CMS NOTES de la SMC - Canadian Mathematical Society · activities. Although there have been ... . Bien que la ... Ouvrez vite vos agendas et

CMS

NOTESde la SMC

Volume 31 No. 6 October / octobre 1999

In this issue / Dans ce numero

Editorial . . . . . . . . . . . . . . . . . . . . . 2

Men as Trees Walking . . . . . . . . 3

Research Notes. . . . . . . . . . . . . . . 8

1999 Jeffery-Williams Lecture 9

Du bureau du directeur admin-istratif . . . . . . . . . . . . . . . . . . . . 15

Awards / Prix . . . . . . . . . . . . . . . . 15

The CORS Corner . . . . . . . . . . . 17

Meetings / ReunionsCMS Winter 1999 MeetingReunion d’hiver 1999 de laSMC . . . . . . . . . . . . . . . . . . . . . . .18

Fields Symposium. . . . . . . . . . 21

Third European Congress ofMathematics. . . . . . . . . . . . . . . 23

Minutes of the Annual GeneralMeeting . . . . . . . . . . . . . . . . . . . 23

Committees at Work . . . . . . . . . 25

Call for Nominations / Appel deCandidatures . . . . . . . . . . . . . . 27

Calendar of events / Calendrierdesevenements. . . . . . . . . . . . . 34

FROM THEEXECUTIVE

DIRECTOR’S DESK

Graham Wright

Make Math Count

When this year’s fund raising cam-paign Make Math Countwas devel-oped, it was hoped to obtain a higherlevel of support and recognition forthe Society’s research and educationalactivities. Although there have beensome disappointments, there have beena number of successes.

The support received from the hostuniversities and the three research in-stitutes has enabled us to expand theprogrammes for our semi-annual meet-ings and, consequently, the number ofparticipants has increased significantly.This very positive trend should con-tinue for our future meetings.

The 1999 Winter Meeting in Mon-treal not only features an excellent sci-entific programme but will also include

the first CMS Job Fair. Informationon this Job fair is included in this is-sue of the CMS NOTES. It is hoped tocontinue this important cooperative ef-fort between the host university, the re-search institutes, the private sector andthe Society and for the Job Fairs to takeplace at each CMS semi-annual meet-ing. I am pleased to report that thesites for future meetings are confirmedto 2004 and I am confident that eachmeeting director will continue to pro-vide an excellent scientific programme.

In the September NOTES, DarylTingley reported on the CMS AwardsBanquet that took place in June, thesupport received from corporations,governments and by the CMS mem-bership in our competition activities,and that it was nice to see mathemat-ical talent getting recognition from themedia. The increased coverage and in-terest in our various educational activ-ities is very encouraging.

I am particularly encouraged withthe interest and support for the CMSprogramme of math camps. AlthoughSummer and Winter IMO TrainingCamps have taken place for a num-ber of years, the CMS wanted to de-velop a programme of regional and na-tional math camps that would have awider scope and broader appeal. Theregional camps are mathematics en-richment camps primarily for Canadianstudents in grades 8 through 10 whilethe national camp is mainly for youngerCanadian students with at least two

(see MATH–page 14)

OCTOBER/OCTOBRE CMS NOTES

CMS NOTESNOTES DE LA SMC

The CMS Notesis published bythe Canadian Mathematical Society(CMS) eight times a year (February,March, April, May, September, Oc-tober, November, and December).

Editors-in-Chief

Peter Fillmore; S. Swaminathan

Managing Editor

Robert W. Quackenbush

Contributing Editors

Education: Edward [email protected]: Monique [email protected]:Noriko Yui;James D. [email protected]

Editorial Assistant

Caroline Baskerville

The Editors welcome articles, lettersand announcements, which shouldbe sent to theCMS Notesat:

Canadian Mathematical Society577 King EdwardP.O. Box 450, Station AOttawa, Ontario, Canada K1N 6N5Telephone: (613) 562-5702Facsimile: (613) 565-1539E-mail: [email protected]@cms.math.caWeb site: www.cms.math.ca

No responsibility for views expressed byauthors is assumed by theNotes, the ed-itors or the CMS. The style files used inthe production of this volume are a mod-ified version of the style files producedby Waterloo Maple Software,c©1994,1995.

ISSN: 1193-9273

c©Canadian Mathematical Society 1999

EDITORIAL

Peter Fillmore

In this issue of the NOTES we arepleased to bring you printed versionsof two of the plenary lectures at the St.John’s meeting: Ed Barbeau’s “Menas trees walking” and John Friedlan-der’s Jeffery-Williams lecture "Primevalues of polynomials". Both talkswere enthusiastically received and weare glad to make them available for fur-ther study. We hope to do more of thisin future and would welcome reader re-action.

The conference season is approach-ing and many of us have begun consid-ering our choices. Here is a brief se-lection. The upcoming CMS meetingin Montreal is full of interest, precededimmediately by the CRM’s 30th an-niversary celebration and followed bythe first CMS Job Fair. Beyond that wecan choose from a number of meetingsthat include special World Mathemati-cal Year 2000 features.

In January we have the JointMathematics Meetings in Washing-ton DC. A united congress of Que-bec mathematical organizations andgroups takes place at Laval in May,and in June follows the MATH 2000meeting at McMaster, organized bythe CMS and a number of otherCanadian mathematically-oriented so-cieties. Just prior to that is the two-day Symposium on the Legacy of JohnCharles Fields, at the Fields Institutein Toronto. July brings the 3rd Eu-ropean Congress of Mathematicians

(Barcelona) and ICMI-9 (Tokyo), andAugust the AMS meeting "Mathemat-ical Challenges of the 21st Century" inLos Angeles.

More information about theseevents can be found in our calendar sec-tion, as well as at the WMY2000 web-site <http://wmy200.math.jussieu.fr>.This list, admittedly of just a few high-lights, is surely an impressive testa-ment to the vitality of the global mathe-matical community as we approach thenew millenium. Why not start planningyour participation now?

Nous sommes heureux de vous présen-ter dans ce numéro de NOTES deuxdes conférences plénières de la réunionde St-John’s, celle d’Ed Barbeau et deJohn Friedlander (“Men as trees walk-ing” et ”Prime values of polynomi-als”). Elles furent toutes deux accueil-lies avec beaucoup d’enthousiasme etc’est avec plaisir que nous vous don-nons l’occasion ici de les relire et de lesapprofondir. Nous espérons renouvelercette expérience dans l’avenir et ac-cueillerions avec plaisir vos commen-taires à ce propos.

La période intensive des con-férences approche à grands pas etplusieurs se penchent sur les choix àfaire. Nous vous en présentons iciun bref aperçu. La prochaine réunionde la SMC à Montréal s’annonce fortintéressante. Elle se trouve précédéede la célébration du trentième anniver-saire du CRM et suivie du premier Car-refour emploi de la SMC. Entre cesdeux événements, il y aura un vastechoix de réunions qui porteront sur l’an2000, année internationale des mathé-matiques.

En janvier prochain se tiendront lesréunions mathématiques communes àWashington, DC. Un congrès réunis-sant les organismes et groupes math-ématiques du Québec aura lieu àl’Université Laval, en mai; il est suivi,en juin, de la rencontre MATH 2000,à McMaster, organisée par la SMC etd’autres sociétés à teneur mathéma-tique du Canada. Tout juste aupara-vant, il y a leSymposium sur l’héritage

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NOTES de la SMC OCTOBER/OCTOBRE

de John Charles Fields, au Fields Insti-tute de Toronto. En juillet, ce sera letroisième Congrès européen de math-ématiques, à Barcelone et en août, àLos Angeles, aura lieu la réunion del’AMS sous le thème “Les défis math-ématiques du 21e siècle”.

On peut trouver tous les ren-seignements sur ces activités dans lasection Calendrier ainsi qu’au siteWeb de l’an 2000, année interna-tionale des mathématiques, à l’adressehttp://wmy200.jussieu.fr. Bien que laliste ci-dessus soit fort partielle, elle té-

moigne de façon convaincante de la vi-talité de la communauté mathématiqueinternationale, au seuil du nouveau mil-lénaire. Ouvrez vite vos agendas etcommencez dès maintenant à planifiervotre participation!

Men as Trees WalkingEd Barbeau, University of Toronto

This article is based on a plenary talkgiven on May 29, 1999 at the SummerMeeting of the Canadian MathematicalSociety in St. John’s, NF.

Ed Barbeau

“Men as trees walking” – this wasthe phrase that came into my mind as Iexamined the draftPrinciples and Stan-dards of School Mathematics(1) re-cently issued by the National Coun-cil of Teachers of Mathematics. Thephrase is from the Bible (Mark 8:22-26), where a curious miracle is de-scribed. A blind man touched by Christcould at first see only “men as treeswalking” and required a second touchin order to see clearly.

One has the same feeling about thePrinciples and Standards; while it isgenerally praiseworthy, it is hard tobring the document into a clear focus. Itcontains a mixture of recommendationsabout topics, processes, teaching stylesand general philosophy, but, as a lec-turer of first-year university students, Idid not get a clear sense of what I wouldbe able to count on from the students in

front of me. It seems that there is sucha plethora of ideas put forward that per-haps a second healing touch is neededto tease out clearly the main threads.

The central issue became clear tome one evening as I watched Rob Buck-man on TV Ontario present a documen-tary on alternative medicine. Manypeople were alienated from traditionalmedicine which they found too reduc-tionist and narrowly focussed. What-ever the specifics, alternative medi-cal regimes were attractive becausethey were holistic, proceeded from abroader worldview and significantly in-volved the patient in the diagnosis andchoice of treatment. Cases that mightseem identical to a traditional doc-tor might receive quite different treat-ments for patients in different environ-ments. Surely something like this is be-hind the pressure for reform in educa-tion. Teachers and students are reactingagainst a curriculum which seems to bereduced to a list of topics and processes,against an imposed canon robbing stu-dents and teachers of their autonomy.Whatever the details might be, we wantstudents to be intimately involved inan educational process that cares aboutwho they are and what characteristicsthey bring to the mathematics class.

But the charge that traditional ed-ucation (as traditional medicine) hasconsistently lacked the human touch isfar too stereotypical; it has resulted ina number of ghosts that have hauntedrecent educational reform. Let us raisea few of them.

Ghost 1: failure, streaming,elitism. There is no doubt that school

was a brutal experience for many stu-dents in the past, but the fact remainsthat success in mathematics depends ona certain level of ability and applica-tion. To deny students the opportu-nity to fail is also to deny them theopportunity to enjoy success. This isnot a call to ignore students who arefloundering, but rather to create condi-tions that do not neglect the imperativesof learning mathematics and the possi-bility for achievement, but do providesupport in which these might be real-ized and students can move on in con-fidence. Many students now are uncer-tain about what they can do or shouldknow; even good students are deniedthe chance to demonstrate what theyare capable of. While there is muchthat can be done to make mathematicsmore generally accessible, it needs tobe recognized that the subject is oftendifficult and beyond the ability or inter-est of some segment of the population.

Ghost 2: the syllabus, list of pre-scribed topics, facts and procedures.The charge here is that having alist istoo confining and leads to an emaciatedmathematics that is reduced to a disem-bodied set of facts and routines. It isnot clear why this shouldnecessarilybe the case. In fact, the criticism seemsto be misplaced; it should be directedat matters of design. In the hands of ateacher familiar with mathematics, thesyllabus can serve as a set of markersand goals that frame what will happenin the class. An uncertain or ignorantteacher will hold to the list without anyregard to connections and deeper un-derstanding.

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Ghost 3: drill, rote learning andmemorization. Memorization was animportant part of traditional education,and many cultures put a great deal ofemphasis on what children shouldre-member. Children seem to have goodmemories, and it seems foolish notto take advantage of this. But theyneed to be taught to evaluate what isworth memorizing, how mastery can bereached and how they can exploit thecoherence of mathematics to leveragetheir knowledge of a few facts into flu-ency over a larger domain. The mostable children can do this naturally; oth-ers need to have the issue explicitly ad-dressed, so there is an underlying eq-uity issue involved here for studentswithout natural talent.

Ghost 4: arithmetic.This is a wordthat seems to have fallen upon hardtimes in the curriculum stakes. Butit is through arithmetic that most ordi-nary citizens engage the connection be-tween mathematics and the world, andlack of numeracy can present a severehandicap. Arithmetic has come to sym-bolize mindless memorization and ma-nipulation. We need to detach it fromthis calumny and exalt it to the level ofmathematical richness that it deserves.

Ghost 5: paper-and-pencil algo-rithms. This also is tarred with thebrush of drudgery, and the advent ofmodern technology has provided de-tractors of traditional calculation witha pretext for summarily discarding it.The issue is really how traditional arith-metic should be handled in a moderncurriculum. Perhaps we need to seethem more as an additional means ofaccustoming students to working withfigures or as examples of algorithms.Long division seems to be particularlysuitable for indicating how one canmove from the idea of tallying a con-tinued subtraction to a mechanical al-gorithm which is fast and accurate; onecan point out that it was a human inven-tion and replaced a method that was de-cidedly inferior (the “scratch method”).What has changed is that the impor-tance of paper-and-pencil methods as

a practical technique is reduced and wenow have alternatives for reaching chil-dren encountering difficulties.

Ghost 6: word problems.Tradi-tional word problems are criticized asbeing artificial, but one can argue thatthis is precisely the point of most wordproblems. They provide a toy situationin which certain points about interpre-tation, and formulation can be made.The question again is not whether weshould retain word problems, but howappropriate they are to the situation andhow we plan to move beyond them.

Ghost 7: authoritarianism. Thisghost has two manifestations, depend-ing on whether you are referring tothe teacher, who, we are told, shouldbe the “guide on the side” rather thanthe “sage on the stage”, or to the sub-ject itself in which pupils are oppressedby the tyranny of “one right answer”.Without disputing the advantages ofthe more open and friendly classroomsthat modern students enjoy, it remainsthe case that a teacher’s effectivenessdepends on what she knows and thatsometimes students need to submergetheir egos and pay attention to whatshe has to say. In the same way, itseems mischievous to deny the powerof mathematical certainty, especiallygiven the diversity of the ways in whichone might think about concepts andapproach problems. (One feels thatthe work of Godel and Lakatos, how-ever lauded by serious mathematicians,have had a particularly pernicious ef-fect on some mathematics educatorsseduced by the vamps of relativism.)There are many ways to encourage theindividuality of pupils without permit-ting them to believe that black is white.

All of these Ghosts are the traces ofessential components of mathematicaleducation in the past which had betterbe part of the future as well. Childrenare going to either succeed or fail atany worthwhile task, and the questionis how humanely the failure is handledand whether pupils are held back or ad-vanced for frivolous reasons. Mathe-matics is an hierarchical subject and we

need to spell out what students need tomaster at each stage; the issue of thesyllabus is one of design and focus. Wecannot deny the need for practice; thequestion is whether the student has thestrategies and perspective to learn andmemorize efficiently and effectively.Arithmetic and the standard algorithmsare as important as ever; what we needis to be sure that they are put into theproper context and conceptual frame-work.

