FROM THE PRESIDENT-Elect’s DESK Anthony To-Ming Lau University of Alberta, Edmonton IN THIS ISSUE DANS CE NUMÉRO the mathematical sciences Australia. About 500 par- Editorial ....................................2 throughout the Pacific Rim ticipants are expected to region. PRIMA is a loose attend, and there will be Book Reviews: Knots association of mathemati- plenary speakers, special and Links ..................................4 cal organizations being sessions and prizes for out- Book Review: Meromorphic co-ordinated by Alejandro standing young mathemati- Functions and Linear Algebra ..6 Adem, Canada Research cians. The scientific com- Chair in Topology at the mittee consists of: Rafael Brief Book Reviews ..................7 This is my first column as University of British Colum- Benguria (Catholic Univer- President-Elect of the Cana- bia, and Deputy Director sity, Chile), Phil Broadbridge Education Notes .......................9 dian Mathematical Society, of PIMS. Its membership (Australian Mathematical and I would like to thank all falls into three categories: Sciences Institute, Austra- News from Departments of you and my colleagues institutes, departments of lia), Kung-Ching Chang Nouvelles du départment.......11 for warm support. I will do mathematical sciences and (Peking University, China), Call for Nominations - 2008 my best to serve the math- national math societies with Michael Cowling (University CMS Excellence in Teaching ematical community during initial support from MSRI of New South Wales, Aus- Award / Appel de mises en the next four years. I look and PIMS. I am excited that tralia), José Antonio de la candidature - Prix d’exellence en forward to working with this will be an opportunity Peña (UNAM & CONACYT, enseignement 2008 ................13 President Tom Salisbury and for the Canadian mathemat- Mexico), David Eisenbud the Executive team, our staff, ical community to interact [CHAIR] (UC Berkeley, USA), Call for Sessions – CMS Winter and all volunteers of the with colleagues of the Pacific Ivar Ekeland (University of 2008 Meeting / Appel de mathematical society. Rim. British Columbia, Canada), sessions – Réunion d’hiver 2008 Yasha Eliashberg (Stanford de la SMC ...............................15 I would like to take this Activities of PRIMA will University, USA), Masaki opportunity to introduce include: Pacific Rim Con- Kashiwara (RIMS, Kyoto Uni- Call for Proposals - 2008 David to the mathematical com- gress, summer schools for versity, Japan), Hyo Chul Borwein Distinguished Career munity a new and exciting graduate students, exchange Myung (Korea Institute of Award / Appel de mises en mathematical association, of scholars, co-advising Advanced Study, Korea) candidature - Prix David-Borwein de mathématicien émérite PRIMA. Two years ago, on and joint Ph.D. programs, Tatiana Toro (University of pour l’ensemble d’une carrière October 24-25, 2005, I was expanded collaborative re- Washington, USA). 2008 ......................................15 privileged to be invited to a search groups, PRIMA lec- meeting of representatives tures, sharing benefits of In 2007, the Winter Meeting CMS Winter 2007 Meeting of a number of leading math- expertise among institutes of the Canadian Mathemati- Réunion d’hiver 2007 .............17 ematical institutions in the and promotion of mathe- cal Society will be hosted Pacific Rim including Austra- matics in developing coun- by the University of Western Deuxième congrès Canada- lia, China, Korea, Columbia, tries. Ontario in London, Ontar- France 2008 / Second Canada- Mexico, Japan, Chile, New io, December 8-10. Also France Congress 2008 ............20 Zealand, Singapore, Hong On the PRIMA website the Second Canada-France Kong, USA, PIMS, MSRI and you Congress will take place at Call for Nominations - 2008 www.primath.org BIRS. A resolution was made will find a long list of events UQAM in Montreal, June Doctoral Prize / Appel de to establish an organiz- in 2007. At the present a 1-6, 2008. I look forward mises en candidature - Prix de ation to be known as the big focus of PRIMA activity is to meeting many of you Doctorat 2008 ........................22 Pacific Rim Mathematical preparing for the First PRIMA there. Nécrologie / Obituary .............23 Association (PRIMA) with congress, July 13-17, 2009, the mission to promote and hosted by the University of Rates and Deadlines facilitate the development of New South Wales, Sydney, Tarifs et échéances .................27 français page 21 VOLUME 39 NO. 6 OCTOBER/OCTOBRE 2007 by Robert Dawson EDITORIAL Saint Mary’s University, Halifax Some Polygonal Oddities Here’s a truly strange result, due to But was Fermat correct? Well, 28 more Fermat numbers have Martin Gardner. Compute the first now been successfully tested for primality - and Fermat struck several rows of Pascal’s Triangle modulo out each time. (Some have been tested indirectly - they are 2: known to be composite but no factor has been found.) If there are indeed no more Fermat primes, Gardner’s result continues 1 to hold up to 4294967295 (32 binary digits, all ones) - and 1 1 then stops forever. 