CONTENTS EDITORIAL TEAM EUROPEAN MATHEMATICAL …Contact: V. Villani or A. Bodin, e-mail:...

NEWSLETTER No. 38

December 2000

EMS News: Agenda, Editorial, Edinburgh Summer School, London meeting .................. 3

Joint AMS-Scandinavia Meeting ................................................................. 11

The World Mathematical Year in Europe ................................................... 12

The Pre-history of the EMS ......................................................................... 14

Interview with Sir Roger Penrose ............................................................... 17

Interview with Vadim G. Vizing .................................................................. 22

2000 Anniversaries: John Napier (1550-1617) ........................................... 24

Societies: L�Unione Matematica Italiana ................................................... 26

The Price Spiral of Mathematics Journals .................................................. 29

Digital Models and Computer Assisted Proofs ............................................ 30

Forthcoming Conferences ............................................................................ 31

Recent Books ............................................................................................... 35

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

telephone: (+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 February 2001. 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:[email protected]

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: ([email protected]). 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: [email protected]

ASSOCIATE EDITORSSTEEN MARKVORSENDepartment of Mathematics Technical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: [email protected] CIESIELSKIMathematics Institute Jagiellonian UniversityReymonta 4 30-059 Kraków, Polande-mail: [email protected] QUINNThe Open University [address as above]e-mail: [email protected]

SPECIALIST EDITORSINTERVIEWSSteen Markvorsen [address as above]SOCIETIESKrzysztof Ciesielski [address as above]EDUCATIONVinicio VillaniDipartimento di MatematicaVia Bounarotti, 256127 Pisa, Italy e-mail: [email protected] PROBLEMSPaul JaintaWerkvolkstr. 10D-91126 Schwabach, Germanye-mail: [email protected] ANNIVERSARIESJune Barrow-Green and Jeremy GrayOpen University [address as above]e-mail: [email protected] [email protected] andCONFERENCESKathleen Quinn [address as above]RECENT BOOKSIvan Netuka and Vladimir Sou³ekMathematical InstituteCharles UniversitySokolovská 8318600 Prague, Czech Republice-mail: [email protected] [email protected] OFFICERVivette GiraultLaboratoire d�Analyse NumériqueBoite Courrier 187, Université Pierre et Marie Curie, 4 Place Jussieu75252 Paris Cedex 05, Francee-mail: [email protected] UNIVERSITY PRODUCTION TEAMLiz Scarna, Barbara Maenhaut

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CONTENTS

EMS December 2000

EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

EXECUTIVE COMMITTEEPRESIDENT (1999�2002)Prof. ROLF JELTSCHSeminar for Applied MathematicsETH, CH-8092 Zürich, Switzerlande-mail: [email protected]. ANDRZEJ PELCZAR (1997�2000)Institute of MathematicsJagellonian UniversityRaymonta 4PL-30-059 Krakow, Polande-mail: [email protected]. LUC LEMAIRE (1999�2002)Department of Mathematics Université Libre de BruxellesC.P. 218 � Campus PlaineBld du TriompheB-1050 Bruxelles, Belgiume-mail: [email protected] (1999�2002)Prof. DAVID BRANNANDepartment of Pure Mathematics The Open UniversityWalton HallMilton Keynes MK7 6AA, UKe-mail: [email protected] (1999�2002)Prof. OLLI MARTIODepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlande-mail: [email protected] ORDINARY MEMBERSProf. BODIL BRANNER (1997�2000)Department of MathematicsTechnical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: [email protected]. DOINA CIORANESCU (1999�2002)Laboratoire d�Analyse NumériqueUniversité Paris VI4 Place Jussieu75252 Paris Cedex 05, Francee-mail: [email protected]. RENZO PICCININI (1999�2002)Dipartimento di Matematica e ApplicazioniUniversità di Milano-BicoccaVia Bicocca degli Arcimboldi, 820126 Milano, Italye-mail: [email protected]. MARTA SANZ-SOLÉ (1997�2000)Facultat de MatematiquesUniversitat de BarcelonaGran Via 585E-08007 Barcelona, Spaine-mail: [email protected]. ANATOLY VERSHIK (1997�2000)P.O.M.I., Fontanka 27191011 St Petersburg, Russiae-mail: [email protected] SECRETARIATMs. T. MÄKELÄINENDepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlandtel: (+358)-9-1912-2883fax: (+358)-9-1912-3213telex: 124690e-mail: [email protected]: http://www.emis.de

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EMS NEWS

EMS December 2000

2001

15 FebruaryDeadline for submission of material for the March issue of the EMS NewsletterContact: Robin Wilson, e-mail: [email protected]

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

4-6 MayEMS Workshop, Applied Mathematics in Europe, Berlingen, Switzerland Contact: R. Jeltsch, e-mail: [email protected]

11-12 MayEMS working group on Reference Levels in Mathematics: Conference on Mathematics at Age 16 in Europe (venue to be announced)Contact: V. Villani or A. Bodin, e-mail: [email protected] or [email protected]

15 MayDeadline for submission of material for the June issue of the EMS NewsletterContact: Robin Wilson, e-mail: [email protected]

9-25 JulyEMS Summer School at St Petersburg (Russia)Asymptotic combinatorics with applications to mathematical physicsOrganiser: Anatoly Vershik, e-mail: [email protected]

19-31 August EMS Summer School at Prague (Czech Republic)Simulation of fluids, and structures interactionsOrganiser: Miloslav Feistauer, e-mail: [email protected]

24-30 AugustEMS lectures at the University of Malta, in association with the 10thInternational Meeting of European Women in MathematicsLecturer: Michèle Vergne (Ecole Polytechnique, Palaiseau, France)Title: Convex polytopesContact: Dr. Tsou Sheung Tsun, e-mail: [email protected] lectures will also be given at University of Rome, jointly arranged by�Tor Vergata� and �Roma Tre�, at dates to be announced. Contact: Maria Welleda Baldoni, e-mail: [email protected]

1-2 SeptemberEMS Executive meeting in Berlin (Germany)

3-6 September1st EMS-SIAM conference, Berlin (Germany)Organiser: Peter Deuflhard, e-mail: [email protected]

30 SeptemberDeadline for proposals for 2002 EMS Lectures.Contact: David Brannan, e-mail: [email protected]

30 SeptemberDeadline for proposals for 2003 EMS Summer SchoolsContact: Renzo Piccinini, e-mail: [email protected]

20021-2 JuneEMS Council Meeting in Oslo (Norway)

EMS AgendaEMS Committee

The EMS and cooperation in mathematics Cooperation in science, and in mathemat-ics in particular, can be discussed from sev-eral points of view � first as exchange ofscientific information, then as cooperationin joint research projects or other kinds ofscientific activity (organisation of confer-ences, schools, etc.), and finally as socialcooperation between different communi-ties. All three types of cooperation are veryimportant for Eastern European countries.Below I discuss them and give my view onthe role of the EMS in encouraging them. I have spent many years organising the StPetersburg Mathematical Society, as amember of the committee (1970-78), asVice-president (1978-97) and as President(1997-now), so I can present some conclu-sions about the role of local societies in themathematical life in Russia, looking in par-ticular at the activities of the Moscow andSt Petersburg Mathematical Societies.

Mathematical work is mainly very indi-vidual (in contrast to the experimental andtechnical sciences), so the tendency to sep-aration is rather strong. The principal roleof the local mathematical societies inRussia is to provide regular meetings withinteresting talks, discussions, information,etc. In mathematics departments in theWest (for example, in the US), part of thisrole is played by regular colloquiums,because most mathematicians areemployed in the universities. In contrast,for many reasons, professional mathemati-cians in Russia were (and are) dispersedover various institutes and centres, some-times without any mathematical environ-ment. So the meetings of local mathemati-cal societies play the role of scientific cen-tres, and provide almost the only opportu-nity for mutual discussions on mathematicsand the place for exchanging information.

Another role of mathematical societies ispublishing activity: some of our societieshave their own journals or proceedings. Tomy mind this is important, but less so thanthe reason above, because there are manyjournals and many possibilities for publica-tion. A much more important function oflocal societies is to represent its communi-ties in other scientific organisations andinternational societies.

The last has a direct connection with therole of the EMS and other internationalorganisations, but before I discuss this con-nection I want to emphasise the new role of

the internet and electronic media whichstep-by-step are changing the type of coop-eration between individual mathematiciansand has forced a change in the daily behav-iour of scientists.

Let me give an example. As everybodyknows, Russia has fewer computers thanthe West and access to the internet is not

common. Nevertheless, these new forms ofcommunication between people haveraised all discussions on the transmissionof scientific and administrative informa-tion to a completely new level. In ourmathematical society, almost all of the 350members use the internet, and so all infor-mation about meetings, special and regu-lar lectures, new books, jobs, prizes, prob-lems, and even discussions on special areas(such as education) can be propagated bye-mail or the web. The role of local mathe-matical societies now becomes a little dif-ferent from before. The societies now havenew duties � to present information aboutmathematical life to individual members,to help them avoid long trips on the inter-net in order to find links, web-pages, etc.,and to support fast contact with other soci-eties.

As an aside, I believe that the EMS stilldoes not fully use the possibilities of theinternet. Doing so could help to solve someproblems that have appeared with individ-ual members, or to discuss other urgentquestions. One could even vote on issuesvia the internet. The EMS�s web-page isstill too short.

As with all kinds of progress, there are

both positive and negative aspects to theinternet. To some extent, cooperation andexchange of information using the internetdeprive us of more vivid forms of commu-nication. But at the same time we have (inour country) no other way of finding need-ed information, especially because inrecent years there has been a decrease inthe subscriptions on our journals, in buy-ing mathematical books, and so on. Underthese circumstances it is very important forus to have access to MatSciNet,Zentralblatt, and other such systems.Ultimately we will need to use the internetinstead of printed matter, because it will beimpossible to maintain enough subscrip-tions, even for the main journals andbooks, in our libraries.

The EMS as a mediator between Eastand WestThe creation of the EMS in the early 1990shas had several consequences, especiallyfor Eastern Europe after the collapse of theSoviet block. The original goal was toestablish an organisation that could unifycooperation between mathematical com-munities in the different European coun-tries.

I think that it is wrong to compare theEMS with the American MathematicalSociety, because their functions and rolesare very different. First of all, unlike theEuropean situation, the US has no local(state) mathematical societies. In contrast,Russia has no national mathematical soci-ety (although we tried to organise one inthe 1980s!), but we have about ten localones (Moscow, St Petersburg, Kazan�,Voronez, Niznii Novgorod, Ural,Novosibirsk, etc.). Also each new state,such as the Baltic states Ukraina,Belorussia, Armenia and Georgia, has atleast one mathematical society. Some ofthese are now very active, while others havemany difficulties but try to keep going.

Cooperation between former SovietUnion states, as well as cooperation withother countries, now helps with these prob-lems. I believe that one of the essentialroles of the EMS is to assist former IronCurtain mathematical communities tobecome incorporated in the European andWorld mathematical communities.

This is not only a question of financialhelp � indeed, I don�t even think that it�sthe main thing. A more delicate problem

EDITORIAL

EMS December 2000 3

EditorialEditorialAnatoly Vershik (St Petersburg)

Member of EMS Executive Committee (1997-2000)

President of St Petersburg Mathematical Society

is to mediate with European organisationsand other communities. In order to dothis, it is important to understand betterthe situation in the scientific life of EasternEurope. I will mention at least two seriouscurrent problems: scientific cooperationand the survival of mathematical schools.

It is useful to recall how the activities ofmost mathematicians were previously sup-pressed by official institutions (the prob-lems of having a job, defending a thesis,travelling abroad, having contact withWestern colleagues, and so on), especiallyfor some categories of mathematicians.Even admittance to the main mathematicaldepartments was forbidden to many peo-ple before the 1990s, and few Soviet math-ematicians were able to participate atmathematical congresses and conferences:even Fields Medallists and invited speakerswere denied permission to go to the ICMs!

The situation has now changed drastical-ly. At the Zurich, Berlin and BarcelonaCongresses there were hundreds of partic-ipants from Russia. But we are now facedwith new problems. On the one hand thereare now no serious obstacles to goingabroad and having contact with colleaguesfrom the West; indeed, many mathemati-cians from Russia and the former SovietUnion, as well as many emigrants of the1970 and 1980s, now have permanent ortemporary positions in the West, and thismust simplify and intensify contact andcooperation. On the other hand the prob-lem is how to make this cooperation moreefficient and, most importantly, how topreserve mathematics in the Eastern coun-tries.

Moreover, there are now many specialgrants for Eastern Europe, such as thoseorganised by the Sorosz Foundation, theAMS and Promatematika (France). Theywere rather small, but well organised.There are also a few local grants inWestern countries such as Germany andHolland which are given to mathemati-cians from both East and West � a greatand disinterested form of support that hasprovided a good illustration of the solidar-ity of mathematicians. At the same timethere have been many complaints aboutthe INTAS-system from Brussels; forexample, one of the INTAS grants finishedtwo years ago but participants fromMoscow, St Petersburg and NizniiNovgorod did not obtain their salaries andnobody from Brussels answered their e-mail messages. I think that one of the rolesof EMS and its East European Committeeis to help with this. It is important tounderstand that it is still very difficult forEastern Europeans to communicate withbureaucrats from the EC.

I believe that there must be more jointresearch teams in various areas, as well asmore visits to Russia from the West; thereare now fewer visits than during the yearsof stagnation. But during the last decadethe mathematical community in Russiaand the Eastern European countries hasundoubtedly started to return step-by-stepto World community, although they havemade only the first few steps.

Another very serious problem relates to

visas for visits to other countries. I under-stand that this is a question for bureaucra-cy at the very highest level, but my impres-sion is that the scientific community can atleast raise the question. The proceduresfor obtaining visas to many countries isvery complicated and humiliating, andremains one of the worst Soviet legacies.Invitations from universities to respectablescientists (including young ones) must begiven preference and must be freed fromsuch procedures.

But the main problem is still the prob-lem of how to prevent decay in our mathe-matics. The traditions of the Russian math-ematical schools are distinguished and dif-ferent from the West. It is completelywrong to say, as I have heard many times(especially from some former Russians),that there are now no serious mathemati-cians in Russia � we have many outstand-ing mathematicians and most seminarsand schools are still active.

But what is true is that we are in a criti-cal situation, and the essential question isabout young mathematicians. Russia had,and still has, an excellent mathematicaleducation in the elementary and highschools, and particularly in the specialmathematical schools. So we still haveenough young and talented people whowant to study mathematics. But the miser-able stipends awarded to students (under-graduate and graduate) as well as some liv-ing difficulties have forced most studentswho have already finished university eitherto drop mathematics for other things(computing or business) or to go abroad.My colleagues and I have received manyletters from the West requesting us to sendour former students to other countries forgraduate school or postdoctoral positions.Indeed, the students from Russia have ahigh reputation.

In a sense, the brain drain is a naturalthing. But we must pay attention to the factthat the collapse of Russian mathematicswould be catastrophic for world mathemat-ics. In order to prevent this disaster weneed to keep at least some young mathe-maticians in our community. If most ofthem leave just after finishing at university,it is bad for both sides. It is clear why it isbad for the Russian mathematical schools,but it is now clear � and we have some sta-tistics � that in general they will not stay inmathematics in the West either. Theyarrive without sufficient grounding fromtheir Russian mathematical school, so theyneed to start their education from thebeginning. At the same time difficultiesarising from their first period abroadforces many of them to go to computercentres or banks.

There are many solutions to this para-dox. First, it is possible to establish com-mon graduate schools � say, between a uni-versity in Russia and another in Germany,with two advisors whose areas are close toeach other, so that a student can sharehis/her time between the two countries.Alternatively, one could establish a fewspecial sufficiently high stipends for ourgraduate students enabling them to makeshort visits to a western university. The

absence of such special programmes forshort visits by young mathematicians is amajor deficiency of our interrelations. Wemust try to correct this � for example, werecently held a special conference foryoung mathematicians from Moscow, StPetersburg and Stockholm on dynamicalsystems and combinatorics. It would begood if such meetings could become a fre-quent occurrence in Europe.

In July 2001 we will hold the firstEuropean summer school in Russia, whichshould provide opportunities for contactsbetween young mathematicians (see EMSNewsletter 37 or the website: http://www.dmi.ras.ru/EIMI/2001/emschool/index.html).

At the round table during the BarcelonaCongress I suggested the establishment of30-40 stipends (from UNESCO, the EC,UNTAS, etc.) to be awarded to the bestRussian (and other Eastern European)graduate students, so that they can spendtime in their countries and can devotethemselves to mathematics for 2-3 years,without having to search for a job. In ourdramatic situation this gives a chance forour mathematics to survive during a diffi-cult period.

In conclusion, I wish to say that the roleof EMS and EC should be more construc-tive in all these aspects. My impression isthat recently we have concentrated toomuch on technical questions, such as linksbetween EMS and other organisations,institutes, and so on. It is more importantto work with local societies to encouragecontacts and interrelations by organisingappropriate European conferences.

It is also very important to pay muchmore attention to the organisation of thefour-yearly European MathematicalCongresses. In future, they must be morebalanced � both geographically, and bysubject area � and more original, and mustprovide a real forum for all Europeanmathematicians. It is important to have abetter financial base for the EMS, and Ithink there are possibilities for this. Thereis a similar question about the EMS councilmeetings which take place each two years.They must be more widely based and lesstechnical; it is better for the EMS ExecutiveCommittee to discuss and solve such tech-nical problems previously.

EDITORIAL

EMS December 20004

EMS Committee forWomen and Mathematics

CorrectionThe September issue of the EMSNewsletter contained a short account ofthe EMS Committee for Women andMathematics. Unfortunately, theauthor�s name and e-mail addressgiven there were incorrect, for whichthe Editor apologises. The contactdetails of the author and CommitteeChair are: Emilia Mezzetti,Dipartimento di Scienze Matematiche,Università di Trieste, Via Valerio 12/1,34127 Trieste, Italy; e-mail: [email protected]

Victor M. Buchstaber (e-mail: [email protected]) graduated from theMoscow State University (MSU) in 1969 and went on to postgraduate study there,with advisors Sergei P. Novikov and Dmitri B. Fukhs. He received a Ph.D. in 1970and a Dr.Sc. in 1984. He has been Research Leader of the Topology Division ofthe Steklov Mathematical Institute of the Russian Academy of Science, Professorin Higher Geometry and Topology at the MSU, and Head of the MathematicalModelling Division at the National Scientific and Research Institute for Physical,Technical and Radio-technical Measurements. He has been on the Council of the Moscow Mathematical Society, Deputy Editor-in-Chief of Uspekhi Mat. Nauk, and Head of the Expert Committee inMathematics in the Russian Foundation for Basic Research.

Statement: I represent the Moscow Mathematical Society (MMS), one of the oldest inEurope (1864). I believe that the EMS must play an important role in developing anddeepening the relations between its corporate members, leaning on the best achievements ofthe national mathematical societies. The eminent achievements of the MMS over the past60 years have undergone a period of rapid fruitful development with a world-wide reputa-tion.Thinking over the experience of the past, the following approaches to organising the life ofa mathematical society seem the most significant:� strong relations between mathematical schools working in different directions, mainte-

nance of generation succession in mathematics, and involvement of new young talent;� stimulating interest in modern achievements of mathematics, while nourishing a love for,

and respect towards, its classical results. The importance of this approach can be demon-strated by the bright and deep applications of classical Abelian function theory and alge-braic geometry to the top modern problems of mathematical physics;

� raising an interest in the sciences related to mathematics, considering them both as spheresof application and as important motive forces and grounds for further development. As aconvincing example, ideas from physics, especially quantum field theory, have affected themodern state of mathematics.

I see my participation in the work of the EMS Executive Committee, in connection with pro-moting and putting into life these approaches.

Mina Teicher (e-mail: teicher:macs.biu.ac.il) is Chair of the Mathematics andComputer Science Department and Director of the Emmy Noether ResearchInstitute for Mathematics (Minerva Center) at Bar-Ilan University in Ramat-Gan, Israel. She received her Ph.D. from Tel-Aviv University for a thesis entitled�Factorization of birational morphisms between 4-folds�. Since then, herresearch interests have developed into geometry and topology, group theory,artificial vision, and mathematical models in brain research. She has travelledwidely, spending the year 1981-82 at the Institute for Advanced Study,Princeton, and paying short-term visits to countries ranging from China, Japanand Tibet, to India and South Africa.

Statement: The EMS should acquire the financial means to enable it to support a broadspectrum of activities on a large scale, and should work to enhance governmental andpublic attitudes towards mathematics. To achieve this we should:� establish The Society of Friends of the EMS to promote donations, public awareness,

government contacts, etc.;� strengthen the scientific relationship with European-based industries and encourage

their financial investment in basic research via the EMS (and individual institutions);� convince Ph.D. students to join the EMS;� work, through governmental and other means, to influence the EU to develop new pro-

grammes better suited to mathematics and to non-governmental organisations like theEMS.

Concerning Mathematics Education, we should work towards:� a unified curriculum based on the current advanced high-school European pro-

grammes;� programmes for the identification and education of especially talented high school stu-

dents (where they do not already exist), especially in underprivileged regions.

EMS NEWS

EMS December 2000 5

New Members New Members of the Executive Committeeof the Executive Committee

At the Council Meeting in Barcelona, Victor Buchstaber and Mina Teicher were elected to the Executive Committee, and Marta Sanz-Solé was re-elected. A mini-biography of Marta Sanz-Solé appeared in EMS Newsletter 32; biographies and statemnts of the othersappear below. Thanks were given to Andrzej Pelczar and Anatoly Vershik who leave the Committee after several years of service.

Present: Rolf Jeltsch (President, in theChair), David Brannan (Secretary), OlliMartio (Treasurer), Bodil Branner, DoinaCioranescu, Luc Lemaire, Andrzej Pelczar,

Renzo Piccinini, Marta Sanz-Solé andAnatoly Vershik; (by invitation) VictorBuchstaber, Tuulikki Mäkeläinen, DavidSalinger, Mina Teicher and Robin Wilson;and (by invitation to a portion of the meet-ing) Chris Lance, Ari Laptev, AndersLindquist, Ulf Persson and Bernd Wegner.Apologies were received from CarlesCasacuberta.

The President thanked the LondonMathematical Society for its hospitality.

Officers� reportsThe President reported that VolkerMehrmann had moved to the TechnicalUniversity of Berlin, allowing closer coop-eration with Bernd Wegner. He had visit-ed Nigeria�s first national mathematicalcentre in July; IMPA in Rio de Janeiro forthe Latin American Congress; theUniversity of California at Los Angeles forthe opening of its Institute of Pure andApplied Mathematics; and Budapest forthe Rényi Institute celebrations. On behalf

of the President, Bodil Branner and BerndWegner had attended a meeting in Lecce,Italy, on Information Science andLibraries in Mathematics; Luc Lemaire

had attended a celebration of the FrenchEMS Prizewinners in Paris; and Jean-Pierre Bourguignon had represented theEMS at the RPA event on 11 November inPortugal.

It was agreed to accept an offer fromOxford University Press (OUP), who pub-lish the journal Interfaces and free bound-aries, of a discounted price of US$65instead of US$90, to members of the EMS;the arrangements will be publicised in theNewsletter and in EMIS. A section in EMISwill shortly be started outlining EMS mem-bers� membership benefits, such as theOUP and International Press discountoffers.

The Secretary reminded Committeemembers of the Executive Committee (EC)that the EMS covers the expenses of ECmembers and others invited to EMS meet-ings, for expenses incurred in connectionwith the meeting if they cannot be coveredfrom other sources, but that the local hostsociety pays all local expenses of EC mem-

bers and others invited to EMS meetings.The Treasurer reported that there are

now few complications with collecting cor-porate member fees, and that the Society

now has a well-functioning system forinvoicing for advertisements in theNewsletter.

MembershipEMS membership currently stands at around2000. It was agreed that membership dri-ves are needed in various countries.

Several possible new Corporate Memberapplications seemed to be coming into thepipeline. The benefits of the EMS having acontact person with each corporate mem-ber were also emphasised.

The reciprocity agreement with theAustralian Mathematical Society had nowbeen signed, and it was hoped soon toagree a reciprocity agreement with theCanadian Mathematical Society. The possibil-ity of the EMS forming reciprocity mem-bership agreements with further societieswas also discussed and approved. TheEMS routinely exchanges the EMSNewsletter with those of its reciprocity soci-eties.

EMS NEWS

EMS December 20006

Executive Committee meeting London, 11-12 November 2000

Tuulikki Mäkeläinen and David Brannan

Matters agreed by electronic votingsince the previous EC MeetingLaurent Guillopé was elected as Chair ofthe Data Base Committee for the period2001-2004; Christian Houzel was electedas the EMS representative on the Abelbicentennial conference programme committee;the reciprocity agreement with theAustralian Mathematical Society wasaccepted; Bernd Wegner was elected asChair for the Electronic publishing committeefor the period 2001-2004; and the meetingParallel processing and applied mathematics2001 (PPAM 01) was accepted as an EMS-SIAM satellite meeting.

European Congresses of Mathematicians(ECM)A lengthy discussion was held of a possiblesite for 4ecm, The Fourth European Congressof Mathematicians, in summer 2004. It washoped to be able to finalise the site selec-tion by the end of 2000. It was noted thatthe dates of the meeting must be carefullycoordinated with those of ICME (The 10thInternational Congress of MathematicsEducation), which will be held inDenmark, in the week of 4-11 July 2004.

There was a brief discussion of the com-position and operation of the ScientificCommittee, and of how different aspectscould be taken properly into considera-tion. The Executive Committee felt itimportant that ECMs attract all activemathematicians in Europe, not just the topmathematicians and new mathematicians.Various ideas to improve the working ofthe various committees set up for theCongress by the EMS were discussed, andthe importance of close collaborationbetween local mathematicians and the var-ious committees was emphasised.

Among the topics raised in the interest-ing discussion of a site for 4ecm were:� why the ECM should be held in a partic-

ular location;� what was the purpose of the ECMs?;� possible benefits to the local mathemat-

ics community;� the possibility of many satellite meet-

ings;

� having an accent on young people;� the differences between an ECM and an

ICM, including mini-symposia, roundtables, etc.; it was thought that localorganisers should be encouraged tothink widely as to the actual format ofthe whole event;

� the possibility of involving the variousEuropean Union �networks�.The Executive Committee wanted

Scientific Committees to choose a wide rangeof topics for speakers; to interest as manypeople as possible across both pure andapplied mathematics; and to discusswhether the lectures should be shorterthan in the past, in order to accommodatemore lectures and to avoid listeners losinginterest after a while.

The Committee agreed that there is aclear need for establishing rules for theEMS Prizes for 2004 and later; for the tim-ing of the Prize Committee�s activities; forthe working of the Prize Committee andselection of candidates, including the agelimit (currently 32), gender balance, geo-graphical distribution and definition of�European� in this context; for the balancebetween pure and applied mathematics;and for the call for nominations andtimetables. The identity of the Chairmanof the Prizes Committee will be knownpublicly from the start, and an open invita-tion for nominations for prizes will be pub-licised.

The Committee expressed its thanks tothe local organisers of 3ecm, especiallyMarta Sanz-Solé, for an efficient andfriendly organisation of the 3ecm inBarcelona in July 2000. The first volume ofthe Proceedings will include the plenarylectures, section lectures, mini-symposiaand presentations of the prizewinners. Thesecond volume will consist of material fromthe round tables, including contributionsfrom panellists and discussions. TheProceedings are planned to come out inthe first part of 2001.

David Brannan and Mina Teicher wereappointed as an ad hoc Committee to pre-pare a set of rules and a schedule for theoperation of the Prize Committee for EMS

Prizes for 2004 and later; they were askedto report to the March meeting of theExecutive Committee.

Stop Press: It has just been announced thatthe European Congress in 2004 will be heldin Stockholm (Sweden).

Council Meeting in Barcelona on 7-8July 2000There was a discussion of various possiblechanges for the following Council meeting in2002. Among the topics were:� should elections be held for individual

members� delegates in any case?� should a Committee member (or its

Chair) present the report of each EMSCommittee, in order to stimulate a dis-cussion on topics of interest to dele-gates?

� how could EMS activate people betweenmeetings?

� should delegates be encouraged to starta discussion?

� projects where EMS is a partner, likeLIMES and EULER (see below), shouldbe presented; in Oslo a presentation onthe proposed EMS publishing house wassuggested.

� highlights of the past two years shouldbe put forward;

� of the two days of the Council, perhapsone day could be for business mattersand the second day for discussions, or aseminar for planning the future.The French delegation to the Barcelona

Council meeting had expressed the wishfor EMS to have more interaction with cor-porate members, and the ItalianMathematical Union had also expressed awish for more frequent exchange of infor-mation. It was agreed to discuss these mat-ters at Kaiserslautern in spring 2001.