These Ghosts are accompanied by anumber of Sirens that drive a lot of ed-ucational reform. Like the ghosts, thesirens also speak to important aspectsof the mathematics education. Here aresome of them.

Siren 1: problem solving and in-vestigation. This siren calls us awayfrom the ghost of drill, of dry and unil-luminating exercises. There is nothingwrong in wanting our children to solveproblems and explore mathematical sit-uations, but we can run into serious dis-tortions if we do not take the trouble toground them mathematically and psy-chologically.

Any problem we present to chil-dren should be carefully analyzed forits mathematical content and appropri-ateness to the maturity of the pupils.Strategic decisions need to be madeas to what mathematics has to be pre-sented beforehand as background andwhat can be brought out in the analysisof the problem. Let me give an exam-ple that I have used with students andprospective teachers.

Let ABCD be a unit square and letE and F be the respective midpointsof sides BC and CD. The three linesegments AE, AF and BF partition thesquare into five regions. It is requiredto determine the area of each region.

There is actually quite a lot in this.At what point should this example beintroduced, before or after the pupilssee the area formula (half-base-times-height) for triangles? This will governhow they might approach the problem.If the students are to try a structuralrather than a formulaic approach, they

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might need at least implicit understand-ing of isometries and might need to un-derstand that areas of nonoverlappingsets add, that areas are invariant underrigid transformations and that they varyas the square of the factor of a dilata-tion. Should some of this be discussedahead of time, or can it emerge as theproblem is covered? If the students ex-ploit the similarity of the two subtrian-gles of ABE, they may need to knowthat AE is perpendicular to BF; whattools can they be expected to deploy?Will students who assign letters to thefive regions have the necessary alge-braic understanding to proceed, or isthis a nice vehicle to introduce them tothis approach? Finally, will the pupilsbe able to negotiate the fractions? Arethe fractions appropriately expressed invulgar or in decimal form; why? Whenwe take all of this into consideration,this example could take quite a bit oftime to do properly - has this been an-ticipated by the teacher?

The problem-solving approach tothe curriculum has a great deal to besaid for it, but it does require imagi-nation by the teacher to envisage whatmight happen and, importantly, whatcan possibly go wrong. It is risky,and we should be sure that teachers areequipped to accept the risks.

Siren 2: relevance; real-life.Math-ematics was seen to be alienating be-cause of the artificiality of what wewere asking students to do. We shouldmake reference to the daily lives of thepupils and concern ourselves with whatthey will need to succeed in their latercareers. But what passes for relevanceoften raises the question “for whom?”.It seems that children often have thephony relevance of a tedious arithmeticproblem dressed up with clowns andcookies or are brought into the worldof adult concerns in the name of rele-vance. Play is a part of the world ofa child, and the unadulterated world ofmathematics gives lots of opportunityfor this. I have never been aware thatsome nice number, geometric, topo-logical or combinatorial novelties have

been beyond the pale for the childrenI have had the pleasure of presentingmathematics to.

Siren 3: patterns.No mathemati-cian can gainsay the important of asense of structure in doing mathemat-ics, and the ability to recognize patternis an important part of this. What hasbeen forgotten in a lot of modern ed-ucational reform is that the power ofpatterning is seen in its use in analyz-ing and generalizing mathematical sit-uations. Much of what passes for pat-terning is a kind of teacher “guess-my-rule” game and rather ad hoc examples.There is no excuse for this. It is hard toprogress through the curriculum with-out finding natural opportunities to ex-ploit patterns, and teachers need to bealert to this and make them explicit totheir pupils at the right time. One as-pect that seems to be neglected is givingthe students the mathematical voice todescribe and analyze patterns.

Siren 4: data analysis.We do nothave to look very far to realize howmuch of our daily lives is governed by aflood of data of one sort or another. Sothere is certainly a duty to help pupilsbecome number-wise and to interpretwhat they read astutely. But the dan-ger is that we do not just cover a lotof canned techniques and introduce jar-gon, which may in some cases stand inthe way of good discussion and analy-sis.

Siren 5: technology.First, let meagree that through modern technologywe are undergoing changes in our livesthat are at least as profound as thoseinduced by the invention of the print-ing press and the Industrial Revolution.But any revolution, no matter how per-vasive, is not completely disjoint fromthe past. Many of the issues raisedby technology are old ones, and thecharge of misuse and mindlessness canbe and has been leveled against Arabicnumeration, algebraic symbolism, log-arithms, slide rules, mechanical addingmachines and numerical algorithms ofall types whether executed on paper oron a bench with pebbles. Certainly,

the modern computer has greatly ex-panded both the range of the mathe-matics wecando as well as the math-ematics wewant to do; our curricu-lum should reflect this. But we do notneed idolatry. Technology istherepartof our environment, as are books andpens, and a general purpose of edu-cation is to produce students who canunderstand and work within their envi-ronment. There are core mathematicalissues that are independent of technol-ogy, but can be greatly informed by ouruse of technology - it is in this spirit thatwe should embrace the use of calcula-tors and computers in our classrooms.

All of the sirens speak to importantaspects of the modern mathematics cur-riculum, but they all involve subtle is-sues and bring the risk of being trivial-ized. The big question is whether wewill have teachers in front of our chil-dren who will handle the issues in a sen-sitive, moderate and intelligent way.

In designing a curriculum, we needto keep both the Ghosts and Sirens inmind, for each of them not only speak toimportant goals but also carry a warn-ing of distortion and counterproductiv-ity. There are a number of factors thatwe need to be cognizant of.

1. For a body of mathematics, Ibelieve that there are three stages thata learner must pass through,initiation,formalizationandconsolidation. In thestage of initiation, the learner encoun-ters the ideas in a somewhat haphaz-ard way, feeling her way about, explor-ing; there has to be some motivation,some reason that the person is inter-ested in the material at hand. At theformalization stage, the ideas are drawntogether and organized; at this point,the pupil should learn proper concepts,processes and conventions. There mayindeed be a lot of artificiality; the pay-off should be a growth in the knowl-edge and mathematical power of thelearner. In the consolidation stage, thelearner reflects upon what has gone be-fore, contextualizing it, detecting re-lationships and connecting it to othermaterial. It may be only here that the

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learner truly appreciates the reasons forwhat she has been taught before. Tra-ditional education has emphasized thesecond of these stages while modernpractice seems to focus on the first andthird without the buttress of the secondstage. A good curriculum allows for allthree stages to occur. The first signifiesto the pupil that the subject mattercouldbelong to them, the second provides thetools toallow it to belong to them andthe third ensures that itdoesbelong tothem.

2. Have a strong focus and a slen-der core for each course in the curricu-lum. Make sure that context is estab-lished, the purpose of the material be-comes clear, there is enough depth tosupport its assimilation and allow stu-dents develop the necessary skills andunderstanding to proceed.

3. Have one or two attainable goalsfor each year; plan to achieve themandmove on. This means that the sortof sterile spiralling that now occurs inschooling should cease, but does notpreclude returning to previous materialto inform and consolidate it.

4. Have enforceable entry re-quirements for secondary and tertiarycourses. No teacher should be asked todeal with a student who does not havereasonable prerequisites for the mate-rial to be studied. There is an impor-tant pedagogical purpose in requiringstudents to review and organize in theirminds the work of several months or ayear; it is this process that makes pur-poseful curriculum progress possible.

5. Change pace. The diversityof the mathematical enterprise shouldbe reflected in our curriculum, whetherwith respect to subject matter, level ofdiscourse or application. There aremany ways in which people can thinkabout or do mathematics. Therefore,we need space to help students findtheir mathematical voices and texturetheir ways of thinking about and doingmathematics.

6. Orchestrate the material. In-crease the level of complexity judi-ciously, make sure that foundational

material is covered and the student ispsychologically prepared for what is tofollow. We need to analyze in muchmore detail what students are asked todo; this is where members of the Uni-versity community can be particularlyhelpful. Too often problems are thrownat pupils with little appreciation of whatneeds to be in place to embark uponthem. A sound curriculum needs bothexercises and problems of many differ-ent intensities.

This is one area in which thePrin-ciples and Standardsseems to be par-ticularly weak. There are a number ofexamples thrown in, that upon closeranalysis seem to involve a great dealthat the proponent might not be awareof. On page 92 is a set of instruc-tions for constructing a golden rectan-gle. This seems as though it might bepretty heavy sledding for a typical stu-dent, and any teacher who embarks onthis without careful thought is wadinginto a treacherous swamp. Why shouldpupils be interested in a golden sec-tion, or in ways of constructing sucha thing? Will most pupils muster thenecessary level of concentration to ne-gotiate the ten-stage unmotivated setof instructions and understand why itworks? This example is meant to illus-trate the connectedness of mathemat-ics, but it seems to me to require such alevel of maturity and mastery of somebasic algebra that it could easily spinout of control in the hands of any butthe most adept teacher.

7. Meet different needs. What-ever reason we can give for teachinganything at all applies to mathematics.Some mathematics is taught so that stu-dents will be able to accept the priv-ileges and responsibilities of citizen-ship, some to situate them in the richculture that they are entitled to inherit,some to provide recreation and addi-tional options towards a full and re-warding life, and some for professionalpreparation. These needs can some-times be met by the same piece of math-ematics, but all should inform the cur-riculum that we set.

8. Foster sound practice and mentalattitudes. How well students performin mathematics seems to depend on theworldview that they bring. In design-ing a curriculum, we should encouragean attitude of mind that involves the fol-lowing:

(a) Awareness of structure; apprecia-tion and exploitation of symmetry;(b) Flow of ideas, analysis and reason-ing;(c) Corroborative quality of mathemat-ics; inner consistency;(d) Shifting of perspective;(e) Checking and monitoring one’swork;(f) Organizing and polishing one’swork;(g) Mathematical register; expressingideas appropriately; the place of heuris-tic and formalism;(h) Sense of context;(i) Attainment of power through under-standing, reasoning and technical facil-ity;(j) Visualization; appropriate imaging;(k) Appreciation of symbolic represen-tation: numeration, algebra, diagrams;(l) Making distinctions: classifica-tion, equivalence, isomorphism, con-gruence;(m) Grasping the interplay betweenconcrete and abstract; progression tohigher order structures that in turn be-come “concrete”.

9. Finally, we come to the subjectmatter itself. The centrality of arith-metic and algebra must be affirmed. Nostudent can succeed at the secondarylevel without a good grasp of arith-metic, nor succeed at the tertiary levelwithout a good algebraic grounding.But these are not all. Geometry andcombinatorics are also indispensible.While students should see how math-ematics can be applied in different ar-eas, it is not clear to me that this shouldnecessarily occur in the mathematicsclass. Science, shop, civics, geographyand music are areas in which impor-tant mathematical concepts can be con-veyed in an appropriate and powerfulsetting. And we should not forget how

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many important mathematical conceptsand processes underly some recreationsand puzzles. In fact, if the elemen-tary and secondary curricula are well-designed, one could devote the middleschool years largely to recreational andcultural issues as a way of consolidat-ing what should have been learned atthe elementary level and preparing theground for a more sophisticated highschool program in which symbolism,algorithm, analysis and reasoning haveimportant roles.

I would like to close with twocourses that might be given at the lev-els of grades 9 and 10. The first is de-signed to give a systematic introduc-tion to algebra and the second to ge-ometry. It is assumed that studentshave gone through an initiation stagein both areas, that they have some fa-miliarity with formulae and the use ofletters to represent numerical quanti-ties, and that they have had the oppor-tunity to play around with geometricobjects either through tactile or com-puter models. While technology isnot explicitly mentioned in either syl-labus, it is understood that it can playa large and appropriate role. Thesesyllabi would be accompanied by re-sources for the teacher that lay out al-ternative approaches as well as use-ful exercises, problems and investiga-tions, and that make explicit the peda-gogical and mathematical goals that arebrought to light through the material.

Course 1: Linear and quadraticfunctions

1. The linear equationax = b.Problems that lead to an equation of thisform.

2. The equation of the straight line;slope; various forms.

3. The solution of a system of twolinear equations in two unknowns us-ing numerical, algebraic and graphicaltechniques. Consistency of two linearequations.

4. Factoring difference of squares.

5. The quadratic: factoring overreals and integers, completion of thesquare, quadratic formula, discrim-inant and its significance; complexnumbers (real quadratic having com-plex conjugate roots); relation of sumand product of roots to coefficients; firstand second order differences; the fac-tor and remainder theorem (for quadrat-ics).

6. Graphing of quadratic functions;comparing the graphs of the functionsf(x), f(x− c), f(ax), f(x) + c; sim-ilarity of all graphs of quadratic func-tions (role of completion of square)

This is the entire prescription fora full-year course. In some classes,it may be necessary for the teacher tocover only this material and nothingelse, spending the time and introduc-ing whatever additional materials arenecessary to ensure that students be-come fluent in the essential compo-nents. Normally, one would hope thatthe teacher would extend the materialin some way, either through the intro-duction of applications, extended in-vestigations and additional topics. Itmay be that the more able students inthe class are given modules to work onindependently possibly with the sup-port of additional pamphlets or com-puter software. I do not see much pointin introducing algebra tiles to the classat large - they amount to an additionalcode interpolated between the studentand the standard code - but they maybe useful for individual students whohave trouble getting an initial grasp ofalgebra.

Other topics that might be intro-duced are: motion of projectiles, his-tory of solving equations, linear andquadratic diophantine equations, Pell’sequation, quadratic residues, first andsecond order recursions (characteristicequation), analysis of the dynamicalsystemx → λx(1 − x), polynomialsof degree exceeding 2; complex num-bers (conjugates, modulus, geometricinterpretation of sum, product and in-verse, roots of unity up to the fourth);calculus of finite differences applied to

polynomials interpolation and extrapo-lation; linear inequalities.

Course 2: Geometry of trianglesand circles.

1. Triangles: congruence theo-rems; ambiguous case; pythagoreantheorem; similarity.

2. Trigonometry: six standard ra-tios and basic identities (cos2 + sin2 =1; sec2 = 1 + tan2); sine, cosine andtangents of angle sums and differences;double angle formulae; conversion for-mulae between sums and products; lawof sines; law of cosines

3. Circles: subtended angles;cyclic quadrilaterals; tangent theorems

4. Coordinate geometry: circles;area of triangles; family of lines pass-ing through pair of intersecting lines;circles passing through intersection ofa pair of circles; radical axis; simpleloci problems

Again, the teacher would have thediscretion of reinforcing these core top-ics for students encountering difficulty,amplifying the material through appli-cations and investigations, or movingto additional topics. These could in-clude the following: complex numbers(use in deriving geometric and trigono-metric results); vector geometry; stat-ics; loci (conjectures, verification andproof); solid geometry; advanced Eu-clidean geometry; transformations andtheir uses (isometries and similarities;composition of isometries; isometrydetermined by three points; isometriesgenerated by reflections)

The modern school, even at thesecondary level, has to serve studentsall across the spectrum of intelligence,ability, motivation and interest, oftenin the same class. The only way thatthe system can cope with this situa-tion is that students are trained early tobecome autonomous is choosing theirgoals and more self-propelled in ac-cessing the resources needed to learnwhat they need to know and in seekingvalidation for their knowledge. Anysystem of education predicated on the

7


teacher being the sole source of knowl-edge and evaluation for the student isbound to fail; the more able and mo-tivated will fall far short of their po-tential and others will not be able to re-ally engage the material. Students mustfirst reflect within themselves what theyvalue and how far they are willing and

able to go in achieving their goals. Asuccessful school regime depends oncertain beliefs and characteristics ofstudents as well as the knowledge andexperience of teachers; unless we rec-ognize the need to start with this, thenour principles and standards will be-come an exercise of merely trying to

stave off the most trenchant critics fromall sides and leave us seeing “men astrees walking”.

