The John L Synge Award was established to honour John Lighton Synge, Fellow of the Royal Society and of the Royal Society of Canada. Synge was one of the first mathematicians working in Canada to obtain international recognition by research in mathematics. He was for some years Head of the Department of Applied Mathematics at the University of Toronto, later a senior professor at the Dublin Institute of Advanced Studies. The Prize, established in 1986, is given at irregular intervals for outstanding research in any branch of the mathematical sciences. Some preference is given to candidates whose age is not over 40 in the year of the award. The award consists of a diploma and a cash prize of $1,500. James Arthur was the first recipient of the John L Synge Award.

This prize, named in honour of Ralph Lent Jeffery and Lloyd Williams, is awarded by the Canadian Mathematical Society. The recipient must be a member of the Canadian mathematical community. The award presentation takes place at the Canadian Mathematical Society Summer Meeting and a plenary lecture given by the recipient. In 1993, the Canadian Mathematical Society Summer Meeting was held 15-19 August. The Jeffery-Williams Lecture was delivered by James G Arthur, University of Toronto, on Trace formulas and automorphic representations, on Tuesday, 17 August at 7:00 p.m.

The prize recognises exceptional achievement in the mathematical sciences. It was established by the Centre de recherches mathématiques and The Fields Institute as the CRM-Fields prize in 1994. James Arthur was awarded the 1997 prize and gave the prize lecture Harmonic Analysis and Trace Formulas at the Centre de recherches mathématiques on 4 April 1997. His abstract was as follows:-

Harmonic analysis could be interpreted broadly as a general principal which relates analytic and geometric objects. Examples occur throughout many areas of mathematics. In group theory, the geometric objects are conjugacy classes, the analytic objects are irreducible characters, and the two can be related by means of trace formulas. We shall give a general introduction to trace formulas, and their applications to group representations and number theory.

The Henry Marshall Tory Medal is awarded by the Royal Society of Canada. The prize was established in 1941 and the first award was made to John Lighton Synge in 1943. It is a biennial award:-

The medal is awarded for outstanding research in a branch of astronomy, chemistry, mathematics, physics, or an allied science, carried out mainly in the eight years preceding the date of the award, but all the research of the candidate is taken into account. The gold plated silver medal is offered every two years if there is a suitable candidate.

The 1997 award was made to James Arthur.

This award was made by the Natural Sciences and Engineering Research Council of Canada with the first award being made in 1991. James Arthur was given this award in 1999. He was the last to receive it under this name as it was renamed the Gerhard Herzberg Canada Gold Medal for Science and Engineering to honour Gerhard Herzberg, winner of the 1971 Nobel Prize in Chemistry. The Natural Sciences and Engineering Research Council gives the following description of the award:-

The Gerhard Herzberg Canada Gold Medal for Science and Engineering is awarded annually to an individual whose body of work, conducted in Canada in the natural sciences or engineering, has demonstrated persistent excellence and influence. It celebrates Canada's most outstanding scientists and engineers, raising public awareness about the major contributions that Canada's top researchers make to science and technology, and to improving the lives of Canadians.

The Natural Sciences and Engineering Research Council gave the following citation for the award to James Arthur.

1999 Winner James Arthur.

In mathematics, the quest for what could be called a grand unified theory is regarded by many as the most challenging frontier of all. The work of Dr James G Arthur in this direction is internationally recognised for its outstanding advances.

Dr Arthur's developments in automorphic forms and representation theory - in particular, his innovative "trace formula" - have opened new approaches to the challenges posed by the "Langlands program," an ambitious and far-reaching theoretical mathematical model. Developed some 30 years ago by Canadian-born mathematician Robert Langlands, the model's ultimate goal is to link two great streams of classical mathematics: analysis, which deals with how phenomena such as planetary motion vary with respect to time; and algebra, which deals with the unchanging world of integers and prime numbers. The model has created a vision of a unified mathematical world in which mathematical disciplines previously believed independent will prove to be related in completely unexpected and astonishing ways.