The next Council meeting will be held onSaturday-Sunday 1-2 June 2002 in Oslo,Norway, with the first session starting at 10a.m. on 1 June.

Changes of EMS StatutesThe Committee discussed various items ofthe EMS Statutes and EMS By-laws thatseemed to require change, noting that anychange in the Statutes need the approvalof the Finnish authorities, but that changesin the By-laws do not require suchapproval.

Among the topics were: the possibility ofallowing mathematics departments tobecome EMS members; the notion of aPresident-elect and a Past President; theneed for gender and geographic and pure-applied balance in the EMS; the idea ofOfficers having 2-year terms, not 4-yearterms; whether the President needed to bea Council delegate; how to expel EMSmembers who do not pay their dues; allow-ing reciprocity membership; the possibilityof joining EMS via the EMS-Zentralblattscheme; and omitting Articles 5.10 and5.8.

Andrzej Pelczar, David Brannan, OlliMartio and Mina Teicher were elected toan ad hoc committee to formulate thechanges needed to the EMS Statutes.

EMS NEWS

EMS December 2000 7

EMS President Rolf Jeltsch (left) and London Mathematical Society President Martin Taylor

EMS ProjectsThe Committee decided to organise ameeting of its member societies (especiallythose with a strong interest in appliedmathematics), applied mathematics soci-eties outside EMS, European Union math-ematics networks, and some influentialindividual European applied mathemati-cians in spring 2001 to increase the visibil-ity and acceptance of EMS among the appliedmathematics community, to involve them inshaping future EMS policy, and to helpmake them feel at home within the EMS.

The EMS had received encouragementfrom several sources for the creation of apublishing house and preparations had pro-ceeded both by e-mail and at meetings inZurich and London. The Committeedecided that a foundation should be creat-ed to be the legal owner of the publishinghouse, called the European MathematicalFoundation, �EMF�, with its seat inSwitzerland. Swiss law places no restric-tions on the nationalities of the personsinvolved; in Switzerland a foundation canhave tax-free status; it will be a non-profitorganisation; and the Statutes must beaccepted by the Swiss authorities. ThePublishing House will be a legal body sep-arate from the EMS. The EMS logo will beused for the EMF, but inserting the abbre-viation EMF instead of EMS.

The Committee decided to commit10000 euros to be the founding capital ofthe European Mathematical Foundation.The tasks of the EMF will be to establishand run the publishing house; any surpluscould be used to support the work of theEMS.

Rolf Jeltsch and Jean-PierreBourguignon had attended the meeting ofthe Zentralblatt Consultative Committee inBerlin at the end of October. It was notedthat management of the subscribers� listhas now been moved to the editorial office,and that the price of Zentralblatt is belowthe price of Mathematical Reviews. TheJahrbuch project has now a coverage of70%, of which 40% has been edited byexperts so far.

It was agreed to send a paper to EU com-missioner M. Busquin describing the impor-

tance of the Zentralblatt/MATH database asa Large European Infrastructure.

The LIMES project [Large Infrastructurein Mathematics � Enhanced Services: fordetails, see EMS Newsletter 37 or the web-site www.emis.de/projects/LIMES] startedofficially in April 2000, and a meeting hadbeen held to divide the tasks: dataimprovement, input structure and nationalaccess nodes. The EMS is a supervisingbody for the project. The director of theproject is Michael Jost; Bernd Wegner andRolf Jeltsch are the Scientific Directors.The partners are: FIZ Karlsruhe(Zentralblatt-MATH, Berlin) (Coordinator);Cellule de Coordination DocumentaireNationale pour les Mathématiques;Eidetica; Coordinamento SIBA, Universitàdegli Studi di Lecce; Danmarks TekniskeVidencenter & Bibliotek; Universidade deSantiago de Compostela; HellenicMathematical Society; TechnischeUniversität Berlin; and the EuropeanMathematical Society. There would be aworkshop in December for the partnersand editorial units, and a later meeting inCopenhagen would be held in April 2001.

The 2000 Mathematics SubjectClassification is a joint project of Zentralblattand Mathematical Reviews. It was agreedthat the EMS should be one owner of thecopyright to the classification, the otherbeing the American Mathematical Society.The reason for ownership of the copyrighthaving to be made clear was in order toavoid abuse of the classification, not in anyway to limit its free usage.

The President reported that he hadsigned the papers for EMS involvement inZentralblatt for Didactics of Mathematics.

Bernd Wegner made a brief presenta-tion to the Executive Committee of theEULER Project (European Libraries andElectronic Resources in MathematicalSciences � for details, see the websitewww.emis.de/projects/EULER) and gave ademonstration of EULER during the lunchbreak. Its current server is in Göttingen. Itspurpose is to provide access via a searchengine (�EULER�) to access various webresources, including OPAC, databases,preprints, e-journals, and WWW cata-

logues using a common metadata profilemethod for providing a homogeneousaccess to heterogeneous resources. Theproject has developed a metadata makerwith a de-duplication facility, and has test-ed a beta version. The project had receivedvery high marks from its reviewers.

The partners in the Euler Project were: TheState Library of Lower Saxony and theUniversity Library of Göttingen; the J.Hadamard Library, University of Orsay;the Centrum voor Wiskunde enInformatica library, Amsterdam; theUniversity of Florence; the library of theInstitut de Recherche MathématiqueAvancée, University of Strasbourg;NetLab, the Research and DevelopmentDepartment at Lund University Library;MathDoc Cell, Grenoble; FIZ Karlsruhe;Zentralblatt für Mathematik; EMS; and theDepartment of Mathematics of theTechnical University of Berlin.

The EULER Project had formally termi-nated in September 2000. The ExecutiveCommittee felt that there is a need for aproduct like EULER; that EULER pro-vides good tools; that effort is needed forfurther development � e.g., to providesearchable data and to become more userfriendly; and it decided that the EMSshould join the consortium to continuework on the EULER Project � as a �spon-soring partner�, rather than a source ofmanpower or finance.

It was reported that the EU-fundedReference Levels Project will have a finalmeeting in in May 2001 (a report of theworking group was nearly ready), and thatthe contract on TOME [Test ofMathematics for Everybody] would besigned on 16 November 2000.

EMS CommitteesIt was decided that the Electronic PublishingCommittee will be chaired by Bernd Wegnerin 2001-2004; and that the other membersof the committee will be: Slawomir Cynk,Laura Fainsilber, Aviezri Fraenkel, EvaBayer-Fluckiger, Laurent Guillopé, HvedriInassaridze, Michael Jost, Jerry L. Kazdan,Volker Mehrmann, Peter Michor, AndrewOdlyzko, Colin Rourke, LaurentSiebenmann, Jan Slovak and DavidWilkins. The committee will be responsiblefor all aspects of electronic publication,and will develop a new remit.

The composition of the EducationCommittee will be discussed at the Marchmeeting of the Executive Committee.

It was reported that Laurent Guillopéhad accepted to serve as chair to theDatabase Committee, with the term 2001-2004. The Executive Committee agreedthat the other members of the DatabaseCommittee should be: FranciscoMarcell�an, Alberto Marini, SteenMarkvorsen, Peter Michor, MarekNiezgodka and Bernd Wegner.

It was agreed to invite ERCOM members[Committee on European ResearchCentres of Mathematics] to write on theirweb home pages that they are members ofthe EMS, to use the EMS logo there too,and to keep the EMS fully informed of dis-cussions between themselves and the EU.

EMS NEWS

EMS December 20008

Bernd Wegner addresses the Committee on the Euler project

The purpose of ERCOM is to enable mem-ber institutions to discuss matters of mutu-al interest, to enable them to approachfunding bodies (such as the EuropeanUnion) with a wider scientific and geo-graphic base than individual institutionscan, to facilitate the exchange of informa-tion, etc.

It was agreed to add Georg Bock, TsouSheung Tsun and Doina Cioranescu to themembership of the Committee on DevelopingCountries.

It was agreed to add George Jaiani andVictor Buchstaber to the membership ofthe Committee on Support for EasternEuropean Mathematicians (CSEEM). Theannual budget of the CSEEM was 10000euros, and they had supported aroundthirty mathematicians in 2000. It was feltthat there was a continuing need toimprove contact between the committeeand mathematicians in Russia � but thetotal resource was of necessity limited. Theidea that EMS member societies shouldhelp to disseminate information on theCommittee�s activities was welcomed.

The Committee received a report on ameeting for Large Infrastructures, held inStrasbourg, and noted that mathematics isincluded in six different programmes,going across several DGs. It considered aset of possible future projects with theEuropean Union, including discussion ofthe Sixth Framework Programme.

It was agreed that a draft of the ExecutiveCommittee agenda should be circulatedbeforehand to committee chairs, with aninvitation to them to suggest agenda itemsand supply discussion papers � possiblyaround three weeks ahead of an EC meet-ing; and that Committee Chairs should beadded to the Newsletter mailing list if theyare not already EMS individual members.

Diderot Mathematical Forums (DMF)The Committee held a general discussionof its Diderot Mathematical Forum pro-gramme, including items such as whetherthey actually worked well, whether it wouldbe easier to set them up with only twosimultaneous sites rather than three, thecritical dependence on local organisers,and the need for the dates/locations ofDMFs to be advertised well and ahead oftime. Plans for the Fifth DiderotMathematical Forum, probably onTelecommunications, were moving ahead.

Summer SchoolsThe EMS has two summer schools plannedfor 2001. The Prague Summer School hasreceived funding from the EuropeanScience Foundation. The St PetersburgSummer School (which will be held at theEuler Institute) has received support fromthe US National Science Foundation, theAmerican Mathematical Society, andCNRS (France). AMS cooperation in gain-ing support swiftly and smoothly from NSFfor the summer school had been muchappreciated. The EMS had given a 5000euro guarantee to the St PetersburgSummer School.

Applications for Summer Schools in2002 in Brasov and in Israel (on Geometry

and Coding) were accepted.The composition of the

Summer Schools Committee wasrecalled as: R. Piccinini(Chair 2000-2003), C. Broto,C. Casacuberta, D. Cioran-escu and R. Fritsch; M.Teicher was added to thislist.

EMS LecturesIt was agreed to inviteMichèle Vergne to give theEMS Lectures in 2001, possi-bly in Rome and Malta.

The lecture notes of NigelCutland (1997 EMS Lecturer)on Loeb Measures in Practice:Recent Advances will appearsoon as Lecture Notes inMathematics 1751, publishedby Springer-Verlag.

Relations with variousinstitutions and organisa-tionsIt was reported that theVenice office of UNESCO hasawarded the EMS a grant ofUS$25000 to support variousEMS activities in 2000.

Plans are in hand for thejoint EMS-SIAM Conference tobe held in Berlin on 2-6September 2001; full detailswill appear in the EMSNewsletter. There will be a reduction in theconference fee for the meeting for EMSmembers. The International Conferenceon Stochastic Programming in Berlin on25-31 August 2001 (with about 150 partic-ipants) was awarded the status of a satellitemeeting of the conference.

The Committee received a report on theactivities of the Banach International Centerfrom its representatives on the Center�sScientific Committee: F. Hirzebruch; M.Sanz-Solé; D. Wallace (1998-2001).

The Society had been informed of theintention of establishing an Institute forScientific Information at the University ofOsnabrück, devoted to mathematics-relatedactivities, especially in support of theMPRESS project. The institute is to haveboth institutional and individual members.It was decided that the EMS should beinvolved in the plans for the Institute, andthat Rolf Jeltsch should attend the found-ing meeting of the Institute on 30November 2000, if possible.

Rolf Jeltsch was appointed as the EMSrepresentative to ICIAM [InternationalCouncil of Industrial and AppliedMathematics] during 2000-2003.

The Publicity Officer reported on the suc-cess of the EMS booth at 3ecm inBarcelona, commenting that the practiceof sharing a booth with Zentralblatt shouldbe repeated; the Executive Committeethanked Tuulikki Makelainen and MrsMartio for their invaluable efforts instaffing the EMS booth in Barcelona. It wasagreed to produce a number of high-qual-ity posters with the EMS logo, for decora-tion of EMS booths at various future meet-

ings. The article that David Salinger hadwritten for the LMS newsletter should besent to all EMS corporate members, forthem to adapt to local situations.

At the GAMM annual meeting in Zurichon 12-15 February 2001, EMS will share abooth with Zentralblatt.

EMS NewsletterThe contents of the Newsletter wereapplauded by the Committee.

There had been repeated requests tohave the Newsletters on EMIS. The articlesof the 1999 issues of the Newsletter wouldshortly be sent as text files to EMIS.

The Committee noted that the questionraised at the Barcelona Council meetingabout the uneven distribution of the bookreviews of different publishers was beingstudied by the Editor-in-Chief.

Future meetings of the ExecutiveCommitteeThe following outline schedule wasapproved:10-11 March 2001: ITWN Kaiserslautern1-2 September 2001: Berlin, prior to theEMS-SIAM Conference.

EMS NEWS

EMS December 2000 9

Title page of Nigel Cutland�s book (see EMS Lectures, above)

Reciprocity arrangementFollowing the reciprocity arrangementsigned in July with the AmericanMathematical Society (see EMSNewsletter 37, page 8), a further reci-procity arrangement was signed inShanghai on 21 October 2000 with theAustralian Mathematical Society.

The European Mathematical SocietySummer School on New Geometric andAnalytic Methods in Inverse Problems was heldin Edinburgh, Scotland, from 24 July to 2August. It was combined with a meeting onRecent Developments in the Wave Field andDiffuse Tomographic Inverse Problems, from3-5 August. Both meetings were supportedby the European Commission and theLondon Mathematical Society. TheOrganising Committee consisted ofProfessors Yaroslav Kurylev (Lough-borough University, UK), Brian Sleeman(University of Leeds, UK) and ErkkiSomersalo (Helsinki University ofTechnology, Finland). The conference wasorganised in collaboration with ICMS atHeriot-Watt University.

Why geometry and analysis?Inverse problems constitute an active andincreasing field of applied mathematics.Roughly speaking, in inverse problems theaim is to retrieve information of inaccessi-ble quantities based on indirect observa-tions. A typical inverse problem is aninverse boundary value problem of a par-tial differential equation, where the objec-tive is to reconstruct the unknown coeffi-cient functions of the equation in adomain, based on a knowledge of theboundary values of its solutions.Application areas of such problems includemedical imaging, geophysical soundingand remote sensing.

The whole area of inverse problems isfar too wide to be covered in any singlesummer school or meeting. In the presentone, the focus was on modern geometricand analytic methods applied to inverseproblems. The role of differential geome-try in inverse problems is becomingincreasingly significant as more complexsystems are studied.

To get an idea, one can consider theinverse conductivity problem. In physicalterms, the goal is to reconstruct the electric

conductivity of a body by injecting electriccurrents into the body and measuring thevoltages at the surface. In 1980, AlbertoCalderón published a groundbreakingarticle in which he formulated the mathe-matical problem, and since then, consider-

able progress has been achieved in themathematical research of this problem.Despite the efforts, several aspects of thisproblem are still open, in particular whenthe conductivity is allowed to be anisotrop-ic (direction dependent). It turns out thatthe anisotropic problem can essentially berephrased in terms of differential geome-try: can one reconstruct the Riemannianmetric of a manifold from the knowledgeof the Cauchy data of the Laplace-Beltramioperator? This rephrasing, of course,brings the well-developed machinery ofRiemannian geometry to our disposal.Currently, new ideas and techniques aresought in the direction of differentialgeometry to treat this and other anisotrop-ic inverse problems.

The need for new ideas coming fromharmonic analysis and control theory isalso recognised among the inverse prob-lems community. Almost every boundarymeasurement carries inherently a bound-ary control problem: inversion techniquesoften rely on ideas such as focusing ofwaves or, more generally, sounding bywaves of prescribed form. The need torecover discontinuities and other singular-ities of the coefficient functions requirestechniques for treating partial differentialequations with non-smooth coefficients.These are only a few of the problems ofinterest in this field. The emphasis of theEMS Summer School programme was onharmonic analysis and control theory ofpartial differential equations.

One of the most important motivations

for the choice of the summer school topicwas to bridge the gap between the realmsof pure and applied mathematics. It is vitalfor the high quality of mathematicalresearch in Europe that the young genera-tion of applied mathematicians have the

most advanced tools at their disposal; it isequally important for pure mathematiciansto have some insight into the possibilitiesin applied areas of their research.

ParticipantsThe summer school lecturers wereProfessors Victor Isakov (University ofKansas, USA), Dmitrii Burago (Penn State,USA), Vladimir Sharafutdinov (Novosi-birsk, Russia), Lassi Päivärinta (Universityof Oulu, Finland), Anders Melin(University of Lund, Sweden), GuntherUhlmann (University of Washington,USA), Alexander Kachalov (SteklovInstitute, St. Petersburg, Russia) and DrMatti Lassas (University of Helsinki,Finland). The participants were mostlygraduate students and young researchersfrom the EU area and associated countries.The summer school and the conferencealso attracted a number of first-rateresearchers on inverse problems from allover the world, the total number of partic-ipants amounting to over 70. The ideabehind arranging the summer schooltogether with a conference was to give thestudents and young researchers a view ofhow the newly acquired ideas work in cur-rent mathematical research. The writtenmaterial of the summer school, as well as aselection of the invited talks, will be pub-lished as lecture notes.

Erkki Somersalo teaches at the Institute ofMathematics, Helsinki University ofTechnology.

EMS NEWS

EMS December 200010

EMS Summer School in EdinburEMS Summer School in EdinburghghErkki Somersalo

Erkki Somersalo and Jari Kaipio

Heriot-Watt University lecture theatre

The first joint meeting of the AmericanMathematical Society and theMathematical Societies of Denmark,Finland, Iceland, Norway and Sweden tookplace in Odense, Denmark, from 13-16June 2000. It was simultaneously the twen-ty-third in a sequence of ScandinavianCongresses of Mathematicians which start-ed in Stockholm in 1909.

Plenary lectures were delivered byTobias Colding (New York), Nigel J.Hitchin (Oxford), Johan Håstad(Stockholm), Elliott Lieb (Princeton),Pertti Mattila (Jyväskylä), Curtis McMullen(Harvard), Alexei Rudakov (Trondheim),Karen K. Uhlenbeck (Austin) and Dan-

Virgil Voiculescu (Berkeley). In additionthere were more than 100 lectures intwelve special sessions.

Titles and abstracts for (almost) all thelectures are available through the confer-ence home page: http://www.imada.ou.dk/$\sim$hjm/AMS.Scand.2000.html

There were 269 registered participants,roughly 25% of whom came from theUnited States. Another 25% were Danes,while the rest of Scandinavia accounted for19%. The remaining 31% representedtwenty different countries, with France andGermany as the major contributors.

ECMI2000, the 11th Conference of ECMI(The European Consortium forMathematics in Industry) was held inPalermo (Torre Normanna) from 26 to 30September 2000. There were about 270participants from 24 countries, with morethan 200 speakers, including nine plenarylectures, 37 mini-symposia (many organ-ised in collaboration with industry) andabout 50 contributed talks. About 10% ofthe participants were industrial delegates.Everybody felt that an ECMI conferenceprovides an excellent forum for discussingthe role of mathematics in (and for) indus-try at a European level.As usual, the conference was organisedaround industrial themes, as they are theECMI�s Special Interest Groups. The focustopics of the conference were micro-elec-tronics, glass, polymers, composite materi-als, fuel pipelines, finance, biomedical,ecosystems, multi-body dynamics, informa-tion and communication technologies,automatic differentiation and sensitivityanalysis, scientific computing and visuali-sation. Most of the mini-symposia showedunique examples of scientific coordinationand collaboration at a European level forhigh quality research within the ECMISpecial Interest Groups. Selected paperswill appear in the newly established ECMIsubseries of Springer volumes onMathematics in Industry. At the conference all appeared happy andenthusiastic; encouraged by the excellentenvironmental setting, clusters of peoplecould often be seen discussing future col-laboration. Particularly relevant was thelarge participation of young scientists,most of whom had been involved in theECMI�s educational activities � in particu-lar, the modelling weeks that have greatlycontributed to establishing long-lastingbridges between European students. Mostof those who had participated in thesemodelling weeks were among the speakersand mini-symposium organisers. Highlights of the conference included the�Alan Tayler lecture� delivered by HelmutNeunzert, the �Wacker prize lecture� deliv-ered by Carl F. Stein, a student fromGotheborg, and the ECMI honorary mem-bership offered to Carlo Cercignani.A delicious 9-course social dinner wasorganised in a princely palace in Palermo,after an excursion to the marvellous cathe-dral of Monreale, where a fusion ofByzantine, Arab and Norman architectureshows how Sicily was one of the fundamen-tal melting pots of modern Europe.The next ECMI conference will take placein 2002 in Latvia; information about it willbe available on the ECMI web page:http://www.ecmi.dk.

The Proceedings of 3ecm were distributed toparticipants in preliminary form as a com-pact disc, and will be published in two vol-umes by Birkhäuser in its Progress inMathematics series. Articles are also accessi-ble as �pdf� files on the 3ecm website(www.iec.es/3ecm). There will also be abook of Proceedings of the Round Tables, edit-ed jointly by the Catalan MathematicalSociety and CIMNE. The videos exhibitedduring the Congress are distributed bySpringer-Verlag in their VideoMath series,and are also available in DVD format.The earlier list of Invited Lecturers omit-ted Nicolas Burq: Lower bounds for shape res-onance width of Schrödinger operators.The full list of participants at the mini-symposium on Mathematical Finance: Theoryand Practice (Chair: Hélyette Geman) was:M. A. H. Dempster, Stanley Pliska, DilipMadan, Ernst Eberlein, Tomas Bjork, TonVorst, Ezra Nahum and Rainer Schobel.The talk by Charles H. Bennett in themini-symposium on Quantum Computingwas held by video-conference. The talk byHans Föllmer had the title Probabilisticaspects of financial risk.The Chair of the round table How toincrease public awareness of mathematics wasVagn Lundsgaard Hansen; sadly, FelipeMellizo died the week before the Congress.Jean-Pierre Bourguignon was a panellist atthe last round table, not Rolf Jeltsch.Rafael de la Llave should be deleted fromthe list of panellists of the round tableBuilding networks of cooperation in mathemat-ics. He was in fact the Chair of the roundtable The impact of new technologies on math-ematical research, in which BrunoBuchberger was a panellist.

Journal of theEuropean

MathematicalSociety

Volume 3, Number 1 of JEMS contained:D. Mucci, A characterization of graphs whichcan be approximated in area by smooth graphsG. Bouchitté and G. Buttazzo,Characterization of optimal shapes and massesthrough Monge- Kantorovich equationM. Harris and A. J. Scholl, A note on trilin-ear forms for reducible representations andBeilinson�s conjectures

NEWS

EMS December 2000 11

Joint AMS-Scandinavia

meetingJune 2000

Hans J. Munkholm

Chair of the Organising

Committee

11th ECMI Conference

(ECMI2000)Vincenzo Capasso, President of ECMI

Update on

3ecmBarcelona

The World Mathematical Year is coming to anend and it is time to look back and ask ourselves:what did we accomplish? what did we learn?how should we proceed? The celebration of themathematical year has taken place all around

the globe, for the language of mathematics iscommon to all peoples, and mathematics is inde-pendent of nations, religions and races. Forgood measure, it should therefore be said thatwhen I focus this article mainly on what hashappened in Europe, my intention is not toneglect the rest of the world but only to find away of selecting events where European mathe-maticians have had the most direct opportunityto influence what has taken place during theyear.

Conferences dedicated to the WMYDuring the year 2000, several internation-al conferences have been dedicated to theWMY. In most cases, the conferenceswould have taken place independently ofthis special occasion. Nevertheless, theyhave helped to make WMY2000 visible tomathematicians and in many cases theycontributed to making mathematiciansaware of the need for communicatingmathematics to the public, by arrangingdiscussions on this topic during the confer-ences. In particular, 3ecm, the ThirdEuropean Congress of Mathematics, whichtook place in Barcelona, Spain, from 10 to14 July, contained a well attended RoundTable on Raising Public Awareness ofMathematics (RPA).

A very special conference was arrangedin Granada, Spain, from 3 to 7 July, as asatellite conference to 3ecm. This confer-ence was one of the main projects of theEMS-committee of the WMY and had the

idea of bringing Europeans and Arabstogether in the old city with the famousAlhambra, castle of the Moorish kings, todiscuss the historical perspectives of bothcultures to our present mathematicalknowledge. It was a magnificent confer-ence, and included a visit to the Alhambra,guided by the Spanish mathematicianRafael Pérez-Gómez who in the mid-1980sestablished that all the seventeen planarcrystallographic groups are represented inthe fascinating tilings at the Alhambra.

A short list of accomplishmentsIn almost all European countries at leastone poster has been produced, motivatedby the WMY, giving suitable links to placeswhere further information about the yearcan be found. In several countries a seriesof posters were produced, usually based onideas submitted for the poster competi-tion, arranged by the EMS and collected inthe ems-gallery at http://www.mat.dtu.dk/ems-gallery/. Series of posters have beenproduced in Belgium, Portugal, Spain,Italy, France, Germany, the UK, and othercountries, while sets of postcards based onideas from the EMS poster competitionwere produced in Denmark, France andGermany. Stamps related to WMY 2000were issued in the following Europeancountries: Belgium, Croatia, the CzechRepublic, Hungary, Italy, Luxembourg,Monaco, Slovakia, Spain and Sweden.Mathematical exhibitions and workshopshave been presented in Denmark, Finland,France, Germany, Italy, Portugal, Spain,

Sweden and the UK. Mathematical lec-tures for the general public have been pre-sented in all countries, and there havebeen many mathematical articles in news-papers and magazines.

It has been interesting to observe thestrongly varying degree to which it hasbeen possible to catch the interest of radioand television in various European coun-tries. In most countries it has been very dif-ficult indeed � in fact close to impossible �with a few notable exceptions, like France.It might be useful, although difficult, tostudy at the European level whether thereis a connection between the general statusof mathematics in society and schools inthe various countries and the willingness to

present mathematics in the media. Howcan we otherwise explain the rather largevariations?

Two projects funded by the EuropeanCommissionTo help in raising public awareness of sci-ence and technology in Europe, theEuropean Commission has funded a pro-posal in mathematics. The contract hasthree partners: the EMS, represented by itsWMY-committee, with a coordinating role,and actual deliveries to be produced by ateam based in Paris, and a UK-team basedin Bangor. The proposal consisted of twoprojects for presentation in connectionwith the European Science andTechnology Week (ESTW) from 6 to 12November 2000. More information can befound at http:/www.cpm.sees.bangor.ac.uk/rpamath/.

As a result of the first project, twelveposters were presented during the ESTWin a series with the title Les mathematiques duquotidien (Mathematics in daily life), atlocations in France, the UK and Denmark,with Paris as the central city. The produc-tion of such posters has been of consider-able interest to schools, where teachersfind it increasingly difficult to motivateand inspire pupils.

One result of the second project has

WMY NEWS

EMS December 200012

The WThe World Mathematical Yorld Mathematical Year ear in Eurin Europeope

Vagn Lundsgaard Hansen (Chair of the EMS committee for the WMY)with the assistance of Ronnie Brown and Mireille Chaleyat-Maurel

WMY2000 stampfrom Monaco

Robin Wilson as Edmond Halley at aWMY2000 lecture to schoolchildren at

London�s Royal Institution.

been the production by the Bangor team ofa booklet on Presenting Mathematics to thePublic, which was distributed to all partici-pants of the 3ecm with a CD-Rom of JohnRobinson�s Symbolic Sculptures, donated byEdition Limitée. The major work has beenthe production by the Bangor team of aCD-Rom Raising Public Awareness of

Mathematics, which mainly presentsMathematics and Knots, and SymbolicSculptures. This was launched officially atmeetings in Obidos (Portugal) on 11November, and at Bangor (UK) on 16November. The above-mentioned posterswere also on exhibit at these occasions,together with posters in the LondonUnderground produced by the IsaacNewton Institute, Cambridge.

What did we learn?First of all, many more professional math-ematicians have now gained first-handexperience that presenting mathematics tothe public is non-trivial and difficult. TheEMS Committee of the WMY has beendeeply impressed by the dedicated workdone by many individuals in severalEuropean countries, who have taken timeout of their usual positions to work for theWMY project. It is important that we allwork together in such endeavours and helpwith encouragement and by supplyinggood ideas. But everyone should be awareof the fact that realising a good idea usual-ly needs hard work. Among the benefits ofsuch efforts is that it enriches, and putsinto perspective, your future work, notonly as a mathematical educator but prob-ably also as a research mathematician.