Reference1. Principles and Standards for

School Mathematics: Discussion Draft(October, 1998) National Council ofTeachers of Mathematics

RESEARCH NOTESNoriko Yui and James Lewis

Royal Society of CanadaNew Fellows

Four mathematicians have beenelected to the Academy of Science ofthe Royal Society of Canada.

Michel Fortin - Departement demathematiques et de statistique, Uni-versite Laval

Michel Fortin jouit d’unereputation internationale qui ledonne pour l’un des pluseminentsspecialistes de l’analyse numeriquede sa generation et l’un des plusgrands mathematicians appliques quele Quebec ait produits. Il est chef defile d’une vast communaute quebecoisede l’analyse numerique. Ses travauxse demarquent par cette faculte qu’ilpossede de jeter des ponts entre latheorie et le pratique ainsi qu’entre sci-entifiques de disciplines differents.Fortin a apporte une contributionpregnantea la theorie deselementsfinis mixtes et hybrides ainsi qu’audeveloppment de methodes lagrang-iennes augmentees. Ce travail estresume dans deux textes influents dontil fut le cosignitaire,Mixed and HybridFinite Element Methodset AugmentedLagrangian Methods.

Michael C. Mackey - Centre for Non-linear Dynamics in Physiology andMedicine, McGill University

Michael C. Mackey is a distin-guished applied mathematician whohas made significant contributions todifferential-delay equations and to aspectrum of mathematical topics inphysiology and physics. He has madeseminal contributions to biomathemat-ics and biophysics, especially in the ar-eas of membrane ion transport, dynam-ical diseases, cell replication and thedynamics of neural systems. Mackeyproposed the “Mackey-Glass” delaydifferential equation for deterministicchaos and introduced the concept of"dynamical diseases". His widely citedbook (with L. Glass),From Clocks toChaos: the Rhythms of Life, showedthe biological community how nonlin-ear dynamics and chaos theory can beused to help understand physiologicalsystems.

Ian Fraser Putnam - Department ofMathematics and Statistics, Universityof Victoria

Ian Fraser Putnam has made out-standing contributions to the theory ofdynamical systems. He has shown thatthe orbits of such systems can be com-pletely determined by algebraic meth-ods - methods of which he was in-strumental in the development. Put-nam’s analysis of the structure of theC*-algebra associated with a minimalhomeomorphism of the Cantor set led

to an overarching scheme for the clas-sification of all amenable C*-algebras- the so-called Elliott program. Put-nam’s application of the C*-algebraclassification to determine the orbitstructure of the Cantor set dynamicalsystem, in terms of the quite simple C*-algebra invariant, was a striking inno-vation. It has changed the face of dy-namical systems theory.

Maciej Zworski - Department ofMathematics, University of Toronto

Maciej Zworski is the world’s lead-ing mathematician under the age offorty in the difficult and fundamentalarea of mathematics connecting par-tial differential equations, mathemati-cal physics and applied mathematics.His research provides works of refer-ence and starting points for other math-ematicians. The amazing progress inthe understanding of resonances in thelast ten years is due largely to Zworsky.He settled a famous problem formu-lated by the physicist Regge thirty yearsago. Zworski also found the precise lo-cation of the shadow boundary in thediffraction of linear oscillatory wavesby a convex boundary, thus provinga long standing conjecture of Kellerand Rubinow. This refines work dat-ing back to the last century and was notknown even on the level of formal ex-pansions.

8


1999 JEFFERY-WILLIAMS LECTURE

Prime Value of PolynomialsJohn B. Friedlander, University of Toronto

This is a transcription of the author’s Jeffery-Williams Lecturegiven at the CMS Summer Meeting in St. John’s, Newfound-land in June of this year.

John Friedlander

ABSTRACT: Consider an irreducible polynomial with inte-ger coefficients. It is an old and interesting problem to try toprove whether or not the polynomial is a prime number forinfinitely many integer values of the variable(s). One expectsthat, although there are counter-examples, this will normallybe the case. Nevertheless, despite the long history of the ques-tion, relatively few cases have been settled in the affirmative.In this talk we survey some of that history and then go on todescribe a number of the ideas behind recent joint work withHenryk Iwaniec leading to further progress.

This lecture is concerned with two of the oldest objectsof study in number theory, polynomials and prime numbers.Let f ∈ Z[X1, X2, . . . , Xr] be a polynomial inr variablesand having integer coefficients. We shall consider the basicquestion

Problem 1 Does there exist an infinity ofr–tuplesn =(n1, . . . , nr) ∈ N

r of positive integers such thatf(n)is a prime number?

In the few cases where we are able to show that there arewe shall consider also the more refined question

Problem 2 Can we find an appropriate asymptotic formulafor the counting function

∑0<f(n)≤x,f(n) prime

1 ?

Polynomials in One VariableWe begin with what should be the most basic and inter-

esting case, polynomials in a single variable. To start at thevery beginning we take the polynomialf(n) = n; in otherwords we try to count the set of all primes. Here the firstproblem is answered already in Euclid: There exist infinitelymany prime numbers.

The proof is so short that we give it here. Suppose to thecontrary there are only finitely many, sayp1, . . . , pN and con-sider the integerp1 . . . pN +1 formed by taking their productand then adding one. This integer is not divisible by any ofthepj since that division leaves a remainder of one. On theother hand, by the fundamental theorem of arithmetic, everyinteger exceeding one is a non-empty product of primes. Thiscontradicts the assumption thatp1, . . . , pN exhaust the list ofprimes and the proof is complete.

Having answered the first problem in this case we now turnto the second problem. It took about two millennia to makethis leap. In the late eighteenth century it was conjectured byLegendre that

π(x) ∼ x

log x,

a statement that is today known as the prime number theorem.Hereπ(x) denotes the number of primes up tox,

π(x) .=∑p≤x

1 ,

log is the natural logarithm, and the asymptotic symbolf(x) ∼ g(x) meanslimx→∞

f(x)g(x) = 1.

Independently but perhaps a little later Gauss also con-jectured the prime number theorem, this time in the formπ(x) ∼ lix wherelix is the ‘logarithmic integral’ which wewrite as

lix .=∫ x

2

dt

log t.

It is easy to see from repeated integration by parts thatlix ∼ x

log x so that the two forms of the prime number theo-rem are equivalent. However it turns out that the logarithmicintegral is a much closer approximation toπ(x).

It took about another century for the proof of the primenumber theorem to materialize. The first important steps inthis direction were taken by Chebyshev who introduced somebeautiful elementary ideas and proved amongst other things

9


thatπ(x) had the expected order of magnitude, in other wordsfor certain positive constantsc1, c2 and allx ≥ 2 we have

c1x

log x< π(x) < c2

x

log x.

We write this as xlog x π(x) x

log x .The next step was taken by Riemann in an important paper

of 1860. Euler had already recognized the relevance for primenumber theory of what is today called the Riemann zeta func-tion and had deduced some of its basic properties but it wasRiemann who took the step of considering it as a function ofa complex variable (which is truly crucial) and who mappedout a strategy whereby the properties of this function in thecomplex plane would lead to the proof of the prime numbertheorem (PNT). This programme was not so easy to completehowever and it was only in 1896 and by adding ingeniousideas of their own that Hadamand and de la Vallee-Poussinsucceeded in independently providing complete proofs of thePNT.

Before leaving the prime number theorem we state, partlybecause we shall need the relevant definitions in any case,two other equivalent forms of PNT. We mention in passingthat when an analytic number theorist says that two such re-sults are ‘equivalent’ it simply means that it is much easier toprove that they imply each other than it is to prove either oneof them. The definition of ‘much’ is left to the reader. In thisvague sense the PNT is equivalent to the statement that

∑n≤x

µ(n) = o(x)

whereµ is the Mobius function which is supported on the‘squarefree’ positive integers (those without repeated primefactors) and for these, it satisfiesµ(n) = (−1)ν(n), ν(n) be-ing the number of distinct prime factors ofn. The statementthatf(x) = o(g(x)) means thatlimx→∞ f(x)/g(x) = 0.

The other equivalent form of PNT we wish to mentiondeals with the von Mangoldt functionΛ(n) defined to belog pfor any prime powern = pj , j ≥ 1 and zero elsewhere. Then

ψ(x) .=∑n≤x

Λ(n) ∼ x .

It is this form of PNT which occurs most naturally in thecontext of zeta-function theory.

Linear PolynomialsAfter the polynomialf(n) = n the next natural target is

the general linear polynomialf(n) = an+bwhere of coursewe takea > 0. These are just the arithmetic progressions.In case the greatest common divisor(a, b) of a andb exceedsone then there can be only one prime in the progression sinceif (a, b) is divisible by some prime then so are all integersan + b and no other prime value can occur. If on the otherhand(a, b) = 1 so that the integers are ‘relatively prime’ then

there is no obvious reason why there might not be infinitelymany primes. And in fact there are. In 1837 in work that has astrong claim to being the birthplace of analytic number theory(perhaps also of representation theory) Dirichlet proved thatprovided(a, b) = 1 there are infinitely many primesan+b. Inany case the result qualifies as this author’s favourite theoremin mathematics.

Problem 1 having been solved for linear polynomials foralready sixty years, it took Vallee-Poussin not very long tocombine his ideas for PNT with those of Dirichlet to set-tle problem 2 in the general case of arithmetic progressions.Specifically he showed for(a, b) = 1,

∑n,an+b≤x

Λ(an+ b) ∼ 1ϕ(a)

x

a.

Hereϕ(a) denotes Euler’s function, the number of reducedresidue classes moduloa, i.e. the number ofb, 1 ≤ b ≤ awith (a, b) = 1. Thus each of the eligible arithmetic progres-sions of modulusa receives asymptotically the same numberof primes.

Although problems 1 and 2 have thus been settled thequestion of the size of the error term in the asymptotic for-mula is still rather poorly understood. This remains a cen-tral problem in the subject and includes within it one of themost famous conjectures in mathematics, the (generalized)Riemann Hypothesis.

Higher DegreeWe turn next to single variable polynomials of higher de-

gree. Here it is easy to see that in order to take an infinitelymany prime values such a polynomial must be irreducible(over the rationals) and primitive (i.e. not having all coeffi-cients divisible by the same prime). One might think that,as in the linear case, these conditions are also sufficient andthat any irreducible polynomial with relatively prime coef-ficients should represent infinitely many primes. That thesituation is not quite so simple is exemplified by the polyno-mial f(n) = n2 + n+ 2 which at first glance might appear areasonable candidate but when re-written as

f(n) = n(n+ 1) + 2

is visibly always an even integer. Nevertheless we expect thisto be exceptional and that the result holds for most irreduciblepolynomials. However it is not known to hold for any singlepolynomial (of degree> 1). It is not even known that thereexists such a polynomial.

An impressive example is given by the ‘Euler-Rabinowitz’polynomialn2 +n+41 which is prime for each of the valuesn = 0, 1, 2, . . . , 39. Despite getting off to such a good start inlife it is not known to be prime infinitely often. There are in-teresting connections between the prime values ofn2 +n+aand the arithmetic of the quadratic number fieldQ(

√1 − 4a)

but that is another story.

10


The simplest looking examplen2 + 1 constitutes one offour famous problems mentioned in a lecture by Landau asproblems for the twentieth century. The others were the Gold-bach conjecture (that every even integer≥ 4 is the sum of twoprimes), the twin prime problem (that there are infinitely manypairs of primes differing by two), and finally the problem ofshowing that between every pair of consecutive squares thereis a prime. Despite Landau in posing four such problemshaving been apparently less demanding than had Hilbert inhis famous lecture about the same time, this has turned outafter all not to be the case. One hundred years later none ofLandau’s four problems have been solved.

Conjectures however have been easier to come by, in thecase of problem 2 as well as problem 1. Thus, for the poly-nomialn2 + 1 Hardy and Littlewood conjectured that

∑n,n2+1≤x

Λ(n2 + 1) ∼ x12

∏p>2

(1 − χ(p)

p− 1

)

whereχ = χ4 is the Dirichlet character of modulus fourwhich is supported on the odd positive integers, defined to be+1 at primes of the form4m+1, to be−1 at primes4m− 1,and extended to all odd integers by multiplicativity.

This conjecture is just a special case of more generalconjectures due to Hardy-Littlewood to Schinzel, and toBateman-Horn which deal with more general polynomials andalso to simultaneous primality of more than one polynomial.The twin prime problem which deals with the simultaneousprimality of n andn+ 2 is an example of the latter.

Restricting ourselves to the case of a single polynomialwe have the following. Letf ∈ Z[X] be irreducible withpositive leading coefficient. Define the ‘volume’

V (x) = Vf (x) =∑

n,f(n)≤x

1 .

Then

Conjecture:∑

n,f(n)≤x

Λ(f(n)) = cfV (x) + o(V (x))

where

cf =∏p

(1 − ν(p)

p

) (1 − 1

p

)−1

with ν(p) the number of roots off(n) ≡ 0 (modulo p). Itis not even so obvious that the above infinite product

∏p

is

convergent but in fact it turns out that it does.. This followsfrom the ‘prime ideal theorem’ which is the generalization ofthe prime number theorem to algebraic number fields. Nev-ertheless, it is possible for the constantcf to vanish in caseν(p) = p for some primep. Thus recall our earlier example

f(n) = n2 + n+ 2 = n(n+ 1) + 2 which is always even sothatν(2) = 2.

More than One VariableHaving seen limited success dealing with polynomials in

one variable it might seem foolhardy to further complicatematters by adding more variables. Surprisingly the problemgets easier. Eventually, it is no longer a problem about primes.Thus for example consider the polynomial

m2 + n2 + r2 + s2 .

By a famous theorem of Lagrange we obtain every positiveinteger, and so of course every prime.

More generally we have, as first conjectured by Waring,a similar result for higher powers, specifically

Theorem (Hilbert). For eachk ∈ N, there existsg(k) ∈ N

such that every positive integer is the sum ofg(k) k–th pow-ers.

In fact with enough variables we can even construct . . .

Theorem (Matijasevic, 1977). There is a functionf ∈Z[X1, X2, . . . , X10] whose positive range is just precisely theset of primes.

