NEWSLETTER No. 37

September 2000

EMS News : Agenda, Editorial, 3ecm report, Summer Schools ...................... 2

Mathematical Modelling in the Biosciences (Philip Maini) .......................... 16

EMS Poster Competition ............................................................................... 19

Interview with Martin Grötschel ................................................................... 20

Interview with Bernt Wegner ........................................................................ 24

A stamp for World Mathematical Year ..........................................................27

Societies: The London Mathematical Society ................................................ 28

Problem Corner ............................................................................................. 30

Forthcoming Conferences .............................................................................. 32

Recent Books .................................................................................................. 36

Designed and printed by Armstrong Press LimitedCrosshouse Road, Southampton, Hampshire SO14 5GZ, UK

phone: (+44) 23 8033 3132; fax: (+44) 23 8033 3134Published by European Mathematical Society

ISSN 1027 - 488X

NOTICE FOR MATHEMATICAL SOCIETIESLabels for the next issue will be prepared during the second half of November 2000. Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department ofMathematics, P.O. Box 4, FIN-00014 University of Helsinki, Finland; e-mail:makelain@cc.helsinki.fi

INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTERInstitutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat,Department of Mathematics, P. O. Box 4, FI-00014 University of Helsinki, Finland, or by e-mail: (makelain@cc.helsinki.fi). Please include the name and full address (with postal code), tele-phone and fax number (with country code) and e-mail address. The annual subscription fee(including mailing) is 60 euros; an invoice will be sent with a sample copy of the Newsletter.

EDITOR-IN-CHIEFROBIN WILSONDepartment of Pure MathematicsThe Open UniversityMilton Keynes MK7 6AA, UKe-mail: r.j.wilson@open.ac.uk

ASSOCIATE EDITORSSTEEN MARKVORSENDepartment of Mathematics Technical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: s.markvorsen@mat.dtu.dkKRZYSZTOF CIESIELSKIMathematics Institute Jagiellonian UniversityReymonta 4 30-059 Kraków, Polande-mail: ciesiels@im.uj.edu.plKATHLEEN QUINNThe Open University [address as above]e-mail: k.a.s.quinn@open.ac.uk

SPECIALIST EDITORSINTERVIEWSSteen Markvorsen [address as above]SOCIETIESKrzysztof Ciesielski [address as above]EDUCATIONVinicio VillaniDipartimento di MatematicaVia Bounarotti, 256127 Pisa, Italy e-mail: villani@dm.unipi.itMATHEMATICAL PROBLEMSPaul JaintaWerkvolkstr. 10D-91126 Schwabach, Germanye-mail: PaulJainta@aol.com ANNIVERSARIESJune Barrow-Green and Jeremy GrayOpen University [address as above]e-mail: j.e.barrow-green@open.ac.ukand j.j.gray@open.ac.uk andCONFERENCESKathleen Quinn [address as above]RECENT BOOKSIvan Netuka and Vladimir Sou³ekMathematical InstituteCharles UniversitySokolovská 8318600 Prague, Czech Republice-mail: netuka@karlin.mff.cuni.czand soucek@karlin.mff.cuni.czADVERTISING OFFICERVivette GiraultLaboratoire d�Analyse NumériqueBoite Courrier 187, Université Pierre et Marie Curie, 4 Place Jussieu75252 Paris Cedex 05, Francee-mail: girault@ann.jussieu.frOPEN UNIVERSITY PRODUCTION TEAMLiz Scarna, Barbara Maenhaut, TobyO�Neil

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CONTENTS

EMS September 2000

EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

EXECUTIVE COMMITTEEPRESIDENT (1999�2002)Prof. ROLF JELTSCHSeminar for Applied MathematicsETH, CH-8092 Zürich, Switzerlande-mail: jeltsch@sam.math.ethz.chVICE-PRESIDENTSProf. ANDRZEJ PELCZAR (1997�2000)Institute of MathematicsJagellonian UniversityRaymonta 4PL-30-059 Krakow, Polande-mail: pelczar@im.uj.edu.plProf. LUC LEMAIRE (1999�2002)Department of Mathematics Université Libre de BruxellesC.P. 218 � Campus PlaineBld du TriompheB-1050 Bruxelles, Belgiume-mail: llemaire@ulb.ac.beSECRETARY (1999�2002)Prof. DAVID BRANNANDepartment of Pure Mathematics The Open UniversityWalton HallMilton Keynes MK7 6AA, UKe-mail: d.a.brannan@open.ac.ukTREASURER (1999�2002)Prof. OLLI MARTIODepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlande-mail: olli.martio@helsinki.fi ORDINARY MEMBERSProf. BODIL BRANNER (1997�2000)Department of MathematicsTechnical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: branner@mat.dtu.dkProf. DOINA CIORANESCU (1999�2002)Laboratoire d�Analyse NumériqueUniversité Paris VI4 Place Jussieu75252 Paris Cedex 05, Francee-mail: cioran@ann.jussieu.frProf. RENZO PICCININI (1999�2002)Dipartimento di Matematica e ApplicazioniUniversità di Milano-BicoccaVia Bicocca degli Arcimboldi, 820126 Milano, Italye-mail: renzo@matapp.unimib.itProf. MARTA SANZ-SOLÉ (1997�2000)Facultat de MatematiquesUniversitat de BarcelonaGran Via 585E-08007 Barcelona, Spaine-mail: sanz@cerber.mat.ub.esProf. ANATOLY VERSHIK (1997�2000)P.O.M.I., Fontanka 27191011 St Petersburg, Russiae-mail: vershik@pdmi.ras.ruEMS SECRETARIATMs. T. MÄKELÄINENDepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlandtel: (+358)-9-1912-2883fax: (+358)-9-1912-3213telex: 124690e-mail: makelain@cc.helsinki.fiwebsite: http://www.emis.de

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EMS NEWS

EMS September 2000

2000

22-27 SeptemberEURESCO Conference at Obernai, near Strasbourg (France):Number theory and Arithmetical Geometry: Motives and ArithmeticOrganiser: U. Jannsen, Regensburg (Germany), e-mail: euresco@esf.orgEURESCO Conference at San Feliu de Guixols (Spain):Geometry, Analysis and Mathematical Physics: Analysis and Spectral TheoryOrganiser: J. Sjöstrand, Palaiseau (France), e-mail: euresco@esf.org

30 SeptemberDeadline for proposals for 2001 EMS LecturesContact: David Brannan, e-mail: d.a.brannan@open.ac.ukDeadline for proposals for 2002 EMS Summer SchoolsContact: Renzo Piccinini, e-mail: renzo@matapp.unimib.it

11-12 NovemberExecutive Committee Meeting in London (UK), at the invitation of theLondon Mathematical Society.

15 NovemberDeadline for submission of material for the December issue of the EMSNewsletterContact: Robin Wilson, e-mail: r.j.wilson@open.ac.uk

2001

15 FebruaryDeadline for submission of material for the March issue of the EMSNewsletterContact: Robin Wilson, e-mail: r.j.wilson@open.ac.uk

10-11 MarchExecutive Committee Meeting in Kaiserslautern (Germany) at the invitationof the Fraunhofer-Institut für Techno- und Wirtschafts Mathematik.

9-20 JulyEMS Summer School at St. Petersburg (Russia)Asymptotic combinatorics with applications to mathematical physicsOrganiser: Anatoly Vershik, e-mail: vershik@pdmi.ras.ru

19-31 August EMS Summer School at Prague (Czech Republic)Simulation of fluids and structures interactionsOrganiser: Miloslav Feistauer, email: feist@ms.mff.cuni.cz

3-6 September1st EMS-SIAM conference, Berlin (Germany)Applied Mathematics in our Changing WorldOrganiser: Peter Deuflhard, e-mail: deuflhard@zib.de

EMS AgendaEMS Committee

How I came to this point �I obtained my PhD at the University ofHelsinki in 1967. My thesis adviser was O.Lehto, who later became the secretary ofthe IMU. The thesis dealt with quasi-con-formal mappings and PDEs, a subject thatbecame more important ten years later.The choice was a lucky one. At the end of1970s I started to study quasi-regular map-pings � non-homeomorphic quasi-confor-mal mappings, the counterparts of analyt-ic functions in higher-dimensionalEuclidean spaces. At that time Y. G.Reshetnyak in Novosibirsk had done pio-neering work on these mappings, but ourgroup at the University of Helsinki (J.Väisälä, S. Rickman and myself) gave newgeometric insight to these mappings. Ourstudies continued the tradition of the func-tion-theoretic school in Finland (E.Lindelöf, F. and R. Nevanlinna and L.Ahlfors).

In 1972 I became an associate professorat the University of Helsinki. In the sameyear I organised my first internationalmeeting, a Nordic Summer School devot-ed to quasi-conformal mappings; amongthe speakers were L. Ahlfors, F. W.Gehring and A. Selberg. The meeting wasone of the first new types of summer courseorganised in Scandinavia, but since thenthey have become commonplace. Therewere no proceedings, but later I did myshare: I have now edited nine conferenceproceedings.

During the period 1969-90 I spent someyears abroad. The most enjoyable visitswere at the University of Michigan in 1974and 1985. Other places include theMoscow State University, the NorwegianInstitute of Technology and the Technionin Haifa, Israel. The latest visit was the fallterm in 1999 at the Mittag-LefflerInstitute, where I took care of the scientificactivities.

In the late 1970s I again began to studynon-linear PDEs. They originated from thetheory of quasi-regular mappings, but thepotential-theoretic aspects turned out to bemore useful than I expected. In 1980 Ibecame a professor of mathematics at theUniversity of Jyväskylä in central Finland,where I stayed for 13 years. In Jyväskylä Ihad a number of good students who laterbecame professors in Finland and abroad.This year I was given an honorary PhDdegree (my third) at the University ofJyväskylä.

In 1993 I moved from Jyväskylä to

Helsinki. I was looking forward to lessadministrative duties in Helsinki than inJyväskylä. This was not going to be so. InJyvaskyla I had served as Head of theDepartment for ten years and short peri-ods as a Dean of the Faculty. After a cou-

ple of years I became Head of theDepartment in Helsinki and other admin-istrative duties started to pile up: I was amember of the Scientific Council of theAcademy of Finland as well as thePresident of the Scientific Academy ofFinland. These were not the onlyheadaches. Recent years have been diffi-cult in the Finnish universities: the newfinancial system, based on the annual �pro-duction� of MSc and PhD, has changed andincreased the duties of heads of depart-ments. The �output� is supposed toincrease and the budget to decrease eachyear. The theory is that people are moreproductive under stress. The theory iseffectively applied to heads of depart-ments.

Treasurer of the EMS �Aatos Lahtinen proposed that the office ofthe EMS be established in Helsinki andthis happened in due course. After servingfull terms as treasurer of the EMS heretired in 1998 and it was necessary to finda replacement. There is no requirementthat the treasurer must be from Finland.However, the office is in Helsinki and for agood reason: the Finnish State pays thesalary of the office secretary and so this

heavy financial burden has been removedfrom the EMS budget. TuulikkiMäkeläinen has taken excellent care of theoffice from the very beginning. Thus Ithought that since I can enjoy the benefitsof a professional secretary, the treasurer�sjob would not take much of my time. Evenso, it took some time before AatosLahtinen was able to persuade me to takethe job.

I have now served as the EMS treasurerfor a couple of years. My expectations havebeen correct, in the sense that routinefinancial matters (fees etc.) do not takemuch of my time. The EMS budget hasreached a certain stationary level. Sincethe number of countries in Europe hasincreased, the same has happened to thenumber of mathematical societies and sothere are new potential corporate mem-bers for EMS. Also, the number of indi-vidual members has increased steadily,although the number could grow muchfaster. This seems to depend on the effortsof the national mathematical societies.The membership fee for an individualmember is low, 15 euros, and he or shereceives the Newsletter, worth 11 euros.There are other benefits as well; for exam-ple, the conference fee for the 3ecm inBarcelona was less for EMS members thanfor other participants.

On the other hand, the treasurer isinvolved in many other duties of theExecutive Committee and the EMS in gen-eral. The activities have enlarged consid-erably and so has the committee work. Iam a member of the Support of EastEuropean Mathematicians Committee, thePublishing House Committee, theWorking Team for the Reference Levels inSchool Mathematics Education in Europe,etc. I feel that I get nothing done proper-ly. I hate meetings but have come to theconclusion that life is not intended to beenjoyable.

and beyond ...I am looking forward to the EMS taking acentral role in mathematics education inEurope. New initiatives and bold steps areneeded. The Chairman of the EMSEducation Committee, V. Villani, has juststepped down and Tony Gardiner hasreplaced him.The project �Reference Levels in SchoolMathematics Education in Europe�, spon-sored by the EMS and the EU ScienceProgramme, is chaired by Antoine Bodin

EDITORIAL

EMS September 2000 3

EditorialEditorialOlli Martio (Helsinki)

EMS Treasurer

and the final report will be ready at theend of 2000. The report will consist ofnational reports describing the school sys-tem in European countries, focusing onthe 16-year-old age group; this age coin-cides, roughly speaking, with the end ofcompulsory education in most EU coun-tries. The project has collected informa-tion in mathematics curriculum from aplurality of resources, such as official bod-ies, textbooks, teacher education, workingconditions, etc. I hope to see an up-to-datedescription of mathematics education inEurope.

In Europe a great variety of mathematicseducation is offered at the elementarylevel. This is of great benefit compared, forexample, to the USA where the compulso-ry school curriculum is quite uniform; itwould be an error to try to create strictstandards in Europe. On the other hand,there should be an information system thatprovides the opportunity to learn from thebest experiences in teaching mathematicsin Europe. This is badly lacking. The sys-tem should also include the mobility ofmathematics teachers and students insideEU countries. I think that it is the time tocreate such a system; the money could bevery well spent.

Good teachers produce good mathe-maticians and other users of mathematics.

For 30 years I have been working on gym-nasium and high-school teacher educationin Finland. I must admit that the standardhas not improved; the following is a shortdescription of the situation in Finland.

Although the teacher salaries are notbad in Finland, the profession seems tolack those attributes that make it desirablefor young people. There are so manyother professions today for those interest-ed in mathematics in Finland that a careerof teaching mathematics is not fashionable.For young people, future career prospectsfor teachers are not very good � once ateacher, always a teacher.

The Finnish Ministry of Education hascomputed the number of mathematicsteachers needed to replace retired teach-ers, and the Finnish universities almostproduce this number. However, it hasturned out that this number is totally inad-equate. There are two underlying facts.First, schoolteachers retire at a youngerage than expected. A retirement peak isjust now taking place, as a high percentageof the teachers lie in the age range of 60 to65. The second reason is that a consider-able number of young teachers in mathe-matics take teaching jobs outside conven-tional schools. The most important task ofinformation technology firms is now toteach people to use their systems, and

teachers who can understand the basics areneeded. The majority of new mathematicsteachers now have computer science astheir second subject. Ordinary schools alsoexpect this, instead of physics and chem-istry, the traditional second subjects. This,together with high-technology industry,consumes the best of the new mathematicsteachers.

The Ministry of Education in Finlandhas not been able to provide any counter-measures to these developments. In gen-eral, the job market for mathematicians isgood in Finland: a recent survey showedzero unemployment, which is strange sincethere are always some mathematicians whoseem to have difficulty in finding theirplace in society. Most mathematics stu-dents find at least temporary employmentbefore finishing their studies. For teacherstudents this has been very easy. Naturallythe average study time has increased, andthe Ministry of Education blames the uni-versities for this.

Finland is not alone with this problem,although the problems vary from oneEuropean country to another. I havefound that those who are responsible forfuture development are often poorlyinformed on solutions and methods usedelsewhere. They do not listen enough tomathematicians.

EDITORIAL

EMS September 20004

Maths Quiz2000

Jaume AguadeEvery mathematician in the world is invit-ed to take part in a unique mathematicalcontest in which competitors will answer asmany challenging mathematical questionsas possible. We call it the Maths Quiz 2000,and it is a contribution of the CRM(Barcelona) to the celebrations of WorldMathematical Year.

We are talking about a question-and-answer global and live competition. All thecompetitors (individuals or small teams)will compete in real time through theInternet. The contest will start at 12 mid-day Greenwich Mean Time on the 17October 2000 and will last exactly 24hours, without interruption.

Beyond the thrill of the challenge, thewinners will be rewarded with two sorts ofprizes. The five highest scoring competi-tors will each receive workstations courtesyof SUN Microsystems, the main sponsor ofthe competition. On the other hand, tospur on all the players, even those farbehind the leaders, we will be giving awaythroughout the competition a considerablenumber of book tokens valued at 100Euros which have been offered byBirkhäuser Verlag to be used in the pur-chase of books from this publisher.

So mark the 17th of October in yourdiaries, find a small group of colleagues,and visit the site of the Maths quiz 2000:http://www.mq2000.org

More World MathematicalYear Stamps

In EMS Newsletters 35 and 36 we featured three stamps issued to commemorateWorld Mathematical Year 2000. Further stamps have now been issued bySlovakia, the Czech Republic, Spain and Croatia.

The Slovakian stamp features the Slovak mathematicians Juraj Hronec andStefan Schwarz with a lattice diagram. The Czech stamp features a statement ofFermat�s last theorem, with the equation xn + yn = zn cancelled by �Andrew Wiles1995� across the equals sign. The Spanish stamp features Julio Rey Pastor, a dom-inant figure in Spain and later in Argentina. The Croatian stamp features theBlanusa graph, the smallest 3-regular graph (other than the Petersen graph) withchromatic index 4, discovered by D. Blanu�a in 1946.

Slovakia Czech Republic

Spain Croatia

Present were Rolf Jeltsch (President, in theChair), Andrzej Pelczar (Vice-President),David Brannan (Secretary), Olli Martio(Treasurer), Bodil Branner, DoinaCioranescu, Renzo Piccinini, Marta Sanz-Solé and Anatoly Vershik, and (by invita-

tion) Robin Wilson, Tuulikki Mäkeläinen,David Salinger and Carles Casacuberta.Apologies were received from Luc Lemaire(Vice-President).

The President thanked the CatalanMathematical Society for organising themeeting and for its hospitality.

Officers� ReportsThe President reported briefly on his pasttravels. He had visited Hanoi, where hehad set up a �Zentralblatt consortium�, andthe Clay Millennium Prizes Event inFrance, and was going to visit the NigerianMathematical Society at the end of July.The EMS was now a partner in DREAMS,a project for raising public awareness inmathematics with J.-F. Rodrigues as itsmain organiser.

The EMS Lectures in Zürich had beenexcellent in content and well attended.

Council 2000 The EMS auditors, H.-J. Munkholm and L.Reich were no longer delegates, so newEMS auditors had to be elected. The pro-fessional auditors could continue.

The Council meeting in 2002 wasplanned to be in connection with the Abelmeeting (2-8 June 2002). The proposalshould be to hold it on Friday 31 May andSaturday 1 June.

EMS CommitteesThe strategy paper prepared for the EU bythe EMS Group on Relations withEuropean Institutions had been wellreceived. The new Commissioner,Philippe Busquin, had been very positive.

There were three special points to be dis-cussed: the (�large project�) Grid, largecomputers (net) and modelling, and theGenome project.

Claude Lobry had prepared a basicworking plan for the EMS Developing

Countries Committee. The Committeewas awarded funding of 2000 euros tocover the 12-month period from 1 July2000.

An EMS Committeee on �Raising PublicAwareness of Mathematics� was estab-lished.

ProjectsThe EC was still looking into the questionof establishing an EMS Publishing House,and would discuss this with interested par-ties at the Barcelona Congress.

Reciprocity AgreementsAn EMS-AMS reciprocity agreement wouldprobably be signed on 13 July. Furtherreciprocity agreements were in thepipeline.

Future Executive Committee meetingsThe next meeting will be held in theLondon Mathematical Society office, DeMorgan House, 57-58 Russell Square,London WC1B 4HS, on 10-12 November2000. The following meeting will be heldat ITWM in Kaiserslautern, on 9-11 March2001.

EMS NewsletterEC members reported that they thoughthighly of the June EMS Newsletter. Variousarticles were solicited by its Editor forfuture issues. EMS members are alwaysencouraged to offer articles to the EMSNewsletter Editor, Robin Wilson [r.j.wilson@open.ac.uk].

David Brannan

EMS NEWS

EMS September 2000 5

EMS Executive Committee meetingEMS Executive Committee meetingBarcelona, 6 July 2000

Photos: David Salinger Andrzej Pelczar, Marta Sanz-Solé and Robin Wilson

Bodil Branner, Rolf Jeltsch and Tuulikki Mäkeläinen

The 2000 meeting of the Society�s Councilwas held in the Residence Hall forResearchers of the Institute for CatalanStudies in Barcelona, Spain. Around 60delegates of members were present,together with members of the ExecutiveCommittee and some invited guests,including the Presidents of the AmericanMathematical Society and of the AfricanMathematical Union. The Presidentexpressed the thanks of the delegates tothe local organisers of the meeting,Sebastià Xambó-Descamps, Marta Sanz-Solé and Javier Martínez de Albèniz.

Address of the President, Rolf JeltschThe President observed that the EMS is ahighly active society with some 50 corpo-rate members and approximately 1900individual members. Unfortunately thenumber of individual members was muchtoo small considering the more than10000�20000 mathematicians in Europe.

The EMS was financially sound but stillnot strong enough to do the job it envi-sioned, namely:� to be active in the EU and on the supra-

national level;� to provide professional service to its

members;� to help less privileged colleagues, e.g.

from Eastern Europe, developing coun-tries, young mathematicians.

It often had problems caused by its ratherlimited funds � he gave two examples, thefunding of EMS summer schools and thedifficulty of finding volunteers to carry outsome rather routine but vital tasks (e.g.maintaining the web sites).

He ended with the statement that EMSwas now 10 years old. In his opinion it hadgained a very high visibility and he wasproud of what had been achieved.

Reports from Executive Committee mem-bersThe President, Rolf Jeltsch (ETH,Switzerland), reported that he had tried tokeep contact with the member societiesand the individual members at least on ayearly basis by writing an annual letter atthe change of the year.

His main aims had been:� to enlarge the membership base of EMS byvarious approaches, such as reaching outfor applied mathematicians, attractingmore corporate members in applied math-ematics like ESMTB, cooperating withSIAM in a major conference in Berlin on 2-6 September 2001, instituting the FelixKlein Prize, changing the name of the�Committee on Applications ofMathematics� to �Applied MathematicsCommittee�, having more articles onapplied mathematics in the Newsletter, andmaking it possible for Europeans livingoutside Europe to join (via reciprocity

agreements with the AmericanMathematical Society, the AustralianMathematical Society and perhaps theCanadian Mathematical Society).� to make the EMS more visible by increasingrelations with other societies, EMS joiningIMU and ICIAM as an Associate Member,contacts with AMS, SIAM and the ClayMathematical Institute, involvement inraising public awareness of mathematics(via the EMS WMY2000 Committee and anew EMS committee on RPA), influencing

the European Commission for example, byinfluencing the definition of the 5th and6th Framework Programmes), and sup-porting the Zentralblatt MATH LIMESproject and the EULER project.

He ended with the observation that hehad been one and a half years in office andit had been satisfying to work together witha great team of devoted colleagues.

The Secretary, David Brannan (OU,England), reported on the discussions atthe four Executive Committee meetingssince the previous Council meeting inBerlin in 1998.

The Treasurer, Olli Martio (Helsinki,Finland), reported on various financialmatters. He informed the Council that theincome of the EMS still consisted mainly ofmembership fees, which showed only asmall increase. Expenses connected toindividual members were rather high, sothat raising the number of individualmembers would not raise income consider-ably. He had no special concern for mem-bership fees from corporate members.The positive financial outcome was partlydue to many Executive Committee mem-bers being supported by local sources,though he had occasional concerns overthe financing of various EMS SpecialProjects. A reserve fund of 70000 euro hadbeen collected by the Society and could beused to allay unexpected costs.

The Financial Statements and Auditors�Reports for 1998 and 1999 were accepted.The budgets for the years 2001 and 2002were approved, with no major changescompared to previous years and nochanges for membership dues.

László Márki (Budapest, Hungary) andJohn Hubbuck (Aberdeen, Scotland) wereelected as Auditors for the years 2001-2002. P. Kaasalainen, with K. Soikkeli asDeputy, was elected as the professionalAuditor for the years 2001-2002.

The recently-retired Publicity Officer,Mireille Chaleyat-Maurel (Paris, France), waswarmly thanked for her long and devotedservice to the EMS. Council noted thatDavid Salinger (Leeds, England) hadrecently succeeded to the post.

3ecm 2000The Chair of the Organising Committee,Sebastià Xambó-Descamps, reported brieflyon the Barcelona Congress. It was expect-ed to have an attendance of about 1600persons from more than 80 countries. Thebudget of the congress was sound, thoughit had been raised from 90 million pesetasto 109.6, and it was balanced. The organ-isers had been able to give 10 million pese-tas towards prizes and 195 million pesetasas grants to young researchers. The grantswere more than 300 in number.

For a successful organisation the supportof people and institutions had been need-ed and they had been secured. TheCongress had dedicated Organising andExecutive Committees.

The Catalan Mathematical Societywished the Council a pleasant stay inBarcelona, both during the Council andthe Congress.

Membership of EMSThe European Mathematical Trust had can-celled its membership, because it had

EMS September 20006

EMS NEWS

EMS Council meetingEMS Council meetingBarcelona, 7 - 8 July 2000

photos: Peter Legisa

ceased to exist. The Royal SpanishMathematical Society expressed the wish toraise its Class from 1 to 2, and this wasagreed. The European Society forMathematical and Theoretical Biology hadapplied for membership in Class 1, andthis was approved.

The Executive Committee had proposeda reciprocity agreement with the AmericanMathematical Society. The draft agreementincluded the sentence: �This exchangeagreement shall in no way interfere withexisting reciprocity agreements betweenthe AMS and corporate members of theEMS.� The reciprocity agreement wasunanimously accepted.

The Executive Committee had proposedthat the Council authorise the ExecutiveCommittee to negotiate reciprocity agree-ments in the spirit of the agreementreached with AMS, on the basis that theseagreements should be enacted on a provi-sional basis until they were ratified at aCouncil meeting. This authorisation wasunanimously given.

Executive Committee electionsBodil Branner (DTU, Copenhagen,Denmark) was elected unanimously to beVice-President for the years 2001-2004.

Four candidates had been nominated forelection to three vacancies for Members-at-Large. Following statements by all thecandidates, a secret ballot was held andVictor Buchstaber (Moscow, Russia), MartaSanz-Solé (Barcelona, Spain) and MinaTeicher (Ramat Gan, Israel) were declaredelected as Members at Large of theExecutive Committee for the years 2001-2004.

Detailed review of the various SocietyactivitiesThe following reports from Officers andCommittees were received and accepted,in some cases after a short discussion:� Publications: Publications Officer, Carles

Casacuberta, (Barcelona, Spain) This cov-ered the Journal of the EuropeanMathematical Society and electronic publi-cations.

� Newsletter: Newsletter Editor-in-Chief, RobinWilson (OU, England) The Newsletternow contained more feature articles,more reports and more on local soci-eties.

� Applied Mathematics Committee� Developing Countries Committee� Education Committee� Electronic Publishing Committee� ERCOM Committee� European Database Committee� Group on Relations with European

Institutions� Special Events Committee� Summer Schools Committee� Support of East European

Mathematicians Committee� Women and Mathematics Committee� World Mathematical Year 2000

CommitteeCouncil noted too that the ExecutiveCommittee had just created a newCommittee for Raising Public Awareness inMathematics, with Vagn Lundsgaard

Hansen in the Chair.

European Congress of Mathematics:ECM4 in 2004One bid had been received to host ECM4.However a number of problems hadappeared, and Council decided to empow-er the Executive Committee to investigatematters further and to choose anothervenue for ECM4 if necessary.

A report on the finally agreed locationwill appear in a forthcoming Newsletter.

EMS Council Meeting in 2002The Norwegian Mathematical Societyinvited EMS Council to Oslo in 2002, priorto the Abel conference of 2-8 June 2002.The invitation was accepted with pleasure,and the dates were preliminarily fixed at31 May- 1 June 2002.

MATH-NETMartin Groetschel presented a brief out-line of the German project MATH-NET.Council encouraged collaboration betweendifferent electronic search systems � forinstance, between EULER and MATH-NET.

ICIAMCouncil was informed that the EMS hadbeen accepted as a large associate memberof ICIAM, the InternationalCouncil/Congress of Industrial andApplied Mathematics.

