FROM THE PRESIDENT’S DESK Tom Salisbury (York) IN THIS ISSUE DANS CE NUMÉRO Editorial/Éditorial .....................2 Treasurer NSERC and the Institutes Another remarkable term of As of this writing (July), Call for Committee Nominations service has ended, that of there has been progress on Appel de candidatures Arthur Sherk (Toronto). the issue of NSERC funding pour les comités .......................3 Arthur has been Treasurer for the mathematics research Message du président ..............5 of the Society for 13 years, institutes. As those of you during which time he has who completed a survey Book Review: Brauer Type served with integrity, care, NSERC conducted will know, Embedding Problems ...........7 Much has happened over the and devotion. Throughout that the Centre de recherches summer, including the very period Arthur’s decisions - be mathématiques (CRM), the Book Review: successful CMS Summer they good news or bad - have Fields Institute, and the Pacific It’s About Time ......................9 Meeting in Calgary, and been delivered with fairness Institute for the Mathematical progress regarding NSERC and good humour. Between Sciences (PIMS) have NSERC Brief Book Reviews ................11 funding of the mathematics new revenue expected from grants expiring March 31, CMS Awards Annoucements/ research institutes. But I’d like fundraising, and the restraint 2008. The most recent institute Lauréats des prix de la SMC ..13 to start this report by thanking Arthur has shown in managing funding competitions took Eddy Campbell (Memorial) our expenses, the immediate place under the umbrella of Education Notes .....................17 whose term as President of the fiscal health of the Society NSERC’s global reallocations Society ended June 30. seems assured. I’m sure all process. That mechanism no News from Departments ........19 the CMS Presidents who have longer exists, and NSERC has Eddy’s fairness, good sense, worked with him join me been in discussions with the News from the Institutes .......20 and enthusiasm have served in applauding and thanking Institutes for some time about the CMS well for a long time, Arthur for his outstanding a replacement, that could also Winter Meeting 2006 / Réunion but especially over the last two d’hiver 2006 ...........................21 contributions, to be formally be used to evaluate proposals years as President. Among his recognized at the 2006 Winter for new institutes. NSERC’s many successes, I’ll single out Call for Sessions: CMS-MITACS CMS meeting. website now has details of his response to the budgetary this program - the Major Joint Conference 2007 / Appel difficulties the CMS recently Arthur leaves us in good de sessions : Congrès conjoint hands, as we welcome David Resources Support (MRS) found itself in. While program - a replacement for MITACS-SMC ...........................30 implementing cost savings, Rodgers as our new Treasurer. David is by training a chemist, the existing Major Facilities Call for Proposals - 2006 Eddy also realized that the but has served the Society Access (MFA) program. Endowment Grant / Appel de real solution was to build an ably for years in the area To provide community propositions - concours du fond effective fundraising capacity of electronic services and input into this discussion, de dotation 2006 ...................32 within the CMS, and he led us promptly and effectively publishing. In his new capa- a liaison committee was Treasurer’s report / Rapport du towards that goal. As he steps city as Treasurer, we will draw struck, consisting of the trésorier ..................................33 down, it is gratifying to note on that experience as well Institute directors and that these efforts have started as his extensive background other stakeholders, along The Canadian Open Math to bear fruit, in large part due in private business. I thank with representatives of Challenge ................................34 to Eddy’s vision and energy. David for his generosity in mathematical sciences It has been a great personal continuing his service to the societies, department chairs, IMO Report .............................36 pleasure working with Eddy CMS, and I look forward to and the associated NSERC working with him. Grant Selection committees Obituary ..................................37 this past year, and I’d like to express the Society’s thanks (GSCs). Community Calendar of events to him for the superb job he consultations took place Calendrier des événements ....39 has done. Français page 5 continued page 4 VOLUME 38 NO. 5 by Robert J. MacG. Dawson EDITORIAL St. Mary’s University A FAMOUS TIME-WASTER UNE FAMEUSE PERTE DE TEMPS Are you familiar with that famous time- Connaissez-vous la conjecture de Collatz, ce fameux problème waster, the Collatz problem? Pick a qui est garanti de vous