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CMS NOTES De La SMC CMS NOTES de la SMC Volume 34 No. 2 March / mars 2002 In this issue / Dans ce numero´ DU BUREAU DE LA memeˆ le placement des futurs doc- PRESIDENTE´ ELUE´ teurs et boursiers postdoctoraux dans 2001 CMS Jeffery Williams les entreprises actives en recherche et Prize Lecture .............. 3 developpement.´ D’autre part beau- coup de departements´ se retrouvent Fermat’s Last Tango in Berke- a` un point tournant et renouvellent ley......................... 5 ou renouvelleront une partie impor- Book Review: Stochastic Pro- tante de leur corps professoral d’ici cesses, Physics and Geome- quelques annees´ alors que de nom- try ........................ 8 breux collegues` prendront leur retraite dans un avenir proche. Book Review: Stamping La communaute´ mathematique´ Through Mathematics ..... 9 canadienne a fait de grands efforts ces dernieres` annees´ pour redorer l’image Awards / Prix ................ 10 Christiane Rousseau des mathematiques´ aupres` du grand public et pour nouer des liens entre les Book Review: Uncle Petros and differents´ ordres d’enseignement. Goldbach’s Conjecture ..... 13 (see page 7 for English translation) L’enseignement et la formation Le paysage mathematique´ s’est con- CRM-Fields Prize 2002 ....... 14 de personnel hautement qualifie´ de- siderablement´ enrichi au pays ces meurent au cœur de notre mission. Nos dernieres` annees.´ La communaute´ Research Notes ............... 16 departements´ reunissant´ professeurs et mathematique´ a maintenant trois in- etudiants´ sont un milieu de vie tout au- Book Review: Mathematics stituts mathematiques´ de premier tant qu’un milieu de formation. Nous Unlimited ................. 18 calibre se concertant dans la pro- sommes heureux d’accueillir ces je- grammation des annees´ thematiques.´ unes qui decident´ de venir etudier´ Education Notes ............. 20 Les contacts se sont accentues´ et en mathematiques´ et en sciences les liens resserres´ entre les branches mathematiques´ et de travailler avec eux Book Review: Best Approxima- des sciences mathematiques´ qui a` leur preparer´ un avenir et a` leur ou- tion in Inner Product Spaces 25 s’enrichissent mutuellement. La vrir le monde. Notre mission de for- recherche multi-disciplinaire prend mation ne s’arreteˆ pas a` la formation Call for Sessions / Appel de un essor sans prec´ edent´ en particulier communications ........... 29 de chercheurs. Plusieurs de nos mem- du cotˆ e´ de la biologie mathematique´ bres sont activement impliques´ dans ou informatique. L’arrivee´ de BIRS la formation des maˆıtres au secondaire Call for Nominations / Appel de favorisera dans le futur l’organisation Candidatures .............. 31 ou encore al’` el´ ementaire.´ En memeˆ de rencontres de recherche permettant temps que nous travaillons a` faire re- aux idees´ et aux theor´ emes` de germer. Calendar of events / Calendrier connaˆıtre l’importance de notre disci- Les contacts et collaborations entre la des ev´ enements´ ............. 33 pline au niveau des organismes subven- communaute´ mathematique´ universi- tionnaires nous assistons al’` erosion´ Rates and Deadlines / Tarifs et taire et les chercheurs industriels se ´ Ech´ eances´ ................. 35 sont multiplies,´ favorisant par le fait (voir PRESIDENTE–page 28) MARCH/MARS CMS NOTES EDITORIAL Those of you who may be tempted to CMS NOTES try your hand at this are invited to keep the Notes in mind as a possible show- NOTES DE LA SMC case for your efforts! The CMS Notes is published by the Canadian Mathematical Society (CMS) eight times a year (February, ***** March, April, May, September, Oc- tober, November, and December). Editors-in-Chief Les périodiques comme celui-ci, qui ne Peter Fillmore; S. Swaminathan se consacrent pas principalement à la [email protected] recherche, sont un aspect intéressant de la culture mathématique. Ce type de Managing Editor publication est en pleine effervescence, Graham P. Wright à la fois en nombre et en qualité. Au Contributing Editors cours des derniers mois, on a d’ailleurs Education: Edward Barbeau; Harry repris dans les Notes d’excellents arti- White cles issus de telles sources. Ce mois- [email protected] Peter Fillmore ci, il nous fait plaisir de vous présen- Meetings: Monique Bouchard ter des critiques d’un roman (de la [email protected] Gazette de la Société mathématique Research: Noriko Yui An interesting aspect of mathematical d’Australie) et d’une comédie musicale [email protected] culture is the periodical, such as this (du bulletin Emissary du MSRI). On Photo: Paul Milnes one, not primarily devoted to research. s’étonnera sans doute que de tels ar- [email protected] Such publications appear to be flour- ticles soient bien faits, mais (comme ishing, both in number and in qual- Editorial Assistant certains le diront dans le cas du