Congress FeesThe delegates of the French MathematicalSociety brought forward the followingmotion accepted by the French NationalCommittee for Mathematics (CNFM) con-cerning high fees for congresses:

The CNFM observes that registration fees for

international conferences in mathematics tend torise steadily. The Committee is concerned by sucha trend, which makes it every year more difficultto support the participation of mathematiciansasking for its financial help.

Following some discussion and reword-ing, Council agreed on the following EMSCouncil resolution:

In relation to the CNFM statement, EMSCouncil expresses its concern over the high feescharged by some conferences. Scientific meetings

are an essential component in the developmentof Science and Mathematics, and should there-fore always be considered as scientific events andnot as commercial activities. In particular, theirorganisers should always make special efforts tofacilitate participation by younger mathemati-cians. It is also unacceptable that a scientistshould be unable to attend a scientific conferencesolely because it has a high conference fee.

African Mathematical UnionThe AMU President, Jan Persens, thankedthe Council for the opportunity to havebeen able to attend and to learn about theEMS. He extended his thanks to EMS andto the London Mathematical Society,whose grants enabled him to attend. AMUhad invited presidents of other organisa-tions for its 2000 Congress and thePresident of EMS had participated.

He called for collaboration, mentioningsummer-winter schools, workshops, spe-cialised conferences in sub-regions, gradu-ate students� courses, external examina-tions, sharing teaching materials, possiblylecture note series at post-graduate level,and researchers� visits. In 2004 the AMUcongress would be held in Tunisia, whereEuropean mathematicians would be mostwelcome.

History of the EMC and of the EMSIt was agreed that EMS should take steps toensure that at least a short History of theEuropean Mathematical Council was writtenand published in a suitable place fairlysoon, and that key persons were inter-viewed or persuaded to write their mem-oirs.

The President reported that theExecutive Committee had received a man-uscript of The History of the EMS, written byDavid Wallace (Strathclyde, Scotland), and

had decided to store it in the EMS archivesfor the moment.

Closing remarksFinally, the President expressed his thanksto all participants for attending. Specialthanks were given to the Catalan hosts ofthe meeting, especially to Sebastià Xambó-Descamps, Marta Sanz-Solé and JavierMartínez de Albèniz.

David Brannan

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EMS September 2000 7

The motto of the Third EuropeanCongress of Mathematics (3ecm) wasShaping the 21st Century. 3ecm was one ofthe main mathematical events of 2000, ayear designated by the InternationalMathematical Union and UNESCO asWorld Mathematical Year (WMY2000).

The event was organised by the Societat

Catalana de Matemàtiques (CatalanMathematical Society, or SCM), of theInstitut d�Estudis Catalans (IEC); particu-larly important roles were played by thePresident of the IEC, Manuel Castellet, andthe President of the SCM, Sebastià Xambó-Descamps. The IEC comprised: SebastiàXambó-Descamps (Chairman), CarlesCasacuberta (in charge of information andpublications), Julià Cufí (finances), Rosa M.Miró-Roig (programming and activities),Marta Sanz-Solé (organisational secretary),and Ferran Puerta and Marta València(infrastructure). They were assisted bytheir mathematical colleagues in all theuniversities in the Barcelona area, and overa hundred students assisted delegates invery many ways during 3ecm.

The Scientific Committee comprised: SirMichael Atiyah (Chairman), Vladimir I.Arnold, Robert Azencott, FabrizioCatanese, Ildefonso Díaz, Antti Kupiainen,Jack van Lint, Colette Moeglin, A. F. M.Smith, Johannes Sjöstrand, DomokosSzász, Stanislaw L. Woronowicz and DonZagier. The Prize Committee comprised:Jacques-Louis Lions (Chairman), NogaAlon, Werner Ballmann, Jan Derezinski,Maxim Kontsevich, Eduard Looijenga,Angus Macintyre, David Nualart, A. N.Parshin, Ragni Piene, Itamar Procaccia,Mario Pulvirenti, Rolf Rannacher, CarolineSeries, Vladimir Sverák and DanVoiculescu. The Round Table Committee

comprised: Miguel de Guzmán(Chairman), Andrey Bolibrukh, Heinz W.Engl, Juan José Manfredi, Carles Perelló,Tomás Recio, Zbigniew Semadeni andVinicio Villani.

Around 1250 mathematicians assembledin Barcelona for the Opening Ceremonyon 10 July. All the 3ecm activities took

place in the Palau de Congressos ofBarcelona, close to the Olympic Stadium.There was an Opening Reception in theGaleria of the Palau de Congressos and awonderful Congress Dinner in the gardensof the Palau de Pedralbes. The weatherwas excellent throughout the congress, andthe local organisation superb. It was anoutstanding success, thoroughly enjoyed byeveryone who participated. Lectures weregiven by those nine Prize-winners who werepresent in Barcelona.

There were nine Plenary Lectures:Robbert Dijkgraaf � The mathematics of M-

theory

Hans Föllmer � Probabilistic applications offinancial risk

Hendrik W. Lenstra, Jr. � Flags and latticebasis reduction

Yuri I. Manin � Moduli, motives, mirrorsYves Meyer � The role of oscillations in some

non-linear problemsCarles Simó � New families of solutions in N-

body problemsMarie-France Vignéras � Local Langlands

correspondence for GL(n, Qp), modulo l ≠ pOlerg Viro � Dequantization of real algebraic

geometry on a logarithmic paperAndrew Wiles � Galois representations and

automorphic forms

There were also twenty-nine InvitedLectures, in parallel sessions:Rudolf Ahlswede � Advances on extremal

problems in number theory and combinatoricsFrançois Baccelli � Flow control in stochastic

networksVolker Bach � Mathematical theory of matter

and radiationVivianne Baladi � Spectrum and statistical

properties of chaotic dynamicsJoaquim Bruna � Sampling in complex and

harmonic analysisXavier Cabré � A conjecture of De Giorgi on

symmetry for elliptic equations in Rn

Peter J. Cameron � The random graph andits relations

Zoé Chatzidakis � Difference fields: model the-ory and applications to number theory

Ciro Ciliberto � Geometric aspects of polyno-mial interpolation in more variables and ofWaring�s problem

Gianni Dal Maso � The calibration method forfree discontinuity problems

Jan Denef � Geometry of arc spaces of algebra-ic varieties

Barbara Fantechi � Stacks for everybodyAlexander B. Givental � Some aspects of sym-

plectic field theoryAlexander Goncharov � Multiple ζ-values,

Galois groups and geometry of modular vari-eties

EMS NEWS

EMS September 20008

3ecm in Bar3ecm in Barcelonacelona10-14 July 2000David Brannan

Rolf Jeltsch (EMS) & Felix Browder (AMS) signing the EMS-AMS Reciprocity Agreement, July 13th 2000

3ecm Round Table �Shaping the 21st Century�, July 14th 2000

Alexander Grigor�yan � Heat kernels onmanifolds, graphs and fractals

Michael Harris � Local Langlands correspon-dence and vanishing cycles on Shimura vari-eties

Kurt Johansson � Random growth and ran-dom matrices

Konstantin M. Khanin � Burgers turbulenceand dynamical systems

Pekka Koskela � Sobolev spaces and quasicon-formal mappings on metric spaces

Steffen L. Lauritzen � Probabilistic networksand the mathematics of local computation

Nicholas S. Manton � Understandingskyrmions using rational maps

Ieke Moerdijk � Models for the leaf space of afoliation

Eric M. Opdam � Recent developments in thetheory of hypergeometric functions

Thomas Peternell � Contact manifolds inalgebraic geometry

Alexander Reznikov � Analytic topologyHenrik Schlichtkrull � Harmonic analysis on

reductive symmetric spacesBernhard Schmidt � Sums of roots of unity

and finite geometryKlaus Schmidt � Algebraic Zd-actionsBálint Tóth � Self-interacting random motions

Ten Mini-Symposia provided a wide rangeof talk on varied topics:Computer algebra (Chair, Wolfram Decker):

Manuel Bronstein, Gaston Gonnet, Gert-Martin Greuel, Erich Kaltofen, HendrikW. Lenstra Jr. and Tomás Recio

Curves over finite fields and codes (Chair,Gerard van der Geer): Noam Elkies,Arnaldo Garcia, Gerard van der Geer,Andrew Kresch, Christian Maire,Henning Stichtenoth and Chaoping Xing

Free boundary problems (Chair, José

Francisco Rodrigues): Giovanni Belletini,Klaus Deckelnick, Irina V. Denisova,Gonzalo Galiano, Harald Garcke,Josephus Hulshof, Régis Monneau,Henrik Shahgholian and José MiguelUrbano

Mathematical Finance: Theory and Practice(Chair, Hélyette Geman): Tomas Bjork,M. A. H. Dempster, Ernst Eberlein, DilipMadan, Ezra Nahum, Stanley Pliska,Rainer Schobel and Ton Vorst

Mathematical finance: theory and practice(Chair, Hélyette Geman): Tomas Björk,Ernst Eberlein, Dilip B. Madan, StanleyR. Pliska and Rainer Schöbel

Mathematics in modern genetics (Chair, PeterDonnelly): David J. Balding, PeterDonnelly, Alison Etheridge, WarrenEwens and Augustine Kong

Quantum chaology (Chair, Sir MichaelBerry): Sir Michael Berry, EugeneBogomolny, Monique Combescure,Christopher Howls, Jonathan Keating,Jens Marklof, Zeév Rudnick and AndréVoros

Quantum computing (Chair, SanduPopescu): Charles H. Bennett, RichardCleve, Nicolas Gisin, Sandu Popescu,Rolf Tarrach and Umesh Vazirani

String theory and M-theory (Chair, MichaelR. Douglas): Duiliu-Amanuel Diaconescu,Jaume Gomis, Chris M. Hull, AlbrechtKlemm, José M. F. Labastida, MarcosMarino, Nikita Nekrassov, ChristophSchweigert and Angel M. Uranga

Symplectic and contact geometry andHamiltonian dynamics (Chair, Mikhail B.Sevryuk): Paul Biran, Yuri V. Chekanov,Hansjörg Geiges, Viktor L. Ginzburg,Alberto Ibort, Àngel Jorba DietmarSalamon and Vladimir M. Zakalyukin

Wavelet applications in signal processing(Chair, Andrew T. Walden): Richard G.Baraniuk, Peter F. Craigmile, PatrickFlandrin, Emma McCoy, Vasily Strelaand Andrew T. Walden

Poster sessions took place over three suc-cessive afternoons. In parallel with thesewas a range of interesting and oftencolourful video exhibitions with the follow-ing titles: Multicoloured fractals, Quadro � acompact soap bubble of genus 4, Geodesics andwaves, Platonic solids, Knot energies,Interpolation of triangle hierarchies, Polytopesections and surfaces, Flows and holonomy, Theoptiverse, Distorted space, CMC � pictures ofconstant mean curvature, Numerical simula-tions of unstable detonations, On Gorgonne�sproblem, Soap bubbles and The dynamics of therabbit.

There were seven interesting RoundTables, in which panellists gave preparedstatements and then engaged in discus-sions with each other and the audience:Mathematics teaching at the tertiary level:

Vladimir Tikhomirov (Chair), DeborahHughes-Hallet, John Mason and RobertMattheij

The impact of mathematical research on indus-try and vice-versa: Irene Fonseca (Chair),Mark Davis and Martin Grötschel

How to increase public awareness of mathemat-ics: Vagn Lundsgaard Hansen (Chair),Jean-Michel Kantor, Carlo Sbordone andJorge Wagensberg

What is mathematics today?: ZbigniewSemadeni (Chair), Pilar Bayer, JaimeCarvalho de Silva and Henri Lombardi

Building networks of co-operation: FriedrichHirzebruch (Chair), Eva Bayer, BodilBranner and Anatoly Vershik

The impact of new technologies on mathematicalresearch: Rafael de la Llave (Chair), BrunoBuchberger, Sergei Matveev and MikaSeppala

Shaping the 21st Century: Miguel deGuzmán (Chair), Alexandra Bellow, Jean-Pierre Bourgignon and Helmut Neunzert

On two afternoons there were sessions onMathematical Software, in which the fol-lowing participated: Anna Maria Bigattiand Lorenzo Robbiano (CoCoA), HarmutHäfner (LINSOL), Mario Essert (Internet),Jordi Guàrdia and Jesús Montes (Newton),Torsten Adolph (DFEM), Bob Matthews(Math Type), Gerd-Martin Greuel(Singular), Alexander Makarenko (societalproblems), Immaculada Fortes(Grammarkov), Peter Dräxler and RainerNörenberg (CREP), Arkadii Kim (Time-delay system), Francis Sergeraert (Kenzo),Konrad Polthier (Java view), LancelotPecquet (Magma) and Ewgenij Gawrilow(Polymake).

Overall the programme for the five dayswas very full and exciting, but with manyopportunities for everyone to sit quietly totalk with colleagues as well as to enjoy avaried diet of excellent talks. All partici-pants were agreed that the arrangementsand the whole meeting had been excellent,and were really sad to leave Barcelonaafter 3ecm!

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EMS September 2000 9

Tuulikki Mäkeläinen at the EMS Booth at 3ecm

The opening of the Third EuropeanCongress of Mathematics in Barcelonagives me special satisfaction in my doublecapacity of mathematician and ofPresident of the Institut d�EstudisCatalans. The Institut d�Estudis Catalans(Institute for Catalan Studies) is theNational Academy of Catalonia and theparent body of the Catalan MathematicalSociety, organiser of this Congress, which Ihope will be to the satisfaction of all of you.Catalonia is a small country in surface areaand population, a country which has neverhad a strong mathematical tradition. Ifyou consult the Zentralblatt für Mathematikor the Mathematical Reviews for the 1970syou will notice that the probability of find-ing an article by a Catalan mathematician,even by a Spanish one, is effectively lessthan any epsilon. But now, 25 years later,there is an abundance of Catalan mathe-matical research published in prestigiousjournals, and there are outstandingCatalan mathematicians in all areas.

This reality is demonstrated by a recentstudy carried out by the Institut d�EstudisCatalans into mathematical research inCatalonia. The ratio between the numberof articles published in Catalonia and thetotal population, scaled by the GrossNational Product, is similar to the ratio inNorway, Great Britain or Germany; fur-thermore, 8.2% of all mathematical articlesare published in the top quality journals.This situates Catalonia, according to thesecriteria, among the most scientifically sig-nificant countries of the world.

Doubtless this situation is the fruit notonly of the labours of recent generations ofCatalan mathematicians, but also of theefforts of our whole society and of therecent entry of the country into the currentdemocratic framework. But this alonewould not be sufficient. It must also bebecause the famous phrase from Cicero�sTusculanae suits us perfectly: �Nature hasplaced in our spirit an insatiable desire toknow the truth�. A truth which this coun-try in its thousand year history has alwayssought, with more or less success, andalmost always with limited resources. Atruth which the Catalan countries, scientif-ically, have always sought looking towardsEurope, a Europe of which we were animportant nucleus in the Middle Ages,before a sharp decline, and of which wehave been steadily forming a more integralpart since the beginning of the twentiethcentury. The Institut d�Estudis Catalans,founded in 1907, is an academic, scientificand cultural institution which has been an

important factor in this cultural and scien-tific integration.

The aim of the Institut d�EstudisCatalans is to advance research, both in sci-ence and in all areas of Catalan culture. Itcontributes to the planning, co-ordination

and realisation of research in various fieldsof science, technology and the humanities.It works to stimulate the progress and gen-eral development of Catalan society and.on occasion, serves as a consultancy forpublic bodies and institutions. The Institutd�Estudis Catalans is divided by broadly-defined subject areas into five sections(including the Academy of the Catalan lan-guage) each of which has a maximum of 21full Fellows; associated to it there are also25 scientific societies, one of which is theCatalan Mathematical Society, with a totalof more than nine thousand members alto-gether.

Let me now give a brief review of ourmathematical history in Catalonia.

Did you know that Gerbert d�Orlhac, amonk from the Catalan Pyrenees who inthe year 1000 was better known as PopeSylvester II, made an in-depth study of theQuadrivium, transformed calculation meth-ods throughout Europe two centuriesbefore Fibonacci, and built an original aba-cus?

Did you know that the Majorcan RamonLlull, one of the most important writers inthe Catalan language of the thirteenth cen-tury, already understood that the Earthwas a sphere and wrote Ars Combinatoriaand Art de Navegar, a book unsurpasseduntil the sixteenth century, which describesthe astrolobe and refers to the use of themagnetic compass? Alexander vonHumboldt maintained over 100 years agothat these scientific advances were trans-mitted from Catalonia via otherMediterranean nations to the rest of the

civilised world.Did you know that in 1375 the Majorcan

Jew Abraham Cresques drew the worldmap known as the Catalan Atlas, a repre-sentation of the known world which wasessential for the navigators of the epoch?

Did you know that the second book ofarithmetic ever printed (after the anony-mous arithmetic text of Treviso) wasSumma de l�art d�Aritmetica by FrancescSantcliment, printed in Catalan inBarcelona in 1482? And that the firstmathematics book printed in Spain was theSpanish translation of Santcliment�s book?

Did you know that the Valencian JosepChaix, author of works on differential andintegral calculus, carried out with PierreMechain in 1793 the calculations to mea-sure the meridian arc between thePyrenees and Barcelona?

I do not wish to bore you with this histo-ry lesson, but I must mention two out-standing contemporary researchers fromour country: Lluís Santaló, born in Gironaand emigrated to Argentina, pioneer ofintegral geometry, stereology and geomet-ric probability, who cannot attend this con-ference due to his delicate state of health.And Ferran Sunyer i Balaguer, possibly thegreatest Catalan mathematician through-out the 1950s and 60s, born a tetraplegic,friend of Hadamard, Mandelbrot andMalliavin, among others, whose theoremon polynomials you can read on the standof the Catalan Mathematical Society and inwhose honour the Institut d�EstudisCatalans awards each year a prize for amonograph which best expounds and sum-marises advances in an active area of math-ematics.

But the greatest change, the integrationof Catalan mathematicians to the ideas andscientific trends, both European and inter-national, has happened in the last twentyor twenty-five years, when some of us madecontact with André Lichnerowicz, BenoEckmann or Paul Malliavin, to choose justthree truly significant names of Europeanmathematicians. The mathematics depart-ments of the universities of Barcelonastarted to become active research centresand with great effort and dedication wereable to gradually break through the scien-tific and cultural isolation to which we hadbeen subjected by a dictatorial governmentand a closed society.

At the beginning of the 1980s theCatalan universities, that of Barcelona, theAutonomous University and thePolytechnical University, began to discoverthat they could compete in high quality

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EMS September 200010

3ecm :3ecm : Opening AOpening AddrddressessManuel Castellet

President of the Institut d�Estudis Catalans

mathematical research. The various math-ematics departments have intensified thetraining of doctoral students and the pro-duction of research work. But to do sothey had to keep in touch with the mathe-matical research being carried out in thedeveloped world.

Thus the Institut d�Estudis Catalans, inview of these changes, created in 1984 theCentre de Recerca Matemàtica, the onlyresearch institute in Spain, with the objec-tive of facilitating contact between Catalanmathematicians and the European andinternational scientific elite; an institutefor visiting professors and youngresearchers, whose aim is to stimulategrowth in quality and quantity of Catalanmathematical research, organising spe-cialised research semesters, awarding post-doctoral grants, inviting outstandingresearchers, organising seminars, confer-ences, advanced courses, etc. The Centrede Recerca Matemàtica, which is a memberof ERCOM (European Research Centreson Mathematics) has been and continuesto be an infrastructure at the service of allCatalan mathematicians to facilitate accessto the most advanced scientific trends andthe incorporation of our local communityin the international mathematical commu-nity.

It is not an isolated fact, then, that thereare five mathematical events being held inBarcelona this year with the support of theEuropean Commission: this LargeEuroconference which we inauguratetoday, two Euroconferences on Login andon Algebraic topology, a PhDEuroconference on Complex dynamics,and a Euro Summer School on Quantumgroups. Next year we celebrate the newcentury with six Euroevents organised bythe Centre de Recerca Matemàtica: aEuroconference on Combinatorics andgraph theory, a PhD Euroconference onAlgebraic topology, and four EuroSummer Schools on Modular forms,Hamiltonian systems, Riemannian geome-try and Algebraic topology.

Neither is it an isolated fact that eightresearch groups in Catalonia (four at theUniversitat Autònoma, one at thePolitècnica and three at the Universitat deBarcelona) form part of various researchtraining networks of the ImprovingHuman Research Potential programmeand that two of the groups, those ofOperator Theory and of AlgebraicTopology, both at the UniversitatAutònoma, are the only mathematicsgroups in all of Spain which have beenselected as Marie Curie Training Sites.The Centre de Recerca Matemàtica willhost five Marie Curie postdoctoral fellowsduring the next academic year.

For the first time in Catalonia, and inSpain, the words of Konrad Knopp, pro-nounced at the 1927 inaugural lecture ofthe University of Tübingen, are beingunderstood: �Mathematics is the basis ofall knowledge and contains all other cul-ture�. Certainly our world is more com-plex every day and at the same time morefragile and unstable. Mathematics has anincreasingly decisive role to play in the

management of complex systems, be theytechnological, financial or social, in thecentury we now enter. Mathematics willincreasingly be an instrument of powerwhich is sometimes dangerously underesti-mated.

I think all of you are aware that thesewords are not intended to convince mathe-maticians, but rather the political andadministrative authorities that we havewith us here at the presidential table.Catalonia is prepared to face the complex-ity of the twenty-first century. In recentyears a real effort has been made towardstechnological renovation and measureshave been taken which have encouragedspectacular economic developments andensured good levels of social welfare. Onemust also take into account, however, thatthe intellectual resources of a country areat least as important, or even more impor-tant, than the material resources. Ouradministrators and governors should notforget that mathematics is the cheapest ofall the sciences: it takes little more thanbrains, and networks for communication.And our country, rich in brains, must pro-tect the second aspect, strengthening, forexample, the two networks of mathemati-cal communication that we have: theCentre de Recerca Matemàtica and theCatalan Mathematical Society.

I will close this parenthesis addressed toour authorities, and return to the desire ofthe Catalan mathematicians for a place inthe European and international scientificcommunity. It istrue that there havealways been inter-nationally recog-nised Catalanresearchers insome scientificareas and in recentyears this recogni-tion has extendedto areas where pre-viously we had nopresence, as is thecase for mathemat-ics; it is also truethat our researchgroups are begin-ning to be visible inthe literature. Butit is one thing to beknown and recog-nised individuallyand quite anotherfor the country tobe recognised inthis way. Withoutthe first we couldnot achieve the sec-ond, but the lattermust always be ourobjective. We wantBarcelona, andCatalonia, to beknown internation-ally, not just forGaudí or Barça.We want to berecognised also as ascientific communi-

ty, as a mathematical community, and thisis the role of the Catalan MathematicalSociety.

The Catalan Mathematical Society is ascientific society that, in a country with apopulation of six million, has 1000 mem-bers and acts on two different fronts: as ameeting ground for all Catalan mathe-maticians, organising conferences, courses,publishing periodically a Butlettí and aNotíces in Catalan; and on the other handas a nucleus for direct connection withmathematical institutions of other coun-tries and especially with the EuropeanMathematical Society.

The participation of the CatalanMathematical Society in the EuropeanMathematical Society has been intensesince the outset, collaborating on theCouncil and on the Executive Committee,and ultimately presenting its candidaturefour years ago now to host this ThirdEuropean Congress of Mathematics, whichwas awarded to us at the Council meetingin Budapest.

We want not just to offer you goodorganisation, a welcoming city and coun-try, a good climate and lots of sun; all thiswe do offer; but on top of all these ingre-dients we hope that you, the Europeanmathematicians, and by extrapolation allthe international mathematical communi-ty, take note that Catalonia is a countrywhich is mathematically developed andprepared to face the complexities of whichI spoke to you a moment ago.

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EMS September 2000 11

The Felix Klein PrizeThe Felix Klein prize has been established bythe European Mathematical Society and theendowing organisation, the Institute forIndustrial Mathematics in Kaiserslautern. It isawarded to a young scientist or a small groupof young scientists (normally under the age of38) for using sophisticated methods to give anoutstanding solution, which meets with thecomplete satisfaction of industry, to a concreteand difficult industrial problem. The prize ispresented every four years at the EuropeanCongresses of Mathematics.

The prize committee consists of six membersappointed by agreement of the EMS and theInstitute for Industrial Mathematics inKaiserslautern. For the 2000 Congress, theywere: Helmut Neunzert (Kaiserslautern)[Chair], Heinz Engl (Linz), Andras Frank(Budapest), Horst Loch (Mainz), OlivierPironneau (Paris), John Ockendon (Oxford)

The winner was David C. Dobson (USA)David C. Dobson started hiswork on the diffraction ofelectromagnetic waves fromperiodic structures, when hewas a postdoctoral student atthe famous Institute ofMathematics and itsApplications of Professor

Avner Friedman. The HoneywellTechnology Center had posed the prob-lem to model and analyse the diffractionand to develop appropriate numericalalgorithms. In a next step an optimalshape design problem for phase lenseswas solved. The fact that he used a�Fraunhofer approximation� was not thereason to give him a prize endowed by theFraunhofer Institute for Mathematics;what convinced the committee that heshould be the first prize winner, was thathe used rigorous and sound mathematicalmethods in a quite tricky way for a prob-lem which Honeywell states to be of veryhigh industrial importance.

The FThe Ferran Sunyer erran Sunyer i Balaguer Pi Balaguer PrizerizeFerran Sunyer i Balaguer (1912-1967) was aself-taught Catalan mathematician who, inspite of a serious physical disability, was veryactive in research in classical mathematicalanalysis, an area in which he acquired interna-tional recognition. Each year, in honour of thememory of Ferran Sunyer i Balaguer, theInstitut d�Estudis Catalans awards an interna-tional mathematical research prize bearing his

name, open to all mathematicians. This prizewas awarded for the first time in April 1993.For further information on the Ferran Sunyer iBalaguer Foundation, see Website:http//crm.es/info/ffsb. htm

The Prize committee for the 2000 Congressconsisted of: Hyman Bass (Michigan), PilarBayer (Barcelona), Antonio Córdoba (Madrid),Paul Malliavin (Paris), Alan Weinstein(Berkeley)

The winners were Juan-Pablo Ortega(Spain) and Tudor Ratiu (Romania)

The Institutd � E s t u d i sCatalans hasawarded theFerran Sunyeri BalaguerPrize toProfessors Juan

Pablo Ortega and Tudor Ratiu for theirmonograph entitled Hamiltonian SingularReduction.

The awarded monograph deals with thereduction of hamiltonian systems using itssymmetries. In the case of smooth systemsthere is a standard formulation of thisprocedure due to Marsden and Weinstein.In the presence of singularities, severalapproaches exist to the reduction proce-dure. The main contribution of theauthors is to show that all these approach-es are in some sense equivalent, by givinga sort of universal model for singularreduction. The monograph is self-con-tained and will be of great use toresearchers in this area.

Professor Tudor Ratiu, professor at theÉcole Polytechnique Fédérale deLausanne, is a well-known specialist in thefield. Professor Juan-Pablo Ortega com-pleted in 1998 a PhD thesis in the area atthe University of California at Santa Cruzand he is presently at the ÉcolePolytechnique Fédérale de Lausanne aswell.

EMS prizesThe EMS prizes were established by theEuropean Mathematical Society. They aremeant to recognise excellent contributions inmathematics by young researchers, not olderthan 32 years. The prizes are presented everyfour years at the European Congress ofMathematics.

The prize committee is appointed by theEMS. It consists of about fifteen international-ly recognised mathematicians covering a largevariety of fields. For the 2000 Congress they

were: Jacques-Louis Lions (Paris) [Chair],Noga Alon (Tel Aviv), Werner Ballmann(Bonn), Jan Derezinski (Warsawa), MaximKontsevich (Bures-sur-Yvette), EduardLooijenga (Utrecht), Angus Macintyre(Edinburgh), David Nualart (Barcelona),Aleksei Parshin (Moscow), Ragni Piene (Oslo),Itamar Procaccia (Tel Aviv), Mario Pulvirenti(Roma), Rolf Rannacher (Heidelberg),Caroline Series (Warwick), Vladimir Sverák(Praha) and Dan Voiculescu (Berkeley)The winners were:

Semyon Alesker (Israel) Semyon Alesker has con-tributed greatly both to theasymptotic theory of convexityand to classical convexity theo-ry. His most significant work is

on valuations (additive functionals) onconvex bodies and it has remodelled acentral part of convex geometry.