How do we proceed?The main issue of the WMY has been toraise the public awareness of mathematics.This important task cannot be restricted toone particular year; it is an on-goingprocess which will take time and needsconstant attention. To continue the workbegun by its committee of the WMY, theExecutive Committee of the EMS hasappointed a new committee with theacronym RPA; further information can befound at http://www.mat.dtu.dk/persons/

Hansen_Vagn_lundsgaard/rpa.htmlA major concern for the new EMS

Committee is to ensure a continuing pres-ence for mathematics at future ESTWs.This needs further support for the meet-ings involved in the planning and prepara-tion of proposals to the EuropeanCommission. The experience gained fromthe first contract with the EuropeanCommission has been very valuable. TheRPA-committee will be eagerly looking forgood ideas that can form the basis forfuture proposals.

One of the first tasks of the RPA-com-mittee will be to arrange a competition forthe best newspaper article on mathematics.The competition will be announced at thebeginning of 2001, allowing authors ofarticles published in national newspapersduring WMY 2000 to submit the best oftheir work for the competition.

The RPA-committee of the EMS will alsoinvestigate the possibilities for establishinga web-site containing eps-files of posters(graphics only, but with open fields wheretexts in various languages can be added),possibly in connection with the ems-gallery. From the experiences gained inconnection with the production of postersduring the WMY, it is clear that such anarchive will be of considerable value tofuture producers of mathematical posters.

It is also planned to create a web-sitecontaining a collection of short mathemat-ical web-stories directed towards secondaryschool pupils and the general public.

Altogether, the EMS Committee of the WMYfinds that the year 2000 has been a terrific yearfor mathematics. Some of the results obtainedmay not seem impressive right at this moment,but as with many things related to mathematics,the products have lasting value and will haveimpact in years to come.

WMY NEWS


Two of John Robinsons mathematical sculptures

EMS Vice-President Luc Lemaire with European Commissioner Philippe Busquin and a WMY2000 poster that appeared in fifty Brussels under-ground stations, in the context of the RPA programme funded by the European Commission and co-organised by the EMS. The picture was taken ata science festival at the Université Libre de Bruxelles.

There is nothing more untrustworthy than aneyewitness, except another eyewitness.

(Sir Walter Raleigh, 1552-1618)Sir Walter Raleigh was one of thefavourites of Elizabeth I. During the reignof James I he fell from grace and was final-ly imprisoned in the Tower of London.To pass the time Raleigh started to write ahistory of the world. There is a story thatone day while writing he heard noises oftumult from the courtyard. Two otherprisoners had started a violent fight whichcaused others to join in. Later the sameday Raleigh tried to find out what hadhappened and interviewed several fellowprisoners who had been present. Everyonehad a different opinion on the course ofevents. Raleigh, frustrated, tore his man-uscript to pieces. (He changed his mindafterwards and started again!)

I have been asked to tell some of myreminiscences on the pre-history of ourSociety. While making an honest try I amaware that I am only one of the fellow pris-oners of Sir Walter Raleigh.

PreludeThe earliest attempt to form a EuropeanSociety for mathematicians that I amaware of took place in the 1978 ICM inHelsinki. A meeting was organised to con-sider the foundation of a Federation ofEuropean Mathematical Societies. Thepossibility of founding a European Societywith individual membership instead wasalso discussed, and it was admitted thatthe latter would give mathematicians amore immediate feeling of being membersof the European MathematicalCommunity. However, a federation waspreferred because it would involve consid-erable less organisation and expense.

One purpose of such a federation wouldbe to provide a forum for exchange ofinformation and for common action � inparticular, the following were mentioned:the coordination, planning and publicityof conferences; fellowships and exchangevisits; the sponsoring of research centres;the foundation of a Federation Newsletter;the fostering of cooperation betweenmathematical societies in matters ofEuropean interest.

During the discussions it turned outthat, despite the positive attitude towardssuch an attempt, it was not possible torealise it. The only outcome was an agree-

ment to form an informal body called theEuropean Mathematical Council (EMC),under the chairmanship of ProfessorMichael Atiyah.

After Helsinki this informal body met inOberwolfach (1980), in Warsaw (1982 and

1983) and again in Oberwolfach (1984).Representatives of some twenty societiesparticipated at these meetings.Mathematicians from the USSR and DDRwere present as observers. True, the ECMprovided a forum for the exchange ofinformation, but because of the informali-ty, common action was scarce. The EMCcarried out biannual surveys of prices ofEuropean mathematical journals, in thehope that these could be used to reduce,or at least maintain, prices of commercial-ly published journals. The Council, how-ever, had no effective means towards thisaim.

The most imposing action initiated bythe EMC was beyond doubt the Euromathproject. It was a very ambitious idea for agigantic mathematical database, togetherwith a sophisticated system of data trans-fer, storing and editing. The databasewould contain all the mathematical knowl-edge in Europe, from preprints to mono-graphs and reviews. In the 1980s thisdream was too huge to be realised � even

on a greatly reduced scale, the project wasso large that it was separated from theEMC to become an independent project,and a legal entity called the EuropeanMathematical Trust was founded for itsupkeep.

This Euromath project proceeded withsubstantial financing from the EuropeanCommunity, but not at the pace that hadbeen expected. An Euromath Centre wasfounded, by a Danish subvention, inCopenhagen. After many delays the pro-ject produced its first (and only) product,the Euromath editor. A widespread criti-cism was that this editor did not containany essentially new features. It did notachieve any substantial success, and littleby little the project was closed. As far as Iunderstand, the parent EuropeanMathematical Trust does not exist anymore.

The idea of Euromath was attractive andworth trying, but it was before its time. Infact, in spite of the internet and the greatprogress with EMIS, we are still far awayfrom the goal.

LibliceI attended the meeting of the EMC inNovember 1986 at Liblice, near Prague.In addition to such normal items as�Survey of journal prices� and �Report ofthe databank committee (Euromath)�, itcontained �Future activities and structureof the EMC�. In introducing this item theChairman, Sir Michael Atiyah, pointed outthat the present informal Council was notintended to be a permanent structure; forinstance, the financial contributions to theEMC were on a voluntary basis and there-fore small and erratic. Many Societies donot have their own means, and so have tojustify the use of their funds to theirGovernment; this makes the formation ofa more formal association desirable. Sucha body should also include the MoscowMathematical Society. This formal bodyshould have links with the InternationalMathematical Union (IMU), but not bedependent on it.

There followed a lively discussion in apositive atmosphere. It was commentedthat physicists and biochemists alreadyhave European organisations and that onewould similarly benefit mathematicians.The new association must be so flexiblethat the participation of all EuropeanCountries would be possible. TheEuropean Mathematical Trust would beentirely separate, but its relation to themain body should be clarified. A form offederation would allow the formation ofsubgroups performing independent activ-ities. An alternative to this Federation ofEuropean Mathematical Societies could bea European Mathematical Society withindividual members.

It was decided that this formalisation ofstructure needed a more profound investi-gation before any decisions could betaken. A small committee was appointedto consider the subject, under the chair-manship of Sir Michael Atiyah. A draft ofthe committee�s conclusions should besent to the ECM member societies as soon

EMS December 200014

FEATURE

The PThe Prre-history e-history of the EMSof the EMS

Aatos Lahtinen

as possible. A discussion of the commit-tee�s proposals would take place at thenext meeting of the ECM in 1988.

Suggestions of the committeeThe considerations of the committee weresent out in May 1988. It had decided thatthe best way to proceed was to formulatecertain principles for discussion at thenext ECM meeting.

The unanimous view of the committeewas that the association would be aFederation of National MathematicalSocieties. However, the membershipshould be defined more flexibly, becausesome countries have several mathematicalsocieties while others have none. Also, itshould be possible for institutional mem-bers to join. The membership fee would

be related to the size and resources of thesocieties, but each society would still haveonly one vote at the Council, which was thesupreme authority.

The Federation would be formally regis-tered in some country, but the committeemade no suggestion as to the location.They only pointed out three importantfactors:� the legal arrangement for charities in

the country;� the stability and convertibility of its cur-

rency;� the initial availability of mathematicians

in that country prepared to assist in theestablishment of the Federation.

For the problem of paying membershipfees with non-convertible currencies, thecommittee suggested setting up a specialEast European Secretariat.

OberwolfachThe next meeting of the EMC took placein Oberwolfach, 15-17 October 1988. In

addition to the normal two days, theChairman Michael Atiyah had reserved anextra day for the meeting. His insight wasright as we used all three days, from morn-ing to evening and even more. The focusat the meeting was on the future structureof the EMC.

The question was considered seriously.After a lively debate there was generalagreement that the time was now ripe forrevival of the idea, from ten years previ-ously, on closer cooperation betweenmathematicians in Europe, based on alegally accepted structure. There was,however, no mutual agreement on theform of the cooperation. In particular, theFrench mathematicians opposed a federa-tion of national societies. They wanted tofollow the structure of the American

Mathematical Society and found aEuropean Mathematical Society of indi-vidual members.

There was a long and animated debateon the pros and cons of these two alterna-tives. It became clear that the supportersof each model were not giving up but weredigging in. Thus it was not possible tocome to a unanimous resolution alongthese lines, while a non-unanimous deci-sion was out of the question. The only wayleft was to find some kind of a compro-mise.

The first agreement was reached on thequestion on the legal form of the new asso-ciation. The EMC decided to give it alegal status by registering it under the lawof some European country. The nextquestion was which country, and this wastied up with the debate on the form of theassociation. Each of the mathematicallyeminent countries France, Germany andGreat Britain was firmly behind its ownconcept and none of them was suggested.Possible alternatives had apparently not

been thought of beforehand.In this open situation I tried to forward

the process by suggesting to MichaelAtiyah that Finland might be willing tohost this new association. I assured himthat Finland would fulfil the three condi-tions of the committee, by giving a shortdescription on the process needed inFinland with a rough timetable and esti-mation of costs. Fortunately I was aware ofthe legal requirements in Finland and Icould certify that the legalisation would bea fairly simple, and not expensive, proce-dure.

Sir Michael Atiyah apparently saw possi-bilities in this proposal because he pre-sented the case to the council, where it wasreceived with interest. After a short dis-cussion the Council decided to accept thesuggestion and to place the seat of thefuture association in Helsinki. This agree-ment somehow opened the deadlock wewere in and progress was also made inother directions. As a compromise it wasdecided that the future association shouldhave both corporate and individual mem-bers. The way the power of decision wouldbe distributed between these two cate-gories was resolved only in principle; thedetails would be settled at the foundationmeeting. The question about the nameraised a long debate, too. Finally theCouncil accepted the name EuropeanMathematical Society.

The Council authorised me to take careof the formal registration of the Society inFinland. In practice, this contained alsothe formulation of a draft of the Statutes ofthe new Society, according to the guide-lines presented and to the requirements ofFinnish law. The next Council meeting in1990 in Poland would make the final deci-sion on the founding of the Society, on itsStatutes and on the principles of theactions the Society would undertake.

The response in FinlandMy suggestion that the legal office of theSociety should be in Finland was at myown initiative; the Finnish mathematicalcommunity had not discussed the questionof the location at all. I therefore returnedto Helsinki with some anxiety. It was,however unnecessary. The FinnishMathematical Society approved my actionand promised its support for the project.

The Chancellor of the University ofHelsinki, Olli Lehto, promised the sup-port of the University. All the officials Icontacted at the Ministry of Education,Justice and Home Secretary took a verypositive attitude and saw no difficulties inthe foundation of the Society as a FinnishLearned Society. I also learned in theprocess that the Government was prepar-ing a Bill for a new law on associations.This would make the registration of theEuropean Mathematical Society even easi-er.

Thus by October 1988 I could alreadywrite a short report to Michael Atiyah,confirming that Finnish mathematicianswere ready to take on the task and carry itto a conclusion. The foundation processcould be initiated.


FEATURE

Sir Michael Atiyah signing the official charter of the Foundation in M¹dralin, 1990. Others fromleft to right: Fritz Hirzebruch, Lászlo Marki, Aatos Lahtinen, Jean-Marc Deshouillers, AndrzejPelczar and Chris Lance. (Photo Courtesy of Prof. Ivan Ivansic.)

Preparing the StatutesPreparing the Statutes appeared to be aniterative process in the sum of two sub-spaces. One was the mathematical com-munity and the other the civil servantswho were taking care of the legal registra-tion. The iteration was complicated by themulti-dimensionality of both subspaces.In fact, it soon turned out to be impossibleto prove that the iteration would con-verge.

The procedure started with the firstdraft composed by Atiyah in December1988, with the Statutes of EuropeanPhysical Society as a model. The mostessential points were:� the members of the Society could be

individuals and organisations otherthan mathematical societies;

� the supreme authority of the Society wasthe Council;

� each member society had one, two orthree Council delegates;

� the number of delegates of other organ-isations and individual members wasrestricted to at most 25% of the totalnumber of Council delegates; the draftdid not specify how to elect the Council.

This draft was sent for comments to math-ematical societies. Simultaneously I triedto specify the requirements of the Finnishlaw. A problem was that the registrationwould be under the future law, and nolawyer was yet willing to make interpreta-tions.

Anyway, in June 1989 we had the seconddraft of the Statutes ready. There was asmall meeting in Oxford in July, with M.Atiyah, J-M. Deshouilles, W. Schwarz, J.Valenca, J. Wright and myself. Based onthe received comments and legal require-ments, we iterated it to the third draftwhich was accepted by all present. In par-ticular, the 25% upper limit on the otherCouncil delegates remained intact. Wealso produced the first draft by-laws.There it was stated that the Council iselected by postal vote where (for example)the number of delegates representingindividual members was restricted to atmost four. There was also a mention ofthe Special Secretariat in Prague (by theoffer of Professor Kufner). The essentialfactors seemed to us to be fairly complete.

Armed with a Finnish translation of thethird draft Statutes, I approached thedepartment of the Ministry of Justice thattook care of the registration of associa-tions, and asked their opinion. They werehelpful, but not happy with our draft. Thelawyer in charge complained that our styleand formulations differed so much fromthe Finnish praxis that it was difficult tosay whether the Statutes were acceptableor not. The most serious point was thelawyer�s demand that the Council must beelected by a General Assembly of all mem-bers. I pointed out the difficulties ofarranging such an assembly in aEuropean-scale society, but it resultedonly to a poor compromise where theAssembly may elect the Council by a postalvote without actually meeting. Anotherdifficulty was that the lawyer was not will-

ing to accept that the President would bethe chairman of both the ExecutiveCommittee and the Council.

I made the alterations that the lawyerdemanded and so by November 1989 wehad the fourth draft of the Statutes whichwould be acceptable to the Ministry butwhich were in some places more cumber-some than the third draft. At the sametime we raised the 25% upper limit on theother Council delegates to 40%. In theBy-laws we raised the upper limit of dele-gates representing individual members to5, and we also added a statement that theupper limit should be reconsidered whenthe Society had 2000 individual members.The Eastern European Secretariat wasremoved from the Statues and the By-laws,because Professor Schwabik from Pragueconsidered that the rapid changes inEastern Europe had made it unnecessary.

This fourth draft was sent once again tothe Societies. Not unexpectedly, manyquestioned the sense in having anAssembly that does not meet. After a dis-cussion with Atiyah I approached theMinistry of Justice once more. By coinci-dence there was a different official incharge. He took our criticism seriouslyand admitted that we could as well have adirect postal vote of the Council without afictitious Assembly.

I was then quite happy to produce inJune 1990 the fifth draft, in which theAssembly was deleted. Also the number ofdelegates of individual members inCouncil was redefined by the formulamin{(n�1)/300)+1, 2C/5}, where C is thetotal number of Council delegates. Ibelieved that the convergence of theStatues had essentially taken place, andthat only minor adjustments were needed.I was wrong.

KyotoAs a final check before the foundationmeeting in October 1990, a meeting wasarranged in August in conjunction withthe ICM 1990 in Kyoto. It seemed appro-priate that the process that had started atICM 1978 would essentially also end at anICM. Michael Atiyah was not present inKyoto, and therefore I chaired the meet-ing. There was general contentment atthe deletion of the Assembly, and itseemed that the Mathematical Societieswere ready to accept this version as final.Unexpectedly the French delegatesannounced that they were not content andthat they would not recommend accep-tance of the Statutes. They opposed theupper limit of Council delegates electedby individual members and said that thesedelegates must be able to have a majorityin the Council. They also raised the pointthat individual members from �poor�countries may not be able to pay even amodest fee. The first point came as a sur-prise to me because I had understood thatalso the French Mathematical Society hadaccepted the upper limit.

In due course I consulted Atiyah. Wedecided to treat the latter problem byadding a new By-law giving the ExecutiveCommittee the authority to waive tem-

porarily the fee of any member. It wasdecided that the more serious problem onthe upper limit should be left to the foun-dation meeting.

M¹dralinThe foundation meeting of the EuropeanMathematical Society, which started as thelast meeting of the EuropeanMathematical Council, was held inMadralin, near Warsaw, on 27-29 October1990. The meeting of the EMC waschaired, as always, by Michael Atiyah.There was a lively discussion on the pur-pose and modes of action of the forthcom-ing Society. Concerning the Statutes therewere several comments most of which wereof a technical nature. As expected, theonly principal question was raised by theFrench mathematicians. They repeatedwhat they had already said in Kyoto �namely, that they would accept only a soci-ety of individual members and not one ofSocieties. In particular, they wanted theupper limit on delegates of individualmembers to be removed.

In the discussion they did not get sup-port. However, the unanimous accep-tance of the Statutes was consideredimportant, and therefore a small ad hoccommittee of Michael Atiyah, Jean-PierreBourguignon, Fritz Hirzebruch, LászloMarki and myself was set up to try to nego-tiate a compromise. We convened atlunchtime. After some attempts it wasagreed to propose a package of four reso-lutions:� the member societies should encourage

their members to become individualmembers of the EMS and should becommitted to collect individual mem-bership fees for the EMS;

� the statutes of EMS will be reconsideredwhen the individual membership of theEMS has reached 4000;

� in the formula defining the number ofdelegates for individual members thedenominator 300 should be replaced by100;

� in the same formula the upper limit2C/5 should be retained.

The council accepted the first three reso-lutions unanimously and the fourth by 28votes to 12. However, when the Frenchwere still unhappy with the upper limit, itwas decided to replace it by 2C/3.

After some minor items the Chairmanproposed that the Council should formal-ly establish the European MathematicalSociety with its seat in Helsinki, which wasagreed. Professor Atiyah was accepted asthe first individual member of the Society.The official charter for the foundation(written in Finnish, of course) was signed,a toast was raised, and we had theEuropean Mathematical Society.

What happened after that is another story.A part of this story, told by David Wallace,can be read at the address http://turn.to/EMSHISTORY99. It cannot be more thana part, because the story of our Societycontinues and continues forever, I hope. Vivat, crescat, floreat!

FEATURE

EMS December 200016

Roger Penrose was formerly Rouse BallProfessor at Oxford University, and currentlyholds the position of Gresham Professor ofGeometry in London.

This interview is in two parts � the secondpart will appear in the March 2001 issue.

When did you first get interested in mathe-matics?From quite an early age � I remembermaking various polyhedra when I wasabout 10, so I was certainly interested inmathematics then � probably earlier, but itbecame more serious around the age of 10.

Are there other mathematicians in yourfamily?Yes, my father was a scientist � he becamea professor of human genetics, but he hadbroad interests and was interested in math-ematics � not on a professional level, butwith abilities and genuine interests inmathematics, especially geometricalthings. I also have an older brother Oliverwho became a professor of mathematics.He was very precocious � he was two yearsolder than I was, but four years ahead inschool. He knew a lot about mathematicsat a young age and took a great interest inboth mathematics and physics; he did adegree in physics later on. My mother alsohad an interest in geometry; she was med-ically trained as my father was.

Did you have good teachers at school?I did have at least one teacher who wasquite inspiring. I found his classes inter-esting, although maybe not terribly excit-ing.

Where did you go to school?I was at school in Canada between the agesof 8 and 13. I don�t know that I got muchmathematics interest from there. Then Iwas back in England at the age of 14.

But you were born in England?Yes. We went over to the US just beforethe War. My father had a job in London(Ontario) at the Ontario hospital, where helater became the Director of PsychiatricResearch. He was interested in mentaldisease and its inheritance, the sort ofthing that he became particularly experton later. So the question of inheritanceversus environmental influence were ofgreat interest to him.

In fact I was born in Colchester in Essex� it�s an old Roman town, possibly the old-est town in England. My father took on aproject called the Colchester survey, whichhad to do with trying to decide whetherenvironmental or inherited qualities weremore important in mental disease. The

conclusion he came to was that the prob-lem was much more complicated than any-body had thought before, which is proba-bly the right answer.

This was before going to Canada?Yes. Then we went over first to the USwhen it started to become clear there wasgoing to be a war. He had this opportuni-ty to work overseas and he took it.

And when did you return to England?Just after the War, in 1945. I went to

University College School in Londonwhere I became more and more interestedin mathematics, but I didn�t think of it as acareer. I was always the one who was sup-posed to become a doctor, but I rememberan occasion when we had to decide whichsubjects to do in the two final years. Eachof us would go up to see the headmaster,one after the other, and he said �Well, whatsubjects do you want to do when you spe-cialise next year�. I said �I�d like to do biol-ogy, chemistry and mathematics� and hesaid �No, that�s impossible � you can�t dobiology and mathematics at the same time,we just don�t have that option�. Since I hadno desire to lose my mathematics, I said�Mathematics, physics and chemistry�. My

parents were rather annoyed when I gothome; my medical career had disappearedin one stroke.

Where did you go to university?I went to University College London formy undergraduate degree. My father wasprofessor there and so I could go therewithout paying any fees. My older brotherhad also been there as an undergraduateand he then went to Cambridge to do aPh.D. in physics. I went to Cambridgeafterwards to do my Ph.D. in mathematics.

I was mainly a pure mathematician inthose days. I�d specialised in geometryand went to Cambridge to do research inalgebraic geometry, where I worked underWilliam Hodge.

A contemporary who was also starting atthe same time was Michael Atiyah, wholater won the Fields Medal and becamePresident of the Royal Society, Master ofTrinity College, Cambridge, and the firstdirector of the Isaac Newton Institute.When you first become a research studentyou�ve no idea who the other people are.It took me a while to realise there wassomething special about him. So it was abit intimidating, I remember, at the begin-ning.

INTERVIEW


Interview with Sir RInterview with Sir Roger Poger Penrenrose ose part 1

Interviewer: Oscar Garcia-Prada

I worked with Hodge for only one year,because he decided that the kind of prob-lems I was interested in were not in his lineof interest. I then worked under JohnTodd for two years, but during that periodI became more and more interested inphysics, largely because of my friendshipwith Dennis Sciama who rather took me

under his wing. He was a good friend ofmy brother�s, and I think I made some-thing of an impression on him when I vis-ited Cambridge and asked him some ques-tions about the steady-state universe whichI don�t think he�d quite thought about. Sohe thought it was worth cultivating myinterest in physics.

So was he one of the most influential peopleyou came across?He was very influential on me. He taughtme a great deal of physics, and the excite-ment of doing physics came through; hewas that kind of person, who conveyed theexcitement of what was currently going onin physics � it was partly Dennis Sciamaand partly lectures that I attended �on theside� when I was in my first year.

I remember going to three courses, noneof which had anything to do with theresearch I was supposed to be doing. Onewas a course by Hermann Bondi on gener-al relativity which was fascinating; Bondihad a wonderful lecturing style whichmade the subject come alive. Another wasa course by Paul Dirac on quantummechanics, which was beautiful in a com-pletely different way; it was just such a per-fect collection of lectures and I reallyfound them extremely inspiring. And thethird course, which later on became veryinfluential although at the time I didn�tknow it was going to, was a course onmathematical logic given by Steen. Ilearnt about Turing machines and aboutGödel�s theorem, and I think I formulatedduring that time the view I still hold � thatthere is something in mental phenomena,something in our understanding of mathe-matics in particular, which you cannotencapsulate by any kind of computation.That view has stuck with me since thatperiod.

You�ve worked in many areas, but let mestart with your 1960s work on cosmology.With Stephen Hawking you discovered thesingularity theorems that won you both theprestigious Wolf prize. What are these the-

orems about, and what do they say aboutspace-time?Well, singularities are regions of space-time where the laws of physics break down.The main singularity one hears about isthe big bang, which represents the origin ofthe universe. Now cosmological modelswere introduced in accordance with theEinstein equations of general relativity,which describe curvature of space-time interms of the matter content. The equa-tions determine the time-evolution of theuniverse. You apply these equations to avery uniform universe, which is what peo-ple did originally, assuming that the uni-verse is homogenous and isotropic, inaccordance with the standard models thatare used to describe cosmology on a largescale. If you extrapolate Einstein�s equa-tions backwards you find that at the verybeginning was this moment where the den-sity became infinite and all matter was con-centrated in a single place. The big bangrepresents the explosion of matter awayfrom this � in fact, the whole of space-timeoriginated in this single event.

Some people used to worry about this, asI did, because it represents a limit to whatwe can understand in terms of knownphysical laws. The same situation aroselater when people started to worry aboutwhat happens to a star which is too massiveto hold itself apart and singularities arise.Back in the 1930s Chandresekar showedthat a white dwarf star, which is a reallyconcentrated body, can have the mass ofthe sun, or a bit more. We know that suchobjects exist � the companion of Sirius isthe most famous one � but if such a bodyhas more than about one-and-a-half timesthe mass of the sun then, as Chandresekarshowed, it cannot hold itself apart as awhite dwarf and will continue to collapse;nothing can stop it. A white dwarf is basi-cally held apart by what�s called electrondegeneracy pressure � this means that theelectrons satisfy an exclusion principlewhich tells you that two electrons cannot bein the same state, and this implies thatwhen they get concentrated they hold thestar apart. So it�s this exclusion principlein effect that stops a white dwarf star fromcollapsing.

However, what Chandra showed is thatgravity will overcome this if the star is toomassive, and that its electron degeneracypressure cannot hold it apart. This prob-lem occurs again in what�s called neutrondegeneracy pressure, which is again theexclusion principle but now applied toneutrons. What happens is that the elec-trons get pushed into the protons and youhave a star made of neutrons. Those neu-trons hold themselves apart by not beingable to be in the same state. But again theChandrasekar argument comes to bear onthe neutron stars and you find that theyalso have a maximum mass which isn�tbelieved to be much more than that of awhite dwarf. So anything with, say, twicethe mass of the sun would seem to have noresting place and would go on collapsingunless it could throw off some of its mater-ial. But it seems unlikely that in all cir-cumstances it would throw off enough

material, especially if it started with a massof, say, ten times the mass of the sun.

So what happens to it? Round about1939, Robert Oppenheimer and variousstudents of his � in particular, HartlandSnyder � produced a model of the collapseof a body. As an idealisation, they consid-ered a body made of pressureless material,which was assumed to be exactly spherical-ly symmetrical � and they showed that itwill collapse down to produce what we nowcall a black hole.

A black hole is basically what happenswhen a body is concentrated to such asmall size for such a large mass that theescape velocity is the velocity of light, orexceeds the velocity of light, the escapevelocity being that speed at which an objectthrown from the surface of the bodyescapes to infinity and doesn�t ever fallback again. It�s about 25000 miles an hourfor an object on the surface of the earth.But if you concentrate the earth so much,or take a larger body with a mass of, say,twice the mass of the sun and concentrateit down, it reaches the region in which thevelocity will reach the speed of light whenit�s just a few miles across. And then itbecomes a black hole once the escapevelocity exceeds the velocity of light, sothat nothing can escape, not even light.

This is exactly what happened in themodel that Oppenheimer and Snyder putforward in 1939. But it didn�t catch on.Nobody paid any attention to it, least of allEinstein, as far as one knows. I think theview of many people was that if you removethe assumption of spherical symmetry thenthe exact model that Oppenheimer andSnyder had suggested would not be appro-priate, and who knows what would hap-pen? Maybe it would not concentrate intoa tiny thing in the centre, but would justswirl around in some very complicatedmotion and come spewing out again � Ithink this was the kind of view some peo-ple had. And you wonder about whetherassuming that there�s no pressure is thefundamental assumption, because matterdoes have pressure when it gets concen-trated.

This was revived in the early 1960s whenthe first quasars were discovered. Theseextremely bright shining objects seemed tobe so tiny, yet so massive that one wouldhave to worry about whether an object hadactually reached the kind of limits that I�vejust been talking about, where you would-n�t see it if it was really inside what�s calledthe event horizon, and where the escapevelocity exceeds the velocity of light, but ifyou get close to that then very violentprocesses could take place which couldproduce extraordinarily bright objects.When the first quasar was observed, peoplebegan to worry again about whether whatwe now call black holes might not really bethere out in the universe.