Actually this is a refinement of his breakthrough result afew years earlier. The following example of such a polyno-mial which I learned about from Ribenboim’s book of primenumber records is due to Jones, Sato, Wada and Wiens in1976. It is a convenient example for English speaking peo-ple, containing 26 variablesa, . . . , z.(k+2)

1 − [wz + h + j − q]2 − [(gk + 2g + k + 1)(h + j) + h − z]2

− [2n + p + q + z − e]2 − [16(k + 1)3(k + 2)(n + 1)2 + 1 − f2]2

− [e3(e + 2)(a + 1)2 + 1 − o2]2 − [(a2 − 1)y2 + 1 − x2]2

− [16r2y4(a2 − 1) + 1 − u2]2 − [((a + u2(u2 − a))2 − 1)(n + 4dy)2

+ 1 − (x + cu)2]2 − [n + l + v − y]2

− [(a2 − 1)l2 + 1 − m2]2 − [ai + k + 1 − l − i]2

− [p + l(a − n − 1) + b(2an + 2a − n2 − 2n − 2) − m]2

− [q + y(a − p − 1) + s(2ap + 2a − p2 − 2p − 2) − x]2

−[z + pl(a − p) + t(2ap − p2 − 1) − pm]2

.

In general so far we can deal with a number of cases forall of which we have

Vf (x) ≥ x

(log x)B

for someB and all largex. For the simplest looking polyno-mials this usually means we require the number of variablesto be at least as large as the degree. (This is not quite alwaysthe case since for example if the polynomial is linear in one ofthe variables we get a positive proportion of integers simplyby fixing appropriately each of the other variables.)

Two VariablesIf we accept the premise that more variables makes things

easier, that the fewer the variables the more basic and inter-esting the problem, but that one variable seems too difficult, it

11


becomes natural to try next to consider two variables. By theremarks in the previous paragraph we might expect to be ableto deal with linear and quadratic polynomials but not beyondthis.

The case of linear polynomials might well be an exercisein books in elementary number theory. The general polyno-mial of this type looks likeαr+βs+γ whereα, β, γ are fixedintegers. One of the first basic results of elementary numbertheory is the theorem that asr, s run throughZ, αr+βs runsthrough all multiples of the greatest common divisors(α, β).Henceαr+βs+γ runs through all integers in the arithmeticprogressionγ modulo(α, β) and by Dirichlet’s theorem wemerely need to know whetherγ has a non-trivial factor incommon with(α, β), equivalently whetherα, β, γ are all di-visible by the same prime.

The case of quadratic polynomials in two variables is arich subject with a distinguished history. Fermat consideredthe basic case ofr2 + s2. It is easy to see that no prime≡ 3 (mod 4) is the sum of two squares since in fact even nointeger≡ 3 (mod 4) can be. Fermat proved the consider-ably deeper fact that every prime≡ 1 (mod 4) is the sum oftwo squares. Following important subsequent work by Euler,Lagrange and Gauss, the issue was settled for all primitive,irreducible binary quadratic ‘forms’ (i.e. homogeneous poly-nomials)

αr2 + βrs+ γs2 ,

by Dirichlet. For the more general non-homogeneousquadratic polynomialsf the result waited much longer and isdue to Iwaniec (in his undergraduate days).

It is easy to see that for these we need thatf representarbitrarily large odd integers and that we require∂f

∂r , ∂f∂s to be

linearly independent. These two conditions turn out to be suf-ficient. The density of primes represented turns out to dependon a rather technical condition which we do not state. Forone class of polynomials, which includes the forms, the num-ber of primes≤ x has order of magnitudex/ log x while forthe remaining polynomials there are somewhat fewer, namelyorderx/(log x)3/2. A typical example from this class is thepolynomialr2 + s2 + 1.

Higher DegreeWe now describe some more recent work, joint with H.

Iwaniec, in which we solve problems 1 and 2 for the polyno-mial x2 + y4. For this polynomial one hasVf ∼ cx

34 so this

is a much thinner set than those that had been treated earlier.The work appeared a few months ago in late 1998 in Annalsof Mathematics.

Theorem (F-I): There are infinitely many primes of theform r2 + s4. Moreover

∑∑r>0 s>0,r2+s4≤x

Λ(r2 + s4) ∼ 4πκx

34

whereκ =∫ 10 (1 − t4)

12 dt.

Corollary:∑∑

r>0 s>0,r2+s4=p<x1 ∼ 4πκ

x34

log x .

It is interesting to compare this asymptotic formula withthe well-known corresponding one forr2 + s2. To obtain thelatter one merely replacesx

34 by x andt4 by t2. The elliptic

integralκ now becomes the area of14 of the unit disc namely

π4 which cancels with the other factor4π . The two formulastell us that the ‘probability’ of a given integerr2 + s2 beingprime is the same when we are told thats is a square as it iswhen we are told thats is not a square.

Although our work was restricted to the single polyno-mial r2 + s4 there are a number of possible extensions, withvarying chances for success.

(A) The values of the polynomialr2 + s4 are treated inour work as the special valuesϕ(r, s2) of the binaryquadratic formX2 +Y 2. One can attempt to similarlytreat the prime values ofϕ(r, s2) for ϕ a general binaryquadratic form. It seems highly likely that this willsucceed but there are lengthy details to be checked.

(B) More problematic is the possible extension to primevalues ofϕ(r, s3) or evenϕ(r, s4). In this case there issome hope but some of the remaining obstacles seemextremely serious. The case ofar2 + bs6 would be ofparticular interest in connection with discriminants ofelliptic curves.

(C) By contrast prime values ofr4 + s4 seem hopelesslyout of reach at this time.

Sieve MethodsWe sketch a little bit about the ideas behind the proof,

which is based on a sieve method.Let A = (an) be a sequence of non-negative real num-

bers. Often in practice this is just the characteristic functionof an interesting set of positive integers (and occasionally oneignores the distinction between the set and its characteristicfunction). In our examplean will be the number of represen-tations ofn in the formn = r2 + s4. Our goal is to study thesum

S =∑n≤x

anΛ(n) .

We writeΛ(n) =∑d|n

λd where this means the sum is over the

positive integers which dividen. Any function on the posi-tive integers (i.e. arithmetic function) can be so written. Thisfollows from the ‘Mobius inversion formula’. In our case itturns out thatλd = −µ(d) log d whereµ is Mobius functiondefined early in the lecture. If we inject this formula forΛinto that forS we obtain a double sum. It is a well-known

12


mathematical axiom that all double sums are given to you inthe wrong order so we interchange this obtaining

S(x) =∑x≤x

λdAd(x)

whereAd(x) =

∑n≤x,n≡0(mod d)

an

and in particularA(x) = A1(x) is the total mass of the se-quence up tox.

To see what is involved here it is useful to recall the sieveof Eratosthenes wherein one wants to count the primes up tox. One begins with the set of positive integers up tox andcasts out all multiples of two, of three, and so on. If one wantsto count the number of remaining integers one needs first tocount the number deleted at each step, the number divisibleby two, by three, by a general integerd.

In general we assume that we can write

Ad(x) = g(d)A(x) + rd(x)

whereg is a ‘nice’ function andr (= remainder) is small atleast on average overd. We shall need a number of assump-tions about the functiong but these are not the central issue.Think of g(d) = 1

d as the prototype of such functions. It isthe one that occurs in the last example mentioned. Half ofthe integers up tox are even, about1d of them are divisible byd. In generalg behaves like a probability. Indeed we assume0 ≤ g(p) < 1 for every primep and thatg is ‘multiplicative’in the sense that(m,n) = 1 ⇒ g(mn) = g(m)g(n). Thereare other more technical assumptions, for example

∑p≤x

g(p) log p ∼ log x .

In the case of the prototypeg(d) = 1d this assumption is a

theorem, proved by Chebyshev. There are a few other similarassumptions we do not mention. More important is the axiomaboutr: We assume

∑d<D

|rd(x)| ≤ A(x)(log x)−1998 (R)

for someD with x23+ε < D < x .

Here of course 1998 is chosen for historical purposes, themain point being thatA(x) represents essentially the trivialbound and we need to save from this only some number oflogarithms. What is more important is that we needD to beas large as possible.

We remark that in practice it is impossible to obtainD > xor evenD > A(x) the mass of the sequence. In the case ofour problem we were inspired by the following result.

Theorem (Fouvry-Iwaniec): Let ε > 0. Then for the se-quencean = number of representations ofn = r2 + s4,assumption (R) holds withD = x

34+ε.

In view of our previous remark this theorem is, apart from theε, best possible.

The above assumptions lead to what might be called the‘classical’ sieve. Over the years we have learned that it is notpossible to produce primes from the classical sieve due to aphenomenon known as the ‘parity problem’ first pointed outby Selberg and described very precisely by Bombieri in hiswork on the ‘asymptotic sieve’.

To illustrate this problem we consider the simplest caseof what are known as the Selberg counter-examples. We take

A = n ≤ x,Ω(n) evenwhenΩ(n) counts the number of prime factors ofn, multiplic-ity included. ThusA obviously contains no prime numbers atall. But it can be shown that this sequence satisfies all of theclassical sieve axioms and withD in (R) essentially as largeas possible.

To get around this problem we add on a new sieve axiom:∑m

∣∣∣ ∑N<n≤2N,mn≤x

µ(mn)amn

∣∣∣ ≤ A(x)(log x)−1998 (B)

for all N satisfyingx−δ√D < N < x

12 (log x)−β for some

β > 0, δ > 0.A good sign is provided by the fact that the new axiom

(B) is not satisfied by the sequence in the Selberg example,since the Mobius functionµ does not change sign onA but isequal to one for all squarefreemn. Hence we have at least achance to get primes.

And in fact we do.

Theorem (F-I): Under the ‘above’ assumptions∑n≤x

anΛ(n) ∼ HA(x)

where

H =∏p

(1 − g(p))(1 − 1

p

)−1.

Of course our assumptions have not been quite precisely statedhere and any reference should be to the formal statements inthe paper.

The potential of this sieve method is very good indeed.It has the capability to settle such important questions as theGoldbach conjecture, the twin prime conjecture, etc.! Thereis however a problem, and quite a huge one at that.

How do we prove that the assumption (B) holds for therelevant sequence? Even a proof of (R) with such a highlevel D is quite a serious issue. Thus for example for thetwin prime or Goldbach problems the fact that (R) holds withD = x

12 −ε follows from the famous Bombieri-Vinogradov

13


theorem while it would also follow even withD = x1−ε froma well-known conjecture of Elliott and Halberstam.

In the case of our example (R) holds withD = x34 −ε

by the above mentioned theorem of Fouvry and Iwaniec andhence we need to obtain (B) for allN > x

38 −δ for someδ. We

are actually able to obtain it for allN > x14+ε for every fixed

ε > 0 (and of courseN bounded above as in the statement of(B)). The proof of this is long and technical (over 80 pages)and we cannot hope to discuss it here. We do mention one outof a number of aspects which seem of independent interestand that come up along the way.

Byproduct on Jacobi SymbolsLet us recall the Legendre symbol, defined forp an odd

prime anda an integer not divisible byp as

(ap

)=

1 if a ≡ square(mod p)−1 else

so that in the canonical projection ofZ onto the field ofp

elements those integersa with(

ap

)= 1 are just those which

project onto non-zero squares.The Jacobi symbol is a natural linear extension of the Leg-

endre symbol defined forb =∏pp

αj

j odd and(a, b) = 1 by

(ab

)=

∏j

(apj

)αj

.

These symbols are examples of Dirichlet characters and itis an important oft-studied problem to show that sums

∑(ab

)asa (or b, or both) run through a given set (often an interval)

are small, i.e. asymptotically half of them give+1, and half−1.

We have

Theorem (Fermat): If p ≡ 1(mod 4) is prime thenp =a2 + b2 wherea > 0, b > 0, a even,b odd in a uniquefashion.

Thus to each such prime we can associate in a naturalfashion a Jacobi symbol

(ab

). Sums over these are much

more complicated than sums over intervals. We were able toshow

Theorem (F-I):

∑p≤x,p=a2+b2

(ab

) x

7677 .

Since the trivial bound is of orderxlog x it follows that asymp-totically half of the time the symbol is 1 and half−1.

ConclusionWe have earlier been lamenting how little is known about

the most basic problems concerning prime values of polyno-mials. On the other hand, just one hundred years ago (well,make that one hundred and ten) we did not know how to provethe prime number theorem. The sieve had hardly advancedat all beyond its ancient Eratosthenian beginnings. So maybewe should not be so pessimistic. It would be nice to knowwhat the next one hundred and ten years will bring.

(MATH–continued from page 1)

years remaining in high school and withthe potential to compete at the mathe-matical olympiad level.

The Imperial Oil Charitable Foun-dation has agreed to be the title spon-sor for the 1999 Regional and NationalMath Camps and, as they stated in arecent media release, “we’re proud tosupport the Esso Math Camps as theyprovide an opportunity for Canadianstudents to explore the many uses ofmathematics in today’s society and, inaddition, these camps help students topursue careers in high technology, sci-ence and business”.

The 1999 Esso National MathCamp took place at the University ofWaterloo from June 19 to 25. The Uni-versity of Calgary hosted an Esso re-gional camp from July 17 to 23 and theUniversity of Western Ontario hosted

one from August 10 to 12. DalhousieUniversity will be hosting a regionalmath camp this fall. The national campin Waterloo and the regional camp inCalgary were both residence campslasting about a week while the camp atthe University of Western Ontario wasa day camp and the regional camp atDalhousie University will be a seriesof weekend camps. Naturally, the costsassociated with the camp depends uponthe format followed.

These camps provide an excellentopportunity for students to spend timeat the host university, have fun do-ing mathematics, establish some veryuseful contacts with fellow students aswell as get to know some other math-ematics teachers. The camps that havetaken place have been very successfuland the CMS is delighted that so manystudents have written expressing theirappreciation and commenting on how

much they enjoyed them. I would liketo thank all of those who have helpedmake the 1999 Esso Math Camps a re-ality and for all their efforts to providethe participants with a rewarding expe-rience and an excellent learning envi-ronment.

As part of World Math Year 2000,and with further support from the pri-vate sector and host universities, theCMS wishes to expand the programmeso that math camps can take place inmore provinces and cities. Informa-tion on those camps that have takenplace anda model for a regional mathcampcan be found at the CMS web site(http://www.cms.math.ca/MathCamps/).This programme has many benefitsfor all of those involved and I invitecolleagues to contact me if they areinterested in participating.

14


DU BUREAU DU DIRECTEUR ADMINISTRATIFPour que les mathématiques, ça compte

(see page 1 for the English version)

Au moment de la mise en oeuvrede la campagne de souscription “Lesmathématiques, ça compte!”, nousavions bon espoir d’atteindre un niveauélevé de soutien et de reconnaissancedes activités dans les domaines de larecherche et de l’éducation de notreSociété. Notre déception sur certainsplans ne doit pas nous empêcher denous réjouir de nos succès.