pro- ity, and in recent months the Notes has Caroline Baskerville fesseur Johnson) encore davantage que reprinted excellent articles from several The Editors welcome articles, letters l’on rédige même ce genre d’article. of them. This month we are pleased Serait-ce un indice que nos efforts de and announcements, which should to present reviews of a novel–from be sent to the CMS Notes at: sensibilisation et de vulgarisation com- the Australian Mathematical Society’s mencent à porter fruit? Canadian Mathematical Society Gazette–and a musical–from MSRI’s 577 King Edward Emissary. It is perhaps surprising that Quoi qu’il en soit, il est clair que P.O. Box 450, Station A such things are done well, but (as one la partie est encore loin d’être gagnée. Ottawa, Ontario, Canada K1N 6N5 might say, with Dr. Johnson) even Dans le même numéro de Emissary, Telephone: (613) 562-5702 more surprising that they are done at on peut lire le compte rendu d’un ate- Facsimile: (613) 565-1539 all. Is this an indication that our efforts lier sur les médias tenu au MSRI l’été E-mail: [email protected] at outreach are beginning to bear fruit? Website: www.cms.math.ca dernier. Les journalistes et mathémati- However that may be, it is clear that ciens présents avaient des priorités pour No responsibility for views expressed by much remains to be done. The same le moins divergentes, et il est ressorti authors is assumed by the Notes, the ed- issue of Emissary carries a report of que « les médias ne considéraient pas la itors or the CMS. The style files used in a media workshop which took place promotion des mathématiques comme the production of this volume are a mod- at MSRI last summer. The reporters l’une de leurs priorités ». La morale ified version of the style files produced and mathematicians present had rather de cette histoire : si nous voulons que by Waterloo Maple Software, c 1994, different priorities, and the point was les médias parlent de nous, c’est à nous 1995. made that "the media doesn’t see pro- de rendre nos recherches intéressantes ISSN: 1193-9273 (Print); moting mathematics as one of its re- et surtout intelligibles pour les journal- 1496-4295 (Online) sponsibilities". The corollary is that if istes et leurs lecteurs. Si vous avez en- we want media coverage, it’s up to us vie d’écrire des articles grand public, c Canadian Mathematical Society 2002 to make our research interesting and in- songez à publier le fruit de vos efforts telligible to reporters and their readers. dans les Notes! 2 NOTES de la SMC MARCH/MARS 2001 CMS JEFFERY-WILLIAMS PRIZE LECTURE Mahler’s measure, hyperbolic geometry and the dilogarithm David W. Boyd, University of British Columbia √ 3 3 m(1 + x + y)= L(χ , 2), 4π −3 (2) where 1 1 1 ∞ −3 1 L(χ ,s)=1− + − + ...= . −3 2s 4s 5s n ns n=1 This result strongly suggests that L1 L2. Probably L1 L2 L3 ..., but as far as is known these could all be equalities! A conjecture of Chinburg In 1984, Chinburg asked whether (2) could be generalized to other conductors, i.e. for each negative discriminant −f David Boyd is there is a polynomial Pf (x, y) ∈ Z[x, y] and a non-zero rf ∈ Q such that The author was sponsored in part by a grant from NSERC. This is a short description of my Jeffery–Williams lec- m(Pf )=rf df ? ture given in June, 2001 at the CMS Summer meeting at the University of Saskatchewan. (3) I wish to thank the Canadian Mathematical Society for Here we have used the abbreviation the honour of being chosen as the Jeffery–Williams lecturer √ f f for 2001. I also wish to thank my good friend and collabo- df = L(χ−f , 2). rator Fernando Rodriguez Villegas without whose help and 4π encouragement this work would not have been done. More generally, given a number field F of degree n with Mahler’s Measure r2 =1(i.e. with a single pair of complex embeddings), a Given a non-zero polynomial P ∈ Z[x1,...,xn] the log- natural generalization of df is arithmic Mahler measure is defined as 3/2 3|disc(F )| ζF (2) 1 1 Z = , F 2n−3 2n−1 2πiθ1 2πiθn 2 π m(P )= ··· log P (e ,..., e ) dθ1 ···dθn 0 0 ζF denoting the Dedekind zeta function of F . Then df = (1) Z √ . One can generalize Chinburg’s question and ask: By using Jensen’s formula, the one variable Mahler mea- Q( −f) P ∈ Z[x, y] r > 0 sure has a simple expression in terms of the roots of the can one find F and rational F so that polynomial and so has been the object of study for many m(PF )=rF ZF ? years. In 1934, Lehmer [L] asked (in essence) whether 0 is L = {m(P (x)) : P ∈ Z[x]} a limit point of 1 . The au- (4) thor’s 1979 Coxeter–James lecture [Bo1], was devoted to Ray [R] constructed examples to settle (3) for f = showing that this question leads naturally to the study of 3, 4, 7, 8, 20 and 24, the example for 7 being particularly re- L = {m(P (x ,...,x ):P ∈ Z[x ,..., x ]} n 1 n 1 n and to markable.
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