So I began thinking about this problemand the whole question of whether theassumption of exact spherical symmetrycould be circumvented, using techniquesof a topological nature which I had startedto develop for quite other reasons. Whatpeople had normally done would be just to

EMS December 200018

INTERVIEW

Roger Penrose and Dennis Sciana at the 1962Relativity Conference in Warsaw

solve complicated equations, but that�s notvery good if you want to introduce irregu-larities and so on, because you simply can�tsolve the equations. So I looked at thisfrom a completely different point of view,which was to look at general topologicalissues: could you obtain a contradictionfrom the assumption that the collapsetakes place without any singularities?Basically what I proved was a theoremwhich was published in 1965 in PhysicalReview Letters, where I showed that if a col-lapse takes place until a certain conditionholds (a qualitative condition which Icalled the existence of a trapped surface), thenyou would expect some type of singularity.What it really showed is that the space-timecould not be continued, it must come to anend somewhere � but it doesn�t say whatthe nature of that end is, it just says thatthe space-time cannot be continued indef-initely.

Can you test this theory in our universe?Well, the first question is: do black holesexist? They are almost a theoretical conse-quence of the kind of discussion I�ve justreferred to. Then Stephen Hawking camein as a beginning graduate student work-ing with Dennis Sciama, and he took offfrom where I�d started, introducing someother results mainly to do with cosmologyrather than black holes. Later we put ourresults together and showed that singular-ities arise in even more general situationsthan we had individually been able to han-dle before.

Now there is a big assumption here towhich we still don�t know the answer. It�scalled cosmic censorship, a term I introducedto emphasise the nature of this hiddenpresumption, that is often tacitly made.Cosmic censorship asserts that �naked sin-gularities� do not occur. We know from thesingularity theorems that singularities ofsome kind occur at least under appropri-ate initial conditions that are not unrea-sonable � but we don�t know that those sin-gularities are necessarily hidden fromexternal view. Are they clothed by what wecall a horizon, so you can�t actually seethem? With a black hole you have thishorizon which shields that singularity fromview from the outside. Now it�s conceiv-able that you could have these naked sin-gularities, but they�re normally consideredto be more outrageous than black holes.The general consensus seems to be thatthey don�t happen, and that tends to be myview also. If you assume that they don�toccur, then you must get black holes. Soit�s a theoretical conclusion that if you havea collapse of a body which is beyond a cer-tain size, then you get black holes.

Now one type of system thatastronomers have observed is where thereis a double star system, only one memberof which is visible. The invisible compo-nent is taken to be a black hole � Cygnus X-1 was the first convincing example. It�s anX-ray source, and what is seen is a bluesupergiant star which is in orbit aboutsomething; the �something� is invisiblethrough a telescope, but seems to be thesource of the X-rays. Now the X-rays

would come about if material is draggedinto a tiny region and gets heated in theprocess of being dragged in; the materialprobably forms a disc, which is the normalview people have. The material getsdragged off the companion star, the bluesupergiant star, and it spirals into the hole,in the standard picture. It gets hotter andhotter until it reaches X-ray temperature,which is the source of these X-rays, andthat�s what�s seen.

Now it doesn�t tell you that this object isactually a black hole, but the dynamics ofthe system are such that the invisible com-ponent has to be much too massive to beeither a white dwarf or a neutron star,because of the Chandrasekar argument,and so on. So the evidence is indirect:what one knows is that there is a tiny high-ly concentrated object which seems to bedragging material into it, and from theneighbourhood of which one sees X-rays.Also gamma ray sources seem to be blackhole systems, and there may now be manyother examples, other double star systemsor black holes in galactic centres. Indeed,there is convincing evidence for a very con-centrated dark object at the centre of ourown galaxy, of the order of something likea million solar masses.

It seems to be a standard phenomenonthat galaxies may have these highly con-centrated objects which we believe to beblack holes in their centres. Some galaxies

may have large ones, and quasars arebelieved now to be galaxies which have attheir centres objects that are muchbrighter than the entire galaxy, so all yousee is this central region which is extraor-dinarily bright. It�s bright because it hasdragged material into it, and it gets extra-ordinarily hot and spews out in certaindirections at nearly the speed of light. Yousee examples of things where jets come outof centres of galaxies and things like this.But all this evidence is indirect. It�s notthat one knows that black holes are there,it�s just that the theory tells us that thereought to be black holes there and the the-ory fits in very well with the observations.But most observations do not directly saythat those are black holes, although there�s

impressive recent evidence of materialbeing swallowed by one without trace.There�s also another potential possibilityof the direct observation of a black hole:when I say �direct�, it�s more because thetheory of black holes is so well developedthat one knows very closely what the geom-etry should be. There�s a geometry knownas a Kerr geometry which seems to be theunique endpoint of a collapsed object toform a black hole, and this geometry hasvery interesting specific properties. Someof these could be tested to see whetherthese concentrated objects that we knoware there really conform with the Kerrgeometry. That would add much moredirect evidence for black holes, but it�ssomething for the future.

What would be the most striking physicalimplications of the singularities here?What the singularities tell us is that thelaws of classical general relativity are limit-ed. I�ve always regarded this as a strengthin general relativity. It tells you where itsown limitations are. Some people thoughtit was a weakness of the theory because ithas these blemishes, but the fact that itreally tells you where you need to bring inother physics is a powerful ingredient inthe theory.

Now what we believe is that singularitiesare regions where quantum theory andgeneral relativity come together, where

things are both small and massive at thesame time. �Small� is where quantumeffects become important, and �massive� iswhere general relativity becomes impor-tant. So when you get the two things hap-pening together, which is what happens insingularities, then the effects of both gen-eral relativity and quantum mechanicsmust be considered together.

Now this applies in the big bang and inthe singularities in black holes, and itwould also apply if the whole universe wereever to collapse, although that is just a con-glomeration of all the black holes into onebig black hole. There�s one thing I findparticularly interesting, however, which isthe stark contrast between the big bangand the singularities in black holes. It�s a


INTERVIEW

bit ironic, because in the earlier stages ofthe black hole singularity discussions, theirreasonableness was that we already knowthere�s a singularity in the big bang. It wasargued that the singularities in black holesare just the same as the big bang, but timeis going the other way � so if you have oneyou should have the other. This was quitea plausible kind of argument, but when welook at these things in detail we see thatthe structures are completely different: thestructure that the big bang had was verysmooth and uniform, whereas the struc-ture we expect to find in singularities isvery complicated and chaotic � at a com-pletely different end of the spectrum.

In fact, this is all tied up in a deep waywith the second law of thermodynamics. Thislaw tells us that there is a time asymmetryin the way things actually behave. This isnormally traced far back in time to somevery ordered structure in the very earlystages of the universe � and the more youtrace it back, the more you find that thisordered structure is indeed the big bang.

So what is the nature of that orderedstructure in the big bang, and what is itscause? Well, in relation to what I was justsaying, it is quantum gravity. We believethat this is where quantum theory andgravitational theory come together. Andwhat this tells us � and I�ve been saying thisfor quite a few years but few people seemto pick up on this completely obvious point� is that the singularity structure, which iswhere we see general relativity and quan-tum mechanics coming together most bla-tantly, is time-asymmetrical. So it tells methat the laws involved in quantum gravity,combining quantum theory with generalrelativity, must be time-asymmetrical,whereas the laws we normally see inphysics are time-symmetrical.

It also tells us, it seems to me, that thelaws of quantum mechanics are not justconcerned with applying quantummechanics to general relativity � when I say�just�, it�s a gross understatement becausenobody knows how to do that, but I think itmust be a union between these two theo-ries, giving a new theory of a differentcharacter. It�s not just quantum mechan-ics: quantum mechanics itself will have tochange its structure and it will have toinvolve an asymmetry in time, but I havereason to believe that this is all tied in withthe measurement problem � the collapseof the wave function, the curious featuresthat quantum theory has which make it inmany respects a totally unsatisfying theoryfrom the point of view of a physical pictureor a philosophically satisfying view of theworld. Quantum mechanics is very pecu-liar, because it involves incompatible pro-cedures. My own view is that this is some-thing that we will only understand whenwe�ve brought Einstein�s general relativityin with quantum mechanics and combinedthem into a single theory.

So my view on quantum gravity is quitedifferent from that of most people. Whatmost people seem to say is �Oh, you�ve gotto try and quantise general relativity, andquantise gravitation theory, and quantisespace-time�: to �quantise� means to take the

rules of quantum mechanics as they areand try to apply them to some classical the-ory, but I prefer not to use that word. I saythat the theory we seek involves also achange in the very structure of quantummechanics. It�s not quantising something;it�s bringing in a new theory that has stan-dard quantum theory as a limit. It also hasstandard general relativity theory asanother limit, but it would be a theory thatis different in character from both thosetheories.

Let me come to another aspect of your work.One of your greatest inventions is twistortheory, which you introduced about 30years ago. What is twistor theory?Well the main object of twistor theory is tofind the appropriate union between gener-al relativity and quantum mechanics. Isuppose I had that view over thirty yearsago (actually, 1963) before I talked aboutthis singularity issue and the asymmetry,and so on. I�d already felt that one needsa radically different way of looking atthings, and twistor theory was originallymotivated by such considerations. Sincewe can�t just �quantise�, we need otherguiding principles.

Let me mention two of them. One wasnon-locality, because one knows aboutphenomena in which what happens at oneend of a room seems to depend on whathappens at the other end. These experi-ments were performed about twenty yearsago by Alain Aspect in Paris � all right,those experiments hadn�t been performedwhen I introduced twistor theory, but theoriginal ideas were there already � I meanthe Einstein-Podolsky-Rosen phenomena,which tell you that quantum mechanicssays that you have �entanglements� � thingsat one end of the world seem to be entan-gled with things at the other end. Nowthat�s only a vague motivation: it�s not real-ly something that twistor theory even nowhas a great deal to say about, but it does saythat somehow non-locality is important inour descriptions, and twistor theory (as ithas developed) certainly has features ofnon-locality, over and above those I wasaware of when I started thinking aboutthese ideas.

Originally, rather than having points inspace-time as the fundamental objects, Ithought more in terms of entire light raysas fundamental. The reason for thinkingabout light rays actually came from some-thing quite different, which I regard asperhaps the most important motivationunderlying twistor theory. In the unionbetween quantum mechanics and generalrelativity, I feel strongly that complexnumbers and complex analytic structuresare fundamental for the way that the phys-ical world behaves. I suppose that part ofmy reason for this goes way back to mymathematical training. When I first learntabout complex analysis at university inLondon I was totally �gob-smacked� � it justseemed to me an incredible subject; someof the simplest ideas in complex analysis,such as if a function is smooth then it�s ana-lytic, are properties which I always thoughtwere totally amazing.

What are twistors, and how are they morefundamental than a point in space-time?Well, you see, if I follow the complex analy-sis I can come back to this. First of all,complex analysis is just mathematics, andit�s beautiful mathematics that�s tremen-dously useful in many other areas of math-ematics. But in quantum theory you see itat the root of the subject � for the first timeone sees that it�s really there in nature, andthat nature operates (at least in the smallscale) according to complex numbers.

Now the thing that struck me from quiteearly on � it�s one of the earliest things Idid in relativity � is if you look out at thesky you see a sphere; but if you considertwo observers looking out at the same sky,one of whom is moving with a high speedrelative to the other, then they see a slight-ly transformed sky relative to each other,and the transformation of that sky pre-serves circles and takes angles to equalangles. Now those people who know aboutcomplex analysis know that this is the wayyou look at the complex numbers: youhave infinity as well, and they make asphere � the Riemann sphere � and thetransformations that send that sphere toitself, the complex analytic transforma-tions, are precisely those that send circlesto circles and preserve angles. I was com-pletely struck by this phenomenon, as itseems to me that what you�re doing whenyou look at the sky is you�re seeing theRiemann sphere � they are the complexnumbers out there in the sky, and itseemed to me that that�s the kind of math-ematical connection. It seemed to me sucha beautiful fact, and in a sense the trans-formations of relativity are all contained inthat fact. Surely that means something.We already know that complex numbersare fundamental to quantum theory, andhere we see complex numbers fundamen-tal to relativity when we look at it this way.

My view was to say �all right, don�t thinkof the points you see when you look at thesky; what you are doing is seeing light rays.You and a star in the distance are connect-ed by a light ray, and the family of thingsthat you see as you look at the sky is thefamily of light rays through your eye atthat moment. So the thing with the com-plex structure is light ray space, telling youthat maybe you can see this link betweenspace-time structure and complex num-bers if you concentrate not on points buton light rays instead.

So that was really the origin of twistortheory � well, that�s cheating slightly, but Isuppose one cheats when one gets used toa certain idea, because although these phe-nomena were known to me and I realisedtheir importance, it was something elsethat really steered me in the direction oftwistor theory. It�s a bit technical, but hadto do with complex numbers � all right,you see them in the sky, but you also seethem in all sorts of other places � in solu-tions of Einstein�s equations, and so on �they started to come up when peoplelooked at specific solutions of Einstein�sequations. It turned out very often thatyou could express things very nicely if youused complex numbers, and it suggested

EMS December 200020

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to me that somehow � I had this image likean iceberg, you see � what you see is a lit-tle bit at the top and there�s the rest of itdown underneath which is invisible. It�sreally a huge area where these complexnumbers at the tip poke up through thewater, while the rest of it is underneath.

So these solutions, where one sees thecomplex numbers, seemed just the tip ofan iceberg, and they were really under-neath governing the way that the first-hand structure works. It was a search to tryand find what that complex structure was,and it wasn�t until certain things that arenot appropriate to describe here, butwhich relate to solutions of Maxwell�sequations and Einstein�s equations whichshow you that the space of light rays,although it�s not quite a complex spacebecause it�s got the wrong number ofdimensions � but if you look at the rightstructure you see it as part of anotherstructure, a slightly extended one with sixdimensions, and it produces complexobjective space which is complex projective3-space.

Now with hindsight I can describe thesethings more satisfactorily. Let me put itlike this. When you think of a light ray,that is an idealised photon idealised in aspecific respect, you are just thinking of itas a path through space-time. But youhave to bear in mind that massless parti-cles (photons, in particular) also have spin(they spin about their direction of motion),and if you introduce the spin they alsohave energy. The spin is a discrete para-meter. It�s either left-handed or right-handed, but when the particle has spin,introducing the energy (a continuous para-meter) gives one more degree of freedom.So instead of having just five dimensions oflight rays, you find a six-dimensional spacethat is naturally the complex 3-space. Soyou�ve got the whole thing, the right-hand-ed ones, the light rays and the left-handedones, and they all fit together to form aspace that�s called projective twistor space.

And it seemed to me that once you takethis space as being more fundamental thanspace-time (the main reason being that it�scomplex), it ties in with other things thatI�ve been interested in for years � the useof spinors and how you treat general rela-tivity, things which I�d learnt in Bondi�sand Dirac�s lectures. This notion of spinors,as a way of treating general relativity, wassomething I found to be powerful, but itdidn�t quite do what I wanted, which was toget rid of the points. That was what twistortheory achieved, and it�s still going on.

So how do twistors actually relate to thesesingularity theorems? Do they have any-thing to say about those theorems?The short answer to that question is no �or, not yet. The hope is that they will, butthe subjects have been going off in twoquite different directions. Twistor theoryis motivated by trying to bring general rel-ativity and quantum mechanics together.If it�s successful in that direction, then itwould have something to say about the sin-gularity problem, but at the moment it hasvery little direct bearing on the singularity

problem. I regard it as a very long round-about route, but one needs first to under-stand how Einstein�s general relativity real-ly fits in with twistor theory. Although con-siderable advances have been made, somedating back to twenty years ago, it�s still aquestion mark. We don�t completely knowhow to represent Einstein�s theory in rela-tion to twistors; there are some very strongindications that there�s a good connectionbetween the two, but how one does it is stillnot clear.

So my view is that the major problem intwistor theory is to see how to incorporateEinstein�s theory into the twistor frame-work, and it�s still not complete. What weseem to see is that in the process of incor-porating Einstein�s theory into twistors, wealso have to incorporate ideas of quantummechanics. So my hope is that in bringingclassical general relativity into the scope oftwistor theory, one will also see how quan-tum theory must be made to combine withgeneral relativity, and in that combinationone will see how to deal with singularities,because that is the place where the combi-nation of the two theories comes in. Also,there must be a time asymmetry in the wayit comes together, and that will explain thedifference between the past and the futuresingularities. But all these things arehopes � they�re not something I can donow.

Twistor theory has been tremendously suc-cessful in applications within mathematics,but has it been helpful in understanding thenature of the physical world?Not very much, I would say. It�s rathercurious, but I would say that this is notunique to twistor theory. One sees it inother areas � like string theory, forinstance � where people start with greatambitions to solve the problems of physics,and instead come up with ideas that havehad implications within mathematics; thisis certainly the case with twistor theory, itsapplications and its interest. If you round-ed up all the people who claimed theyworked on twistor theory, you�d find, Iwould think, that a vast majority of themwere mathematicians with no particularinterest in physics � they might be interest-ed in differential geometry, or integrablesystems, or representation theory. Veryfew of them would have physics as their

prime interest, so it�s kind of ironic thathere�s a theory that�s supposed to beanswering the problems of physics, and yetit�s not caught on at all on the physics side.

You mentioned string theory. Are there con-nections between twistor theory and stringtheory?I think there probably are. It�s not some-thing that has been deeply explored, andthe groups of people who work on thesesubjects are more-or-less disjoint. Therehave been some attempts to bring the the-ories together, but I think that the rightvehicle for doing so hasn�t come about yet.I wouldn�t be at all surprised to find that inthe future some more significant linkbetween these two areas is found, but Idon�t see it right now.

These new theories involving p-branes seemto be more suitable, somehow?Well, there is a connection, but I don�tknow how significant it is. I was talking toEd Witten recently and he was tellingabout the 5-branes they�re interested in.But that�s curious, because in work thatMichael Singer did some years ago withAndrew Hodges and me, the suggestionwas made that what one should really belooking at is generalisations of strings.Whenever you see an ordinary string, youshould really think of it as a surface,because it�s a string in time. It�s one-dimensional in time, so that gives you twodimensions. These things are studied verymuch in connection with complex one-dimensional spaces (Riemann surfaces), sothey are in some natural way associatedwith these Riemann surfaces.

Now what we had in mind, which wasmuch more in line with twistor theory, is tolook at a complex three-dimensional ver-sion of this, which we called pretzel twistorspaces; they�re complex three-dimensionalspaces, so they are six real-dimensional,and if you can think of them as branes insome sense, then they are 5-branes. Nowis there a connection between those 5-branes and the 5-branes of the string theo-ry? I just don�t know, and I haven�texplored it. I didn�t mention it to Wittenwhen I talked to him, but there might besomething to explore here. That�s just offthe top of my head, I don�t know, but yes,it might be that there�s a connection there.

INTERVIEW


In 1964 the Russian mathematician Vadim G.Vizing published, in a Siberian journal., apaper with the title �On an estimate of the chro-matic class of a p-graph� (in Russian). Its mainresult is a theorem that today can be found inmost textbooks on graph theory. Vizing is nowone of the best-known names in modern graphtheory. In 1976 he initiated the study of �listcolourings�, a topic that has received muchattention recently.

In October 2000 Vizing visited the University ofSouthern Denmark in Odense, where the follow-ing interview was conducted by Gregory Gutin(Royal Holloway College, University ofLondon) and Bjarne Toft (University ofSouthern Denmark).

Where did you grow up, and where did youget your education?I was born on 25 March 1937, in Kiev inUkraine. After the war, when I was 10, myfamily was forced to move to theNovosibirsk region of Siberia because mymother was half-German. I started tostudy mathematics at the University of

Tomsk in 1954 and graduated from therein 1959.

I was then sent to Moscow to the famousSteklow Institute to study for a Ph.D. Thearea of my research was function approxi-mation, but I did not like it. I asked mysupervisor for permission to do somethingelse, but was not allowed to change. So Idid not finish my degree and returned toNovosibirsk in 1962.

From 1962 to 1968 I spent a happy peri-od at the Mathematical Institute of theAcademy of Sciences in Academgorodoc,outside Novosibirsk. In 1966 I obtained aPh.D. I did not have a formal supervisor,but A. A. Zykov helped me.

Because of the very cold climate I want-ed to move back to Ukraine, but I couldnot get permission to live in Kiev. Afterliving in various provincial towns, I finallyended up in Odessa, where I taught math-ematics at the Academy for FoodTechnology from 1974.

How was life in Academgorodoc in the1960s?It was nice and quiet, and the atmospherewas good. Zykov let me present my resultsin his seminar, and he became my friend.And later the place attracted some verygood students, like Oleg Borodin,Alexander Kostochka and Leonid S.Melnikov.

What made you choose mathematics in thefirst place?Because I was not happy doing anythingelse!

How did you conceive the idea of yourfamous theorem?In Novosibirsk I started to work on a prac-tical problem that involved colouring thewires of a network. To solve the problem Istudied a theorem of Shannon from 1949(that the edges of any p-graph can becoloured in 3d/2 colours). ThroughShannon�s theorem I got interested inmore theoretical questions.

Shannon�s Theorem is best possible forp-graphs in general, but I asked myselfwhat the situation would be for graphswithout multiple edges. I improvedShannon�s bound stepwise. At one point Ihad something like 8d/7, but eventually Iproved the best possible result, d + 1. Thenext step was to consider p-graphs.

I sent the graph result to the prestigiousjournal Doklady, but they rejected it. Thereferee said that it was just a special case ofShannon and not interesting. They didnot understand it. So I published it locally

in Novosibirsk in Metody Diskret. Analiz. Itappeared in 1964 when I had also solvedthe p-graph case. By this time the resulthad already been mentioned in the Westwhen Zykov stated it in the proceedings ofa meeting in Smolenice that was publishedjointly by the Czechoslovak Academy ofSciences and Academic Press.

Did you expect that your result that wouldeventually find its way into almost all bookson graph theory?No! And I did not consider that the topichad reached its final form by 1964. Forexample, I looked for algorithms and hadmany open problems. In 1968 I publisheda paper on �Some unsolved problems ingraph theory� (English translation inRussian Math. Surveys, 1968), summarisingthese and many other problems. Some arenow classical and still unsolved, like thetotal graph colouring conjecture (posedindependently by Behzad).

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EMS December 200022

InterInterview with Vview with Vadim G. Vadim G. Vizingizinginterviewers: Gregory Gutin and Bjarne Toft

Vizing�s Theorem (1964). The edges of agraph with maximum degree d can becoloured in at most d + 1 colours so that notwo edges with a common vertex are colouredthe same. Moreover, the edges of a p-graphwith maximum degree d (where any two ver-tices are joined by at most p edges) can becoloured in at most d + p colours so that notwo edges with a common vertex are colouredthe same.

Vadim Vizing in 1965

Vadim Vizing in 1975

The Total Graph Colouring Conjecture(Vizing 1964, 1968, Behzad 1965). Thevertices and edges of a graph with maximumdegree d can be coloured in at most d + 2colours so that no two adjacent or incidentelements are coloured the same. Moreover,the vertices and edges of a p-graph withmaximum degree d can be coloured in atmost d + p + 1 colours so that no two adja-cent or incident elements are coloured thesame.

What makes a mathematical result out-standing?A mathematician should do research andfind new results, and then time will decidewhat is important and what is not!

What were the most interesting periods inyour scientific life?Definitely my years in Novosibirsk, when Iworked in graph theory. And now, beingable to do research again with time tothink about unsolved problems. TheINTAS grant from the European Unionhas helped. [The INTAS grant is a 3-yeargrant initiated by the Technical Universityof Ilmenau, with participation fromOdense, Nottingham, Odessa andNovosibirsk.]

How?I have retired. My pension is around $70per month. This corresponds almost tomy earlier salary, for which I had to teachup to 20 hours per week. I earned someextra money by writing a mathematicsbook for those wanting to pass a universi-ty entrance exam. The INTAS grant nowgives me $45 extra per month, and itmakes it possible for me to travel andmeet colleagues. Last year in August wehad an interesting meeting inNovosibirsk.

Have you carried out research during youryears in Odessa?In 1976 I stopped my graph theoryresearch and moved to scheduling. I waswriting a habilitation thesis and finished itin 1985. It did not work out, more forpolitical and economical reasons than forscientific. It was partly my own fault. Icould submit it now, but my interests havechanged and I would rather use my timeon something more useful.

In 1995 I was invited to Odense for thefirst time. This motivated me to go back

to graph theory. During my stay I solveda problem with Melnikov that was laterpublished in the Journal of Graph Theory.

What did you like least before Perestroika?At first we had a police state. Then itbecame bureaucratic. It ended in econom-ic failure.

How has life changed in Ukraine afterPerestroika?In general, the direction is positive, butthere are many negative aspects also.

There is only little social protection. Thebureaucratic system has survived, but nowwithout control. This has led to open cor-ruption.

However, the market economy is devel-oping. Consumer goods are easily avail-able if you have money. If you are healthyand energetic you can earn much moreand live better than before. I like the gen-eral development. Of course there aremany mistakes, but the direction is right.

How often have you travelled outside theformer Soviet Union?Three times, all of them to Denmark.Before Perestroika I had many invitations,more than twenty, but I was never allowedto go, not even to other socialist countries.I tried twice, but was stopped. It washopeless. Very few people from theUkraine could go. From Novosibirsk itwas perhaps easier, but from Ukrainealmost impossible, especially if you werenot from Kiev. You were looked uponwith suspicion if you wanted to travel.When I received foreign letters they wereopened by the KGB and afterwards sent tome privately without comment.

What was your relationship to the commu-nist party?I was asked to join, but I never wanted todo so for political and moral reasons. Idid not want to lose the freedom I had. Iam glad that I did not join, even thoughmy life in some ways might have been eas-ier as a member.

What are your research plans?To work on graph-theoretical questions.But great discoveries are not planned. Iwill work and see where I get!


INTERVIEW

Cover page of Discret Analiz.

Vadim Vizing at Odense with Gregory Gutin and Bjarne Toft

First page of Vizing�s 1964 paper

Scotland has produced many creative andinfluential mathematicians � one thinks ofJames Gregorie, James Stirling, ColinMaclaurin and their many successors � butarguably the greatest and most original ofall was the first Scottish mathematician ofinternational renown, John Napier, whowas born 450 years ago. Napier was indeedthe first Scottish mathematician that weknow about, and it is extraordinary that hecreated mathematics of the highest qualityfrom within a country with no other math-ematicians, with no mathematical tradi-tion, and plunged into religious, politicaland social feuding. As his descendant MarkNapier wrote in 1834:

As for Scotland, until Napier arose, it wasonly famed for mists that science could not pen-etrate, and for the Douglas wars, whose baro-nial leaders knew little of the denary systembeyond their ten fingers.

Born in 1550, Napier was the eldest sonin a wealthy and well-connected family whohad been playing an increasingly impor-tant part in Scottish court and civic lifeover the hundred years leading up to hisbirth. His parents Sir Archibald Napierand his wife Janet Bothwell were bothbarely 16 when their son was born, andfrom the start John Napier was living in anatmosphere of political and religious dis-putation and intrigue: the ScottishReformation was in full spate and SirArchibald was strongly on the Protestantside, as his son was to be. This wasn�t mere-ly a theological but also a political-cumconstitutional position, given the swirl ofintrigue surrounding the Catholic QueenMary, James V�s daughter, and herProtestant-inclining son James VI (as hebecame).

At the age of thirteen, young John wassent to the University of St Andrews, wherehe lodged with the principal, JohnRutherfurd, and where he tells us he devel-oped his theological interests and stronglyanti-Papist views. There is no record ofNapier graduating from St Andrews, and itis supposed that he probably went to studyabroad, as was fashionable among youngScots of his generation and class. He maywell have studied in Paris, where he wouldhave had an opportunity to develop hismathematical knowledge, and perhaps inGeneva too, where he could have learnedGreek in a fiercely Protestant environ-ment.

His being out of the country during thelatter 1560s meant that he missed theexcitements at the Scottish court such asthe murder of Queen Mary�s secretaryDavid Rizzio, the murder of the Queen�shusband Lord Darnley, the Queen�s mar-riage to the Earl of Bothwell (the weddingceremony being performed by Napier�suncle the Bishop of Orkney), the forced

abdication of Mary not long afterwards,and the coronation of her son James VIwhich helped mark the Protestantisation ofScotland. The next we hear of Napier him-self is in the early 1570s. His father remar-ried in 1570 (Napier�s own mother haddied shortly after he went to St Andrews),and Napier himself married ElizabethStirling in 1573, receiving the Merchistonestate from his father as part of the wed-ding settlement.

There are five books in Napier�s textualcorpus, which were all first printed inEdinburgh:� Napier�s first and indeed best selling

book in its day was A plaine discovery of thewhole Revelation of St John, published in1593. This anti-papist tract made his repu-tation as a leading theologian, and wentinto numerous editions in many languages. � His next book, which did not appear for

another twenty-one years, was on a quitedifferent subject. Mirifici logarithmorumcanonis descriptio, of 1614, �Descriptio� forshort, was the book that introduced loga-rithms to the world and established hisreputation among mathematicians acrossEurope.� His next book, in 1617, the year he died,

was called Rabdologiae. This was notabout logarithms but about other devicesand means of calculation.

