation lean heavily upon Professor Harish-Chandra's work.
After graduating from Yale with a Ph.D., Arthur was appointed as an Instructor in Mathematics at Princeton University. There he met Dorothy Pendleton Helm who was appointed as an Instructor in Romance Languages at Princeton University in 1970. In [48] she gives this account of her career up to that point:-
Dorothy Pendleton Helm Arthur was born in Louisville, Kentucky, in 1943. After graduating from Sacred Heart Academy in Louisville, she obtained a B.S. in 1964 from Georgetown University's School of Languages and Linguistics, where she studied French, Spanish, and linguistics. She began her graduate studies at Brown University in 1966, completing an A.M. in 1968; her thesis, "The Royalist of Ferney: A Study of Voltaire during the Maupeou Revolution, 1768-74," was directed by Prof Durand Echeverria. After being admitted to candidacy for the Ph.D., she received a Woodrow Wilson Dissertation Fellowship in 1969 and planned to specialise in Ronsard's early sonnets and Renaissance mythology.
James Arthur and Dorothy Pendleton Helm were married in Princeton on 10 June 1972. They had two children and we will pause our biography to say a little about these two boys. James Pendleton Arthur was born in 1974 and is now a poet. He spoke about his upbringing in [36]:-
My dad, a mathematician, raised me to believe that mathematics is beautiful, so maths is a part of my imaginative terrain. In my late 20s I wrote several 11-line poems because I wanted to create poems that couldn't be uniformly divided into couplets, tercets, or quatrains, 11 being a prime number. One of those 11-liners, "On Day and Night," did make it into my first book. From my perspective, "On Day and Night" is about indivisibility, and about the idea that few things in this world are actually as classifiable as they seem.
James and Dorothy Arthur's younger son is David Greig Arthur who became a mathematician and computer scientist working for Google as a software engineer [11]:-
Originally from Toronto, David received his B.S. in mathematics and computer science from Duke University and his PhD in theoretical computer science from Stanford University. Though David's Ph.D. is in computer science, he says his Ph.D. thesis was "100 pages of mathematical proofs, with no computers anywhere to be seen!"
He did perform at the "genius" level in mathematics competitions [11]:-
In 1999 and 2000, David was a member of the Canadian student team to the International Mathematical Olympiad (IMO). In 2000, David received a gold medal and was only 4 points away from a perfect score. Participating in the IMO was an unforgettable experience for David. "The IMO was a chance to see something beautiful, to meet people from across the world, and to excel at something I cared about," he says. Today, David carries on supporting Canada at the IMO, having acted as the Deputy Team Leader for the 2009 and 2011 events. "It is no less inspiring to meet the world's best mathematicians now than it was ten years ago," says David.
After two years at Princeton, James Arthur was appointed as an Assistant Professor of Mathematics at Yale University. His first publication was The Selberg trace formula for groups of F-rank one which was published in the Annals of Mathematics in 1974. The 60-page paper begins:-
An important tool for the study of automorphic forms is a non-abelian analogue of the Poisson summation formula, generally known as the Selberg trace formula. There have been a number of publications on the subject following Selberg's original paper [Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series (1956)], ... . With the exception of Selberg's brief account [Discontinuous groups and harmonic analysis (1962)], however, most authors have restricted themselves to the groups SL(2) and GL(2). In this paper we develop the formula for a wider class of groups.
Most of the methods used in this paper originate with Selberg (see [Discontinuous groups and harmonic analysis (1962)]), including the ideas behind the proof of Theorem 3.2 and the convergence of the integrals in §8. These were described to me by Robert Langlands whom I would like to thank for his encouragement. I am also grateful for the comments of Stephen Gelbart, who read through the original manuscript.
While working as an Assistant Professor of Mathematics at Yale University, Arthur was awarded a Yale University Junior Faculty Fellowship which he held at Institute Des Hautes Etudes Scientifiques, Bures-sur-Yvette, France in 1974-75. During this time he published further major papers including the 30-page paper Some tempered distributions on semisimple groups of real rank one (1974) which begins:-
The Selberg trace formula leads naturally to the study of certain tempered distributions on reductive groups defined over local fields. An important problem is to calculate the Fourier transforms of these distributions. We shall consider this question for the case that the local field is R and the group G is semisimple and has real rank one.
He also published the major 56-page paper The characters of discrete series as orbital integrals (1976). This paper gives his address as Bures-sur-Yvette, France, and gives important generalisations of Harish-Chandra's results. Ernest A Thieleker writes in the review [42]:-
The methods of proof involve generalisations of Harish-Chandra's theory of the differential equations satisfied by invariant integrals. In the course of the proof an explicit formula for the volume function ν is also developed. The proof of this formula depends on a lemma of Langlands whose detailed proof is published here for the first time.
In 1975 Arthur was awarded a Sloan Fellowship which he held at the Institute for Advanced Study, Princeton being a member of the Institute from July 1976 to June 1977. He was promoted to Professor at Duke University in 1976 but returned to Canada in 1978 when he was appointed as a Professor in the Department of Mathematics at the University of Toronto. He has remained at the University of Toronto throughout the rest of his career. In 1987 he was made a "University Professor" and in 2007 he was also named as "Ted Mossman Chair in Mathematics". The appointment to the Ted Mossman Chair in Mathematics is at the level of Professor with tenure, and the Chair holder has to be to be an outstanding mathematician, whose research and teaching make a major contribution to the quality and stature of the department.

