Newsletterfor the Mathematical SciencesInstitutePacific http://www.pims.math.capims@pims.math.capiWinter 2001Vol. 5 Issue 1TheContentsPIMS Prizes for 2000 . . . . . . . . . . . . . . . . . . 1,14-16Math. Sciences at the Grad. Level . . . . . . 1,9–11Director’s Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2MITACS News . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–4Industrial Activities . . . . . . . . . . . . . . . . . . . . . . . . . .5Scientific Activities . . . . . . . . . . . . . . . . . . . . . . . . . . 6Thematic Programme in PDE . . . . . . . . . . . . . . . .6International Activities . . . . . . . . . . . . . . . . . . . . . . 7Strings and D-branes . . . . . . . . . . . . . . . . . .12,13,17Mathematics of Voting . . . . . . . . . . . . . . . . . . . . . . 17The Amazing Number pi . . . . . . . . . . . . . . 18–20,25PIMS Graduate Information Week . . . . . . . . . . 23Women and Mathematics . . . . . . . . . . . . . . . .24–25Mathematics on Stage . . . . . . . . . . . . . . . . . . . . . . . 26Mathematics is Everywhere . . . . . . . . . . . . . . . . . 27Contact List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28PIMS is supported by• The Natural Sciences and EngineeringResearch Council of Canada• The Alberta Ministry of Innovationand Science• The British Columbia Information,Science and Technology Agency• Simon Fraser University, The Univer-sity of Alberta, The University of BritishColumbia, The University of Calgary,The University of Victoria, The Univer-sity of Washington, The University ofNorthern British Columbia and The Uni-versity of Lethbridge.Bob Russell, New PIMS SiteDirector at SFUIn December, Bob Russellwas appointed as the newSite Director for PIMS atSFU. He replaces PeterBorwein, who is currentlyon a sabbatical leave. Bobhas been active in PIMSfor some time. Currently,he is a member of the Lo-cal Organizing CommitteePlease see Bob Russell, page 2.PIMS Awards Its Prizes for 2000George Bluman receives PIMS Education Prize from Martin Taylor (VPResearch, Univ. of Victoria) and Ken Foxcroft (TD Securities).Please see PIMS Awards, page 14.Entering Mathematical Sciencesat the Graduate LevelAddress given by Robert V. Moody, Univ. of Alberta, at the recentPIMS Graduate Information Week.A little while ago I came across a definition of mathematics that Ilike and that seems quite appropriate here. However, let me startwith the usual sort of definition.According to my OxfordDictionary, mathematics is: theabstract deductive science of num-ber, quantity, arrangement, andspace. The authors are obviouslynot content with this since thereis an attached note: Mathematicsis customarily divided into puremathematics, those topics studiedin their own right, and appliedmathematics, the application ofmathematical knowledge in science,Penrose Tiling(Image courtesy of Uwe Grimm.)Please see Mathematical Sciences, page 9.The Paci£c Institute for the Mathematical SciencesDirector’s NotesNassif Ghoussoub, FRSCQuelle anne´e sensationelle pourles mathe´matiques Canadi-ennes. Notre communaute´a re´agi avec vigueur et pas-sion a` l’appel de l’UnionMathe´matique Internationaleet de L’UNESCO. A` com-mencer par la campagne desposters dans le me´tro deMontre´al dirige´e par le Centrede Recherches Mathe´matiques,la grande re´union de la Socie´te´Mathe´matique du Canada a`Hamilton, le de´file´ e´poustouflant des medailles Fields auFields Institute et bien suˆr, the myriad of educationalinitiatives that PIMS spearheaded all year long: TheMathematics is Everywhere campaign on buses inVancouver and Victoria, Hypatia’s Street Theatrestaged at UBC’s Frederic Wood Theater, and the Pi inthe Sky magazine distributed to every school in Albertaand BC.This year is likely to be even more exciting for worldmathematics. Our efforts toward establishing an Interna-tional Research Station for Mathematical Innova-tion and Discovery in the Canadian Rockies are well un-der way. More details will be given in the next newsletterbut I can report that most hurdles regarding internationalpartnerships and funding are being resolved. We expect ajoint announcement from the various partners well beforethe end of this year.A few words to our young future leaders, especially nowthat a change of the guard is already under way. Being amilitant for mathematics in Canada is not a crime and—unlike Hypatia— there is no danger of being crucified.To the contrary, gratifications abound and there are somesecret pleasures.Indeed, there is nothing more rewarding than the en-lightened smile of UBC’s provost, Barry McBride whenyou hit him with a good idea on how to promote inter-disciplinary research through the mathematical sciences.There is nothing more pleasureful than seeing Univ. ofAlberta’s Dean of Science, Dick Peter, a distinguished lifescientist, lecturing policy makers on the importance ofmathematics. There is nothing more gratifying than wit-nessing Univ. of Alberta’s VP-Research, Roger Smith andUniv. of Calgary’s acting VP-research, Keith Archer, bothsocial scientists, passionately and convincingly describingthe “Oberwolfach concept” to the Alberta minister for In-novation and Science so that we can get one going in theRockies. There is nothing more gratifying than watchingNSERC’s Director Danielle Menard and Nigel Lloyd tellingsenior NSF officials about the admiration they have for theefforts and successes of the Canadian mathematical com-munity. We will always be grateful to them for helping usarrange such a meeting and also for accompanying us toWashington DC so as to jumpstart a major Canada-UScollaborative effort.These are true pleasures my friends and these peopleand many others like them should always have the grati-tude of the mathematical research community.These notes are excerpted from the Director’s speech atthe CMS meeting in Vancouver, December 10–12, 2000Bob Russell, PIMS-SFU Site DirectorContinued from page 1.and the Scientific Committee of the PIMS-sponsoredInternational Conference on Scientific Com-putation and Differential Equations for 2001.Furthermore, along with Keith Promislow, he co-sponsors the PIMS PDF Ricardo Carretero. Bobis also a member of the MITACS research team onMathematical Modelling and Scientific Computation.Bob Russell has been Professor of Mathematicsand Computing Science at Simon Fraser Universitysince 1981. He is currently a member of the SFUBoard of Governors and Senate. He is also the Di-rector of the proposed PIMS Centre for ScientificComputing. Bob is also the editor of various jour-nals, including the SIAM Journal on Numerical Anal-ysis, and a past editor for SIAM Journal for ScientificComputing. He is on the IMACS Board of Direc-tors and is currently the Canadian Applied and In-dustrial Mathematics Society’s official representativefor ICIAM.His main area of research is scientific comput-ing, particularly the numerical solution of differentialequations. He has published two books — one in theSIAM Classics series — and over 70 journal articles.Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences2MITACS NewsMITACS Team Wins PIMS Industrial Outreach AwardThe PIMS Industrial Outreach Prize was recently awardedto the research team consisting of Huaxiong Huang (York),Keith Promislow (SFU), John Stockie (New Brunswick)and Brian Wetton (UBC) from the MITACS MathematicalModeling and Scientific Computation Group. In the arti-cle below they describe their research with Ballard PowerSystems.The Mathematics of Fuel Cells:An Overview of the MITACS-BallardPower Systems CollaborationContributed by Brian Wetton and Keith Promislow.Ballard Power Systems is the world leader in the devel-opment of fuel cell technology. Their designs promisecleaner, more efficient power for the automotive industryand for stationary generators for today’s deregulated elec-trical market. The PIMS-affiliated Mathematical Model-ing and Scientific Computation (MMSC) Group is a teamof Math faculty, post-doctoral fellows, graduate studentsand support staff from UBC, SFU, York University, theUniversity of New Brunswick and the University of Cal-gary. The expertise of our group is in the application ofmodeling and scientific computing methods to the area ofmaterial science in its broadest sense. The MITACS NCEhas provided a forum for collaboration between this aca-demic group and Ballard. The current joint project centerson the convective/diffusive transport of reactant gases (hy-drogen and oxygen) in conjunction with condensation andwater management in fuel cell stacks. Mathematical mod-eling captures the interconnections between the importantelements of complicated physical systems, yielding equa-tions amenable to numerical methods. These models arehelping Ballard to improve cell efficiency and durability,while speeding up the design process.The research collaboration with Ballard was high-lighted at the recent AGM meeting of the MITACS NCE.A presentation by Randy Savoie, the Ballard Director ofProduct Commercialization and Quality (and MITACSboard member) described the broad sweep of applicationsof PEM fuel cell technology. His presentation was followedby Keith Promislow (SFU) and John Stockie (Universityof New Brunswick), who outlined the model developmentand presented results of numerical simulations.The first Ballard-MMSC project (now complete) in-volved the modeling, analysis and computation of the flowof reactants through the Gas Diffusion Electrode (GDE),a layer of porous, conducting material (currently carbonfibre paper) on either side of the catalyst and membranein the fuel cell. Mathematical analysis of the models high-lighted the sensitivity of fuel cell performance to certainGDE parameters, giving insight into the performance ofKeith Promislow (left) and Brian Wetton at the 2000 MITACSAnnual General Meeting.John Stockie (left) and Huaxiong Huang at the 2000 MITACSAnnual General Meeting.various possible GDE materials. Numerical computationsfocused on the importance of various geometrical parame-ters (channel width and spacing, etc.).Our initial modeling of gas flow was limited to two com-ponent isothermal gas flows. In this year’s project, themodel and computations have been extended to includethermal effects, multi-component gases, and condensa-tion in the GDE. Condensation and water managementare crucial issues in fuel cells. Sufficient water must bepresent to keep the membrane (typically Nafion) wet. Toomuch water will block pores and prevent gases diffusing tothe catalyst sites. Condensation modeling in porous me-dia is complicated by the capillary pressure, the pressuredifference between gas and liquid phases. The widely usedmodels of capillary pressure for wetting media are notvalid in the teflonated carbon fibre paper of the GDE. Inaddition, the modeling and computation of the movementThe Paci£c Institute for the Mathematical Sciences Winter 20013GDE membrane in a fuel cell.of boundaries between wet and dry zones insidethe GDL present a considerable scientific chal-lenge.We are also pursuing the modeling of watermovement in the graphite channels which containthe flowing reactant gases. Water generated bythe reaction enters the main gas flow channelsand is blown out of the fuel cell. Liquid watermoving in the channels will either be in dropletsor rivulets. A study of rivulet flow in the chan-nel geometry, with the critical dependence on theliquid-solid contact angle properties of the mate-rial, is being undertaken. The motion of droplets in the channel in a shear flow is also being considered, with the goal ofestimating droplet size at shear-off.The Ballard project is just one of the industrial collaborations of the MMSC group. To find out more details aboutthis project and others, please visit our web page www.math.ubc.ca/∼wetton/mmsc.Report on Biophysics and Biochemistry of Motor Proteins WorkshopSponsored by MITACS and PIMS, this workshop tookplace from August 27 to September 1, 2000 at the BanffConference Centre in Alberta, Canada. The main objec-tives were to review the state of the art of the field frommany different vantage points and to stimulate a multidis-ciplinary investigation into the area.The meeting provided a fruitful multi-disciplinary fo-rum for discussions on the biophysics and biochemistry ofmotor proteins. In addition to lectures and communica-tions given by leading molecular and cell biologists, bio-physicists, applied mathematicians, experimental physi-cists and representatives of other disciplines, there werea number of contributed talks and posters presented byother participants.The workshop attracted a distinguished group of morethan 70 researchers from several fields connected to thestudies of motor protein family of molecules. They camefrom premier scientific institutions in North America, Eu-rope, Asia and Australia representing 14 countries on 4continents.The interdisciplinary nature of the gathering was muchin evidence as can be ascertained by inspecting the de-partment names represented by the participants. Theseincluded: mathematics, applied mathematics, biomathe-matics, physics, biophysics, bioengineering, chemistry, bio-chemistry, botany, anatomy, physiology, pathology, oncol-ogy, biotechnology, cytology, pharmacology and biology.The members of the local organizing committee wereR. S. Hodges (Biochemistry, University of Alberta andPENCE), G. de Vries (Mathematical Sciences, Universityof Alberta) and J. A. Tuszynski (Physics, University of Al-berta). The members of the international organising com-mittee were R. Valle´e (University of Massachusetts, USA),D. Astumian (University of Chicago, USA), D. Sackett(NIH, Bethesda, USA) and R. Vale (UC San Francisco,USA).The 26 speakers who gave invited talks were D. Astu-mian (Chicago), S. Block (Stanford), T. Duke (Cambridge,UK), Y. Engelborghs (Leuven, Belgium), H. Flyvbjerg(Risoe, Denmark), E. Frey (Harvard), R. Goldman (North-western), L. S. Goldstein (UC San Diego), R. Hodges (Al-berta), J. Howard (Washington), F. Ju¨elicher (Curie Insti-tute, France), R. Kelly (Boston College), E. M. Mandelkow(represented by A. Hoenger, DESY, Germany), A. Manio-tis (Iowa), A. Mogilner (UC Davis), C. D. Montemagno(Cornell), D. Odde (Minnesota), G. Oster (Berkeley),D. Sackett (NIH), J. Spudich (represented by W. Shih,Stanford), E. Unger (IMB, Germany), R. Valle´e (Mas-sachusetts), R. Vale (represented by S. Rice, UC San Fran-cisco), T. Vicsek (Eo¨tvo¨s, Hungary), R. H. Wade (repre-sented by F. Kozielski, Institute for Structural Biology,France) and T. Yanagida (Osaka, Japan).The invited lectures were videotaped and will beavailable soon for viewing by streaming video atwww.pims.math.ca/video.R. Douglas Martin Speaks atPIMS-MITACS, UBCOn January 25, Doug Martin, Professor of Statis-tics at the University of Washington and Chief Sci-entist in the Data Analysis Products Divisions atMathSoft, spoke in the PIMS-MITACS Computa-tional Statistics and Data Mining Seminar Series atUBC. In his talk he examined the problem of model-ing stock market returns, given that the data is oftennon-Gaussian. He demonstrated how stock marketdata could be better understood by using robust fac-tor models and Trellis graphics methods. Prof. Mar-tin’s lecture may be viewed via streaming video atwww.pims.math.ca/video.Information on the Computational Statisticsand Data Mining Seminar Series is available atwww.pims.math.ca/industrial/2001/mitacs.Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences4Industrial ActivitiesFourth Annual PIMS Graduate IndustrialMathematical Modelling CampUniversity of VictoriaJune 11–15, 2001The format of the fourth PIMS Graduate IndustrialMathematical Modelling Camp (GIMMC) will besimilar to that of previous years. Eight or nine “mentors”will be on site to lead groups of graduate students througha mathematical modelling problem chosen by the men-tor. A maximum of 60 student participants will ensure asmall student/mentor ratio. Confirmed mentors this yearinclude: Chris Budd (Bath, UK, Fluids), John Chadam(Pittsburgh, Finance), Uli Haussmann (UBC, Finance),Tim Myers (Cranfield, UK, Fluids), and Miro Powojowski(Calgary, Statistics).In past years, problems in both discrete and contin-uous optimisation, graph theory, mathematics of finance,stochastic optimisation and applied statistics have beenpresented, to name a few. Detailed reports on prob-lems treated in previous year’s camps may be found atwww.pims.math.ca/publications.Student participants at the GIMMC are expected toalso particpate in the PIMS Industrial Problem Solv-ing Workshop to be held in Seattle the following week.They will be automatically registered for this workshop.Students accepted for the GIMMC will be fully supportedby PIMS during the two weeks of workshops, includingreimbursement for travel expenses.Graduate students wishing to participate should payparticular attention to the application procedure. Oneletter of recommendation will be required, and completedapplications will be reviewed for acceptance on the fol-lowing dates: February 15, March 15 and April15. Once the cap of 60 participants has been reached,we will no longer accept applications, so it is advisableto apply at the earliest possible time. Application tothe camp is open to Graduate students at any US orCanadian University. For more information and detailson the application procedure, please see the webpagewww.pims.math.ca/industrial/2001/gimmc.Fifth Annual PIMS Industrial ProblemSolving WorkshopPIMS at University of WashingtonJune 18–22, 2001Since its inception in the Summer of 1997 with the firstPIMS-IPSW at UBC, the PIMS industrial workshops havebeen a spectacular success. We expect to continue thisyear with an outstanding workshop, and are particularlyexcited to have the IPSW as the Premier PIMS event atour newest PIMS institution, the University of Washing-ton. Please consider joining us for what will certainly bean outstanding week of applied mathematics.This, the fifth PIMS-IPSW, will follow a similar formatto that of previous workshops. Six industrial scientists willpresent real mathematical problems that are current andrelevant to their companies. The workshop participants(both faculty and graduate students) then spend the restof the week working on these unsolved problems with thehelp of a company representative and some selected aca-demic “mentors”.Four full days are reserved for work on the problems.On the fifth day, oral presentations from each group willbe made before the whole assembly. A Conference Pro-ceedings will be compiled and published by PIMS afterthe workshop. Proceedings from previous PIMS-IPSW’smay be viewed at www.pims.math.ca/publications.Graduate student participation is encouraged. Allgraduate students should apply to attend the trainingcamp, PIMS-GIMMC 2001 at the University of Victoriaduring the previous week. Graduate participants at theGIMMC are automatically registered for the IPSW andwill be fully supported for both events.Limited funds in the form of travel reimbursementsand accommodation expenses are available for otherparticipants. Details about the administration of theworkshop, financial support and problem descriptionsmay be found at the main PIMS-IPSW 2001 web site,www.pims.math.ca/industrial/2001/ipsw.PIMS-IAM Joint DistinguishedColloquium SeriesPIMS and the Institute for Applied Mathematics atUBC jointly sponsor six distinguished colloquiumseach year. The speakers for the year 2000–2001 are:Carlo Cercignani (Politecnico di Milano): “KineticModels for Granular Materials; An Exact Solution”,September 13, 2000.David Brydges (Univ. of Virginia): “Gaussian In-tegrals and Mean Field Theory”, September 27, 2000.Linda Petzold (Univ. of California at Santa Bar-bara): “Algorithms and Software for Dynamic Opti-mization with Application to Chemical Vapor Depo-sition Processes”, November 1, 2000.David Baillie (Simon Fraser University): “Compar-ative Genomics”, January 16, 2001.Gunther Uhlmann (Univ. of Washington): “TheMathematics of Reflection Seismology”, March 6,2001.Bengt Fornberg (Univ. of Colorado): “Radial Ba-sis Functions — A future way to solve PDEs tospectral accuracy on irregular multidimensional do-mains”, March 27, 2001.See www.pims.math.ca/industrial/2000/iampims lect for more information.The Paci£c Institute for the Mathematical Sciences Winter 20015Scienti£c ActivitiesPNW String Theory SeminarPIMS at UBCMarch 13, 2000The first Pacific Northwest String Theory Seminar willbe hosted by PIMS at UBC. This one-day meetingwill feature 6 talks focusing on recent developments instring theory. The lectures will be videotaped andmade available on the PIMS streaming video web page,www.pims.math.ca/video.Organizing Committee: Konstantin Zarembo (chair,PIMS-UBC), Gordon Semenoff (UBC) and Sandy Ruther-ford (PIMS).Invited Speakers Include:H. Ooguri (Caltech)W. Taylor (MIT)K. Skenderis (Princeton)A. Peet (Toronto)For further information, please see the webpagewww.pims.math.ca/science/2001/pnwstring or con-tact Konstantin Zarembo 〈zarembo@pims.math.ca〉.¨§¥¦For a complete list of upcoming scientific ac-tivities at PIMS, please visit the webpagewww.pims.math.ca/science.Joint SSC-WNAR-IMS MeetingSimon Fraser UniversityJune 10–14, 2001Contributed by Tim Swartz, SFU.The 29th annual meeting of the Statistical Society ofCanada (SSC) is being held jointly with the Western NorthAmerican Region of the International Biometric Society(WNAR) and the Institute for Mathematical Statistics(IMS). It is expected to attract approximately 500 reg-istrants and will bring together researchers and users ofstatistics and probability from academia, government andindustry. Simon Fraser University is especially pleased tohost the meeting as it celebrates the creation of its newlyformed Department of Statistics and Actuarial Science.The meeting will hold four workshops on the topicsStatistical Genetics, Data Mining, Survey Methods andBeyond MCMC. In addition to invited paper sessions, theorganizers are calling for contributed papers and are hold-ing a poster session. The meeting will host a Job Fair,special WNAR events and various social events.Special thanks are extended to the Pacific Institutefor the Mathematical Sciences, the Centre de RecherchesMathematiques, the Fields Institute, SFU, the Faculty ofScience at SFU and the Department of Mathematics andStatistics at SFU for their support of the conference.More details concerning the conference are available atthe webpage www.math.sfu.ca/∼tim/sscmtg.html.PIMS Thematic Programme on Nonlinear PDEPIMS at UBC, July 2 – August 18, 2001More than 400 researchers from 15 countries will be par-ticipating in the PIMS Thematic Programme for 2001.The program will concentrate on several interrelated topicsoriginating in finance, physics, chemistry, biology and ma-terial sciences as well as in geometry. The common featureof these topics is that they involve the interplay betweennonlinear, geometric and dynamic components of partialdifferential equations. There will be emphasis on: Vis-cosity Methods in Partial Differential Equations,Phase Transitions, Variational Methods in PartialDifferential Equations, Concentration Phenomenaand Vortex Dynamics, as well as Geometric PDE. Anumber of mini-courses will be offered during the thematicprogram.Mini-course Lecturers:Panagiotis Souganidis (Austin): 5 lectures on “Fully non-linear stochastic PDEs”.Craig Evans (Berkeley): 2 lectures on “Hamilton-Jacobi equa-tions and dynamical systems”.Thaleia Zariphopoulou (Austin): 2 lectures on “ViscositySolutions in Finance”.Andrzej Swiech (Georgia Tech): 5 lectures on “Viscosity so-lutions in infinite dimensional spaces and optimal control ofPDEs”.Pierre-Louis Lions (Paris): 3 lectures on topics to be an-nounced.Henri Berestycki (Paris): 4 lectures on “Propagation offronts in excitable media”.David Kinderlehrer (Carnegie-Mellon): 4 lectures on “Top-ics in metastability and phase changes”.Michael Struwe (ETH): 4 lectures on “Concentration prob-lems in two dimensions”.Wei-Ming Ni (Minnesota ): 4 lectures on “Diffusions, cross-diffusions and their steady states”.Fang-Hua Lin (Courant Institute): 4 lectures on “Vortex Dy-namics of Ginsburg-Landau and related equations”.Yann Brenier (Paris): 4 lectures on “Variational problemsrelated to fluid and plasma modelling”.Eric Se´re´ (Paris): 3 lectures on “Variational problems in rel-ativistic mechanics: Dirac-Maxwell equations”.Maria Esteban (Paris): 3 lectures on “Variational problemsin relativistic quantum mechanics: Dirac-Fock equations”.Cliff Taubes (Harvard): 4 lectures on “Pseudoholomorphicgeometry as a tool to study smooth 4-dimensional manifolds”.Gang Tian (MIT): 4 lectures on “Recent progress in ComplexGeometry”.Rick Schoen (Stanford): 4 lectures on “Geometric VariationalProblems”.For further details, please check the webpagewww.pims.math.ca/pde.Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences6International ActivitiesSecond Canada-China Mathematics CongressAugust 20–23, 2001 in Vancouver, BC, CanadaThis initiative builds on the success of the first Congress held at Tsinghua University, Beijing, in August 1999, and isaimed at developing further the collaborative research effort between the two countries. It is sponsored by The 3 × 3Canada-China initiative, the Centre de Recherches Mathe´matiques, the Fields Institute for the Mathematical Sciences,the Pacific Institute for the Mathematical Sciences and the MITACS Network of Centres of Excellence. Funds have beenset aside by the five organisations not only for the actual meeting but also to support the local and travel expenses withinCanada of selected Chinese mathematical scientists who are planning extended visits to Canadian Universities aroundthe dates of the Congress.Organizing Committee: Nassif Ghoussoub (PIMS Directorand National Math. Coordinator for 3x3 Canada-China Initiative),Arvind Gupta (MITACS program leader), Bradd Hart (Director,Fields Institute), Jacques Hurtubise (Director, CRM), K. C. Chang(Peking University), Lizhong Peng (Peking University), Dayong Cai(Tsing Hua University), XingWei Zhou (Nankai University), JiaXingHong (Fudan University)Officers of the Chinese Delegation: Chen Jia-Er, Pres-ident of the Natural Science Foundation of China, Qing Zhou (Of-ficer of mathematical and physical division of NSF of China), ZixinHou (President of Nankai University), Zhiming Ma (President of theMathematical Society of China)Chinese Plenary Speakers:Weiyue Ding (Peking University): Geometric AnalysisJie Xiao (Tsinghua University): AlgebraYiming Long (Nankai University): TopologyXiaoman Chen (Fudan University): Operator TheoryZhiming Ma (Chinese Academy of Sciences): ProbabilitySession SpeakersI. Algebra and Number Theory:Qingchun Tian (Peking University): Number TheoryXingui Fang (Tsinghua University): Group TheoryMingyao Xu (Tsinghua University): Group TheoryJie Xiao (Tsinghua University): AlgebraII. Mathematical Physics and PDE:Yingbo Zeng (Tsinghua University): Math. PhysicsZhangju Liu (Peking University): Math. PhysicsPeidong Liu (Peking University): Math. PhysicsYoujin Zhang (Tsinghua University): Math. PhysicsChengming Bai (Nankai University): Math. PhysicsSongmu Zheng (Fudan University): PDEJiayu Li (Fudan University): Geometric AnalysisWeiyue Ding (Peking University): Geometric AnalysisIII. Probability and Statistics:Guanglu Gong (Tsinghua University): ProbabilityZhi Geng (Peking University): StatisticsYongjin Wang (Nankai University): SuperprocessesTianping Chen (Fudan University): Neuro NetworksZhiming Ma (Chinese Academy of Sciences): ProbabilityIV. Wavelets and their Applications:Xingwei Zhou (Nankai University): WaveletsLizhong Peng (Peking University): Harmonic Analysis and WaveletsHeping Liu (Peking University): Harmonic AnalysisV. Computational, Industrial and AppliedAnalysis:Houde Han (Tsinghua University): Computational MathematicsDayong Cai (Tsinghua University): Computational MathematicsFengshan Bai (Tsinghua University): Computational MathematicsJianwei Hu (Nankai University): Numerical AnalysisRunchu Zhang (Nankai University): Numerical AnalysisYongji Tan (Fudan University): Mathematics in IndustryZongmin Wu (Fudan University): Approximation Theory and DataTreatmentYangfeng Su (Fudan University): Computational MathematicsXunjing Li (Fudan University): Control TheoryVI. Geometry/Topology:Yiming Long (Nankai University): Symplectic TopologyLei Fu (Nankai University): Algebraic GeometryJinkun Lin (Nankai University): Homotopy TheoryXiaojiang Tan (Peking University): Algebraic GeometryHaizhong Li (Tsinghua University): GeometryZhiying Wen (Tsinghua University): Fractal GeomtryShaoqiang Deng (Nankai University): Kahler ManifoldsQing Ding (Fudan University): Differential GeometryVII. Operator Theory/Functional Analysis:Guanggui Ding (Nankai University): Operator TheoryXiaomanChen (Fudan University): Operator TheoryShufang Xu (Peking University): Numerical AlgebraVIII. Mathematical Finance:Jiongmin Yong (Fudan University): Mathematical FinanceDuo Wang (Peking University): Mathematical FinanceIX. ODE and Dynamical systems:Weigu Li (Peking University): ODEZhiming Zheng (Peking University): ODEMeirong Zhang (Tsinghua University): Dynamical SystemsThe Paci£c Institute for the Mathematical Sciences Winter 20017Educational ActivitiesJunior High Math Nights at Mount RoyalCollege, AlbertaJunior high school students, their teachers, and parents areinvited to Mount Royal College for six Monday nights fromJanuary 29 through March 4 to take part in mathematicalexploration activities. The emphasis is on removing themyth that mathematics is a set of facts that are innate tocertain individuals and the myth that mathematics is notan experimental discipline.PIMS Elementary Mathematics Nights inAlbertaThe highly popular Elementary Mathematics Nightsinvolve volunteers from Mount Royal College and the Uni-versity of Calgary who assist the teachers to guide par-ticipants through a variety of activities. Activities suchas map colouring, games on graphs, dominating sets ofgraphs, Fibonacci numbers, binary numbers, patterns inPascal’s triangle, the traveling salesman problem, and fi-nite state automita may be included.The success of these evenings can be directly attributedto the volunteers: Robert Petzold, Jean Springer, LauraMarik, Peter Zizler, Scott Carlson, Charles Hepler, andSharon Friesen. PIMS would like to thank them and ac-knowledge their contribution to these evenings.Current Schedule of Evenings:Feb. 13, Science Alberta School, CalgaryFeb. 22, Sunnyside Community School, CalgaryMay 15, Westmount Elementary School, StrathmoreMath Mania: An Alternative MathEducation EventSir James Douglas Elementary School, Victoria7:00–8:00pm, February 28, 2001At the Math Mania evenings, “fun” methods are usedto teach math and computer science concepts to chil-dren (and adults!) using games and art. This MathMania will feature exciting geometrical models fromstraws and paper, mathematical puzzles, the slingshoteffect of celestial bodies, the guessing game, a sort-ing network, the penny game, the set game, bubblesand more! For more details please see the webpagewww.pims.math.ca/education/2001/mathmania/february28.html.