Le soutien que nous ont accordéles universités hôtes et les trois in-stituts de recherche nous a permisd’augmenter les programmes de nosréunions semestrielles et, ainsi, ac-croître le nombre de participants defacon importante. Nous souhaitons quecette situation, très positive, se pour-suive au moment de nos prochaines réu-nions.

La réunion d’hiver 1999 à Mon-tréal offre non seulement un excellentprogramme scientifique, mais égale-ment, et pour la première fois, un Car-refour emploi. Ce numéro des NOTESprésente des informations à ce pro-pos. Nous espérons que se pour-suive cette importante collaboration en-tre l’université hôte, les instituts derecherche, le secteur privé et la Sociétéet qu’il y ait dorénavant un Carrefouremploi à chaque réunion semestriellede la SMC. Je suis heureux d’annoncerque tous les endroits des prochainesréunions jusqu’en l’an 2004 sont main-tenant déterminés et je suis persuadéque chacun des présidents et organisa-teurs va tout mettre en oeuvre pouroffrir un excellent programme scien-tifique.

Dans le numéro de septembre desNOTES, Daryl Tingley nous a en-tretenus du banquet des lauréats de laSMC de juin dernier ainsi que de tout lesoutien reçu de la part des entreprises,des gouvernements ainsi que des mem-bres de la SMC dans le cadre de nos

divers concours. Cela faisait plaisir devoir les médias accorder tant d’intérêtau talent mathématique. C’est très en-courageant de constater que nos ac-tivités éducatives soulèvent de plus enplus d’intérêt et font l’objet de plus enplus de reportages.

Je suis tout particulièrement ent-housiasmé de l’intérêt et du soutienqu’a reçu le programme de la SMC descamps mathématiques. Bien que lescamps d’entraînement d’été et d’hiverde l’OIM existent depuis plusieurs an-nées, la SMC voulait élaborer un pro-gramme de camps régionaux et na-tional à la portée et à l’attrait encoreplus vastes. Les camps régionaux sontdes camps d’enrichissement destinésprincipalement aux étudiants canadiensde la 8e à la 10e année (de la deux-ième année à la quatrième année du sec-ondaire), tandis que le camp nationals’adresse d’abord aux jeunes étudiantscanadiens à qui il reste encore au moinsdeux ans de secondaire et qui ont lepotentiel nécessaire pour participer auxolympiades mathématiques.

La Fondation philanthropiquePétrolière Impériale a accepté d’êtrele commanditaire en titre des campsnational et régionaux de 1999. “Noussommes fiers d’offrir ce soutien auxcamps mathématiques Esso, car cesderniers permettent aux jeunes Cana-diens d’explorer les multiples usagesdes mathématiques dans la sociétéactuelle”, affirmait-elle récemmentdans un communiqué de presse. “Deplus, ces camps aident les étudiants àpoursuivre des carrières dans les do-maines de la haute technologie, dessciences et de l’administration.”

Le Camp national de mathéma-tiques Esso a eu lieu à l’Université deWaterloo, du 19 au 25 juin. Les Uni-versités de Calgary et de Western On-tario ont respectivement tenu un camprégional Esso du 17 au 23 juillet et du

10 au 12 août. L’Université Dalhousieaura son tour cet automne. Le campnational et le camp régional de Cal-gary duraient environ une semaine etadoptaient tous deux la forme de camp-séjour; celui de Western Ontario, quantà lui, était un camp de jour et, enfin,celui de Dalhousie sera offert en unesérie de camp de fin de semaine. Biensûr, les coûts varient en fonction de laformule adoptée.

Ces camps représentent une excel-lente occasion pour les étudiants devisiter l’université hôte, de s’amuseren faisant des mathématiques, de créerde précieux contacts avec des com-pagnons ou compagnes d’études ainsique faire la connaissance de nouveauxprofesseurs de mathématiques. Lescamps qui ont déjà eu lieu ont remportéun vif succès! La SMC se réjouit deconstater le nombre élevé d’étudiantsqui ont lui écrit pour exprimer leur ap-préciation et leur bonheur. J’adressemes remerciements chaleureux à tousceux qui ont contribué à faire descamps mathématiques Esso 1999 uneheureuse réalité; grâce à leur effortsdévoués, les participants ont vécu uneexpérience enrichissante dans un excel-lent contexte d’apprentissage.

Dans le contexte de l’an 2000,année internationale des mathéma-tiques, et grâce au soutien continudu secteur privé et des universitéshôtes, la SMC souhaite étendre en-core davantage le programme pourqu’un plus grand nombre de provinceset de villes puissent offrir des campsmathématiques. On peut trouver lesinformations sur les camps de cetteannée ainsi qu’un modèle de camprégional au site Web de la SMC(http://www.cms.math.ca/MathCamps/).Tous ceux qui oeuvrent à ce programmey trouvent grand profit et j’invite tousmes collègues à me faire part de leurintérêt à y participer.

15


AWARDS / PRIX

1999 Canada Wide ScienceFair - CMS Awards

The CMS sponsored a set of three Spe-cial Awards at the 1999 Canada WideScience Fair in Edmonton. There weresome wonderful projects at the fair, andwe would have loved to be able to giveout more than one prize in each cate-gory. But each category had one clearwinner. All judges on the CMS teamwere unanimous about the winner, sothat made our jobs relatively easy. Be-low are the project descriptions of thethree prize-winning entries.

Chaos in the Arctic

Junior – Adrian Maler (Grade 8)

The objective was to discover amodel that would predict populationdynamics. I tried to model populationgrowth with the logistic equation and a“coupled logistic equations” model thatI developed.

Palindrome Problems

Intermediate – Katie McAllister(Grade 9)

The problem of this project is toinvestigate palindrome numbers andto look for relationships among palin-dromes in different bases.

In Search of Perfection

Senior – Jocelyn Land-Murphy(Grade 13)

The purpose of my study was toproduce a general formula for gener-ating the factors of perfect numbers.

*****

A few of the particicpants voluntar-ily expressed appreciation for having“real” mathematicians talking to themabout their work. It was clear that theyfelt many other judges did not reallyappreciate and/or understand the work,and their faces lit up when they figuredout we understood what they were talk-ing about. This certainly underscoredthe presence of the CMS at this event!

Order of Canada

Nathan Mendelsohn

Professor Nathan Mendelsohn of theUniversity of Manitoba has beennamed a Member of the Order ofCanada.

Professor Mendelsohn was born in1917 and graduated from the Univer-sity of Toronto in 1942, where he ob-tained the Ph.D. under the supervisionof G. de B. Robinson. He worked asa government research scientist dur-ing the war, moving to Manitoba in1947. He was Head of the Depart-ment of Mathematics and Astronomy1963-1989 and became the university’sfirst distinguished professor in 1989.He was elected a Fellow of the RoyalSociety of Canada in 1956 and wasawarded the Society’s Henry MarshallTory Medal in 1979. His contributionsto the wider mathematical communityhave been manifold, including serviceas President of the CMS 1969-1971. In1995 he received the CMS Award forDistinguished Service.

He has published more than a hun-dred research papers, mainly in the ar-eas of geometry and combinatorics.

University of WaterlooDistinguished Teaching

Award

Ron Scoins

16


Ron Scoins is the recipient ofthe Distinguished Teaching Award for1999 at the University of Waterloo.The award is made by the Univer-sity Senate on nomination by the stu-dents of the instructor, supported by theDean and a university-wide committeeof faculty and students.

Ron Scoins taught secondaryschool mathematics in the Waterloo

region for 13 years. Having taughtall levels of students in all grades andserved as Department Head he broughtan appreciation for students of all typesand interests when he joined the Fac-ulty of Mathematics in 1974.

Since then he has been a lecturer,Director of the Mathematics Cooper-ative Teaching Option, a major de-veloper of the Canadian Mathematics

Competition, and is currently Asso-ciate Dean, External Affairs, in the Fac-ulty of Mathematics.

He is recognized for his exemplaryteaching, his readiness to provide stu-dents with individual assistance at any-time, and his strong support of studentissues and concerns.

The CORS CornerLaura Logan, CORS President

As you may be aware, CORS (Cana-dian Operational Research Society) hasstarted submitting regular articles to theNOTES. The first was in the previousissue and gave an overview of the So-ciety and its activities. It was writtenby Dr. Rick Caron who was Presidentof the Society until early June of thisyear. At our annual conference, whichthis year was held in Windsor, Ontario,I took over the presidency. I am hon-oured to be able to continue this coop-eration between our societies. I wouldlike to take this chance to describe theSociety and its activities in a little moredetail so that you can get to know usbetter.

CORS is a group of people, bothpractitioners and academics, who aretrying to promote the awareness of Op-erational Research and its many appli-cations. There are many aspects to thiswork and this is where I will concen-trate this time:

• CORS represents OperationalResearchers in policy issues in-volving NSERC and SSHRC.This activity is headed up by theacademics within the group.

• CORS works to make studentsaware of Operational Researchand its potential by having stu-dent sections at various univer-sities, awarding CORS Diplo-mas to those students at par-ticipating universities who com-plete a curriculum that providesa strong foundation of OR tech-niques, and funding travel forstudent members to attend theannual conference.

• CORS annual conferences are anexcellent forum for building andmaintaining strong links withinthe Canadian OR community.There is a wide variety of highquality presentations and excel-lent discussions outside the ses-sions. The conference is also thetime for CORS to present its an-nual awards. The conferencesattract a large number of peoplefrom outside Canada and everyfew years are hosted jointly withINFORMS or IFORS (the Amer-ican and International societies,respectively).

• CORS has sections in each ofthe major cities or regions ofthe country. The local sectionshost their own events throughoutthe year so that everyone has achance to get to know the otherpeople in their area.

• CORS has two regular publi-cations. The Bulletin is ournewsletter that is sent to themembers 4 or 5 times a year. Itkeeps the members up to datewith the activities of the societyand the other members. INFORis the technical journal publishedin conjunction with CIPS, theCanadian Information Process-ing Society. It is a well-regardedjournal with a global subscrip-tion base.

There are lots of other projects andservices, but I should leave somethingfor another time. If you have any ques-tions or comments, please feel free tocheck out our Web site at www.cors.caor e-mail me at [email protected].

17

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19


CMS Winter 1999 MeetingRenaissance - Hotel du Parc

Montr eal, QuebecDecember 11 - 13, 1999

Third Announcement

Please refer to the Second Announcement in the Septemberissue of theCMS Notesfor more complete information on thescientific, education and social programmes. This announce-ment features a preliminary timetable and any changes to theprogrammes previously announced. The most up-to-date in-formation concerning the programmes, including scheduling,is available at the following world wide web address:

http://www.camel.math.ca/CMS/Events/

Meeting registration forms and abstract forms for contributedpapers may be found in the September issue of theCMS Notesand at our website.

Programme UpdatesDepartment Chairs’ Luncheon: As the CRM is organizinga meeting of Chairs from Canadian Mathematics Departmentson November 20-21, 1999, the usual December luncheon ofDepartment Chairs is not yet confirmed.

Symposium on History of Mathematics:The list of speak-ers for this symposium include Tom Archibald (Acadia), Ed-ward Barbeau (Toronto), Len Berggren (Simon Fraser), LouisCharbonneau (UQAM), Chongs Suh Chun (Athabasca),Florin Diacu (Victoria), Hardy Grant (Carleton), MinoruHasegawa (Lakehead), Jacques Lefebvre (UQAM), GregoryMoore (McMaster), Richard O’Lander (St. John’s Univer-sity, USA), Dana Schlomiuk (Montreal), Ronald Sklar (St.John’s University, USA), Viena Stastna (Calgary), GeorgeStyan (McGill), and Peter Zvengrow (Calgary).

AcknowledgementsThe support of the following organizations is gratefully ac-knowledged:- Centre de recherches mathematiques- Institut des sciences mathematiques- Laboratoire de combinatoire et d’informatiquemathematique- Network for Computing and Mathematical Modeling- The Fields Institute for Research in Mathematical Sciences- The Pacific Institute for the Mathematical Sciences.The CMS wishes to acknowledge the contribution of the mem-bers of the Meeting Committee for organizing this meetingand presenting these exciting scientific, educational, and so-cial programs.

Meeting CommitteeMeeting Director:Michel Delfour (Montreal)Local Organizing Committee Chair: Veronique Hussin(Montreal).Francois Bergeron, Nantel Bergeron, George Bluman,Monique Bouchard (CMS ex-officio), Martin Goldstein,Michel Grundland, Lucien Haddad, Pierre Hansen, JoelHillel, Jacques Hurtubise (CMS ex-officio), JacquelineKlasa, François Lalonde, Gilbert Laporte, Benoıt Larose,Sabin Lessard, Paul Libbrecht, Wendy MacCaull, ThomasMattman, Angelo Mingarelli, Richard O’Lander, Ivo Rosen-berg, Christiane Rousseau, David Sankoff, Phil Scott, RonaldSklar, Gordon Slade, Graham Wright (CMS ex-officio), MikeZabrocki.

Items also published with this announce-mentTimetable - block schedule

In the next issue of theCMS NotesFourth AnnouncementUpdated Timetable - block schedule

Reunion d’hiver de la SMCRenaissance Hotel du Parc

Montr eal (Quebec)du 11 au 13 decembre, 1999

Troisi eme annonce

Veuillez consulter la deuxieme annonce dans le numero deseptembre desNotes de la SMCpour obtenir de l’informa-tion detaillee sur les programmes scientifique et pedagogique,et les activites sociales. La presente annonce contientl’horaire et tous les changements aux programmes annoncesprecedemment. Vous trouverez l’information la plus recentesur les programmes, y compris les horaires,a l’adresse Web

20


suivante:

http://www.camel.math.ca/CMS/Events/

Un formulaire d’inscription et un formulaire de resume pourcommunications libresetaient inclus dans le numero deseptembre desNotes de la SMCet au site Web.

Changements au programmeLunch des chefs de departements : Le lunch prevu pourles chefs de departements ce n’est pas encore confirme car leCRM organise une reunion des chefs le 20 au 21 novembre1999.

Histoire des mathematiques :Voici la liste des conferenciersinvites: Tom Archibald (Acadia), Edward Barbeau (Toronto),Len Berggren (Simon Fraser), Louis Charbonneau (UQAM),Chongs Suh Chun (Athabasca), Florin Diacu (Victoria),Hardy Grant (Carleton), Minoru Hasegawa (Lakehead),Jacques Lefebvre (UQAM), Gregory Moore (McMaster),Richard O’Lander (St. John’s University, USA), DanaSchlomiuk (Montreal), Ronald Sklar (St. John’s University,USA), Viena Stastna (Calgary), George Styan (McGill), etPeter Zvengrow (Calgary).

RemerciementsNous remercions les organisations suivantes pour leur soutienfinancier- Centre de recherches mathematiques- Institut des sciences mathematiques- Reseau de calcul et de modelisation mathematique

- Laboratoire de combinatoire et d’informatiquemathematique- The Fields Institute for Research in Mathematical Sciences- The Pacific Institute for the Mathematical SciencesLa SMC tienta remercier tous les membres du comite de co-ordination pour l’organisation de la reunion et des activitesscientifiques,educationelles et sociales.