Group invariant valuations were studiedsince Dehn�s solution of Hilbert�s thirdproblem, with later contributions byBlaschke and others, and culminating inHadwiger�s celebrated characterisationtheorem for the intrinsic volumes. Thelatter theorem was considered the topresult in this area for almost fifty years.Alesker has now considerably extendedthis theory, obtaining very complete clas-sification results under weaker invarianceassumptions. He approximated (continu-ous) rotation invariant valuations by poly-nomial valuations and characterised thelatter, making use of representations ofthe orthogonal group. This enabled himto extend Hadwiger�s theorem to tensorvalued valuations. In another direction,he solved a problem of McMullen, in amuch stronger form, showing that transla-tion invariant valuations are essentially(up to linear combinations and approxi-mation) mixed volumes. The approach isvia representation theory of the generallinear group and involves a surprisingapplication of D-modules. The newapproach has also opened the way tofiniteness result for valuations with othergroup invariances.

Raphael Cerf (France)Raphael Cerf became knownthrough his results on proba-bility theory. Using a largedeviation principle in theproper topology, he has estab-lished a Wulff construction for

the supercritical percolation model inthree dimensions. This result is a very

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EMS September 200012

PPrizes rizes

awarawarded at 3ecmded at 3ecm

major advance in the subject, and pro-vides the right formulation for the geom-etry of the problem. He has been able tocarry out this programme using a correctmixture of combinatorial arguments, geo-metric ideas and probabilistic tools.

In addition to this research RaphaelCerf has made original contributions ingeneric algorithms. He has solved a cen-tral problem in bootstrap percolation andextended to three dimensions themetastable behaviour of the stochasticIsing model in the limit of low tempera-tures.

Dennis Gaitsgory (USA)Dennis Gaitsgory is one of theleaders in the geometricLanglands correspondenceand related areas. In the mod-ern �geometric� representation

theory one replaces functions by complex-es of constructible sheaves on (infinite-dimensional) algebraic varieties. In thisway many deep structures appears, andclassical results in the theory of automor-phic forms can be seen much more clear-ly.

In his thesis and in the subsequent workwith Braverman, Gaitsgory establishedfundamental properties of Eisensteinseries in the geometric setting. In a recentpaper on nearby cycles, he proposed anextremely elegant construction of the con-volution of equivariant perverse sheaveson so-called affine Grassmannians. Thisimplies that the centre of the affine Heckealgebra coincides with the whole sphericalHecke algebra. Also, it gives the best con-ceptual explanation of the Satake equiva-lence.

Recent work of Gaitsgory relates finitequantum groups and chiral Hecke alge-bras. It is a very important step in theprogram of Beilinson and Drienfield inthe geometric Langlands theory.

Emmanuel Grenier (France)Emmanuel Grenier greatly contributed tothe asymptotic analysis of Euler andNavier-Stokes equations with largeCoriolis force. The simplest case (whenthe equations are set on the unit cube withperiodic boundary conditions) was solvedby Grenier around 1995. Later, in collab-oration with Desjardins, Dormy andMasmoudi, he gave rigorous derivationsof several asymptotic models currentlyused in ocean and atmosphere modelling,or in magnetohydrodynamics.

Grenier obtained both positive and neg-ative important results on the problem ofconvergence of the Navier-Stokes equa-tions to the Euler equations in a domainwith solid boundary conditions. In partic-ular, he showed that the positive results ofCaflisch and Sammartino obtained foranalytic initial data, cannot be extended toSobolev data. He also justified the hydro-static limit of the Euler equations in a two-dimensional infinitesimally thin strip.

Grenier gave a very elegant proof ofconvergence for the semi-classical limit ofthe non-linear Schrödinger equations(before appearance of shocks). He also

obtained, simultaneously with E. Rykovand Sinai, a hydrodynamic limit for theZelodvich adhesion particle model.

Dominic Joyce (UK)Dominic Joyce�s work on theexistence of metrics with spe-cial holonomy is among thebest in Riemannian geometryin the last decade. The ques-

tion of the existence of Riemannian met-rics with special holonomy has a long his-tory beginning with the work of Cartan. Itincludes some of the best work of suchpeople as M. Berger, J. Simons, S. T. Yauand R. Bryant. Using a dazzling display ofgeometry and analysis, Joyce constructedcompact examples in the exceptionalcases where the holonomy is Spin7 and G2the only remaining possibilities, the oth-ers on Berger�s list had been eliminated.Joyce also computed the dimension of thedeformation spaces of such metrics andmany others of their invariants. As aresult he also discovered a totally unex-pected version of mirror symmetry forsuch spaces. Dominic Joyce is one of theleading young differential geometers.

Vincent Lafforgue (France)Vincent Lafforgue�s work is amajor advance in the K-theoryof operator algebras; the proofof the Baum-Connes conjec-ture for discrete co-compact

subgroups of SL3(R), SL3(C), SL3(Qp) andsome other locally compact groups. TheBaum-Connes conjecture predicts the K-theory of the reduced C*-algebra of alocally compact group (and of more gen-eral objects). The conjecture plays a cen-tral role in non-commutative geometryand has far-reaching connections with theNovikov conjecture on higher signaturesin topology, to harmonic analysis on dis-crete groups and the theory of C*-alge-bras. Lafforgue�s result is the first passageof the barrier which property T ofKazhdan has posed for many years in theproof of the Baum-Connes conjecture.The proof involves several remarkabletechnical and conceptual developments,like a bivariant K-theory for Banach alge-bras (versus Kasparov�s by now classicalone for C*-algebras) or establishing theconjecture for various completions of theL1 algebras of the groups.

Michael McQuillan (UK)Michael McQuillan has createdthe method of dynamic dio-phantine approximation whichhas led to a series of remark-able results in complex geome-

try of algebraic varieties. Among theseresults one can mention a new proof ofBloch�s conjecture on holomorphic curvesin close sub-varieties of abelian varieties,the proof of the conjecture of Green andGriffiths that a holomorphic curve in asurface of general type cannot be Zariski-dense, and the hyperbolicity for generichypersurfaces in a projective space P3 ofhigh enough degree (Kobayashi conjec-ture).

Stefan Yu. Nemirovski(Russia)S. Yu. Nemirovski hasobtained several strong resultson topology and complexanalysis. Using modern tech-

niques like the famous Seiberg-Witteninvariants he has solved some old classicalproblems about sub-manifolds in com-plex domains. First, he generalised theThom inequality proved by P.Kronheimer and T. Mrowka. As a veryparticular case he proved that there areno non-constant holomorphic functionsin a neighbourhood of an embedded non-trivial 2-sphere in a complex projectiveplane. Another application of his maintheorems is also very attractive. Supposethat an analytic disc is attached from theoutside to a strictly pseudoconvex domainU in a complex 2-plane; then there is nosmooth disc inside U with the sameboundary. As a corollary one gets that itis impossible to attach an analytic discfrom the outside to a strictly pseudocon-vex domain that is diffeomorphic to aclosed ball.

Paul Seidel (France)Paul Seidel became known through hiswork on symplectic topology. In hisPh.D. thesis he studied the fundamentalquestion of whether symplectic diffeo-morphisms which are diffeotopic to theidentity are also symplectically diffeotopicto the identity. He showed that theanswer is negative in many cases, alreadyin dimension 4. His counter-examplesare generalised Dehn twists, and his proofinvolves Floer homology. In furtherwork, Seidel constructed a natural repre-sentation of the fundamental group of thegroup of Hamiltonian symplectomor-phisms into the quantum cohomologyring. This work was basic for later work ofLalonde, McDuff and Polterovich on thetopology of the group of symplectomor-phisms. There is more to say about otherwork. His latest work is related to mirrorsymmetry, showing his broad horizons.

Wendelin Werner (France)Wendelin Werner hasobtained deep results on sto-chastic processes and, moreprecisely, he has proved anumber of significant results

on Brownian path properties, includingthe shape of Brownian islands andBrownian windings.He has made remarkable contributions tothe study of self-avoiding random walksand the corresponding critical exponents.More specifically, he obtained the firstnon-trivial upper bound of the disconnec-tion exponent, and he developed an ele-gant approach for studying the limitingbehaviour of the non-intersection expo-nents for a great number of independentBrownian motions. Among many otherinteresting works, he constructed, with acollaborator, the so-called true self-repelling motion using an ingeniousmethod involving infinite systems of coa-lescing Brownian motions

EMS NEWS

EMS September 2000 13

Asymptotic combinatorics Asymptotic combinatorics with application to mathematicalwith application to mathematical

physicsphysicsEMS Summer School

Euler International Mathematical Institute, St Petersburg, Russia9 � 22 July 2001

This EMS Summer School will concentrate on recent progress in the asymp-totic theory of Young tableaux and random matrices from the point of viewof combinatorics, representation theory and the theory of integrable systems.This subject belongs both in mathematics and in theoretical physics, and ten-tative participants are mathematicians and physicists. Systematic courses onthese subjects and current investigations will be presented.

OrganisationThe Summer School will take place from 9 to 22 July 2001 at theInternational Euler Institute, St Petersburg, Russia. The Summer schoolalready has financial support from the EMS and the RFBR. Additional infor-mation can be found from emschool@pdmi.ras.ru or from http://www.pdmi.ras.ru/EIMI/2001/emschool/index.html.

Scientific committeeA. Vershik (Chair), E. Brezin, O. Bohigos, P. Deift, L. Faddeev, V. Malyshev. Local Organising CommitteeA. Vershik, Ju. Neretin, K. Kokhas, E. Novikova.

Main speakersP. Biane ENS, Paris, FranceE. Brezin ENS, Paris, FranceP. Deift Univ. of Pennsylvania, USAK. Johansson KTH, Stockholm, Sweden V. Kazakov ENS, Paris, FranceR. Kenyon University Paris-11, Orsay, FranceM. Kontsevich IHES, France A. Lascouix University Marne-la-Valiee, FranceA. Okoun�kov UCB, USAG. Ol�shansky IPPI, Moscow, RussiaL. Pastur University Paris-7, FranceR. Speicher University of Heidelberg, GermanyR. Stanley Massachesetts Institute of Technology, USAC. Tracy UCD, USAH. Widom UCSanta Cruz, USA

The lectures will be devoted to asymptotic combinatorics and its applicationsin the theory of integrable systems, random matrices, free probability, quan-tum field theory, etc. � also those topics concerned with low-dimensionaltopology, QFT, new approach in the Riemann-Hilbert problem, asymptoticsof orthogonal polynomials, symmetric functions, representation theory andrandom Young diagrams.

During the last few years great progress has been made in this direction, andnew links and new problems have appeared. The following old problemshave been solved recently: fluctuation of random Young diagrams and of themaximal eigenvalues of the random Hermitian matrix. A new approach tothe Riemann-Hilbert problem and integrable systems was developed byDeift-Baik-Johansson (based on the Korepin-Izergin-Its approach, papers byTracy-Widom and others). The alternative methods came from asymptotictheory of representations and in particular from studying the Plancherelmeasure. These ideas allowed the calculation of the correlation functions ofthe corresponding point processes (Ol�shansky-Borodin and previous resultsby Kerov-Vershik) and also to application of the boson-fermion correspon-dence (Okoun�kov). The explicit distribution of the fluctuations of the char-acteristics of the diagrams was one of the main results, as well as the precisedistribution of the fluctuations of the eigenvalues of the random matrices.

This progress has initiated a great activity in many topics: the growth of poly-mers, ASEP, and random walks on groups and semigroups. Many perspec-tive problems and applications of the results will be discussed in seminars andround table discussions during the Summer School.

Applied Mathematics inApplied Mathematics inour Changing Wour Changing Worldorld

First Joint EMS-SIAM Conference

Berlin : 2 � 6 September 2001

Organiser: Peter Deuflhard, e-mail: deufl-hard@zib.de

Co-Chairs: Rolf Jeltsch (EMS), Gil Strang(SIAM), Peter Deuflhard (Organiser)

Programme committee:EMS: Heinz Engel, Björn Engquist, StefanMüllerSIAM: David Levermore, Volker Mehrmann,Bill MortonECMI: Vincenço Capasso

Preliminary topics (ordered by applications andnot by mathematical subject):1. MedicineComputational assistance of surgery, therapyplanning, medical imaging methods, hospitalinformation systems, pharmacokinetics, tumorgrowth modelling, artificial organs, immune sys-tem modelling, infectious disease control, epi-demic spreading2. BiotechnologyConformational molecular dynamics, drugdesign, mathematical modelling in polymerisa-tion, biomolecular structural storage schemes,patent recognition and circumvention, compu-tational statistics, sequence alignment, fuzzy rea-soning, density functional theory, ab initio com-putation3. Material sciencesRealistic modelling and simulation of supercon-ductivity, composite materials, phase transitions,crystal growth, hystesis, quantum-classicalapproximation and calculation4. Environmental scienceClimate research, climate impact research,short-term and medium-term meteorology,computational modelling of pollution of air,water and soil, atmospheric chemistry, ozonehole, porous media5. Nanoscale technologyIntegrated optics, optical networks, quantumelectronics and optics, general microwave tech-nology, nanoscale techniques in medicine,porous materials6. CommunicationAnalysis, simulation and optimisation intelecommunication, optical networks, transmis-sion rate optimisation, survivable networks7. TrafficDiscrete and continuous models of traffic flow,traffic on-line control8. Market and financeFinancial mathematics and statistics, optionpricing, derivative trading9. Speech and image recognitionSignal analysis, pattern recognition, wavelets10. Engineering designEngineering design in transport systems for air,land and water, in energy production, distribu-tion and conservation, and in consumer prod-ucts

EMS September 200014

EMS NEWS

The Committee for Women andMathematics of the EMS was set up inJanuary 1991 by the Executive Committee,with Eva Bayer as the first Chairperson.Then Christine Bessenrodt chaired theCommittee from 1995 to 1999 and I wasappointed as the new Chair last October.

The main purpose of the Committee isto work as a fact-finding unit, exposing theproblems and supporting the recognitionof achievements of women in mathematics.The first main project of the Committeewas to gather statistical information aboutthe presence of women among mathemati-cians in the European countries with a lowpercentage of women in mathematics.Moreover, Round Tables on themes relat-ed to Women and Mathematics have beenco-organised by the Committee atInternational Congresses (ECM 1992 and1996; ICM 1994 and 1998).

In her report of 1998, ChristineBessenrodt wrote:

It has often been noted that biographies ofwomen and men in mathematics (and thesciences in general) differ in many respects.A more detailed investigation into a larger

number of CVs would be necessary for this .. . It would be worthwhile to carry such aproject through and initiate a discussion onsome specific issues found in such a com-parison. It is a fact that age limits are stipulated in

many announcements for grants andprizes. But it has often been observed that,maybe because of duties at home, womenare more likely to have a late start in edu-cation or a slower career path. So age lim-its seem to be particularly harmful towomen.

During the last meeting of the EWM(European Women in Mathematics), whichtook place in September 1999 at Loccum,near Hannover, the above themes werediscussed in an informal Round Table,moderated by Laura Tedeschini Lalli. As aconsequence of the discussion, a question-naire was prepared. The text was then dis-cussed in our Committee, which addedsome questions and decided to distribute itas widely as possible to women and men allover Europe. We hope to collect a largenumber of answers, so as to be able to carryout a meaningful analysis of the data andto have a basis for further discussion. Thetext of the questionnaire follows. Pleasefill it in and send it to me, Emilia Mazzetti,Dipartimento di Scienze Matematiche,Università di Trieste, Via Valerio 12/1, 34127Trieste, Italy; e-mail: mazzette@univ.trieste.it

EMS NEWS

EMS September 2000 15

Questionnaire1. Are you male or female? ........................................................................................................................................................................

2. How old are you?..................................................................................................................................................................................

3. What is your nationality? ......................................................................................................................................................................

4. How many children do you have?...........................................................................................................................................................

5. At what age did you complete your Ph.D.? ..............................................................................................................................................

6. How many countries have you studied in/worked in? ...............................................................................................................................

7. What is your mother�s job? ...................................................................................................................................................................

What is your father�s job? ......................................................................................................................................................................

8. Do you have a permanent job?...............................................................................................................................................................

9. How many years after your Ph.D. did you obtain your first permanent job?...............................................................................................

10. How many temporary mathematical jobs have you had? ...........................................................................................................................

11. At what age did you write the paper of which you are most proud (so far)?................................................................................................

12. Have you had any gaps in your mathematical career? ...........................................................................................................................

If so, how long were those gaps? ..........................................................................................................................................................

In your opinion, what were the reasons for those gaps? ............................................................................................................................

............................................................................................................................................................................................................

13. Any further comments? ..........................................................................................................................................................................

............................................................................................................................................................................................................

EMS Committee for WEMS Committee for Women and Mathematicsomen and MathematicsEmilia Mazzetti

The spectacular biotechnological advancesof the past two decades have led to anexplosion of data in the biomedical sci-ences. We have recently completed themapping of the human genome, we cannow determine when in development cer-tain genes are switched on, and we canaccurately follow the fate of single cells.The list is endless. However, we are per-ilously close to falling into the practices ofthe nineteenth century, when biology wassteeped in modes of classification andthere was a tremendous amount of list-making activity. This was recognised byD�Arcy Thompson [1] in his classic workOn Growth and Form, first published in1917. He was the first to develop theoriesas to how certain forms arose, rather thansimply cataloguing different forms, as wasthe tradition at that time.

Of course, we have come a long waysince then. The identification of a genethat causes a certain disease or deformityhas huge benefits for medicine. We mustrecognise, though, that genes only specifythe properties of proteins and cells. It isthe physico-chemical interactions of thesecells that lead to (for example) the devel-opment of structure and form in the earlyembryo. Cell fate can be determined byenvironmental factors, and cells respondto signalling cues. Therefore, a study atthe molecular level alone will not help usto understand how cells interact. Havingdevoted a huge amount of effort to takingHumpty Dumpty apart, we must now findout how to put him together again.

Since the interactions that govern bio-logical processes are highly non-linear andmay be non-local, they must be couched ina language that is designed to compute theresults of such complex interactions. Atthe moment, the only language we have fordoing such calculations is mathematics.Mathematics has been extremely successfulin helping us to understand physics. It isnow becoming clear that mathematics andcomputation have a similar role to play inthe life sciences.

Self-organisationOne of the key puzzles in developmentalbiology is the understanding of how thevast array of spatial patterns and structurewe observe in the animal kingdom emergefrom the almost homogeneous mass ofdividing cells that constitute the earlyembryo. The skeleton, for example, is laiddown during chondrogenesis, when spe-

cialised cells (chondroblasts) condense intoaggregates that lead eventually to boneformation. Butterfly wings exhibit beauti-ful colours and patterns, and many animalsdevelop dramatic coat markings.

In all of these examples, although genesplay a key role, genetics say nothing aboutthe actual mechanisms that produce spatialpattern. The first major advance in thisfield was made by Alan Turing [2]. He wasinterested in morphogenesis � the process bywhich form and structure arise. He con-sidered a system of chemicals reacting anddiffusing, modelled by equations of theform

∂u/∂t = D∇∇2u + f(u, p),where u(x, t) is the vector of chemical con-

centrations at spatial point x and time t, Dis a diagonal matrix of diffusion coeffi-cients, and f models the reaction kinetics,which are functions of the chemical con-centrations and various kinetic parametersp. The problem was completed by impos-ing certain boundary conditions � forexample, periodic if one wants to model acylindrical structure, or zero-flux if onewants to model an impermeable boundary.Turing showed that one could chooseequations of this form which exhibited auniform steady state that was stable in theabsence of diffusion, but became desta-bilised when diffusion was introduced andevolved to a spatially varying state � a spa-tial pattern. This phenomenon is knownas diffusion-driven instability and is an exam-ple of self-organisation or an emergent proper-ty. Assuming that one of these chemicalswas a growth hormone, Turing then postu-lated that points where the concentrationof hormone was highest would growfastest, resulting in spatial structure. Forthis reason, the chemicals were termedmorphogens. More generally, one assumesthat these chemicals activate a gene switchif they breach a threshold value, causingcells to differentiate. This theory thushypothesises that the structures one seesoverlie a pre-pattern in chemical concen-trations.

Although the identification of mor-

EMS September 200016

FEATURE

Mathematical Mathematical modelling modelling

in the biosciencesin the biosciencesPhilip Maini (Oxford)

(a)-(c) Some computed solutions of a Turing reaction-diffusion model. (d)-(g) Typical animal coatmarkings. (h) Photograph of a common genet exhibiting a spotted body and striped tail (from

Murray, 1993, with permission).

(h)

phogens has thus far proved elusive,Turing structures have now been shown toexist in chemical systems. When Turingfirst proposed his theory, he met with somehostility from chemists who were con-vinced that such patterns could not arise inchemical systems. This is a nice exampleof mathematics driving research in otherscientific areas.

A number of theories based on differentbiological hypotheses have since been pro-posed for self-organisation, but many ofthese models rely on the common pattern-ing mechanism of short-range activation,long-range inhibition. It is thus possible tomake predictions that do not rely on spe-cific biological details. One such predic-tion is that a spotted animal with a stripedtail is more likely to occur than a stripedanimal with a spotted tail. This is an exam-ple of a developmental constraint. Thebook by James Murray [3] has an excellentin-depth discussion of this and relatedissues.

It transpires that models of the samegeneral form as that above can exhibit awide variety of patterns, such as propagat-ing fronts, spiral waves, target patterns andtoroidal scrolls. Indeed, the Hodgkin-Huxley model for electrical signalling innerve axons (for which they won the Nobelprize) is of the above form. The mostfamous example in chemistry of patternformation is the Belousov-Zhabotinsky (BZ)reaction, in which bromate ions oxidisemalonic acid in a reaction catalysed bycerium, resulting in sustained periodicoscillations in the cerium ions. If, instead,the catalysts Fe2+ and Fe3+ and phenan-throline are used, the periodic oscillationsare visualised as colour changes betweenreddish-orange and blue. This system hasbeen thoroughly modelled mathematicallyand it emerges that the key to patterninghere is a phenomenon termed excitability.An excitable system is one in which thesteady state is stable to small perturbations,but to large (supra-threshold) perturba-tions, the system undergoes a large devia-tion before coming back to its originalsteady state. During this transient period,the system does not respond to perturba-

tions. Coupling diffusion to this type ofkinetic behaviour allows for waves of activ-ity to propagate through the medium.

Excitable media also play a role in theaggregation of certain amoeboid species,such as the cellular slime mouldDictyostelium discoideum (Dd), which hasserved as an important model paradigmbecause it is simple enough to allow exper-imentation, yet sufficiently sophisticated toexhibit many physico-chemical processesthat are similar to those observed in high-er organisms. Under starvation condi-tions, these amoebae signal each other viathe chemical messenger cyclic AMP, result-ing in the propagation of spiral waves ofthe chemical. The amoebae move up gra-dients of cyclic AMP, resulting in the for-mation of aggregations. The formation ofaggregates seems to be a vital componentof the Dd life cycle, as it appears to be nec-

essary to enable the cells to differentiateinto a spore type that can survive harshconditions. This species has been exten-sively studied theoretically and the model-ling has resulted in crucial biological

insights that could not have been gainedeasily in any other way.

This example illustrates the power ofmathematics as the universal scientific lan-guage because, although the details of thebiology underlying Dd aggregation arevery different from the chemistry underly-ing the BZ reactions, the resulting mathe-matical equations are very similar, so thatinsights gained in one field can be trans-ferable to another, seemingly very differ-ent field.

Medical applicationsIntriguingly, the heart is another exampleof an excitable system, allowing electricalactivity to propagate across its surface asthe signal for the heart to beat. Manyheart abnormalities arise as the result ofdisturbances to this wave propagation, andthese have been studied using simple mod-

els of the above general form. Moresophisticated models have been developedthat allow one to predict what effect a sin-gle gene mutation will have on the globaldynamics of the heart. We are now enter-

EMS September 2000 17

FEATURE

Development of spiral waves leading to cell streaming in a mould for slime mould aggregation.

Philip Maini

ing the realm of the �virtual human�, inwhich even surgical procedures may befirst tested in virtual reality � after all, wedo not allow commercial airline pilots tofly a plane until they have completed sev-eral hours of training on a flight-simulator,yet we are happy to let surgeons loose onour brains without equivalent training! Insilico drug-testing is already approachingreality and is attracting a lot of interestfrom pharmaceutical companies. Thereduction in the costs (presently some450m euros) of bringing a drug to marketby the use of good models is beginning tomotivate such companies to invest in mod-elling research.

Recently, the multi-national pharmaceu-ticals giant Hoffman-LaRoche approachedDenis Noble and his colleagues in theDepartment of Physiology, Oxford, to helpwith a problem that arose during theapproval process for one of their drugs bythe US Food and Drug Administration.The FDA had noticed a glitch in the elec-

tro-cardiograms of people taking the drug,and clinicians had concluded that the drugwas dangerous. Noble applied the drug tohis virtual heart model and found that itdeveloped the same glitch. He found thatthe glitch was not a sign of major malfunc-tion and the drug was approved [4].

These types of models are highly com-putational, as they involve intricate molec-ular details linked to cellular activity. Ascomputing power increases daily, morecomplicated models can be analysed. Suchmodels also throw up interesting mathe-matical questions, such as how can oneproperly model the interactions of process-es occurring over many different length

and time scales.The second biggest killer in the devel-

oped world, after heart disease, is cancer.Despite the huge proliferation of experi-mental data and clinical treatments, therehas been no decrease in the death ratesdue to the most common cancers. This islargely because there is still no basic con-sensus models of tumour growth and sur-vival, metastasis (the process wherebypotentially fatal secondary tumours areformed from a primary tumour), tumourangiogensis (whereby nutrients are divert-ed to the tumour), and extra-cellularmatrix breakdown by tumour cells. One ofthe challenges of the next decade is todevelop mathematical models that clarifythese fundamental processes and that canpredict new strategies of clinical therapy.At present this area is attracting a lot ofresearch, and modelling is being used toaddress such problems as effective drug-delivery strategies and ways of decreasingangiogenesis.

BioinformaticsAs technology continues to advance, wemust develop ways to exploit and interpretthe flood of data, not drown in it.Probability theory, statistics, stochasticityand high-powered computing will play animportant role in the rapidly emergingfield of bioinformatics. Some of theimportant questions here are pattern find-ing, data mining and pattern recognition,to name but a few. Presently a major aimis to understand how protein structuresform, and how specific protein structuredetermines function. Here, there may be arole for topology, geometry, electrostaticsand mechanics.

SummaryWe are entering the post-genomic period,and it is clear that mathematics has anever-increasing role to play in biomedi-cine. In 1997 the International Union ofPhysiological Sciences set up the PhysiomeCommission to �promote anatomically andbiophysically based computational model-ling for analysing integrative function interms of underlying structure and molecu-lar mechanisms�. More recently, theresearch councils have set up programmesfunding research at the interface betweencomputation, mathematics and the life sci-ences.

Mathematical biology is a rapidly grow-ing subject and the number of full-timeuniversity faculty engaged in this type ofresearch is increasing. The subject areaitself has expanded enormously and theabove represents a very brief review. Otherareas of active research include neural net-works, neurophysiology, immunology, epi-demiology and ecology. It is clear that the

life sciences will continue to pose exciting,novel and challenging problems for math-ematicians.

References1. D. W. Thompson, On Growth and Form (abridgedversion), Cambridge University Press, 1992.2. A. M. Turing, The chemical basis of morphogenesis,Phil. Trans. Roy. Soc. (London) B 327 (1952), 37-72.3. J. D. Murray, Mathematical Biology, Springer-Verlag,1993.4. Cyberheart, New Scientist No. 2178, 20 March 1999.

Philip Maini [maini@maths.ox.ac.uk] is at theCentre for Mathematical Biology, OxfordUniversity.

FEATURE

EMS September 200018

Time evolution of multiple re-entrant waves in a 3D model of the dog heart (courtesy of Denis Noble).

POSTER COMPETITION

EMS September 2000 19

In EMS Newsletter 34 (December 1999) we gave the results of the EMS Poster Competition, where we invited competitors to design posters for useduring World Mathematical Year 2000. Here are some further entries that have appeared in Denmark.

EM

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Your research belongs to domains that werefor a time neglected by mathematicians.How do you now see the situation?I started as a pure mathematician but gotvery interested in applied mathematics �more precisely applications of mathemat-ics � and now my field of research is opti-misation and discrete mathematics, bothfrom the theoretical and the practical side.Both these areas were looked down on bymathematicians in general. Combinatoricsand graph theory were considered trivialsubjects. Optimisation was, from anorganisational point of view, not part ofmathematics: it belonged to industrialengineering in the US, and in Germanyoptimisation was a branch of operationsresearch within economics faculties. I dothink that these areas are truly mathemati-cal subjects. Giving them away to otherdisciplines is a big mistake, because theirheart � their real content � is mathemati-cal. These areas have contributed a lot tothe development of mathematics in therecent years. Their own progress mainlycomes from mathematics, but also fromcomputer science. At my University (TUBerlin), a large portion of the student bodynow studies these areas, writes diploma orPhD theses and gets good jobs in industry� on the basis of this training.

