CMS NOTES De La SMC

CMS NOTES de la SMC

Volume 31 No. 6 October / octobre 1999

In this issue / Dans ce numero´ FROM THE the first CMS Job Fair. Information EXECUTIVE on this Job fair is included in this issue of the CMS NOTES. It is hoped to Editorial ...... 2 DIRECTOR’S DESK continue this important cooperative ef- fort between the host university, the re- Men as Trees Walking ...... 3 search institutes, the private sector and the Society and for the Job Fairs to take Research Notes ...... 8 place at each CMS semi-annual meeting. I am pleased to report that the sites for future meetings are confirmed 1999 Jeffery-Williams Lecture 9 to 2004 and I am confident that each meeting director will continue to pro- Du bureau du directeur admin- vide an excellent scientific programme. istratif ...... 15 In the September NOTES, Daryl Tingley reported on the CMS Awards Awards / Prix ...... 15 Banquet that took place in June, the support received from corporations, The CORS Corner ...... 17 Graham Wright governments and by the CMS membership in our competition activities, Make Math Count Meetings / Reunions´ and that it was nice to see mathemat- CMS Winter 1999 Meeting When this year’s fund raising cam- ical talent getting recognition from the Reunion´ d’hiver 1999 de la paign Make Math Count was devel- media. The increased coverage and in- SMC...... 18 oped, it was hoped to obtain a higher terest in our various educational activ- level of support and recognition for ities is very encouraging. Fields Symposium ...... 21 the Society’s research and educational I am particularly encouraged with the interest and support for the CMS Third European Congress of activities. Although there have been programme of math camps. Although Mathematics ...... 23 some disappointments, there have been a number of successes. Summer and Winter IMO Training The support received from the host Camps have taken place for a num- Minutes of the Annual General universities and the three research in- ber of years, the CMS wanted to de- Meeting ...... 23 stitutes has enabled us to expand the velop a programme of regional and na- programmes for our semi-annual meet- tional math camps that would have a Committees at Work ...... 25 ings and, consequently, the number of wider scope and broader appeal. The participants has increased significantly. regional camps are mathematics en- Call for Nominations / Appel de This very positive trend should con- richment camps primarily for Canadian Candidatures ...... 27 tinue for our future meetings. students in grades 8 through 10 while The 1999 Winter Meeting in Mon- the national camp is mainly for younger Calendar of events / Calendrier treal not only features an excellent sci- Canadian students with at least two des ev´ enements´ ...... 34 entific programme but will also include (see MATH–page 14) OCTOBER/OCTOBRE CMS NOTES

EDITORIAL (Barcelona) and ICMI-9 (Tokyo), and August the AMS meeting "Mathemat- CMS NOTES ical Challenges of the 21st Century" in NOTES DE LA SMC Los Angeles. More information about these The CMS Notes is published by events can be found in our calendar sec- the Canadian Mathematical Society tion, as well as at the WMY2000 web- (CMS) eight times a year (February, site . March, April, May, September, Oc- This list, admittedly of just a few high- tober, November, and December). lights, is surely an impressive testa- Editors-in-Chief ment to the vitality of the global mathematical community as we approach the Peter Fillmore; S. Swaminathan new millenium. Why not start planning Managing Editor your participation now? Robert W. Quackenbush Peter Fillmore Nous sommes heureux de vous présen- Contributing Editors ter dans ce numéro de NOTES deux In this issue of the NOTES we are des conférences plénières de la réunion Education: Edward Barbeau pleased to bring you printed versions de St-John’s, celle d’Ed Barbeau et de [email protected] of two of the plenary lectures at the St. John Friedlander (“Men as trees walk- Meetings: Monique Bouchard John’s meeting: Ed Barbeau’s “Men ing” et ”Prime values of polynomi- [email protected] as trees walking” and John Friedlan- als”). Elles furent toutes deux accueil- Research: Noriko Yui; der’s Jeffery-Williams lecture "Prime lies avec beaucoup d’enthousiasme et James D. Lewis values of polynomials". Both talks c’est avec plaisir que nous vous don- [email protected] were enthusiastically received and we nons l’occasion ici de les relire et de les Editorial Assistant are glad to make them available for fur- approfondir. Nous espérons renouveler Caroline Baskerville ther study. We hope to do more of this cette expérience dans l’avenir et ac- in future and would welcome reader re- cueillerions avec plaisir vos commen- The Editors welcome articles, letters action. taires à ce propos. and announcements, which should The conference season is approach- La période intensive des con- be sent to the CMS Notes at: ing and many of us have begun consid- férences approche à grands pas et Canadian Mathematical Society ering our choices. Here is a brief se- plusieurs se penchent sur les choix à 577 King Edward lection. The upcoming CMS meeting faire. Nous vous en présentons ici P.O. Box 450, Station A in Montreal is full of interest, preceded un bref aperçu. La prochaine réunion Ottawa, Ontario, Canada K1N 6N5 immediately by the CRM’s 30th an- de la SMC à Montréal s’annonce fort Telephone: (613) 562-5702 niversary celebration and followed by intéressante. Elle se trouve précédée Facsimile: (613) 565-1539 the first CMS Job Fair. Beyond that we de la célébration du trentième anniver- E-mail: [email protected] can choose from a number of meetings saire du CRM et suivie du premier Car- [email protected] that include special World Mathemati- refour emploi de la SMC. Entre ces Web site: www.cms.math.ca cal Year 2000 features. deux événements, il y aura un vaste In January we have the Joint choix de réunions qui porteront sur l’an No responsibility for views expressed by Mathematics Meetings in Washing- 2000, année internationale des mathé- authors is assumed by the Notes, the ed- ton DC. A united congress of Que- matiques. itors or the CMS. The style files used in bec mathematical organizations and En janvier prochain se tiendront les the production of this volume are a mod- groups takes place at Laval in May, réunions mathématiques communes à ified version of the style files produced and in June follows the MATH 2000 Washington, DC. Un congrès réunis- c by Waterloo Maple Software, 1994, meeting at McMaster, organized by sant les organismes et groupes math- 1995. the CMS and a number of other ématiques du Québec aura lieu à Canadian mathematically-oriented so- l’Université Laval, en mai; il est suivi, ISSN: 1193-9273 cieties. Just prior to that is the two- en juin, de la rencontre MATH 2000, day Symposium on the Legacy of John à McMaster, organisée par la SMC et Charles Fields, at the Fields Institute d’autres sociétés à teneur mathéma- c Canadian Mathematical Society 1999 in Toronto. July brings the 3rd Eu- tique du Canada. Tout juste aupara- ropean Congress of Mathematicians vant, ilyaleSymposium sur l’héritage

2 NOTES de la SMC OCTOBER/OCTOBRE de John Charles Fields, au Fields Insti- On peut trouver tous les ren- moigne de façon convaincante de la vi- tute de Toronto. En juillet, ce sera le seignements sur ces activités dans la talité de la communauté mathématique troisième Congrès européen de math- section Calendrier ainsi qu’au site internationale, au seuil du nouveau mil- ématiques, à Barcelone et en août, à Web de l’an 2000, année interna- lénaire. Ouvrez vite vos agendas et Los Angeles, aura lieu la réunion de tionale des mathématiques, à l’adresse commencez dès maintenant à planiﬁer l’AMS sous le thème “Les déﬁs math- http://wmy200.jussieu.fr. Bien que la votre participation! ématiques du 21e siècle”. liste ci-dessus soit fort partielle, elle té-

Men as Trees Walking Ed Barbeau, University of Toronto

This article is based on a plenary talk front of me. It seems that there is such was a brutal experience for many stu- given on May 29, 1999 at the Summer a plethora of ideas put forward that per- dents in the past, but the fact remains Meeting of the Canadian Mathematical haps a second healing touch is needed that success in mathematics depends on Society in St. John’s, NF. to tease out clearly the main threads. a certain level of ability and applica- The central issue became clear to tion. To deny students the opportu- me one evening as I watched Rob Buck- nity to fail is also to deny them the man on TV Ontario present a documen- opportunity to enjoy success. This is tary on alternative medicine. Many not a call to ignore students who are people were alienated from traditional floundering, but rather to create condi- medicine which they found too reduc- tions that do not neglect the imperatives tionist and narrowly focussed. What- of learning mathematics and the possi- ever the specifics, alternative medi- bility for achievement, but do provide cal regimes were attractive because support in which these might be real- they were holistic, proceeded from a ized and students can move on in con- broader worldview and significantly in- fidence. Many students now are uncer- volved the patient in the diagnosis and tain about what they can do or should know; even good students are denied Ed Barbeau choice of treatment. Cases that might seem identical to a traditional doc- the chance to demonstrate what they tor might receive quite different treat- are capable of. While there is much “Men as trees walking” – this was ments for patients in different environ- that can be done to make mathematics the phrase that came into my mind as I ments. Surely something like this is be- more generally accessible, it needs to examined the draft Principles and Stan- hind the pressure for reform in educa- be recognized that the subject is often dards of School Mathematics (1) re- tion. Teachers and students are reacting difficult and beyond the ability or inter- cently issued by the National Coun- against a curriculum which seems to be est of some segment of the population. cil of Teachers of Mathematics. The reduced to a list of topics and processes, Ghost 2: the syllabus, list of pre- phrase is from the Bible (Mark 8:22- against an imposed canon robbing stu- scribed topics, facts and procedures. 26), where a curious miracle is de- dents and teachers of their autonomy. The charge here is that having a list is scribed. A blind man touched by Christ Whatever the details might be, we want too confining and leads to an emaciated could at first see only “men as trees students to be intimately involved in mathematics that is reduced to a disem- walking” and required a second touch an educational process that cares about bodied set of facts and routines. It is in order to see clearly. who they are and what characteristics not clear why this should necessarily One has the same feeling about the they bring to the mathematics class. be the case. In fact, the criticism seems Principles and Standards; while it is But the charge that traditional ed- to be misplaced; it should be directed generally praiseworthy, it is hard to ucation (as traditional medicine) has at matters of design. In the hands of a bring the document into a clear focus. It consistently lacked the human touch is teacher familiar with mathematics, the contains a mixture of recommendations far too stereotypical; it has resulted in syllabus can serve as a set of markers about topics, processes, teaching styles a number of ghosts that have haunted and goals that frame what will happen and general philosophy, but, as a lec- recent educational reform. Let us raise in the class. An uncertain or ignorant turer of first-year university students, I a few of them. teacher will hold to the list without any did not get a clear sense of what I would Ghost 1: failure, streaming, regard to connections and deeper un- be able to count on from the students in elitism. There is no doubt that school derstanding.

3 OCTOBER/OCTOBRE CMS NOTES

Ghost 3: drill, rote learning and a practical technique is reduced and we need to spell out what students need to memorization. Memorization was an now have alternatives for reaching chil- master at each stage; the issue of the important part of traditional education, dren encountering difficulties. syllabus is one of design and focus. We and many cultures put a great deal of Ghost 6: word problems. Tradi- cannot deny the need for practice; the emphasis on what children should re- tional word problems are criticized as question is whether the student has the member. Children seem to have good being artificial, but one can argue that strategies and perspective to learn and memories, and it seems foolish not this is precisely the point of most word memorize efficiently and effectively. to take advantage of this. But they problems. They provide a toy situation Arithmetic and the standard algorithms need to be taught to evaluate what is in which certain points about interpre- are as important as ever; what we need worth memorizing, how mastery can be tation, and formulation can be made. is to be sure that they are put into the reached and how they can exploit the The question again is not whether we proper context and conceptual frame- coherence of mathematics to leverage should retain word problems, but how work. their knowledge of a few facts into flu- appropriate they are to the situation and These Ghosts are accompanied by a ency over a larger domain. The most how we plan to move beyond them. number of Sirens that drive a lot of ed- able children can do this naturally; oth- Ghost 7: authoritarianism. This ucational reform. Like the ghosts, the ers need to have the issue explicitly ad- ghost has two manifestations, depend- sirens also speak to important aspects dressed, so there is an underlying eq- ing on whether you are referring to of the mathematics education. Here are uity issue involved here for students the teacher, who, we are told, should some of them. without natural talent. be the “guide on the side” rather than Siren 1: problem solving and in- Ghost 4: arithmetic. This is a word the “sage on the stage”, or to the sub- vestigation. This siren calls us away that seems to have fallen upon hard ject itself in which pupils are oppressed from the ghost of drill, of dry and unil- times in the curriculum stakes. But by the tyranny of “one right answer”. luminating exercises. There is nothing it is through arithmetic that most ordi- Without disputing the advantages of wrong in wanting our children to solve nary citizens engage the connection be- the more open and friendly classrooms problems and explore mathematical sit- tween mathematics and the world, and that modern students enjoy, it remains uations, but we can run into serious dis- lack of numeracy can present a severe the case that a teacher’s effectiveness tortions if we do not take the trouble to handicap. Arithmetic has come to sym- depends on what she knows and that ground them mathematically and psy- bolize mindless memorization and ma- sometimes students need to submerge chologically. nipulation. We need to detach it from their egos and pay attention to what Any problem we present to chil- this calumny and exalt it to the level of she has to say. In the same way, it dren should be carefully analyzed for mathematical richness that it deserves. seems mischievous to deny the power its mathematical content and appropri- Ghost 5: paper-and-pencil algo- of mathematical certainty, especially ateness to the maturity of the pupils. rithms. This also is tarred with the given the diversity of the ways in which Strategic decisions need to be made brush of drudgery, and the advent of one might think about concepts and as to what mathematics has to be pre- modern technology has provided de- approach problems. (One feels that sented beforehand as background and tractors of traditional calculation with the work of Godel¨ and Lakatos, how- what can be brought out in the analysis a pretext for summarily discarding it. ever lauded by serious mathematicians, of the problem. Let me give an exam- The issue is really how traditional arith- have had a particularly pernicious ef- ple that I have used with students and metic should be handled in a modern fect on some mathematics educators prospective teachers. curriculum. Perhaps we need to see seduced by the vamps of relativism.) Let ABCD be a unit square and let them more as an additional means of There are many ways to encourage the E and F be the respective midpoints accustoming students to working with individuality of pupils without permit- of sides BC and CD. The three line figures or as examples of algorithms. ting them to believe that black is white. segments AE, AF and BF partition the Long division seems to be particularly All of these Ghosts are the traces of square into five regions. It is required suitable for indicating how one can essential components of mathematical to determine the area of each region. move from the idea of tallying a con- education in the past which had better There is actually quite a lot in this. tinued subtraction to a mechanical al- be part of the future as well. Children At what point should this example be gorithm which is fast and accurate; one are going to either succeed or fail at introduced, before or after the pupils can point out that it was a human inven- any worthwhile task, and the question see the area formula (half-base-times- tion and replaced a method that was de- is how humanely the failure is handled height) for triangles? This will govern cidedly inferior (the “scratch method”). and whether pupils are held back or ad- how they might approach the problem. What has changed is that the impor- vanced for frivolous reasons. Mathe- If the students are to try a structural tance of paper-and-pencil methods as matics is an hierarchical subject and we rather than a formulaic approach, they