Comite de CoordinationPresident et coordinateur: Michel Delfour (Montreal)Presidente du Comite local: Veronique Hussin (Montreal).Francois Bergeron, Nantel Bergeron, George Bluman,Monique Bouchard (SMC ex-officio), Martin Goldstein,Michel Grundland, Lucien Haddad, Pierre Hansen, JoelHillel, Jacques Hurtubise (SMC ex-officio), JacquelineKlasa, François Lalonde, Gilbert Laporte, Benoıt Larose,Sabin Lessard, Paul Libbrecht, Wendy MacCaull, ThomasMattman, Angelo Mingarelli, Richard O’Lander, Ivo Rosen-berg, Christiane Rousseau, David Sankoff, Phil Scott, RonaldSklar, Gordon Slade, Graham Wright (SMC ex-officio), MikeZabrocki.

Documents publies avec cette annonceHoraire et programme

Dans le prochain numero desNotes de laSMCQuatrieme annonceHoraire et programmea jour

SYMPOSIUM ON THE LEGACY OF JOHN CHARLES FIELDS:A Canadian Contribution to World Mathematical Year 2000

BackgroundThe United Nations Educational Scientific and Cultural Organization (UNESCO) and the International Mathematical Union

(IMU) have declared the year 2000 to be World Mathematical Year (WMY-2000). Countries around the world will be celebratingachievements in mathematics, in support of the three declared aims of WMY-2000:

The Three Aims of World Mathematical Year 2000:

• Meeting the great challenges of the 21st century

• Promoting mathematics as a key for development

• Raising the public image of mathematics

Across Canada, activities are being planned which will integrate these three aims of WMY-2000. These activities will raiseawareness of Canadian achievements in mathematics and inspire young Canadians to pursue higher studies in mathematics,

21


which will qualify them for rewarding careers in the new millennium. A centerpiece of these Canadian celebrations is a world-class Symposium, organized by the Fields Institute, which will stir pride in our unsung Canadian hero in mathematics: JohnCharles Fields.

The Legacy of John Charles Fields

All Canadians have a right to be proud of Canada’s contributions to the world of mathematics. This project will raiseawareness of the Canadian visionary John Charles Fields and his exceptional legacy to the world of mathematics. In additionto being one of the original research mathematicians in Canada, he established the world’s highest award for achievement inmathematics, now known internationally as the Fields Medal. It is often called the “Nobel Prize of Mathematics”, since Nobeldid not establish a prize for mathematics, but it has the same high standard of excellence. Four Fields Medals are awarded ateach International Mathematical Congress, held at four year intervals (most recently in Berlin 1998, the next in Beijing 2002).The Fields Medal is struck by the Canadian Mint, of Canadian gold, and shows the head of the ancient Greek mathematicianArchimedes on the face.

In the movieGood Will Hunting , Robin Williams’ character explains that his friend the MIT Professor has won the FieldsMedal the highest award for achievement in mathematics. Many viewers assume that this award was invented for the story line.Very few are aware of the Canadian origins of the Fields Medal.

An International Symposium of Fields Medallists

A Symposium on the Legacy of John Charles Fields will be held at the Royal Ontario Museum in Toronto, June 8-9, 2000, incelebration of WMY–2000. This is the first time a Symposium of Fields Medallists has been attempted, anywhere. It will bringtogether some of this century’s leading thinkers to explain their medal-winning work and its impact on our world. The FieldsMedallists will include: P-L. Lions (Paris), Vaughan Jones (Berkeley), Alain Connes (France), David Mumford (Harvard),Sir Michael Atiyah (Oxford/Edinburg), Stephen Smale (Berkeley/Hong Kong) and others to be confirmed. The program willfeature public lectures of interest to teachers and students of mathematics. The high quality of this event will attract otherleading mathematicians from abroad. It has received the IMU imprimatur as an official WMY– 2000 event, and it is listed intheir official newsletter and web-site. This Symposium will have a high profile, with extensive media coverage which will raisepublic awareness of mathematics and its significance.

Highlights of the Symposium on the Legacy of John Charles Fields

• The Banquet Address will be given by Sir Michael Atiyah, the only person with both a Fields Medal and a knighthood.

• A documentary video, for television or schools, is planned on the life and times of John Charles Fields, his legacy and itsimpact through the work of the Fields Medallists.

• The Symposium will include both scientific and public lectures by Fields Medallists and an historical lecture on the lifeand times of John Charles Fields.

• The Symposium is closely coordinated with the JMATH 2000 Meeting, June 10-13, 2000 in Hamilton, which will bringtogether for the first time the Canadian Mathematical Society, the Canadian Applied and Industrial Mathematics Society,plus four other important Canadian conferences in the mathematical sciences.

• The Fields Medallists will be encouraged to extend their stay in Canada before or after the Symposium, to visit and workwith leading Canadian researchers.

• Sponsors are being sought to award Visiting Fellowships to talented young mathematicians from third world countries sothey can join this Canadian celebration, meet and work with Canadian and international mathematicians, and take newinsights home with them.

• The Symposium is being coordinated with other Canadian WMY-2000 activities, including classroom projects and museumevents.

22


3rd EUROPEAN CONGRESS OF MATHEMATICSBarcelona 10–14 July 2000

FIRST ANNOUNCEMENT

The organizing committee is pleased to announce that theThird European Congress of Mathematics will take place fromMonday 10 through Friday 14 July 2000. It is organised bythe Societat Catalana de Matematiques, under the auspices ofthe European Mathematical Society.PLENARY SPEAKERSRobbert Dijkgraaf (Universiteit van Amsterdam, NL)Hans Follmer (Humboldt-Universitat zu Berlin, DE)Hendrik W. Lenstra, Jr. (University of California at Berkeley,USA, and Universiteit Leiden, NL)Yuri I. Manin (Max-Planck-Institut fur Mathematik, Bonn,DE)Yves Meyer (Ecole Normale Superieure de Cachan, FR)Carles Simo (Universitat de Barcelona, ES)Marie-France Vigneras (Universite de Paris 7, FR)Oleg Viro (Uppsala Universitet, SE, and POMI St. Peters-burg, RU)Andrew J. Wiles (Princeton University, USA)

SCIENTIFIC PROGRAMMEThe programme of the Congress includes nine plenary lec-tures, thirty invited lectures in parallel sessions, lectures givenby the EMS prize winners, mini-symposia, round table dis-cussions, and poster sessions. As in previous European Con-gresses, a number of prizes will be awarded to mathemati-cians under the age of 32. Mini-symposia are a new featureof the 3ecm; several current, interdisciplinary topics will be

selected by the Scientific Committee. Short communicationsby participants will be possible in the form of posters. Demon-strations of mathematical software, video and multimedia arealso planned.COMMITTEESThe Scientific Committee is chaired by Sir Michael Atiyah(University of Edinburgh, GB).The Prize Committee is chaired by Jacques-Louis Lions(College de France, FR).The Round Table Committee is chaired by Miguel de Guzman(Universidad Complutense de Madrid, ES).The Organizing Committee is chaired by Sebastia XamboDescamps (Universitat Politecnica de Catalunya, ES).

PRE-REGISTRATIONTo receive the Second Announcement and information aboutthe 3ecm by e-mail, please pre-register via the one of thefollowing:

Societat Catalana de Matematiques,Institut d’Estudis Catalans,Carrer del Carme, 47, E-08001 Bacelonatel: +34 93 270 1620fax: +34 93 270 1180e-mail: [email protected] site: http://www.iec.es/3ecm or http://www.si.upc.es/3ecm/

Editor’s Note: The balance of the First Announcementmay be found at the Congress web site.

DRAFT Pending approval

MINUTES OF THE ANNUAL GENERAL MEETING

Memorial University of NewfoundlandSt. John’s, Newfoundland

May 30, 1999

The meeting opened at 4:00 p.m. with 30 members inattendance.

1. Adoption of the agendaThe agenda was adopted, with the addition of the follow-

ing item:

• Under Other Business, the CUMC will be discussed.

2. Minutes of the previous meeting

G–99–1 MOTION (Hyndman/Williams)That the minutes of the General Meeting, held De-

cember 13, 1998 at the Holiday Inn Kingston Waterfront,Kingston, Ontario, be accepted. Carried Unanimously

3. Matters ArisingThere were no matters arising from the minutes.

4. President’s ReportKane announced that the first Endowment Grants Compe-

tition will take place in the fall of 1999, with at least $30,000of funding. Under normal conditions, grants would be a max-imum of $5,000.

Robert Quackenbush of the University of Manitoba hasbeen appointed Managing Editor for the Society for a three-year term, beginning July 1, 1999. The revised job descriptionfor the Executive Director focuses more on development andfund raising.

Kane reported that the Government Policy Committee had

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been dissolved and its tasks redistributed. He also reportedon the establishment of the new Student Committee.

Task Forces and Review Committees continue their con-sultations. The final task force will focus on Office Strategiesfor the Executive Office.

Everyone is invited to participate in the Math 2000 activi-ties, specifically the two CMS meetings being held in Hamil-ton in June and Vancouver in December. To reflect the specialnature of the event, the June meeting has been named “Math2000”.

5. Nominating Committee ReportWright presented the Tellers’ Report. He proposed a vote

of thanks to all the outgoing members of the Executive andBoard of Directors.

G–99–2 MOTION (Nominating Committee)That the Tellers’ Report for the 1999 Election be ac-

cepted. Carried Unanimously

6. Treasurer’s ReportSherk presented the Treasurer’s Report and noted that

there was a good surplus. There was some discussion re-garding whether the required financial statements could beproduced by a Certified General Accountant, as opposed to aChartered Accountant.

G–99–3 MOTION (Board of Directors)That the Audited Statement for the financial year

ended December 31, 1998 be accepted.Carried Unani-mously

G–99-4 (Board of Directors)That the Treasurer’s Report for the financial year

ended December 31, 1998 be accepted.Carried Unani-mously

G–99–5 MOTION (Board of Directors)That the firm of Raymond Chabot Grant Thornton be

appointed as auditors for the financial year ending De-cember 31, 1999. Carried Unanimously

The market value of the investments has now reachedthe level of $1,500,000 and this makes the new EndowmentGrants Competition possible.

7. Executive Director and Secretary’s ReportWright announced that the Executive Office would be

closed for two weeks in August.

8. Reports from CommitteesG–99–6 MOTION (Board of Directors)That the Annual Report to the Members be accepted,

as amended. Carried Unanimously

Education Committee: The 1998 activities are describedin the Annual Report to Members. Orzech expanded on Edu-cation Committee activities and initiatives since the last reportto the members.

The CMS adjudicated and presented three prizes at theCanada Wide Science Fair held in Edmonton this May.

The education session at this meeting not only presenteda slate of excellent talks but was very successful in drawing asizeable complement of school teachers.

As an outcome of its deliberations at the current meeting,the Education Committee will recommend to the CMS Exec-utive that the CMS appoint a Camel Education Page Editor.The Committee is also working on creating a registry of ed-ucation speakers to be a resource in planning math educationactivities, including CMS and outreach activities. Further, theCommittee will convey to the membership, at a later date, thename of recipient of the 1999 Adrien Pouliot award.

Electronic Services Committee:David Bates visited theCommittee to consult for the Task Force on Finances and FundRaising. Goodaire reported that Camel might serve as a con-duit between students looking for goods and companies. Hewas looking for input.

Loki Jorgenson, the Camel Director, is heading up a revi-sion of the Camel site. Many improvements are planned.

Finance Committee: The Finance Committee was verypleased to see the CMS reach the goal set for the beginning ofthe Endowment Grants Programme. Regarding investments,the CMS has adopted a passive management approach. In-vestments have been moved to Toronto Dominion Quantita-tive Capital.

Fund Raising Committee: The Committee is overseeingthree campaigns. The first focuses on government sponsor-ship. Last year, 8 ministries provided funds. The second cam-paign focuses on corporate sponsorships. Wright has devotedmuch time on this, with good results. Of special note here isthe new partnership with Imperial Oil on the math camps. Thethird campaign focus on membership. The AMS reciprocityagreement is being implemented in 2000 and will no doubtbe at the center of the 2000 campaign. Initiatives to attractnew membership include offering two-year free membershipto new faculty members. Chairs will be asked to nominatenew faculty members.

Human Rights Committee: The Board recently ap-proved a Statement on the Employment of Young Mathe-maticians. This will be circulated to departments and facultyassociations.

International Affairs Committee: There was no reportfrom this Committee.

Mathematical Olympiads Committee: The Commit-tee oversees several competitions. The details on this year’sCOMC were reported at the December meetings. Tingley re-ported that the 1999 CMO was written by 81 students in lateApril. Next year, the CMO will be housed at Simon FraserUniversity.

Tingley announced the 1999 IMO team. This year’s IMOwill be held in Romania in July.

The Correspondence Programme continues to be coordi-nated by Ed Barbeau.

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In 1998 a National Math Camp was held in Waterloo inJune. This year, a similar one will be organized by Tom Grif-fiths, Peter Crippin and Richard Hoshino.

Several regional camps are being held across the country.Next year, as part of the 2000 activities, it is hoped to expandthis to even more sites.

Corporate participation in the Olympiads programme isincreasing steadily. Sun Life has been a long standing con-tributor and has recently increased its donation significantly.Imperial Oil has now become the Title Sponsor for the EssoMath Camps. Tingley invited members to review the materi-als on Camel.

Nominating Committee: The Committee was pleasedto announced that members had been appointed to the newlyestablished Endowment Grants Committee and Student Com-mittee. Other upcoming positions are being considered by theCommittee.

Publications Committee: The Committee confirmed theappointment of Robert Quackenbush as the new ManagingEditor. The work on transferring the Advanced Book Seriesfrom Wiley to Springer is almost complete. The work onthe interior redesign of the journals is now complete and theproduction schedules are almost back on track.

The AMS has asked the CMS to change the direction ofthe Conference Proceedings Series. The Publications Com-mittee is considering this matter.

Research Committee:Kane announced the sites of thefuture semi-annual meetings for the next three years.

Women in Mathematics Committee: Hyndman re-ported that the Committee continues to increase the numberof sites on Camel. The Directory of Women in Mathemat-

ics now includes 58 names and she encouraged organizers torefer to it when seeking speakers.

A poster is being designed to present the Krieger-Nelsonwinners.

Student Committee: Charbonneau reported that theCUMC was held directly preceding this meeting and he en-couraged departments to get their students involved. The newStudent Committee will be trying to inform the students acrossCanada of what is available to them and to help them not feelisolated.

A new web page is being developed but current infor-mation on CUMC is available at www.cumc.math.ca. TheStudent Committee is looking for a webmaster and invitedsuggestions.

It was suggested that Camel might include a directory ofgraduating students.

At the Math 2000 meeting, the CUMC programme and theCMS programme will provide more information so that mem-bers of the CMS might attend CUMC talks. The conferenceposter will also be distributed on a wider basis.