In 1982 Arthur was awarded an E W R Steacie Memorial Fellowship by the Natural Sciences and Engineering Research Council of Canada. These fellowships were awarded:-
... to recognise early stage academic researchers in the natural sciences and engineering and to enhance their research capacity so that they could become leaders in their field and inspire others.
In 1987 Arthur received the John L Synge Award. This was the first of many awards and prizes: for details of twelve of these, see THIS LINK.

Arthur has twice been an invited speaker at the International Congress of Mathematicians. The first was in 1983 in Warsaw when he addressed 'Section 8: Lie groups and representations' giving the talk The trace formula for noncompact quotient. His talk begins:-
In papers in 1956 and 1962, Selberg introduced a trace formula for a compact, locally symmetric space of negative curvature. There is a natural algebra of operators on any such, space which commute with the Laplacian. The Selberg trace formula gives the trace of these operators. Selberg also pointed out the importance of deriving such a formula when the symmetric space is assumed only to have finite volume. Then the Laplace operator will have continuous as well as discrete spectrum; it is the trace of the restriction of the operator to the discrete spectrum that is sought. Selberg gave such a formula for the quotient of the upper half plane by SL(2,Z)SL(2, \mathbb{Z}). Selberg also suggested how to extend the formula to any locally symmetric space of rank 1. Spaces of rank 1 are the easiest noncompact ones to handle for they can be compactified in a natural way by adding a finite number of points. I have recently obtained a trace formula for spaces of higher rank. I shall illustrate the formula by looking at a typical example.
His second invitation to address the International Congress of Mathematicians was in Berlin in 1998. At this Congress he addressed 'Section 7. Lie Groups And Lie Algebras' giving the talk Towards a Stable Trace Formula. The Abstract states:-
The paper is a report on the problem of stabilising the trace formula. The goal is the construction and analysis of a stable trace formula that can be used to compare automorphic representations on different groups.
For a brief overview of Arthur's research contributions up to 2003, see THIS LINK.