� Two years after Napier�s death, in 1619,his son Robert brought out from hismanuscripts a companion work, as itwere, to the 1614 Descriptio, calledMirifici logarithmorum canonis constructio,�Constructio� for short, which explainedhow logarithm tables were constructed.

� Finally, 220 years later, another descen-dant, Mark Napier, edited more of hispapers under the name of De arte logisti-ca (1839).

[First editions of the Descriptio, Rabdologiaeand De arte logistica, as well as early editionsof the other two, were in the TurnerCollection at Keele University, UK, beforethat university secretly sold off the collec-tion to a second-hand book dealer for amess of pottage (see EMS Newsletter 31,pp.10-12, and 32, pp.14-15.)]

Napier�s fame in his own day was as theauthor of A plaine discovery of the wholeRevelation of St John. This remarkable best-seller explains such pressing issues as justwhy the Pope is the Antichrist and how weknow that judgement day will fall between1688 and 1700. It is worth more attentionfrom historians of mathematics than it hasreceived, if less for its conclusions then forthe process by which he reaches andexplains those conclusions.

Given the assumption that the text of thebook of Revelation contains predictionsabout the subsequent course of human his-tory � which is not an unfair inferencefrom the opening words:

The revelacion of Jesus Christe, which Godgave unto him, for to shewe unto his servauntes

thynges which must shortly come to passe . . .Happy is he that redith, and they that heare thewordes of the prophesy . . . [Revelation Chapter1, Tyndale translation]� and given that in the succeeding 1500 orso years some of the predicted events musthave happened, then this gives clues abouthow to match up the language of predic-tion with the historic record. So whatNapier was seeking to establish was a func-tion, if you like, between two continua: thehistoric time-line from the time of Christonwards, and the narrative time-line of StJohn�s vision as presented in theApocalypse which is being mapped onto it.To evaluate the functional correlation, hehad to make considered judgements aboutwhat trumpets are, what seals are, whatcandlesticks are, and so on, the conclusionsof which he presented in a series of 36numbered propositions. Once the functionis established, from the information aboutthe past which you have, you are then in aposition to use the correlation to work outthe things you don�t know � in particular,the date of the last judgement.

I�ve described Napier�s procedure in hisPlaine discovery in this functional way inorder to point up the similarities with whathe was later doing in constructing loga-rithms. Napier constructed logarithmsthrough considering two moving points, Pand L, say, moving along a finite and aninfinite line respectively, in such a way thatwhile L is moving at constant speed, inarithmetical progression, P is moving geo-metrically, it�s slowing down, in the origi-nal construction, its speed being propor-tional to the distance it still has to go. Thenhe defined the logarithm of the distance Phad still to go as the distance the otherpoint L has travelled. The idea that multi-plication of terms in a geometric progres-sion correspond to addition of terms in anarithmetical progression had long beenfamiliar, from Greek times if not earlier.The fresh insight that Napier brought wasto situate this in two continuous move-ments � he even uses the word �fluxion� atone point � so that he could make infer-ences about one from what happened onthe other.

So in very broad terms, both the Plainediscovery and Napier�s construction of loga-rithms involve functional relationshipsbetween two continua, using informationfrom one to make deductions about theother. It might be ill advised to push thisparallel too far, but both are examples ofNapier�s overwhelming characteristic, hislateral thought in the service of makingcalculations easier. In some ways he was acomputationalist, a calculator, even morestrongly and more pervasively than he wasthe inventor of logarithms.

Part of his subsequent success and fameechoing down the ages is due to luck. It wasamazing good fortune, which he could nothave anticipated, that logarithms turnedout in the course of the century after his

ANNIVERSERIES

EMS December 200024

2000 Anniversaries2000 AnniversariesJohn Napier (1550-1617)

John Fauvel

ANNIVERSERIES


death to be not only a calculating devicefor astronomers and navigators � doublingthe life of astronomers, as Laplaceremarked � but also central to the devel-opment of mathematics itself. Already bythe 1650s and 1660s it was becoming clear

that logarithms were much deeper mathe-matical objects than their initial motivationmight suggest, relating to the area mea-sure of hyperbolas, and thus a vital tool forthe integral calculus, as well as beingthought of as an infinite series, whichopened up another great swathe of mathe-matical analysis. The fact that we still teachthe logarithm function to young peoplewho wouldn�t have a clue how to use loga-rithm tables or even what they are for,indicates how Napier�s invention has tran-scended its original use and purpose.

Napier�s genius was fundamentally thatof an amazingly gifted and innovative cal-culator. His all-pervading interest in calcu-lating showed itself especially strongly inhis remarkable little book Rabdologiaewhich appeared in 1617, the year of hisdeath. This described three inventions foraiding calculations, the so-called Napier�srods, the promptuary, and a chessboard abacus.

Napier�s rods, or Napier�s bones, are a phys-ical realisation of an old method of multi-plying numbers, known since the middleages in Europe and maybe in India long

before that. Thus the concept was old butits physical realisation was new, demon-strating Napier�s lateral technologicalthinking. Rabdologiae explains how to con-struct the rods as well as how to use themfor multiplication and division, taking

square roots and cube roots, and doing therule of three. These became very popularand there are still many sets of rods, gen-erally in wood or ivory, in our museums.

The promptuary was a more complicatedand more powerful modification of therods, enabling ready handling of muchlarger numbers. It uses flat cards ratherthan rods, but with rather similar markingsand factorings. It is sufficiently sophisticat-ed that it has been called �the first calculat-ing machine�, though it�s not quite amachine as we usually understand theterm, its operation depending on quite alot of manual manipulation. The onlyknown example of a promptuary from thetime of Napier is in the ArcheologicalMuseum in Madrid, and was only recog-nised for what it is in the last twenty years.

The third of Napier�s calculationaldevices, his chessboard abacus, is the mostinnovative and of greatest conceptualinterest, even though he described it as�more of a lark than a labour�. The funda-mental insight is that multiplication ofbinary numbers is more straightforward

than multiplication in base 10, which ofcourse computers got around to realising350 years later. So in Napier�s proceduredecimal numbers are converted into bina-ry, the operation is carried out (multiplica-tion, division or whatever) and then theresult is converted back into a decimalnumber. Notice two things. One is that thistransformation of base is really quite radi-cal and innovative � no-one else had donethis kind of thing before. The other thingyou might notice is that the process of con-verting into different numbers, carryingout your operation and then coming back,is structurally the same as the logarithmicprocedure; and indeed, one might argue,of his theological procedures.

Why do we remember John Napier? Hisdeep significance may be that, along withothers of his time, Napier was a central fig-ure in the transformation of the mediaevalinto the modern world-view, in a very spe-cific way arising from his deep concern forcomputation and calculating effectiveness.We know the immediate context of loga-rithms and why they were taken up sowidely and so rapidly: the need for ways ofdoing mathematical calculations wasbecoming evident to the navigators andothers who were beginning to lay the foun-dations for the British imperial adventure.For some years, a century or more, it wasincreasingly clear that European expan-sion, geographically, in military engineer-ing, in terms of trade and business prac-tices, was predicated upon better mathe-matical skills. Napier happened to beworking at a time when the idea of quan-tification was settling deep into the mind-set of the movers and shakers ofRenaissance Europe, and supplied a num-ber of justifications for considering thathow you handle and compute with num-bers is a really important issue. In someways there was nothing else like this con-ceptual revolution in the applicability ofmathematics to the world until the statisti-cisation of inquiries in the 19th century.BibliographyBaron, Margaret, �John Napier�, Dictionary of scientificbiography ix, Charles Scribner�s Sons, New York, 1974,609-613. Bryden, D. J., Napier�s bones: a history and instructionmanual, Harriet Winter Publications, 1992.Fauvel, John, �Revisiting the history of logarithms�,Learn from the masters! (ed. F. Swetz, J. Fauvel et al.),Mathematical Association of America, 1995, 39-48.Knott, Cargill Gilston (ed.), Napier tercentenary memorialvolume, Edinburgh, 1915.Macdonald, William Rae, A catalogue of the works of JohnNapier, pp.101-169 of his translation of Napier�sConstructio, Edinburgh, 1889. Napier, John, Rabdology (tr. W. F. Richardson), MITPress, 1990.Sherman, Francis, Flesh and bones: the life, passions andlegacies of John Napier, Napier Polytechnic, 1989.

John Fauvel is Senior Lecturer in Mathematicsat the Open University, UK. This article isbased on a lecture to commemorate the 450thanniversary of John Napier�s birth, given atNapier University, Edinburgh, on 1 December2000. The event was organised jointly byNapier University and the International Centrefor Mathematical Sciences, Edinburgh.

The UMI (Italian Mathematical Union)was established in 1922. On 31 March ofthat year, Salvatore Pincherle, an eminentmathematician from Bologna, circulated aletter to all Italian mathematicians inwhich a possible national mathematicalsociety was proposed. In July a tentativeissue of the future Bollettino was published.About 180 mathematicians supported theproposal, and in December the first meet-ing was held and the first of the Society�sby-laws received its approval. The UMImembership swelled to 400 by 1940 andcurrently stands at over 2700. The regis-tered office remains in Bologna, in theDepartment of Mathematics.

The goals of the UMI were to communi-cate the nature of the mathematical sci-ences and how mathematics contributes tosociety, and to promote the understandingof mathematics by publishing high-levelpapers, by encouraging the output of high-quality expositions for members and stu-dents at all levels, and by organising meet-ings in order to stimulate the productionand the exchange of ideas.

The formation of the UMI was inspiredby the existence and the increasing growthof other mathematical societies, such as theSociété Mathématique de France (1872),the Deutsche Mathematik Vereinigung

(1891), the American Mathematical Society(1891) and, particularly, the InternationalMathematical Union (1920).

Before long, distinguished members,among them Luigi Bianchi and VitoVolterra, strongly supported Pincherle�sinitiative and they were presenting papersin the first issues of the formative journal.

One of the major achievements of theUMI during this period was the organisa-tion of the meeting of the InternationalMathematical Union (IMU) in Bologna in1928. Not only was the organisation car-ried out effectively, but a difficult politicalquestion was successfully confronted.Salvatore Pincherle, elected IMU

President in 1924, had successfully workedto get together all those with a keen inter-est in mathematics, despite their nationali-ty, by overcoming a consequence of theFirst World War. A strong difference ofopinion existed between French mathe-maticians and the mathematicians of othercountries, mainly concerning the invitationto the German delegation. In the event,around 840 mathematicians assembled inBologna, and besides 336 people fromItaly there were 76 mathematicians fromGermany, 56 from France and 52 from theUnited States. The opening address was inLatin.

Currently the UMI organises a generalmeeting every four years, with a wide turn-out of mathematicians from Italy and over-seas, in one of the cities where there is amathematics department. Under theinfluence of Pincherle, the Bollettinodell�Unione Matematica Italiana was found-ed: at the beginning there were two sec-tions devoted to �Short communications�and �Abstracts of papers published in otherjournals, Letters to the Society, News fromthe members, Book reviews, etc.�. In 1939a special section devoted to the history ofmathematics and to mathematical educa-tion was introduced.

Since then the Bollettino has increasedsubstantially the number of papers of out-standing scientific quality. At presentthere are two sections of the Bollettino(Sections A and B) in which expositorypapers and high-level scientific papers arepublished.

Recent years have seen a huge increasein the diversity of activities in which the

Unione is involved, from education, to pop-ularisation, to institutional policy. TheUMI has encouraged, and will continue todo so, the participation of research mathe-maticians in the reform of mathematicseducation at university level and at anyhigh-school level. The UMI, in coopera-tion with the MPI (Italian Department ofEducation) and the MURST (ItalianDepartment of University and Research),supports efforts to review and reform theundergraduate mathematics curriculum inresponse to current changes in the world.

Traditionally the UMI supports, as aninstitutional task, the Committee for

SOCIETIES

EMS December 200026

LL�Unione Matematica Italiana�Unione Matematica ItalianaGiuseppe Anichini

Four presidents: (from left to right) A. Figà-Talamanca, C. Pucci, E. Magenes and C. Sbordone

Mathematics Education, the Committeefor Research and the Education ofMathematics in Engineering, and theCommittee for the Mathematical Olympicgames.

Prizes and awardsSince the sixties the Unione has sought topromote and reward mathematicalachievements, mainly for young people, bymeans of prizes and awards. Currentlyfour prizes are awarded by the UMI.These prizes are awarded on the recom-mendation of a Committee especiallyappointed by the Ufficio di Presidenza(Officers of the Society) of the UMI.

The Premio Renato Caccioppoli was estab-lished in 1960 with a donation by his fam-ily in memory of Renato Caccioppoli, lateProfessor of Mathematical Analysis at theUniversity of Napoli. This prize is award-ed every four years in recognition of anoutstanding contribution to mathematicalanalysis by an Italian mathematician of nomore than 38 years old. The prizeamounts to ten million Italian lira (approx-imately 5200 euros).

The Premio Giuseppe Bartolozzi was estab-lished in 1969 with a donation by ProfessorFederico Bartolozzi and his family in mem-

ory of his son Giuseppe. This prize isawarded every two years in recognition ofan outstanding contribution to mathemati-cal research by an Italian mathematician ofno more than 33 years old. The prizeamounts to three million Italian lira(approximately 1500 euros).

The Premio Franco Tricerri was estab-lished in 1995, using funds collected bycolleagues, friends and students of FrancoTricerri, Professor of Geometry at theUniversity of Firenze, who tragically diedin a plane accident in China in 1994. Thisprize is awarded every two years in recog-nition of an outstanding contribution todifferential geometry by a graduate of notmore than 3 years� standing in mathemat-ics or physics. The prize amounts to twomillion Italian lira (approximately 1000euros).

The Premio Calogero Vinti was establishedin 1998 with a donation by the family andformer students of Calogero Vinti,Professor of Mathematical Analysis at theUniversity of Perugia. This prize is award-ed every four years in recognition of anoutstanding contribution to mathematicalanalysis by an Italian mathematician of nomore than 40 years old. The prizeamounts to eight million Italian lira

(approximately 4100 euros).

PublicationsApart from the Bollettino, several otherpublications are produced by the UMI.Since 1974 the Notiziario dell�UMI hasappeared. This presents news of interest,about meetings, prizes and awards, educa-tion, Ph.D. achievements, and so on.There are ten issues each year, with sup-plements on special occasions.

The Unione Matematica also edits a seriesof Quaderni: a series of textbooks for youngresearchers aimed at arguments outsidethe usual mathematical path to the Ph.D.degree; a series of monographs covering awide range of subjects in mathematics; aseries of Opere dei Grandi Matematici includ-ing all (or a selection) of the papers of well-known Italian mathematicians. Amongthese have been Felice Casorati, PaoloRuffini, Luigi Bianchi, Leonida Tonelli,Ulisse Dini, Giuseppe Peano, GregorioRicci-Curbastro, Vito Volterra, ErnestoCesaro, Corrado Segre, Guido Fubini,Giuseppe Vitali, Renato Caccioppoli andSalvatore Pincherle.

The structure and the membersThe UMI has an Executive Committee offour elected members: the President, theVice-President, the Treasurer and theSecretary. The Scientific Committee con-sists of 19 members: those of the ExecutiveCommittee and 15 other elected members;elections take place every four years. TheScientific Committee often nominates spe-cial committees for specific reasons (math-ematics education, Publications, the teach-ing of non-degree-level mathematics, etc.).Of the 2700 members, many are universityresearchers, while many others are school-teachers or belong to industries or to pub-lic research centres.

Finally we recall the Presidents of theUnione.

After Salvatore Pincherle, the foundingPresident, the Presidents of the UMI havebeen Luigi Berzolari, Enrico Bompiani,Giovanni Sansone, Alessandro Terracini,Giovanni Ricci, Guido Stampacchia,Enrico Magenes, Carlo Pucci, VinicioVillani, Alessandro Figà Talamanca andAlberto Conte; the current President isCarlo Sbordone. In addition, EnricoBompiani (1952-75) and Carlo Pucci (from1995) were appointed HonoraryPresidents by plenary meetings of themembers.

The author is very indebted to the fol-lowing papers for information:Carlo Pucci, L�Unione Matematica Italianadal 1922 al 1944: documenti e riflessioni,Symposia Mathematica XXVII, IstitutoNazionale di Alta Matematica FrancescoSeveri, Roma, 1992, 187 pages.Giovanni Sansone, Le attivit dell�UnioneMatematica Italiana nel primo cinquanten-nio della sua fondazione, Bollettino UMI,suppl. fasc. 2, Bologna, 1974, 8 pages.

Giuseppe Anichini is Professor of MathematicalAnalysis in the Engineering Faculty of Firenze,Italy, and has been Secretary of the UnioneMatematica Italiana since 1988.

SOCIETIES


Professor Carlo Pucci, Honorary President

In this article we report on the EULERproject, which has developed a web basedsearch engine for distributed mathematicalsources. The main features of this EULERprototype are uniform access to differentsources, high precision of information, de-duplication facilities, user-friendliness andan open approach enabling participationof additional resources. We describe thefunctionalities of the EULER enginereport on the transition from the proto-type developed in the project to a consor-tium-based service in the internet.

Aims and achievementsThe aim of the EULER project was to pro-vide a system for strictly user-oriented,integrated-network-based access to mathe-matical publications. The period for theproject terminated in September 2000.The EULER system has been designed tooffer a �one-stop shopping site� for usersinterested in Mathematics. An integrationof all types of relevant resources has beentaken into account: bibliographic databas-es, library online public access catalogues,electronic journals from academic publish-ers, online archives of pre-prints and greyliterature, and indexes of mathematicalInternet resources. They have been madeinteroperable, using common Dublin Corebased metadata descriptions. A commonuser interface, called the �EULER engine�,assists the user in searching for relevanttopics in different sources in a single effort.As a matter of principle, the EULER sys-tem has been designed as an open,scaleable and extensible information sys-tem. Mathematicians and librarians frommathematics in research, education andindustry will be the main users andproviders of such an enterprise.

EULER is an EMS initiative and espe-cially focuses on real user needs. The pro-ject has been funded by the EuropeanUnion within the programme �Telematicsfor Libraries�. Standard, widely used andnon-proprietary technologies such asHTTP, SR/Z39.50 and Dublin Core (DC)are used. Common resource descriptionsof document-like objects enable inter-operability of heterogeneous resources.One of the main achievements of the pro-ject is the development of a DC-basedmetadata structure that can be used as acommon target into which the metadata ofthe given resources could be converted.

At distributed servers, multi-lingualEULER service interfaces are provided asentry points to the EULER engine, offer-ing browsing, searching, some documentdelivery and user support (help texts, tuto-rial, etc.). The interface is based on com-mon user-friendly and widely used webbrowsers (public domain or commercial),such as Netscape. The multi-lingual userinterface has the common features of every

good Internet service and a self-explainingstructure. Users have one single entrypoint to start their information search; thisentry point contains browsing indices ofauthors and keywords, form-based search-es for authors, titles and other relevant bib-liographic information, and a selection ofdifferent information sources. Good de-duplication facilities enable to display theavailability of the same item at differentsites within the same record.

The partners of the EULER projectThe currently accessible contents in theEULER prototype are provided by thepartners of the project. This groupincludes libraries from all over Europe,which represent several different types oflibraries: the State Library of LowerSaxony and University Library ofGöttingen and the J. Hadamard library(University of Orsay) represent librarieswith a national responsibility for collectingall publications in pure mathematics; thelibrary of the Centrum voor Wiskunde enInformatica (Amsterdam) represents thetypical research library of a nationalresearch centre; the University of Florencerepresents a typical university library withits distributed department libraries; thelibrary of the Institut de RechercheMathématique Avancée (University ofStrasbourg) represents a typical library ofan important mathematical institute.

In addition, a partner specialised in dig-

ital libraries and net-based information isrepresented by NetLab, the Research andDevelopment Department at LundUniversity Library: they give a large set ofclassified internet resources, complement-ing a similar collection, the �Math Guides�,organised by the Göttingen Library.MathDoc Cell (Grenoble), as a nationalcentre for coordination and resource-shar-ing of mathematics research libraries inFrance, contributes also in giving metada-ta of its national indexes on preprints andthesis. Via the partner FIZ Karlsruhe,Zentralblatt MATH provides a part of its

contents as a freely accessible resource inthe project. The EMS provides itsElectronic Library of Mathematics as aresource, distributed through its EMIS sys-tem of Internet servers; scientific co-ordi-nation of this library is currently organisedwith the Department of Mathematics of theTechnical University of Berlin, the finalpartner of the EULER project.

The EULER serviceBased on the structure described above, asubgroup of the current EULER partnersdecided to develop a service from theEULER prototype. It had been guaran-teed during the project work that theEULER engine and other tools could beinstalled at new sites, and thus the groupwas able to go on with the current offerfrom EULER. It is expected that allresources made accessible during the pro-ject phase will remain open for the service,even if the corresponding partner cannotcontribute further work to run the service.The current members of the group careabout improvements to the EULER offerand handle software problems potentiallycoming up with new partners. In particu-lar, the administration of access control(for resources beyond free metadata) is achallenge for the future. This will lead toan essential improvement of the documentdelivery facilities, and will motivate scien-tific publishers to support the system andmake their contents accessible throughEULER.

But also as a free search portal EULERwill provide a very useful gateway to math-ematics. The present aim is to get morelibraries interested in participating inEULER. This means that they shouldmake their catalogues accessible by provid-ing the metadata of their holdings. TheEULER engine will include their resourcein the searches made by the user, and userswill get a bigger choice of providers wherethey may ask for a copy of the documentsthey are interested in. The de-duplicationcheck will provide them with comprehen-sive lists of where to find the documents (abook, journal article or other source), andhaving a good coverage of the mainlibraries in Europe, they will probably get areference for where they will be offered theresource at very low cost.

Several additional libraries are alreadyinterested in discussing participation inthe EULER service. A definite decisionwill take some time and the preparation ofthe metadata possibly even more, but mostof these first contacts are very promising.The first bricks of a European catalogue ofmathematics resources in the libraries havebeen installed, while others will be added.

Access to EULER is available throughEMIS, the Information Service of theEuropean Mathematical Society on theweb (with 39 servers world-wide, seewww.emis.de).

Laurent Guillopé (e-mail: [email protected]) is at the Université deNantes; Bernt Wegner (e-mail: [email protected]) is at the Technische UniversitätBerlin.

EULER PROJECT

EMS December 200028

The EULER project: achievements and continuation

Laurent Guillopé (Nantes) and Bernd Wegner (Berlin)

Bernt Wegner with Einstein

The material given here was presented by theauthor in a talk at the annual assembly of thechairmen of German Math departments(KMathF) in May 2000, and also in a talk atthe DMV Jahrestagung in September 2000,both in Dresden; see the website http://w w w . m a t h e m a t i k . u n i - b i e l e f e l d .de/\char126rehmann/BIB/

In common with scientists of other disci-plines, many of us mathematicians are,concerned about the rapid price increasefor scientific journals. Recently a majorreduction in the library budget for theMathematical Library at BielefeldUniversity forced me, as the personresponsible for our departmental library,to take some measures to decide whichjournals we should cancel. Since manydepartments are in a similar position, Ithink it may be useful to publicise theinformation that I gathered for my depart-ment.

As a first step I listed all the mathemati-cal journals at Bielefeld, in a table on theWeb, including the publishers, the 1998price, and also some information from thecitation index ISI: http://www.isinet.com/,such as the number of citations of thatjournal and the average impact of eacharticle, insofar as this information wasavailable to us.

With a little perl script, I made this listmore transparent by ordering it withrespect to various data: by publisher, byprice, or by ISI number, so by a mouseclick, I could locate the most expensivejournals, those with the strongest impactfactor, and so on. I then made this listpublic, not only to my department, but alsoto some colleagues and librarians world-wide, on the website http://www.mathe-matik.uni bielefeld.de/\char126rehmann/BIB/,and received a good response. In particu-lar, I learned that at the same time theAmerican Mathematical Society (AMS) hadcollected data on about 250 journals,including their respective numbers ofpages and prices for the years 1994-99: seethe website . Since that table was not verytransparent at a first glance I decided, withpermission from the AMS, to extract thedata in a similar way as I did with theBielefeld list, using some perl script to docomputations of derived data, such as theprice per page and the price increases overthe years, and also to sort the table accord-ing to these data.

These tables contained some surprises. Ilearned that these 250 journals published323,786 pages of refereed mathematics in1999. I also learned to my great surprisethat many journals had an average annualprice increase of 15% or more during thelast five or six years. This inflation of 15% or

more per year was during a time, when, in thewestern world, the average price inflation wasusually below 2% or so. And this is true both forthe price increase per volume and for the priceincrease per page!

My conclusion is that mathematiciansare funny consumers: they buy the materi-al which they produce themselves frompeople � the commercial scientific publish-ers � who do nothing other than distributethat material at prices that increase beyondany reasonable measure. Not only that:mathematicians work hard for the publish-ers, usually without pay, by acting as theireditors, collecting and refereeing thematerial written by their colleagues, and asauthors, by perfectly typesetting theirmanuscripts, leaving almost nothing to dofor the publishers but count their profits.

We have a really strange situation: itseems that serious people are willing toaccept such price differences. For exam-ple, consider the following information(provided by the publishers themselves): in 1999 �Inventiones Mathematicae� published2894 pages for US$2760, a price per page ofUS$0.95, while �Annals of Mathematics� pub-lished 2294 pages for US$220, a price per pageof US$0.10.

I chose these particular journals, since Ithink that they have a similar reputation.But when I mention these figures to col-leagues, many are surprised by the drasticprice difference. This is not an isolated sit-uation: checking the tables will show youseveral similar cases.

A typical pattern might occur to youwhen you scan these tables. Journals thatare cheap are very often produced bylearned societies or by universities, whileexpensive journals are produced by privatepublishers. (Using the word �produced�here is often an abuse, since I pointed outabove that the production is essentiallydone by the mathematicians themselves,while the publisher just does the distribu-tion.)

Another fact might strike you. Whateveryou might think about citation indices andimpact factors, at least they don�t seem toprovide any arguments for preferringhigh-priced journals above others. If theysuggest anything, it seems to be the oppo-site: if you click on the list ordered by�impact� (see the website http://www.mathem a t i k . u n i - b i e l e f e l d . d e / \ c h a r 1 2 6rehmann/BIB/impact.html, you will find atthe top many journals run by learned soci-eties or universities and offered at moder-ate prices.

This situation is no longer acceptable.So what is to be done? It is certainly nec-essary for us all to become better acquaint-ed with the facts concerning scientific pub-

lication: every mathematician should knowmore about journal prices. For that pur-pose I will, in accordance with the AMS,annually update the price tables as soon asnew data is available, and I hope this willhelp others to make the right decisionsconcerning their local library budget.

We also should take appropriate deci-sions ourselves when acting as author, ref-eree or editor, asking ourselves: Why arewe submitting to an expensive journal?Why are we refereeing for it? And if youare an editor, why are you not taking anymeasures to produce the journal by your-self?

Meanwhile, there are successful journalsrun by mathematicians themselves, such asGeometry and Topology, The Electronic Journalof Combinatorics, or DOCUMENTA MATH-EMATICA, just to mention a few amongmany. For example, I proved that, usingthe facilities of DOCUMENTA MATHE-MATICA it was possible to produce a seri-ous work such as the ICM98 Proceedings inshorter time, with better quality, and formuch less money, than most of the earlierproductions of ICM proceedings. To dosuch things is not hard nowadays, sincemany electronic tools are at our fingertips:I gave a public description of that produc-tion process at the Berkeley Workshop onThe Future of Mathematical Communication:1999, including financial details and thetechnical tools used: see the websiteh t tp : / /www.ma thema t i k .un i - b i e l e f e l d .de/\char126rehmann/EP/index.html. And Iguess most of our colleagues workingactively in the area of publication are will-ing to share their experiences and knowl-edge in order to support similar projectsby others.

PRICE SPIRAL


The PThe Price Spiral of Mathematics Journalsrice Spiral of Mathematics Journalsand What to Do About Itand What to Do About It

Ulf Rehmann (Bielefeld)

2001 anniversariesThe following mathematicians haveanniversaries during 2001.If you would like to write an anniver-sary article about any of them, pleasecontact the Editor.

Muhammad al-Tusi (b. 1201)Girolamo Cardano (b.1501)Pierre de Fermat (b. 1601)E. W. Tschirnhaus (b. 1651)

Mikhail Ostrogradsky and JuliusPlücker (b. 1801)

Carl Jacobi (d. 1851)Charles Hermite and Peter Guthrie

Tait (d. 1901)Richard Brauer, P. S. Novikov, I. G.

Petrovsky and A. Tarski (b. 1901)

The first collection of reviewed electronicgeometry models is available online at thenew Internet server http://www.eg-models.de[1]. This archive is open for any geometerto publish new geometric models, or tobrowse this site for material to be used ineducation and research. Access to the serv-er is free of charge.