But while the circumstantial evidence for a fundamental and absolute relationship between the two streams of mathematics is striking, the mathematical explanation remains elusive. Its pursuit has become a major field of mathematical endeavour around the world. When it is finally achieved, the knowledge that comes from the Langlands program will represent a fundamental ordering principle in mathematics and beyond. Some mathematicians believe that it will eventually explain phenomena not yet understood about the basic forces of nature.

Dr Arthur's trace formula, developed in the early 1980s, has become mathematicians' most powerful tool in this pursuit. A deep and highly complex equation, one side of the trace formula deals with explicit geometric information, while the other side contains the more elusive spectral information that is at the heart of the Langlands program.

Dr Arthur and others have been able to complete part of the Langlands program by using the geometric side of the trace formula to illuminate the spectral side. The formula is a far-reaching illustration of the basic duality between geometric and spectral objects that runs throughout all of mathematics and physics. It is analogous to the particle-wave duality of elementary particles in quantum mechanics. A more concrete analogy is the relationship between the shape of a musical instrument (its geometry) and the sound it produces (determined by the spectrum of its sound waves).

After completing the trace formula, Dr Arthur went on to create what have come to be known as "Arthur packets." These packets enable mathematicians to deal with previously inexplicable anomalies in the energy levels (eigenvalues) that are part of the spectral information on the analytic side of the trace formula. Dr Arthur recognised that the anomalies have certain universal properties that allow them to be systematically analysed. Arthur packets resolve the apparent inconsistencies and place all energy levels in the Langlands program on an equal footing.

These and many other significant insights have brought Dr Arthur numerous prestigious honours. An Elected Fellow of both the Royal Society of Canada (1980) and the Royal Society of London (1992), he was named "University Professor" at the University of Toronto in 1987. He was awarded the Synge Award in 1987 and the Henry Marshall Tory Medal of the Royal Society of Canada in 1997. He has also received the CRM-Fields Institute Prize in Mathematics (1997), the E W R Steacie Memorial Fellowship (1982) from NSERC, and the Sloan Fellowship (1975-77) from the Sloan Foundation.

Dr Arthur is a professor in the Department of Mathematics at the University of Toronto, where his teaching skills and ability to inspire Canada's next generation of mathematicians is legendary. In the words of one of his peers, Dr Arthur's "accomplishments, his current research and his vision of the future establish him as one of the outstanding mathematicians of the world."

The Wilbur Lucius Cross Medal for Alumni Achievement, is an award by the Yale University Graduate School Alumni Association to recognise:-

... distinguished achievements in scholarship, teaching, academic administration, and public service ...

Two of the four awards made in 2000 were to mathematicians: James Arthur and Evelyn Boyd Granville.

The G de B Robinson Prize is awarded by the Canadian Mathematical Society. They give the following citation for the 2003 award to James Arthur.

CMS 2003 G de B Robinson Prize - Dr James Grieg Arthur, University of Toronto.

The G de B Robinson Award was inaugurated to recognise the publication of excellent papers in the Canadian Journal of Mathematics and the Canadian Mathematical Bulletin and to encourage the submission of the highest quality papers to these journals. The first award was presented for papers that appeared in the Canadian Journal of Mathematics in 1994-1995. The 2003 G de B Robinson Prize is awarded to Professor James Arthur of the University of Toronto for the paper, "A Note on the Automorphic Langlands Group", Canadian Mathematical Bulletin 45 (2002), 466-482. The paper addresses an explicit conjecture of Langlands on the existence of an extension of the absolute Galois group that would serve as a universal group in the theory of automorphic forms. The author presents a possible candidate for this universal group and a possible candidate for the complexification of Grothendieck's motivic Galois group. Besides making a fundamental contribution to a central area, this paper is exceptionally lucid and inspiring in its presentation. As such, it represents the ideal that G de B Robinson Prize was designed to acknowledge.