Arthur has been elected to leading Academies and Societies. He has been elected: a Fellow of the Royal Society of Canada (1980); a Fellow of the Royal Society of London (1992); a Foreign Honorary Member of the American Academy of Arts and Sciences (2003); a Fellow of the American Mathematical Society (2012); a Foreign Associate of the National Academy of Sciences (US) (2014); and a Fellow of the Canadian Mathematical Society (2019). When elected to the National Academy of Sciences, his research interests were given as follows [23]:-
James Arthur works in the area of mathematics called automorphic representations. These objects date back to the nineteenth century, but in recent years have come to occupy a central place in mathematics known as the Langlands program. The conjectures of Robert Langlands that define the programme are still largely unresolved. Roughly speaking, they assert that concrete spectral data wrapped up in automorphic representations govern fundamental processes from quite different domains of mathematics. One of the primary tools for analysing automorphic representations has been a complex trace formula that relates this spectral data to various kinds of geometric data. It was introduced in special cases by Selberg in 1956, and was then established for general groups by Arthur over the later period from 1975 to 2000. Arthur has used the trace formula to classify certain automorphic representations and establish some cases of the Langlands conjectures. He is now working on the central premise of the Langlands program known as the principle of functoriality.
In 2003 Arthur was nominated by Robert Langlands to become president of the American Mathematical Society in February 2005. In this nomination we learn about Arthur's talents other than research. For example, Langlands writes [34]:-
Beginning in 1997, and recently appointed for a second five-year term to end in 2007, he has been an Academic Trustee at the Institute for Advanced Study, responsible, in particular, for explaining mathematics and mathematicians to the other members of the Board of Trustees, who are largely drawn from the world of business and finance. He always carefully arms himself with a knowledge of the broad spectrum of scientific activities of the Institute's School of Mathematics and the details of its yearly programs, but his best weapon when articulating our needs and achievements before the Board has perhaps been his conviction of the importance of defending the place of mathematics in the academic world and in society as a whole. I suspect at the same time that he simply enjoys cultivating the art of persuasion.
In the nomination, Robert Pritchard, the former President of the University of Toronto is quoted regarding Arthur's work on many university committees [34]:-
Jim has served on many of the University's most important special committees ... Why is he chosen? Because he ... personifies our highest aspirations, has superb academic and scholarly judgement, has very high standards, is utterly reliable, is highly courteous, is practical and not just a theorist, always acts in a principled way, and conducts himself with dignity in all situations. He is a very special person quite apart from being a superb mathematician. He is extremely considerate of others and listens hard to competing views. He's fair and will always work to do the right thing.
After being elected as President of the American Mathematical Society, Arthur was interviewed before taking up the role. He spoke about the public interest in mathematics [18]:-
There is a sympathetic curiosity about mathematics. People don't know what mathematicians do, but they somehow have the feeling that mathematics has mystery, power, and beauty. I think people who are not mathematicians have a sense of that. We should stoke their curiosity. We already do so, but I think we can do more. This is probably the most important aspect of strengthening mathematics in the United States and around the world. To have public sympathy and interest in mathematics would help just about everything that concerns us as mathematicians. It would persuade more talented young people to go into mathematics, because mathematics would be regarded with interest, and maybe even awe, by their parents and their friends, and would seem to be a worthy cause to spend one's life pursuing. It would help funding for mathematics, because voters would have a sympathy for it. Also, public appreciation of the beauty of mathematics would help encourage good people to become high school teachers.
In the 2014 interview [47] Arthur spoke about the importance of having good mathematics teachers:-
If teachers are not comfortable with maths and do not really understand it, they are not going to like it. And if children don't like it, they will certainly not want to think about it. Thinking about maths is extremely important for students, not just in the half hour or hour of a maths class but at other times as well, just because it is interesting. We need teachers who love maths, and who are free to communicate it in their own way. In my opinion, this is by far the most important thing. Students love to be taught by someone with a passion for a subject. Mathematics can seem beautiful and mysterious and powerful to anyone, but especially to impressionable children. They might laugh at an eccentric teacher with a passion for something as arcane as mathematics, but that is part of the deal. Once students plug in, they will start thinking about maths out of curiosity, and that is how they become strong.
Arthur was very active in the mathematical community outside the University of Toronto. He had many editorial roles: on the Editorial Advisory Board of the Journal für die reine und angewandte Mathematik, Göttingen, from 1985; on the Editorial Advisory Board of the Journal of the American Mathematical Society (1988-2002); an Associate Editor for the Canadian Journal of Mathematics and the Canadian Mathematical Bulletin (1986-1991); on the Editorial Board of International Mathematics Research Notices, Duke University (1991-2000); and as the Editor of the World Directory of Mathematicians (1997-98, 2001-2002). He served on numerous committees including: the Executive Committee of the International Mathematical Union (1991-1998); the NSERC Grant Selection Committee for Mathematics (1988-1991); the Academy Fellowship Review Committee of the Royal Society of Canada (1991-1993); the National Program Committee of the American Mathematical Society (1989-1991); the Committee on Committees of the American Mathematical Society (1991-1992); the Steering Committee for Centre de Recherches Mathematiques, University of Montreal (1989-1992); the Nominating Committee of the Fields Institute (1992-1996); the Selection Committee for 2002 Fields Medals, International Mathematical Union; and the Search Committee for President of the Clay Mathematics Institute (2003). Of the many duties that he undertook let us mention: Convenor for Mathematics Division, Royal Society of Canada (1988-1990); and the Academic Trustee for Mathematics for the Institute for Advanced Study, Princeton (1997-2007).

