## Professor James G. Arthur

### Department of Mathematics

Appointed a University Professor in 1987Professor James Arthur has published 56 research papers, comprising over two thousand pages in mathematical journals of the highest reputation. He is regarded as one of two or three leading mathematicians in the world in the central fields of representation theory and automorphic forms. In addition to being an outstanding scientist, Professor Arthur is an excellent teacher and has a distinguished record of service to both the University and the mathematics community.

Representation theory is the study of the deeper aspects of symmetry. The notion of symmetry plays a prominent role in mathematics, and indeed throughout much of science. It is a fundamental unifying force. Representation theory probes the hidden mathematical properties of symmetry in much the same way that spectroscopy analyzes hidden physical properties of light and matter. Automorphic forms is the branch of representation theory that relates symmetry with arithmetic and number theory. According to a general philosophy of R. Langlands of Princeton, automorphic forms hold the key to unifying vast areas of mathematics, some of which date back several centuries. The Langlands programme is a stunning blueprint for relating arithmetic and algebra on the one hand, with analysis and spectral theory on the other.

Over the past thirty years, Professor Arthur has been a leader in the quest to take the Langlands programme from the realm of conjecture to actual mathematical realization. In so doing, he made a number of fundamental discoveries, which have had a major impact on mathematical research. In a series of papers that spanned two decades, he was able to construct the general trace formula, a mathematical equation of great 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, in terms of what are now known as Arthur packets. In the early 1990s, he found a local version of the trace formula that had been conjectured by D. Kazhdan.

Professor Arthur is highly sought after as a lecturer on the international scene. His oral and written exposition of mathematics is clear and inspiring. He is a dedicated mentor of young faculty and graduate students. His active role in the International Mathematical Union, which organizes the International Congress of Mathematicians every four years, has brought Canada to greater prominence on the world mathematical stage.

Professor Arthur has achieved many distinctions in his career. 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 CRM/Fields Institute Prize and the Henry Marshall Tory Medal in 1997. In 1999 he received the Canada Gold Medal for Science and Engineering from NSERC, making him the only mathematician to have won Canada's top award in science. Professor Arthur has twice been an invited lecturer for the International Congress of Mathematicians. 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 in recognition of his achievements in mathematics.