Dr Arthur obtained his B.Sc. and M.Sc. at the University of Toronto and went on to Yale University, under the supervision of Professor R P Langlands, for his Ph.D. He was instructor at Princeton, Assistant Professor at Yale as well as Professor at Duke University in the 1970s. In 1978, he joined the University of Toronto as a Professor in the department of Mathematics.

Among the honours Dr Arthur has received are: Elected Fellow of American Academy of Arts and Science (2003); Honorary Doctorate, University of Ottawa (2002); Whittemore Lectures, Yale University (2001); Guggenheim Fellowship (2000); Wilbur Lucius Cross Medal, Graduate School, Yale University (2000); Canada Gold Medal for Science and Engineering, NSERC (1999); Faculty Award of Excellence, University of Toronto (1999); Henry Marshall Tory Medal, Royal Society of Canada (1997); CRM/Fields Institute Prize (1997); Jeffery-Williams Lecturer, Canadian Mathematical Society (1993); Aisenstadt Chair, CRM, University of Montreal (1992); Elected Fellow of the Royal Society of London (1992); Synge Award - Royal Society of Canada (1987); E W R Steacie Memorial Fellowship (1982); Elected Fellow of the Royal Society of Canada (1980); Sloan Fellowship - Institute for Advanced Study, Princeton (1975-77).

The Killam Prize is awarded by the Canada Council for the Arts. Five prominent scholars in the fields of natural sciences, philosophy, music, health sciences and geotechnical and geo-environmental engineering were honoured with the 2004 Killam Prizes, Canada's most distinguished annual awards for outstanding career achievements in engineering, natural sciences, health sciences, social sciences and humanities. The citation for James Arthur is as follows.

James G Arthur - Natural Sciences (University of Toronto)

James G Arthur is regarded as one of the leading mathematicians in the world in the fields of representation theory and automorphic forms. Representation theory is the study of the deeper aspects of symmetry. Automorphic forms is the branch of representation theory that relates symmetry with arithmetic and number theory. Professor Arthur is specifically interested the Langlands programme - a blueprint for relating arithmetic and algebra with analysis and spectral theory.

Over the past 30 years, Professor Arthur has made many fundamental discoveries, which have had a very significant impact on mathematical research. In a series of papers that span two decades, he was able to construct the general trace formula, a mathematical equation of enormous power that had been sought by mathematicians since the 1950s. His joint work with L Clozel, which appeared in Annals of Mathematics Studies, solved a critical comparison problem for trace formulae on different groups. He has introduced a remarkable conjectural classification of automorphic representations, now known as Arthur packets.

Born in Toronto, James Arthur received B.Sc. and M.Sc. from the University of Toronto. He received his Ph.D. from Yale University. He then spent nine years in the United States at Princeton University, Yale University, and Duke University, and he returned to the University of Toronto in 1979 as professor.

Professor Arthur has achieved many distinctions. Elected as a Fellow of the Royal Society of Canada in 1980 and the Royal Society of London in 1992, he became the first recipient of the Synge Award of the Royal Society of Canada in 1987. He was awarded the Centre de Recherches Mathématiques-Fields Institute Prize and the Henry Marshall Tory Medal in 1997. In 1999 he received the Canada Gold Medal for Science and Engineering from Natural Sciences and Engineering Research Council of Canada, the only mathematician to have done so. He was awarded the Wilbur Lucius Cross Medal from the Graduate School of Yale University and a Guggenheim Fellowship in 2000. In 2002, he received an honorary doctorate from the University of Ottawa.

James G Arthur was Wolf Prize Laureate in Mathematics 2015. The citation states that the award of the prize to Arthur was:-

... for his monumental work on the trace formula and his fundamental contributions to the theory of automorphic representations of reductive groups.

The Wolf Foundation gives the reasons for the award.

James G Arthur: Wolf Prize Laureate in Mathematics 2015.