Arthur was awarded an honorary degree by the University of Ottawa in 2002. His Profile for the award states [7]:-
James Arthur is regularly cited as perhaps the best and most immediately influential mathematician currently active in Canada. His work stands at the forefront of a bold initiative to unify the diverse branches of pure mathematics. Through more than three decades as a mathematician, Arthur charted a course for himself with way stations spanning the globe.
Let us end our biography by giving the Abstract or the three Virginia Mathematics Lectures (i) L-functions and Number Theory, (ii) The Trace Formula and Automorphic Forms, and (iii) Beyond Endoscopy and Functoriality which he gave 14-16 November 2016 [20]:-
Number theory is founded on the basic properties of integers and prime numbers. But its study these days is increasingly leading us to the far reaches of some of the most diverse and powerful areas of mathematics. Nowhere is this more apparent than in the Langlands program, which represents a profound unifying force for mathematics.

We shall try to introduce the Langlands program through the theory of L-functions. These are infinite series that look like the famous Riemann zeta function, except that they have nontrivial coefficients. The information that goes into the coefficients is in fact very interesting, and gives an elegant way of organising fundamental data from number theory, representation theory and algebraic geometry. The Langlands program postulates deep relationships among different L-functions, and hence also the data in their coefficients.

We shall discuss these matters, and explain how they are part of the theory of automorphic forms. We shall then describe the trace formula, which has led to important results in the classification of automorphic representations. If time permits, we shall also say something about Beyond Endoscopy, a proposal by Langlands for attacking the central conjecture of the subject known as the Principle of Functoriality.

References (show)

