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.

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.

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.

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.

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.

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!"

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.

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.

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.

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.

... 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 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 $SL(2, \mathbb{Z})$. 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.

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.

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.

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.

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.

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.

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.

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.

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.