July 24, 2014

2005 CRM-Fields Prize Lecture
October 27, 2005 -- 3:30 p.m.

David Boyd
University of British Columbia

Explicit formulas for Mahler's measure

Explicit formulas for Mahler's measure

If P is a polynomial in n variables, its Mahler measure, m(P) is defined to be the average of log|P| integrated over the
product of n circles. This quantity appears naturally as an entropy in certain discrete dynamical systems and as a rate of growth in many other situations. When n = 1, there is a classical formula of Jensen that expresses m(P) in terms of the zeros of P, but for n > 1 there is no such general formula. In the late 1970's, Smyth proved some intriguing formulas for a few polynomials of 2 and 3 variables that showed that m(P) can sometimes be related to special values of Dirichlet L-functions. Recently, starting from an insight of Deninger, formulas have been proved and conjectured for infinite families of polynomials in 2, 3 and more variables relating the value of m(P) to special values of L-functions of various kinds including L-functions of elliptic curves, Hecke L-functions and L-functions defined by modular forms. We will present a varied selection of some of these formulas and explain how some of them are proved.

January 2005 -
The directors of the Centre de recherches mathématiques (CRM) of l'Université de Montréal, François Lalonde, and the Fields Institute for Research in Mathematical Sciences, Barbara Keyfitz, are pleased to announce the awarding of the CRM-Fields Prize for 2005 to Professor David Boyd in recognition of his exceptional achievement and work in analytic number theory.

The Centre de recherches mathématiques and The Fields Institute established the CRM-Fields prize in 1994 to recognize exceptional research in the mathematical sciences. The recipient is chosen by a selection committee made up of members of the Advisory Committee of the CRM and the Scientific Advisory Panel of the Fields Institute.

David Boyd, this year's recipient, is one of Canada's leading number theorists. He has made seminal contributions to analytic number theory, noteworthy among which are his explorations of the deep connections between the Mahler measure of polynomials and special values of their associated L-functions.

Professor Boyd received his B.Sc. from Carleton University in 1963, and his M.A. and Ph.D. from the University of Toronto in 1964 and 1966. He has taught at the University of Alberta and the California Institute of Technology, and has been at UBC since 1971 where he is currently Full Professor. He is a winner of the E.W.R. Steacie prize, a Fellow of the Royal Society of Canada, and has won both the Canadian Mathematical Society's Coxeter-James and Jeffery-Williams prize lectures. His service to the Canadian mathematical community includes terms as vice-president of the Canadian Mathematical Society, chair of the NSERC Mathematics grant selection committee, and Acting Director of the Pacific Institute for the Mathematical Sciences.

Previous recipients of the prize are H.S.M. (Donald) Coxeter, George A. Elliott, James Arthur, Robert V. Moody, Stephen A. Cook, Israel Michael Sigal, William T. Tutte, John B. Friedlander, John McKay, Edwin Perkins, and Donald A. Dawson.

Janvier 2005. -
Le directeur du Centre de recherches mathématiques(CRM) de l'Université de Montréal, M. François Lalonde et la directrice du Fields Institute for Research in Mathematical Sciences, Mme Barbara Keyfitz, sont fiers d'annoncer que le prix CRM-Fields 2004 est octroyé au professeur David Boyd en reconnaissance de l'excellence de sa contribution à la recherche en théorie analytique des nombres.

Ce prix a été créé par les deux centres en 1994 afin de souligner l'excellence de recherches en sciences mathématiques. Le récipiendaire est choisi par un comité de sélection formé à partir du Comité consultatif du CRM et du Comité aviseur scientifique du Fields Institute.

Le récipiendaire de cette année, David Boyd, est une figure dominante de la théorie des nombres au Canada. Il a contribué de façon déterminante au développement de la théorie analytique des nombres, notamment dans l'exploration des connexions profondes entre la mesure de Mahler des polynômes et les valeures particulières des fonctions L associées.

Le professeur Boyd a obtenu un baccalauréat ès sciences de l'Université Carleton (1963) et ses diplômes de M.A. et Ph.D. de l'Université de Toronto (1964 et 1966 respectivement). Il a enseigné à l'Université de l'Alberta et au California Institute of Technology avant de se joindre à l'Université de Colombie-Britannique en 1971, où il est présentement professeur titulaire. Il a reçu le Prix E.W.R. Steacie et est membre élu de la Société Royale du Canada. Il est récipiendaire des prix Coxeter-James et Jeffery-Williams de la Société mathématique du Canada. Parmi ses contributions à la communauté mathématique canadienne, mentionnons qu'il a été vice-président de la Société mathématique du Canada, président du comité de sélection du CRSNG en mathématiques, et directeur par intérim du Pacific Institute for the Mathematical Sciences.

Les professeurs H.S.M. (Donald) Coxeter, George A. Elliott, James Arthur, Robert V. Moody, Stephen A. Cook, Israel Michael Sigal, William T. Tutte, John B. Friedlander, Edwin Perkins, John McKay, et Donald A. Dawson ont été les récipiendaires précédents du Prix CRM-Fields.

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