The geometry models in this archivecover a broad range of mathematical topicsfrom geometry, topology and, to someextent, numerics. Examples are geometricsurfaces, algebraic surfaces, topologicalknots, simplicial complexes, vector fields,curves on surfaces, convex polytopes, andin some cases, experimental data fromfinite element simulations.

All models in this archive are reviewedby an international team of editors. Thecriteria for acceptance follow the basicrules of mathematical journals and arebased on the formal correctness of the dataset, the technical quality, and the mathe-matical relevance. This strict reviewingprocess ensures that users of the EG-Models archive obtain reliable and endur-ing geometry models. For example, theavailability of certified geometry modelsallows for the validation of numericalexperiments by third parties. All modelsare accompanied by a suitable mathemati-cal description. The most important mod-els will be reviewed by the Zentralblatt fürMathematik.

We are advocating the construction andsubmission of digital geometric modelsfrom various areas of mathematics. Theadvantages of these digital models gobeyond those of the classical plaster shapesand dynamic steel models of earlier days.At the end of the 19th century severalmathematicians felt the need to handlephysically the geometric objects theythought about. In particular, Felix Kleinand Hermann Amandus Schwarz inGöttingen built many models of curves,surfaces and mechanical devices for teach-ing and other educational purposes.

What are the main reasons for today�smathematicians to construct digital modelsof geometric shapes and make them avail-able via the EG-Models server? There areobvious educational aspects, as for the his-torical models, and the means of interac-tive visualisation are definitively useful forscientific purposes, too.

But the focus of this article is another,somewhat different, view. Nowadays com-puter generated or assisted proofs entervirtually all areas of mathematics, and stillthe majority of the mathematicians arereluctant to accept the validity of suchresults. On the one hand, it seems some-what strange to abstain completely fromusing tools such as the computer for doingmathematics, disregarding, maybe, aes-

thetic arguments. On the other hand, theinherent property of a proof is its verifia-bility; that is, verifiable by someone who issufficiently trained. But this very propertyof a proof might be challenged in individ-ual cases, where a computer is involved tosolve a task too arduous or too tiring forany human. We are not going to raise thegeneral question about the development ofthe mathematical culture, but we dobelieve that the installation of a server formathematical models can help to improvethe transparency of computer assistedproofs. For instance, think of a proof thatis established by a computer constructionof some complicated geometric shape. Astandardised description, independentlychecked by experts and available to every-one, would provide anenormous potential forvalidation.

Using the digitalmodels, interestedmathematicians canverify the claims ontheir own, using appro-priate software of theirchoice. Moreover,once there is a modelavailable, it is possibleto perform one�s owncomputational experi-ments on this data set.This could be a numer-ical evaluation as wellas a search for anotherproperty yet to beanalysed for thismodel.

Each model comeswith a detailed descrip-tion that identifies theauthor, explains themathematical purpose,and includes referencesto other sources ofinformation. Eachmodel has a uniqueidentification numberfor unambiguous cita-tion. Each model isequipped with quali-fied metadata informa-tion, and therefore, thearchive can besearched via special-ized search enginessuch as those fromEMIS and MathNet /.Each model itself isrepresented by a mas-ter file from a fixed setof file formats, includ-ing XML formats spec-ified by DTDs. Byrestricting the data for-

mats we want to ensure that the server�sinformation can be kept up to date on atechnical level. Additional files in arbitraryformats are welcome for explanatory pur-poses.

The Electronic Geometry Models Serveropened in November 2000.

References 1. Electronic Geometry Models, http://www.eg-models.de2. Udo Hertrich-Jeromin, Isothermic cmc-1Cylinder, Electronic Geometry Models, No.2000.09.038, DarbouxSphere_Master.jvx.3. Michael Joswig and Günter M. Ziegler, Aneighborly cubical 4-polytope, ElectronicGeometry Models, No. 2000.05.003,C45_Master.poly.

DIGITAL MODELS

EMS December 200030

Digital models and computerDigital models and computer-assisted pr-assisted proofsoofsMichael Joswig and Konrad Polthier

Darboux transform of a spherical discrete isothermic net [2].Given the data it is easy to verify that this describes an isother-mic surface. Additionally, it can be checked that this surface hasdiscrete constant mean curvature.

Schlegel diagram of a cubical 4-polytope whose graph is iso-morphic to the graph of the 5-dimensional cube [3].

Please e-mail announcements of European confer-ences, workshops and mathematical meetings ofinterest to EMS members, to [email protected]. Announcements should be written in astyle similar to those here, and sent as MicrosoftWord files or as text files (but not as TeX inputfiles). Space permitting, each announcement willappear in detail in the next issue of the Newsletterto go to press, and thereafter will be briefly noted ineach new issue until the meeting takes place, with areference to the issue in which the detailedannouncement appeared

8-18: ICMS Instructional Conference onNonlinear Partial Differential Equations,Edinburgh, UKInformation:Web site: http://www.ma.hw.ac.uk/icms/currrent

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

February � July: Random Walks specialsemester, Vienna, AustriaScope: the semester will be dedicated to vari-ous problems connected with stochasticprocesses on geometric and algebraic struc-tures, with an emphasis on their interplay,and also on their interaction with theoreticalphysicsTopics: some of the focal points are: proba-bility on groups; products of random matricesand simplicity of the Lyapunov spectrum;boundary behaviour, harmonic functions andother potential theoretic aspects; Brownianmotion on manifolds; combinatorial and spec-tral properties of random walks on graphs;random walks and diffusion on fractalsMain speakers: Alano Ancona (Paris), MartineBabillot (Orléans), Martin Barlow(Vancouver), Itai Benjamini (Rehovot), Robvan den Berg (Amsterdam), DonaldCartwright (Sydney), Davide Cassi (Parma),Thierry Coulhon (Cergy), Bernard Derrida(Paris), Persi Diaconis (Stanford), StevenEvans (Berkeley), Alex Furman (Chicago),Hillel Furstenberg (Jerusalem), RostislavGrigorchuk (Moscow), Geoffrey Grimmett(Cambridge), Yves Guivarch (Rennes), DavidHandelman (Ottawa), Pierre de la Harpe(Genève), Frank den Hollander (Nijmegen),Barry Hughes (Melbourne), Felix Izrailev(Puebla), Michael Keane (Amsterdam), YuriKifer (Jerusalem), Gregory Lawler (Durham),

February 2001

January 2001

Joel Lebowitz (New Brusnwick), FrancoisLedrappier (Paris), Russ Lyons(Bloomington), Gregory Margulis (NewHaven), Fabio Martinelli (Rome), StansilavMolchanov (Charlotte), Sergei Nechaev(Paris), Amos Nevo (Haifa), Yuval Peres(Jerusalem), Ben-Zion Rubshtein (Beer-Sheva), Laurent Saloff-Coste (Ithaca), OdedSchramm (Rehovot), Jeff Steif (Goeteborg),Toshikazu Sunada (Sendai), Domokos Szasz(Budapest), Balint Toth (Budapest), AnatoliVershik (St. Petersburg), George Willis(Newcastle, NSW) Programme: there will be two separate mainperiods of activity in the first(February/March) and in the second(May/June/July) halves of the semester. Thefirst period will start with a two-week work-shop with the general theme Random Walksand Statistical Physics, 19 February - 2 March2001. Towards the end of the second periodthere will be another two-week workshop withthe general theme Random Walks and Geometry,25 June - 6 July 2001 Organising committee: Vadim A.Kaimanovich (Rennes, France), Klaus Schmidt(Vienna, Austria), Wolfgang Woess (Graz,Austria)Site: Erwin Schrödinger InstituteInformation:e-mail: [email protected] Web site: http://www.esi.ac.at/Programs/rwalk2001.html

15-16: Workshop on Fractional BrownianMotion: Stochastic Calculus andApplications, Barcelona Speakers include: Coutin, Hu, Memin,Mishura, Nualart, Oksendal, Qian, Russo,Valkeila, ZaehleSite: Facultat de Matemàtiques, Universitat deBarcelonaInformation: Web site: http://orfeu.mat.ub.es/~gaesto/welcome.htm

15-19: Analytic Methods of Analysis andDifferential Equations (AMADE-2001),Minsk, BelarusTopics: integral transforms and special func-tions; differential equations and applications;integral, difference, functional equations andfractional calculus; real and complex analysisMain speakers: P. Adler (France), A.B.Antonevich (Belarus), A.E. Barabanov(Russia), H. Begehr (Germany), V.I. Burenkov(UK), L. de Castro (Portugal), I.V. Gaishun(Belarus), Yu.V. Gandel (Ukraine), H.-J.Glaeske (Germany), R. Gorenflo (Germany),V.V. Gorokhovik (Belarus), V.I. Gromak(Belarus), V.A. Il�in (Russia), N.A. Izobov(Belarus), N.K. Karapetyanz (Russia), A.Karlovich (Mexico), A.A. Kilbas (Belarus), V.Kiryakova (Bulgaria), V.I. Korzyuk (Belarus),M. Lanza de Cristoforis (Italy), A. Laurincikas(Lithuania), F. Mainardi (Italy), L.G.Mikhailov (Tadzhikistan), V.V. Mityushev

(Poland), E.I. Moiseev (Russia), A.M.Nakhushev (Russia), Yu.V. Obnosov (Russia),Ja.V. Radyno (Belarus), F. Rebbani (Algeria),M. Reissig (Germany), O.A. Repin (Russia),M. Stojanovich (Yugoslavia), J.J. Trujillo(Spain), N.A. Virchenko (Ukraine), Vu KimTuan (Kuwait), N.I. Yurchuk (Belarus), L.A.Yanovich (Belarus), P.P. Zabreiko (Belarus),E.I. Zverovich (Belarus)Languages: Russian, English Call for papers: the deadline for one-pageabstracts is 1 January; see Web site below Programme committee: V.I. Korzyuk(Belarus), L.A. Aksent�ev (Russia), V.I.Burenkov (UK), P. Butzer (Germany), R.Gorenflo (Germany), V.I. Gromak (Belarus),V.A. Kakichev (Russia), V.S. Kiryakova(Bulgaria), G.S. Litvinchuk (Portugal), O.I.Marichev (USA), S.A. Minyuk (Belarus), Yu.V.Obnosov (Russia), Ya.V. Radyno (Belarus),V.N. Rusak (Belarus), S. Rutkauskas(Lithuania), H.M. Srivastava (Canada), J.J.Trujillo (Spain), M.A. Sheshko (Poland), N.A.Virchenko (Ukraine), L.A. Yanovich (Belarus),P.P. Zabreiko (Belarus), E.I. Zverovich(Belarus). Organizing committee:Academician I.V. Gaishun (Belarus),Academician V.A. Il�in (Russia), A.V. Kozulin(Belarus), A.A. Kilbas (Belarus), M.V.Dubatovskaya (Belarus), S.V. Rogosin(Belarus), H. Begehr (Germany), H.-J.Glaeske (Germany), V.V.Gorokhovik(Belarus), N.A. Izobov (Belarus),N.K. Karapetyants (Russia), A. Kufner(Czech), M. Lanza de Cristoforis (Italy), P.A.Mandrik (Belarus), V.V. Mityushev (Poland),E.I. Moiseev (Russia), M. Saigo (Japan), S.G.Samko (Portugal), A.A. Sen�ko (Belarus), N.I.Yurchuk (Belarus) Proceedings: to be published in IntegralTransform and Special Functions and in Proc.Inst. Math. (Minsk)Information: e-mail: [email protected] Web site: http://amade.virtualave.net

19-23: New Trends in Potential Theory andApplications, Bielefeld, Germany

25-1 March: NATO Advance ResearchWorkshop: Application of AlgebraicGeometry to Coding Theory, Physics, andComputation, Eilat, IsraelInformation: e-mail: [email protected] page: http://www.mat.uniroma2.it/~cilibert/workshop.html[For details, see EMS Newsletter 37]

18-24 Geometric Analysis and Index TheoryConference, Trieste, ItalyInformation:Web site:http://www.sissa.it/~bruzzo/ncg2001/ncg2001.html

26-29: Numerical Methods for FluidDynamics, Oxford, UKAim: to bring together mathematicians, engi-neers and other scientists in the field of com-putational fluid dynamics, to review recentadvances in mathematical and computational

March 2001

CONFERENCES


FForthcoming conferorthcoming conferencesencesCompiled by Kathleen Quinn

techniques for modelling fluid flows Topics: all areas of CFD but with particularemphasis given to adaptivity, biomedical mod-elling and innovative methods in CFDInvited speakers: include: M.J. Baines(Reading), T.J. Barth (NASA Ames), J.-D.Benamou (INRIA-Rocquencourt), F. Brezzi(Pavia), S.M. Deshpande (IISC-Bangalore), C.Farmer (Geoquest), D. Kr�ner (Freiburg), R.LeVeque (Washington), D. Noble (Oxford), R.Rannacher (Heidelberg), P.L. Roe (Michigan),S.J. Sherwin (Imperial-London), E. Süli(Oxford), N.P. Weatherill (Swansea) Programme: invited lectures, 20-minute con-tributed talks and poster sessions. These willbe selected mainly, but not exclusively, on thebasis of their likely contribution to the abovethemesOrganiser: this is the seventh internationalconference on CFD organised by the ICFD(Institute for Computational Fluid Dynamics),a joint research organisation at theUniversities of Oxford and ReadingOrganising committee: M.J. Baines(Reading), M.B. Giles (Oxford), M.T. Arthur(DERA, Farnborough), M.J.P. Cullen(ECMWF), M. Rabbitt (British Energy)Prize: a feature of the meeting will be thethird award of �The Bill Morton Prize� for apaper on CFD by a young research worker.The Prize papers will be presented by theauthors at a special session of the Conferenceand the prize will be presented at theConference dinnerInformation: contact Mrs B. Byrne, OxfordUniversity Computing Laboratory, WolfsonBuilding, Parks Road, Oxford OX1 3QD, UK, tel: +44-1865-273883, fax: +44-1865-273839 e-mail: [email protected] Web site: http://web.comlab.ox.ac.uk/ oucl/people/bette.byrne.html

26-29: Quantum Field Theory,Noncommutative Geometry and QuantumProbability Workshop, Trieste, ItalyWeb site: http://www.sissa.it/~bruzzo/ncg2001/ncg2001.html

2-6: Lévy Processes and Stable Laws,Coventry, UKInformation:Web site: http://science.ntu.ac.uk/msor/conf/Levy/

7-9: 16th British Topology Meeting,Edinburgh, UKInformation:Web site: 9-12: 53rd British MathematicalColloquium, Glasgow, ScotlandSponsors: The Edinburgh MathematicalSociety, the Glasgow Mathematical JournalTrust and the London Mathematical SocietySpecial sessions: partial differential equations(Jean-Yves Chemin, Pierre Collet, EmmanuelGrenier, John Toland) and modular forms(Kevin Buzzard, Ernst-Ulrich Gekeler, JacquesTilouine)Plenary speakers: Henri Berestycki (Paris),Michel Broué (Paris), Henri Darmon(Montreal), Clifford Taubes (Harvard)Other speakers: Nikolaos Bournaveas, Rob de

April 2001

Jeu, Kenneth Falconer, Cameron Gordon,Alexander Ivanov, Mark Jerrum, Paul Martin,Steffen König, Oleg Kozlovski, Ian Leary, RanLevi, James McKee, Viacheslav Nikulin, RobinWilsonRegistration: £30 before 26 February, £40afterwardsInformation: Department of Mathematics,University of Glasgow, Glasgow G12 8QWe-mail: Web site: http://www.maths.gla.ac.uk/bmc200115-21 : Spring School in Analysis, Pasekynad Jizerou, Czech Republic Theme: Banach spaces Main speakers: Joram Lindenstrauss (TheHebrew University of Jerusalem), IsraelGideon Schechtman (The Weizmann Instituteof Science, Rehovot, Israel), Yoav Benyamini(The Technion, Haifa, Israel), Gilles Lancien(Université de Franche-Comte, BesanconCedex France), W. B. Johnson (not yet con-firmed, Texas A&M University, United States) Language: English Organising committee: Jaroslav Lukes, JanRychtar (Czech Republic) Grants: probably support for a limited num-ber of students Deadlines: for reduced fee, 15 January; forsupport, 15 JanuaryInformation:e-mail: [email protected] Web site: http://www.karlin.mff.cuni.cz/katedry/kma/ss/apr01/ss.htm

27-2 June: Spring School in Analysis:Function Spaces and Interpolation, Pasekynad Jizerou, Czech Republic Theme: function spaces and interpolation Topics: function spaces, interpolation,rearrangement estimates, Sobolev inequalities,K-divisibility, Calderon couples, extrapolation Main speakers: A. Cianchi (University ofFlorence, Italy), M. Cwikel (Technion, Haifa,Israel), M. Milman (Florida AtlanticUniversity, USA) Language: English Organizing committee: Jaroslav Lukes, LubosPick (Czech Republic) Lecture notes: notes containing main talks tobe published Grants: probably support for a limited num-ber of students Deadlines: for reduced fee, 15 February; forsupport, 15 February Information:e-mail: [email protected] Web site: http://www.karlin.mff.cuni.cz/katedry/kma/ss/jun01/ss.htm 28-1 June: Harmonic morphisms and har-monic maps, Marseille, FranceAim: to gather specialists four years after thefirst international conference on harmonicmaps and harmonic morphisms, brest97Main speakers: (to be confirmed) P. Baird(France), R. Bryant (USA), F.E. Burstall (UK),J. Eells (UK), B. Fuglede (Danemark), S.Gudmundsson (Sweden), F. Helein (France),S. Ianus (Romania), D. Kotschick (Germany),L. Lemaire (Belgium), P. Li (USA), M.Micallef (UK), C. Negreiros (Brazil), Y.Ohnita (Japan), Y.L. Ou (China), F. Pedit(USA), Z. Tang (China), B. Watson (Jamaica),

May 2001

J.C. Wood (UK)Programme: one-hour lectures and thirty-minute talksCall for papers: prospective speakers shouldcontact M. Ville (e-mail below) Programme committee: J. Eells (Cambridge);L. Lemaire (Brussels); J.C. Wood (Leeds)Organising committee: M. Ville (EcolePolytechnique), E. Loubeau (Brest), S.Montaldo (Cagliari). Sponsors: CIRM, MinistriesProceedings: will be submitted to a publisherSite: Centre International de RencontresMathématiques, LuminyGrants: for information on financial support,contact M. Ville (e-mail below) Deadlines: no deadline but limited number ofseatsInformation: e-mail: [email protected] Web-site: http://beltrami.sc.unica.it/harmor/

4-9: Fractals in Graz 2001, Analysis-Dynamics-Geometry-Stochastics, Graz,Austria [Loosely linked to a special semester on ran-dom walks at the Erwin Schrödinger Institutein Vienna. For further information seehttp://www.esi.ac.at/Programs/rwalk2001.html]Theme: fractals Scope: analysis on fractals, fractals in dynam-ics, geometry of fractals, stochastic processeson fractals Aim: to bring together researchers from vari-ous mathematical areas who share a commoninterest in fractal structures, with open-mind-edness to interaction between different fieldsinside and outside the fractal world. The sub-title of the conference gives the range of top-icsMain speakers: Martin Barlow (Canada),Thierry Coulhon (France), Kenneth Falconer(UK), Hillel Furstenberg (Israel), Ben Hambly(UK), Jun Kigami (Japan), Takashi Kumagai(Japan), Michel Lapidus (USA), AndrzejLasota (Poland), Michel Mendès-France(France), Robert Strichartz (USA), AlexanderTeplyaev (Canada) Language: English Organising committee: Martin Barlow(Vancouver), Robert Strichartz (Ithaca), PeterGrabner (Graz), Wolfgang Woess (Graz) Site: Technical University of Graz Information:e-mail: [email protected] Web site: http://finanz.math.tu-graz.ac.at/~fractal/

18-22: Fourth European Conference onElliptic and Parabolic Problems: Theory,Rolduc, Netherlands[The former Pont-à-Mousson meeting is nowsplit into two conferences. This one is devotedto more theoretical aspects; the other, withmore emphasis on applications, takes place inGaeta, Italy, 24-28 September 2001] Topics: besides elliptic and parabolic issues,topics include geometry, free boundary prob-lems, fluid mechanics, evolution problems ingeneral, calculus of variations, homogeniza-tion, control, modelling and numerical analy-sis

June 2001

CONFERENCES

EMS December 200032

Invited speakers: include C. Bandle (Basel),H. Beirão da Veiga (Pisa), X. Cabré(Barcelona), P. G. Ciarlet (Paris), M. Escobedo(Bilbao), H. Farwig* (Darmstadt), M. Fila(Bratislava), D. Hilhorst* (Orsay), D.Kinderlehrer (CMU), Yan-Yan Li (Rutgers),F.H. Lin (New York), S. Luckhaus (Leipzig),H. Matano (Tokyo), U. Mosco (Rome), J.C.C.Nitsche (Minneapolis), F. Otto (Bonn), M.Padula (Ferrara), P. Pedregal* (Ciudad Real),L.A. Peletier (Leiden), J.F. Rodrigues*(Lisbon), C.J. van Duijn (Amsterdam); * organisers of thematic sessions Organising committee: J. Bemelmans(Aachen), B. Brighi, A. Brillard (Mulhouse),M. Chipot (Zurich), F. Conrad (Nancy), I.Shafrir (Haifa) V. Valente (IAC, Rome), G.Vergara-Caffarelli (Rome)Programme: in addition to the main lecturesparallel sessions of short communications willbe organized. Deadline: for submission of abstracts, 1 April Note: The division between theory and appli-cations will not be enforced, but a theoreticalsubject will certainly have a greater audiencein Rolduc, and an applied one a greater audi-ence in Gaeta Information:e-mail: [email protected],[email protected] Web site: http://www.math.unizh.ch/rolducgaeta

19-22: Computational Intelligence, Methodsand Applications (CIMA 2001), Bangor, UKInformation:e-mail: [email protected];[email protected];[email protected] site: http://www.icsc.ab.ca/cima2001.htm[For details, see EMS Newsletter 37]

25-29: Cmft2001, Computational Methodsand Function Theory, Aveiro, Portugal Theme: the various aspects of interaction offunction theory and scientific computation;other areas from complex variables (includinggeneralisations such as quaternions, etc.),approximation theory and numerical analysisare also covered. Aim: to assist in the creation and mainte-nance of contact between scientists fromdiverse cultures; there is a strong effort toencourage the participation of highly quali-fied scientists who normally have only limitedopportunity to attend international confer-ences Organisers: H. Malonek (Aveiro, Portugal),N. Papamichael (Nicosia, Cyprus), StRuscheweyh (Würzburg, Germany), E. B. Saff(Tampa, USA)Note: anyone interested in being invitedshould send the following details by ordinarymail or e-mail: name, affiliation, address,phone/fax/e-mail, please send me the SecondAnnouncement, I intend to submit a commu-nication (yes or no)Information: contact: H. R. Malonek,Departamento de Matematica Universidadede Aveiro, Portugal tel./fax: +351-234-370359 /+351-234-382014 e-mail: [email protected] Web site: http://event.ua.pt/cmft2001/

1-6: Eighteenth British CombinatorialConference, Brighton, United KingdomInformation:e-mail: [email protected] 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]

4-6: MathFIT workshop: The Representationand Management of Uncertainty inGeometric Computations, Sheffield, UKInformation:Web site: http://www.shef.ac.uk/~geom2001/

5-7: British Congress of MathematicsEducation, Keele, UKInformation:Web site: http://www.bcme.org.uk8-13: Second Workshop on Algebraic GraphTheory, Edinburgh, Scotland Main topics: the new fullerenes, eigenspacetechniques, generalisations from distance-reg-ular graphs, topological considerationsKey speakers: N.L. Biggs (London School ofEconomics), P.J. Cameron (Queen Mary &Westfield College), D. Cvetkovic (Belgrade),P.W. Fowler (Exeter), M.A. Fiol (Barcelona),W. Haemers (Tilburg), P. Hansen (Directeurdu GERAD, Montreal), B. Mohar (Ljubljana),B. Shader (Wyoming)Information: e-mail: [email protected] Web site: http://www.ma.hw.ac.uk/icms/current/graph/index.html

9-22: European summer school: Asymptoticcombinatorics with application to mathemat-ical physics, St Petersburg, RussiaAim: to observe the recent progress in theasymptotic theory of Young tableaux and ran-dom matrices from the point of view of com-binatorics, representation theory and theoryof integrable systems. Systematic courses onthe subjects and current investigations will bepresentedScientific committee: E. Brezin (ENS,France), O. Bohigos (Orsay, France), P. Deift(U.Penn, USA), L. Faddeev (POMI, Russia), V.Malyshev (INRIA, France), A. Vershik (POMI,Russia, Chair)Main speakers: P. Biane (Paris), E. Brezin(Paris), P. Deift (USA), K. Johansson(Stockholm), V. Kazakov (Paris), R. Kenyon(Orsay), M. Kontsevich (France), A. Lascoux(France), A. Okoun�kov (USA), G. Ol�shansky(Moscow), L. Pastur (Paris), R. Speicher(Heidelberg), R. Stanley (MIT), C. Tracy(USA), H. Widom (USA) Topics: asymptotic combinatorics and itsapplications in the theory of integrable sys-tems, random matrices, free probability, quan-tum field theory, etc. Also those topics con-cerned with low-dimensional topology, QFT,new approach in Riemann-Hilbert problem,asymptotics of the orthogonal polynomials,symmetric functions, representation theoryand random Young diagramsLocal organising cmmittee: A. Vershik, Ju.Neretin., K. Kokhas., E. Novikova Site: International Euler Institute

July 2001 Language: EnglishSponsors: RFBR, EMS, NATO, local fundsInformation:e-mail: [email protected] site: .pdmi.ras.ru/EIMI/2001/emschool/index.html

15-20: Algorithms for Approximation IVInternational Symposium, Huddersfield, UK[in celebration of the 60th Birthdays ofClaude Brezinski, Maurice Cox and JohnMason] Theme: approximation theoryAim: to provide an opportunity for exchangeof ideas about current theoretical and practi-cal research on approximationTopics: radial basis functions, splines, rationalapproximation, computer-aided geometricdesign, shape preserving methods, wavelets,support vector machines and neural networks,non-linear approximation, spectral methods,orthogonal polynomials, approximation on asphere, special functions, applicationsMain speakers: M. Buhmann (Germany),M.G. Cox (UK), K. Driver (South Africa), M.Floater (Norway), T. Goodman (UK), W. Light(UK), C.A. Micchelli (USA), L. Nielsen(Denmark), G. Plonka (Germany), T. Poggio(USA), L.L. Schumaker (USA), G.A. Watson(UK)Special sessions: splines, wavelets, orthogonalpolynomials and pade approximation, inte-grals and integral equations, the mathematicsand statistics of metrology, mathematicalmodelling methods in medicineProceedings: to be published as a specialissue of The Journal of Numerical Algorithms byKluwer Academic/Plenum PublishersProgramme committee: C. Brezinski(France), M. Buhmann (Germany), T.Goodman (UK), T. Lyche (Norway), L.L.Schumaker (USA), G.A. Watson (UK). Organising committee: I.J. Anderson (UK),J.C. Mason (UK), D.A. Turner (UK), M.G.Cox (UK), A.B. Forbes (UK), J. Levesley (UK),W. Light (UK)Sponsors: European Office Of AerospaceResearch And Development, LondonMathematical Society, Software Support ForMetrology (National Physical Laboratory,Department Of Trade And Industry)Site: School of Computing and Mathematics,University of Huddersfield, Queensgate,Huddersfield, HD1 3DH, UKGrants: a limited amount of money will beavailable as grants for bona fide research stu-dents and people from less advantaged coun-triesDeadlines: for abstracts, 31 December 2000(please e-mail the symposium committee atthe address below if you require an extendeddeadline); for registration, 15 June (late regis-tration will be allowed, but will incur a tenpercent surcharge)Note: registration forms are available at theWeb site belowInformation: e-mail: [email protected] Web-site: Http://Helios.Hud.Ac.Uk/A4a4/

23-27: 20th IFIP TC 7 Conference onSystem Modelling and Optimization, Trier,GermanyScope: IFIP TC7 promotes applications, thedevelopment of new techniques and theoreti-

CONFERENCES


cal research in all areas of system modellingand optimisation. Each biennial conferencebrings together TC7 working groups and awide scientific and technical community, whoshare information through lectures and dis-cussionsMain speakers: A. Ben-Tal (Haifa), O.L.Mangasarian (Madison), K.-H. Hoffmann(Bonn), J.-S. Pang (Baltimore), F. Jarre(Düsseldorf), R. Rackwitz (München), C.T.Kelley (Raleigh), R. Schultz (Duisburg), K.Kunisch (Graz), P.L. Toint (Namur) Language: English Deadlines: for submission of extendedabstracts, 31 December 2000 Organising committee: E. Sachs (Chair),Universitaet Trier FB IV - Mathematik Information:e-mail: [email protected] site: http://ifip2001.uni-trier.de