¨§¥¦For a complete list of upcoming educationactivities at PIMS, please visit the webpagewww.pims.math.ca/education.Greater Vancouver Regional Science FairUniversity of British ColumbiaApril 5–7, 2001At the upcoming Greater Vancouver Regional Science Fair,PIMS will supply judges, mathematical expertise, andprizes for the mathematics component of the Fair. Bypromoting mathematics within the Science Fair context,PIMS encourages students to develop a feel for the adven-ture of a self-directed exploration in longer-term projectsin the mathematical sciences.More information on the Greater Vancouver RegionalScience Fair is available at www.sciencefairs.bc.ca/regions/gvrsf/Vancouver info.html.Changing the Culture 2001SFU at Harbour CentreMay 11, 2001The Fourth Annual Changing the Culture conference, or-ganized and sponsored by PIMS, has as its theme Writ-ing, Speaking and Thinking Mathematics. The conferencewill explore connections between numeracy and literacy,mathematics and language, mathematics and literature,and how we can use language to teach mathematics.The conference is free, but registration is required be-cause space is limited.For further information please see the webpagewww.pims.math.ca/education/CtC/CtC01.htmlor contact Malgorzata Dubiel 〈dubiel@math.sfu.ca〉.Third Annual PIMS Elementary GradesMath ContestUniversity of British ColumbiaMay 26, 2001This contest — open to students in Grades 5 to 7 — giveselementary school students an opportunity to experiencemathematics as an exciting sport! Like theMathCountscompetition, which is available only to high school stu-dents, it determines the winners right on the spot — afterthree problem-solving rounds, the last of which is a seriesof mathematical duels among the top performers in thedifferent grades. Even for those who do not make it to thetop it is instructive, inspiring and entertaining.The event is organized by PIMS under the guidanceof Dr. Cary Chien of David Thompson Secondary Schoolin Vancouver, in collaboration with the BC Association ofMathematics Teachers and volunteers from Lower Main-land schools of all levels. Details can be found on ourwebsite at www.pims.math.ca/elmacon.Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences8Entering Mathematical Sciences atthe Graduate LevelContinued from page 1.Robert Moody speaking at the PIMSGraduate Information Week.technology, andother areas. Themain parts of puremathematics arealgebra (includ-ing arithmetic),analysis, geometry(including topol-ogy), and logic.Applied mathe-matics includesthe various formsof mechanics(hydromechanics,elasticity, etc.), statistics, and many other newer areas ofapplication to computing, economics, biological sciences,etc. I am sure that you can think of lots of thingsthat could be added on: the mathematics of finance,actuarial mathematics, control theory, dynamical systems,relativity theory, crystallography, mathematical physicsincluding particle physics, quantum mechanics, andstring theories, operations research, information theory,encryption, and so on and on. In other words, it is veryhard to define mathematics.In a recent article by the famous mathematician DavidMumford, he recalls the definition that Davis and Hershgive of mathematics: Mathematics is the study of men-tal objects with reproducible properties. This is so muchnicer: rather than simply saying what present mathemat-ical knowledge consists of, it opens the doors for the pos-sibilities of what it might also be. And this I think isexactly the right position for any one entering graduatestudies in the Mathematical Sciences. Certainly there ismuch to learn that is now the standard repertoire of work-ing mathematicians and scientists, but this is also the timewhen you have a chance to do something totally new andperhaps totally unexpected. If we match this definitionwith the usual definition: science is the study of physi-cal objects with reproducible properties then we see howinevitably mathematics and science are linked.This is something that I would like to emphasize. Oneof the most persistent and tiresome debates is over the rela-tive importance of pure and applied mathematics. Neitherwould be very much without the other, though individu-als obviously differ in taste and style. Let’s look at someexamples.David Mumford himself is an interesting example. Hebecame famous (Fields Medalist) as an algebraic geometer,and several of his books in this area are classics. However,in mid-career he switched his interest to the mathematicsof vision and has established a name for himself in thisdifficult area of biological mathematics. The article that Iquote from (which, by the way, is in a very interesting bookof essays: Mathematics: Frontiers and Perspectives, AMSPubl.) is entitled, “The dawning of the age of stochas-ticity.” His point, stressing the importance of stochastic(random) processes, is something that I also find myselfcoming to appreciate.The subject of fractals is relatively new — somethinglike 25 years. Its father was Benoit Mandelbrot, a mathe-matician at IBM. Fractals are not intrinsically so difficult.What was hard was the realization that self-similarity atdifferent scales is manifest all over Nature: branching trees,spiral shells, decaying mountains, rivers and rivulets, incrystallization of materials, Brownian motion, in the mor-phology of plants, animals, and minerals. This could havebeen noticed by anyone at any time. The strange thing isthat up to Mandelbrot no one really saw it. More fascinat-ing is that the mathematics used to describe it comes fromsome of the most arcane parts of point set topology: Can-tor sets, Hausdorff dimensions, and self-similar measures.It seems certain that Nature is at least as discontinuousas it is continuous (think of quantum mechanics) and thatthis new mathematics will have more importance, not lessas time goes by. What’s more fractals are strongly linkedto chaos theory which is another of those “obvious” phe-nomena of Nature that no one noticed until a few yearsago!My third example is from my own experience. The il-lustration at the beginning of this article is a Penrose tiling(that’s Sir Roger Penrose, famous for his books on themind and artificial intelligence). The origins of this tilingare totally unexpected. The logician Hao Wang had askedthe question of whether or not an algorithm existed which,given any finite number of polygon tile shapes, could de-cide whether or not the plane could be tiled without gapsor overlaps with copies of these tiles, i.e. could a machinefigure out this out for us? He could not prove or disprovethis, but he did prove that it would be true (an algorithmwould exist) if whenever a set of tiles could tile the planein some way then it could also tile it in a periodic way. Healso conjectured that this was true.Now that would mean that aperiodic sets of tiles wouldNOT exist. It is hard to believe that up to the 1960’s theanswer to this was not known (or even thought about).Wang’s student Berger finally proved that in fact Wang’sconjecture was false! — and then went on to constructan aperiodic tiling set with some 20,000 or so tile types.Rafael Robinson reduced it to 6 and showed that the tilingproblem was equivalent to the famous Halting Problem ofTuring. Penrose worked on the problem and reduced it to2!This is piece of pure mathematics. However in 1984new forms of solid state metallic materials were discoveredthat shook the world of crystallography: purely diffractivematerials with symmetries that cannot arise from periodic(crystallographic) arrangements of atoms. I don’t thinkthat you will disagree that there is something profoundlymathematical about the picture below which is the diffrac-tion image of a real quasicrystal.The Paci£c Institute for the Mathematical Sciences Winter 20019Diffraction image of the quasicrystalAl70Co11Ni19.(Image courtesy of Conrad Beele.)The amazing thingis that the Pen-rose tiling is alsopure point diffractive,something that math-ematicians mightnever have guessed,even though it is apurely mathematicalconstruct of Fourieranalysis. (It was acrystallographer whofirst suggested it). Soa whole new area ofaperiodic order wascreated.Diffraction image of Penrose tiling.(Image courtesy of Moritz Hoeffe.)What are the pri-mary mathematicaltools? Discrete ge-ometry (whose ori-gins, as its namesuggests come fromthe measurement ofland), point set topol-ogy (a product ofpure mathematics),Fourier analysis (de-veloped from physi-cal ideas around heat,sound and waves),harmonic analysis —especially in the gen-eral context on locally compact Abelian groups (more puremathematics), dynamical systems (originated by Poincare´in his study of the orbits of the planets), ergodic theory(developed in statistical mechanics to study the statisti-cal properties of large ensembles), C∗-algebras (developedout of quantum mechanics and analysis), algebraic-numbertheory (developed out of pure interest in numbers and inparticular Fermat’s Last Theorem), and stochastic pro-cesses (developed originally from the study of games ofchance). And don’t forget the original inputs from logic,algorithms, and solid-state physics.Amazing, isn’t it?I have spent a lot of time talking on this. What is mypoint?Quite simply that mathematics is an evolving processthat is incredibly rich, incredibly linked with all of the restof science, technology, and human culture. That there aremathematical things around us that have not been discov-ered, not because they are deep, but because no one hashad the eyes to see them or the mind to think on them.You should keep your mind wide open. Don’t narrowyour vision too quickly; don’t shut doors on this and thattype of mathematics. Take a variety of subjects. Whatevermathematics you study will likely come in useful.In fact I would go further: in graduate school you havea chance to learn all sorts of up-to-date mathematical ideasfrom real experts. Unlikely as it may seem to you, youhave one huge advantage over these experts. Though theyprobably know more about their own subject than youever will, you will also be learning from your variety ofclasses things that they do not know, or have completelyforgotten. Each person has a collection of knowledge andideas different from every other. You may see connectionsthat no one else has even a chance to see. Never forgetthis, however discouraged you may get.Also in graduate school you will have more freedom tothink and to do your own thing than you will have againuntil you retire.Finally, the concentration and energy that you canbring to bear on problems is almost optimal at your age.(Older mathematicians are more wily and use intuitionbuilt out of many years of experience to make up for theirdeclining brain power).Put this all together and you realize that during thesegraduate years you actually have the potential to makevery significant contributions. I mean VERY significant.Do not underestimate the possibilities that are there. Ofcourse significant discovery is also a matter of chance andbeing in the right place at the right time, but there is nodenying the importance of a prepared and open mind. Thefact is that there things like fractals all around us for thosewith the right eyes to see them.So much for the heroic side. But not everyone canbe a hero. What about fears and doubts about enteringgraduate school?Certainly research is not everyone’s thing. It can belonely and very frustrating. Most ideas will not work out.The slick proofs of classes do not magically appear; theyare only the product of many years of refinement. Re-search is not like solving homework problems! Researchersgenerally learn to live with problems for long periods at atime. One prominent researcher told me: “it is not thatthe highs are so high and the lows are so low, but that thehighs are so short and the lows are so long”. One word ofadvice: keep several problems going at once. It helps tobe able to get off a problem that is not going anywhere.Here are some suggestions. A Master’s Degree is a goodway to find out what you like to do and what is not for you.You will have to take a bunch of core courses. These makea great foundation for you whatever you do that involvesmathematics, even if you only use them indirectly. I don’tthink that you need to worry about this knowledge beinguseless or too refined and so on. As much as you can, tryto see how the different courses that you take fit together.Mathematics is not a collection of disjoint subjects.If you want to test your tolerance and enthusiasm forresearch, take the thesis option. There is much to be saidfor it. Anyone who completes a thesis has proven thatthey can undertake and complete a major task of orga-nization and presentation. This is a valuable asset and Ithink employers of all types also recognize the logisticaland problem solving skills that are acquired in the pro-Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences10cess. You will also most likely learn some brand of TEX,which is a good thing. If you are fortunate, you will havea publishable paper out of it too.If you are in the straight mathematical side of things, Iwould encourage you to also get some background in statis-tics and probability theory, as well as some background incomputer science. Not only will you find these useful inyour mathematical endeavours, you will find that they aregood items at job interview time. The same applies tothose of you in computer science or in statistics. Takesome mathematics classes. It is amazing how the storeof techniques and problem-solving abilities developed inmathematics can help in unexpected ways.Some people skip over the Masters degree and gostraight to the Ph.D., but I think it is better not to bein a big rush.In fact, I have different thoughts on the Ph.D. If youare not certain of your ambitions, I think that it may notbe the best use for 3 or 4 years of your time. At thislevel you are becoming a real expert in some part of themathematical sciences. Unless you see some career goal atthe end of it, this knowledge may not be of further directvalue to you. However, if you are thinking of a career inacademia, a Ph.D. is essential. It is also a fact that it isvery hard to come back to graduate school after you havebeen out of it for a few years.Other thoughts about the world of mathematics. Thereis no doubt that there are many kinds of mathematicianand many levels of ability. The best research mathemati-cians are awesome and can be intimidating. However,mathematics is a huge endeavour of human culture. Peo-ple find many important and satisfying niches: education,writing and exposition, public dissemination, research, or-ganization, as well as in the numerous branches of knowl-edge to which mathematics can be applied.Persistence, patience, diligence, and a little daring canmake up for lack of genius: I can assure you that I amliving proof of this!This is a good time to be going into graduate schoolin the Mathematical Sciences. Mathematics of all flavoursis flourishing and is well supported here in Canada. Theapplications of mathematics to other areas of science havenever been more diverse, more profound, nor more publically recognized.It is true that mathematics is a hard task-master. Hersecrets are reluctantly revealed. However, she will neverfail you as a source of wonderful and beautiful ideas. Overthe ages, mathematics has attracted some of the finestminds the world has known. Unlike almost every otherbranch of science, these ideas never fade with time.I can tell you that I still love the subject after work-ing non-stop in it for 42 years, and if I were to be go-ing to university today, I would be studying mathematics.The difference is that I would try to learn as many differ-ent parts of it (including mathematical physics and somestochasitics) as I could.Tudor S. Ratiu: PIMS Distinguished Lecture at the University of VictoriaOn January 12, Professor TudorS. Ratiu (Ecole PolytechniqueFederale de Lausanne) gave alecture on “Variational Prin-ciples, Groups, and Hydrody-namics” as part of the PIMSDistnguished Lecture Series atthe University of Victoria. Inhis lecture, he presented Hamil-tonian systems whose configu-ration space is a Lie group.The systems discussed were allgeodesic flows of certain metricswith a number of common characteristics. He began withthe classical example introduced by Euler: a free rigidbody whose configuration space is a proper rotation group.Then he discussed a homogeneous incompressible fluid flowwhose configuration space is a group of volume preserv-ing diffeomorphisms of a smooth manifold. The Arnold-Ebin-Marsden program for the analysis of the equationsof motion was also presented. Another example was theKorteweg-de Vries equation whose configuration space isthe Bott-Virasoro group. Generalization of