9. Other BusinessEd Barbeau made a plea for articles for the Education

Section of the Notes.G–99–7 MOTION (Wright/Charbonneau)That the Annual General Meeting express a special

vote of thanks to Katherine Heinrich for her tireless workduring her six years on the Executive Committee.Carried Unanimously

10. AdjournmentThe meeting adjourned at 5:20 p.m.

COMMITTEES AT WORK

ICMI UpdatePeter Fillmore, Dalhousie University

Chair, International Affairs Committee

The recent appearance of ICMI Bulletin No.46 (June 1999)provides an opportunity to remind readers of the importantwork ICMI is doing and of Canadian involvement in that work.Further information on topics touched on in this report may befound at http://elib.zib.de/imu/icmi/bulletin/, which containsthe complete text of the Bulletin.

What is ICMI? The International Commission on Mathe-matical Instruction is the arm of the IMU (International Math-ematical Union) that is concerned with matters educational.A new executive committee was elected at the IMU GeneralAssembly in August 1998, including Hyman Bass (Columbia)as president and Bernard Hodgson (Laval) as secretary. Thismeans in particular that the Secretariat is now located at

Laval University, where for example the ICMI Bulletin willbe edited and produced for the next four years. Canada’s newrepresentative to ICMI is Eric Muller (Brock).

What does ICMI do? Its most important activitiesare the International Congresses on Mathematical Educa-tion (ICME) and the ICMI Studies. The congresses areheld at four-year intervals, between the International Con-gresses of Mathematicians (ICM). The next, ICME-9, willtake place July 31 to August 6, 2000, at the Nippon Con-ference Centre, near Tokyo. The first announcement is avail-able at http://www.ma.kagu.sut.ac.jp/icmi9/ or by e-mail [email protected]. Readers may recall that a verysuccessful earlier congress, ICME-7, was held at Laval andchaired by Bernard Hodgson.

Nine ICMI studies have been completed and a furtherthree are in progress. A typical study consists of a study

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conference followed by a study volume. The volumes arepublished by Kluwer, the most recent being [Perspectives onthe Teaching of Geometry for the 21st Century] (1998). Anew study was recently launched, on the topicThe Future ofthe Teaching and Learning of Algebra].

Other ICMI activities include four affiliated study groups(history, women, psychology and competitions) and the Soli-darity Programme (to assist the development of mathematicseducation in countries where there is a need that justifies in-ternational assistance).

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CALL FOR NOMINATIONS / APPEL DE CANDIDATURESEditors-in-Chief - Canadian Mathematical Bulletin

Redacteurs-en-chef - Bulletin canadien de mathematiques

The term of office of the present Editors-in-Chief of the Can-dian Mathematical Bulletin will end December 31, 2000. ThePublications Committee of the CMS now invites nominationsfor the next Editors-in-Chief to serve a five year term.

Applications should consist of a formal letter of applica-tion and include the following:

• A curriculum vitae

• An expression of views of the publication indicating ifany changes in direction or policy are contemplated

• Since editorial responsibilities often necessitate a less-ening of responsabilities in an individual’s normalwork, applicants should indicate that they have the sup-port of their university department and, in particular, oftheir head of department.

The Publications Committee will communicate its recom-mendation to the Executive Committee of the CMS in April2000. Any input from the mathematical community concern-ing this important selection process is welcome.

Applications (with supporting material) and/or commentsshould be sent to the address below. The deadline for the re-ceipt of applications isNovember 15, 1999.

Le mandat des redacteurs-en-chef actuels du Bulletin cana-dien de mathematique prendra fin le 31 decembre 2000. Le

Comite des publications de la SMC sollicite des mises encandidatures pour les prochains redacteurs-en-chef pour unmandat de cinq ans.

Les mises en candidature doivent inclure une lettreformelle et leselements suivants:

• Un curriculum vitae

• L’expression de votre opinion sur la publication indi-quant si des changements de directions ou de politiquessont envisages

• Puisque les responsabilites de redaction necessitentsouvent une reduction dans la charge normale de tra-vail, les candidats devraient indiquer qu’ils(elles) ontl’appui de leur departement et en particulier, de leurchef de department.

Le Comite des publications transmettra ses recomman-dations au Comite executif de la SMC en avril 2000. Lescommentaires de la communaute mathematique au sujet decette importante selection sont bienvenus.

Les mises en candiatures (avec material a l’appui)et/ou commentaires devraientetre acheminesa l’adresse ci-dessous. L’echeance pour la reception des mises en candida-ture est le15 novembre 1999.

Editors-in-Chief - CMS Notes / Redacteurs-en-chef - Notes de la SMC

The term of office of the present Editors-in-Chief ofthe CMS Noteswill end December 31, 2000. The Publica-tions Committee of the CMS invites applications for the nextEditor(s)-in-Chief to serve for a five year term.

Applications should consist of a formal letter of applica-tion with curriculum vitae.

The Publications Committee will communicate its rec-ommendation to the Executive Committee of the CMS inApril 2000. Applications and/or comments should be sent,by November 15,1999to the address below.

Le mandat du redacteurs-en-chef actuels desNotes de laSMCprendra fin le 31 decembre 2000. Le Comite des publi-cations de la SMC sollicite les mises en candidature pour leprochain redacteurs-en-chef pour un mandat de cinq ans.

Les mises en candidature doivent inclure une lettreformelle ave curriculum vitae.

Le Comite des publications transmettra ses recommenda-tion au Comite executif de la SMC en avril 2000. Les candi-datures et/ou commentaires devraientetre achemines, avantle 15 Novembre 1999a l’adresse ci-dessous.

Address for Nominations / Addresse de mise en candidatures:James A. MingoChair / President

CMS Publications Committee / Comite des publications de la SMCDepartment of Mathematics and Statistics

Queen’s UniversityKingston, Ontario K7L 3N6

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THE INSTITUT DES SCIENCES MATH EMATIQUES ETLE CENTRE DE RECHERCHES MATH EMATIQUES

JOINT POST DOCTORAL FELLOWSHIP / BOURSES POSTDOCTORAL CRM-ISM

The Institut des sciences mathématiques (ISM) and the Centre de recherches mathématiques (CRM) are inviting applicationsfor their joint postdoctoral fellowship program starting in approximately September 2000. The annual stipend is Cdn. $32, 000for one year, renewable for a second year.

The ISM coordinates the graduate programs of six Québec universities (Concordia, Laval, McGill, Sherbrooke, Universitéde Montréal and UQAM). More than 200 faculty members participate in its ten programs: Algebra and Number Theory,Analysis, Combinatorics, Algebraic Computation and Algorithms, Nonlinear Dynamics, Geometry and Topology, Applied andComputational Mathematics, Mathematical Physics, Probability: Theory and Applications, Decision Theory and MathematicalStatistics, and Category Theory and Applications.

The CRM is a national research center in the mathematical sciences. Its ongoing areas of research include: algebraand combinatorics, analysis, differential equations and approximation theory, geometry and topology, numerical analysis,optimisation and multidisciplinary research, mathematical physics, probability and statistics, and dynamical systems. Eachyear, the CRM organizes a wide range of events attracting participants from around the world. The main themes for 2000-2001are Mathematical Methods in Biology and Medicine, and Symplectic Geometry and Topology and Gauge Theory. In 2001-2002the main themes will be Groups and Geometry. However, high-quality applications in all fields of interest to the CRM or to theISM are welcome.

Applications must arrive at the ISM byFriday, January 7, 2000. The following documents are required: a curriculumvitae, a statement of research, and three letters of recommendation. Please indicate in your application which ISM program bestrepresents your research interests. An e-mail address and fax number (if available) must be provided with all correspondence.Candidates are encouraged to contact the professors with whom they would like to work.

Applications must be sent to the address given below:

L’Institut des sciences mathématiques (ISM) et le Centre de recherches mathématiques (CRM) sollicitent des candidaturesdans le cadre de leur programme conjoint de bourses postdoctorales débutant approximativement en septembre 2000. Lesbourses sont d’une valeur de 32 000 $ CAN pour un an, renouvelables pour une deuxième année.

L’ISM coordonne les programmes d’études supérieures de six des universités québécoises (Concordia, Laval, McGill,Sherbrooke, Université de Montréal et UQAM). Plus de 200 professeurs et professeures participent à ses dix programmes:Algèbre et théorie des nombres, Analyse, Combinatoire et calcul algébrique, Dynamique non linéaire, Géométrie et topologie,Mathématiques appliquées et calcul scientifique, Physique mathématique, Probabilités: théorie et applications, Théorie de ladécision et statistique, et Théorie des catégories et applications.

Le CRM est un centre national de recherche en sciences mathématiques. Les recherches qu’on y poursuit portent entre autressur les domaines suivants: l’algèbre et la combinatoire, l’analyse, les équations différentielles et la théorie de l’approximation, lagéométrie et la topologie, l’analyse numérique, l’optimisation et les recherches multidisciplinaires, la physique mathématique,les probabilités et la statistique, et les systèmes dynamiques. Le CRM organise annuellement un large éventail d’activitésscientifiques impliquant une participation internationale. Les thèmes principaux de l’année 2000-2001 seront d’une part lesméthodes mathématiques en biologie et en médicine et d’autre part la topologie et la géométrie symplectiques ainsi que la théoriede jauge. Les thèmes de l’année 2001-2002 porteront sur les groupes et la géométrie. Cependant, toute candidature méritoiretouchant à un domaine d’intérêt du CRM ou de l’ISM sera bienvenue.

Les candidatures doivent parvenir à l’ISM (adresse ci-dessous) au plus tardle vendredi 7 janvier 2000. Les documentssuivants doivent être joints: un curriculum vitae, un résumé des intérêts de recherche, et trois lettres de recommandation. Vousêtes prié d’indiquer sur votre demande lequel des programmes de l’ISM représente le mieux vos intérets de recherche. Uneadresse électronique et un numéro de télécopieur (si disponible) doivent être inclus avec toute correspondance. Les personnesintéressées sont encouragées à contacter les professeurs avec qui elles aimeraient travailler. Les candidatures doivent êtreenvoyées à:

Professor François Lalonde, Director / DirecteurInstitut des sciences mathématiques

Case postale 8888, Succursale Centre-VilleMontréal (Québec), Canada, H3C 3P8

Fax / Télécopieur : (514) 987-8935 E-mail / Courrier électronique : [email protected]

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UNIVERSITY OF TORONTO – TORONTO, ONTARIODEPARTMENT OF MATHEMATICS

Tenure-Stream Appointment in Algebra, Number Theory and Geometry

The University of Toronto solicits applications for a tenure-stream appointment in the fields of Algebra, Number Theory andGeometry. Preference will be given to researchers in arithmetic geometry.

The appointment is at the downtown (St. George) campus at the level of Assistant Professor, to begin July 1, 2000. Candidatesare expected to have demonstrated excellence in both teaching and research after the Ph.D.; in particular, a candidate’s researchrecord should show clearly the ability to make significant original and independent contributions to Mathematics. Salarycommensurate with experience.

Applicants should send their complete C.V. including a list of publications, a short statement describing their researchprogramme, and all appropriate material about their teaching. They should also arrange to have at least four letters of referencesent directly to:

Search CommitteeDepartment of Mathematics

University of Toronto100 St. George Street, Room 4072

Toronto, Canada M5S 3G3

At least one letter should be primarily concerned with the candidate’s teaching. In addition, it is recommended thatapplicants submit the electronic application form which is available from our World Wide Web Employment Opportunitiespage: http://www.math.toronto.edu/jobs/

To ensure full consideration, this information should be received byDecember 1, 1999.In accordance with its Employment Equity Policy, the University of Toronto encourages applications from qualified women

or men, members of visible minorities, aboriginal peoples and persons with disabilities.


Tenure-Stream Appointment in Applied Mathematics

The Department of Mathematics, University of Toronto solicits applications for a tenure-stream appointment for a mathe-matician working in the area of Applied Mathematics.

The appointment is at the downtown (St. George) campus at the level of Assistant Professor, to begin July 1, 2000. Candidatesare expected to have demonstrated excellence in both teaching and research after the Ph.D.; in particular, a candidate’s researchrecord should show clearly the ability to make significant original and independent contributions to mathematics. Salarycommensurate with experience.






To ensure full consideration, this information should be receivedby December 1, 1999.In accordance with its Employment Equity Policy, the University of Toronto encourages applications from qualified women

or men, members of visible minorities, aboriginal peoples and persons with disabilities.

29



Tenure-Stream Appointment in Applied Mathematics – Computational Science

The Department of Mathematics, University of Toronto solicits applications for a tenure-stream appointment for a mathe-matician working in the area of Applied Mathematics (Computational Science).

The appointment is at the downtown (St. George) campus at the level of Assistant Professor, to begin July 1, 2000. Candidatesare expected to have demonstrated excellence in both teaching and research after the Ph.D.; in particular, a candidate’s researchrecord should show clearly the ability to make significant original and independent contributions to mathematics. Salarycommensurate with experience.






To ensure full consideration, this information should be receivedby December 1, 1999.In accordance with Canadian immigration requirements this advertisement is directed to Canadian citizens and to permanent

residents of Canada. In accordance with its Employment Equity Policy, the University of Toronto encourages applications fromqualified women or men, members of visible minorities, aboriginal peoples and persons with disabilities.

UNIVERSITY OF TORONTO – TORONTO, ONTARIODEPARTMENT OF MATHEMATICSLimited Term Assistant Professorships

The Department invites applications for one or more limited term Assistant Professorships which may, subject to budgetaryapproval, become available at the St. George (downtown), Scarborough or Erindale campus, for a period of one to three years,beginning July 1, 2000. Duties consist of teaching and research, and candidates must demonstrate clear strength in both.Preference will be given to candidates with recent doctoral degrees. Salaries commensurate with qualifications.

Applicants should send their complete C.V. including a list of publications, a short statement describing their researchprogramme, and all appropriate material about their teaching. They should also arrange to have at least three letters of referencesent directly to:




At least one letter should be primarily concerned with the candidate’s teaching. In addition, it is recommended thatapplicants submit the electronic application form which is available on our World Wide Web Employment Opportunities page:http://www.math.toronto.edu/jobs/

To ensure full consideration, all information should be receivedby December 1, 1999.Further information about academic positions in the Department of Mathematics is available on the World Wide Web by

accessing the above URL.In accordance with Canadian immigration requirements, this advertisement is directed to Canadian citizens and to permanent

residents of Canada. In accordance with its Employment Equity Policy, the University of Toronto encourages applications fromqualified women or men, members of visible minorities, aboriginal peoples and persons with disabilities.

30


UNIVERSITY OF TORONTO – SCARBOROUGH, ONTARIOPHYSICAL SCIENCES DIVISION

Tenure-Stream Appointment in Mathematics

The Physical Sciences Division, University of Toronto at Scarborough invites applications for a tenure-stream appointmentin Mathematics. Preference will be given to candidates with interests in the areas of Analysis or Applied Mathematics.