4 NOTES de la SMC OCTOBER/OCTOBRE might need at least implicit understand- been beyond the pale for the children the modern computer has greatly ex- ing of isometries and might need to un- I have had the pleasure of presenting panded both the range of the mathe- derstand that areas of nonoverlapping mathematics to. matics we can do as well as the math- sets add, that areas are invariant under Siren 3: patterns. No mathemati- ematics we want to do; our curricu- rigid transformations and that they vary cian can gainsay the important of a lum should reflect this. But we do not as the square of the factor of a dilata- sense of structure in doing mathemat- need idolatry. Technology is there part tion. Should some of this be discussed ics, and the ability to recognize pattern of our environment, as are books and ahead of time, or can it emerge as the is an important part of this. What has pens, and a general purpose of edu- problem is covered? If the students ex- been forgotten in a lot of modern ed- cation is to produce students who can ploit the similarity of the two subtrian- ucational reform is that the power of understand and work within their envi- gles of ABE, they may need to know patterning is seen in its use in analyz- ronment. There are core mathematical that AE is perpendicular to BF; what ing and generalizing mathematical sit- issues that are independent of technol- tools can they be expected to deploy? uations. Much of what passes for pat- ogy, but can be greatly informed by our Will students who assign letters to the terning is a kind of teacher “guess-my- use of technology - it is in this spirit that five regions have the necessary alge- rule” game and rather ad hoc examples. we should embrace the use of calcula- braic understanding to proceed, or is There is no excuse for this. It is hard to tors and computers in our classrooms. this a nice vehicle to introduce them to progress through the curriculum with- All of the sirens speak to important this approach? Finally, will the pupils out finding natural opportunities to ex- aspects of the modern mathematics cur- be able to negotiate the fractions? Are ploit patterns, and teachers need to be riculum, but they all involve subtle is- the fractions appropriately expressed in alert to this and make them explicit to sues and bring the risk of being trivial- vulgar or in decimal form; why? When their pupils at the right time. One as- ized. The big question is whether we we take all of this into consideration, pect that seems to be neglected is giving will have teachers in front of our chil- this example could take quite a bit of the students the mathematical voice to dren who will handle the issues in a sen- time to do properly - has this been an- describe and analyze patterns. sitive, moderate and intelligent way. ticipated by the teacher? Siren 4: data analysis. We do not In designing a curriculum, we need The problem-solving approach to have to look very far to realize how to keep both the Ghosts and Sirens in the curriculum has a great deal to be much of our daily lives is governed by a mind, for each of them not only speak to said for it, but it does require imagi- flood of data of one sort or another. So important goals but also carry a warn- nation by the teacher to envisage what there is certainly a duty to help pupils ing of distortion and counterproductiv- might happen and, importantly, what become number-wise and to interpret ity. There are a number of factors that can possibly go wrong. It is risky, what they read astutely. But the dan- we need to be cognizant of. and we should be sure that teachers are ger is that we do not just cover a lot 1. For a body of mathematics, I equipped to accept the risks. of canned techniques and introduce jar- believe that there are three stages that Siren 2: relevance; real-life. Math- gon, which may in some cases stand in a learner must pass through, initiation, ematics was seen to be alienating be- the way of good discussion and analy- formalization and consolidation. In the cause of the artificiality of what we sis. stage of initiation, the learner encoun- were asking students to do. We should Siren 5: technology. First, let me ters the ideas in a somewhat haphaz- make reference to the daily lives of the agree that through modern technology ard way, feeling her way about, explor- pupils and concern ourselves with what we are undergoing changes in our lives ing; there has to be some motivation, they will need to succeed in their later that are at least as profound as those some reason that the person is inter- careers. But what passes for relevance induced by the invention of the print- ested in the material at hand. At the often raises the question “for whom?”. ing press and the Industrial Revolution. formalization stage, the ideas are drawn It seems that children often have the But any revolution, no matter how per- together and organized; at this point, phony relevance of a tedious arithmetic vasive, is not completely disjoint from the pupil should learn proper concepts, problem dressed up with clowns and the past. Many of the issues raised processes and conventions. There may cookies or are brought into the world by technology are old ones, and the indeed be a lot of artificiality; the pay- of adult concerns in the name of rele- charge of misuse and mindlessness can off should be a growth in the knowl- vance. Play is a part of the world of be and has been leveled against Arabic edge and mathematical power of the a child, and the unadulterated world of numeration, algebraic symbolism, log- learner. In the consolidation stage, the mathematics gives lots of opportunity arithms, slide rules, mechanical adding learner reflects upon what has gone be- for this. I have never been aware that machines and numerical algorithms of fore, contextualizing it, detecting re- some nice number, geometric, topo- all types whether executed on paper or lationships and connecting it to other logical or combinatorial novelties have on a bench with pebbles. Certainly, material. It may be only here that the

5 OCTOBER/OCTOBRE CMS NOTES learner truly appreciates the reasons for material is covered and the student is 8. Foster sound practice and mental what she has been taught before. Tra- psychologically prepared for what is to attitudes. How well students perform ditional education has emphasized the follow. We need to analyze in much in mathematics seems to depend on the second of these stages while modern more detail what students are asked to worldview that they bring. In design- practice seems to focus on the first and do; this is where members of the Uni- ing a curriculum, we should encourage third without the buttress of the second versity community can be particularly an attitude of mind that involves the fol- stage. A good curriculum allows for all helpful. Too often problems are thrown lowing: three stages to occur. The first signifies at pupils with little appreciation of what (a) Awareness of structure; apprecia- to the pupil that the subject matter could needs to be in place to embark upon tion and exploitation of symmetry; belong to them, the second provides the them. A sound curriculum needs both (b) Flow of ideas, analysis and reason- tools to allow it to belong to them and exercises and problems of many differing; the third ensures that it does belong to ent intensities. (c) Corroborative quality of mathemat- them. This is one area in which the Prin- ics; inner consistency; 2. Have a strong focus and a slen- ciples and Standards seems to be par- (d) Shifting of perspective; der core for each course in the curricu- ticularly weak. There are a number of (e) Checking and monitoring one’s lum. Make sure that context is estab- examples thrown in, that upon closer work; lished, the purpose of the material be- analysis seem to involve a great deal (f) Organizing and polishing one’s comes clear, there is enough depth to that the proponent might not be aware work; support its assimilation and allow stu- of. On page 92 is a set of instruc- (g) Mathematical register; expressing dents develop the necessary skills and tions for constructing a golden rectan- ideas appropriately; the place of heuris- understanding to proceed. gle. This seems as though it might be tic and formalism; 3. Have one or two attainable goals pretty heavy sledding for a typical stu- (h) Sense of context; for each year; plan to achieve them dent, and any teacher who embarks on (i) Attainment of power through under- and move on. This means that the sort this without careful thought is wading standing, reasoning and technical facil- of sterile spiralling that now occurs in into a treacherous swamp. Why should ity; schooling should cease, but does not pupils be interested in a golden sec- (j) Visualization; appropriate imaging; preclude returning to previous material tion, or in ways of constructing such (k) Appreciation of symbolic represen- to inform and consolidate it. a thing? Will most pupils muster the tation: numeration, algebra, diagrams; 4. Have enforceable entry re- necessary level of concentration to ne- (l) Making distinctions: classifica- quirements for secondary and tertiary gotiate the ten-stage unmotivated set tion, equivalence, isomorphism, con- courses. No teacher should be asked to of instructions and understand why it gruence; deal with a student who does not have works? This example is meant to illus- (m) Grasping the interplay between reasonable prerequisites for the mate- trate the connectedness of mathemat- concrete and abstract; progression to rial to be studied. There is an impor- ics, but it seems to me to require such a higher order structures that in turn be- tant pedagogical purpose in requiring level of maturity and mastery of some come “concrete”. students to review and organize in their basic algebra that it could easily spin 9. Finally, we come to the subject minds the work of several months or a out of control in the hands of any but matter itself. The centrality of arith- year; it is this process that makes pur- the most adept teacher. metic and algebra must be affirmed. No poseful curriculum progress possible. 7. Meet different needs. What- student can succeed at the secondary 5. Change pace. The diversity ever reason we can give for teaching level without a good grasp of arith- of the mathematical enterprise should anything at all applies to mathematics. metic, nor succeed at the tertiary level be reflected in our curriculum, whether Some mathematics is taught so that stu- without a good algebraic grounding. with respect to subject matter, level of dents will be able to accept the priv- But these are not all. Geometry and discourse or application. There are ileges and responsibilities of citizen- combinatorics are also indispensible. many ways in which people can think ship, some to situate them in the rich While students should see how math- about or do mathematics. Therefore, culture that they are entitled to inherit, ematics can be applied in different ar- we need space to help students find some to provide recreation and addi- eas, it is not clear to me that this should their mathematical voices and texture tional options towards a full and re- necessarily occur in the mathematics their ways of thinking about and doing warding life, and some for professional class. Science, shop, civics, geography mathematics. preparation. These needs can some- and music are areas in which impor- 6. Orchestrate the material. In- times be met by the same piece of math- tant mathematical concepts can be con- crease the level of complexity judi- ematics, but all should inform the cur- veyed in an appropriate and powerful ciously, make sure that foundational riculum that we set. setting. And we should not forget how

6 NOTES de la SMC OCTOBER/OCTOBRE many important mathematical concepts 5. The quadratic: factoring over polynomials interpolation and extrapo- and processes underly some recreations reals and integers, completion of the lation; linear inequalities. and puzzles. In fact, if the elemen- square, quadratic formula, discrim- tary and secondary curricula are well- inant and its significance; complex Course 2: Geometry of triangles designed, one could devote the middle numbers (real quadratic having com- and circles. school years largely to recreational and plex conjugate roots); relation of sum cultural issues as a way of consolidat- and product of roots to coefficients; first 1. Triangles: congruence theo- ing what should have been learned at and second order differences; the fac- rems; ambiguous case; pythagorean the elementary level and preparing the tor and remainder theorem (for quadrat- theorem; similarity. ground for a more sophisticated high ics). 2. Trigonometry: six standard ra- school program in which symbolism, 6. Graphing of quadratic functions; tios and basic identities (cos2 + sin2 = algorithm, analysis and reasoning have comparing the graphs of the functions 1; sec2 = 1 + tan2); sine, cosine and important roles. f(x), f(x − c), f(ax), f(x)+c; sim- tangents of angle sums and differences; ilarity of all graphs of quadratic func- double angle formulae; conversion for- I would like to close with two tions (role of completion of square) mulae between sums and products; law courses that might be given at the lev- This is the entire prescription for of sines; law of cosines els of grades 9 and 10. The first is de- a full-year course. In some classes, 3. Circles: subtended angles; signed to give a systematic introduc- it may be necessary for the teacher to cyclic quadrilaterals; tangent theorems tion to algebra and the second to ge- cover only this material and nothing 4. Coordinate geometry: circles; ometry. It is assumed that students else, spending the time and introduc- area of triangles; family of lines pass- have gone through an initiation stage ing whatever additional materials are ing through pair of intersecting lines; in both areas, that they have some fa- necessary to ensure that students be- circles passing through intersection of miliarity with formulae and the use of come fluent in the essential compo- a pair of circles; radical axis; simple letters to represent numerical quanti- nents. Normally, one would hope that loci problems ties, and that they have had the oppor- the teacher would extend the material Again, the teacher would have the tunity to play around with geometric in some way, either through the intro- discretion of reinforcing these core top- objects either through tactile or com- duction of applications, extended in- ics for students encountering difficulty, puter models. While technology is vestigations and additional topics. It amplifying the material through appli- not explicitly mentioned in either syl- may be that the more able students in cations and investigations, or moving labus, it is understood that it can play the class are given modules to work on to additional topics. These could in- a large and appropriate role. These independently possibly with the sup- clude the following: complex numbers syllabi would be accompanied by report of additional pamphlets or com- (use in deriving geometric and trigono- sources for the teacher that lay out al- puter software. I do not see much point metric results); vector geometry; stat- ternative approaches as well as use- in introducing algebra tiles to the class ics; loci (conjectures, verification and ful exercises, problems and investiga- at large - they amount to an additional proof); solid geometry; advanced Eu- tions, and that make explicit the peda- code interpolated between the student clidean geometry; transformations and gogical and mathematical goals that are and the standard code - but they may their uses (isometries and similarities; brought to light through the material. be useful for individual students who composition of isometries; isometry have trouble getting an initial grasp of determined by three points; isometries Course 1: Linear and quadratic algebra. generated by reflections) functions Other topics that might be intro- The modern school, even at the duced are: motion of projectiles, his- secondary level, has to serve students 1. The linear equation ax = b. tory of solving equations, linear and all across the spectrum of intelligence, Problems that lead to an equation of this quadratic diophantine equations, Pell’s ability, motivation and interest, often form. equation, quadratic residues, first and in the same class. The only way that 2. The equation of the straight line; second order recursions (characteristic the system can cope with this situa- slope; various forms. equation), analysis of the dynamical tion is that students are trained early to 3. The solution of a system of two system x → λx(1 − x), polynomials become autonomous is choosing their linear equations in two unknowns us- of degree exceeding 2; complex num- goals and more self-propelled in ac- ing numerical, algebraic and graphical bers (conjugates, modulus, geometric cessing the resources needed to learn techniques. Consistency of two linear interpretation of sum, product and in- what they need to know and in seeking equations. verse, roots of unity up to the fourth); validation for their knowledge. Any 4. Factoring difference of squares. calculus of finite differences applied to system of education predicated on the