this is theCamassa-Holm equation, and Prof. Ratiu discussed itspossible choices of configuration spaces. The averaged in-compressible Euler flow and recent results about it werepresented. Finally, Prof. Ratiu showed that the Euler-Poincare equation could be an abstract tool that easilyenables one to recognize such systems.Prof. Ratiu received his B.Sc. and M.A. at the Uni-versity of Timisoara, Romania, and his Ph.D. at the Uni-versity of California, Berkeley. His main area of researchis Hamiltonian systems. He held visiting positions atprestigious research centres such as Max Planck Institutefor Mathematics (Germany), Mathematical Sciences In-stitute at Cornell University, Erwin Schrodinger Institutefor Mathematical Physics (Austria), Institute des HautesEtudes Scientifiques (France), Isaac Newton Institute forMathematical Sciences in Cambridge (England). His con-tribution to mathematics was recognized by numerousawards and prizes. Among other honors, he was an A. P.Sloan Foundation Fellow in 1984–1987, a Humbold Prizewinner in 1997, and won a Ferran Sunyer i Balaguer Prizefor the year 2000. Prof. Ratiu is an editor for 5 researchjournals. He has published over 100 scientific papers and6 books, among which is the well-known Introduction toMechanics and Symmetry, written with J. E. Marsden in1994.The lecture was videotaped and is available in stream-ing video from www.pims.math.ca/video.The Paci£c Institute for the Mathematical Sciences Winter 200111Strings and D-branesContributed by K. Zarembo, PIMS-PDF.String theory emerged in the early 70’s, but was widelyappreciated only fifteen years later, after it was shown toprovide a consistent theory of quantum gravity and to becapable of unifying all of the fundamental forces in nature.Since then, string theory has undergone several periods ofintensive development. During the breakthrough of themid 80’s, sometimes referred to as the “first superstringrevolution”, supersymmetric string theory was establishedas a basis for the unification of fundamental forces. The“second superstring revolution”, that began few years ago,has led to the discovery of unexpected and profound linksbetween gravity and other types of interactions. Recentdevelopments in string theory have shed new light on someold problems in theoretical physics, most notably on theblack hole entropy problem and on strong coupling dy-namics in gauge theories.The modern theory of fundamental interactions is geo-metric in nature and is based on certain symmetry princi-ples. The symmetry that underlies electromagnetic, weakand strong interactions is gauge invariance. In its sim-plest form pertinent to electromagnetic interaction, gaugeinvariance is equivalent to the statement that the phaseof the wave function of a charged particle is not observ-able. The possibility of choosing this phase arbitrarily ateach point requires a compensating vector field Aµ. Froma mathematical point of view, Aµ defines a connectionin the principal bundle U(1). For a physicist, the fieldAµ describes massless particles, photons, which transmitelectromagnetic interactions. Theories of weak and stronginteractions are gauge theories based on non-abelian Liegroups. In principle the gauge fields of these interactionscould also describe massless particles, but certain dynam-ical effects make weak gauge bosons massive and perma-nently confine carriers of strong interaction inside hadrons.The symmetry principle responsible for gravity is dif-ferent. Gravity is essentially a consequence of general co-variance, the independence of the physics on the choice ofthe coordinate system in space-time. General covarianceis in many respects similar to gauge symmetry, but thereis a fundamental difference between gravity and all otherinteractions in that gravity is related to the geometry ofspace-time itself – the gravitational field is the space-timemetric gµν .A unification of gauge interactions does not require anynew principles beyond gauge symmetry. One can imagine(and there are indications that this may indeed be true)that gauge groups of individual interactions are subgroupsof a unique Lie group of a unified theory. A unificationthat would include gravity is more tricky and possibly willrequire a new fundamental principle beyond what we cur-rently know. The mathematical grace of string theoryand its success in solving several puzzles encountered inthe quantization of gravity make string theory the mostpromising idea in the search for such a principle.In string theory, a point-like particle is substituted byan object extending in one dimension, a string. Fundamen-tal strings can vibrate much like the strings in a violin. Infact, the motion of a violin string and of the fundamentalstring are described by the same differential equation, butoscillations of the fundamental string are constrained by acondition that they do not transmit energy and momen-tum along the string. Also, the ends of the string in aviolin are fixed, while the ends of the fundamental openstring can move freely, or the ends of the strings can beglued together to form a closed string. Upon quantization,the string oscillations lead to an infinite set of discrete en-ergy levels. A string in a definite quantum state of itsinternal motion will behave like a particle with a definitemass and spin, so different particle species in string theoryarise as different vibrational modes of a unique object, thestring. Typical masses of string excitations are supposedto be large and, in many cases, it is enough to consideronly massless states. Remarkably, massless modes of anopen string behave exactly like photons or, in a slightlymore complicated setting, like non-abelian gauge bosons,while the spectrum of massless modes of a closed stringcontains a graviton. Therefore, the theory of closed stringsis essentially a theory of quantum gravity.String theory appears to be very restrictive and thesimplest versions of it suffer from various instabilities. Theconsistency of string theory uniquely fixes space-time di-mensionality and requires supersymmetry, the extension ofPoincare invariance that unifies particles of different spinsand statistics. Superstring theory can only be well de-fined in 9+1 (9 space, 1 time) dimensions, 6 of which aresupposed to be compactified and invisible because of theirvery small size. There are five distinct superstring theoriesin ten dimensions, but it has become clear that all of themare related by certain symmetry transformations and, infact, they are different pieces of a more general structureknown as M-theory.a bFig. 1: String world sheet (a) vs particle world line (b).The interaction of strings can be introduced in a purelygeometric manner. When a string propagates in time, itsweeps out a two-dimensional surface, a world-sheet, justlike a particle’s trajectory forms a world-line (Fig. 1). Fora free closed string, the world-sheet is a cylinder, topolog-ically a sphere with two holes. A sphere with four holesthen naturally describes the string scattering: two incom-ing strings join together to form an intermediate one-stringVol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences12state which then splits into outgoing strings (Fig. 2). Thusstrings interact via elementary processes of joining andsplitting. It is possible to develop a systematic string per-turbation theory similar to the Feynman diagram tech-nique in quantum field theory by allowing world-sheets ofall possible topologies.Fig. 2: 2→ 2 string scattering.Similar rules for the interaction of open strings have asimple but surprising consequence: even if one studies thescattering of only open strings, one can see closed stringsemerging in intermediate states. This means that openstrings will interact gravitationally, since closed stringstransmit gravitational forces. Therefore, gravity is an in-evitable consequence of string theory. As an illustration,consider 2 → 2 scattering. This process, in particular,can proceed via splitting into two intermediate stringswhich then annihilate. The intermediate state forms anopen string loop, but, alternatively, it can be viewed as aworld sheet of a closed string, so the closed string exchange(roughly speaking, gravitational interaction) and scatter-ing via the intermediate open strings (roughly speaking,gauge interaction) are two equivalent descriptions of thesame process (Fig. 3). Many of the recent achievements instring theory rely on this simple observation, sometimesreferred to as channel duality.baFig. 3: An illustration of channel duality: the same amplitudeis open string loop in one channel and a closed string exchangein another channel.The idea of channel duality has proven especially fruit-ful in the context of D-branes, stringy analogs of blackholes. A D(irichlet)p-brane is a p-dimensional dynami-cal object in closed string theory. The world volume ofa D-brane is a (p + 1)-dimensional hypersurface in ten-dimensional space, on which strings can begin and ter-minate. Coordinates of the string world-sheet that areperpendicular to the D-brane satisfy Dirichlet boundaryconditions, as opposed to the Neumann boundary condi-tions for an open string which can freely move in all of ten-dimensional space-time. In a sense, D-brane dimensionallyreduces open string theory. Open string excitations can-not leave the world volume of a D-brane because of theDirichlet boundary conditions, so the theory of Dirichletstrings becomes effectively (p + 1)-dimensional. At ener-gies too small to excite higher string modes, this theorycan be well approximated by (p + 1)-dimensional gaugefield theory. In particular, the world-volume theory of aD3-brane is a certain field theory in (3 + 1) dimensions.aopen stringpair creationclosed stringemissionbFig. 5: A D-brane is a hypersurface on which strings can endor begin (a). The world sheet on an infinite open string endingon a D-brane can simultaneously be interpreted in terms of aclosed string emission (b).How can D-branes be identified with black holes? Thereasoning follows from the space-time picture of the cre-ation and the subsequent annihilation of a pair of openstrings with one end attached to the D-brane. By chan-nel duality, this process can alternatively be interpretedas the emission of a closed string. Consequently, D-branescan emit closed string states and, in particular, gravitons.Anything capable of emitting gravitons carries mass (grav-itational charge), so D-branes have tension, mass per unitvolume. More careful analysis shows that they also carryanalogs of electric or magnetic charges. From the point ofview of classical gravity, which well approximates stringtheory at large distances, Dp-branes have δ-functionalmass and charge distributions extending in p dimensions.Such a distribution will curve space and from a large dis-tance will look like a charged black p-brane, an extendedanalog of a black hole. The two entirely different pictures,that of the black p-brane in (9 + 1) dimensions and theworld-volume picture based on (p+ 1)-dimensional gaugetheory, are complementary to one another. This comple-mentarity in many cases was shown to imply tight linksbetween gravity and gauge theories and, indeed, the studyof D-branes uncovered many unexpected interrelations.The idea that has attracted, perhaps, the most atten-tion was put forward by J. Maldacena in 1997 and ex-ploited some earlier work on the comparison between thegravitational and world-volume pictures of a D3-brane.According to Maldacena’s conjecture, classical gravity act-ing on the direct product of the five-dimensional sphereand the five-dimensional Anti-de-Sitter (AdS) space is ex-actly equivalent to the (3+1)-dimensional gauge theory ofa D3-brane in the limit when gauge interactions becomeinfinitely strong and therefore very complicated from theusual point of view.Please see Strings, page 17.The Paci£c Institute for the Mathematical Sciences Winter 200113PIMS Awards Prizes for 2000Continued from page 1.The first annual PIMS Prizes in Education, Research andIndustrial Outreach were awarded on December 10 at aBanquet held at the UBC University Centre. The prizes,valued at $3000 each, were donated by the Toronto Do-minion Bank Financial Group and TD Securities.ThePIMS Education Prize rewards individuals whohave played a major role in encouraging activities whichhave enhanced public awareness and appreciation of math-ematics, as well as fostering communication among vari-ous groups and organizations concerned with mathemat-ical training at all levels. The review committee for thePIMS Education Prize was Michael Lamoureux (chair ofthe committee and PIMS Deputy Director), Florin Diacu(PIMS-UVic Site Director), Arvind Gupta (MITACS Pro-gramme Leader), Bryant Moodie (PIMS-UA Site Director)and Dale Rolfsen (PIMS-UBC Site Director).The PIMS Education Prize was awarded to GeorgeBluman, who is the chair of the UBC Math Department.George Bluman’s lifetime commitment to mathematics ed-ucation in British Columbia, both in the public school sys-tem and at the University of British Columbia, make himan outstanding recipient for the PIMS Education Prize.Many aspects of his activities were highlighted by his nom-inees, including: providing stimulating mathematics expe-riences for students, through the Euclid contest and var-ious school workshops; supporting math teachers in theschools; working to raise and maintain high standards inthe school system; developing a healthy dialogue with theBC Ministry of Education; encouraging math students atUBC to pursue careers in teaching; and encouraging astrong commitment to teaching at UBC.Typical of his activities and impact is his over-twenty-year involvement with the Euclid contest as the BC andTerritories organizer for this high school enrichment con-test in mathematics. George supports the idea that theEuclid contest is an event every Math 12 student shouldbe able to enter and, in doing so, feel a sense of accom-plishment. Beyond organizing the contest, he has devel-oped three levels of School Workshop programs which givestudents (elementary, junior high, and senior high) theopportunity to participate in problem solving workshopswith university faculty and students. BC enjoys the high-est level of participation, per capita, in the Euclid contestand its universities benefit from the excellent preparationthese students receive through the program. In the wordsof the nominators, much of the BC success in Euclid canbe directly attributed to George’s efforts.George personally knows most of the mathematicsteachers from around the province and uses this networkto provide a dialogue between the BC secondary schoolsystem and the universities. He has been tracking highschool students’ performance at university for over twentyyear, and often makes personal phone calls or writes tohigh schools to give suggestions on how to improve theirstudents’ performance. Again, his nominators attest toHugh Morris (Chairman of the PIMS Board of Directors) con-gratulates George Bluman (right).Terry Gannon (left) receives the PIMS Research Prize fromPIMS Director Nassif Ghoussoub (centre), Ken Foxcroft (TDSecurities) and Indira Samarasekera (VP Research, UBC).the positive impact his work has had on the designs, andsuccesses, of their mathematics program. The scope andmagnitude of his service to mathematical education overthe past twenty years is phenomenal.In his comments after receiving the award, GeorgeBluman states that, “It is not easy for mathematiciansto be involved in educational activities. Education is-sues are often very sensitive with many different (oftenunfairly stereotyped) ‘conflicting’ groups and interests—pontificating university professors, strict union mentalitiesof teachers, anxious students and parents, scandal-seekingmedia, politicking Ministries of Education paying little at-tention to common sense and giving lip-service to the opin-ions of informed teachers and professors. It is essential thatall such special interest groups trust each other and stopbickering for the common good. After all we should wantour students to have the best education possible withinour means.”