This does not mean that applied mathe-matics should dominate mathematics. Weneed the right balance between these areas� and we must have people leaning moreto the theoretical and more to the practicalside. The role of persons �in between� isimportant.

The focus of my own research is combi-natorial optimisation, a domain that isbooming in industry since we are now in aposition to solve really large models.

Could you be more specific about these mod-els?Let me give you some examples that areparticularly interesting at present.Telecommunications is an industry that isvery competitive due to the recent world-wide deregulation. As a result, competitorsin the market need to reduce costs andprovide new and better service: this imme-diately means optimisation. So my grouphas been developing models to design net-works that have certain properties: theysurvive certain failures, provide a certainservice, and are cost-effective. Anotherexample is the assignment of channels toantennas in mobile communication. Togive you a specific instance: in one system,we were able to reduce interference by 80-

90% by inventing and solving appropriatemodels. These were based initially oncolouring theory for graphs but � due toadditional side constraints � they got moreand more complex and we had to developthe theory further in order to solve modelsof this type.

The second area of great importance ispublic transportation: in Berlin, for exam-ple, we help to optimise the public trans-portation systems by finding the least num-ber of buses that Berlin needs in order tosupply all lines serving the city. We deter-mine the optimal assignment of drivers tobuses and approach similar problems.These tasks lead to extremely large opti-misation problems; for instance, the bus-scheduling problem results in an integermulti-commodity flow problem with about100 million variables. So we need to devel-op methods that solve problems of thisextremely large size. This combinesresearch in computer science, hardwaretechnology and mathematics. It leads toacceptable models that resolve and help todesign plans that are practically feasible,that are used in practice, and that save tensof millions of Deutschmarks.

What typically happens in a practicalapplication is that the mathematical mod-els that arise have not been investigatedsufficiently. So we have to look at thesemodels from the theoretical side and con-tinue developing whatever mathematics is

relevant. There are always two-wayexchanges between theory and practice,and this combination is very attractive andinteresting.

Many mathematicians have difficulty posi-tioning themselves and their students vis-à-vis industrial demands. How do youimprove this situation? Do you see an evo-lution in this respect?I try to position myself and my students,and I do help my colleagues if they ask, butI cannot give advice to everybody in thisrespect. I believe we need theoreticalresearch �in its own right�, research that isnot linked with real applications; this is animportant part of our culture and I don�tthink all mathematicians have to positionthemselves with respect to industry. It isnecessary, however, for mathematics as afield to be aware of its interfaces with othersciences and industry. If we don�t want tobecome an ivory-tower science, we mustthink about the way we teach, how we com-municate with our environment, and inparticular, about how we support industry.Industry, after all, is what we live from.Mathematics can contribute a lot to it, andit should.

It is a �global challenge� to find the rightway to handle problems in industry mathe-matically. There are so many areas wheremathematics can significantly contribute toprogress in industry. We need on-going

EMS September 200020

INTERVIEW

Interview with Martin GrötschelInterview with Martin Grötschel(Berlin)

Interviewer: Jean-Pierre Bourguignon (Bures-sur-Yvette)

discussions with industrialists through jointprojects. For me, this is the only sensibleway to learn about new technologies and tofind out how we can contribute. New tech-nologies are real challenges for mathemat-ics, often involving mathematical questionswe never thought about before. If we con-tinue addressing only the hard problemsthat mathematicians of former centuriesleft open, then we may overlook the chal-lenges of our own time. So, sometimes it isa good idea to put aside the hard questionsof the past and look at the (maybe equallyhard) problems of our day.

Let me mention information technology,biotechnology, telecommunication, trans-portation, and manufacturing processes.All of these topics have mathematical com-ponents: control of robots or of logistic sys-tems, production planning in general,design of new materials, the understand-ing of mechanisms in biosystems: these areareas where we really don�t know yet whatis going on. So mathematics as a whole hasto think about how it can develop themathematics involved in order to help thespecialists understand their (typically verycomplex) systems. This is not the job ofindividual mathematicians, but must bethe task of the whole community. We mustreach out into these areas and start puttingresearch effort into such projects, for thelong-term survival of our discipline as arich domain with active research.

For myself, I have currently chosen sev-eral areas of applications, but they changewhenever new technologies arise, andwhenever new challenges come up. In myresearch institute, the Konrad-Zuse-Zentrum, we do research; we are not a fac-

tory or a software provider. Of course, wedo produce software, but we only work onproblems where no standard solution isavailable and where no company in theworld has software of the type that is need-ed. In some cases this has led to spin-offcompanies that develop and extend someof the software further. I think that thisentrepreneurial aspect is an important newdevelopment in mathematics.

To change topic, you chaired the organisingcommittee of the Berlin InternationalCongress of Mathematicians and presidedover it. ICM98 was recognised by all par-ticipants as a great success. At a distancenow, can you share with us your feelingsabout the experience?We were delighted to hear that many par-ticipants really enjoyed the Congress.Organising ICM�98 was a lot of work, notonly for myself. Many colleagues, stu-dents, assistants, secretaries, etc. wereinvolved and contributed significantly.Working together, having to rely on eachother�s efforts stimulated a new spirit ofcooperation within the mathematical com-munity in Berlin, and more broadly, inGermany.

In addition to deepened personal rela-tions, there were many other spin-offs. Afew examples: the Video Math Festivalinspired Springer to start a new series onmathematical videos; the general publiclectures at the Urania were highly success-ful, and there will be a book containingthese talks. Enzensberger�s lecture wasbroadly publicised (e.g. printed in theFrankfurter Allgemeine Zeitung the weekendafter) and brought mathematics into the

literature pages of journals and newspa-pers. The ICM was an event we wanted tostage for both mathematicians and thegeneral public.

There is always the question as towhether such big meetings are really nec-essary. Certainly for research, specialisedmeetings are much more appropriate: youmeet your closest colleagues and learn thelatest news. But there is space for bigmeetings that provide surveys andoverviews, and where you can learn aboutprogress elsewhere and get ideas fromother fields. I enjoy attending such meet-ings. They are excellent opportunities toobtain the newest information presentedby the best person in a field in a didacti-cally pleasant form for non-specialists.

I think the speakers at ICM 98 in gener-al did a very good job. Many participantsI talked to felt well served by the speakersand the organisation. Also, internationalcommunication is extremely important,and I was particularly happy to learn frompeople from Eastern Europe and develop-ing countries how they benefited fromICM�98. It was an efficient means to meetpeople from other parts of the world andmake new contacts. For those who work inrich countries at top institutes, this is alltrivial: they can travel at any time, any-where. But this is not so for many excel-lent mathematicians around the world,and ICMs play an important role here.

ICMs do have a role in the future, Ithink. The question is: should a particularcity (like Berlin) make all these enormousefforts, just as a service for society or forthe mathematical community? When thefirst thoughts about organising an ICM in

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ZIB building, Konrad-Zuse Zentrum

Berlin were voiced, Berlin was divided intotwo cities. Time passed by, the two becameone, and we thought it might be a goodidea to invite the world to Berlin and toGermany. There are still bad feelingsabout Germany (that we experienced insome letters we received), but I think wegave answers by our special events, exhibi-tions and the lectures about the Nazi peri-od. So, we did try to present the �newGermany�. We also felt that research inmathematics has not had the publicity itdeserves in Germany, so we thought ofusing ICM as a vehicle to carry our mes-sage.

You made big efforts to make the congress amedia event in Germany. Were they suc-cessful? Have the relations between mathe-maticians and the German press changed?I think they have changed considerably.We thought for a long time about how topresent ICM to journalists. In the end, theorganising committee hired a professionaljournalist with a Masters degree in mathe-matics to produce material for the press.Mathematicians are not good at writingreadable texts for non-mathematicians, sohe would be in charge of the interfacebetween us and the press. A journalistwould read his articles, get interested, andthen interview mathematicians, do TVinterviews, and the like. From theTechnical University�s press office cameother good ideas on how to involve thepress, make announcements at the righttime, and so on.

Our press conference was attended byalmost 50 journalists, which is very unusu-al: seven TV stations and more than twen-ty radio stations came. We participated inat least ten major TV programmes andgave uncountably many radio interviewsworld-wide. The TU press office collectedabout 300 pages of published newspapermaterial about ICM�98. I think this was areal success story, at least within the math-ematical world. At the same time journal-ists got excited about mathematics.Interviews usually started with a statementby the journalist that he/she was very poorin mathematics. With simple examples weshowed how one can get excited aboutmathematics; and this worked.

The only bad experience was with DerSpiegel. The other papers were veryfavourable and there was ICM informationalmost daily in each of the five majorBerlin newspapers. Without public eventsthis would not have been possible. All thepresentations in the Urania drew newspa-per interest. It still keeps on � for exam-ple, just last week I had a phone call froma German TV station that wants to featurethe Hilbert problems. Journalists reallyenjoyed seeing where mathematics hidesin everyday life, and all those I met saidthat it is something worth reporting on.

Of course we were lucky that the Fermatproblem had been solved shortly before,and that Wiles was present and was a won-derful interviewee. Journalists loved him:certainly a superstar helps. I really thinkthat lots of articles that now appear aboutmathematics would never have appeared

before. To give another example, an arti-cle of mine in the German edition ofScientific American generated a lot ofrequests from readers, and they are nowreprinting it in a special issue; this has alsohappened to several of my colleagues.Making mathematics a public event andfamiliarising our fellow scientists withwhere mathematics is, was one goal wereally achieved with ICM 98: many of mycolleagues in engineering, chemistry, med-icine, etc., heard about mathematics,looked at TV programmes, and becameexcited about the new mathematics that isdeveloping.

Recently the German federal governmentintroduced programmes to enhance elec-tronic means of communication. Throughthe FachInformationsZentrum (Karlsruhe)it supported the Zentralblatt fürMathematik, which as Zentralblatt-Math isbecoming a European � the EMS joining ascopyright holder being one sign. You have

been involved in some of these actions.Where do you stand on them?This is a very complicated topic. First, pro-jects like this were not started by theGerman government, but by mathemati-cians and physicists. I fear that in the nearfuture we will no longer have easy andcheap access to the literature, since pricesare sky-rocketing and nobody can buyjournals any more. The paradox is that weproduce more publications but have accessto fewer of them. This concerns everybodyaround the world.

I (and many colleagues) believe that wehave to do something about this, and thatthe remedy should be easy now that we cando things electronically. Building a worldlibrary is in the background but is notachievable. So my idea is to change ourpublication system in the long run. Thedifficulty is that we do not exactly knowhow to, what the economics of the systemare, and who does what for what price. Mysuggestion is to experiment, see how

EMS September 200022

INTERVIEW

Martin Grötschel at the ICM�98, Berlin

things go, and then let the market decidewhat is to be done. It would be a disasterto stop all printed journals now and goelectronic right away; there must be a tran-sition period. For me, journals should beelectronically available for many reasons.One particular reason is the exceptionalsearch features that these media provide.But you should also be able to print articlesout. At the moment we are far away fromthat service, so we have to try some stepsthat involve different actions.

This is not an issue that concerns onlymathematics: all sciences face it, and solu-tions to this problem will not only apply tomathematics but to all sciences. The firstpriority is to join the other sciences, since95% of what we do is not topic-specific.For the other 5% that are particular tomathematics we must find our own solu-tions. In Germany four scientific societies(mathematics, chemistry, physics, andcomputer sciences) started a consortiumon electronic information and communcia-tion. About ten disciplines are now part ofit: psychology, education, biology, sociolo-gy, etc., have representatives who discusswhat should be done for the future. Formathematics we decided to start with aninformation system for mathematicians,for which we developed a prototype inGermany with the aim of making it global-ly available and transforming it into aninternational framework. This is calledMATH-NET.

The Executive Committee of theInternational Mathematical Union hasdecided to develop MATH-NET as a glob-al information and communication systemfor mathematics, and I do hope that theEMS is willing to play an important role init.

While building up MATH-NET we mustlink it with existing systems, such asZentralblatt-MATH, MathSci, and tradition-ally produced journals. We have to negoti-ate with publishers about what they makeavailable for the communication system.The German government have supportedactivities that help the different societies towork together. Workshops of all kindshave been organised and have been suc-cessful in helping us to learn from eachother, and in improving the understand-ing of the scientific communication in gen-eral. Emphasis was put on how we cancope with our differences.

The German government have leanedvery strongly towards developing thesemarkets commercially. The idea was thatmathematicians develop prototypes thatpublishers take over and make moneyfrom; but there was strong resistance tothis within the mathematics community:that�s why we should go our own way. Iwant to convince my colleagues that a rea-sonable way to progress is to get national,European and international mathematicalsocieties to work with scientific societiesworld-wide to set up a user-friendly systemwith clear interfaces that can be under-stood by all � free of charge.

A main task, indeed, is setting up inter-faces so that we can communicate witheach other easily. This is what the future of

electronic information and communicationis about: a few central navigation and coor-dination systems and agreements of mutu-al support; data to be provided locally, andownership and rights to be local; the sys-tem itself to be decentralised; the centralcomponents to have the aim of guiding usto the individual data and seeing the glob-al picture. We should not cut ourselves offfrom what publishers are doing: they pro-vide a service of value, they referee, theyjudge. All participants in the marketshould take part in this publicationsforum.

An important aspect is to have librariesworking with us. Libraries guarantee long-term access to data from the past, theyhave existed for three thousand years ormore, and they do a good job. It is notclear what the job of a librarian will be infifty years. Of course, books will not disap-pear, but we have to shape the future bylisting what we want, and what the special

needs are for our research. I think we willincreasingly lean to the electronic side andto the service that electronic informationsystems can offer. This is likely to be par-ticularly true for research journals; it maynot be true for books, but books may besupported by CD-ROMs, on-line supple-ments, etc.

The future is unclear. There are majordevelopments at present, and in ten yearsthe winners may be visible, but we must beready to move now. We cannot wait for thecommercial market to tell us what we haveto use. I don�t believe that monopolies willarise, because mathematicians are toointelligent and too strong to accept that.Many will fight against them.

Many mathematicians now realise that,unless a high price is paid, long-term accessto mathematical data is not guaranteed ifthey are stored electronically. Who shouldpay for this? How can long-term accessibil-ity be guaranteed?Right now, I cannot say who will pay, but inthe long run it will be the State through itssupport of research in universities,libraries, etc. But let me be a bit more spe-cific.

First, it is a fairy tale that printed publi-cations have long-term survival. I guessthat 80% of the books printed in the 17thor 18th century have disappeared com-pletely; we don�t know them any more.And I think this is even more so in music:all music pieces that have ever been com-posed, where are they? They were writtenand printed, but most have completely dis-appeared. So printing on paper is noguarantee for survival.

Secondly, if we print everything we pro-duce now, we won�t have enough space tostore it all, so I see no way to use paper inthe long run as the main medium of stor-age. Next, paper also deteriorates, and wenow know that texts printed on acid paperdisintegrate. We can learn a lesson fromthe library in Alexandria. They had peo-ple who copied the papyri of those days;every hundred years, a papyrus was copiedsimply because it was disintegrating.

So two thousand years ago they had copymechanisms already in place. This is whatwe should also do, and I think it is verysimple. Since we know that electronic stor-age devices such as discs or tapes will dis-integrate, we must learn to copy them sys-tematically. I personally don�t believe thatarchiving is a problem; the problem is toorganise the copying. And �copying�means also copying the software necessaryto retrieve and to read the data. If wedon�t want to do this, we must find mecha-nisms to transform them into new formats,as some �dialects� may not be readable inthe future. This means active archiving:passive archiving will not be enough. Justputting things somewhere was not effectivewith books, because rats or mice sometimesate them or water or fire destroyed them,and the same is true here.

However, active archiving may meanthat we have to be selective. We shouldprobably not copy everything that is avail-able. So, to concentrate on mathematics,we may consider it reasonable to look atthe body of literature produced within ayear and decide what should be archived.Alternatively, we could simply archiveeverything, but who will then be responsi-ble for its maintenance? So I think we willhave a problem of choice and selection,because we want to have institutions thatwill be responsible for active archiving.

Here, of course, some publishers will say�We are the active archivists�, but I don�tthink this is a long-term solution. Whoguarantees that Elsevier, for example, willstill exist next year? They could be boughtby Microsoft or somebody else and com-pletely change their policies. So archivingmust be outside the commercial sector.This task should be the responsibility ofpublic enterprises as part of the conserva-tion of our culture. It has to be supportedby the State, just as the State supportslibraries, museums, and the like. I imaginethat the Deutsche Bibliothek and othernational libraries could be active archivers,while institutions like FachInformations-Zentrum or a similar agency run by theState could say �We archive for certain sub-jects�. We then immediately have a prob-lem of the management of rights: can FIZ-Karlsruhe actively archive journals pub-lished by Springer, Elsevier, AcademicPress or the AMS? Here are difficult copy-right issues that I want to see resolved. Itis all a matter of organisation. I don�tthink it is extremely urgent, but in the nextten years we need to come up with schemesthat solve the archiving problem in the wayI describe. Costs must clearly be coveredby the State, and libraries should beresponsible for taking up the task

INTERVIEW

EMS September 2000 23

You have contributed more than 25 years ofdedicated work to the Zentralblatt MATHinformation service. How did you getinvolved? And how did it evolve?At the beginning of the 1970s a colleagueof mine at TU Berlin asked me to serve asa consultant for Zentralblatt für Mathematik(the traditional title of Zentralblatt MATH),for all areas dealing with topology.Though my main area of research interestis geometry, I became quite fascinated bythe possibility of getting a comprehensiveview of the developments in these neigh-bouring subjects, and also by the chance tocontribute to the flow of scientific informa-tion. Rather soon, the subjects I had tohandle also included global analysis, Liegroups, topological groups and classicalmechanics.

The Editors-in-Chief at that time wereUlrich Güntzer (for the office in WestBerlin) and Walter Romberg (for the onein East Berlin). Due to a call to Tübingen,and for several other reasons, Güntzer lefthis position in 1974 and I was asked by theHeidelberg Academy of Sciences tobecome his successor. The joint Germanco-operation on Zentralblatt stopped in1978 as a consequence of the reorganisa-tion of the government-supported infor-mation and documentation activities in thewestern part of Germany. The partner inEast Berlin was more-or-less forced to takethe stronger involvement of a non-acade-mic institution like FIZ Karlsruhe[Fachinformationszentrum Karlsruhe] inthe editing of Zentralblatt as a reason tocancel the co-operation. The editorial stafffrom East Berlin found some other occu-pation in the Academy of Sciences of theGDR.

The development of my activities atZentralblatt could easily cover a full articleby itself, so let me just mention some of themilestones. The main achievements alwaysdeveloped from taking up the challengesof new situations and of new frameworksinto which the Zentralblatt had to adjust, inorder to guarantee its survival as a leadingworld-wide reviewing service. To be a bitmore specific, these included:� the administrative transition ofZentralblatt to FIZ Karlsruhe, where allpublicly supported information and docu-mentation activities in mathematics,physics, astronomy, computer science,nuclear engineering and aviation engi-neering had to be concentrated;� cancelling the co-operation from theeditorial office in East Berlin, as alluded toabove;

� unsuccessful efforts to arrange a mergerbetween Zentralblatt and MathematicalReviews on the basis of a fair distribution ofwork and income;� permanent discussions to orchestrate the

role of Zentralblatt as an �author� and that ofSpringer-Verlag as the author�s publisher,in a more transparent and efficient waywith respect to costs; � a successful approach towards extendingthe editorial basis of Zentralblatt in theEuropean domain and establishing aninternational involvement for the scientificsupervision of this service, as is now doneby the EMS.

These developments were accompaniedby: � technological changes concerning edit-ing and usage of the service: installation ofthe searchable database in different net-works and on different platforms (this hadalready started in 1978); � transition from obsolete to new editorialtools such as TeX, offers on CD-ROM,adjustments of the web-service to therapidly evolving needs of the internetusers, and redesign of the service from asearchable database to a comprehensiveportal for mathematical literature withconvenient user facilities.

Clearly, all these activities led to a reduc-tion of my duties as a subject editor forZentralblatt, though I succeeded in becom-ing in charge of my favourite area (differ-ential geometry) rather soon, and I lookedafter this until quite recently.

You are currently expanding theZentralblatt MATH project into developingcountries. What is the philosophy behindthis?The web-service increased the possibilitiesof offering Zentralblatt to more users withconvenient subscription conditions. Toarrange this turned out to cost quite a lotof my time, but being at the disposal of asmany mathematicians as possible shouldbe one of the main goals of Zentralblatt, andthis has to be taken care of from the�author�s� side, for the benefit of the math-ematical community.

We should strive towards a situationwhere every mathematician can useZentralblatt to get information about, andaccess to, the research literature in mathe-matics. Previously this was a privilege forthose living in a country with good finan-cial support for libraries, because subscrip-tion rates to both Zentralblatt andMathematical Reviews became higher andhigher.

Assuming that mathematicians in richercountries accept that these rates could, orshould, be cut down (even drastically) fordeveloping countries, the idea is then tofigure out case by case what these mathe-matical communities can afford, and tomake corresponding fair offers. However,as an isolated action this will not helpmuch. In particular, we should also takecare of the parallel task of establishingrobust and reliable document deliveryfacilities for the developing countries; thishas to be done in close co-operation withthe big libraries in Europe. Also, care mustbe taken about sponsoring the initialinstallation of the technical infrastructureso as to enable them to read the electronicproducts.

I know that this is not in accordance withthe policy of some commercial publishers.The mathematical community and col-leagues with high reputation should putpressure on them to make these thingsmore-or-less freely available to mathemati-cians in developing countries.

How do you see your relationships withother projects, such as the EULER project?Zentralblatt is currently associated with fivemain projects: EMIS (the EuropeanMathematical Information Service),EULER (EUropean Libraries andElectronic Resources in mathematical sci-ences), ERAM (Electronic ResearchArchive in Mathematics), LIMES (LargeInfrastructures in Mathematics-EnhancedServices) and MPRESS (Mathematical

INTERVIEW

EMS September 200024

Interview with Bernd WInterview with Bernd Wegner egner (Zentralblatt MATH)

Interviewer: Steen Markvorsen (Lyngby, Denmark)

PREprints Search System). The common feature is that they are all

designed to improve access to informationon mathematics and communicationbetween mathematicians. ERAM andMPRESS extend the information service tosources which are not covered byZentralblatt, like preprints and the litera-ture before 1931. ERAM and EMIS pro-vide easily accessible articles which can belinked with the Zentralblatt database. EMIS(the website of the EMS) and its close-to-40mirrors provide an excellent environmentfor using these services world-wide; itextends the information provided byZentralblatt in several different directions,and includes (for example) a conferencecalendar and a directory of mathemati-cians. EULER connects this with informa-tion on library holdings and mathematicswebsites. Finally, LIMES has just recentlybeen designed to improve these offers (inparticular, those of Zentralblatt) and toestablish them as a large facility for thedaily work of the mathematician in nation-al and regional networks [see EMSNewsletter 36, pp. 8-9].

Concerning the use of Zentralblatt MATH,you have expressed the opinion that theuser-flow-statistics should not be made pub-lic. Could you explain?This is not exactly my favourite question,but let me answer it this way. My onlyreservation is actually the same as for sta-tistics concerning primary publications inmathematics. A large part of the sub-scription rate is justified by making the ser-vice available at any time to almost every-body. Moreover, those who are using theseservices, use them at quite different levels.Hence uncommented statistics lead to mis-interpretations and improper usage of thefigures. The SCI [Science Citation Index]is the best example of such an unfortunatedevelopment in mathematics. I have noobjection to revealing such figures if theyare commented on properly.

Concerning the interface with electronicjournals, what do we do about the archivingproblem? Is it special to mathematics?Could the Zentralblatt MATH databaseplay a role in solving it?The archiving problem is a general prob-lem for all electronic products, thoughthere may be particular difficulties withmathematics because of the coding of for-mulas, the addition of animations and thelinks to mathematical software. Some pub-lishers propose that the high price for elec-tronic products is a result of their duty totake precautions for the future archiving ofthe contents, but I do not believe that mostof them really will reinvest money into thearchiving once a change of technologyneeds major work to keep the old electron-ic offers readable. Back volumes of math-ematical journals do not earn muchmoney, and moreover, the publishers cur-rently seem to have a short life period onlybefore being merged, swallowed or con-verted into a totally commercial business,which typically then mainly takes care ofthe shareholder values.

In my opinion the only institutions whocare about reliable archiving are the biglibraries. This was their obligation withpaper publications, and they will have todevelop skills for the archiving of electron-ic stuff also. But at present, libraries haveto catch up with the transition of theiractivities to the usage of electronic mediaand they are confronted with a fundamen-tal problem which makes archiving impos-sible for them. In contrast to the timewhen they delivered printed journals, thepublishers (at least the bigger ones) arenow reluctant to give the content of theelectronic journals to libraries. The role ofthe library is simply to take care of the pay-ment of the subscription and to install thelinks from their clients to the correspond-ing publisher. Here pressure will have tobe come from authors, readers and librari-ans so that the contents of journals will beavailable at a set of selected libraries rep-resenting the interest of all subscribers.

At present I do not see any role forZentralblatt in this discussion, but havingonce installed a system of referencelibraries, Zentralblatt should provide appro-priate links to their holdings for those whohave the license for accessing these jour-nals.

Some publishers argue (correctly, I guess?)that what they mostly sell are the �fancywrappings�, including some search enginesand web-portals etc. to their respective pub-lishing houses. This is where they add valueand is what we pay for. Shouldn�t we behappy about this?As long as no rough figures are given con-cerning what is really needed by the pub-lishers to create and maintain these addedvalues, the whole discussion will take placewithout any verifiable background andeverybody can propose what they happento have in mind.

There is no question that the main valueof an article consists of its scientific con-tent. If publishers tell us that the reason

for the high prices are the fancy wrap-pings, then they should give us the chanceto decide what we really want. My experi-ence is that these added values do notautomatically have to be expensive.However, if my estimate of the costs iswrong, we should simply claim the possi-bility of getting the content without theseexpensive additions. The main addedvalue to a paper is organised by the editorsand contributed by the referees. Theseservices will not change for electronic jour-nals and, in all cases I know, have beencompletely voluntary and � up to a mar-ginal cost � without any expense on thepublishers� side.

The reviewing philosophy of Zentralblattis strictly oriented towards scientific con-tent. Hence, as with Mathematical Reviews,we require that articles reviewed inZentralblatt must have been subject to peer-reviewing before. Both services are theonly ones in mathematics that are compre-hensive.

There are linked offers by some publish-ers providing free or charged access to bib-liographic data and summaries of theirjournals. These are called databases, ofcourse, and they also have modest searchfacilities and metadata, but they are ofcomparably low quality. The main aim ofthese offers is clearly to increase the visibil-ity of the products of one given publisheror a group of publishers. After all, if thepurpose were really to increase the naviga-tion possibilities for the user, then thesepublishers should have cared more aboutmaking the links fully comprehensive.Moreover, the recent initiative for the DOI[Digital Object Identifier] would have beenarranged in such a way that small publish-ers could also benefit from it. At present,most major commercial publishers in sci-ence and two or three society publishersshare this facility.

Projects like EULER and other portalsdeveloped by libraries in co-operation withZentralblatt will soon be more appropriate

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for getting comprehensive entrances tomathematical literature, and will providethe user with more convenient search facil-ities by investing more efforts in the pro-duction of the metadata.

It seems to me that review-databases wouldfit in very nicely with cross-reference andDOI initiatives � It would be highly desirable to have cross-referencing between all mathematical jour-nals, and not only between those that canpay for the DOI. Links to and fromZentralblatt and digital object identifiers ona less commercial level may provide achance to extend this linking to a wider setof journals. At present, Zentralblatt tries toimplement as many links as possible, offer-ing to publishers the opportunity to linkthe references sections in their mathemati-cal journals with the corresponding entryin our database. This will be freely accessi-ble for the readers of these journals. Butunique identifiers of electronic productswill be an essential requirement to estab-lish this linking as a reliable tool.

Why don�t we have more survey articles?The problem with surveys is a very old one.The reputation of surveys is generallylower, and the work to be done is consid-erably higher, than the production of aresearch article. Electronic media willopen new perspectives for surveys, hand-books and encyclopaedic publications. Ihope that such articles will come up soonas living projects, supported by a team andupdated regularly. Electronic publicationprovides many new degrees of freedom forsuch publications.