faire gaspiller votre temps. Choisissez positive integer, any positive integer. un entier positif, n’importe lequel. S’il s’agit d’un nombre pair, Now, if it’s even, divide it by two; if it’s divisez-le par deux; s’il est impair, multipliez-le par trois et odd, multiply by three and add one. Now ajoutez-lui le nombre 1. Maintenant, recommencez le processus repeat this process, and keep on going et continuez jusqu’à ce que les suites commencent à se répéter, until the numbers start to cycle, or until ou jusqu’à ce que l’un de vos collègues vous rappelle à l’ordre your colleagues remind you that you en mentionnant que vous étiez attendu à une réunion il y a plus were meant to be in a meeting ten minutes ago. Ready? Let’s de dix minutes. Bon, vous êtes prêt? Commençons avec 7 qui start with 7. Next we get 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, donne 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 4, 2, 1, 4... 5, 16, 4, 2, 1, 4... (OK, you can stop now!) (Oui, vous pouvez arrêter maintenant!) The question is, do we always end up in the same cycle, or La question est de savoir si nous obtenons toujours la même are there other ways that one of these sequences can end? suite, ou s’il y a d’autres façons dont les suites peuvent se Nobody knows, although much research has been done on the terminer. Personne ne le sait, malgré que l’on ait beaucoup subject. It is known that if there is a sequence that doesn’t étudié la question. Tout ce que nous savons, c’est que, s’il reach the “standard ending” it must have pretty large num- existe une suite qui n’atteint pas la « fin usuelle », elle doit bers in it (no term less than ten quadrillion). Graphs of the être composée de nombres pas mal énormes (de l’ordre de pas number of steps needed to reach 1 show intriguing fractal tex- moins de dix mille milliards). Les graphiques de la quantité tures (for these, and other facts, see the Wolfram MathWorld NOTES DE LA SMC CMS NOTES “Collatz Problem” page http://mathworld.wolfram.com/ Les Notes de la SMC sont publiés par la Société The CMS Notes is published by the Canadian CollatzProblem.html.) But what’s really going on? mathématique du Canada (SMC huit fois l’an Mathematical Society (CMS) eight times a (février, mars, avril, mai, septembre, octobre, year (February, March, April, May, September, Most intriguingly, John H. Conway proved in 1972 that a slight novembre et décembre). October, November and December). generalization of this problem is formally undecidable. All RÉDACTEURS EN CHEF EDITORS-IN-CHIEF that is needed is that, instead of choosing one of two linear Robert J. MacG. Dawson; S.Swaminathan Robert Dawson, Srinivasa Swaminathan functions to create the next term based on the current number’s [email protected] [email protected] parity, we choose based on a larger set of residue classes. It’s actually not too hard to see why this ought to work. RÉDACTEURS-GÉRANT MANAGING EDITOR Graham P. Wright Graham P. Wright Most of us are familiar with the “Turing machine”, a [email protected] [email protected] hypothetical computer with an infinite set of one-bit registers that can simulate any other computer. It may be shown that RÉDACTION CONTRIBUTING EDITORS Éducation : Edward Barbeau this is equivalent to an “abacus” or “register machine” that has Education: Edward Barbeau [email protected] [email protected] a finite number of registers each capable of holding an arbitrary Réunions : Gertrud Jeewanjee Book Reviews: Peter Fillmore natural number. The instruction set of an abacus is very simple. [email protected] [email protected] Critiques littéraires: Peter Fillmore A register may be increased; or you may attempt to decrease it, Meetings: Gertrud Jeewanjee [email protected] [email protected] branching to a new instruction if it is already zero. And that’s Recherche : Vacant Research: Vacant it! A clever simulation argument due to Minsky shows that any [email protected] [email protected] abacus is equivalent to one with only two registers. Assistante à la rédaction : Editorial Assistant: Susan Latreille Susan Latreille Those two registers, in turn, may be represented by the number Note aux auteurs : indiquer la section choisie The Editors welcome articles, letters and an- pour votre article et le faire parvenir au Notes nouncements, which can be sent to the CMS of 2’s and 3’s respectively in the prime factorization of a single de la SMC à l’adresse postale ou de courriel Notes at the address below. ci-dessous. natural number. But the residue mod 6 tells us if either of these No responsibility for the views expressed by is equal to 0, and linear functions like those in the original Les Notes de la SMC, les rédacteurs et la SMC authors is assumed by the CMS Notes, the ne peuvent être tenus responsables des opin- Collatz problem can increase or decrease those powers.