“PIMS is to be congratulated for taking a sincereVol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences14From left, Brian Wetton, Huaxiong Huang, and Keth Promis-low receive the PIMS Industrial Outreach Prize from MurrayMargolis (Powerex Corp.) and Ken Foxcroft (TD Securities).NSERC Director, Danielle Menard speaking at the PIMSAwards Ceremony.From left are Shahid Hussain (Telus Corp.), Frieda Granot(Dean of Graduate Studies, UBC), Indira Samarasekera (VPResearch, UBC), Danielle Menard (Director, NSERC) andCharles Lamb (Math, UBC).Indira Samarasekera (VP Research, UBC) congratulates PIMSResearch Prize winner Terry Gannon (left).interest in Education with its various Education activitiesincluding the recognition of those involved through thisAward.”He drew particular attention to the semi-annual pub-lication of Pi in the Sky, prizes for Math projects inScience Fairs, the support of elementary school Math ac-tivities such as the PIMS Elementary Grades MathContest, theMathematics is Everywhere poster cam-paign, the Senior Undergraduate Industrial MathWorkshop, and theGraduate Industrial Math Mod-eling Camp.“All of the above are new initiatives and continuing ac-tivities which certainly would not have happened withoutthe existence of PIMS. Moreover PIMS is very fortunateto have the services of Klaus Hoechsmann for developingand promoting its educational activities. We all now knowthat Klaus is also a budding playwright from his very well-written and PIMS-sponsored play Hypatia which shouldbe performed for students in schools around the world.”The PIMS Research Prize is given for a partic-ular outstanding contribution to the mathematical sci-ences, disseminated during the past five years. Nomina-tions for the Research Prize were adjudicated by the PIMSScientific Review Panel, the members of which are chairNassif Ghoussoub (PIMS Director), David Boyd (UBC),David Brillinger (Berkeley), Ron Graham (UCSD), Al-istair Lachlan (SFU), Richard Karp (Berkeley), BernardMatkowsky (Northwestern), Robert Moody (Univ. of Al-berta), Nicholas Pippenger (UBC), Ian Putnam (Univ. ofVictoria), Gordon Slade (UBC), and Gian Tian (MIT).The Research Prize was awarded to Terry Gannonof the Dept. of Mathematical Sciences, University of Al-berta. Terry’s accomplishments cover two separate direc-tions, both of which have won him international recogni-tion. The first accomplishment is his work on the “Moon-shine Conjectures”, which concern a fantastic connectionThe Paci£c Institute for the Mathematical Sciences Winter 200115between the representations of the Monster Group andcertain classes of modular forms. Richard Borcherds wasawarded the Fields Medal in 1998 for his proof of these con-jectures. However, Borcherds’ proof contained one partthat was non-conceptual and had to be shown by bruteforce computation. Terry provided a conceptual argumentto replace this computation. The second and more exten-sive of Terry’s accomplishments concerns the classificationof two-dimensional conformal field theories. The probleminvolves determining all modular invariants which can beconstructed from characters of the representations of theunderlying affine Kac-Moody Lie algebras. The first suc-cess in classifying two-dimensional conformal field theorieswas the A-D-E classification of Capelli, Itzykson and Zu-ber for affine-SU(2). In 1994, Terry discovered a solutionto the affine-SU(3) problem and has since made enormousadvances towards a solution of the general problem.In describing his research, Terry states, “My bias as amathematician is toward breadth. Most mathematicians,it seems, try to strike oil by drilling deep wells. This strat-egy makes a lot of sense. But actually I’m more drawntowards half-completed bridges and wobbling fences. Thetheory in those places is relatively undeveloped, so there’sa lot of basic results still open. And I get a little restlessstaying too long in one place.”“Some of my work which attracted a little attentionwas in an area called Monstrous Moonshine. It was noticedthat 196 884 –the first interesting coefficient of a function(the j-function) important to classical number theory–equals 1 + 196 883, the sum of the first two dimensions ofrepresentations of a very special symmetry (the Monstergroup). The second, third,... coefficients of that functionwere likewise related to the higher dimensions. The chal-lenge was to explain what that classical number theoryhad to do with this newly discovered symmetry. A bridgehad to be built! Borcherds did most of the work, and forthis was awarded a Fields Medal in 1998. He showed thatthere’s a new and very complicated algebraic structure (avertex operator algebra) whose symmetry is that Monstergroup, and whose ‘graded dimension’ is the j-function. Ifwe twist the graded dimension by various elements of theMonster, we get other special functions (Hauptmoduls) ofclassical number theory. The best known way to show thisis by a theorem I found with Chris Cummins.”“But much of my work thus far has occurred near a cer-tain wobbling fence separating math from physics. Stringtheory, or more precisely, conformal field theory (CFT),was created by physicists for their own shady purposes, butits impact has been far greater in math. Five of the twelveFields medals awarded in the 1990s were to men whosework directly concerned CFT (namely, Drinfeld, Jones,Witten, Borcherds, Kontsevich). I’ve tried to clarify someof the algebra and number theory in CFT, but mostly I’vebeen working towards the classification of all CFTs re-lated to a class of infinite symmetries called Kac-Moodyalgebras. These CFTs seem to be the fundamental ones,and their classification is uncovering unexpected (and un-explained!) links with other areas of math. I hope tocomplete this classification within the next couple years.”“Research for me is something like chasing squirrels.As soon as you spot one and leap towards it, it darts away,zigging and zagging, always just out of reach. If you’re alittle lucky, you might stick with it long enough to see itclimb a tree. You’ll never catch the damn squirrel, but it’lllead you to a tree. Chasing squirrels is a way to find trees!In math, the trees are called theorems. The squirrels arethose nagging little mysteries we write at the top of manysheets of paper. We never know where our question willtake us, but if we stick with it, it’ll lead us to a theorem,and to our next paper. That I think is what research inmath is like.”“Receiving the PIMS Research Prize has been enor-mously significant for me personally, and surprisingly hum-bling. Recognition from our peers is notoriously rare forthose of us near the beginnings of our careers, and now Ihave some expectations other than my own to live up to(yikes!). Validating and supporting research is the biggestrole PIMS can play, in my view. The PIMS post-doc pro-gram is wonderful, and the plan for an Oberwolfach-styleinstitute is really very exciting. But one thing which is stillquite disappointing in western Canada is the intellectualisolation of the universities from each other. For instance,Calgary and Edmonton are only 3 car-hours apart and yetit’s exceptionally rare when one of us gives a talk at theother university. I wonder if PIMS could actively encour-age more of these grassroots interactions, e.g. by supplyingeach local PIMS office with funds whose sole purpose is toinvite other westerners to give colloquium talks. Maybethis could help build more of a western mathematical sci-ences community.”The PIMS Industrial Outreach Prize recognizesindividuals who have employed mathematical analysis inthe resolution of problems with direct industrial, economicor social impact. The review panel for this prize waschaired by the MITACS Programme Leader Arvind Gupta(SFU). The other members of the panel were Don Denney(Syncrude, Inc.), Shahid Hussain (Telus Corp.), MurrayMargolis (Powerex Corp.), Brian Seymour (UBC) and RexWestbrook (Univ. of Calgary).The prize was awarded to Dr. Huaxiong Huang (York),John Stockie (University of New Brunswick), KeithPromislow (SFU) and Brian Wetton (UBC). This team ofresearchers are part of the PIMS-affiliated MathematicalModeling and Scientific Computation Group in MITACS.They are working with Ballard Systems, the world leaderin hydrogen fuel cell design, to develop models to helpBallard improve the efficiency and durability of fuel cells.Using parabolic poles, they modeled the reactant gasflow through the Gas Diffusion Electrode (GDE), a layerof porous, conducting material on either side of the cata-lyst and membrane in the fuel cell. Mathematical analysisof the models highlighted the sensitivity of fuel cell per-formance to certain GDE parameters, giving insight intothe performance of various possible GDE materials.Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences16The Mathematics of VotingContributed by Florin Diacu,PIMS Site Director, University of VictoriaDid your vote in the recent federal election convey yourwill? Think of the ballot. You had to say: I like candidatex and I reject all the others. You may have wished to makey your second choice or tell that z was unacceptable. Un-fortunately you couldn’t. Our system is not that flexible.Let us improve it then. But how?Donald SaariIn a recent lecture given atthe University of Victoria, Don-ald Saari, a distinguished math-ematician from the University ofCalifornia at Irvine, addressed thisissue and showed the advantagesand drawbacks of different democ-racies. Is there a best voting sys-tem, and if so, how good is it?The problem is difficult. An entirebranch of mathematics is research-ing it today. Let us follow some ideas and see how we canuse them.The beginnings of voting are lost in history. Writtensources attest to the existence of voting procedures in An-tiquity and all through the Middle Ages. Confusion inchoosing the right system was common. In 1130, for ex-ample, the ambiguity of voting led to the election of twoPopes, an event that created a rift within the CatholicChurch.In 1770 the French mathematician Jean-Charles Borda(1733–1799) proposed a new rule. He asked that votersassign points according to their ranking of the candidates.For example, in a 3 candidate election, the first ranked ona ballot received 3 points, the second obtained 2, and thethird got 1. The candidate with more points won.But we could use different point rules: 6 for the firstplace, 5 for the second, and 0 for the third; or 10 for thefirst, 2 for the second, and 1 for the third. In fact ourpresent system gives 1 for the first place and 0 for theothers. Which one is better?Though we can see some pros and cons in each case,it is hard to choose the best. But mathematicians foundthe answer. They have bad and good news for us. Thebad news: Borda’s count 3,2,1 is not ideal; it can stilllead to distorted results. The good news: within the pointmethod, the Borda count is by far the best. Moreover, ourvoting rule 1,0,0 is the worst; it gives the least amount ofinformation about what voters want and can yield resultsthat speak against the people’s will.This becomes clear from the following examples. In1970 the centre-right candidate Buckley won the New Yorksenate election even though more than 60% of the voteswent for either of the two centre-left candidates. A lessobvious but even more disturbing case is the recent Bush-Gore race. If those voting for Nader could have made Goretheir second choice (which is a reasonable assumption), thedemocrats would have won without trouble, as the popularvote suggests.There are more complicated systems. The run-offmethod, for example, uses the 1,0,0 point rule in com-bination with several rounds of vote. Only a more than50% support makes the winner. Otherwise the last candi-date is dropped and the vote is repeated. An alternativeis to exclude all but the first two candidates and vote asecond time.This system, however, has its flaws too. The first ver-sion takes too long to be efficient in a national election,whereas the second can bring weird outcomes. In the 26November 2000 election for the Romanian Presidency, thismethod led to a run-off between a left wing extremist anda right wing one. The centre vote had been split amongseveral candidates.The only simple and efficient method that in mostcases expresses the will of the majority is the Borda count.Ranking the candidates and assigning a balanced rule ofpoints, as in the 3,2,1 example, would make our electionsfairer. Further, information on the mathematics of votingmay be found in the following references:D. G. Saari, Basic Geometry of Voting (Springer Verlag, 1995).D. Black, The Theory of Committees and Elections (CambridgeUniversity Press, 1958).Streaming video of an earlier lecture given by D. Saariat the PIMS Opening Meeting (University of Vic-toria, October 4, 1996) is available on the webpagewww.pims.math.ca/video.Strings and D-branesContinued from page 13.The AdS/CFT (CFT stands for ‘Conformal FieldTheory’) correspondence was generalized to accom-modate some other types of gauge theories, and a va-riety of dynamical phenomena in gauge theories wereshown to have their counterpart in gravity. Never-theless, many questions in the gauge theory/gravitycorrespondence still remain open. In particular, it isnot known how far the AdS/CFT correspondence canbe extended beyond the conformally invariant gaugetheories and the strong coupling limit. The study ofthese issues may lead to further insights on the inter-play between gauge theories and gravity.References:B. Greene, The Elegant Universe: Superstrings, hidden di-mensions, and the quest for the ultimate theory (Norton,1999).M. B. Green, J. H. Schwarz and E. Witten, SuperstringTheory (Cambridge Univ. Press, 1987).J. Polchinski, String Theory : Vol. 1, An Introduction tothe bosonic string and Vol. 2, Superstring theory and be-yond (Cambridge Univ. Press, 1998).O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri andY. Oz, “Large N field theories, string theory and gravity,”Physics Reports, 323, 183 (2000).The Paci£c Institute for the Mathematical Sciences Winter 200117The Amazing Number piContributed by Peter Borwein,Dept. of Math. and Statistics, Simon Fraser UniversityThe history of pi parallels virtually the entire history ofMathematics. At times it has been of central interest andat times the interest has been quite peripheral (no punintended). Certainly Lindemann’s proof of the transcen-dence of pi was one of the highlights of nineteenth centurymathematics and stands as one of the seminal achieve-ments of the millennium (very loosely this result says thatpi is not an easy number). One of the low points was theIndiana State legislatures attempts to legislate a value ofpi in 1897: an attempt as plausible as repealing the law ofgravity.The amount of human ingenuity that has gone intounderstanding the nature of pi and computing its digits isquite phenomenal and begs the question “why pi?”