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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.
PRESS RELEASE
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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