The appointment will be at the level of Assistant Professor, effective on or after July 1, 2000. The successful candidate willbe cross-appointed to the graduate Department of Mathematics, University of Toronto, on the downtown (St. George) Campus.

Candidates are expected to have demonstrated excellence in both teaching and research after the Ph.D.; in particular,a candidate’s research record should show clearly the ability to make significant original and independent contributions tomathematics. Salary commensurate with experience.

Applicants should send their complete C.V. including a list of publications, a short statement describing their proposedresearch programme, and all appropriate material about their teaching background. Applicants should also arrange for at leastfour letters of reference, including at least one primarily concerned with the candidate’s teaching abilities and experience.

All correspondence should be sent to:Professor J.C. Thompson

Chair, Physical Sciences DivisionUniversity of Toronto at Scarborough

1264 Military TrailScarborough, Ontario

M1C 1A4

In addition, it is recommended that applicants submit the electronic application form at http://www.math.utoronto.ca/jobs/.Candidates who apply also for other appointments with the Department of Mathematics, University of Toronto need only

submit one application to that Department but should indicate clearly that they wish to be considered for this appointment aswell.

To ensure full consideration, this information should be receivedby December 1, 1999.

Any inquiries about applications should be addressed to:Professor R.-O. Buchweitz

Associate Chair HiringDepartment of Mathematics


Toronto, Ontario M5S 3G3

In accordance with Canadian immigration requirements this advertisement is directed to Canadian citizens and to permanentresidents of Canada. In accordance with its Employment Equity Policy, the University of Toronto encourages applications fromqualified women or men, members of visible minorities, aboriginal peoples and persons with disabilities.

CMS MEMBERSHIP ...

The 2000 Membership Notices havebeen mailed. Don’t forget to renewyour membership.

ADHESION A LA SMC ...

Les avis d’adhesion 2000etait postes.N’oubliez pas de renouveller votreadhesion.

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UNIVERSIT E McGILL UNIVERSITY – MONTR EAL, QUEBECDEPARTMENT OF MATHEMATICS AND STATISTICS

DEPARTEMENT DE MATH EMATIQUES ET DE STATISTIQUES

The Department of Mathematics and Statistics of McGill Uni-versity invites applications for a tenure track position in statis-tics at the assistant professor level.

A Ph.D. degree in statistical science is essential. Preferredareas of specialization are computational statistics, samplesurveys and time series analysis, although not exclusively so.Preference will be given to applicants with a strong theoreticalbackground in statistics, whose work is driven by applications.

The appointment is to begin July 1, 2000.

Applicants are expected to have demonstrated the capacity forindependent research of excellent quality. Selection criteriainclude research accomplishments, as well as potential con-tributions to the research interests of the Department and to itseducational programs at both the undergraduate and graduatelevels.

Applications, with a curriculum vitae, a list of publications, aresearch proposal, an account of teaching experience and thenames, phone numbers and e-mail addresses of at least fourreferences (with one addressing the teaching record) shouldbe sent to:

Professor K. GowriSankaran, ChairDepartment of Mathematics and Statistics

McGill University805 Sherbrooke Street West

Montreal, Quebec, Canada H3A 2K6

Candidates must arrange to have the letters of recommenda-tion sent directly to the above address. Candidates are alsoencouraged to include copies of up to 3 selected publicationswith their application.

To ensure full consideration, applications must be received byNovember 30, 1999,although the search will continue untilthe position is filled.

McGill University is committed to equity in employment andin accordance with Canadian immigration requirements, pri-ority will be given to Canadian citizens and permanent resi-dents of Canada.

Le departement de mathematiques et de statistique del’Universite McGill cherchea pourvoir un poste en statis-tiques au niveau de professeur adjoint, menanta la perma-nence.

Un doctorat en statistique est essentiel. Les domaines pri-oritaires sont l’echantillonnage, la statistique informatiqueou l’analyse des series chronologiques; ces priorites ne sontpas exclusives. La preference sera accordee aux candidatsayant une forte formation theorique en statistiques et dont lestravaux sont motives par des applications.

La date d’entree en fonction sera le premier juillet 2000.

Les candidats devront avoir demontre leur capacite de menerabien une recherche independante et de haut niveau. Parmi lescriteres de selection des candidats figurent leurs realisationsen recherche, ainsi que leurs contributions potentielles auxactivites de recherche du departement eta ses programmesd’enseignementa tous les cycles.

Les demandes, comprenant un curriculum vitae, une liste depublications, un aperçu des projets de recherches, une descrip-tion de l’experience acquise en enseignement et les noms,numeros de telephone et adresses electroniques d’au moinsquatre repondants (dont un pourra commenter les qualites d’enseignant du candidat) doiventetre envoyeesa :

Professeur K. GowriSankaran, DirecteurDepartement de mathematiques et statistique

Universite McGill805, rue Sherbrooke ouest

Montreal (Quebec) Canada H3A 2K6

Les candidats doivent demandera leurs repondants d’envoyerleurs lettres de recommandation directementa l’adresse ci-dessus. Ils sontegalement invitesa inclure en annexea leurdemande des copies de trois de leurs publications au plus.

Pour etre prises pleinement en consideration, les demandesdevrontetre reçuesle 30 novembre 1999au plus tard. Lesrecherches se poursuivront jusqu’a ce que le poste soit comble.

L’Universite McGill souscrit a l’equite en matiere d’emploiet, conformement a la legislation canadienne en matiered’immigration, accorde la priorite aux citoyens canadiens etaux residents permanents du Canada.

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UNIVERSIT E McGILL UNIVERSITY – MONTR EAL, QUEBECDEPARTMENT OF MATHEMATICS AND STATISTICS

DEPARTEMENT DE MATH EMATIQUES ET DE STATISTIQUES

Consultant en statistique / Statistical Consultant

Le departement de mathematiques et statistique del’Universite McGill esta la recherche d’un consultant pour co-gerer son nouveau service de consultation statistique (SCS).Dans un premier temps ce poste est prevu pour une periode detrois ans. L’entree en fonction est prevue le 1er janvier 2000.Titre : Charge d’enseignementFonctions : S’occuper, avec un autre consultant du Centrede calcul de McGill, de l’exploitation quotidienne du servicede consultation statistique (SCS), sous la supervision du di-recteur du service. Animer des seances d’information sur lecampus sur les services offerts par le SCS. Offrir des consul-tations aux professeurs etetudiants et executer des analysesstatistiques, rediger des rapports et collaborera la redactiondes demandes de subvention et autres projets de recherche.Administrer la facturation des services. Donner au moinsun cours de statistique de 1er cycle par an. Participeral’organisation d’un cours de consultation statistique au niveaudes 2e/3e cycles. Aider lesetudiants de 2e/3e cycle en statis-tique dans leur these ou projet, au besoin. Animera l’occasiondes ateliers sur les logiciels et les statistiques appliquees. Lireles publications sur les statistiques, participera des confer-ences scientifiques de manierea pouvoir offrir des conseilsajour a l’ensemble de la clientele.Qualifications : Maên statistique est imperative. Au moinsdeux ans d’experience dans un poste de statisticien appliqueou d’analyste de donnees. Connaissance des logiciels statis-tiques standards, des methodes multivariees et univarieesparametriques et non parametriques. La connaissance de lamethode des series chronologiques est un atout. Bon sens dela communication orale etecrite et des relations interperson-nelles.Date limite : Veuillez faire parvenir votre candidature avantle1er novembre 1999.La selection se poursuivra neanmoinsjusqu’a ce que le poste soit pourvu. Les candidats doivent de-mandera trois repondants d’envoyer leurs lettres de recom-mandation directementa l’adresse ci-dessous.

Veuillez faire parvenir votre candidaturea :Professeur K. GowriSankaran, Directeur

Departement de mathematiques et statistiqueUniversite McGill

805, rue Sherbrooke ouestMontreal (Quebec) Canada H3A 2K6

Courriel: [email protected]

L’Universite McGill souscrit a l’equite en matiere d’emploiet, conformement a la legislation canadienne en matiered’immigration, accorde la priorite aux citoyens canadiens etaux residents permanents du Canada.

The Department of Mathematics and Statistics at McGill Uni-versity seeks a person to co-manage a newly established sta-tistical consulting service. The initial three-year appointmentis to begin no later than January 1, 2000.Title: Faculty LecturerDuties: Be responsible, together with a co-consultant fromthe McGill Computing Centre, for the day-to-day running ofthe Statistical Consulting Service (SCS), under the supervi-sion of the Director of the SCS. Provide on-campus informa-tion sessions on the services offered by the SCS. Consult withclients including both faculty and students and, where agreedupon, perform statistical analyses, write reports, and collab-orate on grant proposals and other research projects. Help toadminister the billing of clients. Teach at least one undergrad-uate course in statistics per year. Assist in the running of agraduate statistical consulting course. Help graduate studentsin statistics with their theses or projects, where appropriate.Give occasional software and applied statistics workshops.Read the current statistics literature and attend scientific con-ferences in order to provide clients with up-to-date advice.Qualifications: A Master’s degree in statistical science isessential. At least two years experience as an applied statisti-cian and data analyst. Familiarity with the standard statisticalsoftware packages and with both parametric and nonparamet-ric univariate and multivariate methods. Knowledge of timeseries methods an asset. Effective oral, written, communica-tion, and interpersonal skills.Deadline: For full consideration applications should be re-ceived byNovember 1, 1999,although the search will con-tinue until the position is filled. Candidates must arrangefor three letters of recommendation to be sent directly to theaddress below.

Send your application to:Professor K. GowriSankaran, Chair

Department of Mathematics and StatisticsMcGill University

805 Sherbrooke Street WestMontreal, Quebec, Canada H3A 2K6

Email: [email protected]

McGill University is committed to equity in employment andin accordance with Canadian immigration requirements, pri-ority will be given to Canadian citizens and permanent resi-dents of Canada.

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CALENDAR OF EVENTS / CALENDRIER DES EVENEMENTS

OCTOBER 1999 OCTOBRE 1999

11–24Workshops for Mathematics and its Applications (Uni-versity of Minnesota, Minneapolis,MN)http://www.ima.umn.edu/reactive/fall/msl.html

16–17West Coast Operator Algebra Symposium (Universityof Victoria, Victoria, BC)www.math.uvic.ca/ wcoas

NOVEMBER 1999 NOVEMBRE 1999

14–18International Conference on Mathematics Educationinto the 21st Century (Cairo, Egypt)Dr. A Rogerson: [email protected]

20–21Canadian Department Chairs’ Meeting, CRM, [email protected]

29–Dec. 3Group Theory and Computation (University ofSydney, Australia)http://math.auckland.ac.nz/conference/groups-11-1999

DECEMBER 1999 DECEMBRE 1999

2–5 The Future of Mathematical Communication, 1999,MSRI (Berkeley, California)http://www.msri.org/activities/events/9900/fmc99/

11–13 CMS Winter Meeting / Reunion d’hiver de la SMC(Universite de Montreal)http://cms.math.ca/CMS/Events/

JANUARY 2000 JANVIER 2000

19–22Joint Mathematics Meetings, including the 106th An-nual Meeting of the AMS (Washington DC),a WMY2000 eventwww.ams.org/meetings/

MARCH 2000 MARS 2000

6–10 Fourth International Conference on Operations Re-search (Havana, Cuba)[email protected]

MAY 2000 MAI 2000

5–7 Unified Congress of Mathematical Associations andGroups of Quebec (Universite Laval), a WMY2000 [email protected]

JUNE 2000 JUIN 2000

Canadian Mathematics Education Study Group Meeting(UQAM, Montreal)Dates to be announced

4–7Annual Meeting of the Statistical Society of Canada (Ot-tawa, Ontario)Andre Dabrowski: [email protected]

4–8 Canadian Annual Operator Algebra Symposium(Fields Institute, Toronto, Ontario)[email protected];[email protected]

8–9Symposium on the Legacy of John Charles Fields(The Royal Ontario Museum, Toronto); a WMY2000 eventwww.fields.utoronto.ca

10–13 MATH 2000 (McMaster University, Hamilton, On-tario)Paticipating Societies include the Canadian Mathematical So-ciety (CMS), the Canadian Applied and Industrial Mathemat-ics Society (CAIMS), the Canadian Operational Research So-ciety (CORS), the Canadian Symposium on Fluid Dynamics(CSFD), the Canadian Society for the History and Philosophyof Mathematics (CSHPM) and the Canadian UndergraduatesMathematics Conference (CUMC). A WMY2000 eventMonique Bouchard: [email protected]

12–15Integral Methods in Science and Engineering (Banff,Alberta)[email protected]

JULY 2000 JUILLET 2000

10–14 Third European Congress of Mathematics(Tokyo/Makuhari)www.ma.kagu.sut.ac.jp/ icme9/

11–2541st International Mathematical Olympiad (Korea)

31–Aug 7International Congression the Teaching of Mathe-matics ICME-9 (Barcelona, Spain)[email protected]; http://www.si.upc.es/3ecm/

AUGUST 2000 AOUT 2000

7–12AMS Meeting (Los Angeles); a WMY2000 eventwww.ams.org/meetings/

SEPTEMBER 2000 SEPTEMBRE 2000

22–24American Mathematical Society Central Section Meet-ings (University of Toronto)http://www.ams.org/meetings/


10–12 CMS Winter Meeting / Reunion d’hiver de la SMC(University of British Columbia, Vancouver, B. C.)Monique Bouchard: [email protected]

JUNE 2001 JUIN 2001

CMS Summer Meeting / Reunion d’ete de la SMC(University of Saskatchewan, Saskatoon, Saskatchewan)Monique Bouchard: [email protected]

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Canadian Mathematics Education Study Group Meeting(University of Alberta, Edmonton)

Annual Meeting of the Statistical Society of Canada(Vancouver, British Columbia)


CMS Winter Meeting / Reunion d’hiver de la SMC(York University, Toronto, Ontario)Monique Bouchard: [email protected]

JUNE 2002 JUIN 2002

CMS Summer Meeting / Reunion d’ete de la SMC(Universite Laval, Quebec, Quebec)Monique Bouchard: [email protected]

AUGUST 2002 AOUT 2002

20–28International Congress of Mathematicians,(Beijing, China)[email protected]; http://icm2002.org.cn/


CMS Winter Meeting / Reunion d’hiver de la SMC(University of Ottawa / Universit e d’Ottawa,Ottawa, Ontario)Monique Bouchard: [email protected]

JUNE 2003 JUIN 2003

CMS Summer Meeting / Reunion d’ete de la SMC(University of Alberta, Edmonton, Alberta)Monique Bouchard: [email protected]


CMS Winter Meeting / Reunion d’hiver de la SMC(Simon Fraser University, Burnaby, British Columbia)Monique Bouchard: [email protected]

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If undelivered, please return to:Si NON-LIVRE, priere de retournera :CMS Notes de la SMC577 King Edward, C.P. 450, Succ. AOttawa, Ontario, K1N 6N5, Canada