The general trace formula

Name: Professor James Arthur

Position: Ted Mossman Chair in Mathematics, held since 2009

Affiliation: Department of Mathematics

Education: PhD, Yale University

Areas of Expertise:  Representations of Lie groups, automorphic forms

Publications

– “An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimura varieties," Clay Mathematics Proceedings (2005)

– "A stable trace formula III. Proof of the main theorems," Annals of Mathematics (2003)

– “Unipotent automorphic representations: conjectures,” Astérisque (1989)

Major Awards & Honour

2007     Academic Trustee for the Institute for Advanced Studies at Princeton

2007     President of the American Mathematical Society

1999     NSERC Canada Gold Medal for Science and Engineering (only mathematician to date)

1997     Henry Marshall Tory Medal

1992     Fellow of the Royal Society of London

1980     Fellow of the Royal Society of Canada

Can  geometry unlock the secrets of arithmetic?

During the last 30 years, a few pioneers have worked in the vanguard of representation theory, a branch of mathematics which uses linear transformations of vector spaces to analyze abstract algebraic structures. This translation of one field of mathematics into the terms of another has opened up a range of new approaches to some of the most fundamental problems in modern mathematics, and it has led to important advances in the study of combinatorics, geometry, number theory, probability theory, quantum mechanics and quantum field theory.

University Professor James Arthur, the Ted Mossman Chair in Mathematics, has made a number of essential contributions to representation theory. In a series of landmark papers written since the early 1980s, he constructed the general “trace formula” — sought since the 1950s — which in turn gave rise to important refinements of theory. Arthur also created a conjectural classification of automorphic forms — a part of representation theory which connects symmetry to arithmetic and number theory — commonly known as “Arthur packets.” This classification is proving indispensable in the ongoing development of mathematical theories put forward by Arthur’s fellow Canadian mathematician Robert Langlands.

The Langlands program is an ambitious roadmap for the ways in which arithmetic and algebra can be fused with analysis and spectral theory. In a tribute to Langlands, Professor Peter Sarnak of Princeton University wrote that Langlands’ work “led him to what has become one of the holy grails of the subject: the Principle of Functoriality. The theory of Eisenstein series, combined with the general noncompact adelic trace formula developed by Arthur over many years, is among the most powerful tools that we have today in the theory of automorphic forms. A list of well-known results whose proof relies on the trace formula would cover many pages.”

The trace formula has facilitated progress on several parts the Langlands program. Recently, proof of the “Fundamental Lemma” has paved the way for further breakthroughs in the theory of automorphic forms, many of them due entirely to Arthur’s fundamental work on the trace formula.

The Ted Mossman Chair in Mathematics was established in 1996 through a generous gift from James Mossman in memory of his father.

Story by Brendan de Caires
Photo: Liam Sharp