7 OCTOBER/OCTOBRE CMS NOTES teacher being the sole source of knowl- able to go in achieving their goals. A stave off the most trenchant critics from edge and evaluation for the student is successful school regime depends on all sides and leave us seeing “men as bound to fail; the more able and mo- certain beliefs and characteristics of trees walking”. tivated will fall far short of their po- students as well as the knowledge and Reference tential and others will not be able to re- experience of teachers; unless we rec- 1. Principles and Standards for ally engage the material. Students must ognize the need to start with this, then School Mathematics: Discussion Draft ﬁrst reﬂect within themselves what they our principles and standards will be- (October, 1998) National Council of value and how far they are willing and come an exercise of merely trying to Teachers of Mathematics

RESEARCH NOTES Noriko Yui and James Lewis

Royal Society of Canada Michael C. Mackey is a distin- to an overarching scheme for the clas- New Fellows guished applied mathematician who sification of all amenable C*-algebras has made significant contributions to - the so-called Elliott program. Put- Four mathematicians have been differential-delay equations and to a nam’s application of the C*-algebra elected to the Academy of Science of spectrum of mathematical topics in classification to determine the orbit the Royal Society of Canada. physiology and physics. He has made structure of the Cantor set dynamical Michel Fortin -Departement´ de seminal contributions to biomathemat- system, in terms of the quite simple C*- mathematiques´ et de statistique, Uni- ics and biophysics, especially in the ar- algebra invariant, was a striking inno- versiteLaval´ eas of membrane ion transport, dynam- vation. It has changed the face of dy- Michel Fortin jouit d’une ical diseases, cell replication and the namical systems theory. reputation´ internationale qui le dynamics of neural systems. Mackey Maciej Zworski - Department of donne pour l’un des plus eminents´ proposed the “Mackey-Glass” delay Mathematics, University of Toronto specialistes´ de l’analyse numerique´ differential equation for deterministic Maciej Zworski is the world’s lead- de sa gen´ eration´ et l’un des plus chaos and introduced the concept of ing mathematician under the age of grands mathematicians´ appliques´ que "dynamical diseases". His widely cited forty in the difficult and fundamental le Quebec´ ait produits. Il est chef de book (with L. Glass), From Clocks to area of mathematics connecting par- file d’une vast communautequ´ eb´ ecoise´ Chaos: the Rhythms of Life, showed tial differential equations, mathemati- de l’analyse numerique.´ Ses travaux the biological community how nonlin- cal physics and applied mathematics. se demarquent´ par cette faculte´ qu’il ear dynamics and chaos theory can be His research provides works of refer- possede` de jeter des ponts entre la used to help understand physiological ence and starting points for other math- theorie´ et le pratique ainsi qu’entre sci- systems. ematicians. The amazing progress in entifiques de disciplines differents.´ Ian Fraser Putnam - Department of the understanding of resonances in the Fortin a apporte´ une contribution Mathematics and Statistics, University last ten years is due largely to Zworsky. pregnante´ alath` eorie´ des el´ ements´ of Victoria He settled a famous problem formu- finis mixtes et hybrides ainsi qu’au Ian Fraser Putnam has made out- lated by the physicist Regge thirty years developpment´ de methodes´ lagrang- standing contributions to the theory of ago. Zworski also found the precise lo- iennes augmentees.´ Ce travail est dynamical systems. He has shown that cation of the shadow boundary in the resum´ e´ dans deux textes influents dont the orbits of such systems can be com- diffraction of linear oscillatory waves il fut le cosignitaire, Mixed and Hybrid pletely determined by algebraic meth- by a convex boundary, thus proving Finite Element Methods et Augmented ods - methods of which he was in- a long standing conjecture of Keller Lagrangian Methods. strumental in the development. Put- and Rubinow. This refines work dat- Michael C. Mackey - Centre for Non- nam’s analysis of the structure of the ing back to the last century and was not linear Dynamics in Physiology and C*-algebra associated with a minimal known even on the level of formal ex- Medicine, McGill University homeomorphism of the Cantor set led pansions.

8 NOTES de la SMC OCTOBER/OCTOBRE

1999 JEFFERY-WILLIAMS LECTURE Prime Value of Polynomials John B. Friedlander, University of Toronto

This is a transcription of the author’s Jeffery-Williams Lecture Polynomials in One Variable given at the CMS Summer Meeting in St. John’s, Newfound- We begin with what should be the most basic and inter- land in June of this year. esting case, polynomials in a single variable. To start at the very beginning we take the polynomial f(n)=n; in other words we try to count the set of all primes. Here the first problem is answered already in Euclid: There exist infinitely many prime numbers. The proof is so short that we give it here. Suppose to the contrary there are only finitely many, say p1,...,pN and consider the integer p1 ...pN +1formed by taking their product and then adding one. This integer is not divisible by any of the pj since that division leaves a remainder of one. On the other hand, by the fundamental theorem of arithmetic, every integer exceeding one is a non-empty product of primes. This contradicts the assumption that p1,...,pN exhaust the list of primes and the proof is complete. Having answered the first problem in this case we now turn John Friedlander to the second problem. It took about two millennia to make this leap. In the late eighteenth century it was conjectured by ABSTRACT: Consider an irreducible polynomial with inte- Legendre that ger coefficients. It is an old and interesting problem to try to x prove whether or not the polynomial is a prime number for π(x) ∼ , log x infinitely many integer values of the variable(s). One expects that, although there are counter-examples, this will normally a statement that is today known as the prime number theorem. be the case. Nevertheless, despite the long history of the ques- Here π(x) denotes the number of primes up to x, tion, relatively few cases have been settled in the affirmative. In this talk we survey some of that history and then go on to . X π(x) = 1 , describe a number of the ideas behind recent joint work with p≤x Henryk Iwaniec leading to further progress. This lecture is concerned with two of the oldest objects log is the natural logarithm, and the asymptotic symbol of study in number theory, polynomials and prime numbers. f(x) f(x) ∼ g(x) means limx→∞ =1. Let f ∈ Z[X1,X2,...,Xr] be a polynomial in r variables g(x) and having integer coefficients. We shall consider the basic Independently but perhaps a little later Gauss also con- question jectured the prime number theorem, this time in the form π(x) ∼ li x where li x is the ‘logarithmic integral’ which we Problem 1 Does there exist an infinity of r–tuples n = write as Z r . x dt (n1,...,nr) ∈ N of positive integers such that f(n) li x = . is a prime number? 2 log t

In the few cases where we are able to show that there are It is easy to see from repeated integration by parts that x we shall consider also the more refined question li x ∼ log x so that the two forms of the prime number theorem are equivalent. However it turns out that the logarithmic Problem 2 Can we find an appropriate asymptotic formula integral is a much closer approximation to π(x). for the counting function It took about another century for the proof of the prime X number theorem to materialize. The first important steps in 1? this direction were taken by Chebyshev who introduced some 0

9 OCTOBER/OCTOBRE CMS NOTES that π(x) had the expected order of magnitude, in other words there is no obvious reason why there might not be infinitely for certain positive constants c1,c2 and all x ≥ 2 we have many primes. And in fact there are. In 1837 in work that has a x x strong claim to being the birthplace of analytic number theory c <π(x) 1). It is not even known that there After the polynomial f(n)=n the next natural target is exists such a polynomial. the general linear polynomial f(n)=an+b where of course An impressive example is given by the ‘Euler-Rabinowitz’ we take a>0. These are just the arithmetic progressions. polynomial n2 + n +41which is prime for each of the values In case the greatest common divisor (a, b) of a and b exceeds n =0, 1, 2,...,39. Despite getting off to such a good start in one then there can be only one prime in the progression since life it is not known to be prime infinitely often. There are in- 2 if (a, b) is divisible by some prime then so are all integers teresting connections between the prime values of n√+ n + a an + b and no other prime value can occur. If on the other and the arithmetic of the quadratic number field Q( 1 − 4a) hand (a, b)=1so that the integers are ‘relatively prime’ then but that is another story.

10 NOTES de la SMC OCTOBER/OCTOBRE

The simplest looking example n2 +1constitutes one of f(n)=n2 + n +2=n(n +1)+2which is always even so four famous problems mentioned in a lecture by Landau as that ν(2)=2. problems for the twentieth century. The others were the Gold- bach conjecture (that every even integer ≥ 4 is the sum of two More than One Variable primes), the twin prime problem (that there are infinitely many Having seen limited success dealing with polynomials in pairs of primes differing by two), and finally the problem of one variable it might seem foolhardy to further complicate showing that between every pair of consecutive squares there matters by adding more variables. Surprisingly the problem is a prime. Despite Landau in posing four such problems gets easier. Eventually, it is no longer a problem about primes. having been apparently less demanding than had Hilbert in Thus for example consider the polynomial his famous lecture about the same time, this has turned out m2 + n2 + r2 + s2 . after all not to be the case. One hundred years later none of Landau’s four problems have been solved. By a famous theorem of Lagrange we obtain every positive Conjectures however have been easier to come by, in the integer, and so of course every prime. case of problem 2 as well as problem 1. Thus, for the poly- More generally we have, as first conjectured by Waring, nomial n2 +1Hardy and Littlewood conjectured that a similar result for higher powers, specifically X Y Theorem (Hilbert). For each k ∈ N, there exists g(k) ∈ N 2 1 χ(p) Λ(n +1)∼ x 2 1 − g(k) k p − 1 such that every positive integer is the sum of –th pow- n,n2+1≤x p>2 ers. where χ = χ4 is the Dirichlet character of modulus four In fact with enough variables we can even construct . . . which is supported on the odd positive integers, defined to be Theorem (Matijasevic, 1977). There is a function f ∈ +1 at primes of the form 4m +1,tobe−1 at primes 4m − 1, Z[X1,X2,...,X10] whose positive range is just precisely the and extended to all odd integers by multiplicativity. set of primes. This conjecture is just a special case of more general Actually this is a refinement of his breakthrough result a conjectures due to Hardy-Littlewood to Schinzel, and to few years earlier. The following example of such a polyno- Bateman-Horn which deal with more general polynomials and mial which I learned about from Ribenboim’s book of prime also to simultaneous primality of more than one polynomial. number records is due to Jones, Sato, Wada and Wiens in The twin prime problem which deals with the simultaneous 1976. It is a convenient example for English speaking peo- primality of n and n +2is an example of the latter. ple, containing 26 variables a,...,z. Restricting ourselves to the case of a single polynomial 2 2 f ∈ Z[X] (k+2) 1 − [wz + h + j − q] − [(gk +2g + k +1)(h + j)+h − z] we have the following. Let be irreducible with − [2n + p + q + z − e]2 − [16(k +1)3(k +2)(n +1)2 +1− f 2]2 positive leading coefficient. Define the ‘volume’ − [e3(e +2)(a +1)2 +1− o2]2 − [(a2 − 1)y2 +1− x2]2 X − [16r2y4(a2 − 1)+1− u2]2 − [((a + u2(u2 − a))2 − 1)(n +4dy)2 V (x)=Vf (x)= 1 . +1− (x + cu)2]2 − [n + l + v − y]2 n,f(n)≤x − [(a2 − 1)l2 +1− m2]2 − [ai + k +1− l − i]2 2 2 Then − [p + l(a − n − 1) + b(2an +2a − n − 2n − 2) − m] 2 2 − [q + y(a − p − 1) + s(2ap +2a − p − 2 p − 2) − x] Conjecture: −[z + pl(a − p)+t(2ap − p2 − 1) − pm]2 . X In general so far we can deal with a number of cases for Λ(f(n)) = cf V (x)+o(V (x)) all of which we have n,f(n)≤x x V (x) ≥ f (log x)B where −1 Y ν(p) 1 B x c = 1 − 1 − for some and all large . For the simplest looking polyno- f p p p mials this usually means we require the number of variables to be at least as large as the degree. (This is not quite always with ν(p) the number of roots of f(n) ≡ 0(modulo pQ).It the case since for example if the polynomial is linear in one of is not even so obvious that the above infinite product is the variables we get a positive proportion of integers simply p convergent but in fact it turns out that it does.. This follows by fixing appropriately each of the other variables.) from the ‘prime ideal theorem’ which is the generalization of Two Variables the prime number theorem to algebraic number fields. Nev- If we accept the premise that more variables makes things ertheless, it is possible for the constant cf to vanish in case easier, that the fewer the variables the more basic and inter- ν(p)=p for some prime p. Thus recall our earlier example esting the problem, but that one variable seems too difficult, it