Through license agreements we now haveaccess to electronic journals from otherfields, journals that we previously did noteven know existed. Will this access changethe directions of mathematical research,and hence maybe also of the reviewingprocess?Being able to read journals from otherfields more easily will have only a modestimpact on the development of mathemati-cal research. This mainly concerns jour-nals at the borderline of mathematics,journals that are only partially covered byZentralblatt. If mathematicians develop agrowing interest in these journals, thenthis may lead to the decision that they willbe handled in their entirety, even thoughsome articles may just marginally deal withmathematics. But this development onlyneeds the availability of subscriptions tothe electronic versions of these specificjournals. That the access will be open forthe whole campus, as it had been for theprinted version, should be taken for grant-ed. So there is still no need to enter intosuch kinds of licensing agreements, wherea customer is more-or-less pressed to buy awhole bunch of journals from the samepublisher just in order to get a fair sub-scription rate.

How do you find time for other activities,such as differential geometry?Finding time for this gets harder and hard-

er. It is also very difficult to maintain myprivate activities, such as tennis and sail-ing. There are two commitments whichalways force me to care about my geomet-ric research � the more challenging is myresearch students, the more relaxing con-sists of my participation at conferences ongeometry. But the latter have increasinglyhad to be reduced in favour of generalmathematical meetings and conferenceson information and communication in sci-ence.

How did you become interested in geome-try? Mathematics was one of my three favouritetopics in the natural sciences � chemistryand astronomy were the other two. Thechance to graduate in mathematics in areasonable time seemed to be the best, so Ichose mathematics, and I enjoyed thestudy. There were two reasons for thechoice of geometry. I use a lot of geomet-ric intuition for my mathematical thinking� though I do not forget about the need forprecise mathematical arguments.Moreover, during my studies and for mythesis, Kurt Leichtweiss was my main advi-sor. He was a reliable counterpart for sug-gesting research topics and judging therelevance and the correctness of myresearch. More importantly, he did not tryto force my work into the directions of hisown main interests, but left the preferencesto me. Such conditions and possibilitiescannot be found everywhere.

Differential geometry is strongly represent-ed at TU Berlin. Could you tell some of thehistory behind the Berlin school and maybe

speculate on the future and importance ofdifferential geometry?TU Berlin was founded as a polytechnichigh-school with main activities in engi-neering science. Differential geometryand descriptive geometry were importantmathematical subjects needed there, sothere has been a chair of geometry forquite a long time at TU; Christoffel held

the chair for some time. When the capaci-ties of the universities were extended inthe 1960s and 1970s, geometry was electedas a favourite subject at TU; this led to anexpansion in the positions available. Atpresent we have five professors in thegroup, though not all of them do researchin differential geometry exclusively.

I think that classical differential geome-try will play an important role in futuremathematics, due to its applications toother sciences � in particular, the engi-neering sciences. Graphics tools will givenew possibilities for handling objects fromdifferential geometry and for providingnew insights into their structure. As a com-panion, discrete geometry will receivemore attention, because the use of com-puters makes solutions to problems acces-sible, which could not have been obtainedwith the human brain alone. There will belinks between both subject areas.

The development of differential geome-try will also be subject to the problems thatwe face in mathematics in general. Thereare different reasons for the reduced moti-vation of young people to study mathemat-ics and care about mathematical research.The main reason is that there is moremoney in business: those who graduate inmathematics prefer to accept a better paidjob in industry. The lack of people in thisarea is a current matter of discussion, andin some countries in Eastern Europe inparticular is a very urgent problem; acade-mic jobs are among the worst paid there.If you think of the many good geometerswho have been educated in EasternEuropean countries, there is one pertinentquestion: who will be their successors?

Nevertheless, I am optimistic that therewill be no reduction in mathematicalresearch and training on a world-widelevel. I am also confident that ZentralblattMATH will have an ever-increasing num-ber of interesting papers to review in thefuture � there is no likelihood of a reduc-tion of this (net-)work within our mathe-matical community.

INTERVIEW

EMS September 200026

FEATURE

EMS September 2000 27

The year 2000 was proclaimed �WorldMathematical Year 2000� by UNESCO. Onthat occasion, the Belgian Post Officerecently released a special stamp, entitled�Mathematical formulas and the Ishangobone�. Here is a description of what can beseen on the tiny graphical composition, ofabout the standard Belgian stamp size �that is, 27.66 mm by 40.20 mm.

The upper right and left corners men-tion the price of the stamp in BelgianFrancs (17BEF), and equivalently in thenew European currency, the Euro (0.42Euro). These amounts (less than 0.40US$) are mentioned at both sides of theindication Belgique-België of the country ofissue, printed in French and Dutch, thetwo official languages of Belgium. ThePost Philately Department sells the stampseparately or in sheets of 30. It can beobtained at the Centre Monnaie, B-1000Brussels, Belgium (or through e-mail, atphilately@ philately.post.be).

The stamp, as presented in a Belgian catalogueof the Post Philately Department.

The composition carries another identi-fication, that of the designer, ClotildeOlyff. This graphic artist and typographer�metamorphoses� letters in her artwork, toanalyse the alphabet and to simplify andstructure it into geometrical forms. Herletter sets and typographic games havebeen released in the US, Canada andGermany. This time, she turned her atten-tion to mathematical symbols.

The background of the stamp is mainlyblue, except for the white part on the topabove the serrated line. �Blue is the color ofhope�, confessed Olyff, and this of coursewas willingly accepted as a very good rea-son for using it on a stamp about mathe-matics. A circle, in yellow, surrounds theformulas. It is a pity that the stamp is notsquare, since then it could have referred tothe 2000-year-old mathematical problemof �squaring the circle�. A red Gauss curvecrosses the drawing, producing a ratherpeculiar effect on the complete 30-stampsheets when they all seem linked to eachother. This normal curve emphasises the

growing impact of applied mathematics,and in particular of statistics.

Clotilde Olyff, the designer of Belgium�s mathe-matical stamp

The two formulas are shown in green.Above is Stokes� formula in its unifiedform: ∫xdω = ∫∂xω. The expression wasquoted by André Weil, in his autobiogra-phy The Apprenticeship of a Mathematician, asthe initial mathematical reason for startingthe Bourbaki group (see the outlined textand [2]). It goes without comment that thereference below, xn + yn = zn, points toFermat�s last theorem, proved about fiveyears ago by Andrew Wiles.

Finally, on the bottom of the stamp, inthe postage margin, there are white marks,in a group of three and a group of six lines.It is not a new kind of bar code, althoughthis misinterpretation would not have beenan unfortunate one. The marks refer tothe Ishango bone, which is one of the old-est mathematical finds. Prof. J. deHeinzelin discovered the bone, now pre-served at the Royal Belgian Institute forNatural Sciences, in Africa (see [1]). Itspresence on the stamp emphasizes theimportance of mathematics for Africa, andthis was one of the goals UNESCO had setin declaring 2000 the �World MathematicalYear�. This expression stands, in English,in the lower part of the stamp, as to under-line the statements made by the formulas.

Two special seals were issued on the occasion ofthe release of the stamp. One carried a repro-duction of the formulas, and was issued in fourBelgian cities, during an advanced sale. Thesecond, called Cancellation �Date of issue�,showed some Ishango marks.

References1. D. Huylebrouck, The Bone that began the Space

Odyssey, The Mathematical Intelligencer 18 (fall 1996), p. 56.2. A. Weil, The Apprenticeship of a Mathematician, VitaMathematica Series, Birkhäuser Verlag, Basel � Boston� Berlin, 1992.

F. Dumortier, University of Limburg, B-3950Diepenbeek, Belgium;D. Huylebrouck, Sint-Lucas Architecture,Paleizenstraat 42, 1030 Brussels, Belgium.

A stamp for the WA stamp for the World Mathematical Yorld Mathematical YearearF. Dumortier and D. Huylebrouck

Weil associated the birth of Bourbaki toStokes� formula in his autobiography (see[2]). In the English translation of the origi-nal French monograph, he declared the fol-lowing (pages 99-100):

Awaiting me upon my return toStrasbourg were Henri Cartan and thecourse on �differential and integral calculus�,which was our joint responsibility � here Ihad escaped being saddled with �generalmathematical�. We were increasingly dissat-isfied with the text traditionally used for thecalculus course. Because Cartan was con-stantly asking me the best way to treat agiven section of the curriculum, I eventuallynicknamed him the Grand Inquisitor. Nordid I, for my part, fail to appeal to him forassistance. One point that concerned himwas the degree to which we should generalizeStokes� formula in our teaching.

This formula is written as follows∫b(X)ω = ∫X dω,

where is a differential form, d is its deriva-tive, X its domain of integration and b(X)the boundary of X. There is nothing difficultabout this if for example X is the infinitelydifferentiable image of an oriented sphereand if is a form with infinitely differentiablecoefficients. Particular cases of this formulaappear in classical treatise, but we were notcontent to make do with these.

In his book on invariant integrals, ElieCartan, following Poincaré in emphasizingthe importance of this formula, proposed toextend its domain of validity.Mathematically speaking, the question wasof a depth that far exceeded what we were ina position to suspect. Not only did it bringinto play the homology theory, along withRham�s theorems, the importance of whichwas just becoming apparent; but this ques-tion is also what eventually opened the doorto the theory of distributions and currentsand also to that of sheaves. For the timebeing, however, the business at hand forCartan and me was teaching our courses inStrasbourg. One winter day toward the endof 1934, I thought of a brilliant way ofputting an end to my friend�s persistent ques-tioning. We had several friends who wereresponsible for teaching the same topics invarious universities. �Why don�t we gettogether and settle such matters once and forall, and you won�t plague me with your ques-tions any more?� Little did I know that atthat moment Bourbaki was born.

Despite its name, the LondonMathematical Society (LMS) has, almostsince its foundation, served as the nationalsociety for the British mathematical com-munity. Its establishment in 1865 madeBritain one of the first countries in theworld to have such an organisation,prompting and/or influencing the forma-tion of several other national societies

world-wide, such as the SociétéMathématique de France (1873), theCircolo Matematico di Palermo (1884), theAmerican Mathematical Society (1888),and the Deutsche Mathematiker-Vereinigung (1890).

Before its creation, other mathematicalsocieties had existed in Britain, such as theManchester Society, founded in 1718, andthe Oldham Society of 1794. But thesewere very much local bodies, more akin toworking men�s clubs than learned societies.Moreover, although they flourished for atime, none survived; even the famousSpitalfields Mathematical Society of EastLondon [1], which dated from 1717, wasforced to dissolve in 1845, due to a rapiddecline in membership.

The formation of the LMS was inspiredby the increasing need for specialised sci-entific outlets during the nineteenth centu-ry. In Britain, this resulted in the founda-tion of national societies specifically devot-ed to geology (1807), astronomy (1820),statistics (1834) and chemistry (1841).

What was to become the British mathe-matical society arose from a chance remarkin a conversation between two former stu-dents of University College London in thesummer of 1864. During a discussion ofmathematical problems, it occurred to

them that �it would be very nice to have aSociety to which all discoveries inMathematics could be brought, and wherethings could be discussed, like theAstronomical [Society]� [4, p.281]. Thetwo young men were Arthur CowperRanyard and George Campbell DeMorgan, the son of one of the most influ-ential British mathematicians of the day.

Augustus De Morgan was the foundingprofessor of mathematics at UniversityCollege, which he had single-handedlyestablished as the home of advanced math-

ematical education in London. Consciousof the key role the Professor�s reputationcould play in attracting members to theSociety, it was agreed �that George shouldask his father to take the chair at the firstmeeting� [4, p.281].

Agreeing to this, the senior De Morganapparently insisted that their tentative titleof �The London University MathematicsSociety� be changed, first to the �UniversityCollege Mathematical Society�, and then,in order to widen the scope of the society�smembership, to the �LondonMathematical Society�.

The newly retitled society held its inau-gural meeting at University CollegeLondon on Monday 16 January 1865, withAugustus De Morgan as its first presidentgiving the opening address. The venuewas an appropriate one, for the Society�srejected title of �University CollegeMathematical Society� was still more accu-rate at this stage: of the 27 founding mem-bers, no fewer than 26 were, or had been,associated in some way with the College.Within months, the Society had attractedover sixty new members from around thecountry, including many of the leadingBritish mathematicians of the nineteenthcentury, such as Arthur Cayley, JamesJoseph Sylvester, Henry John StephenSmith, George Salmon, William KingdonClifford and James Clerk Maxwell.

Right from the Society�s inception, it hadbeen intended that papers presented atmeetings should be printed and circulatedamong the members. However, theincrease in the cost of printing and distrib-ution caused by the sharp increase inmembership resulted in the first volume ofthe Society�s Proceedings, covering the peri-od from January 1865 to November 1866,containing just 11 of the 37 papers pre-sented during that time. The need forfinancial retrenchment was to be a charac-teristic feature of the Society�s formativeyears.

Even an increase in the members� sub-

FEATURE

EMS September 200028

The London Mathematical SocietyThe London Mathematical SocietyA brief history

Adrian Rice

Arthur Ranyard George De Morgan

University College London

scription from its initial ten shillings (50p)to one guinea (£1.05) in November 1867proved insufficient, and by 1873 theSociety found its balance in the red for thefirst time, due to its ever-increasing publi-cation of papers. The Society was thusfaced with surviving on an even tighterbudget than before. Money-saving ideas,such as reducing subscriptions to certainjournals, cutting back on printing, andcharging members for copies of papers,were seriously considered until the appliedmathematician Lord Rayleigh (1842-1919)made a generous bequest of £1,000 in1874. This gift was gratefully accepted, thefinancial pressure was relieved, and theLMS was thus saved from what could haveresulted in its early demise.

Before long, distinguished memberswere presenting papers that would previ-ously have graced the pages of the RoyalSociety�s Philosophical Transactions or theProceedings of the Cambridge PhilosophicalSociety. De Morgan�s influence may havegiven the Society much needed initialmomentum, but it was papers by laterpresidents such as Cayley, Sylvester,Rayleigh and Smith which placed it on alevel with other scientific societies. Whathad begun life as a simple club of DeMorgan�s ex-students at University Collegehad been transformed into the nationalbody for research-level mathematicians[7].

There was now no obstacle to the publi-cation of papers, which increased substan-tially from this point. By 1900, over 900papers had been published in theProceedings, with the volume of pages forthat year exceeding 700. The next centu-ry saw a massive expansion in the Society�spublication activities. In 1926, under theinfluence of G. H. Hardy, the driving fig-ure of the Society for the first half of thetwentieth century, the Journal was founded,followed by the Bulletin in 1969. Morerecently, in keeping with the times, 1997saw the launch of a purely electronic jour-

nal, the LMS Journal of Computation andMathematics. Joint publication ventureshave included the Journal of AppliedProbability (from 1964), Nonlinearity (from1988), and the History of Mathematics bookseries (from 1989).

As soon as its finances permitted, theLMS sought to promote and reward math-ematical achievement by means of prizesand awards. Its premier award, the DeMorgan Medal (awarded every threeyears), was endowed through subscriptionsof members in honour of the Society�s firstpresident. Initially awarded in 1884 toArthur Cayley, subsequent medallists haveincluded J. J. Sylvester, Felix Klein,Bertrand Russell, G. H. Hardy, and themost recent recipient, the number theoristRobert Rankin [6]. Other prizes regularlyawarded by the LMS are two BerwickPrizes, up to four annual Whitehead Prizesand the Pólya Prize, all instituted in com-memoration of important figures in theSociety�s history.

The Society has also been instrumentalin bringing the work of mathematiciansfrom overseas to the attention of a Britishaudience. Its first initiative was the estab-lishment of the honorary foreign membercategory in 1867, of which the first recipi-ent was the French geometer MichelChasles. Later honorary members includ-ed Kronecker, Poincaré, Cantor andHilbert. More recently, to honour thework of G. H. Hardy, not just in mathe-matics, but in the internationalisation ofthe subject, a Hardy Lectureship (recentlyrenamed the Hardy Fellowship) was set upto enable distinguished overseas mathe-maticians to visit the United Kingdom foran extended period to work and exchangeideas. By means of this scheme, such dis-tinguished individuals as Ahlfors, Dynkin,Bombieri, Tits and Feit have visited theUK as Hardy lecturers in the last 25 years.As this brief survey has shown, since itsfoundation in 1865, the LMS has grownsteadily in size and importance. From an

initial size of 27, its membership swelled to250 by 1900, and currently stands at overtwo thousand. From a college club, it hasgrown into a learned society, was incorpo-rated as a limited company in 1894, andwas finally granted a Royal Charter in1965. Although originally intended as apurely research-based organisation, recentyears have seen a huge increase in thediversity of activities in which the Society isinvolved � from education to popularisa-tion to government policy. But perhapsthe most significant recent developmenthas been its move in 1998 to its first exclu-sive headquarters (see EMS Newsletter 31).The impressive building, appropriatelyrenamed De Morgan House, is a fittingreflection of its status as Britain�s nationalmathematical society � a far cry from themodest collection of University Collegealumni who gathered for their first meet-ing 135 years ago.

Further reading1. J. W. S. Cassels, The Spitalfields MathematicalSociety, Bull. London Math. Soc. 11 (1979), 241-258.2. E. F. Collingwood, A century of the LondonMathematical Society, J. London Math. Soc. 41 (1966),577-594.3. H. Davenport, Looking back, J. London Math. Soc.41 (1966), 1-10.4. S. E. De Morgan, Memoir of Augustus De Morgan,Longmans, Green and Co., London, 1882.5. J. W. L. Glaisher, Notes on the early history of theSociety, J. London Math. Soc. 1 (1926), 51-64.6. A. C. Rice, The origin of the De Morgan medal,London Math. Soc. Newsletter 240 (1996), 20-22.7. A. C. Rice, R. J. Wilson and J. H. Gardner, From stu-dent club to national society: the founding of theLondon Mathematical Society in 1865, HistoriaMatematica 22 (1995), 402-421.8. A. C. Rice and R. J. Wilson, From national to inter-national society: the London Mathematical Society,1867-1900, Historia Mathematica 25 (1998), 376-417.

Adrian Rice is an assistant professor of mathe-matics at Randolph-Macon College, Virginia,USA

FEATURE

EMS September 2000 29

Augustus De Morgan Arthur Cayley G. H. Hardy

116 Let x be a natural number. Prove that x is a perfect square if and only if for every natural number y there is a natural number zsuch that x + yz is also a perfect square.

117 Let x, y and z be positive numbers satisfying xyz = 1. Show that (x9 + y9)/(x6 + x3y3 + y6) + (y9 + z9)/(y6 + y3z3 + z6) + (z9+ x9)/(z6 + z3x3 + x6) ≥ 2.

118 Which five integers have the following property: if we add in different ways four of these numbers we obtain 21, 25, 28 and 30?119 A rectangular sheet of dimensions 12 × 10 is cut into two pieces of equal area. What is the minimum length of the line of inter-

section?120 For real numbers a, b, c, d, consider the two sets

A = {x : x2 + a|x| + b = 0} and B = {x : [x]2 + c[x] + d = 0 },where [x] denotes the integer part of x. Prove that if the intersection of A and B has exactly three elements then a cannot be aninteger.

121 Given a line l in the plane and three circles with centres A, B, C tangent to l and pairwise externally tangent to one another, showthat triangle ABC has an obtuse angle and find all possible values for this angle.

Solutions to some earlier problems

104 Choose a point P on the bisector of an angle with vertex A. A line through P meets the arms of the angle in points B and C. Show that 1/AB + 1/AC is constant as the line rotates about P.

Solution by J. N. Lillington, Winfrith Technology Centre, Dorchester, UK; also solved by Niels Bejlegaard, Stavanger (Norway), André Brémond,Avignon (France), Dr. M. Dzamonja, UEA, Norwich, UK, and Dr. Z. Reut, London.

Let ∠BAC = 2α = constant, and let BC make an arbitrary angle δ with AP. By the sine rule for the triangles ABP, ACP we have

AP/sin δ = AP/sin [π � (δ + α)] = AP/sin (δ + α) and AC/sin δ = AP/sin [π � (π � δ + α)] = AP/sin (δ � α).

Hence: 1/AB + 1/AC = 1/(AP sin δ).[sin (δ + α) + sin (δ � α)] = 2 sin δ cos α /(AP sin δ) = 2cos α/AP independent of δ(using sin x + sin y = 2 sin [(x + y)/2] cos [(x � y)/2]).

105 Find all non-negative integers x, y, z which satisfy the equation 3 (x + y + z) = xyz .

Combined solution by Niels Bejlegaard and J. N. Lillington.

In the last corner I sang the praises ofRomania�s pioneering work in identifyingand breeding rising generations of mathe-maticians and scientists. This path of suc-cess has headed upwards for decades now,and remains unbroken. It is no wonderthat last summer Romania hosted theInternational Mathematics Olympiad forthe fourth time and attained fourth placeamong 81 participating nations.

The secret ingredient of these amazingachievements is probably a sound under-pinning in the lengthy practice of spottingmathematically able teenagers. Aside fromthe National Mathematics Olympiad thatacts as the flagship of home competitions,there is a close network of local and region-al contests. One of these is the DistrictMathematics Competition, a cross between awarm-up and the pinnacle of the NationalRomanian Mathematics Olympiad, and istherefore not formally part of the nationalevent. Nonetheless, this kind of contest isuseful for students who want to competefor the best solutions to somewhat unwieldy

problems. Such encounters have becomevery popular in the last twenty years,because they provide a proper training forthe real thing, a showdown with the bestcompetitors in the national finale.

The competing teams are chosen fromthe first-rate mathematical wizards at dis-trict level. The problems posed at thisevent and its rules are similar to those inthe final stage of the nation-wideOlympiad. As always, the irrepressiblejournal Gazeta Matematica provides an inex-haustible fund of challenging problems.Every offshoot from these peculiar contestsis named after a Romanian mathematicianof outstanding merit: according to VasileBerinde, Dean of the Faculty of Science,the prototype is designated The GheorgeTiteica contest. It was initiated in 1979 by ahandful of mathematicians from theUniversity of Craiova in order to stimulateinterest in mathematics in grade 7 � 12 stu-dents from Oltenia, a province in thesouthern part of Romania. Soon after, fur-ther competitors from around the country

and from outside (Moldova, Greece,Bulgaria and Yugoslavia) joined in thiscompetition.

The most striking difference between thestructures of the Titeica contest and theNational Olympiad is the terraced configu-ration of the former. Each of the threerounds has two levels. The prelude to thisintellectual bout is open to all registeredparticipants at the second stage, the teamcontest, where only teams of three mem-bers engage in combat. And another pecu-liarity is that only three age-groups racethe others (at level 1 grades 7 � 8, at level 2grades 9 � 10, and at level 3 grades 11 �12). The reason for this division into threeclasses is two-fold, for one thereby obtainsa pick of individual contestants as well asteam-mates. Both individual and teamprizes are awarded.

That completes the Corner for this quar-ter. Please send me your solutions as wellas contest materials, and propose problemsfor readers to solve. Wherever possible,proposals should be accompanied by asolution, references and other insights, tohelp the editor. The problems can rangefrom elementary to advanced, from easy todifficult, and original ones are particularlysought. So, please submit any interestingproblems you come across, especially thosefrom (problem) books and contests that arenot easily accessible � but other interestingproblems may be acceptable provided thatthey are not well known and referencesgive their provenance

PROBLEM CORNER

EMS September 200030

PPrroblem Corneroblem CornerContests from Romania, part 2

Paul Jainta

Given 3 (x + y + z) = x y z (*), suppose without loss of generality that x ≥ y ≥ z ≥ 0 (**)Trivially, (0, 0, 0) is a solution. Otherwise, suppose that x ≥ y ≥ z ≥ 1; then (*) and (**) imply that x y z = 3 (x + y + z) = 9 ≥ y z (***).We seek solutions first with z = 1.

Because 3 (x + y + z) > 3x, 1 ≤ y ≤ 3 is impossible, by (*). Thus y = 4 yields 3(x+5) = 4x, and so x = 15. Similarly, y = 5 yields x = 9 and y = 6 yields x = 7; and 7 = y = 9 contradicts assumption (**). Therefore, (15, 4, 1), (9, 5, 1) and (7, 6, 1) are the only possible solutions for z = 1.

Now consider z = 2.y = 2 yields x = 12, y = 3 yields x = 5, and y = 4 gives no integer solution for x.So (12, 2, 2) and (5, 3, 2) are the only possible solutions for z = 2.

Finally, consider z = 3.z = 3 yields y = 3 and x = 3; thus, (3, 3, 3) is the only possible solution for z = 3.

106 Let (x, y, z) be the coordinates of a point on the surface of a sphere with centre 0 and radius 1. Find the greatest and least values of the expression xy + xz + yz.

Solution by Niels Bejlegaard. Also solved by J. N. Lillington and Dr. Z. Reut.

We have (x + y + z)2 = (x2 + y2 + z2) + 2 (xy + xz + yz) and (x + y + z)2 ≤ 3 (x2 + y2 + z2).So 1 + 2 (xy + xz + yz) ≤ 3 and xy + xz + yz ≤ 1.Otherwise, we have x2 + y2 + z2 ≥ 0 which leads to 1 + 2 (xy + xz + yz) ≥ 0 and xy + xz + yz ≤ �0.5. Thus �0.5 ≤ xy + xz + yz ≤ 1.

107 The towns in a country are connected by one-direction roads. For any two towns, one can be reached from the other. Prove that there is a town from which all the other towns can be reached.

Solution by Oren Kolman, London; also solved by Niels Bejlegaard, Dr. M. Dzamonja and J. N. Lillington

We apply induction on the number n of towns in the country C. For n = 1 or 2, the assertion is trivial. For n > 2, consider the country C\{t} with one town t removed. This depleted country of n � 1 towns still satisfies the induction hypothesis, since its towns are connected by one-way roads, andhence there is a town t0 in C\{t} which can be reached from every town in C\{t}. If t0 can also be reached from t, then t0 is as required; if t0 cannot be reached from t, then the one-way road connecting t0 and truns from t0 to t, and so t is as required.

108 Find all polynomials p(x) for which p(x2) = (p(x))2 for all x. Use the result to determine which polynomials fulfil the condition p(x2 � 2x) = (p(x � 2))2.

Solution by J. N. Lillington; also solved by Niels Bejlegaard.

Let p(x) = a0 + a1x + a2x2 + ... + anxn . Suppose p(x) contains a term arxr+asxs, where the coefficients between ar and as are 0, ar and as are non-zero, and r < s.Then p(x2) contains arx2r + asx2s , but [p(x)]2 also contains 2arasxr+s with 2r < r + s < 2s. Since this is impossible, there can be only one non-zero coefficient in the expression of p(x). Thus, p(x) = xn (n = 0, 1, 2, ... ) or p(x) = 0.Now, we write p(x) in the form p(x) = b0 + b1(1 + x) + b2(1 + x)2+ ... + bnxn and apply the same argument as above. Suppose p(x) contains a term br(1 + x)r+bs(1 + x)s, where the coefficients between br and bs are 0, br and bs are non-zero, and r < s.Because 1 + (x2 � 2x) = (x � 1)2, it follows that p(x2 � 2x) contains br(x � 1)2r+ bs(x � 1)2s

and [p(x � 2)]2 also contains 2brbs(x � 1)r+s, with 2r < r + s < 2s. Since this is impossible, there is only one non-zero coefficient in p(x). Thus the polynomial p(x) has the form (1 + x)n, for arbitrary n.

109 Find all positive real numbers c such that the sequence a0, a1, a2, ... , where a0 = 1, ai = cai�1, i = 1,2, ... , is bounded.

Solution by Niels Bejlegaard; also solved by Dr. M. Dzamonja, J. N. Lillington and Dr. Z. Reut

We make use of the fact that a monotone and bounded sequence is convergent. For any 0 < c < 1, the sequence 1, c, cc, c(cc), ... , is bounded by M = 1. We therefore assume that c > 1, and monotonicity is assured. To find the maximum value of c for which the sequence is bounded, we consider the function f(x) = cx � x. Using ai = cai�1, we can write the expression for f(x) as f(ai) = cai� ai = ai+1 � ai. Since lim ai = lim ai+1 = a, we have f(a) = 0, so ca = a and hence c = a1/a. But a1/a = e1/a lna, so we can differentiate the function g(a) = (ln a)/a to obtain the required maximum value of (1 � ln a)/a2.This expression vanishes for a = e, so for 1 ≤ c ≤ e1/e we obtain the upper bound M = e. Thus, the sequence is bounded for all 0 ≤ c ≤ e1/e.