Arthur's development of the trace formula for reductive groups is a monumental mathematical achievement. It generalises the Selberg trace formula for SL(2) from 1956. In his work, Arthur introduced many major tools in non-commutative harmonic analysis on general reductive groups. Building on the work of Langlands, Shelstad, Kottwitz, Waldspurger and others, Arthur obtained the trace formula in stable form. Using the Fundamental Lemma proved by Ngo, Arthur's work culminated in his description, as envisioned by Langlands, of the structure of automorphic representations of classical groups (symplectic groups and quasi-split special orthogonal groups). Some of the highlights are the functoriality associated to the standard representation, the multiplicity formulas in the discrete spectrum, the classification of the expected counter-examples to the generalised Ramanujan conjecture, and the description of local L-packets and global A-packets. Arthur's work had an enormous impact. For example, it had been a central tool in Lafforgue's proof of the Langlands correspondence for function fields. Recently it has been used by Clozel, Harris, Taylor and others in constructing Galois representations associated to automorphic forms via p-adic methods. Arthur's ideas, achievements and the techniques he introduced will have many more deep applications in the theory of automorphic representations, and the study of locally symmetric spaces. Arthur's work is a mathematical landmark that will inspire future generations of mathematicians.

University of Toronto announced Wolf Prize Award to James Arthur.

Jessica Lewis, a writer with the Faculty of Arts & Science at the University of Toronto wrote the article Wolf Prize in Mathematics awarded to University Professor James Arthur. Here are some extracts.

Wolf Prize in Mathematics awarded to University Professor James Arthur.

University Professor James Arthur, the Ted Mossman Chair in Mathematics, has been awarded the highly prestigious Wolf Prize in Mathematics.

Presented by the Wolf Foundation in Israel, the prizes have been given since 1978 in the categories of agriculture, chemistry, mathematics, medicine, physics and arts and are considered by many to be precursors to the Nobel Prize in the fields in which the Nobel is awarded.

The award to Arthur marks only the second time that a Canadian has won the mathematics prize. The first to win it was Robert Langlands, who coincidentally was Arthur's graduate supervisor at Yale from 1968 to 1970. Only 11 Canadians have won a Wolf Prize in any of its six categories - U of T's Nobel Prize-winning chemist John Polanyi won it in 1982 in his respective field.

"I am honoured and proud to be awarded the Wolf Prize in Mathematics," said Arthur. "I am in awe as I look back at the names of mathematicians who have won the prize in years past. Recognition of this sort is of course wonderful for me. But I hope that it might also give inspiration to younger mathematicians who are working to build the future of the subject.

"I would be very happy if it helps to bring increased attention to the outstanding work now being done in Canada by many mathematicians and many other scientists."

"This is a magnificent, global tribute to a man whose contributions have been recognised many times over by an array of accolades. It is a reason to celebrate, for all those who value the advancement of knowledge," said U of T President Meric Gertler.

"We applaud Jim's achievements and leadership in research and training," said Kumar Murty, chair of the department. "This kind of international award is a wonderful testament, not only to the efforts of a particular researcher, but to the high standards of the department as a whole. It gives a kind of forward momentum and I am hopeful that we can work together to take full advantage of it and propel the department to even greater heights."

The Steele Prizes were established by the American Mathematical Society in 1970 in honour of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein. Osgood was president of the American Mathematical Society during 1905-1906, and Birkhoff served in that capacity during 1925-1926. The prizes are endowed under the terms of a bequest from Leroy P Steele. There are three categories, one of which is Lifetime Achievement. This is awarded:-

... for the cumulative influence of the total mathematical work of the recipient, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through Ph.D. students.

James Arthur was awarded the 2017 Leroy P Steele Prize for Lifetime Achievement. The American Mathematical Society lists the citation, Arthur's biography, and his response.

Citation for Lifetime Achievement: James Arthur.

The 2017 Steele Prize for Lifetime Achievement is awarded to James Arthur for his fundamental contributions to number theory and harmonic analysis, and in particular for his proof of the Arthur-Selberg trace formula.