11 OCTOBER/OCTOBRE CMS NOTES

PP 3 becomes natural to try next to consider two variables. By the 4 x 4 Corollary: r>0 s>0,r2+s4=p0 s>0,r2+s4≤x π turns out that d where is Mobius¨ function R defined early in the lecture. If we inject this formula for Λ 1 4 1 κ = (1 − t ) 2 dt S where 0 . into that for we obtain a double sum. It is a well-known 12 NOTES de la SMC OCTOBER/OCTOBRE mathematical axiom that all double sums are given to you in Theorem (Fouvry-Iwaniec): Let ε>0. Then for the se- 2 4 the wrong order so we interchange this obtaining quence an = number of representations of n = r + s , 3 +ε X assumption (R) holds with D = x 4 . S(x)= λdAd(x) In view of our previous remark this theorem is, apart from the x≤x ε, best possible. The above assumptions lead to what might be called the where X ‘classical’ sieve. Over the years we have learned that it is not Ad(x)= an possible to produce primes from the classical sieve due to a n≤x,n≡0(mod d) phenomenon known as the ‘parity problem’ first pointed out by Selberg and described very precisely by Bombieri in his A(x)=A (x) and in particular 1 is the total mass of the se- work on the ‘asymptotic sieve’. x quence up to . To illustrate this problem we consider the simplest case To see what is involved here it is useful to recall the sieve of what are known as the Selberg counter-examples. We take of Eratosthenes wherein one wants to count the primes up to x. One begins with the set of positive integers up to x and A = {n ≤ x, Ω(n)even} casts out all multiples of two, of three, and so on. If one wants to count the number of remaining integers one needs first to when Ω(n) counts the number of prime factors of n, multiplic- count the number deleted at each step, the number divisible ity included. Thus A obviously contains no prime numbers at by two, by three, by a general integer d. all. But it can be shown that this sequence satisfies all of the D In general we assume that we can write classical sieve axioms and with in (R) essentially as large as possible. Ad(x)=g(d)A(x)+rd(x) To get around this problem we add on a new sieve axiom: X X −1998 where g is a ‘nice’ function and r (= remainder) is small at µ(mn)amn ≤ A(x)(log x) (B) least on average over d. We shall need a number of assump- m N0 δ>0 the one that occurs in the last example mentioned. Half of , . 1 A good sign is provided by the fact that the new axiom the integers up to x are even, about d of them are divisible by d. In general g behaves like a probability. Indeed we assume (B) is not satisfied by the sequence in the Selberg example, µ A 0 ≤ g(p) < 1 for every prime p and that g is ‘multiplicative’ since the Mobius¨ function does not change sign on but is mn in the sense that (m, n)=1⇒ g(mn)=g(m)g(n). There equal to one for all squarefree . Hence we have at least a are other more technical assumptions, for example chance to get primes. And in fact we do. X g(p) log p ∼ log x. Theorem (F-I): Under the ‘above’ assumptions p≤x X anΛ(n) ∼ HA(x) 1 n≤x In the case of the prototype g(d)= d this assumption is a theorem, proved by Chebyshev. There are a few other similar where assumptions we do not mention. More important is the axiom Y 1−1 H = (1 − g(p)) 1 − . about r: We assume p X p −1998 |rd(x)|≤A(x)(log x) (R) Of course our assumptions have not been quite precisely stated dx level D is quite a serious issue. Thus for example for the or even D>A(x) the mass of the sequence. In the case of twin prime or Goldbach problems the fact that (R) holds with 1 −ε our problem we were inspired by the following result. D = x 2 follows from the famous Bombieri-Vinogradov

13 OCTOBER/OCTOBRE CMS NOTES theorem while it would also follow even with D = x1−ε from are small, i.e. asymptotically half of them give +1, and half a well-known conjecture of Elliott and Halberstam. −1. 3 −ε In the case of our example (R) holds with D = x 4 We have by the above mentioned theorem of Fouvry and Iwaniec and 3 p ≡ 1(mod 4) p = 8 −δ Theorem (Fermat): If is prime then hence we need to obtain (B) for all N>x for some δ.We 2 2 1 +ε a + b where a>0, b>0, a even, b odd in a unique are actually able to obtain it for all N>x4 for every fixed ε>0 (and of course N bounded above as in the statement of fashion. (B)). The proof of this is long and technical (over 80 pages) Thus to each such prime we can associate in a natural a and we cannot hope to discuss it here. We do mention one out fashion a Jacobi symbol b . Sums over these are much of a number of aspects which seem of independent interest more complicated than sums over intervals. We were able to and that come up along the way. show Byproduct on Jacobi Symbols Let us recall the Legendre symbol, defined for p an odd Theorem (F-I): a p X prime and an integer not divisible by as a 76 x 77 . a 1 a ≡ square(mod p) b = if p≤x,p=a2+b2 p −1 else x Z p Since the trivial bound is of order log x it follows that asymp- so that in the canonical projection of onto the field of totically half of the time the symbol is 1 and half −1. a elements those integers a with p =1are just those which project onto non-zero squares. Conclusion We have earlier been lamenting how little is known about The Jacobi symbol is a naturalQ linear extension of the Leg- αj the most basic problems concerning prime values of polyno- endre symbol defined for b = pj odd and (a, b)=1by p mials. On the other hand, just one hundred years ago (well, Q αj a a make that one hundred and ten) we did not know how to prove b = p . j j the prime number theorem. The sieve had hardly advanced These symbols are examples of Dirichlet charactersP and it at all beyond its ancient Eratosthenian beginnings. So maybe a is an important oft-studied problem to show that sums b we should not be so pessimistic. It would be nice to know as a (or b, or both) run through a given set (often an interval) what the next one hundred and ten years will bring.

(MATH–continued from page 1) one from August 10 to 12. Dalhousie much they enjoyed them. I would like University will be hosting a regional to thank all of those who have helped years remaining in high school and with math camp this fall. The national camp make the 1999 Esso Math Camps a re- the potential to compete at the mathe- in Waterloo and the regional camp in ality and for all their efforts to provide matical olympiad level. Calgary were both residence camps the participants with a rewarding expe- The Imperial Oil Charitable Foun- lasting about a week while the camp at rience and an excellent learning envi- dation has agreed to be the title spon- the University of Western Ontario was ronment. sor for the 1999 Regional and National a day camp and the regional camp at As part of World Math Year 2000, Math Camps and, as they stated in a Dalhousie University will be a series and with further support from the pri- recent media release, “we’re proud to of weekend camps. Naturally, the costs vate sector and host universities, the support the Esso Math Camps as they associated with the camp depends upon CMS wishes to expand the programme provide an opportunity for Canadian the format followed. so that math camps can take place in students to explore the many uses of These camps provide an excellent more provinces and cities. Informa- mathematics in today’s society and, in opportunity for students to spend time tion on those camps that have taken addition, these camps help students to at the host university, have fun do- place and a model for a regional math pursue careers in high technology, sci- ing mathematics, establish some very camp can be found at the CMS web site ence and business”. useful contacts with fellow students as (http://www.cms.math.ca/MathCamps/). The 1999 Esso National Math well as get to know some other math- This programme has many beneﬁts Camp took place at the University of ematics teachers. The camps that have for all of those involved and I invite Waterloo from June 19 to 25. The Uni- taken place have been very successful colleagues to contact me if they are versity of Calgary hosted an Esso re- and the CMS is delighted that so many interested in participating. gional camp from July 17 to 23 and the students have written expressing their University of Western Ontario hosted appreciation and commenting on how 14 NOTES de la SMC OCTOBER/OCTOBRE

DU BUREAU DU DIRECTEUR ADMINISTRATIF Pour que les mathématiques, ça compte

(see page 1 for the English version) divers concours. Cela faisait plaisir de 10 au 12 août. L’Université Dalhousie Au moment de la mise en oeuvre voir les médias accorder tant d’intérêt aura son tour cet automne. Le camp de la campagne de souscription “Les au talent mathématique. C’est très en- national et le camp régional de Cal- mathématiques, ça compte!”, nous courageant de constater que nos ac- gary duraient environ une semaine et avions bon espoir d’atteindre un niveau tivités éducatives soulèvent de plus en adoptaient tous deux la forme de camp- élevé de soutien et de reconnaissance plus d’intérêt et font l’objet de plus en séjour; celui de Western Ontario, quant des activités dans les domaines de la plus de reportages. à lui, était un camp de jour et, enfin, recherche et de l’éducation de notre Je suis tout particulièrement ent- celui de Dalhousie sera offert en une Société. Notre déception sur certains housiasmé de l’intérêt et du soutien série de camp de fin de semaine. Bien plans ne doit pas nous empêcher de qu’a reçu le programme de la SMC des sûr, les coûts varient en fonction de la nous réjouir de nos succès. camps mathématiques. Bien que les formule adoptée. Le soutien que nous ont accordé camps d’entraînement d’été et d’hiver Ces camps représentent une excel- les universités hôtes et les trois in- de l’OIM existent depuis plusieurs an- lente occasion pour les étudiants de stituts de recherche nous a permis nées, la SMC voulait élaborer un pro- visiter l’université hôte, de s’amuser d’augmenter les programmes de nos gramme de camps régionaux et na- en faisant des mathématiques, de créer réunions semestrielles et, ainsi, ac- tional à la portée et à l’attrait encore de précieux contacts avec des com- croître le nombre de participants de plus vastes. Les camps régionaux sont pagnons ou compagnes d’études ainsi façon importante. Nous souhaitons que des camps d’enrichissement destinés que faire la connaissance de nouveaux cette situation, très positive, se pour- principalement aux étudiants canadiens professeurs de mathématiques. Les suive au moment de nos prochaines réu- de la 8e à la 10e année (de la deux- camps qui ont déjà eu lieu ont remporté nions. ième année à la quatrième année du sec- un vif succès! La SMC se réjouit de La réunion d’hiver 1999 à Mon- ondaire), tandis que le camp national constater le nombre élevé d’étudiants tréal offre non seulement un excellent s’adresse d’abord aux jeunes étudiants qui ont lui écrit pour exprimer leur ap- programme scientifique, mais égale- canadiens à qui il reste encore au moins préciation et leur bonheur. J’adresse ment, et pour la première fois, un Car- deux ans de secondaire et qui ont le mes remerciements chaleureux à tous refour emploi. Ce numéro des NOTES potentiel nécessaire pour participer aux ceux qui ont contribué à faire des présente des informations à ce pro- olympiades mathématiques. camps mathématiques Esso 1999 une pos. Nous espérons que se pour- La Fondation philanthropique heureuse réalité; grâce à leur efforts suive cette importante collaboration en- Pétrolière Impériale a accepté d’être dévoués, les participants ont vécu une tre l’université hôte, les instituts de le commanditaire en titre des camps expérience enrichissante dans un excel- recherche, le secteur privé et la Société national et régionaux de 1999. “Nous lent contexte d’apprentissage. et qu’il y ait dorénavant un Carrefour sommes fiers d’offrir ce soutien aux Dans le contexte de l’an 2000, emploi à chaque réunion semestrielle camps mathématiques Esso, car ces année internationale des mathéma- de la SMC. Je suis heureux d’annoncer derniers permettent aux jeunes Cana- tiques, et grâce au soutien continu que tous les endroits des prochaines diens d’explorer les multiples usages du secteur privé et des universités réunions jusqu’en l’an 2004 sont main- des mathématiques dans la société hôtes, la SMC souhaite étendre en- tenant déterminés et je suis persuadé actuelle”, affirmait-elle récemment core davantage le programme pour que chacun des présidents et organisa- dans un communiqué de presse. “De qu’un plus grand nombre de provinces teurs va tout mettre en oeuvre pour plus, ces camps aident les étudiants à et de villes puissent offrir des camps offrir un excellent programme scien- poursuivre des carrières dans les do- mathématiques. On peut trouver les tifique. maines de la haute technologie, des informations sur les camps de cette Dans le numéro de septembre des sciences et de l’administration.” année ainsi qu’un modèle de camp NOTES, Daryl Tingley nous a en- Le Camp national de mathéma- régional au site Web de la SMC tretenus du banquet des lauréats de la tiques Esso a eu lieu à l’Université de (http://www.cms.math.ca/MathCamps/). SMC de juin dernier ainsi que de tout le Waterloo, du 19 au 25 juin. Les Uni- Tous ceux qui oeuvrent à ce programme soutien reçu de la part des entreprises, versités de Calgary et de Western On- y trouvent grand profit et j’invite tous des gouvernements ainsi que des mem- tario ont respectivement tenu un camp mes collègues à me faire part de leur bres de la SMC dans le cadre de nos régional Esso du 17 au 23 juillet et du intérêt à y participer.