1 0 1 1 1 1 1 Should we expect more? Well, a well-known theorem on the 1 0 0 0 1 density of primes tells us that in the neighborhood of n, the 1 1 0 0 1 1 density of primes is about 1/log(n). Thus, the probability that 1 0 1 0 1 0 1 a random number selected near 22n is prime is on the order 1 1 1 1 1 1 1 1 of 2-n. And if we select random numbers near 22n for n = 1 0 0 0 0 0 0 0 1 32,33,34,... the probability of getting even one prime is about 1 1 0 0 0 0 0 0 1 1 2-31. Now, of course we’re always picking odd numbers, so 1 0 1 0 0 0 0 0 1 0 1 our chances are better - about 2-30. 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 So, in a sense, we can say that the chance that Gardner’s 1 1 0 0 1 1 0 0 1 1 0 0 1 1 sequence ever finds another constructible number is one in a 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 billion. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (That’s enough triangle. -Ed.) Now read these off as binary numbers: 1, 3, 5, 15, 17, 51, 85, NOTES DE LA SMC CMS NOTES 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, Les Notes de la SMC sont publiés par la Société The CMS Notes is published by the Canadian ... Does this sequence look familiar? It should: these are the mathématique du Canada (SMC huit fois l’an Mathematical Society (CMS) eight times a first few odd n such that the regular n-gon is constructible with (février, mars, avril, mai, septembre, octobre, year (February, March, April, May, September, compass and straightedge! Don’t believe me; check Sloane’s novembre et décembre). October, November and December). Handbook of Integer Sequences. RÉDACTEURS EN CHEF EDITORS-IN-CHIEF It’s not obvious whether we should include 1, representing Robert J. MacG. Dawson; S.Swaminathan Robert Dawson, Srinivasa Swaminathan perhaps a circle with a single dot on the perimeter; but when [email protected] [email protected] we realize that the even-sided constructible n-gons are found by multiplying the odd numbers above by powers of 2, it is RÉDACTEURS-GÉRANT MANAGING EDITOR clear that we need 1 to anchor the mostly-nontrivial sequence Graham P. Wright Graham P. Wright 1,2,4,8,... [email protected] [email protected] A little thought will make it clear why this is true. It may be shown (see, for instance, chapter 14 of Baragar’s excellent A RÉDACTION CONTRIBUTING EDITORS Survey of Classical and Modern Geomtries) that no constructible Éducation : Edward Barbeau Education: Edward Barbeau [email protected] [email protected] number can be a root of a polynomial of odd degree which is Réunions : Gertrud Jeewanjee Book Reviews: Peter Fillmore irreducible over the rationals. From this, it follows (with a little [email protected] [email protected] work) that if the n-gon is constructible, then n has no repeated Critiques littéraires: Peter Fillmore Meetings: Gertrud Jeewanjee odd prime factor. [email protected] [email protected] Recherche : Vacant Research: Vacant Which odd prime factors p are available to us? The complex [email protected] th p-1 [email protected] p roots of 1 satisfy the cyclotomic polynomial w + ... + Assistante à la rédaction : Editorial Assistant: w2 + w + 1 = 0, of degree p-1, and this is irreducible by Susan Latreille Susan Latreille Eisenstein’s criterion. Then p-1 must be a power of 2; so p = Note aux auteurs : indiquer la section choisie The Editors welcome articles, letters and an- 2m + 1 for some m. However, if m has an odd factor, 2m + 1 pour votre article et le faire parvenir au Notes nouncements, which can be sent to the CMS is composite; so an odd prime factor of n must be of the form de la SMC à l’adresse postale ou de courriel Notes at the address below. 2r ci-dessous. 2 +1 - a Fermat number. The first few Fermat numbers are No responsibility for the views expressed by 3,5,17,257,65537,..., and Fermat conjectured based on these Les Notes de la SMC, les rédacteurs et la SMC authors is assumed by the CMS Notes, the that every such number was prime. ne peuvent être tenus responsables des opin- editors or the CMS. The style files used in ions exprimées pas les auteurs.