  1. 1997 CRM-Fields Prize awarded to James Arthur, Centre de Recherches Mathématiques (1997).
  2. 2003 G de B Robinson Prize: James Greig Arthur, Canadian Mathematical Society (2003).
  3. 2003 G de B Robinson Prize - Dr James Grieg Arthur, University of Toronto, Canadian Mathematical Society (2003).
  4. 2017 Leroy P Steele Prize: James Greig Arthur, University of Toronto (2017).
  5. 2017 Leroy P Steele Prizes, Notices of the American Mathematical Society 64 (4) (2017), 311-314.
  6. 2019 Companion of the Order of Canada: James Greig Arthur, University of Toronto (2019).
  7. Arthur, James: Profile, University of Ottawa (2002).
  8. Arthur (nee Scott), Katherine Mary Patricia, The Globe and Mail (6 December 2011).
  9. J G Arthur, Harmonic Analysis of Tempered Distributions on Semisimple Lie Groups of Real Rank One (Ph.D. Thesis) (Yale University, 1970).
  10. J G Arthur, James Arthur: Memories of a Conversation with Yau from Long Ago, Notices of the International Consortium of Chinese Mathematicians 7 (1) (2019), 1.
  11. Career Profile - David Arthur, Canadian Mathematical Society.
  12. J Cogdell and F Shahidi, 2017 Leroy P Steele Prize for Lifetime Achievement, Notices of the American Mathematical Society 65 (6) (2018), 637-645.
  13. Conference on automorphic forms and the trace formula, in honour of James Arthur on the occasion of his 60th birthday, Fields Institute, 13-16 October 2004, University of Toronto (2004).
  14. CRM-Fields Prize Recipient 1997: James Arthur, Centre de Recherches Mathématiques (4 April 1997).
  15. Dr James Arthur: Companion of the Order of Canada, The Governor General of Canada (19 November 2018).
  16. L Farag, James G Arthur appointed as a Companion of the Order of Canada, University of Toronto (27 December 2018).
  17. Helm-Arthur, The Courier-Journal, Louisville, Kentucky (19 December 1971), 141.
  18. A Jackson, Presidential views: Interview with James Arthur, Notices of the American Mathematical Society 52 (3) (2005), 350-352.
  19. A Jackson, Presidential views: Interview with James Arthur, Notices of the American Mathematical Society 54 (2) (2007), 246-248.
  20. James Arthur - Virginia Mathematics Lectures - November 14-16, 2016, Department of Mathematics, University of Virginia (14 November 2016).
  21. James Arthur, University of Toronto.
  22. James Arthur, The Royal Society.
  23. James Arthur, National Academy of Sciences.
  24. James Arthur '62. Old Boy of Distinction Award (2020), Upper Canada College (11 February 2020).
  25. James G Arthur, Trinity College, University of Toronto.
  26. James G Arthur. President 2005-2006, American Mathematical Society.
  27. James G Arthur, Institute for Advanced Study.
  28. James G Arthur to Receive 2017 Steele Prize for Lifetime Achievement, Institute for Advanced Study (10 November 2016).
  29. James G Arthur: Creating a vision of a unified mathematical world, Universities Canada (10 March 2016).
  30. James G Arthur: Wolf Prize Laureate in Mathematics 2015, Wolf Foundation (2015).
  31. James Greig Arthur, International Mathematical Union (2011).
  32. James Greig Arthur, science.ca.
  33. R P Langlands, The Trace Formula and Its Applications: An Introduction to the Work of James Arthur, Canadian Mathematical Bulletin 44 (2) (2001), 160-209.
  34. R P Langlands, Nomination for President Elect: James G Arthur, Notices of the American Mathematical Society 50 (10) (2003), 1303-1305.
  35. J Lewis, Wolf Prize in Mathematics awarded to University Professor James Arthur, U of T News (2 February 2015).
  36. E Phillips, "Poetry as a way of thinking: an interview with James Arthur" by Emilia Phillips, 32poems.com (2015).
  37. Professor James G Arthur, Department of Mathematics, Fields Institute, University of Toronto (2004).
  38. J Siegel-Itzkovich, Wolf Prizes in the sciences and arts presented to nine North Americans, The Jerusalem Post (29 January 2015).
  39. C Sorensen, G Vendeville and P King, U of T faculty, alumni and supporters named to Order of Canada, U of T News (27 December 2018).
  40. The collected works of James G Arthur, Department of Mathematics, University of Toronto.
  41. The James Arthur Conference, Mathematics, University of Toronto (February 2005), 8-9.
  42. E A Thieleker, Review: The characters of discrete series as orbital integrals, by James Arthur, Mathematical Reviews MR0412348 (54 #474).
  43. K D Thomas, The Fundamental Lemma: From Minor Irritant to Central Problem, Institute for Advanced Study Newsletter (Summer 2010).
  44. Three alumni honoured for outstanding achievements, Upper Canada College (11 February 2020).
  45. UCC community members join Order of Canada, Upper Canada College (17 January 2019).
  46. AMS President-Elect, London Mathematical Society Newsletter 322 (2004), 20.
  47. J Lewis, Jim Arthur on the value of learning basic math skills, University of Toronto (31 March 2014).
  48. D P H Arthur, Autour de Cénie: le témoinage épistolaire de Françoise de Graffigny sur son succés théâtral en 1750, Ph. D. Dissertation (Brown University, 2009).
  49. S Gelbart, Review: Simple algebras, base change, and the advanced theory of the trace formula, by James Arthur and Laurent Clozel, Mathematical Reviews MR1007299 (90m:22041).
  50. D Jiang, Review: The endoscopic classification of representations. Orthogonal and symplectic groups, by James Arthur, Mathematical Reviews MR3135650.

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Written by J J O'Connor and E F Robertson
Last Update March 2024