15 OCTOBER/OCTOBRE CMS NOTES

AWARDS / PRIX

1999 Canada Wide Science The problem of this project is to Professor Nathan Mendelsohn of the Fair - CMS Awards investigate palindrome numbers and University of Manitoba has been to look for relationships among palin- named a Member of the Order of dromes in different bases. Canada. The CMS sponsored a set of three Spe- Professor Mendelsohn was born in cial Awards at the 1999 Canada Wide 1917 and graduated from the Univer- Science Fair in Edmonton. There were In Search of Perfection sity of Toronto in 1942, where he ob- some wonderful projects at the fair, and tained the Ph.D. under the supervision we would have loved to be able to give of G. de B. Robinson. He worked as out more than one prize in each cate- a government research scientist dur- gory. But each category had one clear ing the war, moving to Manitoba in winner. All judges on the CMS team 1947. He was Head of the Depart- were unanimous about the winner, so ment of Mathematics and Astronomy that made our jobs relatively easy. Be- 1963-1989 and became the university’s low are the project descriptions of the ﬁrst distinguished professor in 1989. three prize-winning entries. He was elected a Fellow of the Royal Society of Canada in 1956 and was Chaos in the Arctic Senior – Jocelyn Land-Murphy awarded the Society’s Henry Marshall (Grade 13) Tory Medal in 1979. His contributions to the wider mathematical community The purpose of my study was to have been manifold, including service produce a general formula for gener- as President of the CMS 1969-1971. In ating the factors of perfect numbers. 1995 he received the CMS Award for Distinguished Service. ***** He has published more than a hun- A few of the particicpants voluntar- dred research papers, mainly in the ar- ily expressed appreciation for having eas of geometry and combinatorics. “real” mathematicians talking to them about their work. It was clear that they Junior – Adrian Maler (Grade 8) felt many other judges did not really The objective was to discover a appreciate and/or understand the work, University of Waterloo model that would predict population and their faces lit up when they ﬁgured out we understood what they were talk- Distinguished Teaching dynamics. I tried to model population Award growth with the logistic equation and a ing about. This certainly underscored “coupled logistic equations” model that the presence of the CMS at this event! I developed. Order of Canada Palindrome Problems

Intermediate – Katie McAllister (Grade 9) Nathan Mendelsohn Ron Scoins

16 NOTES de la SMC OCTOBER/OCTOBRE

Ron Scoins is the recipient of region for 13 years. Having taught Competition, and is currently Asso- the Distinguished Teaching Award for all levels of students in all grades and ciate Dean, External Affairs, in the Fac- 1999 at the University of Waterloo. served as Department Head he brought ulty of Mathematics. The award is made by the Univer- an appreciation for students of all types He is recognized for his exemplary sity Senate on nomination by the stu- and interests when he joined the Fac- teaching, his readiness to provide students of the instructor, supported by the ulty of Mathematics in 1974. dents with individual assistance at any- Dean and a university-wide committee Since then he has been a lecturer, time, and his strong support of student of faculty and students. Director of the Mathematics Cooper- issues and concerns. Ron Scoins taught secondary ative Teaching Option, a major de- school mathematics in the Waterloo veloper of the Canadian Mathematics

The CORS Corner Laura Logan, CORS President

As you may be aware, CORS (Cana- • CORS works to make students • CORS has sections in each of dian Operational Research Society) has aware of Operational Research the major cities or regions of started submitting regular articles to the and its potential by having stu- the country. The local sections NOTES. The ﬁrst was in the previous dent sections at various univer- host their own events throughout issue and gave an overview of the So- sities, awarding CORS Diplo- the year so that everyone has a ciety and its activities. It was written mas to those students at par- chance to get to know the other by Dr. Rick Caron who was President ticipating universities who com- people in their area. of the Society until early June of this plete a curriculum that provides year. At our annual conference, which a strong foundation of OR tech- this year was held in Windsor, Ontario, niques, and funding travel for • CORS has two regular publi- I took over the presidency. I am hon- student members to attend the cations. The Bulletin is our oured to be able to continue this coop- annual conference. newsletter that is sent to the eration between our societies. I would members 4 or 5 times a year. It like to take this chance to describe the • CORS annual conferences are an keeps the members up to date Society and its activities in a little more excellent forum for building and with the activities of the society detail so that you can get to know us maintaining strong links within and the other members. INFOR better. the Canadian OR community. is the technical journal published CORS is a group of people, both There is a wide variety of high in conjunction with CIPS, the practitioners and academics, who are quality presentations and excel- Canadian Information Process- trying to promote the awareness of Op- lent discussions outside the ses- ing Society. It is a well-regarded erational Research and its many appli- sions. The conference is also the journal with a global subscrip- cations. There are many aspects to this time for CORS to present its an- tion base. work and this is where I will concen- nual awards. The conferences attract a large number of people trate this time: There are lots of other projects and from outside Canada and every • services, but I should leave something CORS represents Operational few years are hosted jointly with for another time. If you have any ques- Researchers in policy issues in- INFORMS or IFORS (the Amer- tions or comments, please feel free to volving NSERC and SSHRC. ican and International societies, check out our Web site at www.cors.ca This activity is headed up by the respectively). academics within the group. or e-mail me at [email protected].

17 OCTOBER/OCTOBRE CMS NOTES &06 '2&725$/ 35,=( &21)e5(1&(6 &HQWUH GH &RQIpUHQFH 6(66,216 6(66,216 6(66,216 ([KLELWV ([SRVLWLRQV /81&+ 'e-(81(5 /81&+ 'e-(81(5 &2))(( %5($. 3$86( &$)e %XUHDX G·LQVFULSWLRQ RXYHUW GH j 5HJLVWUDWLRQ RSHQ IURP DP WR SP /81&+(21 /81&+ '(6 '(/(*$7(6· 3$57,&,3$176 2SHQLQJ 5HPDUNV '$9,' & /$< 0$&,(- =:256., (//,277 + /,(% 0RWV GH ELHQYHQXH ([KLELWV ([SRVLWLRQV &RIIHH ZLOO EH DYDLODEOH LQ WKH ([KLELW DUHD /H FDIp VHUD VHUYL GDQV O·DLUH G·H[SRVLWLRQ ('8&$7,21 3/(1$5< &2;(7(5-$0(6 /(&785( $'5,$12 *$56,$ -,$1 6+(1 3$9(/ $ 3(9=1(5 =+,+21* ;,$ $1'5($6 '5(66 5HQDLVVDQFH+{WHO GX 3DUF 6&+('8/( +25$,5( 5(1$,66$1&( +Ñ7(/ '8 3$5& 0HHWLQJ 6DORQ /DXULHU 6DORQ 'HV 3LQV G DGPLQLVWUDWLRQ %RDUG RI 'LUHFWRUV &RQIHUHQFH &HQWHU 5pXQLRQ GX &RQVHLO &06 'HYHORSPHQW *URXS *URXSH GH GpYHORSSHPHQW 5HQDLVVDQFH+{WHO GX 3DUF $YHQXH GX 3DUF 0RQWUHDO 4XpEHF &DQDGD &06 :,17(5 0((7,1* 5e81,21 '·+,9(5 '( /$ 60& &RQIHUHQFH &HQWHU 6DORQ -HDQQH0DQFH 5pXQLRQ GX &RPLWp H[pFXWLI ([HFXWLYH &RPPLWWHH 0HHWLQJ 7LPH 7KXUVGD\ MHXGL )ULGD\ YHQGUHGL 6DWXUGD\ VDPHGL 6XQGD\ GLPDQFKH 0RQGD\ OXQGL +HXUH 'HFHPEHU GpFHPEUH 'HFHPEHU GpFHPEUH 'HFHPEHU GpFHPEUH 'HFHPEHU GpFHPEUH 'HFHPEHU GpFHPEUH /(&785(6 &RQIHUHQFH &HQWHU

18 NOTES de la SMC OCTOBER/OCTOBRE OLQHDU DOJHEUD GH O·DOJqEUH OLQpDLUH )RUXP RQ WKH WHDFKLQJ RI )RUXP VXU O·HQVHLJQHPHQW e &21)e5(1&(6 &HQWUH GH &RQIpUHQFH %DQTXHW 6(66,216 $VVHPEOpH *pQpUDOH 5HQDLVVDQFH+{WHO GX 3DUF &2))(( %5($. 3$86( &$) 5HFHSWLRQ &DVK EDU 6(66,216 *HQHUDO 0HHWLQJ 6(66,216 38%/,& /(&785( -(11,)(5 &+$<(6 6&+('8/( +25$,5( 5(1$,66$1&( +Ñ7(/ '8 3$5& 5HFHSWLRQ 5HFHSWLRQ LQVFULSWLRQ 5pFHSWLRQ &DVKEDU DQG &DVK EDU EDU SD\DQW HW 6DORQ 'HV 3LQV 5pXQLRQ GX &RQVHLO (YHQLQJ 5HJLVWUDWLRQ 5HQDLVVDQFH+{WHO GX 3DUF 5HQDLVVDQFH+{WHO GX 3DUF $YHQXH GX 3DUF 0RQWUHDO 4XpEHF &DQDGD &06 :,17(5 0((7,1* 5e81,21 '·+,9(5 '( /$ 60& &RQIHUHQFH &HQWHU 6DORQ -HDQQH0DQFH G DGPLQLVWUDWLRQ 5pXQLRQ GX &RPLWp H[pFXWLI 0HHWLQJ ([HFXWLYH &RPPLWWHH 0HHWLQJ %RDUG RI 'LUHFWRUV (YHQLQJ /(&785(6 &RQIHUHQFH &HQWHU

19 OCTOBER/OCTOBRE CMS NOTES

CMS Winter 1999 Meeting Acknowledgements Renaissance - Hotelˆ du Parc The support of the following organizations is gratefully ac- knowledged: Montreal,´ Quebec´ - Centre de recherches mathematiques´ December 11 - 13, 1999 - Institut des sciences mathematiques´ - Laboratoire de combinatoire et d’informatique Third Announcement mathematique´ - Network for Computing and Mathematical Modeling - The Fields Institute for Research in Mathematical Sciences Please refer to the Second Announcement in the September - The Pacific Institute for the Mathematical Sciences. issue of the CMS Notes for more complete information on the The CMS wishes to acknowledge the contribution of the mem- scientific, education and social programmes. This announce- bers of the Meeting Committee for organizing this meeting ment features a preliminary timetable and any changes to the and presenting these exciting scientific, educational, and so- programmes previously announced. The most up-to-date in- cial programs. formation concerning the programmes, including scheduling, is available at the following world wide web address: Meeting Committee

http://www.camel.math.ca/CMS/Events/ Meeting Director: Michel Delfour (Montreal)´ Local Organizing Committee Chair: Veronique´ Hussin (Montreal).´ Meeting registration forms and abstract forms for contributed François Bergeron, Nantel Bergeron, George Bluman, papers may be found in the September issue of the CMS Notes Monique Bouchard (CMS ex-officio), Martin Goldstein, and at our website. Michel Grundland, Lucien Haddad, Pierre Hansen, Joel Hillel, Jacques Hurtubise (CMS ex-officio), Jacqueline Klasa, François Lalonde, Gilbert Laporte, Benoˆıt Larose, Programme Updates Sabin Lessard, Paul Libbrecht, Wendy MacCaull, Thomas Mattman, Angelo Mingarelli, Richard O’Lander, Ivo Rosen- Department Chairs’ Luncheon: As the CRM is organizing berg, Christiane Rousseau, David Sankoff, Phil Scott, Ronald a meeting of Chairs from Canadian Mathematics Departments Sklar, Gordon Slade, Graham Wright (CMS ex-officio), Mike on November 20-21, 1999, the usual December luncheon of Zabrocki. Department Chairs is not yet confirmed. Symposium on History of Mathematics: The list of speak- Items also published with this announce- ers for this symposium include Tom Archibald (Acadia), Ed- ward Barbeau (Toronto), Len Berggren (Simon Fraser), Louis ment Charbonneau (UQAM), Chongs Suh Chun (Athabasca), Timetable - block schedule Florin Diacu (Victoria), Hardy Grant (Carleton), Minoru Hasegawa (Lakehead), Jacques Lefebvre (UQAM), Gregory Moore (McMaster), Richard O’Lander (St. John’s Univer- In the next issue of the CMS Notes sity, USA), Dana Schlomiuk (Montreal), Ronald Sklar (St. Fourth Announcement John’s University, USA), Viena Stastna (Calgary), George Updated Timetable - block schedule Styan (McGill), and Peter Zvengrow (Calgary).

Reunion´ d’hiver de la SMC Veuillez consulter la deuxieme` annonce dans le numero´ de septembre des Notes de la SMC pour obtenir de l’informa- Renaissance Hotelˆ du Parc tion detaill´ ee´ sur les programmes scientiﬁque et pedagogique,´ Montreal´ (Quebec)´ et les activites´ sociales. La presente´ annonce contient l’horaire et tous les changements aux programmes annonces´ du 11 au 13 decembre,´ 1999 prec´ edemment.´ Vous trouverez l’information la plus recente´ sur les programmes, y compris les horaires, a` l’adresse Web Troisieme` annonce

20 NOTES de la SMC OCTOBER/OCTOBRE suivante: - Laboratoire de combinatoire et d’informatique mathematique´ http://www.camel.math.ca/CMS/Events/ - The Fields Institute for Research in Mathematical Sciences - The Paciﬁc Institute for the Mathematical Sciences Un formulaire d’inscription et un formulaire de resum´ e´ pour La SMC tient a` remercier tous les membres du comitedeco-´ communications libres etaient´ inclus dans le numero´ de ordination pour l’organisation de la reunion´ et des activites´ septembre des Notes de la SMC et au site Web. scientiﬁques, educationelles´ et sociales.

