Past Winner
1999 Gerhard Herzberg Canada Gold Medal for Science and Engineering

James Arthur

Mathematics

University of Toronto

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 recognized 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 recognized that the anomalies have certain universal properties that allow them to be systematically analyzed. 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."