PROBLEM CORNER

EMS September 2000 31

i → ∞ i → ∞

Please e-mail announcements of European confer-ences, workshops and mathematical meetings of inter-est to EMS members, to k.a.s.quinn@open.ac.uk.Announcements should be written in a style similar tothose here, and sent as Microsoft Word files or as textfiles (but not as TeX input files). Space permitting,each announcement will appear in detail in the nextissue of the Newsletter to go to press, and thereafterwill be briefly noted in each new issue until the meet-ing takes place, with a reference to the issue in whichthe detailed announcement appeared

1-7: International Workshop and Fall School�Geometric Analysis�, Potsdam, GermanyScope: the meeting will reflect a good part ofthe research activities of the EU research andtraining network �Geometric analysis�Topics: operator algebras, index theory, non-commutative Chern character, pseudodifferen-tial operators on manifolds with singularities,K-theorySpeakers: J.-M. Bismut, A. Connes, M.Karoubi, B.-W. Schulze, N. TelemanProgramme: plenary talks and special sessionson the above mentioned topicsOrganisers: A. Connes, N. Teleman, E.Schrohe, B.-W. SchulzeSponsors: EU research and training networkand the Clay Mathematical InstituteInformation: contact the organisers:N. Teleman, Univ. di Ancona, e-mail: teleman@popcsi.unian.itE. Schrohe, Universität Potsdam, e-mail: schrohe@math.uni-potsdam.deB.-W. Schulze, Universität Potsdam, e-mail: schulze@math.uni-potsdam.de

7-10: International Conference onMathematical Modelling and ComputationalExperiments (ICMMCE), Dushanbe,TajikistanInformation:Web site: http://www.tajnet.com/

8-11: X Congress of YugoslavMathematicians, Belgrade, YugoslaviaThe congress is included in the project �WorldMathematical Year 2000� ofthe International Mathematical Union(http://wmy2000.math.jussieu.fr)Topics: Logic, Algebra, Geometry and topol-ogy, Real and complex analysis, Functionalanalysis and operator theory, Differential equa-tions, Combinatorics and graph theory,Probability and statistics, Numerical analysisand optimisation, Computer science, Appliedmathematics, Teaching, History and populari-sation of mathematicsMain speakers: Invited lectures will be deliv-ered by outstanding Yugoslav and foreignmathematicians � the names will be given inthe second announcementProgramme: Invited lectures, section lectures,short communications, poster sections,round table, mini-symposium, workshops

October 2000

Organising and programme committee: NedaBokan (chair), Sinisa Vrecica, Zoran Ivkovuc,Bosko Jovanovic, Milan Jovanovic, ZoranKadelburg, Miodrag Mateljevic, ZarkoMijajlovic,Vladimir Micic, Pavle Mladenovic, PredragObradovic, Zarko Pavicevic,Gordana Pavlovic-Lazetic, Stevan Pilipovic,Dragoslav Herceg, Stevo Segan, RadojeScepanovicSite: Faculty of Mathematics, Studentski trg 16,Belgrade, YugoslaviaDeadlines: September 25Information: e-mail: matkon10@matf.bg.ac.yufor application: prijave@matf.bg.ac.yufor abstracts: apstrakti@matf.bg.ac.yuWeb site: http://www.matf.bg.ac.yu/matkon10/

15-21: DMV Seminar on the Riemann ZetaFunction and Random Matrix Theory,Mathematisches Institut Oberwolfach,GermanyInformation: Prof. Dr Matthias Kreck,Universität Heidelberg, MathematischesInstitut, Im Neuenheimer Feld 288, 69120Heidelberg, Germany

15-21: DMV Seminar on Motion byCurvature, Mathematisches InstitutOberwolfach, GermanyInformation: Prof. Dr Matthias Kreck,Universität Heidelberg, MathematischesInstitut, Im Neuenheimer Feld 288, 69120Heidelberg, Germany

16-17: Workshop on Non-Newtonian andViscoelastic fluid flows: Mathematical theory, Modelling andApplications, Pisa, ItalyTheme: Applied mathematics, computationalfluid dynamicsMain speakers: M. Deville (Switzerland), R.Glowinski (USA), J. M. Marchal (Belgium), M. Picasso (Switzerland), A.Sequeira (Portugal), A. Fasano (Italy), K. R. Rajagopal (USA)Programme: The workshop will consist of invit-ed lectures by seven international experts inapplied mathematics and computational fluiddynamics � in particular, in the field of non-Newtonian and viscoelastic fluid flows.Language: EnglishOrganising committee: A. Quarteroni(Switzerland/Italy), A. Fasano (Italy), G. Da Prato (Italy)Sponsors: ESF through the program AMIF andthe �Associazione Amici della Scuola Normale�Site: Scuola Normale Superiore di Pisa, Piazzadei Cavalieri 7, 56126 Pisa, ItalyGrants: AMIF grants for European youngresearchersDeadlines: for registration: 30 September (16September if an AMIF grant is requested)Information:

e-mail: amicisns@sns.itWeb site: http://www.sns.it/html/Home/Associazione_Amici_della_Normale/amici10c.html

20-22: One Hundred Years of the JournalL�Enseignement Mathématique, Geneva,SwitzerlandInformation:e-mail: EnsMath@math.unige.chWeb site: http://www.unige.ch/math/EnsMath[For details, see EMS Newsletter 36]

20-23: Singularities in Classical, Quantumand Magnetic Fluids, Coventry, UKInformation:Web site:http://www.maths.warwick.ac.uk/research/programmes/current

23-25 Third International Conference onApplied Mathematics and EngineeringSciences, Casablanca, Morocco Information:Web site: http://www.cimasi.org.ma/

30-3 November: International ConferenceDedicated to M. A. Lavrentev on the Occasionof his Birthday Centenary, Kiev, UkraineInformation:Web site: http://www.imath.kiev.ua/lavconf.htm

30-3 November: Clifford Analysis and itsApplications, NATO Advanced ResearchWorkshop, Praha, Czech RepublicInformation:Web site: http://www.karlin.mff.cuni.cz/~clifford

30-4 November: Evolution Equations 2000:Applications to Physics, Industry, LifeSciences and Economics, Levico Terme(Trento), Italy Information: contact Mr. A. Micheletti,Secretary of CIRM, Centro Internazionale perla Ricerca Matematica Istituto Trentino diCultura 38050 Povo (Trento) tel: (+39)-0461-881628, fax: (+39)-0461-810629 e-mail: michelet@science.unitn.it Web site: http://alpha.science.unitn.it/cirm/ [For details, see EMS Newsletter 36]

12-18: DMV Seminar on ComputationalMathematics in Chemical Engineering andBiotechnology, Mathematisches InstitutOberwolfach, GermanyOrganisers: Peter Deuflhard (Berlin), RupertKlein (Potsdam/Berlin) and Christof Schütte(Berlin)Information: Prof. Dr Matthias Kreck,Universität Heidelberg, MathematischesInstitut, Im Neuenheimer Feld 288, 69120Heidelberg, Germany

12-18: DMV Seminar on CharacteristicClasses of Connections, Riemann-RochTheorems, Analogies with e-Factors,Mathematisches Institut Oberwolfach,GermanyOrganisers: Spencer Bloch (Chicago) andHelene Esnault (Essen)Information: Prof. Dr Matthias Kreck,Universität Heidelberg, MathematischesInstitut, Im Neuenheimer Feld 288, 69120

November 2000

CONFERENCES

EMS September 200032

FForthcoming conferorthcoming conferencesencescompiled, for this issue, by Barbara Maenhaut

Heidelberg, Germany

27-1 December: Foundations of Probabilityand Physics, Vaxjo, Sweden[For details, see EMS Newsletter 36; however thefollowing information has changed.]Invited Speakers: S. Albeverio (Bonn,Germany), H. Atmanspacher (Freiburg,Germany), L. Ballentine (Burnaby, Canada), J.Bricmont (Belgium), A. Holevo (Moscow,Russia), S. Gudder (Denver, U.S.A.), T. Kolsrud(Stockholm, Sweden), P.Lahti (Turku, Finland), H. Narnhofer (Wien,Austria), V. Serdobolskii(Moscow, Russia), J. Summhammer (Wien,Austria), O. Viskov (Moscow, Russia), I.Volovich(Moscow, Russia)Information:e-mail: send abstracts and application forms toRobert Nyqvist <robert.nyqvist@msi.vxu.se>Web site: http://www.msi.vxu.se/aktuellt/konferenser.html

1-3: 2000 WSES International Conference onLinear Algebra and Applications, Vravrona(suburb of Athens), Greece Proceedings: CD-ROM Proceedings plusWSES Press Luxurious BooksDeadline: for submission of papers, 30September Notification: of acceptance/rejection, 30October Information: e-mail: M.Makrynakis, math@worldses.orgWeb site: http://www.worldses.org/wses/math/1-3: 2000 WSES International Conference onNumerical Analysis and Applications,Vravrona, Greece Details: as for above WSES Conference onLinear Algebra1-3: 2000 WSES International Conference onDifferential Equations: Theory andApplications, Vravrona, Greece Details: as for above WSES Conference onLinear Algebra1-3: 2000 WSES International Conference onOptimization and Applications, Vravrona,Greece Details: as for above WSES Conference onLinear Algebra1-3: 2000 WSES International Conference onProbability, Statistics, Operational Research,Vravrona, Greece Details: as for above WSES Conference onLinear Algebra1-3: 2000 WSES International Conference onComputer mathematics - Education,Vravrona, Greece Details: as for above WSES Conference onLinear Algebra1-3: 2000 WSES International Conference onAlgorithms, Discrete Mathematics, Systemsand Control, Vravrona, Greece Details: as for above WSES Conference onLinear Algebra1-3: 2000 WSES International Conference onTopology and Differential Geometry,Vravrona, Greece Details: as for above WSES Conference onLinear Algebra

16-21: Applications of Singularity Theory to

December 2000

Geometry, Liverpool, UKInformation: contact Peter Gibline-mail: pjgiblin@liv.ac.uk

18-20: 5th International Conference onMathematics in Signal Processing, Coventry,UK[For details, see EMS Newsletter 34, but updateddetails are given below]Theme: Recent developments such as non-lin-ear/non-Gaussian signal processing, multi-ratesignal processing, blind deconvolution/signalseparation and broadband systems Aim: To bring together mathematicians andengineers with a view to exploring recentdevelopments and identifying fruitful avenuesfor further researchMain speakers: D. Broomhead (UK), J.Chambers (UK), S. McLaughlin (UK), P.Rayner (UK), P. Regalia (UK), R. Riedi (USA).Organising committee: J. G. Mcwhirter (UK),O. R. Hinton (UK), M. D. MacLeod (UK), M.Sandler (UK), S. McLaughlin (UK), I. K.Proudler (UK) Sponsors: The Institution of ElectricalEngineersSite: University of Warwick, UKInformation: contact Pamela Bye, The Instituteof Mathematics and its Applications, CatherineRichards House, 16 Nelson Street, Southend-on-Sea, Essex SS1 1EF, England, fax: (+44)-(0)1702-354111 e-mail: conferences@ima.org.ac.uk Web-site: http://www.ima.org.uk

5-10: Finite and Infinite Combinatorics,Budapest, HungaryThe special occasion for this meeting is to hon-our the 70th birthdays of Professors Vera T.Sós and András Hajnal. Theme: graph theory, combinatorics, set theo-ryTopics: graph theory, extremal graphs,extremal families of subsets, combinatorialoptimisation, combinatorial number theory,discrepancy theory, infinite combinatorics, settheory, random graphs Main speakers: Noga Alon, Béla Bollobás,Jaroslav Nesetril, Saharon Shelah, MiklósSimonovits, Robert TijdemanLanguage: EnglishOrganising committee: L. Lovász (Co-Chair),G. O. H. Katona (Co-Chair), E. Gyõri, P.Komjáth, I. Z. Ruzsa, G. Y. Katona (Secretary),Á. Kisvölcsey (Secretary). Sponsors: János Bolyai Mathematical Society,Mathematical Institute of the HungarianAcademy of SciencesInformation:e-mail: finf@renyi.hu Web site: http://www.renyi.hu/~finf

8-18: ICMS Instructional Conference inNonlinear Partial Differential EquationsEdinburgh, UKTheme: The aim is to instruct young mathe-maticians on modern methods for non-linearPDE theory. Talks will be presented at threelevels. Around one-third of the total time willbe spent presenting basic methods and apply-ing them to prototypical examples which illus-trate both their power and their limitations.This material will be suitable for students at the

January 2001

start of their postgraduate studies. Talks at theintermediate level will be devoted to refiningthese basic methods to cover more subtle andcomplex situations. The series of moreadvanced lectures will review recent develop-ments and current research.Topics: Modern methods for non-linear PDEtheory, where the motivation comes fromimportant new areas in applied science, includ-ing image processing, materials science, gasand fluid dynamics, and quantum mechanics ofatoms and molecules. Although this set of top-ics has been selected for detailed analysis, themethods are universal in their scope.Main speakers: A. Aftalion (France), L.Ambrosio (France), J. M. Ball (UK), Y. Brenier(France), G. R. Burton (UK), V. Caselles(Spain), M. J. Esteban (France), J. Kristensen(UK), C. Le Bris (France), A. Quarteroni(Switzerland), C. A. Stuart (Switzerland), V.Sverak (USA), J. F. Toland (UK), N. Touzi(France), N. S. Trudinger (Australia)Organisers: J. M. Ball (Oxford), M. J. Esteban(Paris), J. Toland (Bath)Sponsors: Engineering and Physical SciencesResearch Council of the United Kingdom & ECFramework VSite: Heriot-Watt University Campus,Edinburgh, UKGrants: To citizens of the EU or associatedstates who are under 35 years of age (seeWebsite for full details)Information:Mailing address & telephone: The InternationalCentre for Mathematical Sciences, 14 IndiaStreet, Edinburgh EH3 6EZ, UK. tel: (+44)-(0)131-220-1777;fax (+44)-(0)131-220-1053

e-mail: icms@maths.ed.ac.ukWeb site: http://www.ma.hw.ac.uk/icms/current/npde/index.html

28�3 February: 2001 XXI InternationalSeminar on Stability Problems for StochasticModels, Eger, HungaryInformation:e-mail: stabil@math.klte.hu orkolchin@mi.ras.ru Web site: http://neumann.math.klte.hu/~stabilhttp://bernoulli.mi.ras.ru [For details, see EMS Newsletter 36]

11-15: 2001 WSES International Conferenceon Fuzzy Sets & Fuzzy Systems (FSFS �01),Puerto De La Cruz, Tenerife, Canary IslandsTopics: Fuzzy logic, fuzzy sets, fuzzy topologyand fuzzy functional analysis, fuzzy differentialgeometry, fuzzy differential equations, fuzzyalgorithms, fuzzy geometry, fuzzy languages,fuzzy control, fuzzy signal processing, fuzzysubband image coding, VLSI fuzzy systems,approximate reasoning, fuzzy logic and possi-bility theory, fuzzy expert systems, fuzzy sys-tems theory, connectionist systems, learningtheory, pattern classification and clustering,hybrid and knowledge-based networks, artificiallife, fuzzy systems in robotics, fuzzy systems foroperational research, NN training using fuzzylogic, interaction between: neural networks �fuzzy logic � genetic algorithms, fuzzy systemsand non-linear systems, fuzzy systems andchaos and fractals, modelling and simulation,hybrid intelligent systems, oceanic and vehicu-

February 2001

CONFERENCES

EMS September 2000 33

lar engineering, man-machine systems, cyber-netics and bio-cybernetics, relevant topics andapplications, parallel and distributed systems,fuzzy systems and fuzzy engineering for: elec-tric machines, power systems, Real-time sys-tems, information systems, decision supportsystems, discrete event systems, communica-tions, multimedia, educational software, soft-ware engineering, adaptive control, aerospace,special topics, others. Language: EnglishCall for papers:http://www.worldses.org/wses/fsfsScientific committee: Chairs: D. Fogel (USA),N. E. Mastorakis (Greece), E. Oja (Finland), F.Torrens (Spain), M. Makrynaki (Greece).Members: G. Antoniou (USA), H. Arabnia(USA), H.-G. Beyer (Germany), H.-H. Bothe(Denmark), A. L. L. Chye (Singapore), R.Ciegis (Lithuania), P. Corr (Northern Ireland),S. Dlay (UK), M. J. Er (Singapore), J. Fodor(Hungary), K. Hirota (Japan), D. Kalpic(Croatia), D.-S. Kang (Korea), N. Kasabov(New Zealand), R. Kruse (Germany), F. Kurfess(Canada), P Lorenz (France), V. Mladenov(Netherlands), A. Mohamed (Egypt), M.Mohammadian (Australia), F. Murtagh(Northern Ireland), F. Naghdy (Australia), M.Paprzycki (USA), H. Radev (Bulgaria), R.Rojas (Germany), D. Sanchez (Kluwer, USA), T.Whalen (USA), Y. Zhang (USA), H.-J.Zimmermann (Germany), J. Zurada (USA).Sponsors: WSES, IIARD, IMACSDeadline: for paper submission, 30 OctoberInformation:e-mail: fsfs@worldses.orgWeb site: http://www.worldses.org/wses/fsfs

11-15: 2001 WSES International Conferenceon Evolutionary Computations (EC �01),Puerto De La Cruz, Tenerife, Canary Islands Topics: Genetic algorithms (GA), mathematicalfoundations of GA, evolution strategies, geneticprogramming, evolutionary programming,classifier systems, cultural algorithms, simulat-ed evolution, artificial life, learning theory, pat-tern classification and clustering, evolutionarycomputations (EC) in knowledge engineering,evolvable hardware, molecular computing, ECin control theory, EC in signal processing, ECfor image coding, approximate reasoning, ECin robotics, EC for operational research, neuralnetworks training using EC, interactionbetween: neural networks � fuzzy logic � evolu-tionary, computations, EC and non-linear sys-tems theory, modeling and simulation, hybridintelligent systems, EC for electric machines,EC for power systems, EC for real-time sys-tems, EC for information systems, EC for deci-sion support systems, EC for discrete event sys-tems, EC for communications, EC for multi-media, EC for educational software, EC forsoftware engineering, EC for adaptive control,EC for aerospace, oceanic and vehicular engi-neering, global optimisation, man-machine sys-tems, cybernetics and bio-cybernetics, relevanttopics and applications, parallel and distrib-uted systems, special topics, others. Language: EnglishCall for papers:http://www.worldses.org/wses/ecScientific committee, Sponsors, Deadline: asfor above WSES Conference on Fuzzy SetsInformation:e-mail: ec@worldses.org

Web site: http://www.worldses.org/wses/ec

11-15: 2001 WSES International Conferenceon Neural Networks and Applications (NNA�01), Puerto De La Cruz, Tenerife, CanaryIslands Topics: Biological neural networks, artificialneural networks, mathematical foundations ofneural networks, virtual environments, neuralnetworks (NN) for signal processing, connec-tionist systems, learning theory, architecturesand algorithms, neurodynamics and attractornetworks, pattern classification and clustering,hybrid and knowledge-based networks, artificiallife, implementation of (artificial) NN, VLSItechniques for NN implementation, neuralcontrol, NN for robotics, NN for optimisation,systems theory and operational research, NN innumerical analysis problems, NN trainingusing fuzzy logic, NN training using evolution-ary computations, interaction between: neuralnetworks � fuzzy logic � genetic algorithms,NN and non-linear systems, NN and chaos andfractals, modeling and simulation, hybrid intel-ligent systems, NN for electric machines, NNfor power systems, NN for real-time systems,NN in information systems, NN in decisionsupport systems, NN and discrete event sys-tems, NN in communications, NN for multi-media, NN for educational software, NN forsoftware engineering, NN for adaptive control,NN for aerospace, oceanic and vehicular engi-neering, man-machine systems, cybernetics andbio-cybernetics, relevant topics and applica-tions, parallel and distributed systems. specialtopics, others. Language: EnglishCall for papers:http://www.worldses.org/wses/nnaScientific committee, Sponsors, Deadline: asfor above WSES Conference on Fuzzy SetsInformation:e-mail: nna@worldses.org Web site: http://www.worldses.org/wses/nna

19-23: New Trends in Potential Theory andApplications, Bielefeld, GermanyTheme: This conference is meant to be a con-tinuation of a series of international confer-ences on Potential Theory and related fields(as, e.g., the ones in Prague 1987, Amersfoort1991, Kouty 1994, Hammamet 1998). Thebeginning of the new millennium seems appro-priate to reflect on the current developmentsand to specify new promising directions ofresearch in this classical area of mathematics.Emphasis will be given to various applications,in particular, in mathematical physics. Wewould also like to celebrate the 60th birthdayof our colleague and friend, Wolfhard Hansen,on the afternoon of February 22. Topics: differential geometry, Dirichlet forms,fractals, linear and non-linear PDE,Schrödinger operators, Markov processes, sto-chastic analysis.Site: University of Bielefeld, Bielefeld,Germany. Information: V. Metz, Fakultät fürMathematik, Universität Bielefeld, Postfach 1001 31, D-33501 Bielefeld, Germanye-mail: metz@mathematik.uni-bielefeld.de. Web site: http://www.ams.org/mathcal/info/2001_feb19-23_bielefeld.html

25-1 March: NATO Advance Research

Workshop: Application of AlgebraicGeometry to Coding Theory, Physics, andComputation, Eilat, IsraelAim: To present some of the best recent workin algebraic geometry with emphasis on possi-ble relations with and applications to mathe-matical physics, coding theory, industrial andcomputational aspects. Proceedings will bepublished with the contribution of NATO.Scientists from all NATO countries, MiddleEast countries and the former Soviet Union areinvited to participate.Sponsors: NATO Scientific Affairs Division,Emmy Noether Research Institute ofMathematics in Bar-Ilan UniversityNATO Organising committee: C. Ciliberto(Roma Tor Vergata), F. Hirzebruch (MPI,Bonn), R. Miranda (Colorado State Univ.), M.Teicher (Bar-Ilan Univ.).Local organisation: B. Kunyavskii (Bar-IlanUniv.).The following people are planning to partici-pate: E. Arbarello (SNS, Pisa), V. Batyrev(Tübingen), A. Beauville (Paris), J. Bernstein(Tel Aviv), F. Catanese (Göttingen), A. Conte(Turin), R. Donagi (Philadelphia), G. van derGeer (Amsterdam), P. Griffiths (Princeton), V.Iskovskih (Moscow), V. Kaminski (Ramat Gan),M. Kontsevich (Bures-sur-Yvette), V. Kulikov(Moscow), A. Libgober (Chicago), V. Lin(Haifa), R. Livné (Jerusalem), Y. Miyaoka(Kyoto), N. Mok (Hong Kong), D. Mumford(Providence), T. Peternell (Bayreuth), M.Salvetti (Pisa), R. Schoof (Roma Tor Vergata),E. Sernesi (Roma III), Y. T. Siu (Cambridge,USA).Information: e-mail: NATO@macs.biu.ac.ilWeb site: http://www.mat.uniroma2.it/~cilibert/workshop.html

18-24: Workshop on Geometric Analysis andIndex Theory, Trieste, ItalyThis conference is organised by the EuropeanResearch Training Network�Geometric Analysis�, in collaboration with theInternational Centre for Theoretical Physics inTrieste, the Università di Ancona and theInstitut de Mathematiques de Luminy inMarseille.The workshop is dedicated in memory of Prof.Enzo MartinelliOrganisation: ICTP, Trieste, in collaborationwith the University of Ancona, Italia, andInstitut de Mathématiques , Luminy, Marseille Support: European Research TrainingNetwork �Geometric Analysis� and InternationalCentre for Theoretical Physics, Trieste.Organising Committee: Jean-Paul Brasselet(jpb@iml.univ-mrs.fr), G. Landi (landi@math-sun1.univ.trieste.it), N. Teleman(teleman@popcsi.unian.it).Scientific and Advisory Committee: A. Connes(IHES), N. Teleman (Ancona), J.-M. Bismut(Orsay), J. Bruening (Berlin), B. W. Schulze(Potsdam), C. Baer (Hamburg), J. Bellissard(Toulouse), A. Legrand (Toulouse), J. P.Brasselet (Marseille), P. Almeida (Lisbon), R.Nest (Copenhagen), A. Valette (Neuchatel), T.Kappeller (Zürich), B. Bojarski (Warsaw), D.Andrica (Cluj-Napoca), G. Landi (Trieste).

26-29: Workshop on Quantum Field Theory,

March 2001

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EMS September 200034

Noncommutative Geometry and QuantumProbability, Trieste, Italy The main subjects are non-commutative struc-tures (non-commutative geometry and its appli-cations to physics, notably quantum field theo-ry, and quantum probability)Organisation: International School forAdvanced Studies, Trieste, Italy, in collabora-tion with the International Centre forTheoretical Physics.Supported by: Regione Friuli-Venezia Giulia,International School for Advanced Studies,University of Trieste, European RTN�Geometric Analysis�International Advisory Committee: A. Connes(IHES Paris), B. Dubrovin (SISSA), Yu. Manin(MPI Bonn)Scientific Committee: U. Bruzzo, (SISSA), C.Cecchini (Udine), L. Dabrowski (SISSA), A. O.Kuku (Trieste), G. Landi (Trieste), N. Teleman(Ancona)Organising Committee: U. Bruzzo, L.Dabrowski, G. LandiEach of the following scientists will deliver twolectures:L. Accardi (Università Roma Tor Vergata andCentro Vito Volterra), A. Connes (IHES Paris),D. Kreimer (IHES Paris), S. L. Woronowicz(Warsaw University)Moreover some additional 12 one-hour lectureswill be delivered.Information about the Workshop is availablefrom the web page http://www.sissa.it/\~{}bruzzo/ncg2001 /ncg2001.html

27-29: Modelling permeable Rocks,Cambridge, UKTopics: Faults and fractures, modelling tech-niques, quantification and modelling at variousscales, process modelling/pattern formation,quantitative techniques for data collection,model validation, uncertainty modelling.Main Speakers: G. De Marsily (France), D.Sanderson (UK), C. White (USA).Organising committee: A. H. Muggeridge(UK), L. M. Abriola (USA), P. Burgess (UK), A.Galli (France), J. Gomez-Hernandez (Spain) ,L. Holden (Norway), A. D. W. Jones (UK), P.R.King (UK), C. Mijnssen (Netherlands), G.Pickup (UK), C. Ravenne (France), J. J. Walsh(UK).Sponsors: BP Exploration Operating CompanyLimited, Elf Exploration UK plc andSchlumberger Cambridge Research Limited.Site: Churchill College, Cambridge, UKDeadlines: 29 September for abstractsInformation: Conference Officer: Pamela Byee-mail: conferences@ima.org.ukWeb site: http://www.ima.org.uk

9-11 : Modelling in Industrial Maintenanceand Reliability, Greater Manchester, UKAim: To provide a forum for discussion of tra-ditional and innovative modelling approachesto improve the performance of plant, productsand processes. Topics: Maintenance (inspection, preventivemaintenance, replacement, spares), Conditionmonitoring (methodology, decision support),Reliability (design improvement, reliability andrisk analysis, reliability-centred-maintenance),Warranty modelling (methodology, human fac-tors).