. Afterall there are more numbers than one can reasonably con-template that could get a similar treatment. Furthermore,pi is just one of the very infinite firmament of numbers.Part of the answer is historical. It is the earliest and themost naturally occurring hard number (technically, hardmeans transcendental which means not the solution of asimple equation). Even the choice of label “transcenden-tal” gives it something of a mystical aura.What is pi? First and foremost it is a number, between3 and 4 (3.14159...). It arises in any computations involv-ing circles: the area of a circle of diameter 1 or equivalently,though not obviously, the perimeter of a circle of diameter1/2. The nomenclature pi is presumably the Greek letter“p” in periphery. The most basic properties of pi were un-derstood in the period of classical Greek mathematics bythe time of the death of Archimedes in 212 BC.The Greek notion of number was quite different fromours, so the Greek numbers were our whole numbers: 1,2,3... . In Greek geometry the essential idea was not num-ber but continuous magnitude, e.g. line segments. It wasbased on the notion of multiplicity of units and, in thissense, numbers that existed were numbers that could bedrawn with just an unmarked ruler and compass. Therules allowed for starting with a fixed length of 1 and seeingwhat could be constructed with straight edge and compassalone. (Our current notion is much more based on count-ing.) The question of whether pi is a constructible magni-tude had been explicitly raised as a question by the sixthcentury BC and the time of the Pythagoreans. Unfortu-nately pi is not constructible, though a proof of this wouldnot be available for several thousand years. In this contextthere isn’t a more basic question than “is pi a number?”Of course, our more modern notion of number embracesthe Greek notion of constructible and doesn’t depend onconstruction. The existence of pi as a number given by aninfinite (albeit unknown) decimal expansion poses littleproblem.Very early on the Greeks had hypothesized that piwasn’t constructible, Aristophanes already makes fun ofBabylonians 2000? BCE 1 3.125 (3 18 )Egyptians 2000? BCE 1 3.16045 (4( 89 )2)China 1200? BCE 1 3Bible (1 Kings 7:23) 550? BCE 1 3Archimedes 250? BCE 3 3.1418 (ave.)Hon Han Shu 130 AD 1 3.1622 (=√10 ?)Ptolemy 150 3 3.14166Chung Hing 250? 1 3.16227 (√10)Wang Fau 250? 1 3.15555 ( 14245 )Liu Hui 263 5 3.14159Siddhanta 380 3 3.1416Tsu Ch’ung Chi 480? 7 3.1415926Aryabhata 499 4 3.14156Brahmagupta 640? 1 3.162277 (=√10)Al-Khowarizmi 800 4 3.1416Fibonacci 1220 3 3.141818Al-Kashi 1429 14Otho 1573 6 3.1415929Viete 1593 9 3.1415926536 (ave.)Romanus 1593 15Van Ceulen 1596 20Van Ceulen 1610 35Newton 1665 16Sharp 1699 71Seki 1700? 10Kamata 1730? 25Machin 1706 100De Lagny 1719 127 (112 correct)Takebe 1723 41Matsunaga 1739 50Vega 1794 140Rutherford 1824 208 (152 correct)Strassnitzky and Dase 1844 200Clausen 1847 248Lehmann 1853 261Rutherford 1853 440Shanks 1874 707 (527 correct)History of calculating the digits of pi (Pre 20th Century).“circle squarers” in his fifth century BC play “The Birds.”Lindemman’s proof of the transcendence of pi in 1882settles the issue that pi is not constructible by the Greekrules and a truly marvelous proof was given a few yearslater by Hilbert. Not that this has stopped cranks fromstill trying to construct pi.Does this tell us everything we wish to know about pi.No, our ignorance is still much more profound than ourknowledge! For example, the second most natural hardnumber is e which is provably transcendental. But whatabout pi+e? This embarrassingly easy question is currentlytotally intractable (we don’t even know how to show thatpi+e is irrational). The number pi is a mathematical appleand e is a mathematical orange and we have no idea howto mix them.Why compute the digits of pi? Sometimes it is neces-sary to do so, though hardly ever more than the 6 or sodigits that Archimedes computed several thousand yearsago are needed for physical applications. Even far fetchedcomputations like the volume of a spherical universe onlyrequire a few dozen digits. There is also the “Everest Hy-pothesis” (“because its there”). Probably the number ofpeople involved and the effort in time has been similar inthe two quests. A few thousand people have reached thecomputational level that requires the carrying of oxygen– though so far I know of no pi related fatalities. ThereVol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences18has been significant knowledge accumulated in this slightlyquixotic pursuit. But this knowledge could have been de-rived from computing a host of other numbers in a varietyof different bases. Once again the answer to “why pi” islargely historical and cultural. These are good but notparticularly scientific reasons. Pi was first, pi is hard andpi has captured the educated imagination. (Have you everseen a cartoon about log 2 - a number very similar to pi?)Whatever the personal motivations pi has been muchcomputed and a surprising amount has been learnt alongthe way.In constructing the all star hockey team of great math-ematicians, there seems to be pretty wide agreement thatthe front line consists of Archimedes, Newton, and Gauss.Both Archimedes and Newton invented methods for com-puting pi. In Newton’s case this was an application ofhis newly invented calculus. I know of no such calcula-tion from Gauss though his exploration of the Arithmetic-Geometric mean iteration laid the foundation of the mostsuccessful methods for doing such calculations. There isless consensus about who comes next. I might add Hilbertand Euler next (on defense). Both of these mathematiciansalso contribute to the story of pi. Perhaps von Neumannis in goal – certainly he is a candidate for the most versa-tile and smartest mathematician of the twentieth century.One of the first calculations done on ENIAC (one of thefirst real computers) was the computations of roughly athousand digits of pi and von Neumann was part of theteam that did the calculation.One doesn’t often think of a problem like this havingeconomic benefits. But as is often the case with pure math-ematics and curiosity driven research the rewards can besurprising. Large recent records depend on three things:better algorithms for pi; larger and faster computers; andan understanding of how to do arithmetic with numbersthat are billions of digits longThe better algorithms are due to a variety of peopleincluding Ramanujan, Brent, Salamin, the Chudnovskybrothers and ourselves. Some of the mathematics is bothbeautiful and subtle. (The Ramanujan type series listedin the appendix are, for me, of this nature.)The better computers are, of course, the most salienttechnological advance of the second half ot the twentiethcentury.Understanding arithmetic is an interesting and illumi-nating story in its own right. A hundred years ago we knewhow to add and multiply — do it the way we all learnedin school. Now we are not so certain except that we nowknow that the “high school method” is a disaster for mul-tiplying really big numbers. The mathematical technologythat allows for multiplying very large numbers together isessentially the same as the mathematical technology thatallows image processing devices like CAT scanners to work(FFTs). In making the record setting algorithms work,David Bailey tuned the FFT algorithms in several of thestandard implementations and saved the US economy mil-lions of dollars annually. Most recent records are set whennew computers are being installed and tested. (Recentrecords are more or less how many digits can be computedin a day — a reasonable amount of test time on a costlymachine.) The computation of pi seems to stretch themachine and there is a history of uncovering subtle andsometimes not so subtle bugs at this stage.What do the calculations of pi reveal and what doesone expect? One expects that the digits of pi should lookrandom — that roughly one out each ten digits should bea 7, etc. This appears to be true at least for the first fewhundred billion. But this is far from a proof — an actualproof of this is way out of the reach of current mathe-matics. As is so often the case in mathematics some ofthe most basic questions are some of the most intractable.What mathematicians believe is that every pattern possi-ble eventually occurs in the digits of pi — with a suitableencoding the Bible is written in entirety in the digits, as isthe New York phone book and everything else imaginable.The question of whether there are subtle patterns inthe digits is an interesting one. (Perhaps every billionthdigit is a seven after a while. While unlikely, this is notprovably impossible. Or perhaps pi is buried within piin some predictable way.) Looking for subtle patterns inlong numbers is exactly the kind of problem one needs totackle in handling the human genome (a chromosome isjust a large number base 4, at least to a mathematician).I have included two tables giving a chronology of thecomputation of digits of pi. They are from David H. Bai-ley, Jonathan M. Borwein, Peter B. Borwein, and SimonPlouffe, “The Quest for Pi,” (June, 1996) The Mathemat-ical Intelligencer. The first table shows pre 20th centurycomputations and the second shows computations done inthe 20th century. The chronology in these tables is for theproblem of computing all of the initial digits of pi. Thereis also a shorter chronology of computing just a few verydistant bits of pi. The record here is 40 trillion and isdue to Colin Percival using the methods described in thereference above. It is surprising that this is possible at all.I have also included a list of significant mathematicalformulae related to pi. It is taken from Lennert Berggren,Jonathan M. Borwein and Peter B. Borwein, Pi: A SourceBook (Springer-Verlag, 1988)†.This text accompanied an address given at the celebra-tion to replace the lost tombstone of Ludolph van Ceulenat the Pieterskerk (St Peter’s Church) in Leiden on thefifth of July, 2000. A version of this paper appears in theNieuw Archief voor Wiskunde 1 (2000), pp. 254–258.It honours the particular achievements of Ludolph aswell as the long and important tradition of intellectualinquiry associated with understanding the number pi andnumbers generally. The author would like to pay tributeto the mathematical community of the Netherlands on theoccasion of its honouring one of its founding fathers themathematician, Ludolph van Ceulen (1540–1610).†Reproduced by permission of Springer-Verlag, New York.The Paci£c Institute for the Mathematical Sciences Winter 200119Selected Formulae Related to piArchimedes (ca 250 BCE):Let a0 := 2√3, b0 := 3 andan+1 :=2anbnan + bnand bn+1 :=√an+1bn.Then an and bn converge linearly to pi (with an error O(4−n).)Francois Vie`te (ca 1579):2pi =√12√12 +12√12√√√√12 +12√12 +12√12 · · ·John Wallis (ca 1650):pi2 =2 · 2 · 4 · 4 · 6 · 6 · 8 · 8 · · ·1 · 3 · 3 · 5 · 5 · 7 · 7 · 9 · · ·William Brouncker (ca 1650):pi = 41 + 12+ 92+ 252+···Ma¯dhava, James Gregory, Gottfried Wilhelm Leibnitz(1450–1671):pi4 = 1−13 +15 −17 + · · ·Isaac Newton (ca 1666):pi = 3√34 + 24(23 · 23 −15 · 25 −128 · 27 −172 · 29 − · · ·)Machin Type Formulae (1706–1776):pi4 = 4 arctan(15)− arctan(1239 )pi4 = arctan(12) + arctan(13)pi4 = 2 arctan(12)− arctan(17)pi4 = 2 arctan(13) + arctan(17)Leonard Euler (ca 1748):pi26 = 1 +122 +132 +142 +152 + · · ·pi490 = 1 +124 +134 +144 +154 + · · ·pi26 = 3∞∑m=11m2(2mm)Srinivasa Ramanujan (1914):1pi =∞∑n=0(2nn)3 42n+ 5212n+4 .1pi =√89801∞∑n=0(4n)!(n!)4[1103 + 26390n]3964nEach additional term of the latter series adds roughly 8 digits.Ferguson 1946 620Ferguson Jan. 1947 710Ferguson and Wrench Sep. 1947 808Smith and Wrench 1949 1,120Reitwiesner et al. (ENIAC) 1949 2,037Nicholson and Jeenel 1954 3,092Felton 1957 7,480Genuys Jan. 1958 10,000Felton May 1958 10,021Guilloud 1959 16,167Shanks and Wrench 1961 100,265Guilloud and Filliatre 1966 250,000Guilloud and Dichampt 1967 500,000Guilloud and Bouyer 1973 1,001,250Miyoshi and Kanada 1981 2,000,036Guilloud 1982 2,000,050Tamura 1982 2,097,144Tamura and Kanada 1982 8,388,576Kanada, Yoshino and Tamura 1982 16,777,206Ushiro and Kanada Oct. 1983 10,013,395Gosper 1985 17,526,200Bailey Jan. 1986 29,360,111Kanada and Tamura Sep. 1986 33,554,414Kanada and Tamura Oct. 1986 67,108,839Kanada, Tamura, Kubo, et. al Jan. 1987 134,217,700Kanada and Tamura Jan. 1988 201,326,551Chudnovskys May 1989 480,000,000Chudnovskys Jun. 1989 525,229,270Kanada and Tamura Jul. 1989 536,870,898Kanada and Tamura Nov. 1989 1,073,741,799Chudnovskys Aug. 1989 1,011,196,691Chudnovskys Aug. 1991 2,260,000,000Chudnovskys May 1994 4,044,000,000Takahashi and Kanada Jun. 1995 3,221,225,466Kanada Aug. 1995 4,294,967,286Kanada Oct. 1995 6,442,450,938Kanada Jun. 1997 51,539,600,000Kanada Sep. 1999 206,158,430,000History of calculating the digits of pi (20th Century).Louis Comtet (1974):pi490 =3617∞∑m=11m4(2mm)Eugene Salamin and Richard Brent (1976):Set a0 = 1, b0 = 1/√2 and s0 = 1/2. For k = 1, 2, 3, . . . com-puteak =ak−1 + bk−12 sk = sk−1 − 2kckbk =√ak−1bk−1 pk =2a2kskck = a2k − b2kThen pk converges quadratically to pi.Jonathan Borwein and Peter Borwein (1985):Set a0 = 6− 4√2 and y0 =√2− 1. Iterateyk+1 =1− (1− y4k)1/41 + (1− y4k)1/4ak+1 = ak(1 + yk+1)4 − 22k+3yk+1(1 + yk+1 + y2k+1)Then ak converges quartically to 1/pi.David Chudnovsky and Gregory Chudnovsky (1989):1pi =12∞∑n=0(−1)n (6n)!(n!)3(3n)!13591409+n545140134(6403203)n+1/2Each additional term of the series adds roughly 15 digits.Please see Selected Formulae, page 25.Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences20Report on the PIMS Algebra 2000 SummerSchool at the University of AlbertaContributed by Akbar Rhemtulla, Robert Moody, and BruceAllison, Dept. of Math. Sciences, University of Alberta.The PIMS Thematic Programme in Algebra called Alge-bra 2000, took place at the University of Alberta over a4-week period from June 19 to July 14. The programmeconsisted of summer schools and workshops in three areasof algebra: Lie Theory, Group Theory and Representa-tions and the Mathematics of Aperiodic Order. Each areawas featured for 2 (overlapping) weeks. The programmewas very successful and attracted over 100 participants.In the first week, the Lie Theory summer school fea-tured 2 series of introductory talks for graduate students.Stephen Donkin (Queen Mary and Westfield College) gave5 lectures on Algebraic Groups and Arturo Pianzola (Uni-versity of Alberta) gave 5 lectures on Lie Algebras. TheLie Theory workshop in the second week included one-hourtalks by experts from Europe, United States and Canadaon recent developments in the subject. Topics includedvertex operator algebras and various infinite dimensionalgeneralizations of finite dimensional simple Lie algebras.Many of the talks, as well as much informal discussion, fo-cused on the increasing interplay between Lie theory andmathematical physics.The Groups and Representations summer schoolstarted in the second week with introductory lectures forgraduate students given by Peter Kropholler (Queen Maryand Westfield College), Gerald Cliff (University of Al-berta), Alexander Turull (University of Florida). Theworkshop in the following week was devoted to hour longtalks. Half of these were in representation theory and therest in infinite groups. Topics included representations offinite groups, profinite groups and Burnside problems.In the third week of the programme we began the sum-mer school on Aperiodic Order. This school included fourlecture series (a total of 10 lectures):Michael Baake (Universita¨t Tu¨bingen): “Introductionto long-range aperiodic order”Jeffry Lagarias (AT & T Research Labs): “The geome-try of point sets”Boris Solomyak (University of Washington): “Tilingsand dynamical systems”Michael Baake (Universita¨t Tu¨bingen): “Stochastic andother directions”These beautifully prepared talks were of a consistentlyhigh standard. Because the subject is new and has quitediverse mathematical components, these lectures were en-joyed immensely by students and researchers alike. Inaddition, Uwe Grimm gave several hands-on computerdemonstrations illustrating the main features of aperiodictilings.The final week of the programme consisted of a work-shop on Aperiodic Order with 4 to 5 one-hour talks perday. The talks from the participants covered the en-tire spectrum of the subject from the spectral theory,through diffraction, substitution systems, automatic se-quences, random tilings, aperiodic Schro¨edinger operators,and aperiodic approaches to random number generators.Many of the participants in Algebra 2000 took advan-tage of the weekend between their summer school andworkshop to go to Jasper where they stayed in the Pal-isades Science Centre. This retreat provided continued dis-cussions throughout the weekend and helped enormouslyto develop the fine spirit of the entire