5-18 2001 BALTICON 2001, BALTICON2001, Banach algebra theory in context,Krogerup Hojskole, Humlebaek, Denmark [15th in a series of conferences and work-shops]Topics: the emphasis will be on the connec-tions between Banach algebra theory andother areas of mathematics; for instance (list-ed alphabetically), automatic continuity theo-ry, Banach spaces, homological algebra theo-ry, locally compact groups and harmonicanalysis, operator theory, spectral theory,topologyInvited speakers include: G. Dales, J. Esterle,A.Ya. Helemskii, B.E. Johnson, C. Read andG. WillisLocal organising committee: Niels Grønbækand Kjeld Bagger Laursen, both University ofCopenhagen Sponsors: include the MathematicsDepartment of the University of Copenhagen,the Danish Science Research Council, PomonaCollegeCall for papers: all interested are urged tosign up and to submit papersSite: Krogerup Hojskole, approx. 25 milesnorth of CopenhagenDeadlines: for abstracts, 15 February Note: around 60 speakers and contributorsare expected Information:e-mail: [email protected] Web site: http://www.math.ku.dk/conf/balticon2001/

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

12-19: Summer School 2001: Homologicalconjectures for finite dimensional algebras,Nordfjordeid, Norway Topics include: origin of conjectures, resolu-tions and syzygies, homologically finite subcat-egories, some geometrical aspects, infinitely

August 2001

generated modules, derived categories, con-nections to the commutative settingProgramme: the meeting is in two parts: inthe first part the participants lecture on intro-ductory topics; the second part is a workshopwhere specialists in the area lecture on recentresultsWorkshop specialists: Luchezar L. Avramov(USA), Edward L. Green (USA), DieterHappel (Germany), Birge Huisgen-Zimmermann (USA), Bernard Keller (France),Claus M. Ringel (Germany) Organisers: Peter Dräxler ([email protected], Universität Bielefeld),Henning Krause ([email protected], Universität Bielefeld), ØyvindSolberg ([email protected], NTNU,Trondheim)Sponsors: support is provided by the TMRscheme of the EC; further support applied forInformation: contact Øyvind Solberg,([email protected], NTNU, Trondheim) Web sites: http://www.mathematik.uni-bielefeld.de/~sek/summerseries.htmlhttp://www.math.ntnu.no/~oyvinso/Nordfjordeid/

24-30: 10th International Meeting ofEuropean Women in Mathematics, Malta Programme: pure session on Cohomology theo-ries, applied session on Mathematics applied tofinance, interdisciplinary session on The uses ofgeometry, social session on Mathematics outsidethe classroom: cultural differencesInformation: contact Dr Tsou Sheung Tsun(EWM01), Mathematical Institute, 24-29 StGiles, Oxford OX1 3LB, United Kingdom, fax: +44-01865-273583 Web site: http://www.maths.ox.ac.uk/~ewm01/

27-31: Equadiff 10, CzechoslovakInternational Conference on DifferentialEquations and their Applications, Prague,Czech RepublicHonorary presidents: Ivo Babuska, JaroslavKurzweil Topics: ordinary differential equations, par-tial differential equations, numerical methodsand application Language: English Organising committee: Jiri Jarnik (Chair),Bohdan Maslowski (Secretary), Jan Chleboun,Vladimir Dolezal, Eduard Feireisl, MiroslavKrbec, Alexander Lomtatidze, Josef Malek,Pavol Quittner, Milan Tvrdy, JaromirVosmansky Advisory board: H. Amann (Switzerland), D.Arnold (USA), F. Brezzi (Italy), P. Brunovsky(Slovakia), F. Clarke (France), G. Da Prato(Italy), N. Everitt (UK), B. Fiedler (Germany),J. Hale (USA), W. Jaeger (Germany), I.Kiguradze (Georgia), P.L. Lions (France), J.Mawhin (Belgium), P. Raviart (France), K.Schneider (Germany), N. Trudinger(Australia), A. Valli (Italy), W. Wendland(Germany) Site: Charles University of Prague, Faculty ofLaw Notes: 2nd announcement including all formsis available at the Web site Deadlines: for registration, 31 May; forabstracts, 31 March Information:e-mail: [email protected] Web site: www.math.cas.cz/~equadiff/

24-28: Fourth European Conference onElliptic and Parabolic Problems:Applications, Gaeta, Italy[The former Pont-à-Mousson meeting is nowsplit into two conferences. This one is devotedto applications; the other, with more emphasison theory, takes place in Rolduc, Netherlands,18-22 June 2001] Topics: besides elliptic and parabolic issues,topics include geometry, free boundary prob-lems, fluid mechanics, evolution problems ingeneral, calculus of variations, homogenisa-tion, control, modelling and numerical analy-sisInvited speakers include: H. Amann(Zurich), C. Baiocchi (Rome), J. Ball (Oxford),A. Bermúdez (Santiago), M. Bertsch (Rome),C.M. Brauner* (Bordeaux), A. Capuzzo-Dolcetta* (Rome), J. Escher (Hannover), E.Fereisl (Prague), A. Friedman (Minneapolis),G. Geymonat (Montpellier), W. Hackbusch(MIP), A. Henrot* (Nancy), M. Iannelli*(Trento), M. Mimura (Hiroshima), P. Podio-Guidugli (Rome), J. Rubinstein (Haifa), E.Sanchez-Palencia (Paris), S. Sauter* (Zurich),A. Sequeira (Lisbon)* organisers of thematic sessions Organising committee: J. Bemelmans(Aachen), B. Brighi, A. Brillard (Mulhouse),M. Chipot (Zurich), F. Conrad (Nancy), I.Shafrir (Haifa) V. Valente (IAC, Rome), G.Vergara-Caffarelli (Rome)Programme: in addition to the main lecturesparallel sessions of short communications willbe organised. Deadline: for submission of abstracts, 1 AprilNote: The division between theory and appli-cations will not be enforced, but a theoreticalsubject will certainly have a greater audiencein Rolduc, and an applied one a greater audi-ence in Gaeta Information:e-mail: [email protected],[email protected] Web site: http://www.math.unizh.ch/rolducgaeta

16-22: Conference of the AustrianMathematical Society and the DeutscheMathematiker Vereinigung, Vienna, AustriaPlenary speakers: V. Capasso (Milano),M.H.A. Davis (London), I. Ekeland (Paris),W.T. Gowers (Cambridge), M. Kreck(Heidelberg), N.J. Mauser (Vienna), V.L.Popov (Moskau), T. Ratiu (California), D.Salamon (Zürich), G. Teschl (Vienna), J.-C.Yoccoz (Paris), D. Zagier (Bonn), G.M. Ziegler(Berlin) Local organiser: Karl Sigmund (University ofVienna) Site: Technical UniversityInformation: Karl Sigmund, University ofVienna, Institute of Mathematics,Strudlhofgasse 4, 1090 Vienna, tel: +43 1 4277 506 02 or 506 12; fax: +43 14277 9506 e-mail [email protected] Web site: http://www.mat.unive.ac.at/~oemg/Tagungen/2001/

October 2001

September 2001

CONFERENCES

EMS December 200034

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

A. S. Asratian, T. M. J. Denley and R.Häggkvist, Bipartite Graphs and theirApplications, Cambridge Tracts inMathematics 131, Cambridge University Press,Cambridge, 1998, 259 pp., £40, ISBN 0-521-59345-XThis book is devoted to the study of a par-ticular class of graphs. Yet the bookdemonstrates that this is a rich class thatcaptures many important properties ofgraphs in general.

The book is divided into twelve chapterswhich include metric properties (with anappendix: addressing schemes for com-puter networks), connectivity (with anappendix on the construction of linearsuperconcentrators), and expanding prop-erties (with an appendix on expandersand sorters). Curiously, algorithms for theminimum spanning tree problem areincluded in an appendix devoted also tothe travelling salesman problem. Graphtheory is maturing: one day every class ofgraphs will have a book. (jnes)

M. Atkinson, N. Gilbert, J. Howie, S.Linton and E. Robertson (eds.),Computational and Geometric Aspects ofModern Algebra, London MathematicalSociety Lecture Note Series 275, CambridgeUniversity Press, Cambridge, 2000, 279 pp.,£27.95, ISBN 0-521-78889-7 This volume of proceedings contains 18papers, for which it is hard to find any uni-fying description other than the title of theconference. There are papers on grouppresentations, term rewriting, stringrewriting, cancellation diagrams with non-positive curvature, a new proof of the cut-point conjecture for negatively curvedgroups, papers on discontinously cocom-pact actions by isometries, with computa-tions in word-hyperbolic groups and ingroups with exponent 6, and several fur-ther topics.

I will explicitly mention the papers thatexceed 20 pages. Bartholdi and Grigor-chuk present a paper whose expositorypart associates a graded Lie algebra to agroup G with a given N-series, discussesthe questions of growth; the authors thenconstruct two examples of groups of inter-mediate growth that can be used ascounter-examples to a conjecture on thestructure of just-infinite groups of finitewidth. A paper by Huch and Rosebrock isconcerned with two mutually dual smallcancellation conditions that generalise(C6), (C4) (T4), (C3) and (T6); they solvethe conjugacy problem for the groups witha respective presentation. A paper byMadlener and Otto surveys the application

of prefix rewriting to the subgroup prob-lem in combinatorial group theory, and apaper of Mislin and Tolleli is concernedwith periodic Farrell and Tate cohomolo-gies for hierarchically decomposablegroups. Finally, a paper of Nekrashevychand Sushchansky studies automorphismgroups of spherically homogeneous root-ed trees. (ad)

H. Bercovici and C. Foias (eds.),Operator Theory and Interpolation,International Workshop on Operator Theoryand Applications, IWOTA 96, OperatorTheory, Advances and Applications 115,Birkhäuser, 2000, 309 pp., ISBN 3-7643-6229-4The papers in this volume were presentedat the International Workshop onOperator Theory and Applications(IWOTA), held at Indiana University,Bloomington, in June 1996. They repre-sent most of the areas that were discussedat the workshop, with some emphasis onmodern interpolation theory, a topic thathas seen much progress in recent years.Much of the work in this volume is relatedto Béla Sz.-Nagy�s results on interpolationand dilation theory.

The book may serve as an inspiration forfurther research, and can be recommend-ed to researchers and postgraduate stu-dents involved in these fields. (knaj)

F. Bergeron, G. Labelle and P. Leroux,Combinatorial Species and Tree-likeStructures, Encyclopedia of Mathematics andits Applications 67, Cambridge UniversityPress, Cambridge, 1998, 457 pp., £55, ISBN0-521-57323-8This book gives, in English for the firsttime, a thorough presentation of the com-binatorial theory of species (which origi-nated in the work of A. Joyal in 1980).The introductory Chapter 1 explains asso-ciated power series and the operations ofaddition, multiplication, substitution anddifferentiation of species. Chapter 2 intro-duces further operations, weightedspecies, virtual species (enabling speciessubtraction), and molecular and atomicspecies. Chapter 3 is devoted to combina-torial functional equations; among otherthings, Lagrange inversion, iterativemethods, and a useful overview of asymp-totic analysis are presented. Chapter 4deals with unlabelled enumeration andasymmetric structures and gives proofs forsubstitution formulas for weighted species.Chapter 5 presents species on totallyordered sets and combinatorial theory ofdifferential equations.

The book contains more than 350 exer-cises and an extensive bibliography. Theexposition is illuminated by many dia-grams. In the appendix, numerous tablesare given. The book is essential for any-

body who wants to learn more about thispart of combinatorics with quite a strongalgebro-categorical aesthetic appeal. As abonus, it has one of the never boring fore-words by G.-C. Rota. (mkl)

P. Berthelot, D-modules arithmétiques II.Descente par Frobenius, Mémoires de laSMF, 81, Société Mathématique de France,Paris, 2000, 136 pp., FRF 150, ISBN 2-85629-086-8 This is a continuation of the author�s pro-ject of developing foundations of crys-talline/rigid cohomology with coefficientsin terms of �arithmetic D-modules�. Thebasic objects of interest are schemes (resp.formal schemes) smooth over a given baseZ/pnZ-scheme (resp. a p-adic formalscheme).

The volume is devoted to various func-tionality properties of arithmetic D-mod-ules with respect to (a lift of) the Frobeniusmorphism F. The main result (�Frobeniusdescent�) is a far-reaching generalisationof the classical Cartier isomorphism. Thisis used by the author to establish, amongother things, compatibility of F* with vari-ous cohomological operators. (jnek)

A. Böttcher and S. M. Grudsky, ToeplitzMatrices, Asymptotic Linear Algebra, andFunctional Analysis, Birkhäuser, Basel,2000, 116 pp., DM58, ISBN 3-7643-6290-1This text is a self-contained introductionto some problems for Toeplitz matrices onthe border between linear algebra andfunctional analysis. The text looks atToeplitz matrices with rational symbols,and focuses attention on the asymptoticbehaviour of the singular values; thisincludes the behaviour of the norms, thenorms of the inverses, and the conditionnumbers as special cases. The text illus-trates that the asymptotics of several linearalgebra characteristics depend in a fasci-nating way on functional analytic proper-ties of infinite matrices. Many conver-gence results can be comfortably obtainedby working with appropriate C*-algebras,while refinements of these results (forexample, estimates of the convergencespeed) nevertheless require hard analysis.

This book is warmly recommended tobeginners specialising in functional analy-sis and algebra. (knaj)

A. Candel and L. Conlon, Foliations I,Graduate Studies in Mathematics 23,American Mathematical Society, Providence,2000, 402 pp., US$54, ISBN 0-8218-0809-5This is the first volume of a two-volumemonograph on the qualitative theory offoliations. It consists of three parts. Thefirst part The Foundations is designed as anintroduction to the theory of foliated man-ifolds for postgraduate students. Theauthors state that the readers of this partare assumed to have a fairly good back-ground in manifold theory, but I thinkthat the authors do not require any extra-ordinary knowledge. This first part canalso be considered as a necessary prereq-uisite for the next two parts. The secondand third parts have the titles Codimension

RECENT BOOKS


Recent booksRecent booksedited by Ivan Netuka and Vladimír Sou³ek

One and Arbitrary Codimension. These aredevoted to the study of the foliations ofcodimension 1 and to the foliations ofhigher codimension, and this division isquite understandable, because the meth-ods used in these two cases are quite dif-ferent. They can be compared with thetheory of flows on surfaces, and the theoryof flows on manifolds of dimension >2.Already this first volume covers a lot ofmaterial on foliated manifolds, and a greatpart deals with more general foliatedspaces.

In a way, the book is built on examples.This means that the authors first demon-strate various phenomena on examples,and only when the reader understandsthem do they present any systematic theo-ry. The authors pay great attention toexamples, and you can find a large num-ber of them in the book. We also findmany exercises. This is very important,especially in the book of this extent. Theyare well chosen, and will keep the interestof a reader on a high level. The biographyhas 149 items, and goes up to 1999. Thebook is surely not a short introduction intofoliations or a concise survey of foliationtheory, but is a fundamental source foreverybody with a serious interest in folia-tions. (jiva)

R. Cerf, Large Deviations for ThreeDimensional Supercritical Percolation,Astérisque 267, Société Mathématique deFrance, Paris, 2000, 177 pp., FRF 250,ISBN 2-85629-091-4 The aim of the work is to propose amethod for analysing phase separationand coexistence for the three-dimensionalBernoulli percolation model. The mainresults concern the large deviation princi-ples and their application to the Wulffcrystal. The case of a single cluster, as wellas the whole configuration, are consid-ered.

The book is divided into twelve chapterswith the headings: Introduction, The largedeviation principles (LDP), Sketch of theproofs, The model, Surface tension, Thesurface tension, Coarse graining, The cen-tral lemma, Proof of the LDP for a singlecluster, Collections of sets, The surfaceenergy of a Caccioppoli partion, Proof ofthe LDP for the whole configuration.

The large deviation principles are statedin Chapter 2, together with their applica-tion to the Wulff crystal. Chapter 3 is aninformal sketch of the proofs for the singlecluster case. The notation and the modelare introduced in Chapter 4. Importantfacts on the theory of Caccioppoli sets andthe Wulff Isoperimetric Theorem aresummed up in Chapter 6. (mhusk)

S. D. Chatterji and H. Wefelscheid (eds.),Selected Papers. G. C. Young, W. H. Young,Presses Polytechniques et UniversitairesRomandes, Lausanne, 2000, 484 pp.,CHF149, ISBN 2-88074-445-8 In this volume the authors present a selec-tion of 52 of the 215 published articles ofGrace Chisholm Young (1868-1944) andWilliam Henry Young (1863-1941), a com-plete list of which appears next to a brief

chronology of their lives. The mathemati-cal work of the Youngs can be convenient-ly divided into three broad categories: thetheory of real functions, Fourier analysis,and miscellaneous. The bibliography isbased entirely on that of I. Grattan-Guinness in Historia Mathematica 2 (1975),43-58. The authors have grouped the arti-cles according to the year of their publica-tion. The three books of the Youngs fol-low the list of the articles. The two obitu-aries, by G. H. Hardy (1877-1947) and M.L. Cartwright (1900-1998) respectively,give a balanced account of the mathemati-cal work of the Youngs, as viewed by theiralmost-contemporaries. A brief overviewof the totality of their mathematical workfrom a modern viewpoint is given in theessay by Chatterji.

This book should form an ideal resourcefor mathematicians and specialists in thehistory of mathematics. (knaj)

B. Cipra, What�s Happening in theMathematical Sciences: 1998-1999,American Mathematical Society, Providence,1999, 126 pp., ISBN 0-8218-0766-8The contents of the fourth volume in thisseries is well expressed by the titles givenbelow. This lively presentation of anamazingly wide spectrum of happenings inmathematics is impressive. I believe thatthis should be presented to a wide audi-ence even outside mathematics, whichcould be fascinated by the ideas, concepts,and beauty of the mathematical topics.

The contents: A blue-letter day for comput-er chess (the end of the long way to beatKasparov does not mean solving the com-binatorial games problem); A prime of chaos(on quantum chaology and algebraic num-ber theory); Proof by example: a mathemati-cian�s mathematician (on the impact of PaulErdõs); Computers take algebraic geometryback to its roots (algorithmic questions inalgebraic geometry); As easy as EQP (onautomatic theorem proving); Beetlemania:chaos in ecology (on experimental evidencefor chaotic dynamics); From wired to weird(on revolutionary quantum computing);Tales from the cryptosystem (computationalcomplexity and cryptographic systems);But is it math? (mathematics and art:Escher, etc.); Mathematical discovery byHenri Poincaré (Henri Poincaré�sthoughts). (jslo)

C. Corduneanu and I. W. Sandberg(eds.), Volterra Equations andApplications, Stability and Control: Theory,Methods and Applications 10, Gordon andBreach, Amsterdam, 2000, 496 pp., £ 75,ISBN 90-5699-171-XThis volume contains 52 papers out ofmore than 60 presentations of the sympo-sium held at University of Texas atArlington in 1996 to celebrate the 100thanniversary of Vito Volterra�s (1860-1940)publications on integral equations.

It begins with nine invited papersaddressing both history (M. Schetzen,Retrospective of Vito Volterra and hisinfluence on nonlinear system theory, andR. K. Miller, Volterra integral equations atWisconsin) and recent developments (N.

Azbelev, Stability and asymptotic behaviorof solutions of equations with aftereffect,C. T. H. Baker and A. Tang, GeneralizedHalanay inequalities for Volterra function-al differential equations and discretizedversions, P. Clément and G. da Prato,Stochastic convolutions with kernels aris-ing in some Volterra equations, H. Engler,An example of Lp-regularity for hyperbol-ic integro-differential equations, V.Lakshmikantham and A. S. Vatsala, Thepresent status of UAS for Volterra anddelay equations, I. W. Sandberg, Myopicmaps and Volterra series approximation,and O. J. Staffans, State space theory forabstract Volterra operators).

This is followed by 43 contributedpapers addressing a great variety of prob-lems. In particular, they deal with stabili-ty theory, stochastic processes, classicalVolterra equations (also in connection withdynamical systems and blow-up type prob-lems), numerical problems (with attentionto finite-element method and generalisa-tions of known discretisation methods forordinary differential equations), periodicsolutions, control theory (especially opti-mal control), infinite-dimensional systems,integro-differential equations, approxima-tion methods, abstract Volterra operatorsand equations, applied problems inphysics and engineering, and other top-ics.

This volume will be of interest for bothpure and applied mathematicians, as wellas theoretically oriented engineers andgraduate students seeking a broad state-of-the-art insight into Volterra equationsand their applications. (trou)

W. A. de Graf, Lie Algebras: Theory andAlgorithms, North-Holland MathematicalLibrary 56, North-Holland, Amsterdam, 2000,393 pp., US$118, ISBN 0-444-50116-9The theory of Lie algebras has manyexplicit constructions and concrete algo-rithms (Levi decomposition, branchingrules, Hall-Shirshov and Grbner bases,etc.). This book contains a standardcourse in Lie algebras and includes practi-cally all existing algorithms in this theory.The approach simplifies proofs of someimportant theorems and makes themmore transparent and clear. Moreover,since current research on Lie algebrasrequires the use of computers, such anexposition facilitates understanding andpractical use of computational methodsfor solving concrete problems. The authoris also the author of a sub-package Lie alge-bras in the programme package GAP. Thishas enabled him to create a book that willbe useful for experts, as well as for inter-ested researchers from other fields ofmathematics and mathematical physics.(ae)

J. J. Duistermaat and J. A. C. Kolk, LieGroups, Universitext, Springer, Berlin, 2000,344 pp., DM79, ISBN 3-540-15293-8 This book is devoted to the theory offinite-dimensional Lie groups and theirrepresentations, mainly from the differen-tial geometry point of view. Lie algebrasare studied in the first chapter, together

RECENT BOOKS

EMS December 200036

with their relations to Lie groups. Theproof of Lie�s third fundamental theoremon the existence of a simply connected Liegroup with a given Lie algebra is included.Proper actions of groups on manifolds, thecorresponding stratification of manifoldinto orbit types and the related blowing-up process are the main topics of the sec-ond chapter. In the third chapter, theauthors study compact Lie groups andalgebras, their fundamental group, thecorresponding Weyl group and Stiefel dia-grams.

Invariant densities and problems ofinvariant integration are discussed,together with the classical Weyl integra-tion formula. The last chapter presents agood overview of the representation theo-ry of compact Lie groups, including thePeter-Weyl theorem, induced representa-tions, character formulas and real forms ofcomplex representations. There is also anice description of the right regular repre-sentation of Lie groups, the Borel-Weiltheorem and its applications. The bookcan be recommended as a higher levelintroduction to theory of (compact) Liegroups and their representations. (jbu)

Y. Eliashberg and L. Traynor (eds.),Sympletic Geometry and Topology,IAS/Park City Mathematics Series 7, AmericanMathematical Society, Providence, 1999, 430pp., US$69, ISBN 0-8218-0838-9 The seventh volume in this series is devot-ed to various aspects of symplectic topolo-gy and related topics. The individualparts present the contents of the followinglectures: Introduction to symplectic topology byDusa McDuff, Holomorphic curves anddynamics in dimension three by HelmutHofer, An introduction to the Seiberg-Wittenequations on symplectic manifolds by CliffordTaubes, Lectures on Floer homology byDietmar Salamon, A tutorial on quantumcohomology by Alexander Givental, Eulercharacteristics and Lagrangian intersections byRobert MacPherson, Hamiltonian groupactions and symplectic reduction by LisaJeffrey, and Mechanics, dynamics, and sym-metry by Jerrold Marsden.The result is a lively exposition of recentdevelopments in this exciting branch ofmathematics, often starting with quite ele-mentary and introductory facts and reach-ing far beyond standard textbooks, up tosketches of proofs of most recent deepresults. In particular, this volume will beuseful reading for graduate students andexperts. (jslo)

W. Ewald, From Kant to Hilbert. A SourceBook in the Foundations of Mathematics, I,II, Clarendon Press, Oxford, 2000, 648 and690 pp., £50, ISBN 0-19-850537-X, 0-19-850535-3 and 0-19-850536-1This is an excellent collection of carefullyselected and edited classical texts on thefoundations of mathematics. Each text ispreceded by an introduction and notesand a comprehensive bibliography isincluded at the end of each volume. Manytexts appear in a reliable English transla-tion for the first time.

The selection starts with the texts of

Berkeley, MacLaurin and d�Alembert,continues with a selection from Kant andLambert and valuable translations of thetexts of Bernard Bolzano. It then contin-ues with excerpts or complete texts ofGauss, Gregory, De Morgan, Hamilton,Boole, Sylvester, Cayley, Peirce, Baire,Hilbert, Brouwer, Zermelo and Hardy.The book is in some sense complementaryto van Heijenoort�s source book in mathe-matical logic �From Frege to Gdel� (forexample, it contains no texts by Frege,Peano, Russell, or Weyl), and represents atraditional and widely accepted view onthe foundations of mathematics. Thispoint of view is expressed by the very lasttext of this collection, �The architecture ofmathematics� (Bourbaki, 1948). (jmlc)

J. Faraut, S. Kaneyuki, A. Korányi, Qi-keng Lu and G. Roos, Analysis andGeometry on Complex HomogeneousDomains, Progress in Mathematics 185,Birkhäuser, Boston, 2000, 540 pp., DM138,ISBN 0-8176-4138-6 and 3-7643-4138-6 The book is an introduction to severalbasic topics in complex analysis and geom-etry at an advanced graduate level; a cer-tain amount of preliminary knowledge isrequired. It is based on lectures deliveredat the CIAMPA Autumn School in Beijingin 1997, and extended in several interest-ing directions. It consists of five parts writ-ten by different authors. The parts aremore or less independent.

The first part (by J. Faraut) deals withthe theory of function spaces on complexsemi-groups, and gives an overview of thetheory of Hilbert spaces of holomorphicfunctions on complex manifolds endowedwith the action of a (real) Lie group. Themain problems discussed are the decom-position of the Hilbert space into irre-ducible invariant subspaces and a descrip-tion of the reproducing kernel on it. Thesecond part (by S. Kaneyuki) on gradedLie algebras and related geometric struc-tures gives a nice survey of recent resultson semi-simple pseudo-Hermitian sym-metric spaces and Siegel domains. Thethird part (by A. Korányi) presents anintroduction to the theory of holomorphicfunctions on Cartan domains. It is basedon the Harish-Chandra approach arisingfrom the theory of semi-simple groups.The fourth part (by Q. Lu) is devoted tothe study of properties of Laplace-Beltrami operator and various integraltransforms. The last part (by G. Ross) onJordan triple systems contains anotherapproach to study of geometry and analy-sis of Hermitian bounded symmetricdomains.

All contributions are written carefullyand systematically. Let me mention espe-cially Parts 2 and 3 which bear a strongrelation to the geometry and analysis ofinvariant operators for special geometricstructures. (jbu)

J. Francheteau and G. Métivier, Existencede chocs faibles pour des systèmes quasi-linéaires hyperboliques multidimension-nels, Astérisque 268, Société Mathématique deFrance, Paris, 2000, 198 pp., FRF250, ISBN

2-85629-092-2In this work the authors consider weakshocks for systems of conservation laws inany space dimension. The main result is aconstruction on a space-time domain,independent of the parameter ε, of fami-lies of weak solutions uε, discontinuousalong a smooth hypersurface Σε, withjumps of order ε. For a fixed ε, the prob-lem can be recast as a non-linear mixedhyperbolic problem with a free non-char-acteristic boundary, which has been solvedby A. Majda. When ε tends to 0, the fronttends to be characteristic; this induces aloss of stability and regularity. As a conse-quence, the classical non-linear methodsbased on Picard�s iterations and differenti-ations do not apply. To prove suitable apriori estimates and construct the solutionsthe authors use more sophisticated meth-ods, such as the para-differential calculusand Nash-Moser�s type iteration schemes.These results have important applicationsto Euler�s equations of gas dynamics, bothto the full system and the isotropic system.Weak solutions of both two systems areconstructed and compared.

The authors start the book with a niceintroduction, giving a summary of existingliterature, pointing out the generalscheme of proofs, indicating crucial pointsand difficulties that must be overcome andbriefly describing how this can be done.(jkop)

H. Gordon, Discrete Probability,Undergraduate Texts in Mathematics,Springer, New York, 1998, 266 pp., DM68,ISBN 0-387-98227-2This is an undergraduate text designed foran introductory course in probability theo-ry. With a few exceptions, only elementarymathematics is used throughout. Thebook starts with the definition of probabil-ity on discrete sample spaces. It proceedsto discuss combinatorial probability (sam-pling with and without replacement),independence of events and conditionalprobability, and random variables andtheir mean and variance. Independenceof random variables is treated only briefly.One section is devoted to the weak law oflarge numbers. The Poisson distributionas a limit of a sum of independentBernoulli variables, the Stirling formula,and the De Moivre-Laplace theorem (with-out proof) are all treated in one chapter.The rest of the book is devoted to momentgenerating functions, random walks anddiscrete Markov chains.