Changements au programme Comite´ de Coordination Lunch des chefs de departements´ : Le lunch prevu´ pour President´ et coordinateur : Michel Delfour (Montreal)´ les chefs de departements´ ce n’est pas encore confirme´ car le Presidente´ du Comite´ local :Veronique´ Hussin (Montreal).´ CRM organise une reunion´ des chefs le 20 au 21 novembre François Bergeron, Nantel Bergeron, George Bluman, 1999. Monique Bouchard (SMC ex-officio), Martin Goldstein, Michel Grundland, Lucien Haddad, Pierre Hansen, Joel Histoire des mathematiques´ : Voicila liste des conferenciers´ Hillel, Jacques Hurtubise (SMC ex-officio), Jacqueline invites:´ Tom Archibald (Acadia), Edward Barbeau (Toronto), Klasa, François Lalonde, Gilbert Laporte, Benoˆıt Larose, Len Berggren (Simon Fraser), Louis Charbonneau (UQAM), Sabin Lessard, Paul Libbrecht, Wendy MacCaull, Thomas Chongs Suh Chun (Athabasca), Florin Diacu (Victoria), Mattman, Angelo Mingarelli, Richard O’Lander, Ivo Rosen- Hardy Grant (Carleton), Minoru Hasegawa (Lakehead), berg, Christiane Rousseau, David Sankoff, Phil Scott, Ronald Jacques Lefebvre (UQAM), Gregory Moore (McMaster), Sklar, Gordon Slade, Graham Wright (SMC ex-officio), Mike Richard O’Lander (St. John’s University, USA), Dana Zabrocki. Schlomiuk (Montreal), Ronald Sklar (St. John’s University, USA), Viena Stastna (Calgary), George Styan (McGill), et Peter Zvengrow (Calgary). Documents publies´ avec cette annonce Horaire et programme Remerciements Nous remercions les organisations suivantes pour leur soutien Dans le prochain numero´ des Notes de la financier - Centre de recherches mathematiques´ SMC - Institut des sciences mathematiques´ Quatrieme` annonce -Reseau´ de calcul et de modelisation´ mathematique´ Horaire et programme a` jour

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

Background The 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 celebrating achievements 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 raise awareness of Canadian achievements in mathematics and inspire young Canadians to pursue higher studies in mathematics,

21 OCTOBER/OCTOBRE CMS NOTES 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: John Charles 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 raise awareness of the Canadian visionary John Charles Fields and his exceptional legacy to the world of mathematics. In addition to being one of the original research mathematicians in Canada, he established the world’s highest award for achievement in mathematics, now known internationally as the Fields Medal. It is often called the “Nobel Prize of Mathematics”, since Nobel did not establish a prize for mathematics, but it has the same high standard of excellence. Four Fields Medals are awarded at each 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 mathematician Archimedes on the face.

In the movie Good Will Hunting, Robin Williams’ character explains that his friend the MIT Professor has won the Fields Medal 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, in celebration of WMY–2000. This is the first time a Symposium of Fields Medallists has been attempted, anywhere. It will bring together some of this century’s leading thinkers to explain their medal-winning work and its impact on our world. The Fields Medallists 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 will feature public lectures of interest to teachers and students of mathematics. The high quality of this event will attract other leading mathematicians from abroad. It has received the IMU imprimatur as an official WMY– 2000 event, and it is listed in their official newsletter and web-site. This Symposium will have a high profile, with extensive media coverage which will raise public 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 its impact 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 life and times of John Charles Fields. • The Symposium is closely coordinated with the JMATH 2000 Meeting, June 10-13, 2000 in Hamilton, which will bring together 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 work with leading Canadian researchers. • Sponsors are being sought to award Visiting Fellowships to talented young mathematicians from third world countries so they can join this Canadian celebration, meet and work with Canadian and international mathematicians, and take new insights home with them. • The Symposium is being coordinated with other Canadian WMY-2000activities, including classroom projects and museum events.

22 NOTES de la SMC OCTOBER/OCTOBRE

3rd EUROPEAN CONGRESS OF MATHEMATICS Barcelona 10–14 July 2000

FIRST ANNOUNCEMENT selected by the Scientiﬁc Committee. Short communications by participants will be possible in the form of posters. Demon- The organizing committee is pleased to announce that the strations of mathematical software, video and multimedia are Third European Congress of Mathematics will take place from also planned. Monday 10 through Friday 14 July 2000. It is organised by the Societat Catalana de Matematiques,` under the auspices of COMMITTEES the European Mathematical Society. The Scientiﬁc Committee is chaired by Sir Michael Atiyah PLENARY SPEAKERS (University of Edinburgh, GB). Robbert Dijkgraaf (Universiteit van Amsterdam, NL) The Prize Committee is chaired by Jacques-Louis Lions Hans Follmer¨ (Humboldt-Universitat¨ zu Berlin, DE) (College` de France, FR). Hendrik W. Lenstra, Jr. (University of California at Berkeley, The Round Table Committee is chaired by Miguel de Guzman´ USA, and Universiteit Leiden, NL) (Universidad Complutense de Madrid, ES). Yuri I. Manin (Max-Planck-Institut fur¨ Mathematik, Bonn, The Organizing Committee is chaired by Sebastia` Xambo´ DE) Descamps (Universitat Politecnica` de Catalunya, ES). Yves Meyer (Ecole´ Normale Superieure´ de Cachan, FR) Carles Simo´ (Universitat de Barcelona, ES) PRE-REGISTRATION Marie-France Vigneras´ (Universite´ de Paris 7, FR) To receive the Second Announcement and information about Oleg Viro (Uppsala Universitet, SE, and POMI St. Peters- the 3ecm by e-mail, please pre-register via the one of the burg, RU) following: Andrew J. Wiles (Princeton University, USA) Societat Catalana de Matematiques,` Institut d’Estudis Catalans, SCIENTIFIC PROGRAMME Carrer del Carme, 47, E-08001 Bacelona The programme of the Congress includes nine plenary lec- tel: +34 93 270 1620 tures, thirty invited lectures in parallel sessions, lectures given fax: +34 93 270 1180 by the EMS prize winners, mini-symposia, round table dis- e-mail: [email protected] cussions, and poster sessions. As in previous European Con- web site: http://www.iec.es/3ecm or http://www.si.upc.es/3ecm/ gresses, a number of prizes will be awarded to mathematicians under the age of 32. Mini-symposia are a new feature Editor’s Note: The balance of the First Announcement of the 3ecm; several current, interdisciplinary topics will be may be found at the Congress web site.

G–99–1 MOTION (Hyndman/Williams) DRAFT Pending approval That the minutes of the General Meeting, held De- cember 13, 1998 at the Holiday Inn Kingston Waterfront, Kingston, Ontario, be accepted. Carried Unanimously MINUTES OF THE ANNUAL GENERAL MEETING Memorial University of Newfoundland 3. Matters Arising St. John’s, Newfoundland There were no matters arising from the minutes. May 30, 1999 4. President’s Report Kane announced that the ﬁrst Endowment Grants Compe- The meeting opened at 4:00 p.m. with 30 members in tition will take place in the fall of 1999, with at least $30,000 attendance. of funding. Under normal conditions, grants would be a max- 1. Adoption of the agenda imum of $5,000. The agenda was adopted, with the addition of the follow- Robert Quackenbush of the University of Manitoba has ing item: been appointed Managing Editor for the Society for a three- year term, beginning July 1, 1999. The revised job description • Under Other Business, the CUMC will be discussed. for the Executive Director focuses more on development and fund raising. 2. Minutes of the previous meeting Kane reported that the Government Policy Committee had

23 OCTOBER/OCTOBRE CMS NOTES been dissolved and its tasks redistributed. He also reported The education session at this meeting not only presented on the establishment of the new Student Committee. a slate of excellent talks but was very successful in drawing a Task Forces and Review Committees continue their con- sizeable complement of school teachers. sultations. The final task force will focus on Office Strategies As an outcome of its deliberations at the current meeting, for the Executive Office. the Education Committee will recommend to the CMS Exec- Everyone is invited to participate in the Math 2000 activi- utive that the CMS appoint a Camel Education Page Editor. ties, specifically the two CMS meetings being held in Hamil- The Committee is also working on creating a registry of ed- ton in June and Vancouver in December. To reflect the special ucation speakers to be a resource in planning math education nature of the event, the June meeting has been named “Math activities, including CMS and outreach activities. Further, the 2000”. Committee will convey to the membership, at a later date, the name of recipient of the 1999 Adrien Pouliot award. 5. Nominating Committee Report Wright presented the Tellers’ Report. He proposed a vote Electronic Services Committee: David Bates visited the of thanks to all the outgoing members of the Executive and Committee to consult for the Task Force on Finances and Fund Board of Directors. Raising. Goodaire reported that Camel might serve as a con- G–99–2 MOTION (Nominating Committee) duit between students looking for goods and companies. He That the Tellers’ Report for the 1999 Election be ac- was looking for input. cepted. Carried Unanimously Loki Jorgenson,¨ the Camel Director, is heading up a revi- sion of the Camel site. Many improvements are planned. 6. Treasurer’s Report Finance Committee: The Finance Committee was very Sherk presented the Treasurer’s Report and noted that pleased to see the CMS reach the goal set for the beginning of there was a good surplus. There was some discussion re- the Endowment Grants Programme. Regarding investments, garding whether the required financial statements could be the CMS has adopted a passive management approach. In- produced by a Certified General Accountant, as opposed to a vestments have been moved to Toronto Dominion Quantita- Chartered Accountant. tive Capital. G–99–3 MOTION (Board of Directors) That the Audited Statement for the financial year Fund Raising Committee: The Committee is overseeing ended December 31, 1998 be accepted. Carried Unani- three campaigns. The first focuses on government sponsor- mously ship. Last year, 8 ministries provided funds. The second cam- G–99-4 (Board of Directors) paign focuses on corporate sponsorships. Wright has devoted That the Treasurer’s Report for the financial year much time on this, with good results. Of special note here is ended December 31, 1998 be accepted. Carried Unani- the new partnership with Imperial Oil on the math camps. The mously third campaign focus on membership. The AMS reciprocity G–99–5 MOTION (Board of Directors) agreement is being implemented in 2000 and will no doubt That the firm of Raymond Chabot Grant Thornton be be at the center of the 2000 campaign. Initiatives to attract appointed as auditors for the financial year ending De- new membership include offering two-year free membership cember 31, 1999. Carried Unanimously to new faculty members. Chairs will be asked to nominate The market value of the investments has now reached new faculty members. the level of $1,500,000 and this makes the new Endowment Human Rights Committee: The Board recently ap- Grants Competition possible. proved a Statement on the Employment of Young Mathe- maticians. This will be circulated to departments and faculty 7. Executive Director and Secretary’s Report associations. Wright announced that the Executive Office would be International Affairs Committee: There was no report closed for two weeks in August. from this Committee. 8. Reports from Committees Mathematical Olympiads Committee: The Commit- G–99–6 MOTION (Board of Directors) tee oversees several competitions. The details on this year’s That the Annual Report to the Members be accepted, COMC were reported at the December meetings. Tingley re- as amended. Carried Unanimously ported that the 1999 CMO was written by 81 students in late Education Committee: The 1998 activities are described April. Next year, the CMO will be housed at Simon Fraser in the Annual Report to Members. Orzech expanded on Edu- University. cation Committee activities and initiatives since the last report Tingley announced the 1999 IMO team. This year’s IMO to the members. will be held in Romania in July. The CMS adjudicated and presented three prizes at the The Correspondence Programme continues to be coordi- Canada Wide Science Fair held in Edmonton this May. nated by Ed Barbeau.

24 NOTES de la SMC OCTOBER/OCTOBRE

In 1998 a National Math Camp was held in Waterloo in ics now includes 58 names and she encouraged organizers to June. This year, a similar one will be organized by Tom Grif- refer to it when seeking speakers. fiths, Peter Crippin and Richard Hoshino. A poster is being designed to present the Krieger-Nelson Several regional camps are being held across the country. winners. Next year, as part of the 2000 activities, it is hoped to expand Student Committee: Charbonneau reported that the this to even more sites. CUMC was held directly preceding this meeting and he en- Corporate participation in the Olympiads programme is couraged departments to get their students involved. The new increasing steadily. Sun Life has been a long standing con- Student Committee will be trying to inform the students across tributor and has recently increased its donation significantly. Canada of what is available to them and to help them not feel Imperial Oil has now become the Title Sponsor for the Esso isolated. Math Camps. Tingley invited members to review the materi- A new web page is being developed but current infor- als on Camel. mation on CUMC is available at www.cumc.math.ca. The Nominating Committee: The Committee was pleased Student Committee is looking for a webmaster and invited to announced that members had been appointed to the newly suggestions. established Endowment Grants Committee and Student Com- It was suggested that Camel might include a directory of mittee. Other upcoming positions are being considered by the graduating students. Committee. At the Math 2000 meeting, the CUMC programme and the Publications Committee: The Committee confirmed the CMS programme will provide more information so that mem- appointment of Robert Quackenbush as the new Managing bers of the CMS might attend CUMC talks. The conference Editor. The work on transferring the Advanced Book Series poster will also be distributed on a wider basis. from Wiley to Springer is almost complete. The work on the interior redesign of the journals is now complete and the 9. Other Business production schedules are almost back on track. Ed Barbeau made a plea for articles for the Education The AMS has asked the CMS to change the direction of Section of the Notes. the Conference Proceedings Series. The Publications Com- G–99–7 MOTION (Wright/Charbonneau) mittee is considering this matter. That the Annual General Meeting express a special vote of thanks to Katherine Heinrich for her tireless work Research Committee: Kane announced the sites of the during her six years on the Executive Committee. future semi-annual meetings for the next three years. Carried Unanimously Women in Mathematics Committee: Hyndman reported that the Committee continues to increase the number 10. Adjournment of sites on Camel. The Directory of Women in Mathemat- The meeting adjourned at 5:20 p.m.