programme.Report on the Fall 2000 Pacific NorthwestStatistical MeetingContributed by Carl Schwarz,Dept. of Mathematics and Statistics, SFUThe Pacific Northwest Statistical Meeting (PNWSM)for the fall of 2000 was held on November 17 at thenewest member of PIMS — the University of Washing-ton. The PNWSM gratefully acknowledges financial helpfrom PIMS to assist graduate students in attending thetalks.The first of two featured speakers was June Morita,University of Washington-Bothell who spoke on Contri-butions to Statistical Literacy. In her talk, she outlinedsome educational outreach activities being used to assistlocal teachers in understanding many of the new threads ofmathematics and statistics that are entering the curricu-lum. The age-level of the activities ranged from K-12, butsome of the activities could be used at all levels. The activ-ities ranged from edible classroom materials (histograms),to TrashBall (paired experiments), to the probability thata space object would hit the ocean rather than falling onland (survey sampling).The second speaker was Constance van Eeden (Uni-versity of British Columbia), who spoke on “Estimation inRestricted Parameter Spaces: Some History and Some Re-cent Developments”. The addition of seemingly trivial in-formation often makes development of a trivial proceduremuch more difficult. For example, consider the simple casewhere X has a N(θ,1) distribution and θ is to be estimatedwhen one knows that θ ≥ 0. That is, one is looking foran estimator θˆ of θ satisfying θˆ ≥ 0. Finding “good” esti-mators for such situations is a difficult problem. Standardresults, e.g. maximum likelihood estimators become inad-missible, but estimators that do dominate the MLE haverecently been found.The PNWSM are held three times per year rotat-ing among the universities in British Columba, Albertaand Washington state. The next meeting is scheduledfor this spring and will be held at Simon Fraser Univer-sity. For details about the next meeting, please checkwww.stat.sfu.ca/stats.The Paci£c Institute for the Mathematical Sciences Winter 200121Report on the PIMS PDF WorkshopThe third Annual PIMS Postdoctoral Fellow Workshopwas hosted by PIMS-SFU on December 9–10, 2000. Thefirst day of this two-day workshop was held in the PIMSfacility on the SFU main campus and the second day washeld at the SFU Harbour Centre in downtown Vancouver.From left, Jorgen Rasmussen (U. Lethbridge), SandyRutherford (PIMS Sci. Exec. Officer) and Ji-Guang Bao(PIMS-UBC) at the PIMS PDF Workshop.This workshop gives PIMS postoctoral fellows an op-portunity to present their research to each other in aninformal setting. The following seventeen talks were pre-sented over the course of two very long days:Siva Athreya (PIMS-UBC): “Ballistic deposition on a planarstrip”Ji-Guang Bao (PIMS-UBC): “Surfaces with Prescribed Gaus-sian Curvature”Ricardo Carretero (PIMS-SFU): “Bose-Einstein conden-sates: breathing lumps of coherent matter”Wai-Shun Cheung (PIMS-UC): “Introduction to NumericalRange and Its Generalizations”Antal Jarai (PIMS-UBC) “Invasion percolation and the in-cipient infinite cluster”Benjamin Klopsch (PIMS-UA): “Counting Groups — allyour fingers are needed”Luis Lehner (PIMS-UBC): “Numerical Relativity: a labora-tory for General Relativity”Sam Lightwood (PIMS-UVic): “Embedding Theorems ford-Dimensional Symbolic Dynamics, d > 1”Miro Powojowski (PIMS-UC): “Random processes andstochastic PDEs in applications”Jorgen Rasmussen (U. Lethbridge): “Algebraic aspects andstring theory applications”Sujin Shin (PIMS-UVic): “Invariant Measures for PiecewiseConvex Transformations of an Interval”Ladislav Stacho (PIMS-SFU): “On factor d-domatic color-ings of graphs”Joachim Stadel (PIMS-UVic): “The Big and Small N of As-trophysical N-body Simulations”Sumati Surya (PIMS-UBC): “Topology Change and CausalContinuity”Yuqing Wang (PIMS-UBC): “Synchronous phase clusteringin pulse-coupled neurons with spatially decaying excitatorycoupling”Bert Wiest (PIMS-UBC): “Orderability of fundamentalgroups of 3-manifolds”Konstantin Zarembo (PIMS-UBC): “On quantization con-dition for a magnetic charge”Second Pacific Rim Conference onMathematicsContributed by Robert Miura,Dept. of Mathematics, UBCApproximately 150 mathematicians from Australia,Canada, China, France, Hong Kong, India, Japan, Ko-rea, New Zealand, the Philipines, Singapore, Switzerland,Tajikstan, the United States, and Uzbekistan attendedthe Second Pacific Rim Conference on Mathematics onJanuary 4–8, 2001 at Academia Sinica in Taipei, Taiwan.The six main themes of the Conference were Combina-torics, Computational Mathematics, Dynamical Systems,Integrable Systems, Mathematical Physics, and NonlinearPartial Differential Equations.There were 12 one-hour plenary talks, approximatelyforty 45 minute invited talks, and 55 contributed papers.The plenary talks were excellent with each speaker givinga general background for the audience and then present-ing more details later in the talk. The Plenary Speakerswere Ian Affleck (UBC), Craig Evans (UC Berkeley), JoelFeldman (UBC), Genghua Fan (Academia Sinica, China),Alberto Grunbaum (UC Berkeley), Song-Sun Lin (ChiaoTung University, Taiwan), Junkichi Satsuma (Univ. ofTokyo), Leon Simon (Stanford), Stephen Smale (City U,Hong Kong), Gilbert Strang (MIT), Yingfei Yi (GeorgiaTech), and Xuding Zhu (Sun Yat-Sen University, Taiwan).The two plenary speakers from Canada were in theMathematical Physics Session, along with Izabella ÃLaba(UBC), Robert McCann (Toronto), and Gordon Se-menoff (UBC), who were invited speakers. Brian Alspach(Regina) and Rong-Qing Jia (Alberta) were invited speak-ers in the Combinatorics and Computational Mathemat-ics Sessions, respectively. The Canadian Representativeon the Organizing Committee was Robert Miura (UBC).PIMS provided support for the Canadian participants inthe conference.The local organizers from the Institute of Mathematicsat Academia Sinica led by Fon-Che Liu, former Directorof the Institute, and Tai-Ping Liu, current Director, did asuperb job of making sure that all the needs of the con-ference participants were met. All talks were given at theActivity Centre of Academia Sinica where most partici-pants had their accommodations. The lecture rooms wereideal and had a full complement of audio-visual equipment.The Conference Reception, Conference Banquet, and Ex-cusion to an art gallery, temple, pottery town, and a nightmarket were the highlights of the social activities.A committee meeting was held after the Conference Re-ception to discuss the site of the Third Pacific Rim Confer-ence on Mathematics and was attended by representativesfrom Australia, Canada, China, Hong Kong, Japan, Tai-wan, and the United States. It was proposed that the nextConference be held in Vancouver in the summer of 2004under the sponsorship of PIMS. This was accepted enthu-siastically and unanimously by the committee, as well asby the participants after it was announced at the Confer-ence Banquet.Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences22PIMS Graduate Information Week 2001Contributed by John Collins, University of Calgary and Jim Muldowney, University of Alberta.Visiting students attend a lecture at the University of Alberta.The Pacific Institute for the Mathematical SciencesGraduate Student Information Seminar, held atthe Universities of Alberta and Calgary on Jan-uary 9–12, was a great success. Twenty-four topfourth year undergraduates in mathematics, statis-tics, and computer science from universities allacross Canada arrived in Calgary on the Tuesdayafternoon.After a welcoming student/faculty mixer thatevening, the students were treated on Wednesdayto a full program of presentations about graduatestudies at the University of Calgary, including talksby research groups in discrete math, analysis, in-dustrial and collaborative mathematics, the mathfinance lab, the statistical consulting lab, computergraphics, quantum computing, and several others.The Dean of Graduate Studies, James Frideres,outlined some of the many attractions in studyingat Calgary, while the PIMS Deputy Director, Michael Lamoureux, described the advantages of joining the PIMS team ofwestern universities. The departments’ Director of Graduate Programs, John Collins, detailed the scholarship possibilitiesand amenities of each of the programs. Gary MacGillivray gave a presentation on programs at the University of Victoria.At a western-style dinner that evening, the Associate Dean of Science, Robert Woodrow, discussed additional fundingopportunities from the Government of Alberta that make graduate study in the province particularly rewarding forprospective students.After further informative sessions and meetings with faculty members on Thursday morning, the students went by busto Edmonton that afternoon. Dick Peter, Dean of Science, and Peter Steffler, Associate Dean of Graduate Studies, alongwith faculty and graduate students from the departments of Computing Science and Mathematical Sciences welcomedthem to the University of Alberta campus at a banquet at the Faculty Club. Bryant Moodie, PIMS University of AlbertaSite Director, gave a brief account of PIMS and its particular relevance to graduate studies in the mathematical sciences.Friday morning activities were kicked off with a presentation by Bob Moody (University of Alberta) on “GraduateStudies in Mathematical Sciences: 2001”†. Jim Hoover (Univ. of Alberta) talked about “The relationship between theoret-ical computer science and ‘standard’ mathematics”. Presentations on graduate studies at PIMS universities were given byStudents discuss with Akbar Rhemtulla, Chair, Dept. of Math.Sciences, University of Alberta.Denis Sjerve (UBC), Randy Sitter (SFU), Lorna Stewart(Univ. of Alberta) and Jim Muldowney (Univ. of Alberta).After a lunch with local CS and MathSci faculty andgraduate students, the visitors had a full afternoon of smallgroup meetings, interviews and tours scheduled to addresstheir individual interests. Over 100 meetings with local re-searchers and representatives of the other PIMS sites werearranged by PIMS staff.A farewell party and supper was held at the VarsconaHotel on Whyte after which some participants were said tohave explored the attractions of Whyte Avenue late intothe evening.Financial support for the seminar was provided byPIMS and each of the two host universities. Travel andaccommodation for the whole event as well as the Calgaryprogram were arranged by John Collins, Sheelagh Carpen-dale and Marian Miles (PIMS Administrator, Univ. of Cal-gary). Local arrangements in Edmonton were taken careof by Jim Muldowney, Lorna Stewart, Martine Bareil andLina Wang (PIMS Administrator, Univ. of Alberta).†The text of this lecture is reprinted in this newsletter.The Paci£c Institute for the Mathematical Sciences Winter 200123Women and Mathematics: New Posters and Contests in 2001Contributed by Krisztina Va´sa´rhelyiHypatia of Alexandria Ada Lovelace ByronBuilding on the momentum generated by theMathemat-ics is Everywhere poster campaign, PIMS is continuingthe project in 2001 with a new theme and new format.Klaus Hoechsmann’s innovative poster series, blending theappeal of a public contest with advertising on city transit,and the internet, has demonstrated that given the rightapproach, it is possible to rouse interest in the “terminallyunpopular”. The new series will attempt to convey a dif-ferent, yet from a public perspective, equally challengingmessage, utilising the proven formula of the previous se-ries.With the intention of introducing the public, and inparticular young people, to the idea that mathematics is acareer asset, a colourful palette of biographies will be pre-sented monthly. The poster seriesWomen and Mathe-matics will showcase portraits of twelve women who havemade contributions to the broad field of the mathemat-ical sciences. Biographies of famous historical figures aswell as accomplished contemporary mathematicians willbe presented together with women who are well-knownto the public but who are primarily recognised for theirachievements in fields other than mathematics.Mathematics is expanding rapidly beyond its tradi-tional domains. The importance of mathematics in en-gineering, physics, chemistry and computer science for ex-ample is obvious, but mathematical competence is becom-ing increasingly important in other fields including biology,medicine, public health, psychology and even journalism.With the growth of information technologies in all fields,the demand for mathematically trained individuals in thework force will continue to rise. Ironically, mathematicsstill suffers from a bad reputation. Fear and loathing ofthe subject is firmly established already at the elementaryschool level. The attitude that mathematics is a career ob-stacle continues to influence education choices. Girls areespecially susceptible to rejecting a course of study whichfavours mathematical content. The “smart girl” stigmaamong teenagers can be a powerful deterrent.The Women and Mathematics campaign willpresent an alternative, much more positive, image of math-ematics in the lives of women. Mathematics can involvelifelong dedicated research, it can be an enjoyable pursuitand it can represent a valuable tool in a variety of ende-vours. The last point is aptly illustrated by the case ofFlorence Nightingale. She is a prominent figure and rolemodel, widely acknowledged for her achievements in thefields of nursing and public health. Yet her perhaps lesswell known contributions to statistics have been pivotalto her other accomplishements. Possessing mathematicalskills is an asset and can enrich and promote success invirtually all fields. This is one of the main messages of thecampaign.The target audience for this project includes studentsin elementary and secondary schools as well as the gen-eral public of any age or gender. However, by focusing onwomen we want to draw attention to the problem of lowfemale participation in the mathematical sciences. There-fore, a primary and important goal of this project is toreveal to girls the appealing and attractive sides of a lifeinvolving mathematical study or activities.The women featured on the posters represent many lev-els of mathematical pursuit. The inclusion of establishedrole models is an attempt to create a link to other womennot well known to those outside of scientific circles andwhose work is not easily appreciated by non-specialists.Effort will be focused on presenting the topics in a colour-ful and interesting manner, exploiting the advantages ofthe visual medium. The posters will be designed to cap-ture the attention of the observer and invite further explo-ration on the website.Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences24The contest itself will promote internet-based bio-graphical research in addition to problem-solving. A set offive quiz questions will be posted on the contest website.One of the questions will be a mathematics problem, high-lighting the field of involvement of the featured individual.Answers to the remaining biographical questions can befound by searching the web. This approach will hopefullyencourage contestants to read and learn about women inmathematical pursuits. At the end of the month a shortbiography along with the correct answers will be posted onthe web and a prize will be drawn among the correct en-tries. The poster-campaign will be advertised in schools inBC and Alberta to encourage initiatives for class projects.The first two posters in the series will feature Hypatiaof Alexandria and Ada Lovelace Byron. Hypatia was thelast of the Alexandrian scholars and her murder by a fun-damentalists mob in 415 AD is often taken as the onset ofthe Dark Ages. Ada Lovelace Byron is widely regarded asone of the founders of computer science. In the 1840’s shewrote the first computer computer algorithm — an algo-rithm to compute the Bernoulli numbers on the AnalyticalEngine designed (but never built) by Charles Babbage.Izabella ÃLaba of the UBC Department of Mathematicswill oversee the project which will be carried out at thePIMS-UBC site in a a team effort by Heather Jenkins andKrisztina Va´sa´rhelyi, with input from Klaus Hoechsmann.Pi in the Sky: December 2000 issueThe second issue of the PIMS Education Magazine, Pi inthe Sky, features a variety of interesting articles as well aschallenging problems and numerous jokes which will inter-est young and old mathematicians alike. Akbur Rhemtullawrites about Counting with Base Two and the Game ofNum, and Byron Schmuland about the Collector’s Prob-lem. Learn about pi in The Number pi and the Earth’sCircumference by Wieslaw Krawcewicz. How many dig-its in the decimal expansion can you remember? Findout more about triangles in The Anatomy of Triangles byKlaus Hoechsmann. After reading Relating MathematicalIdeas to Simple Observations by Jim G. Timourian, tryyour hand at physics in The Slingshot Effect of CelestialBodies by Florin Diacu.The picture on the cover page is a fragment of apainting by prominent Russian mathematician AnatolyT. Fomenko which was inspired by mathematical ideas.Selected Formulae for PiContinued from page 20.Jonathan Borwein and Peter Borwein (1989):1pi = 12∞∑n=0(−1)n(6n)!