Introduction of L-functions into the theory of automorphic forms began with a conjecture of Ramanujan, its proof by Mordell, and the exploitation of Mordell's ideas by Hecke, who had already had experience with Euler products in the context of Dedekind ζ-functions and related L-functions. Later Selberg introduced methods from the spectral theory of second-order differential equations on a half-line, as well as a form of the Frobenius reciprocity theorem, familiar from the representation theory of finite groups. In the context of discrete subgroups of Lie groups it became known as the Selberg trace formula. For groups with compact quotient, it is hardly more difficult than the Frobenius theorem itself. For groups with quotients of finite volume but not compact, not only its formulation but also its proof required ingenuity and a good deal of skill in the use of the spectral theory.

The first trace formula for general groups was established by Arthur in the 1970s in a series of three papers. Starting from the particular case of SL(2) which had been established by Selberg in 1956, Arthur has built a whole mathematical framework and introduced many major tools in noncommutative harmonic analysis in order to prove the trace formula for a general reductive group. The final result is now called the Arthur-Selberg trace formula. The proof in itself takes sixteen long and difficult papers that Arthur published between 1974 and 1988. This is considered to be a major achievement in mathematics.

As Langlands suggested at the end of the 1960s, the trace formula is a powerful tool for proving the Langlands principle of functoriality, especially in the so-called endoscopic case. For this purpose, one first needs to stabilise the Arthur-Selberg trace formula. Arthur published eight papers between 1997 and 2003 on the stabilization process. Using the stable trace formula and the Fundamental Lemma proved in 2008 by Ngô Bao Châu, Arthur has recently been able to establish the Langlands functoriality for the standard representations of the classical groups (symplectic, orthogonal, and unitary).

As a consequence, he has obtained explicit formulas for the multiplicities in the automorphic discrete spectrum for those classical groups. The Arthur-Selberg trace formula is a central tool in Lafforgue's proof of the Langlands correspondence for function fields.

Arthur's contribution to mathematics is fundamental. His work already has had, is having, and will have an enormous impact on several branches of mathematics. But his service to the mathematical community is also very impressive. Arthur played an important role in shaping the work of several important national and international committees and organisations. All this culminated when he served as President of the American Mathematical Society.

In 1992 Arthur was elected a Fellow of the Royal Society. He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 2003 and a foreign member of the National Academy of Sciences in 2014. In 2015 he was awarded the Wolf Prize in Mathematics.

Biographical Sketch: James Arthur.

James Arthur is a university professor and holds the Ted Mossman Chair in Mathematics at the University of Toronto. He was born in Hamilton, Ontario, in 1944 and received a B.Sc. from the University of Toronto in 1966, an M.Sc. from the University of Toronto in 1967, and a Ph.D. from Yale University in 1970. He then held positions in mathematics at Princeton University, Yale University, and Duke University before returning to the University of Toronto in 1979.

Arthur is a fellow of the Royal Society of Canada, a fellow of the Royal Society of London, a Foreign Honorary Member of the American Academy of Arts and Sciences and a Foreign Associate of the National Academy of Sciences. His various honours and awards include an honorary doctorate at the University of Ottawa in 2002, the Canada Gold Medal in Science and Engineering in 1999, and the Wolf Prize in Mathematics in 2015. He has given several addresses at International Congresses of Mathematicians, including a Plenary Lecture at the congress in Seoul, Korea, in 2014, and he gave a Plenary Lecture at the first Mathematical Congress of the Americas in Guanajuato, Mexico, in 2013. He is presently working on Beyond Endoscopy, a proposal by Robert Langlands for using the trace formula to study the general principle of functoriality.

Arthur has served mathematics in several senior administrative roles. He was a member of the Executive Committee of the International Mathematics Union from 1991 to 1998 and the Academic Trustee for Mathematics on the Board of Trustees of the Institute for Advanced Study from 1997 to 2007. He also served as president of the AMS from 2005 to 2007. He lives in Toronto with his wife, Penny. They have two sons: James, a poet in the creative writing program at Johns Hopkins University; and David, a computer engineer at Google, in Mountain View, California.