April 2001

Organising Committee: P. Scarf (UK), W.Wang (UK), L. Thomas (UK), R. Dekker(Netherlands). Site: University of Salford, GreaterManchester, UK.Deadlines: 31 October 2000 for abstracts Information: Conference Officer: Pamela Bye e-mail: conferences@ima.org.ukWeb site: http://www.ima.org.uk

6-13: SPT2001 � Symmetry and PerturbationTheory, Cala Gonone (Sardinia, Italy)Third of a series after SPT96 (Torino) andSPT98 (Roma)Theme: The interactions between symmetryand perturbation theoryLanguage: EnglishScientific committee: D. Bambusi (Milano), P.Chossat (Nice), G. Cicogna (Pisa), A.Degasperis (Roma), G. Gaeta (Roma), J. Lamb(London), G. Marmo (Napoli), M. Roberts(Warwick), G. Sartori (Padova), F. Verhulst(Utrecht), S. Walcher (Munich), B. Zhilinskii(Dunkerque)Organising committee: G. Gaeta, D. BambusiProceedings: to be publishedSite: Hotel Villaggio Palmassera, Cala Gonone(a small village on the eastern coast ofSardinia), special full board rate for partici-pants but you can stay elsewhere if you wish.Grants: reduced registration fee for studentsand participants from developing countriesDeadlines: No deadline, but limitations on thenumber of participants (pre-registration to startin autumn)Information:e-mail: spttre@tiscalinet.itWeb site: http://web.tiscalinet.it/SPT2001/spt.html

28-1 June: Harmonic Morphisms andHarmonic Maps, Luminy, FranceSite: Centre International de RencontresMathématiques, LuminyInformation: Contact M. Villee-mail: ville@math.poytechnique.fr

28-1 June: Marmonic morphisms and maps,Marseille, FranceLanguage: EnglishScientific committee: James Eells (Cambridgeand Warwick), Luc Lemaire (Brussels), John C.Wood (Leeds).Organising committee: Eric Loubeau (Brest),Stefano Montaldo (Cagliari), Marina Ville(Paris).e-mail: ville@math.polytechnique.frwebsite: http://beltrami.sc.unica.it/harmor/

6-10: 3d International Conference on Large-Scale Scientific Computations, Sozopol,BulgariaInformation:e-mail: scicom01@parallel.bas.bgWeb site: http://copern.bas.bg/Conferences/SciCom01.html

8-10: 2001 Belgian Mathematical Society andDeutsche Mathematiker Vereinigung Meeting,University of Liège, BelgiumInformation: Web site: http://math-www.uni-paderborn.de/

June 2001

May 2001

Liege2001/

19-22: Computational Intelligence, Methodsand Applications (CIMA 2001), Bangor, UKTheme: advanced computingTopics: CIMA 2001 features 4 symposia: Fuzzylogic and applications (FLA 2001), Advances inintelligent data analysis (AIDA 2001),Advanced computing in biomedicine (ACBM2001), Advanced computing in financial mar-kets (ACFM 2001)Main speakers: Jim Bezdek (USA), AndrewWebb (UK)Language: EnglishCall for papers: maximum 7 pages. Pleaseinclude contact details and a short c.v. para-graph. Electronic submissions are encouraged:operating@icsc.ab.ca. Submissions are accept-ed by fax (+1-780-352-1913) or mail (2 copies)at ICSC International Computer ScienceConventions, 5101C-50 Street, Wetaskiwin, AB,T9A 1K1, CanadaProgramme committee: General chair LudmilaKuncheva (UK), Co-chair Tim Porter (UK); Symposia chairs: FLA 2001, Vilem Novak andIrina Perfilieva (Czech Republic); AIDA 2001,Mayer Aladjem (Israel), ACBM 2001, FriedrichSteimann (Germany), ACFM 2001, ChristianHaefke (USA).Organising committee: Jeanny Ryffel, ICSC,(Canada), http://www.iscs.ab.ca; Karen Tallents,University of Wales, Bangor (UK)Sponsors: University of Wales, Bangor; BritishComputer Society (more are expected)Proceedings: all accepted papers will beincluded in the conference proceedings; somepapers will be selected for journal publicationSite: University of Wales, Bangor Deadlines: for submissions, 31 October 2000;notification of acceptance, 31 December 2000;final paper/registration, 15 February 2001Information:e-mail: planning@icsc.ab.ca;operating@icsc.ab.ca;l.i.kuncheva@bangor.ac.ukWeb site: http://www.icsc.ab.ca/cima2001.htm

1-6: Eighteenth British CombinatorialConference, Brighton, UKInformation:e-mail: bcc2001@susx.ac.uk Web sites:http://www.maths.susx.ac.uk/Staff/JWPH/http://hnadel.maps.susx.ac.uk/TAGG/Confs/BCC/index.html [For details, see EMS Newsletter 36]

5-18: Groups St Andrews 2001, Oxford, UKInformation: Groups St Andrews 2001,Mathematical Institute, North Haugh, StAndrews, Fife KY16 9SS, Scotland e-mail: gps2001@mcs.st-and.ac.uk Web site: http://www.bath.ac.uk/~masgcs/gps01/ [For details, see EMS Newsletter 36]

27-31: Equadiff 10, CzechoslovakInternational Conference on DifferentialEquations and Their Applications, Prague,Czech RepublicInformation:Web site: www.math.cas.cz/~equadiff/

August 2001

July 2001

CONFERENCES

EMS September 2000 35

Books submitted for review should be sent to thefollowing address: Ivan Netuka, MÚUK, Sokolovská 83, 186 75Praha 8, Czech Republic.

J. Albree, D. C. Arney and V. F. Rickey, AStation Favorable to the Pursuits of Science:Primary Materials in the History ofMathematics at the United States MilitaryAcademy, History of Mathematics 18, AmericanMathematical Society, London MathematicalSociety, Providence, 2000, 272 pp., US$59,ISBN 0-8218-2059-1This book is devoted to the history of theU. S. Military Academy at West Point, thefirst national military school, founded on16 March 1802. The first part of the firstchapter discusses the history of the depart-ment of mathematics, and the second partdescribes the formation and developmentof the West Point�s Library during the nine-teenth century. The third part describesthe teaching of mathematics and mechanicsat West Point in the nineteenth century; ashort section describes the development ofthe curriculum and the leadership by suchpeople as Sylvanus Thayer, Charles Daviesand Albert Church. The fourth partdescribes the history of the foundation ofthe Catalog of books at West Point, and thefifth part explains the catalogue�s organisa-tion. The last part contains some impor-tant references on the history and develop-ment of West Point.

The second chapter � the major part ofthis book � reveals the rich collection ofmathematical works located at West Point.The printed catalogue of the West Pointcollection contains more than 1300 workspublished between 1496 (Regiomontanus,Epytoma Joanis de mote regio in almagestiiptolomei) and 1917 (Darboux, Principes degéométrie analytique). The collection showsthe strong European influence on the earlyAcademy and includes numerous textbookswritten at West Point. Significant contribu-tions were made in algebra, geometry, cal-culus and descriptive geometry.At the end are four appendices. The first isthe Catalog of 1803, the catalogue of books,maps and charts that belong to the MilitaryAcademy at West Point. The second con-tains thirty photographs of attractive andinteresting title pages, frontispieces andother visual features. Appendices 3 and 4includes the portraits and frontispieces ofthe books in this catalogue.

This book provides an important sourcefor a general audience, as well as for thoselooking for more scholarly information.Both expert and general reader will findsomething of interest. (mnem)

C. D. Aliprantis and K. C. Border, InfiniteDimensional Analysis, Springer, Berlin,1999, 672 pp., DM 119, ISBN 3-540-65854-8

This is the second, completely revisedand enlarged (paperback) edition. Theoriginal motivation for the authors was �topresent and organize the analytic founda-tions underlying modern economics andfinance�, including �the material whichappears only in esoteric research mono-graphs that are designed for specialists�.

Because of the non-homogeneity and alarge amount of the material of this impor-tant book, a detailed description follows.The necessary material on set theory (18pp.), topology (50 pp.), metric spaces (54pp.) and set systems and measurable func-tions (38 pp.) is introduced. This is fol-lowed by material on topological vectorspaces (76 pp.), normed spaces (26 pp.),Riesz spaces (38 pp.) and Banach lattices(30 pp.), and by charges and measures (34pp.), measures and topology (30 pp.), inte-grals (32 pp.), Lp-spaces (28 pp.), the RieszRepresentation theorems (18 pp.) andprobability measures on metric spaces (20pp.). Other topics include spaces ofsequences (30 pp.), correspondences (34pp.), measurable correspondences (30pp.), Markov transitions (34 pp.) andergodicity (13 pp.). The list of referenceshas 307 items.

Most of the theorems are proved(including Tychonoff�s theorem, theChoquet capacitability theorem, theBishop-Phelps Theorem, the Lyapunovconvexity theorem, the Riesz representa-tion theorem and the Michael andKuratowski-Ryll-Nardzewski selection the-orems), but some are stated (with a precisereference) without proof (the Eberlein-�mulian and James theorems, the Stone-Weierstrass theorem, the Radon-Nikodymtheorem and also the Knaster-Kuratowski-Mazurkiewicz theorem, from which fixed-point theorems are deduced).

The book is well organised and writtenin a lucid style. It is a very good �guide� forany non-specialist interested in the topicsincluded. The authors have managed �towrite the book so that it will be useful asboth a reference and a textbook�. (lz)

H. Amann and J. Escher, Analysis I,Grundstudium Mathematik, Birkhäuser, Basel,1998, 445 pp., ISBN 3-7643-5974-9 and 3-7643-5976-5The book under review is the second vol-ume of a course of analysis. In about fourhundred pages it brings together a largeportion of material in a very condensedform. The book contains many exercisesand the authors express their hope thatthe book will be helpful to teachers leadingseminars.

The present volume contains threechapters devoted to integration, to func-tions of several variables and to the curvi-linear integral. This could give the wrongimpression that it is a calculus book, but all

things are done in considerable depth andwith high precision. To give an example,the chapter on integration contains amongother things sections on Fourier series andthe gamma function; the last one containsthe Gauss formula, the product formula for1/Γ, the Stirling formula and the formularelating the gamma and beta functions.The authors broadly use modern tech-niques and do not hesitate to include manythings that traditionally form part ofadvanced courses: for example, the lastchapter contains the Goursat lemma andthe residuum theorem.

The book is a good reference book and auseful companion for teachers; from stu-dents it will demand intensive but reward-ing work. It is highly recommended bookfor mathematically orientated universitylibraries. (jve)

G. A. Anastassiou and S. G. Gal,Approximation Theory. Moduli ofContinuity and Global SmoothnessPreservation, Birkhäuser, Boston, 2000, 525pp., DM 188, ISBN 0-8176-4151-3 and 3-7643-4153-3This monograph, in two parts, is an inten-sive and comprehensive study of the com-putational aspects of the moduli ofsmoothness and the global smoothnesspreservation property.

In Part I, the authors study the compu-tational aspect of almost all moduli of con-tinuity over wide classes of functions,exploiting some of their convexity proper-ties. Numerous applications of approxi-mation theory are presented and exact val-ues of errors in explicit forms are given.

In Part II, the authors study and exam-ine the global smoothness preservationproperty (GSPP) for almost all known lin-ear approximation operators in approxi-mation theory, including trigonometricoperators and algebraic interpolationoperators of Lagrange, Hermite-Féjer andShepard type, also operators of stochastictype, convolution type, wavelets type inte-gral operators and singular integral opera-tors, etc. A great variety of applications ofGSPP to approximation theory, functionalanalysis and computer-aided geometricdesign is provided.

This monograph contains the researchof both authors over the past ten years inthese subjects. It also references most ofthe works of other main researches in theseareas. It is well worth reading, and can bestrongly recommended to researchers andgraduate students involved in approxima-tion theory, applied mathematics, numeri-cal analysis and engineering disciplines.(kn)

G. E. Andrews, R. Askey and R. Roy,Special Functions, Encyclopedia ofMathematics and its Applications 71,Cambridge University Press, 1999, 664 pp.,£55, ISBN 0-521-62321-9Books in the Encyclopedia of Mathematics andits Applications series are designed as ency-clopedic treatments with an emphasis on awide range of applications of the chosentopic. Special functions form a typicalexample of a subject with a long and inter-

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EMS September 200036

Recent booksRecent booksedited by Ivan Netuka and Vladimír Sou³ek

esting history, together with a broad anddeep modern development. The field istoo large now, so a selection has had to bemade. The book concentrates on hyperge-ometric functions and the associatedhypergeometric series, including their var-ious generalisations.

In this book, modern tools of represen-tation theory are not used. The book startswith a description of classical gamma andbeta functions, including their finite fieldsand p-adic analogues. Later, the Selbergmulti-dimensional generalisation of thebeta integral and its finite fields versionsare presented. The book also contains q-extensions of gamma function and betaintegrals. Throughout, the theory devel-oped is used in applications to other partsof mathematics (such as combinatorics andspherical harmonics) and there are a lot ofbeautiful and useful explicit formulas.Each chapter ends with a series of exercis-es. (vs)

M.-C. Arnaud, Le �Closing Lemma� enTopologie C1, Mémoires de la SociétéMathématique de France 74, SociétéMathématique de France, Paris, 1998, 120pp., ISBN 2-85629-071-XThe whole booklet (115 pages) concen-trates on one special problem and its vari-ants. Suppose that f is a diffeomorphism ofa manifold M and suppose that x∈M is apoint which is an accumulation point of itsorbit. Is it possible to find a diffeomor-phism g close to f in the Cj topology suchthat x is a periodic point of g? The answerfor j = 0 is yes, and the proof is simple;nothing is known for j ≥ 2. The case j = 1is the subject of the book. The positiveanswer was proved first by Pugh andRobinson. This book brings a simplerproof and more precise results, and alsostudies new versions of the problem, suchas the case of symplectic vector fields andthe ergodic version of the orbit closinglemma. (vs)

V. I. Arnold, G.-M. Greuel and J. H. M.Steenbrink (eds.), Singularities, Progress inMathematics 162, Birkhäuser, Basel, 1998,458 pp., DM 148, ISBN 3-7643-5913-7This volume contains papers on singulari-ty theory and its applications, written byparticipants of a conference organised atthe Mathematical Institute in Oberwolfachin July 1996 in honour of E. Brieskorn onthe occasion of his 60th birthday. Some ofthese papers are extended version of talkspresented at the conference.

At the beginning of the book, there is achapter written by G. M Greuel presentingan overview of results of E. Brieskorn andhis contribution to the theory of singulari-ties. The following twenty-one contribu-tions are divided according to someaspects of singularity theory into fourparts: Classification and invariants,Deformation theory, Resolution, andApplications (also containing various dif-ferent topics).

The first part includes a contribution byW. Huang and J. Lipman (Differentialinvariants of embeddings of manifolds incomplex spaces). In this paper a reduced

complex space with an r-dimensional con-nected submanifold i: W → V is consid-ered, and its Segre classes are defined.These Segre classes are up to sign C1-invariants of the pair (V, W). Another dif-ferential invariant is the multiplicity of Win V.

In the second part (Deformation theory),there is a paper of E. Shustin: Equiclassicaldeformation of plane algebraic curveswhere sufficient conditions for the exis-tence of a deformation of a plane irre-ducible curve into a curve of the samedegree, genus and class, having only nodesand cusps are given. (jbu)

B. Artmann, Euclid � The Creation ofMathematics, Springer, New York, 1999, 343pp., DM 98, ISBN 0-387-98423-2As the title indicates, the main object ofthis book is Euclid�s Elements, the book atthe core of mathematical education andthe heart of western culture for two thou-sand years.

The book is divided into 31 chapters.The first two, General historical remarksand The contents of the Elements, have anintroductory character. The followingthirteen chapters contain a generaldescription of the contents and structure ofall thirteen books of Euclid�s Elements. Theremaining sixteen chapters, with commontitle �The Origin of Mathematics�, presentsome important and interesting topics,general remarks about typical mathemati-cal procedures, subjects of particular philo-sophical and historical interest, and others(Pythagoras of Samos, squaring the circle,problems and theories, the birth of rigour,incommensurability and irrationality, etc.).In the author�s opinion, the Elements areread, interpreted, and commented uponfrom the point of view of modern mathe-matics.

At the end of the book there are �Notes�where we find hints for further reading,the bibliography and the index. The bookhas 116 illustrations. This book can bewarmly recommended for students, teach-ers and lovers of mathematics. A solidbackground in high school mathematics isnecessary. (jbe)

Y. Benyamini and J. Lindenstrauss,Geometric Nonlinear Functional Analysis1, Colloquium Publications 48, AmericanMathematical Society, Providence, 2000, 488pp., US$65, ISBN 0-8218-0835-4This important monograph, written byleading specialists in the field, is devotedmainly to the study of general uniformly con-tinuous and general Lipschitz mappingsbetween Banach spaces. The authors treatneither �infinite-dimensional topology�(which considers general homeomor-phisms) nor �classical non-linear functionalanalysis� (non-linear differential equations,degree theory, bifurcation results, etc.)which deal mostly with more special non-linear mappings. This study is naturallyconnected with the theory of classificationof Banach spaces with respect to uniformhomeomorphisms and bi-Lipschitz maps.

Besides this theory, the following topicsare covered in detail: retractions, exten-

sions, selections, approximations and fixedpoints of uniformly continuous andLipschitz mappings; Radon-Nikodýmproperty of Banach spaces and its connec-tion to Gâteaux differentiability ofLipschitz mappings (includingChristensen�s Haar null sets and Gaussianand Aronszajn null sets); differentiabilityof convex functions on separable spaces;oscillation of uniformly continuous func-tions on unit spheres of subspaces (relatedto Dvoretzky�s theorem); small perturba-tions of isometries (e.g. Hyers-Ulam prob-lem).

The notes and remarks that follow eachof the seventeen chapters contain interest-ing historical notes and information abouta vast amount of related results (with refer-ences). The bibliography contains 596items. Challenging open problems aredescribed and explained. The vast major-ity of the material appears for the first timein book form, and many quite recent deepresults are proved. A basic knowledge ofreal and functional analysis is necessary,but more specialised topics are explainedin appendices at the end. Volume 2 of thisuseful book will treat, for example, Fréchetdifferentiability of Lipschitz mappings andthe analytic Radon-Nikodym property ofBanach spaces. (lz)

B. Bertram, C. Constanda and A.Struthers (eds.), Integral Methods inScience and Engineering, Research Notes inMathematics 418, Chapman & Hall/CRC,Boca Raton, 2000, 360 pp., £58, ISBN 1-58488-146-1This text is based on lectures presented atthe Fifth International Conference onIntegral Methods in Science andEngineering held at MichiganTechnological University, Houghton,Michigan, 1998. It contains three invitedpapers and 57 contributed papers � a col-lection of papers that address the solutionof mathematical problems arising in vari-ous branches of physics by integral meth-ods in conjuction with various approxima-tion schemes. Written by acknowledgedexperts, these peer-reviewed papers offervaluable insight into recent developmentsin both theory and applications.

Topics and applications include: scatter-ing processes, micromechanics, fluidmechanics, elasticity theory, hydrodynam-ics, plates and shells, acoustics, populationgenetics, optimal control, combustionproblems, fracture theory, inverse prob-lems, non-linear problems, boundary valueproblems at resonance, variational meth-ods, boundary element methods, potentialmethods, wavelet expansions, integralequations and reaction-diffusion system.

The book provides inspiration for fur-ther research and can be strongly recom-mended to applied mathematicians, engi-neers and physicists. (kn)

L.J. Billera, A. Björner, C. Greene, R. E.Simion and R. P. Stanley (eds.), NewPerspectives in Algebraic Combinatorics,Mathematical Sciences Research InstitutePublications 38, Cambridge University Press,1999, 345 pp., £32.50, ISBN 0-521-77087-4

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EMS September 2000 37

Algebraic combinatorics involves the use oftechniques from algebra, topology andgeometry in the solution of combinatorialproblems, or the use of combinatorialmethods to attack problems in these areas.

During 1996-97 the MathematicalSciences Research Institute held a full aca-demic year programme on combinatorics,with special emphasis on the connectionswith other branches of mathematics, suchas algebraic geometry, topology, commuta-tive algebra, representation theory andconvex geometry.

This book represents work done or pre-sented at seminars during the programme.It contains contributions on matroid bun-dles, combinatorial representation theory,lattice points in polyhedra, bilinear forms,combinatorial differential topology andgeometry, Macdonald polynomials andgeometry, enumeration of matchings, thegeneralised Baues problem, andLittlewood-Richardson semigroups.(mloe)

D. van Dalen, Mystic, Geometer, andIntuitionist. The Life of L.E.J. Brouwer,Vol. 1: The Dawning Revolution, ClarendonPress, Oxford, 1999, 440 pp., £75, ISBN 0-19-850297-4�Child and student� is the title of the firstchapter of the biography and the tenthchapter is named �The breakthrough�.Roughly speaking, the period 1920-1930 istreated, and concerns of Brouwer�s lin-guistic and philosophical activities con-nected with the development of his ownintuitionism. Generally, great activity,including moral positions on anti-national-ism and passion for justice, were character-istic for L. E. J. Brouwer.

Mysticism, topology, intuitionism andphilosophy of language are treated asmajor themes of Brouwer�s life, while top-ics such as a quarrel with Lebesgue or arejection of the Denjoy policy are treatedas smaller ones. The reader can find intel-ligible and adequate explanations of anature of mathematical problems and adocumentation of the foundational activityin mathematics in the first third of thetwentieth century, represented, togetherwith Brouwer, by H. Weyl. The book con-cludes with a Bibliography of Brouwer�swritings and the Contents of Volume 2 areincluded. The book is comprehensive andinteresting. (jml)

H. Davenport, The Higher Arithmetic,Cambridge University Press, 1999, 241 pp.,£16.95, ISBN 0-521-63269-2 and 0-521-63446-6This is the seventh edition of the well-known and charming introduction to num-ber theory. In a comparatively small num-ber of pages, the text contains a surpris-ingly wide spectrum of topics. The bookdoes not assume any training in numbertheory, so the book can serve for under-graduate courses. Despite this, the readercan find a formulation of remarkably deepresults (e.g., on elliptic curves). The topicsdiscussed are treated in a transparent way,and each chapter ends with notes givingadditional important information on the

subject. Since the book is well known andhas proved itself since its first appearancein 1952, it can be recommended both forindependent study and as a reference textfor a general mathematical audience. (�p)

P. Deligne, P. Etingof, D. S. Freed, L. C.Jeffrey, D. Kazhdan, J. W. Morgan, D. R.Morrison and E. Witten, Quantum Fieldsand Strings: A Course for Mathematicians,2 vols., American Mathematical Society,Providence, 1999, 723 and 725 pp., ISBN 0-8218-1987-9 and 0-8218-1988-7This book in two volumes consists of writ-ten notes from a programme in quantumfield theory organised by the Institute forAdvanced Study in Princeton in 1996-1997, taken either from the speakersthemselves or by mathematicians in theaudience. It is a concise introduction tothe quantum field theory and perturbativestring theory, with as much emphasis on amathematically satisfying exposition andclarity as possible.

The topics contained include: notes onsupersymmetry; classical field theory;supersolutions; introduction to quantumfield theory; perturbative quantum fieldtheory; index of Dirac operators; renor-malisation groups; dimensional regularisa-tion; conformal field theory; string theory;super space descriptions of super gravity;two-dimensional conformal field theoryand string theory; Kaluza-Klein compacti-fications, supersymmetry and Calabi-Yauspaces; dynamics of quantum field theory;and the dynamics of N = 1 supersymmet-ric field theories in four dimensions.

This book will be helpful to all mathe-maticians and mathematical physicists whowish to learn about the beautiful subject ofquantum field theory. It is, however, not atextbook and assumes some preliminaryknowledge of the topics covered. (mm)

J. D. Dixon, M. P. F. du Sautoy, A. Mannand D. Segal, Analytic Pro-p Groups,Cambridge Studies in Advanced Mathematics,Cambridge University Press, 1999, 368 pp.£37.50, ISBN 0-521-65011-9This book presents a fairly straightforwardaccount of the theory of p-adic analyticgroups. The central topic of the book ishow the group-theoretic properties of apro-p group reflect its status as a p-adicanalytic group.

The book is divided into three parts.Readers seeking a simple introduction topro-p groups and to p-adic analytic groupswill find this in Parts I and II.

The first chapter of Part III (Chapter 10)gives an account of the theory of pro-pgroups of finite coclass; the coclass of agroup G of order pn is n � c, where c is thenilpotency class of G; this theory is centralto the classification of finite pro-p-groups.The next two chapters discuss the dimen-sion subgroup series in finitely generatedpro-p groups and its associated graded Liealgebra. In Chapter 13, sketches of thebeginnings of a theory of analytic groupsover pro-p rings other than the p-adic inte-gers are given, together with a proof of thefollowing theorem of £ubotzky and Shaler(1994): Every R-perfect group satisfies the

Golod-Shafarevich inequality.This book can be highly recommended

to all mathematicians working in grouptheory, p-adic analysis and Lie groups andalgebras. (tk)

M. Fey and R. Jeltsch, HyperbolicProblems: Theory, Numerics, Applications,I, II, International Series of NumericalMathematics 129/130, Birkhäuser, Basel,1999, 502 and 503 pp., ISBN 3-7643-6123-9, 0-8176-6123-9, 3-7643-6087-9 and 0-8176-6087-9This book is the Proceedings of theSeventh International Conference onHyperbolic Problems, held in Zürich inFebruary 1998. It contains, in two vol-umes, more than one hundred papers pre-sented at this conference.

The majority of the papers are con-cerned with non-linear hyperbolic equa-tions of conservation laws. They can bedivided into three groups. A large numberof papers are devoted to mathematicalaspects of conservation laws such as exis-tence, uniqueness, asymptotic behaviour,large-time stability or instability of wavesand structures, various limits of the solu-tions, the Riemann problem, and so on.Approximately the same number of contri-butions are concerned with numericalanalysis � for example, stability and con-vergence of numerical schemes, schemeswith special properties such as shock cap-turing, interface fitting, and high-orderapproximations to multi-dimensional sys-tems. The third group of articles is orien-tated towards a wide range of applications,such as non-linear waves in solids, compu-tational fluid dynamics, combustion, rela-tivistic astrophysical problems, multiphasephenomena and geometrical optics.

The Proceedings give an excellentoverview of the contemporary state of artin the area of hyperbolic problems. It con-tains a large amount of material, and willbe of interest to researchers and studentsof applied mathematics, scientists andengineers engaged in applied mathematicsand scientific computing. (mf)

J. Fresnel, Espaces quadratiques, euclidi-ens, hermitiens, Actualités scientifiques etindustrielles 1445, Hermann, Paris, 1999,315 pp., FF 160, ISBN 2-7056-1445-1The text of this book is divided into threeparts. In the first part, the author gives adetailed introduction to vector spacesendowed with a bilinear form. It containsmany results about the orthogonal group(e.g., theorems about its centre, generatorsand special subgroups). The second part isdevoted to real inner product spaces.From its very beginning, the properties ofassociated orthogonal groups are discussedfrom many points of view. The third partis treated analogously to the second, butcan be viewed as a self-contained introduc-tory text concerning similar questions inthe case where the basic field is the field ofcomplex numbers. The author collectstogether a lot of material, including about150 well-arranged exercises. The quotedliterature has forty items, the majority inFrench; four items are other books of J.

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Fresnel. The book may serve as a veryattractive text, not only for students butalso for non-specialists. (lbe)

W. Haussmann, K. Jetter and M. Reimer(eds.), Advances in MultivariateApproximation, Mathematical Research 107,Wiley-VCH, Berlin, 1999, 334 pp., DM 198,ISBN 3-527-40236-5This volume is based on lectures presentedat the third international conference onmultivariate approximation held at HausBommerholz, the guest-house of theUniversity of Dortmund, from 27September to 2 October 1998. It contains22 articles, the list of conference partici-pants and the scientific programme.

The contributions cover the followingtopics: Interpolation and approximationon spheres and balls (problems of approx-imation theory in discrete geometry;spherical polynomial approximations;spherical designs; hyperinterpolation onthe sphere; cubature formulas on spheres),Harmonic functions (dense vector spacesof universal harmonic functions; best one-sided L1-approximation by harmonic andsubharmonic functions), Wavelets andsplines (multiscale modelling from geopo-tential data; family of orthogonal waveletson the quincunx grid; bivariate splinespaces; interpolation by cubic C1-splines onClough-Tocher splits; box spline orthogo-nal projection), Periodic and monotoneapproximation (monotonic behaviour ofBernstein operators; optimal periodicinterpolation in multivariate periodicHilbert spaces; error estimates for period-ic interpolation of functions from Besovspaces), Fekete points (on the asymptoticsof points which maximise determinants ofthe form det[g(|xi � xj|)]), Numerical sta-bility of fast Fourier transforms, Best one-sided L1-approximation by blending func-tions, Simultaneous approximation in theDirichlet space, Besov regularity for theStokes problem, Average case analysis ofnumerical integration, Weighted K-func-tionals and moduli of smoothness.

The reviewer is convinced that this bookwill form an inspiration for furtherresearch and so help to establish the linksbetween the various communities workingon multivariate approximation. (kn)

J. L. Heilbron: Geometry Civilized,Clarendon Press, Oxford, 2000, 309 pp., £25,ISBN 0-19-850078-5 and 0-19-850690-2This wonderful book is a combination ofmathematics, its history, and the history ofcivilisation and culture (architecture,painting, measuring, astronomy, etc.).Besides an introduction to plane geometryand trigonometry, it contains some fasci-nating historical description.

This book helps us to understand thereason why certain geometrical problemswere studied and the way they were stud-ied. A lot of effort is spent on increasingour understanding of geometry. Theauthor carefully explains solutions of somegeometrical problems, describes the beau-ty and precision of geometrical methodsand proofs (Pythagoras� theorem, the con-struction of perpendiculars, measuring

and constructing angles, constructions fortriangles, facts about the square of thehypotenuse, regular polygons � construc-tions of polygons, tangent, bisector,chords, ovals, etc.).