COMMITTEES AT WORK

ICMI Update Laval University, where for example the ICMI Bulletin will Peter Fillmore, Dalhousie University be edited and produced for the next four years. Canada’s new Chair, International Affairs Committee representative to ICMI is Eric Muller (Brock). The recent appearance of ICMI Bulletin No.46 (June 1999) What does ICMI do? Its most important activities provides an opportunity to remind readers of the important are the International Congresses on Mathematical Educa- work ICMI is doing and of Canadian involvement in that work. tion (ICME) and the ICMI Studies. The congresses are Further information on topics touched on in this report may be held at four-year intervals, between the International Con- found at http://elib.zib.de/imu/icmi/bulletin/, which contains gresses of Mathematicians (ICM). The next, ICME-9, will the complete text of the Bulletin. take place July 31 to August 6, 2000, at the Nippon Con- ference Centre, near Tokyo. The ﬁrst announcement is avail- What is ICMI? The International Commission on Mathe- able at http://www.ma.kagu.sut.ac.jp/icmi9/ or by e-mail from matical Instruction is the arm of the IMU (International Math- [email protected]. Readers may recall that a very ematical Union) that is concerned with matters educational. successful earlier congress, ICME-7, was held at Laval and A new executive committee was elected at the IMU General chaired by Bernard Hodgson. Assembly in August 1998, including Hyman Bass (Columbia) as president and Bernard Hodgson (Laval) as secretary. This Nine ICMI studies have been completed and a further means in particular that the Secretariat is now located at three are in progress. A typical study consists of a study

25 OCTOBER/OCTOBRE CMS NOTES conference followed by a study volume. The volumes are Other ICMI activities include four afﬁliated study groups published by Kluwer, the most recent being [Perspectives on (history, women, psychology and competitions) and the Soli- the Teaching of Geometry for the 21st Century] (1998). A darity Programme (to assist the development of mathematics new study was recently launched, on the topic The Future of education in countries where there is a need that justiﬁes in- the Teaching and Learning of Algebra]. ternational assistance).

26 NOTES de la SMC OCTOBER/OCTOBRE

CALL FOR NOMINATIONS / APPEL DE CANDIDATURES Editors-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- Comite´ des publications de la SMC sollicite des mises en dian Mathematical Bulletin will end December 31, 2000. The candidatures pour les prochains redacteurs-en-chef´ pour un Publications Committee of the CMS now invites nominations mandat de cinq ans. for the next Editors-in-Chief to serve a ﬁve year term. Les mises en candidature doivent inclure une lettre Applications should consist of a formal letter of applica- formelle et les el´ ements´ suivants: tion and include the following: • • A curriculum vitae Un curriculum vitae

• An expression of views of the publication indicating if • L’expression de votre opinion sur la publication indi- any changes in direction or policy are contemplated quant si des changements de directions ou de politiques • Since editorial responsibilities often necessitate a less- sont envisages´ ening of responsabilities in an individual’s normal • work, applicants should indicate that they have the sup- Puisque les responsabilites´ de redaction´ necessitent´ port of their university department and, in particular, of souvent une reduction´ dans la charge normale de tra- their head of department. vail, les candidats devraient indiquer qu’ils(elles) ont l’appui de leur departement´ et en particulier, de leur The Publications Committee will communicate its recom- chef de department.´ mendation to the Executive Committee of the CMS in April 2000. Any input from the mathematical community concern- Le Comite´ des publications transmettra ses recomman- ing this important selection process is welcome. dations au Comiteex´ ecutif´ de la SMC en avril 2000. Les Applications (with supporting material) and/or comments commentaires de la communaute´ mathematique´ au sujet de should be sent to the address below. The deadline for the re- cette importante selection´ sont bienvenus. ceipt of applications is November 15, 1999. Les mises en candiatures (avec material´ a` l’appui) et/ou commentaires devraient etreˆ achemines´ a` l’adresse ci- Le mandat des redacteurs-en-chef´ actuels du Bulletin cana- dessous. L’ech´ eance´ pour la reception´ des mises en candida- dien de mathematique´ prendra ﬁn le 31 decembre´ 2000. Le ture est le 15 novembre 1999.

Editors-in-Chief - CMS Notes / Redacteurs-en-chef´ - Notes de la SMC The term of office of the present Editors-in-Chief of Le mandat du redacteurs-en-chef´ actuels des Notes de la the CMS Noteswill end December 31, 2000. The Publica- SMC prendra fin le 31 decembre´ 2000. Le Comite´ des publi- tions Committee of the CMS invites applications for the next cations de la SMC sollicite les mises en candidature pour le Editor(s)-in-Chief to serve for a five year term. prochain redacteurs-en-chef´ pour un mandat de cinq ans. Applications should consist of a formal letter of applica- Les mises en candidature doivent inclure une lettre tion with curriculum vitae. formelle ave curriculum vitae. The Publications Committee will communicate its rec- Le Comite´ des publications transmettra ses recommenda- ommendation to the Executive Committee of the CMS in tion au Comiteex´ ecutif´ de la SMC en avril 2000. Les candi- April 2000. Applications and/or comments should be sent, datures et/ou commentaires devraient etreˆ achemines,´ avant by November 15,1999 to the address below. le 15 Novembre 1999 a` l’adresse ci-dessous.

Address for Nominations / Addresse de mise en candidatures: James A. Mingo Chair / President´ CMS Publications Committee / Comite´ des publications de la SMC Department of Mathematics and Statistics Queen’s University Kingston, Ontario K7L 3N6

27 OCTOBER/OCTOBRE CMS NOTES

THE INSTITUT DES SCIENCES MATHEMATIQUES´ ET LE CENTRE DE RECHERCHES MATHEMATIQUES´ JOINT POST DOCTORAL FELLOWSHIP / BOURSES POSTDOCTORAL CRM-ISM

The Institut des sciences mathématiques (ISM) and the Centre de recherches mathématiques (CRM) are inviting applications for their joint postdoctoral fellowship program starting in approximately September 2000. The annual stipend is Cdn. $32, 000 for one year, renewable for a second year. The ISM coordinates the graduate programs of six Québec universities (Concordia, Laval, McGill, Sherbrooke, Université de 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 and Computational Mathematics, Mathematical Physics, Probability: Theory and Applications, Decision Theory and Mathematical Statistics, and Category Theory and Applications. The CRM is a national research center in the mathematical sciences. Its ongoing areas of research include: algebra and combinatorics, analysis, differential equations and approximation theory, geometry and topology, numerical analysis, optimisation and multidisciplinary research, mathematical physics, probability and statistics, and dynamical systems. Each year, the CRM organizes a wide range of events attracting participants from around the world. The main themes for 2000-2001 are Mathematical Methods in Biology and Medicine, and Symplectic Geometry and Topology and Gauge Theory. In 2001-2002 the main themes will be Groups and Geometry. However, high-quality applications in all fields of interest to the CRM or to the ISM are welcome. Applications must arrive at the ISM by Friday, January 7, 2000. The following documents are required: a curriculum vitae, a statement of research, and three letters of recommendation. Please indicate in your application which ISM program best represents 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 candidatures dans le cadre de leur programme conjoint de bourses postdoctorales débutant approximativement en septembre 2000. Les bourses 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 la décision 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 autres sur les domaines suivants: l’algèbre et la combinatoire, l’analyse, les équations différentielles et la théorie de l’approximation, la géométrie 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’activités scientifiques impliquant une participation internationale. Les thèmes principaux de l’année 2000-2001 seront d’une part les méthodes mathématiques en biologie et en médicine et d’autre part la topologie et la géométrie symplectiques ainsi que la théorie de jauge. Les thèmes de l’année 2001-2002 porteront sur les groupes et la géométrie. Cependant, toute candidature méritoire touchant 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 tard le vendredi 7 janvier 2000. Les documents suivants doivent être joints: un curriculum vitae, un résumé des intérêts de recherche, et trois lettres de recommandation. Vous êtes prié d’indiquer sur votre demande lequel des programmes de l’ISM représente le mieux vos intérets de recherche. Une adresse électronique et un numéro de télécopieur (si disponible) doivent être inclus avec toute correspondance. Les personnes intéressées sont encouragées à contacter les professeurs avec qui elles aimeraient travailler. Les candidatures doivent être envoyées à:

Professor François Lalonde, Director / Directeur Institut des sciences mathématiques Case postale 8888, Succursale Centre-Ville Montréal (Québec), Canada, H3C 3P8 Fax / Télécopieur : (514) 987-8935 E-mail / Courrier électronique : [email protected]

28 NOTES de la SMC OCTOBER/OCTOBRE

UNIVERSITY OF TORONTO – TORONTO, ONTARIO DEPARTMENT 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 and Geometry. 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. 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 to Mathematics. Salary commensurate with experience. Applicants should send their complete C.V. including a list of publications, a short statement describing their research programme, and all appropriate material about their teaching. They should also arrange to have at least four letters of reference sent directly to:

Search Committee Department of Mathematics University of Toronto 100 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 that applicants submit the electronic application form which is available from our World Wide Web Employment Opportunities page: http://www.math.toronto.edu/jobs/ To ensure full consideration, this information should be received by 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.

UNIVERSITY OF TORONTO – TORONTO, ONTARIO DEPARTMENT OF MATHEMATICS Tenure-Stream Appointment in Applied Mathematics

The Department of Mathematics, University of Toronto solicits applications for a tenure-stream appointment for a mathematician 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. 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 to mathematics. Salary commensurate with experience. Applicants should send their complete C.V. including a list of publications, a short statement describing their research programme, and all appropriate material about their teaching. They should also arrange to have at least four letters of reference sent directly to:

Search Committee Department of Mathematics University of Toronto 100 St. George Street, Room 4072 Toronto, Canada M5S 3G3

29 OCTOBER/OCTOBRE CMS NOTES

UNIVERSITY OF TORONTO – TORONTO, ONTARIO DEPARTMENT OF MATHEMATICS Tenure-Stream Appointment in Applied Mathematics – Computational Science The Department of Mathematics, University of Toronto solicits applications for a tenure-stream appointment for a mathematician 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. 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 to mathematics. Salary commensurate with experience. Applicants should send their complete C.V. including a list of publications, a short statement describing their research programme, and all appropriate material about their teaching. They should also arrange to have at least four letters of reference sent directly to: Search Committee Department of Mathematics University of Toronto 100 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 that applicants submit the electronic application form which is available from our World Wide Web Employment Opportunities page: http://www.math.toronto.edu/jobs/ To ensure full consideration, this information should be received by 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 from qualiﬁed women or men, members of visible minorities, aboriginal peoples and persons with disabilities.

UNIVERSITY OF TORONTO – TORONTO, ONTARIO DEPARTMENT OF MATHEMATICS Limited Term Assistant Professorships The Department invites applications for one or more limited term Assistant Professorships which may, subject to budgetary approval, 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 research programme, and all appropriate material about their teaching. They should also arrange to have at least three letters of reference sent directly to: Search Committee Department of Mathematics University of Toronto 100 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 that applicants 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 received by 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 from qualiﬁed women or men, members of visible minorities, aboriginal peoples and persons with disabilities.

30 NOTES de la SMC OCTOBER/OCTOBRE

UNIVERSITY OF TORONTO – SCARBOROUGH, ONTARIO PHYSICAL SCIENCES DIVISION Tenure-Stream Appointment in Mathematics

The Physical Sciences Division, University of Toronto at Scarborough invites applications for a tenure-stream appointment in 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 will be 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 to mathematics. Salary commensurate with experience. Applicants should send their complete C.V. including a list of publications, a short statement describing their proposed research programme, and all appropriate material about their teaching background. Applicants should also arrange for at least four 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 Division University of Toronto at Scarborough 1264 Military Trail Scarborough, 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 as well. To ensure full consideration, this information should be received by December 1, 1999.

Any inquiries about applications should be addressed to: Professor R.-O. Buchweitz Associate Chair Hiring Department of Mathematics University of Toronto 100 St. George Street, Room 4072 Toronto, Ontario M5S 3G3

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

CMS MEMBERSHIP ... ADHESION´ A` LA SMC ... The 2000 Membership Notices have Les avis d’adhesion´ 2000 etait´ postes.´ been mailed. Don’t forget to renew N’oubliez pas de renouveller votre your membership. adhesion.´

31 OCTOBER/OCTOBRE CMS NOTES

UNIVERSITE´ McGILL UNIVERSITY – MONTREAL,´ QUEBEC´ DEPARTMENT OF MATHEMATICS AND STATISTICS DEPARTEMENT´ DE MATHEMATIQUES´ ET DE STATISTIQUES

The Department of Mathematics and Statistics of McGill Uni- Le departement´ de mathematiques´ et de statistique de versity invites applications for a tenure track position in statis- l’Universite´ McGill cherche a` pourvoir un poste en statistics at the assistant professor level. tiques au niveau de professeur adjoint, menant a` la perma- A Ph.D. degree in statistical science is essential. Preferred nence. areas of specialization are computational statistics, sample Un doctorat en statistique est essentiel. Les domaines pri- surveys and time series analysis, although not exclusively so. oritaires sont l’echantillonnage,´ la statistique informatique Preference will be given to applicants with a strong theoretical ou l’analyse des series´ chronologiques; ces priorites´ ne sont background in statistics, whose work is driven by applications. pas exclusives. La preference´ sera accordee´ aux candidats ayant une forte formation theorique´ en statistiques et dont les The appointment is to begin July 1, 2000. travaux sont motives´ par des applications. La date d’entree´ en fonction sera le premier juillet 2000. Applicants are expected to have demonstrated the capacity for independent research of excellent quality. Selection criteria Les candidats devront avoir demontr´ e´ leur capacite´ de mener a` include research accomplishments, as well as potential con- bien une recherche independante´ et de haut niveau. Parmi les tributions to the research interests of the Department and to its criteres` de selection´ des candidats figurent leurs realisations´ educational programs at both the undergraduate and graduate en recherche, ainsi que leurs contributions potentielles aux levels. activites´ de recherche du departement´ et a` ses programmes d’enseignement a` tous les cycles. Applications, with a curriculum vitae, a list of publications, a research proposal, an account of teaching experience and the Les demandes, comprenant un curriculum vitae, une liste de names, phone numbers and e-mail addresses of at least four publications, un aperçu des projets de recherches, une descrip- references (with one addressing the teaching record) should tion de l’experience´ acquise en enseignement et les noms, be sent to: numeros´ de tel´ ephone´ et adresses electroniques d’au moins quatre repondants´ (dont un pourra commenter les qualites´ d’ enseignant du candidat) doivent etreˆ envoyees´ a:` Professor K. GowriSankaran, Chair Department of Mathematics and Statistics Professeur K. GowriSankaran, Directeur McGill University Departement´ de mathematiques´ et statistique 805 Sherbrooke Street West Universite´ McGill Montreal, Quebec, Canada H3A 2K6 805, rue Sherbrooke ouest Montreal´ (Quebec)´ Canada H3A 2K6 Candidates must arrange to have the letters of recommendation sent directly to the above address. Candidates are also Les candidats doivent demander a` leurs repondants´ d’envoyer encouraged to include copies of up to 3 selected publications leurs lettres de recommandation directement a´ l’adresse ci- with their application. dessus. Ils sont egalement´ invites´ a´ inclure en annexe a´ leur To ensure full consideration, applications must be received by demande des copies de trois de leurs publications au plus. November 30, 1999, although the search will continue until Pour etreˆ prises pleinement en consideration,´ les demandes the position is filled. devront etreˆ reçues le 30 novembre 1999 au plus tard. Les McGill University is committed to equity in employment and recherches se poursuivront jusqu’a ce que le poste soit comble.´ in accordance with Canadian immigration requirements, pri- ority will be given to Canadian citizens and permanent resi- L’Universite´ McGill souscrit a l’equit´ e´ en matiere` d’emploi dents of Canada. et, conformement´ alal` egislation´ canadienne en matiere` d’immigration, accorde la priorite´ aux citoyens canadiens et aux residents´ permanents du Canada.