Response from James Arthur.

I am thrilled and honoured to receive the Steele Prize for Lifetime Achievement. It is a cliché, but true nonetheless, for me to say that I feel humbled to look down the list of past winners. I would like to thank the Steele Prize Committee for selecting me. I would also like to thank the AMS and the many mathematical colleagues in particular who donate their time to serve on prize committees and to participate in the many other activities that do so much to help our subject thrive.

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. These feelings have remained with me throughout my professional life, and they have motivated me more than any specific theorem or result.

I am very grateful to Robert Langlands for his encouragement, both during my time as a graduate student and since then. I am also grateful to him personally and as a member of the larger community for what he has given to mathematics. His mathematical discoveries truly are magical and powerful. They are becoming more widely known among mathematicians today, and I have no doubt that they will bring pleasure and inspiration to many generations of mathematicians to come.

Much of my mathematical life has been connected in one way or another with what has become known as the Arthur-Selberg trace formula. It is now a very general identity that, like other things in mathematics, links geometric objects (such as closed geodesics) with spectral objects (such as eigenvalues of a Laplacian). The trace formula has many different terms, but as we are beginning to understand them now, each of these sometimes arcane quantities (either geometric or spectral) seems to have its own particular role in the larger scheme of things. I have been fortunate that the trace formula has assumed a more central role than might have been imagined earlier. I am excited to think that there is now a well-defined (if also rather imposing) strategy for using the trace formula to attack what is known as the principle of functoriality, the central tenet of the Langlands program.

On 19 November 2018 James Arthur was named to the highest rank of the Order of Canada, one of the country's highest honours. Arthur was named a Companion of the Order for his contributions to contemporary mathematics, notably his "ground-breaking" advancements to the theory of the trace formula, a complex mathematical formula that relates geometric and spectral information.

"I have had scientific recognition in Canada, but I feel very proud - and thrilled - to be recognised more broadly with the Order of Canada," he said.

Arthur was invested on 4 July 2019. The Governor General of Canada writes:-

James Arthur has broadened our understanding of mathematics in its most fundamental form. With exemplary dedication and gifted insight, the University of Toronto professor has made transformative contributions to number theory, notably to its trace formula. Renamed the Arthur-Selberg formula in his honour, it has become an integral tool now used in the practice of primary mathematical areas, the effect of which has been profound. Recipient of the 2015 Wolf Prize, Dr Arthur is renowned for his academic and professional leadership.

James Arthur was given the Old Boy of Distinction Award by Upper Canada College in 2020. The citation for this award reads as follows.

James Arthur '62.

Considered one of the top mathematicians in Canada and the only Canadian to serve as head of the American Mathematical Society, James is the Ted Mossman Chair in Mathematics at the University of Toronto, and has been a professor at U of T since 1978.

As one of the world's leading academics in the field of mathematics, James is a highly sought-after lecturer internationally. His work is centred on the trace formula, and he has made fundamental contributions to the theory of automorphic forms.

James has demonstrated a lifetime of significant achievement and leadership in his field. A dedicated mentor to young faculty and graduate students, he has brought Canada to greater prominence on the world mathematical stage. His commitment to learning and excellence inspires future generations of academics working in the field of mathematics.

In 2015, James was awarded the Wolf Prize in Mathematics from the Wolf Foundation. This prize is considered by many to be the precursor to the Nobel Prize, and marks only the second time it has been won by a Canadian.

James was Head Boy at the College in 1962, and went on to study at the University of Toronto and Yale University. He was elected a Fellow of the Royal Society of Canada in 1981, a Fellow of the Royal Society in 1992, and a Foreign Honorary Member of the American Academy of Arts and Sciences in 2003. In 2012, he became a Fellow of the American Mathematical Society. James was appointed Companion of the Order of Canada in 2018, and elected a Fellow of the Canadian Mathematical Society in 2019.