The history presented here is not onlyrestricted to the European tradition, butsome examples come from the other cul-tures (in particular, Chinese and Indian).Anyone thinking that Euclidean geometryis something for our great-grandparentswould be surprised. The book explainsEuclidean geometry in a most accessibleway, and is beautifully illustrated. Thereare diagrams on nearly each page, illustra-tions from older books, photographs andeight colour plates. (mnem)

N. J. Hitchin, G. B. Segal and R. S. Ward,Integrable Systems: Twistors, Loop Groups,and Riemann Surfaces, Oxford GraduateTexts in Mathematics 4, Clarendon Press,Oxford, 1999, 136 pp., £25, ISBN 0-19-850421-7In September 1997 there was an instruc-tional conference on integrable systems inOxford. This book contains lecture seriesprepared by three experts in the field. Ashort introduction by N. Hitchin (10pages) nicely reviews the main features ofintegrable systems and the principal toolsused in their study. The first part of thebook (N. Hitchin, 42 pages) describes theimportance of algebraic geometry for inte-grable systems. Starting with a very briefintroduction to Riemann surfaces, linebundles, cohomology groups with values insheaves, vector bundles, and Jacobians ofRiemann surfaces, these tools are thenused in a study of Lax pairs for matrix-val-ued polynomials and the correspondingcompletely integrable Hamiltonian sys-tems. The second part (G. Segal, 68 pages)treats integrable systems using methods ofinverse-scattering theory. The mostimportant tools used here are loop groups,restricted Grassmannians and the Birkhofftheorem, and the main examples discussedare Korteweg-de Vries and non-linearSchrödinger equations; relations to themethods of algebraic geometry used in theHitchin paper are explained here. Thelast part (R. Ward, 14 pages) is a shortsummary of the twistor description of self-dual Yang-Mills equations and their vari-ous dimensional reductions (which includemost of the known integrable systems).

The subject of the book is fascinatingand written versions of the lecture seriesare nicely presented and preserve well theinformal spirit of the lectures. This is avery useful book for graduate students andfor mathematicians (or physicists) fromother fields interested in the topic. (vs)

I. M. James (ed.), History of Topology,Elsevier, Amsterdam, 1999, 1056 pp.,US$190.50, ISBN 0-444-82375-1This book contains forty contributionswritten by professional mathematicians orhistorians of mathematics. The title of thebook is a little misleading because almostall of the contributions deal with algebraicor differential or geometric topology; mostof the topics belonging to general topolo-

gy by classification schemes are not includ-ed.

About ten contributions are devoted tosignificant mathematicians of the above-mentioned fields, and about five othersconcern special groups or time (e.g., topol-ogists at conferences, in Hitler�s Germany,or the Japanese school). The remainingcontributions deal with special topics fromthe parts of topology mentioned above �especially manifolds, shape theory, fixed-point theory, topology and physics, homo-topy theory, graph theory, fibre bundles,and several others from algebraic topolo-gy. The contributions are written in such away that non-specialists can read andunderstand them without difficulty. (mh)

G. R. Krause and T. H. Lenagan, Growthof Algebras and Gelfand-KirillovDimension, Graduate Studies in Mathematics22, American Mathematical Society,Providence, 1999, 212 pp., US$39, ISBN 0-8218-0859-1This is the second edition of the authors�fundamental monograph on the Gelfand-Kirillov dimension. The first edition waspublished in 1985 (Research Notes inMathematics 116, Pitman). There arepractically no changes to the original text,only errors have been corrected. In orderto cover the progress of the last fifteenyears in this field, the authors have addeda chapter that sketches the developmentand provides the reader with the relevantreferences. The number of references hasdoubled with respect to the first edition;they comprise even preprints and papersthat were due to appear as the book wentto press, and go up to 1999.

In most cases the appearance of a secondedition signals that the book is good andinteresting. This is certainly the case withthe monograph under review. We can saythat it is very carefully and clearly written,and can be recommended not only to spe-cialists, but to anybody who either intendsto start working in this interesting field orneeds some information concerning theGelfand-Kirillov dimension. For students,it requires only minor prerequisities, main-ly from algebra. For specialists in otherfields, it is quite easy to find the requiredinformation. The book reads well, and theauthors often mention the historical devel-opment of the subject and provide the textwith many interesting examples. They alsopresent various applications, most of themin algebra. There is also a chapter on thegrowth of groups, a notion that has beenapplied in differential geometry. (jva)

N. P. Landsman, Mathematical Topicsbetween Classical and QuantumMechanics, Springer Monographs inMathematics, Springer, New York, 1998, 529pp., DM 124, ISBN 0-387-98318-XThe main theme of this book is a discus-sion of various mathematical tools used fordescription of classical mechanics (Poissonmanifolds, Poisson algebras and their rep-resentations, pure states) and quantummechanics (Jordan algebras, projectiveHilbert spaces, C*-algebras and their rep-resentations, pure states, transition proba-

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EMS September 2000 39

bilities) and a study of the relations amongthem. In particular, quantisations of clas-sical systems and classical limits of quan-tum systems are studied systematically.

The first part of the book describes twobasic tools for decriptions of classical andquantum mechanics: the algebra of observ-ables and the set of pure states of the the-ory. It is shown here how these two differ-ent descriptions are related. The secondpart is devoted to various types of quanti-sation and to classical limits. In the thirdpart, the author describes geometrical con-structions of Poisson algebras and C*-alge-bras. To cover the cases needed in the the-ory of elementary particles, the theory ofLie groupoids and Lie algebroids are used(generalising the case of Lie groups andprincipal fibre bundles). In the last part,an interesting and far-reaching analogy isdescribed between symplectic reduction ofsymplectic manifolds and the Rieffelinduction in the representation theory ofC*-algebras, including typical applicationsin theoretic physics (for example, the the-ory of systems with constraints or θ-vacua).

The book has two special features. Thefirst is the introductory chapter (30 pages)where the content of the book and themain notions and ideas are introduced anddiscussed clearly and with sufficient detail.The second is the last chapter called Notes(50 pages), containing various additions,comments and references. (vs)

S. Lang, Complex Analysis, Graduate Textsin Mathematics 103, Springer, New York,1999, 485 pp., DM 129, ISBN 0-387-98592-1This is the fourth edition of the author�spopular book. As said in the Preface, therequired prerequisites include a two-yearcalculus course and ε-δ-techniques. Thefirst part of the book can serve as a basiccourse in complex function theory and thesecond part as an advanced undergraduateor a graduate course. Power series domi-nate at many places in the book. The con-tents are not essentially changed in thisedition, but new exercises and examplesare included and many minor improve-ments are made. The very understandablestyle of explanation, which is typical forthis author, makes the book valuable forboth students and teachers. (jve)

S. Lang, Math Talks for Undergraduates,Springer, New York, 1999, 121 pp., DM 59,ISBN 0-387-98749-5This new book of its very prolific author isbased on talks delivered by him atEuropean and North American universi-ties. It takes the form of simulated discus-sions: expositions are combined with stu-dents� questions and remarks. The mater-ial is chosen to be accessible with minimalprerequisites and of interest to a broadmathematically oriented audience. Thebook can be used by those who have todeliver similar talks, or even by students instudent-run seminars. The materialincludes, for example, prime numbers,approximation theorems in analysis, har-monic polynomials, and other topics. Thetext is written in a lively light style with an

adequate degree of precision. Even formore advanced readers it provides pleas-ant reading, showing how to popularisemathematics. (jve)

N. Macrae, John von Neumann, AmericanMathematical Society, Providence, 1999, 405pp., US$35, ISBN 0-8218-2064-8This biography is a study of an exception-ally well-balanced man who � withoutpushiness or expenditure on public rela-tions � became the most quietly effectivemathematical mind of the twentieth centu-ry. He was born as Neumann János on 28December 1903 in Budapest. In 1921, hewas sent to become a chemical engineer,first to Berlin and then to Zürich. At thattime he was already intensively working onthe idea of Cantor�s ordinal numbers. Hisfirst postgraduate home was in Göttingen,where there were many brilliant mathe-maticians and physicists around. By theend of 1928, he had published twenty-twomajor papers on mathematics. He wasthen invited to the United States, aroundthe time of the 1929 stock market crash.During the period 1931-37, he visitedPrinceton which had one of the greatestconcentrations of brains in mathematicsand physics ever to be assembled any-where. The book also describes his stay atLos Alamos, connected with the history ofthe A-bomb. John von Neumann died on8 February 1957 in Washington DC from atragically early cancer.

Von Neumann is especially known forhis pioneering work on computer design.More than any other individual, he becamethe creator of the modern digital comput-er. He believed that Darwin, Faraday,Einstein, etc., followed their own innerlight and knew that no organiser, noadministrator, and no institution can domore than furnish conditions favourablefor research. He wrote a series of funda-mental mathematical papers and inventeda notion now called a �von Neumann alge-bra�.

The book also surveys his substantialcontributions to theoretical and appliedphysics, decision theory, set theory, gametheory, meteorology, biology, cybernetics,mathematical economics, and the develop-ment of the atomic and hydrogen bombs.The author also presents many facts aboutJohn von Neumann�s private life. (mk)

P. Mattila, Geometry of Sets and Measuresin Euclidean Spaces, Cambridge Studies inAdvanced Mathematics 44, CambridgeUniversity Press, 1999, 343 pp., £40, ISBN 0-521-65595-1 and 0-521-46576-1This is a new edition of the book publishedin 1995; for a review, see EMS Newsletter18, December 1995, p. 38. (vs)

H. Milbrodt and M. Helbig,Mathematische Methoden derPersonenversicherung, Walter de GruyterCo., Berlin, 1999, 654 pp., DM 134, ISBN 3-11-014226-0As prerequisities for studying the book, theauthors quote a deep knowledge of realanalysis and an introductory course ofprobability theory. The book covers the

traditional scope of life insurance and pen-sion mathematics, using continuous-timeMarkov chains; health insurance and mod-els with random interest rates are not cov-ered.

There are two forms of exposition in thethirteen chapters of this book. In the first,mathematical constructions in the spirit ofreal analysis are prevalent; in the other dis-crete-time models are mostly used to pro-vide practical information. The approachto the decrement models is based on theresidual lifetime under multiple decre-ment causes and for joint lives.Information is presented on the life tableconstruction, including recent statisticaldevelopments. In the chapters on theinsurer�s liabilities, the pure and the officepremiums, and the technical provisions,the reader finds the standard formulastogether with the recursive equations typi-cal for a Markov chain approach. Thiele�sequation, Hattendorf�s theorem and thetheory of Cantelli are dealt with in detail.

A particular feature of the book is its 270exercises, mostly asking the reader toprove a statement complementing the textmaterial. The book includes actuarialtables from a larger set obtainable on theinternet. Some overloading of the textwith symbols, formal definitions andproofs may dissuade some readers from asystematic study of this book. (pm)

D. Mumford, The Red Book of Varietiesand Schemes, Lecture Notes in Mathematics1358, Springer, Berlin, 1999, 304 pp., DM79, ISBN 3-540-63293-XThis is the second edition of a famous andwell-known introduction to algebraicgeometry, written to show that the lan-guage of schemes is fundamentally geo-metrical and clearly expressing the intu-ition of algebraic geometry.

There is an appendix on �Curves andTheir Jacobians� which is an expanded ver-sion of a series of lectures that the authorgave in November 1974 at the Universityof Michigan. It is a nice introduction tothe most beautiful and oldest topics inalgebraic geometry � curves and theirJacobians, and is presented here in a formaccessible to the general mathematicalcommunity. Starting with several possibil-ities for how to describe curves in projec-tive space, the book continues with thedescription of the moduli space of curves;Jacobians and theta functions are thenintroduced in a natural way. At the end,the Torelli theorem and the Schottkyproblem are discussed. The book endswith notes entitled �Guide to the literatureand references� and �Supplementary bibli-ography on the Schottky problem�.

This book can be strongly recommendedto anybody interested in algebraic geome-try and willing to learn about varieties andschemes and their main properties. (jbu)

G. Nakamura, S. Saitoh, J. K. Seo and M.Yamamoto (eds.), Inverse Problems andRelated Topics, Research Notes inMathematics 419, Chapman & Hall/CRC,Boca Raton, 2000, 233 pp., £42.95, ISBN 1-58488-191-7

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This book contains ten papers on the the-oretical aspects of inverse problems and sixpapers on numerical simulation, includingapplications to magnetostatics, memoryreconstruction, and tomo-electrocardiog-raphy.

The contributions comprise reviews andhighly original results by specialists inapplied mathematics, medical engineers,and researchers on inverse problems. Itcan be strongly recommended toresearchers and postgraduate studentsinterested in these fields. (kn)

B. Perrin-Riou, p-adic L-Functions and p-adic Representations, SMF/AMS Texts andMonographs 3, American MathematicalSociety, Providence, 2000, 150 pp., US$49,ISBN 0-8218-1946-1The arithmetical structure of integral val-ues of complex L�functions is fundamen-tal, not only in number theory. Variousconstructions of p-adic L-functions haveappeared since the work of Kubota andLeopoldt on a p-adic analogue of the com-plex Riemann zeta function by p-adicinterpolation. The present volume isdevoted to a recent general approach toconstructions of p-adic L-functions, start-ing directly from p-adic Galois representa-tions.

This book was originally published inFrench (Astérisque 225, 1995), and the pre-sent edition mirrors the progress (theColmez proof of reciprocity law Rec(V),contributions of Benois, etc.) which thefield has undergone in the meantime.The volume consists of four chapters andthree appendices: Construction of themodule of p-adic L-functions without factorat infinity; Modules of p-adic L-functions ofV; Study of values of the module of p-adicL-functions; and p-adic L-functions of amotive. Appendix A contains results inGalois cohomology, Appendix B containsthe weak Leopoldt conjecture, andAppendix C (written jointly with J.-M.Fontaine) describes local Tamagawa num-bers, the Euler-Poincaré characteristic andapplications to functional equations. Eachchapter starts with a summary, making ori-entation in the book more comfortable.The book is written in a concise but read-able style, and can be recommended toreaders interested in this rapidly growingsubject. (�p)

T. J. Rivlin and E. B. Saff (eds.), Joseph L.Walsh: Selected Papers, Springer, New York,2000, 682 pp., DM 249, ISBN 0-387-98782-7In this volume the authors present a selec-tion from the 281 published papers ofJoseph Leonard Walsh (1895-1973), acomplete list of which follows after thepreface. A list of Walsh�s books is included,although no excerpts from them appear inthis volume; this omission is due to a limi-tation on the size of the volume. The listsof papers, books and his Ph.D. students isfollowed by a biography of Walsh and sum-mary of his work by Morris Marden, hisfirst Ph.D. student, and by additionalmaterial by D. V. Widder and W. E. Sewell.

The selected papers have been divided

into seven broad sections: Zeros and criti-cal points of polynomials and rationalfunctions; Walsh functions; Qualitativeapproximation; Conformal Mapping;Polynomial approximation, Rationalapproximation; and Spline functions. Thesections are ordered following the evolu-tion of Walsh�s work. Appended to thesesections are commentaries on his work anda discussion of subsequent developmentsinfluenced by it.

One of Walsh�s papers has attained anunpredictably remarkable after-life. Itmade little impact for almost half a centu-ry and then, unexpectedly, sparked enor-mous activity that continues up to the pre-sent day. The work referred to is �A closedset of normal orthogonal functions�, Amer.J. Math. 45 (1923), 5-24, which introducedwhat are now known as �Walsh Functions�.

The reviewer believes that this bookforms an ideal resource for students andresearchers involved in function theory,numerical analysis, approximations andFourier analysis. (kn)

F. Santosa and I. Stakgold (eds.),Analytical and Computational Methods inScattering and Applied Mathematics,Research Notes in Mathematics 417, Chapman& Hall/CRC, Boca Raton, 2000, 273 pp,£46.99, ISBN 1-58488-159-3Most of the papers in this volume werepresented at the International Conferencein Applied Mathematics, held in memoryof Ralph Ellis Kleinman in November1998 at the University of Delaware; addi-tional papers were solicited from his col-leagues. This multidisciplinary collectionconsists of authoritative overviews and newresults from experts in their fields. Theyfeature up-to-date work on scattering, wavepropagation, partial differential equations,analytical techniques for partial differen-tial equations and applied mathematics,and practical computational techniques forwave problems.

This book is an ideal resource for engi-neers working on electromagnetics andwave propagation and for applied mathe-maticians working on partial differentialequations and inverse problems. (kn)

K. Sato, Lévy Processes and InfinitelyDivisible Distributions, Cambridge Studies inAdvanced Mathematics 68, CambridgeUniversity Press, Cambridge, 1999, 486 pp.,£50, ISBN 0-521-55302-4This book serves as an introduction to thetheory of stochastic processes. In particu-lar, it is intended to provide a comprehen-sive basic knowledge of Lévy processes,additive processes and infinitely divisibledistributions, with detailed proofs. Lévyprocesses are stochastic processes whoseincrements in non-overlapping time inter-vals are independent and their incrementsare stationary; both Wiener process andPoisson processes are typical Lévy process-es. Important classes of stochastic process-es, including classes of Markov processesand semi-martingales, are obtained bygeneralising Lévy processes.

Relaxing the stationary assumption forincrements, we get the class of additive

processes. The distributions of Lévy andadditive processes at any time are infinite-ly divisible; conversely, additive processesare described by systems of infinitely divis-ible distributions. Further importantprocesses, such as stable, semi-stable andself-similar additive processes, areobtained after an additional assumption ofself-similarity. (vb)

M. Schneider and Y.-T. Siu (eds.), SeveralComplex Variables, Mathematical SciencesResearch Institute Publications 37, CambridgeUniversity Press, 2000, 564 pp., £40, ISBN 0-521-77086-6This book contains sixteen papers writtenby participants of the 1995-96 Special Yearin Several Complex Variables, held at theMathematical Sciences Research Institutein Berkeley, California. Most of the con-tributions are surveys or expository papersdescribing results and techniques from dif-ferent parts of the theory of several com-plex variables, while a few articles areresearch articles with expository sections.

We now mention some of the papers inmore detail. There is an article by M. S.Baouendi and L. P. Rotschild (Local holo-morphic equivalence of real analytic sub-manifolds in CN), where local biholomor-phisms that map one real analytic or realalgebraic submanifold of CN into anotherare studied; the notion of Segre sets associ-ated with a point of a real analytic CR sub-manifold of CN is the main ingredient inthis work. In an interesting article by D.Barlet (How to use the cycle space in com-plex geometry), the topological and ana-lytical properties of the space Cn(Z) of n-cycles in the complex manifold Z aredescribed, and some correspondencesdefined on the cycle spaces are presented.Among the papers devoted to a study ofgeometry and toplogy of Kähler manifolds,there are papers by F. Campana and T.Peterneli (Recent developments in theclassification theory of compact Kählermanifolds) and D. Toledo (Rigidity theo-rems in Kähler geometry and fundamentalgroups of varieties). A survey of the mostimportant results in Seiberg-Witten theory,which are related to algebraic or Kähleriangeometry, is given in the contribution byCh. Okonek and A. Teleman (Recentdevelopments in Seiberg-Witten theoryand complex geometry). (jbu)

G. Semenoff and L. Vinet (eds.), Particlesand Fields, CRM Series in MathematicalPhysics, Springer, New York, 1999, 489 pp.,DM 158, ISBN 0-387-98402-XThis book is based on lectures presented tothe CAP-CRM Summer School on Particlesand Physics, held in Banff in 1994. Theycover the topics of integrable theories,matrix models, statistical systems, fieldtheory and their applications to condensedmatter physics, as well as certain aspects ofalgebra, geometry and topology.

The first contribution, by E. Corrigan, isdevoted to integrable Toda field theoriesassociated with affine Dynkin diagrams.The second, written by J. Feldman, H.Knörrer, D. Lehmann and E. Trubowitz,describes a gas of interacting fermions in

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the field of a lattice of magnetic ions. Thethird chapter, by D. S. Fried, explains howquantum groups arise from the path inte-gral in three-dimensional topologicalquantum field theory. In the next chapter,T. Miwa describes a method for computingcorrelation functions of solvable latticemodels. In Chapter 5, A. Morozov pre-sents matrix models from the point of viewof integrable systems. Chapter 6, by A.Niemi, explains the localisation of certainpath integrals in equivariant cohomology.Chapter 7, by S. Ruijsenaars, is devoted tosystems of Calogero-Moser type. The fol-lowing chapter, by M. de Wild Propitiusand F. A. Bais, addresses planar gauge the-ories with broken symmetries. Chapter 9,by A. Cappelli, C. A. Trugenberger and G.R. Zemba, describes a characterisation ofquantum Hall fluids based on the W1+∞algebra. The final chapter, by P. I.Etingof, deals with the spectral theory ofquantum vertex operators for the quantumalgebra u1(sl2). (mm)

M. Stern, Semimodular Lattices: Theoryand Applications, Encyclopedia ofMathematics and its Applications 73,Cambridge University Press, 1999, 370 pp.,£50, ISBN 0-521-46105-7This monograph develops the theory ofthose lattices which are, in some sense orother, close to modular lattices. The firstchapter is an ingenious outline of the relat-ed parts from lattice theory and introducesthe notion of a semimodular lattice as a lat-tice L where, for any a, b ∈ L, the fact that[a ∧ b, a] is a prime interval implies that thetransposed interval [b, a ∨ b] is also prime.In the following chapters, the author pre-sents modular pairs, modular, distributive,standard and neutral elements, and M-symmetric lattices; you will hardly find abetter exposition or source for supersolv-able, strong or balanced lattices.

To complete the picture, let us mentionthat you will also find here semimodularlattices of finite length, local distributivityand modularity (with the Kurosh-Orereplacement property), and many otherinteresting and recent topics. The wholebook makes very nice reading, and the textis very carefully arranged, including com-ments on the history of the subject as wellas indications of papers and books for fur-ther study. It will be useful to anyoneinterested in lattice theory. (lbe)

D. Stirzaker, Probability and RandomVariables: A beginner�s guide, CambridgeUniversity Press, 1999, 368 pp., £45, ISBN 0-521-64297-3 and 0-521-64445-3This textbook on probability theory is writ-ten in a tutorial style, oriented towardsbeginners. The author does not assume adeep knowledge of, or skills in, mathemat-ics.

The basic concepts of probability andchance are first described and explained atan intuitive level. Motivations based onproportions, as well as on relative frequen-cies, are mentioned. After that, there fol-lows a rigorous definition of probability,both in discrete and continuous case. Thetext then proceeds to an explanation of the

crucial notions of probability theory: ran-dom variables, probability distributions,independence, conditional distributions,etc. The last chapter introduces generat-ing functions, the most powerful tool forhandling independent random variables.

The basic topics of mathematical statis-tics are touched on, as well as Bayes� rule,the (weak) law of large numbers, the cen-tral limit theorem and ordered statistics.The text is written in an informal tutorialstyle with an emphasis on motivation andwith a number of illustrative examples. Itis appropriate for beginners and readersinterested in the basic ideas of probabilitytheory. (pl)

J. J. Tattersall, Elementary Number Theoryin Nine Chapters, Cambridge UniversityPress, 1999, 407 pp., £16.95, ISBN 0-521-58503-1 and 0-521-58531-7Despite the fact that this book has morethan 400 pages, it is intended for a one-semester introduction to basic number the-ory, at the level of prospective secondaryschool teachers. The topics covered aredivisibility, prime numbers, perfect andamicable numbers, modular arithmetic,congruences of higher degree, partitions,the Pell equation, binary quadratic forms,continued fractions, p�adic analysis, andapplications in cryptology.

The book is very readable, containing awealth of historical comments that makethe reading of it very interesting, even formore experienced readers. Althoughsome comments are a bit inaccurate (e.g.,the term �Davenport covering� was neverused in the literature, or the claim that therule for Easter was definitely settled at theCouncil of Nicaea), such banal trifles donot affect the positive impression accom-panying the reading of this book. Theauthor writes in the introduction that �his-torical vignettes are included to humanizethe mathematics involved�; this is certainlytrue for novices trying to enter a new field.The sections are accompanied by exercis-es, and answers to selected exercises,together with a satisfactory index and bib-liography, can be found at the end of thebook. It can be recommended to anybodywho wants to learn about elementary num-ber theory and its historical background.(�p)

Sundaram Thangavelu, HarmonicAnalysis on the Heisenberg Group, Progressin Mathematics 159, Birkhäuser, Boston,1998, 191 pp., ISBN 0-8176-4050-9 and 3-7643-4050-9This is a well-written exposition of therecent progress in harmonic analysis onHeisenberg groups, stemming from thebeautiful known results in a Euclidean set-ting. The material in this book is due tothe author and his collaborators, and otherfundamental work is also included. For afull understanding of the theory and itsmotivations, the reader should be familiarwith Euclidean harmonic analysis.

Chapter 1 contains basic material, suchas the definition of Hn, its various repre-sentations, the operator-valued Fourierand Weyl transforms on Hn, the functions

of Hermite and Laguerre type, andincludes theorems of Paley-Wiener typeand the Hardy-type uncertainty principlein the non-commutative setting of Hn. Thefirst two sections of Chapter 2 are based onfundamental Strichartz� results on thespectral analysis of the sub-Laplacian. Theexpansion into eigenfunctions is given andan Abel summability theorem is proved.Further, the Paley-Wiener theorem andsome restriction theorem are proved forspectral projections, Bochner-Riesz meansare considered and, with help from theLittlewood-Paley-Stein theory of g-func-tions, a Mikhlin-Hörmander multipliertheorem for the group Fourier transformon Hn is obtained. Chapter 3 focuses onthe Heisenberg group Hn and the unitarygroup U(n) as a Gelfand pair, and studiesthe Gelfand transform on the (commuta-tive) algebra L1(Hn/U(n)). The Gelfandspectrum is given here by bounded U(n)-spherical functions, and a Wiener-Tauberian theorem and a maximal theo-rem for spherical means are then proved.Chapter 4 gives a sharper maximal theo-rem and a theorem of Tauberian type forspherical means, in the case of the reducedHeisenberg group Hn/Γ, where Γ = {(0,2πk): k∈Z}. Related results on mean peri-odic functions are also included. (mkr)

C. Voisin, Mirror Symmetry, SMF/AMSTexts and Monographs 1, AmericanMathematical Society, Société Mathématique deFrance, Providence, 1999, 120 pp., US$27,ISBN 0-8218-1947-XMirror symmetry is a conjectural dualitypredicting, for a complex manifold X, theexistence of its �mirror image�, a manifoldX1 for which the moduli spaces MX and MX1

of deformations of complex structures dec-orated with �complexified Kähler parame-ters� are isomorphic, with an isomorphismpreserving, in a specific sense, the localproduct structures.

The mirror conjecture is based onexperimental evidence, stemming fromcalculations in physics and geometry; a rig-orous proof is known only for a very nar-row class of examples. This book gives anoverview of various manifestations of themirror symmetry phenomena.

The first chapter introduces the mainsource of examples: Calabi-Yau manifolds(compact Kähler manifolds with trivialcanonical bundle). The second chapterexplains the physical origins of the conjec-ture, formulated as a duality between A-and B-models of the N = 2 supersymmet-ric σ-model. In the next chapter, a corre-spondence between deformations of theHodge structure of a Calabi-Yau threefoldand Gromov-Witten invariants of its mirroris constructed.

In Chapter 4, an involution ∆→∆*on theset of toric Fano varieties is constructed.Mirror symmetry can be then observed as acertain correspondence between hypersur-faces of a toric Fano variety correspondingto the polyhedron ∆ and hypersurfaces ofthe variety corresponding to ∆*.

The central theme of Chapter 5 is thetheory of Gromov-Witte invariants and itsapplications to variations of Hodge struc-

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tures. The last chapter shows how count-ing rational curves on a quintic of the four-dimensional projective space leads to solu-tions of the Picard-Fuchs equation of themirror family.

This book is far from being elementaryand self-contained. Although it assumes abroad preliminary knowledge of algebraicand differential geometry, it might yetgive, even to a non-specialist, some basicorientation in the complicated and rapidlydeveloping world of mirror symmetry.(mm)

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EMS September 2000 43

Contributions wantedContributions are needed for future issues of the EMS Newsletter. Thesecan be feature articles, interviews, information about your mathematicalsociety, letters to the Editor, historical notes, or anything else that youthink will be of interest to Newsletter readers. Contributions fromEastern European countries, or form countries rarely represented in thisNewsletter, are particularly welcome. Please e-mail or send your contribution to the Editor-in-Chief, RobinWilson, at the address given on page 1.