(n!)3(3n)!(A+ nB)Cn+1/2 ,whereA := 212175710912√61 + 1657145277365B := 13773980892672√61 + 107578229802750C := [5280(236674 + 30303√61)]3.Each additional term of the series adds roughly 31 digits.Roy North (1989):Gregory’s series for pi, truncated at 500,000 terms gives toforty places4500,000∑k=1(−1)k−12k − 1= 3.141590653589793240462643383269502884197Only the underlined digits are incorrect.Jonathan Borwein and Peter Borwein (1991):Set a0 = 1/3 and s0 = (√3− 1)/2. Iteraterk+1 =31 + 2(1− s3k)1/3sk+1 =rk+1 − 12ak+1 = r2k+1ak − 3k(r2k+1 − 1)Then 1ak converges cubically to pi.David Bailey, Peter Borwein and Simon Plouffe(1996):pi =∞∑i=0116i( 48i+ 1 −28i+ 4 −18i+ 5 −18i+ 6)The Paci£c Institute for the Mathematical Sciences Winter 200125Mathematics on StageContributed by Klaus HoechsmannIn the spring of 1999, we at PIMS were wondering withwhat kind of special event we should mark the year 2000as the UNESCO World Year of Mathematics. Should wehost yet another great lecture, another workshop or paneldiscussion? The idea of a theatre performance presenteditself.Every practitioner knows that mathematical work,with its twists and turns, its sudden impasses and break-throughs, is highly dramatic, yet can such drama bestaged? There are several excellent recent plays (e.g., TomStoppard’s “Arcadia” and David Auburn’s “Proof”) whichdo revolve around mathematics — using it as a source ofanecdote and speculation, but never quite getting to thesource itself. Within our cultural horizon, we could notfind anything ready-made, so we turned to home cooking.Fortunately, we were able to enlist the guidance and exten-sive collaboration of Ted Galay, who is a mathematicianas well as an experienced playwright.The mathematics presented would have to be simple,precise, elegant, but without the formalisms which are thedreaded heart of “math” to graduates of modern schools.We looked to the past for a mathematical story whichwould meet these criteria. Our protagonist would be Hy-patia, the last leading intellectual of the cosmopolitan cityof Alexandria (Egypt): mathematician, astronomer, andphilosopher, savagely murdered in 415 AD by a funda-mentalist mob.A major proponent of Neoplatonism, she is said to havewandered in crowds of strangers to engage them in philo-sophical debate. The play imagines this outreach effortextended to mathematics itself — i.e., thought withoutideological wrapping. Nonetheless, she gets entangled inthe power struggle between burgeoning Church and mori-bund State. Half of the play deals with this conflict.The other half consists of three (originally four) math-ematical skits — staged by Hypatia and three likemindedfriends as comical street theatre. The first shows the well-known measuring of the earth’s circumference by Eratos-thenes. The third moves from the earth to the heavens —first estimating the distances between earth, moon, sun,and then showing how to gauge interplanetary distances— based on the ideas of Aristarchos. Between those two“applied” skits, the second one shows off the two majoringredients of mathematics: insight and reason — the for-mer in a take-off on Plato’s “Meno” dialogue, the latterin a hands-on proof of the impossibilty of arithmeticallydoubling a square.The third skit is the most ambitious, as it tries to showthe elegant simplicity of ancient thought. How far is themoon? By timing a solar eclipse, you see that it moveseastward by one diameter an hour. Therefore the numberof hours in a sidereal month tells you the length of its cir-cuit — hence also its distance from the earth — in moondiameters. Since a lunar eclipse lasts thrice as long, theearth is three times wider than the moon – and now youhave these distances in terms of earth diameters. Thenfollow Eratosthenes to convert it all to miles.In technical terms, the demonstrations on stage mainlyuse the visually obvious fact that small angles are roughlyproportional to their sines and tangents, and the moresubtle (but visually demonstrable) fact that the slices ofa regular 25-gon are about four times as high as they arewide, whence the ratio of circumference to radius is 254 , afairly good approximation to 2pi. In Hypatia’s time, it wasalready well known (cf. Claudius Ptolemy) that an astron-omy based on combinations of uniform circular motionsis a woefully coarse mathematical model. Its great ad-vantage, however, is that it opens the door to a relativelysophisticated understanding by the mathematical novice— and even allows some choreographic representation.Apart from its length (30 minutes), the most seriousshortcoming of the third skit is the underplaying of themain difference between the systems of Aristarchos andHipparchos: the question is not whether the earth movesor stands still, but whether the other planets move abouta common, observable centre (the sun) or about individ-ual, imaginary points on “deferent” circles. In our nextproduction of the play, we shall try to do this better witha slightly modified script.Even in its present state, the play was warmly receivedby an almost full house at UBC’s Frederic Wood The-atre, on December 10, 2000. Available funds allowed onlya staged reading — with off-book action in the mathe-matical skits, and minimal props and costumes — but anamazingly quick-witted group of professional actors, whowere also eager learners of mathematics, and a directorwhose generosity was matched only by his command ofstage craft.Director Bryan Wade (left) with author Klaus Hoechsmann.For further details, including the complete script,please see www.pims.math.ca/education/drama.Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences26Mathematics is Everwhere Campaign ConcludesContributed by Krisztina Va´sa´rhelyiThe last contest in the Mathematics is Everywhereposter campaign is currently underway with an image ofthe leaning tower of Pisa and a question about calculus.On the buses in January.The contest series has been very successful and itsclosing is marked by the publication of a year 2001 wallcalendar. The calendar, designed by Heather Jenkins,PIMS Communications Officer, was distributed to numer-ous schools and mathematics departments in Canada andUSA. It is a complete collection of the pictures and asso-ciated questions of the poster campaign.PIMS Calendar for 2001.In previous issues of the PIMS Newsletter we ran pro-files of the monthly contests from February to Septem-ber 2000. In October, reflected images of a puppet be-tween two mirrors were used to explore “mappings”. PaigeZanewick from Calgary won that contest. Paige is a 14year old student in Grade 8 who has always enjoyed mathand tries to incorporate it into her life every day. Paigesays that if she had a choice in selecting her courses, shewould choose all those that deal with mathematics. In ad-dition to her love of math, Paige enjoys horse back riding,skating, skiing, tennis, badminton, soccer and surfing thenet. She learnt about the contest from her teacher, whochallenged the students to participate and, as a reward,offered a free period to all if someone from the class wonthe $100 prize. In her own words, Paige thinks “...thatthis contest is a really great way to get interested in math.It challenges people to try their hardest to figure out theanswer to the question”.On the buses in October.Bus passengers in November saw a child with twodifferent-sized cubes, both made out of small blocks.“With how many extra blocks can Carlo make his twosolid cubes into a single big one?” Those curious enoughto follow the links on the contest webpage were treated toa thrilling detective story, spanning over 300 years, whichculminated in the triumphant proof of Fermat’s Last The-orem by Andrew Wiles. The November winner was YakovShklarov of Calgary.On the buses in November.In December an image of bees on a honeycomb (con-tributed by C. Keeling of SFU) was shown to representa packing problem. “If 4% of the distance betweeen cellcentres is wax, how much of the total surface is wax?” Thewinner of the December contest was Vancouver native TomWatson (43), who is employed at the Canada Customs andRevenue Agency. Tom has always liked numbers but notesthat there are many who claim to have hated math inschool because they don’t understand the subject. Whilehe had a natural affinity for mathematics from an early ageon, Tom says that kids may become interested if the topicwas presented in a more exciting way. Tom remembereda teacher from elementary school who regularly gave stu-dents a math problem to solve before recess and the winnerrecieved a chocolate bar or, at the very least, was allowedto leave the classroom first! Tom was alerted to the posterby a relative who spotted it on the bus and he says he findsthe contest a good idea. Tom’s other interests include mu-sic, old movies, puzzles and of course anything that makeshim think.On the buses in December.The Paci£c Institute for the Mathematical Sciences Winter 200127Rita Aggarwala, a leader of tomorrowThe Alberta Science and TechnologyFoundation has selected Rita Aggar-wala as the recipient of their newLeader of Tomorrow Award. Ag-garwala is an associate professor inthe Department of Mathematics andStatistics at the University of Calgary.“It was exciting,” Aggarwala saysof the gala event on October 20 at theShaw Conference Centre, Edmonton,where she was presented the award byLorne Taylor, Minister for Innovationand Science. “It’s great for mathemat-ics and statistics to be recognised; andit was very inspiring to see all the amazing research going on.”At the 11th annual awards ceremony, approximately 20 differ-ent individuals, businesses or research groups were finalists in 10categories including Technology, Industrial Research Innovation,and Public Awareness. In the Leader of Tomorrow categorythere were 2 other finalists, John Doucette, a graduate student atthe University of Alberta who works in the engineering field, andVeer Gidwaney of Calgary who at just 21 is on his third companyproducing business-to-business software.At the age of 29, Aggarwala has already established herself as aleader in her field of research which includes progressive censoring,including its application to warranty development, applied statis-tics and statistical quality improvement. She accelerated throughhigh school and university then joined the math faculty at theUniversity of Calgary as an assistant professor at the age of 24 andwas granted tenure at 28. Her young career has been very produc-tive with numerous publications including a recently co-authoreda book which is aimed at statistical practitioners in research andindustry who need sophisticated scientific tools to analyse data inthe development of their products and services.In her acceptance speech, Rita thanked PIMS for its impacton her own career and on the research life at the University ofCalgary. In particular, she singled out the support of PIMS to theStatistical Consulting and Research Group at the Universityof Calgary. This group was established with the aim of providinggraduate students and other academics-in-training with valuablecomputing, applied research and consulting experience. This isachieved by working with real, current data and design problemsposed by other experimental researchers and industrial organisa-tions who are in search of this type of expertise. Students benefitfrom the these “real-world” problems which are often messier thanthe textbook examples and a few of these have resulted in researchprojects for graduate students.®©ªMailing AddressPIMS Contact ListDirector: Dr. N. GhoussoubAdmin. Asst: Jacquie BurianPhone: (604) 822–9328, Fax: 822–0883Email: director@pims.math.caDeputy Director andUC-site Director: Dr. M. LamoureuxAdmin. Asst: Marian MilesPhone: (403) 220-3951, Fax: 282-5150Email: uc@pims.math.caSFU-site Director: Dr. R. RussellAdmin. Asst: Sadika JungicPhone: (604) 268–6655, Fax: 268–6657Email: sfu@pims.math.caUBC-site Director: Dr. D. RolfsenAdmin. Asst: Leslie MacFaddenPhone: (604) 822–3922, Fax: 822–0883Email: ubc@pims.math.caUA-site Director: Dr. B. MoodieAdmin. Asst: Lina WangPhone: (780) 492–4308, Fax: 492–1361Email: ua@pims.math.caUVic-site Director: Dr. F. DiacuAdmin. Asst: Irina GavrilovaPhone: (250) 472–4271, Fax: 721–8962Email: uvic@pims.math.caUW-site Director: Dr. T. ToroAdmin. Asst: Mary SheetzPhone: (206) 543–1173, Fax: 543–0397Email: uw@pims.math.caScientific Executive Officer: Dr. A. RutherfordPhone: (604) 822–1369, Fax: 822–0883Email: sandy@pims.math.caEducation Coordinator: Dr. K. HoechsmannPhone: (604) 822–3922, Fax: 822–0883Email: hoek@pims.math.caAsst. to the Director: Katrina SohPhone: (604) 822–6851, Fax: 822–0883Email: katrina@pims.math.caPIMS/MITACS Website Manager: Kelly ChooPhone: (250) 472–4927, Fax: 721–8962Email: chook@pims.math.caComputer Systems Manager: Shervin TeymouriPhone: (604) 822–0410, Fax 822-0883Email: shervin@pims.math.caComputer Systems Manager: Brent KearneyPhone: (604) 268-6654, Fax: 268-6657Email: brentk@pims.math.caCommunications Officer: Heather JenkinsPhone: (604) 822–0402, Fax: 822–0883Email: heather@pims.math.caPIMS-MITACS Admin. Asst., UBC: Clarina ChanPhone: (604) 822-0401, Fax: 822–0883Email: clarina@pims.math.caPIMS-MITACS Admin. Asst., UA: Martine BareilPhone: (780) 492-4835, Fax: 492–1361Email: mbareil@ualberta.caNewsletter Editor: A. RutherfordAssistant Editor: H. JenkinsThis newsletter is available on the world wideweb at www.pims.math.ca/publications.Vol. 5, Issue 1 The Paci£c Institute for the Mathematical Sciences28
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Newsletter (Pacific Institute for the Mathematical Sciences) : vol. 5, issue 1, Winter 2001 Pacific Institute for the Mathematical Sciences 2001
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Title | Newsletter (Pacific Institute for the Mathematical Sciences) : vol. 5, issue 1, Winter 2001 |
Alternate Title | Pacific Institute for the Mathematical Sciences newsletter |
Creator |
Pacific Institute for the Mathematical Sciences |
Publisher | Pacific Institute for the Mathematical Sciences |
Date Issued | 2001 |
Subject |
Mathematics--Research--Periodicals |
Genre |
Other |
Type |
Text |
Language | eng |
Date Available | 2014-05-29 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 2.5 Canada |
DOI | 10.14288/1.0085989 |
URI | http://hdl.handle.net/2429/46874 |
Affiliation |
Science, Faculty of Mathematics, Department of Non UBC |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty Researcher Other |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/2.5/ca/ |
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- Full Text
- 46718-1.0085989-fulltext.txt
- Citation
- 46718-1.0085989.ris
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