32 NOTES de la SMC OCTOBER/OCTOBRE

UNIVERSITE´ McGILL UNIVERSITY – MONTREAL,´ QUEBEC´ DEPARTMENT OF MATHEMATICS AND STATISTICS DEPARTEMENT´ DE MATHEMATIQUES´ ET DE STATISTIQUES

Consultant en statistique / Statistical Consultant

Le departement´ de mathematiques´ et statistique de l’Universite´ McGill est a` la recherche d’un consultant pour co- gerer´ son nouveau service de consultation statistique (SCS). The Department of Mathematics and Statistics at McGill Uni- Dans un premier temps ce poste est prevu´ pour une periode´ de versity seeks a person to co-manage a newly established sta- trois ans. L’entree´ en fonction est prevue´ le 1er janvier 2000. tistical consulting service. The initial three-year appointment Titre : Charge´ d’enseignement is to begin no later than January 1, 2000. Fonctions : S’occuper, avec un autre consultant du Centre Title: Faculty Lecturer de calcul de McGill, de l’exploitation quotidienne du service Duties: Be responsible, together with a co-consultant from de consultation statistique (SCS), sous la supervision du di- the McGill Computing Centre, for the day-to-day running of recteur du service. Animer des seances´ d’information sur le the Statistical Consulting Service (SCS), under the supervi- campus sur les services offerts par le SCS. Offrir des consul- sion of the Director of the SCS. Provide on-campus informa- tations aux professeurs et etudiants´ et executer´ des analyses tion sessions on the services offered by the SCS. Consult with statistiques, rediger´ des rapports et collaborer alar` edaction´ clients including both faculty and students and, where agreed des demandes de subvention et autres projets de recherche. upon, perform statistical analyses, write reports, and collab- Administrer la facturation des services. Donner au moins orate on grant proposals and other research projects. Help to un cours de statistique de 1er cycle par an. Participer a` administer the billing of clients. Teach at least one undergrad- l’organisation d’un cours de consultation statistique au niveau uate course in statistics per year. Assist in the running of a des 2e/3e cycles. Aider les etudiants´ de 2e/3e cycle en statis- graduate statistical consulting course. Help graduate students tique dans leur these` ou projet, au besoin. Animer a` l’occasion in statistics with their theses or projects, where appropriate. des ateliers sur les logiciels et les statistiques appliquees.´ Lire Give occasional software and applied statistics workshops. les publications sur les statistiques, participer a` des confer- Read the current statistics literature and attend scientific con- ences scientifiques de maniere` a` pouvoir offrir des conseils a` ferences in order to provide clients with up-to-date advice. jour a` l’ensemble de la clientele.` Qualifications: A Master’s degree in statistical science is Qualifications : Maên statistique est imperative.´ Au moins essential. At least two years experience as an applied statisti- deux ans d’experience´ dans un poste de statisticien applique´ cian and data analyst. Familiarity with the standard statistical ou d’analyste de donnees.´ Connaissance des logiciels statis- software packages and with both parametric and nonparamet- tiques standards, des methodes´ multivariees´ et univariees´ ric univariate and multivariate methods. Knowledge of time parametriques´ et non parametriques.´ La connaissance de la series methods an asset. Effective oral, written, communica- methode´ des series´ chronologiques est un atout. Bon sens de tion, and interpersonal skills. la communication orale et ecrite´ et des relations interperson- Deadline: For full consideration applications should be re- nelles. ceived by November 1, 1999, although the search will con- Date limite : Veuillez faire parvenir votre candidature avant tinue until the position is filled. Candidates must arrange le 1er novembre 1999. La selection´ se poursuivra neanmoins´ for three letters of recommendation to be sent directly to the jusqu’a` ce que le poste soit pourvu. Les candidats doivent de- address below. mander a` trois repondants´ d’envoyer leurs lettres de recommandation directement a` l’adresse ci-dessous. Send your application to: Veuillez faire parvenir votre candidature a:` Professor K. GowriSankaran, Chair Professeur K. GowriSankaran, Directeur Department of Mathematics and Statistics Departement´ de mathematiques´ et statistique McGill University Universite´ McGill 805 Sherbrooke Street West 805, rue Sherbrooke ouest Montreal, Quebec, Canada H3A 2K6 Montreal´ (Quebec)´ Canada H3A 2K6 Email: [email protected] Courriel: [email protected] McGill University is committed to equity in employment and L’Universite´ McGill souscrit a l’equit´ e´ en matiere` d’emploi in accordance with Canadian immigration requirements, pri- et, conformement´ alal` egislation´ canadienne en matiere` ority will be given to Canadian citizens and permanent resi- d’immigration, accorde la priorite´ aux citoyens canadiens et dents of Canada. aux residents´ permanents du Canada. 33 OCTOBER/OCTOBRE CMS NOTES

CALENDAR OF EVENTS / CALENDRIER DES EV´ ENEMENTS´

OCTOBER 1999 OCTOBRE 1999 4–8 Canadian Annual Operator Algebra Symposium (Fields Institute, Toronto, Ontario) [email protected]; 11–24 Workshops for Mathematics and its Applications (Uni- [email protected] versity of Minnesota, Minneapolis,MN) http://www.ima.umn.edu/reactive/fall/msl.html 8–9 Symposium on the Legacy of John Charles Fields (The Royal Ontario Museum, Toronto); a WMY2000 event 16–17 West Coast Operator Algebra Symposium (University www.ﬁelds.utoronto.ca of Victoria, Victoria, BC) www.math.uvic.ca/ wcoas 10–13 MATH 2000 (McMaster University, Hamilton, On- tario) NOVEMBER 1999 NOVEMBRE 1999 Paticipating Societies include the Canadian Mathematical So- 14–18 International Conference on Mathematics Education ciety (CMS), the Canadian Applied and Industrial Mathemat- into the 21st Century (Cairo, Egypt) ics Society (CAIMS), the Canadian Operational Research So- Dr. A Rogerson: [email protected] ciety (CORS), the Canadian Symposium on Fluid Dynamics (CSFD), the Canadian Society for the History and Philosophy 20–21Canadian Department Chairs’ Meeting, CRM, Mon- of Mathematics (CSHPM) and the Canadian Undergraduates treal Mathematics Conference (CUMC). A WMY2000 event [email protected] Monique Bouchard: [email protected] 29–Dec. 3 Group Theory and Computation (University of Sydney, Australia) 12–15 Integral Methods in Science and Engineering (Banff, http://math.auckland.ac.nz/conference/groups-11-1999 Alberta) [email protected]

DECEMBER 1999 DECEMBRE´ 1999 JULY 2000 JUILLET 2000 2–5 The Future of Mathematical Communication, 1999, 10–14 Third European Congress of Mathematics MSRI (Berkeley, California) (Tokyo/Makuhari) http://www.msri.org/activities/events/9900/fmc99/ www.ma.kagu.sut.ac.jp/ icme9/ 11–13 CMS Winter Meeting / Reunion´ d’hiver de la SMC 11–25 41st International Mathematical Olympiad (Korea) (Universite´ de Montreal)´ 31–Aug 7 International Congression the Teaching of Mathe- http://cms.math.ca/CMS/Events/ matics ICME-9 (Barcelona, Spain) [email protected]; http://www.si.upc.es/3ecm/ JANUARY 2000 JANVIER 2000 ˆ 19–22 Joint Mathematics Meetings, including the 106th An- AUGUST 2000 AOUT 2000 nual Meeting of the AMS (Washington DC), 7–12 AMS Meeting (Los Angeles); a WMY2000 event a WMY2000 event www.ams.org/meetings/ www.ams.org/meetings/

MARCH 2000 MARS 2000 SEPTEMBER 2000 SEPTEMBRE 2000 6–10 Fourth International Conference on Operations Re- 22–24 American Mathematical Society Central Section Meet- search (Havana, Cuba) [email protected] ings (University of Toronto) http://www.ams.org/meetings/ MAY 2000 MAI 2000 5–7 Uniﬁed Congress of Mathematical Associations and DECEMBER 2000 DECEMBRE´ 2000 Groups of Quebec (Universite´ Laval), a WMY2000 event 10–12 CMS Winter Meeting / Reunion´ d’hiver de la SMC [email protected] (University of British Columbia, Vancouver, B. C.) Monique Bouchard: [email protected] JUNE 2000 JUIN 2000 Canadian Mathematics Education Study Group Meeting JUNE 2001 JUIN 2001 (UQAM, Montreal) Dates to be announced CMS Summer Meeting / Reunion´ d’et´ edelaSMC´ 4–7 Annual Meeting of the Statistical Society of Canada (Ot- (University of Saskatchewan, Saskatoon, Saskatchewan) tawa, Ontario) Andre´ Dabrowski: [email protected] Monique Bouchard: [email protected]

34 NOTES de la SMC OCTOBER/OCTOBRE

Canadian Mathematics Education Study Group Meeting DECEMBER 2002 DECEMBRE 2002 (University of Alberta, Edmonton) Annual Meeting of the Statistical Society of Canada CMS Winter Meeting / Reunion´ d’hiver de la SMC (Vancouver, British Columbia) (University of Ottawa / Universite´ d’Ottawa, Ottawa, Ontario) DECEMBER 2001 DECEMBRE 2001 Monique Bouchard: [email protected] CMS Winter Meeting / Reunion´ d’hiver de la SMC (York University, Toronto, Ontario) JUNE 2003 JUIN 2003 Monique Bouchard: [email protected]

JUNE 2002 JUIN 2002 CMS Summer Meeting / Reunion´ d’et´ edelaSMC´ (University of Alberta, Edmonton, Alberta) CMS Summer Meeting / Reunion´ d’et´ edelaSMC´ Monique Bouchard: [email protected] (Universite´ Laval, Quebec,´ Quebec)´ Monique Bouchard: [email protected] DECEMBER 2003 DECEMBRE 2003 AUGUST 2002 AOUTˆ 2002 20–28 International Congress of Mathematicians, CMS Winter Meeting / Reunion´ d’hiver de la SMC (Beijing, China) (Simon Fraser University, Burnaby, British Columbia) [email protected]; http://icm2002.org.cn/ Monique Bouchard: [email protected]

RATES AND DEADLINES / TARIFS ET ECH´ EANCES´

Net rates/Tarifs nets Institutional Members Corporate Members Others Membres institutionels Membres organisationnels Autres Full Page $ 200 $ 375 $ 500 1/2 Page $ 120 $ 225 $ 300 1/4 Page $70 $ 130 $ 175 Inserts: maximum 4 pages $ 160 $ 300 $ 400 Surcharges apply for prime locations - contact [email protected] Des supplements´ sont applicables pour des places de choix - communiquer avec [email protected] Issue/Numero:´ Deadline/Date limite: February/fevrier´ December 15 decembre´ March/mars January 15 janvier April/avril February 15 fevrier´ May/mai March 15 mars September/septembre July 15 juillet October/octobre August 15 aoutˆ November/novembre September 15 septembre December/decembre´ October 15 octobre Max. page size/Taille max. des pages: Back page/4e de couverture: 7.5 x 8.5 in./pouces Inside page/page interieure:´ 7.5 x 10 in.pouces

The CMS Notes is mailed in the ﬁrst week of the issue month. Subscription to the Notes is included with the CMS membership. For non-CMS members, the subscription rate is $40 (CDN) for subscribers with Canadian addresses and $40 (US) for subscribers with non-Canadian addresses. Les Notes de la SMC sont postees´ la premiere` semaine du mois de parution. L’adhesion´ a` la SMC comprend l’abonnement aux Notes de la SMC. Le tarif d’abonnement pour les non-membres est de 40 $ CAN si l’adresse de l’abonne´ est au Canada et de 40 $ US si l’adresse est a` l’etranger.´

If undelivered, please return to: Si NON-LIVRE,´ priere` de retourner a:` CMS Notes de la SMC 577 King Edward, C.P. 450, Succ. A Ottawa, Ontario, K1N 6N5, Canada