The strengths of the book are undoubt-edly its exercises (over 400 of them), allwith numerical solutions, and the manyinteresting remarks on the history of prob-ability theory and biographies of impor-tant personalities that are scatteredthroughout the book. On the other hand,important definitions and facts are some-times hidden in the text. (mkul)

T. V. Gramchev and P. R. Popivanov,Partial Differential Equations.Approximate Solutions in Scales ofFunctional Spaces, Mathematical Research108, Wiley-VCH, Berlin, 2000, 155 pp.,

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DM148, ISBN 3-527-40138-5In this book the authors present in a uni-fied form the results of their research overthe last two decades in micro-local analysisof pseudo-differential operators. Thereader is supposed to be familiar withbasic facts from the theory of Gevrey class-es, pseudo-differential operators, micro-local analysis, Fourier integral operatorsand differential geometry. The authorsstudy micro-local properties (solvability,hypo-ellipticity) of pseudo-differentialoperators in Sobolev spaces and Gevreyclasses; construct approximate solutionswith non-classical phase functions andamplitudes; investigate linear differentialoperators with multiple characteristics andquasi-homogeneous operators; and pre-sent applications of Airy operators tooblique derivative problems for hyperbol-ic equations and applications of Gevreyclasses to dynamical systems (approximatenormal forms for pairs of glancing hyper-surfaces). The language of the book isvery general and abstract. Unfortunately,the misprints make the technicallydemanding notions even more difficult toread. The book is suitable for experts inthe field. (efa)

A. Guichardet, Groupes quantiques,Mathématiques, EDP Sciences, Les Ulis, 1995,149 pp., FRF 160, ISBN 2-7296-0564-9 and2-271-05272-6 This book gives an excellent introductioninto the field of quantum groups. It is arelatively subtle volume, but neverthelesscontains a lot of interesting material. Thetext is written with the necessary mathe-matical rigour, which ensures that thebook will be well received by mathemati-cians. On the other hand, its readingrequires no extraordinary mathematicalpreparation, and will be understandable tophysicists.

The book starts with a chapter present-ing some prerequisites from algebra. Itthen continues with a chapter introducingthe main concepts concentrated aroundthe notion of a Hopf algebra. We findhere also the notion of compact quantumgroup, in the sense of Woronowicz, andrelations to the Poisson structures. Thethird chapter deals with formal deforma-tions of the objects introduced in the pre-vious chapter. From the fourth chapterthe author passes to very concrete consid-erations: namely, a chapter about thequantum group Uh sl(2,k), a chapter aboutthe quantum group Uh sl(n+1,k), and avery interesting chapter about deforma-tions of homogeneous spaces.

The book reads very well. One reasonfor this is that it contains many interestingexamples, and hints for further studies aregiven. It can be strongly recommended.(jiva)

L. C. Kannenberg, Geometric Calculus.Giuseppe Peano, Birkhuser, Boston, 2000,150 pp., DM138, ISBN 0-8176-4126-2 and3-7643-4126-2The first edition of Peano�s importantwork on geometrical calculus, �precededby the first operations of deductive logic�,

was published with a small print run in1888, and has never been reissued in itsentirety (only an extract was printed inPeano�s Opera Scelte and Hubert Kennedyincluded an English translation of theIntroduction and Chapter 1 in his SelectedWorks of Giuseppe Peano). Now a completeand reliable translation is available to awider audience.

The preliminary chapter of the book isPeano�s first publication in mathematicallogic: he first develops a calculus of classesand then a calculus of propositions, intro-ducing for the first time modern notation(such as the symbols ∪ and ∩). Of partic-ular interest is his treatment ofGrassmann�s regressive product. ChapterIX represents one of the first attempts toaxiomatise the idea of a linear vectorspace. Comments on two errors, discov-ered and corrected by Honbo Li, areincluded in a short Editorial Note. Thebook (unfortunately) contains no othercomments on this classical text. (jmlc)

A. Khrennikov, Interpretations ofProbability, VSP BV, Utrecht, 1999, 228 pp.,ISBN 90-6764-310-6 The book presents an interesting discus-sion on quantum mechanics from a proba-bility point of view. It is well known thatthe theory of quantum mechanics givesstrange results in some specific situations:the Einstein-Podolsky-Rosen paradox andBell�s inequalities seem to be the mostpopular of these. The author points outthat such difficulties could be caused by aninconvenient measurement of random-ness, and instead of values in the ordinaryinterval [0,1], he proposes the space of p-adic numbers as the most convenientrange for probability employed in quan-tum mechanics.

The book begins with a survey on thenotion of probability. Kolmogorov�s mea-sure-theoretical approach and von Mises�idea on collectives giving frequency prob-ability theory and proportional approachto randomness are introduced and com-pared. After that, the author proceeds torandom principles in quantum mechanics.The Einstein-Podolsky-Rosen paradox isformulated and compared with Bell�sinequality for probabilities as well as forcovariances and with the idea on hiddenvariables. The next two sections are devot-ed to the necessary theory of p-adic num-bers and their calculus. The book con-cludes with a discussion on tests for ran-domness for p-adic-valued probability.

The book is intended as a deep presen-tation of the author�s idea that p-adic-val-ued probability is able to remove, andeven to explain, such difficulties as theEinstein-Podolsky-Rosen Paradox and tovalidate places where the theory produces�negative� probabilities. It will be valuablefor theoretical physicists, especially thoseworking in quantum mechanics and relat-ed fields. On the other hand, it has valuefor mathematicians dealing with probabil-ity theory, since the book is an interestingattempt to use special Banach space-val-ued probability for description of observedphenomena. (pl)

H. Koch, Number Theory. AlgebraicNumbers and Functions, Graduate Studiesin Mathematics 24, American MathematicalSociety, Providence, 2000, 368 pp., US$59,ISBN 0-8218-2054-0 According to the preface, it is the author�sconviction �that an area of mathematicssuch as number theory that has developedover a long period of time can be proper-ly studied and understood if one proceedsthrough this entire development in abbre-viated form, much as an organism recapit-ulates its evolutionary path in abbreviatedform during its embryonic development.From this I derived the concept of allow-ing the reader to take part from chapter tochapter in the historical development ofnumber theory�.

Leaving aside the allusion to a by-now-discredited biological principle, let usexamine the contents of the book in thelight of the author�s intentions. Chapter 1consists of several topics in elementarynumber theory, such as Pythagoreantriples, the (incomplete) history of Zell�sequation and Fermat�s last theorem, con-gruences, quadratic reciprocity law andthe distribution of primes. Chapter 2 isdevoted to elementary theory of orders innumber fields, including Dirichlet�s theo-rem on units, finiteness of class numberand Minkowski�s theorem in the geometryof numbers. Chapters 3 and 4 develop thetheory of Dedekind rings and valuations.These are used in Chapter 5, which treatsfunction fields in one variable over perfectconstant fields, up to the Riemann-Rochtheorem. Chapter 6 is on higher ramifica-tion groups (including Herbrand�s theo-rem) and their applications, such as thedecomposition of prime ideals in cyclo-tomic and Kummer extensions. Chapter7 begins with an introduction of adèlesand idèles and reproduces Tate�sapproach to the functional equation ofHecke L-series, as well as F. K. Schmidt�sproof of the functional equation of thezeta-function of a function field. Analyticproperties of Hecke L-functions are usedin Chapter 8 to prove various distributionresults for prime ideals that generaliseDirichlet�s theorem on primes in arith-metic progressions. Chapter 9 is devotedto the arithmetic of quadratic fields, andtreats the correspondence between classesof binary quadratic forms and ideal classesin quadratic fields, units and class numberformulas. Finally, Chapter 10 gives a briefsurvey of class field theory. There arethree appendices, on elements of divisibil-ity (including the structure theory offinitely generated modules over PID�s andEuclidean domains), on traces, norms anddiscriminants, and on Fourier analysis onlocally compact abelian groups.

The book requires as a prerequisite agood knowledge of basic algebra andGalois theory and is meant to be an intro-ductory text aimed at Ph.D. students innumber theory and related areas.

The brief description of its contentsshows that the book is concerned mainlywith the general theory of number fieldsand function fields. In fact, a significant

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part of the material goes beyond what onewould expect from an introductory text. Adisadvantage of this approach, however, isthe absence of the full-flavoured �concretearithmetic�, regardless of its place in thehistorical development of algebraic num-ber theory. The most significant omis-sions include cubic and biquadratic reci-procity laws, genus theory of quadraticforms, Hilbert symbols, a more detailedanalysis of the class number formula andexamples of zeta-functions of functionfields. For these reasons this book can berecommended to students of number the-ory for its rigour and emphasis on theory,but its study should be complemented byreading other, more �concrete� texts, suchas Borevich and Shafarevich or Ireland-Rosen. (jnek)

I. Lasiecka and R. Triggiani, ControlTheory for Partial Differential Equations:Continuous and Approximation Theories,1: Abstract Parabolic Systems, Encyclopediaof Mathematics and its Applications 74,Cambridge University Press, Cambridge, 2000,644 pp., £75, ISBN 0-521-43408-4 This volume represents a comprehensiveand up-to-date treatment of quadraticoptimal control theory for linear parabol-ic-like partial differential equations (PDEs)over a finite or infinite time horizon andrelated differential (integral) and algebra-ic Riccati equations. A semigroupapproach is systematically used. Besidescontinuous problems, numerical approxi-mation theory is pursued. On an abstractlevel, the controlled system is assumed tohave the form dy/dt = Ay + Bu, with A andB linear possibly unbounded operators,the former generating a C0- (and evenanalytic) semigroup on a Hilbert space,and with the control u being an L2-func-tion in time. The quadratic cost function-al to be minimised then involves still anobservation operator R.

The abstract theory for such problemsapplicable to a broad class of PDEs is pre-sented in Chapters 1 and 2 for the finiteand infinite horizon cases. Chapter 3 pre-sents many PDE illustrations with Dirichletor Neumann boundary control or pointcontrol. This includes the heat equation,the Kelvin-Voight, Kirchhoff, and Euler-Bernoulli equations, and thermo-elasticplates. Chapter 4 provides a detailednumerical approximation, including opti-mal rates of convergence, detailed illustra-tion being then given in Chapter 5.Finally, Chapter 6 returns to an abstractlevel, dealing with a min-max game theo-ry over an infinite time interval.

This thorough very detailed expositionlargely expands the lecture notes of theseexperienced authors, published in 1991 bySpringer-Verlag, and is primarilyaddressed to applied mathematicians andtheoretical engineers interested in optimalcontrol, in particular, of linear distributed-parameter systems, as well as to graduatestudents in this area. This volume will befollowed by optimal control theory forhyperbolic or Petrowski-type PDEs(Volume II) and for hyperbolic-likedynamics and coupled PDE systems

(Volume III). (trou)

K. B. Laursen and M. M. Neumann, AnIntroduction to Local Spectral Theory,London Mathematical Society MonographsNew Series 20, Clarendon Press, Oxford,2000, 591 pp., £75, ISBN 0-19-852381-5 This monograph develops the local spec-tral theory for bounded linear operatorson Banach spaces. Chapter 1 is devoted todecomposable operators. The authorsderive several basic characterisations ofdecomposability, explore the role of thelocal spectrum, and establish the impor-tant connection with the theory of spectralcapacities. Chapter 2 centres around cer-tain characterisations and applications ofBishop�s property ß and the decomposi-tion property δ for bounded linear opera-tors on an arbitrary complex Banachspace; the main goal of this chapter is toshow that property ß describes preciselythe restrictions of decomposable operatorsto closed invariant subspaces, that proper-ty δ characterises the quotients of decom-posable operators by closed invariant sub-spaces, and that there is a complete duali-ty between the two properties. In Chapter3, distinguished parts of the spectrum andtheir relationships to local spectra arestudied, and several important classes ofspectral subspaces are considered; particu-lar emphasis is placed on the relationsbetween the spectra and essential spectraof two operators that are connected witheach other through some intertwiningcondition. Chapter 4 collects essentiallyeverything that is known about the spec-tral theory of multipliers, and particularlyabout convolution operators on group andmeasure algebras. Chapter 5 illustratesthe usefulness of local spectral theory inautomatic continuity. Finally, Chapter 6contains a list of open problems. Themodest prerequisites from functionalanalysis and operator theory that theauthors require are collected in theAppendix. (dmed)

J. M. Lee, Introduction to TopologicalManifolds, Graduate Texts in Mathematics202, Springer, New York, 2000, 385 pp., 138fig., DM69, ISBN 0-387-95026-5 and 0-387-98759-2 This is a first course on topology for post-graduate students, written by an authorwho has evidently great experience inteaching this subject. In order to reducethe prerequisites to the minimum, theauthor has included an appendix in whichhe reviews the necessary notions from settheory, the theory of metric spaces, andgroup theory. Later, we find a specialchapter �Some group theory�, whose aim isto have the necessary algebraic techniquesavailable when studying the fundamentalgroup. Moreover, he assumes no knowl-edge even of general topology, and devel-ops the relevant part of this theory in fulldetail.

The main part of the book is concen-trated around the topology of 2-mani-folds. The author has devoted the wholeintroductory chapter to motivating thenotion of a manifold (of arbitrary dimen-

sion), showing the role played by mani-folds in topology, geometry, complexanalysis, algebra, algebraic geometry, clas-sical mechanics, general relativity andquantum field theory. He then introducesan important tool, simplicial complexes,and presents triangulation theorems formanifolds of dimensions 1, 2 and 3. Usingsimplicial complexes, he describes 1-man-ifolds and gives a complete classification ofcompact 2-manifolds. He introduces thefundamental group, paying much atten-tion to this notion. We find here variousmethods enabling us to compute the fun-damental group, the Seifert-Van Kampentheorem, and covering spaces. Of course,the central objects of interest are the fun-damental groups of compact 2-manifolds.To make the theory of 2-manifolds rela-tively complete, the author introduces thenotions of homology and cohomology.

The book is very carefully written. Inthe text we find many exercises, classifiedas simpler problems, which should not beomitted because some are used later in themain text. At the end of each chapter areproblems that are classified as more diffi-cult. The author has written this course asa first course in topology, and as a prepa-ration for more advanced courses ontopology and differential geometry. Thisis the reason why he did not touch PL-structures or differential structures onmanifolds. It can be used to in full or par-tially for various basic courses on topology.It is especially good that the course is writ-ten in a very clear and attractive way, andwe can expect that it will attract the atten-tion of students. (jiva)

M. Liebeck, A Concise Introduction toPure Mathematics, Chapman & Hall/CRC,Boca Raton, 2000, 162 pp., £17.95, ISBN 1-584888-193This well-written book is based on theauthor�s lectures �Foundations of Analysis�at Imperial College for students in the firstterm of their degree. It contains sectionson number systems, combinatorics, geom-etry, and a basic introduction to analysisand set theory.

The aim of this book is to fill the gapbetween high-school mathematics andmathematics taught at university. In par-ticular, the reader is shown what it meansto prove something rigorously. The bookbegins with topics often taught at highschool, and goes further; for example, itstarts with a naive understanding of realnumbers and ends with least upperbounds and a proof of the existence of thenth root. The author illustrates the theorywith number of exercises.

In order to keep the book easily read-able, a few mathematical proofs are inac-curate; however, these omissions are notexplicitly mentioned, which might confusea thorough reader. This book is easy toread for anyone with a high-school mathe-matics background. (sh)

V. G. Maz�ya and S. B. Poborchi,Differentiable Functions on Bad Domains,World Scientific, Singapore, 1997, 481 pp.,£56, ISBN 981-02-2767-1

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Sobolev spaces of functions whose partialderivatives belong to Lp hold an excep-tional position among spaces of differen-tiable functions. These spaces are welladapted for solving boundary value prob-lems in the theory of partial differentialequations. For these applications, it isimportant to know under what circum-stances the inequalities and theorems onembedding, extension and traces hold.The validity depends on the quality ofdomain. If the boundary is locally repre-sented as an isometric image of a graph ofa Lipschitz function, then the domain isstill relatively good, although it can have�corners�. Most of the material of the bookis devoted to domains with non-Lipschitzsingularities or even to general domains.

The introductory chapter contains aself-contained exposition to the generaltheory of Sobolev spaces. Next, manyexamples of wild domains are shown todemonstrate the failure of basic statementsof the theory when the assumptions ondomain are violated. The second partdeals with parameter-dependent domains.Typically, a family of domains dependingon a small positive parameter ε is consid-ered. The family exhibits a certain degen-eracy as ε tends to 0. The asymptoticbehaviour of norms of extension operatorsand trace operators is then investigated.In the third part, a domain with an inneror outer cusp (peak) is mostly considered.Here, the results that depend on thedomain shape include Friedrichs� inequal-ity, Hardy�s inequality, estimates of theextension operator (also the weightedcase), trace theorems and embedding the-orems (the Sobolev inequality). Anothertype of domains considered are domainsbetween two graphs, of type {[x, y]|φ(x) <y < ψ(x)}, which even for smooth graphsmay have singularities at boundary pointsbelonging to the contact set {[x, y]| φ(x) =y = ψ(x)}. Some results on traces are con-sidered on arbitrary domains.

Specialists in function spaces willalready have this book, as well as others inthe excellent series of books by V. G.Maz�ya. For the same reason, the book iswidely known among experts in boundaryvalue problems for elliptic partial differen-tial equations. Although such equationsare not explicitly studied in the book (withone exception), the theory developedthere is needed for an analysis of suchproblems. However, the book may be use-ful and interesting for mathematiciansworking in other related areas, such as therest of PDE theory, the calculus of varia-tions, numerical analysis and the theory offunctions of several real variables. The�bad domains� are not artificial productsinvented only for counter-examples, andthe emphasis is put on simple shapes withcusps that occur in real life. The book isstrongly recommended to researchers andadvanced students. (jama)

R. J. Y. McLeod and M. L. Baart,Geometry and Interpolation of Curves andSurfaces, Cambridge University Press,Cambridge, 1998, 414 pp., £50, ISBN 0-521-32153-0

This textbook provides an elegant andserious introduction to the basic conceptsand results of (elementary) algebraicgeometry. The computational and algo-rithmic aspects provide the guidelines ofthe exposition, but the synthetic approachis also presented. The result is a pleasantcombination of intuitive and technicalexposition of the material.

The main topics include: simple inter-polation and spline theory, conic sections,an introduction to algebraic projectivegeometry, the theory of algebraic curves(including resultants), the Maclaurin-Bézout theorem, resolutions of singulari-ties and the genus of curves, and the theo-ry of algebraic surfaces. Much space isdevoted to applications and examples.

The book is designed as a text for a gen-uine course on algebraic geometry and itsapplications, and selections for shortercourses are also possible. I believe thatprofessionals seeking applied mathemat-ics, as well as students and researchers, willmake good use of this text. (jslo)

Y. Meyer and R. Coifman, Wavelets.Calderón-Zygmund and MultilinearOperators, Cambridge Studies in AdvancedMathematics 48, Cambridge University Press,Cambridge, 2000, 314 pp., £42,50, ISBN 0-521-42001-6This book is a translation of Ondelettes etopérateuers, Opérateuers de Calderón-Zygmund, by Yves Meyer, and the volumeOpérateuers multilinéaires, by R. R. Coifmanand Yves Meyer. The original numberingof the chapters and of the theorems hasbeen retained.

In this volume the theory of paradiffer-ential operators and the Cauchy kernel onLipschitz curves are discussed, with theemphasis firmly on their connection withwavelet bases. Calderón-Zygmund opera-tors have a special relationship withwavelets and with classical pseudo-differ-ential operators, of which they are aremarkable generalisation. They form thesubject of an independent theory whichthe authors expand completely andautonomously in Chapters 7-11.Multilinear analysis is one of the routesinto the non-linear problems studied inChapters 12-16.

This route is possible only for those non-linear problems with a holomorphic struc-ture, enabling them to be decomposedinto a series of multilinear terms ofincreasing complexity. The multilinearoperators turn out to be the Calderón-Zygmund operators, whose continuity isestablished using the earlier chapters.Wavelets make a final appearance, aseigenfunctions of certain realisations ofparaproducts, in the final chapter, which isdevoted to J. M. Bony�s theory of paradif-ferential operators. The bibliography lists239 items. This book can be strongly rec-ommended to those wishing to learn aboutthe mathematical foundations of operatortheory and wavelets. (knaj)

L. A. Moyé and A. S. Kapadia, DifferenceEquations with Public HealthApplications, Biostatistics: A Series of

References and Textbooks 6, Marcel Dekker,New York, 2000, 392 pp., US$165, ISBN 0-8247-0447-9 Difference equations are a powerful toolfor solving many problems arising inapplications involving health-relatedresearch. The authors start with a generalintroduction to difference equations anddevelop the iterative solution for first-order equations. The main method usedfor solving difference equations is themethod of generating functions. Generalproperties of generating functions aredescribed (scaling, the convolution princi-ple, the use of partial fractions, coefficientcollection) and the role of probability gen-erating functions is emphasised. Amongthe applications of difference equations wefind a model of unusual heart rhythm, therandom walk problem, a model of clinicvisits, run theory, drought prediction andfollow-up losses in clinical trials. The endof the book is devoted to applications ofdifference-differential equations in epi-demiology that are derived from theChapman-Kolmogorov forward equations.

The book is written in an understand-able style for students in biostatistics andfor researchers in this field. It is surpris-ing that such a fundamental concept as thecharacteristic equation is not introduced,specialised to difference equations.Mathematically, the book should be readwith some care; for example, the inter-change of derivative and infinite summa-tion is frequently used, but not discussed.Nevertheless, the book describes someuseful methods for solving differenceequations and can be recommended as asource of interesting examples of applica-tions. (ja)

P. J. Nahin, Duelling Idiots and OtherProbability Puzzlers, Princeton UniversityPress, Princeton, 2000, 232 pp., £15.95,ISBN 0-691-00979-1There are many textbooks on probabilitytheory, but unfortunately, books withinteresting problems and examples fromprobability theory are extremely rare. Thisis such an exception. The author is anexperienced teacher. His collection oftwenty-one puzzles (with solutions) isdesigned for students who are eager to trytheir skills on challenging problems.

The first problem, whose name formsthe title of the book, can be formulated asfollows. Two idiots A and B decide to duel,but they have only one six-shot revolverand only one bullet in it. First A spins thecylinder and shoots at B. If the gun doesnot fire, then B spins the cylinder andshoots at A. The process continues untilone fool shoots the other. What is theprobability that A will win? How manytrigger pulls will occur (on average) beforesomebody wins? Most of the problems areformulated in this style: for example, findthe distribution function and density ofthe length of a walk through a square gar-den if one enters it at a randomly chosenpoint on the border and continues in arandom direction. A few problems are for-mulated mathematically: for example,find the density of the random variable Z

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EMS December 200040

= XY when X and Y are independent vari-ables with rectangular distribution on (0,1). The author proposes simulating theproblems on a computer to check the the-oretical results. The last part of the bookcontains MATLAB programs that servethis purpose. To help understanding ofthe principles of simulation, a short chap-ter on random numbers generators isincluded. This book can be recommendedas inspiration for teachers of introductorycourses on probability theory. (ja)

D. Perrin, Géometrie algébrique,Mathématiques, EDP Sciences, Les Ulis, 1995,301 pp., FRF 240, ISBN 2-7296-0563-0 and2-271-05271-8This is a well-composed first course onalgebraic geometry, based on the author�scourses from 1991-94 at the UniversitéParis Sud (Orsay). Covering the materialshould take approximately 50 hours, and aquarter of this time should be devoted toexercises. The methods are entirely alge-braic, but the author requires from a read-er only a fairly standard knowledge ofalgebra. He very skilfully introduces thereally necessary ideas from commutativealgebra, and has included an appendixMémento d�algébre, where one can find acompact summary of the necessary defini-tions and results with references.

The text in fact represents an introduc-tion into contemporary algebraic geome-try. The principal notion is an algebraicvariety, always endowed with the corre-sponding sheaf. The author�s explicitlystated idea behind the exposition is tostart with problems that can be simply for-mulated, but whose solution is non-trivial.Concerning the important notion of ascheme, in the text we meet only schemesof dimension 0, but, being aware of theimportance of this notion, the authorincludes an appendix Les schémas. In thetext are many exercises and problems,including those used at the examinationsorganised by the author. In summary, themain feature of this book is a good choiceof topics and a very nice presentation ofthem. (jiva)

A. Pietsch and J. Wenzel, OrthonormalSystems and Banach Space Geometry,Encyclopedia of Mathematics and itsApplications 170, Cambridge University Press,Cambridge, 1998, 553 pp., £55, ISBN 0-521-62462-2This voluminous monograph is devoted tothe interplay between orthonormal expan-sions and Banach space geometry. A the-ory of orthonormal expansions with vec-tor-valued coefficients is presented.Besides the classical trigonometric system,other orthonormal systems are considered(Haar and Walsh functions, Rademacherfunctions and Gaussian random variables).Harmonic analysis is a starting point andclassical inequalities and special functionslead to the study of orthonormal systemsof characters on compact Abelian groups.The authors investigate numerical para-meters that can be used to quantify certainproperties of Banach spaces (such as ameasure of non-Hilbertness of the space).

The text yields a detailed insight into con-cepts involving type and co-type of Banachspaces, B-convexity, super-reflexivity, thevector-valued Fourier and Hilbert trans-forms, and the unconditionality propertyfor martingale differences. The list of sec-tions includes ideal norms, operatorideals, Khintchine constants, Sidon con-stants, Riemann, Dirichlet and Parsevalideal norms, Gauss versus Rademacher,the Maurey-Pisier theorem, J-convexity,unconditional norms, super weakly com-pact operators, and uniform convexity anduniform smoothness. The text alsoincludes challenging unsolved problems.

The book is accessible to graduate stu-dents and researchers interested in func-tional analysis and is understandable witha basic knowledge of Banach space theorytogether with a background from realanalysis, probability and algebra. (jl)

B. P. Rynne and M. A. Youngson, LinearFunctional Analysis, SpringerUndergraduate Mathematics Series, Springer,London, 2000, 273 pp., DM59, ISBN 1-85233-257-3This book provides an introduction to theideas and methods of linear functionalanalysis and is addressed mainly at under-graduate students. The opening chapteroutlines the basic ideas from linear alge-bra, metric spaces and Lebesgue measureand integration that are required through-out the book. Further chapters are devot-ed to the fundamental properties ofnormed linear and Hilbert spaces and tolinear transformations between thesespaces (also the open mapping theoremand its equivalent forms). Elementaryproperties of linear operators on Hilbertspaces and of compact operators areincluded in the next chapters. The lastchapter is concerned with applications ofprevious results to integral and differentialequations. More sophisticated theorems,such as the Hahn-Banach theorem, theprinciple of uniform boundedness and thenotions as reflexivity, are not included.

The text includes many exercises withcomplete solutions. The book is under-standable with only standard undergradu-ate linear algebra and real analysis. (jl)

Séminaire Bourbaki 1998/99, exposés850-864, Astérisque 266, SociétéMathématique de France, Paris, 2000, 483pp., FRF 450, ISBN 2-85629-090-6 This volume comprises written versions offifteen lectures at the Bourbaki Seminarduring 1998-99. The topics covered arethe following: recent proofs of the localLanglands Conjecture for GL(n) and ofKepler�s Conjecture on the densest spherepacking in R3; quantum computing; theclassification of simple Lie algebras incharacteristic p > 7; chaotic behaviour ofthe motion of inner planets in the solarsystem; spin glasses; p-adic L-functionsand p-adic integration; Brownian motionwith obstacles; finite subgroups of Liegroups; Thurston�s uniformisation theo-rem; singularities of solutions of non-lin-ear wave equations; holonomy groups; L2-methods in algebraic geometry; reso-

nances and quasi-modes. (jnek)

M. Serfati (ed.), La recherche de la vérité,L�écriture des mathématiques, ACL, Les éditionsdu Kangourou, Paris, 1999, 335 pp., ISBN2-87694-057-4This book presents 10 articles based onthe history of mathematics seminar at theUniversité Paris VII. The texts are writtenwith a view to the historical analysis ofideas and to epistemology and presentactual researches into the history of math-ematics.

The first contribution by the editor isdevoted to the birth of the procedure forsolving cubic (algebraic) equations in 16th-century Italy. The second article, writtenby M. Waldschmidt, is a historical surveyof the theory of transcendental numbersup to 1900 (when Hilbert posed his 7thproblem, whose solution was the source ofa new era for this theory); in particular, itpresents works of Liouville, Cantor,Hermite, Lindemann and Weierstrass onthis topic. In the third article, M. Barbutdescribes an approach for teaching proba-bility to beginners which is based on intu-ition of the economical value: the expect-ed value of a (discrete) random variable isaxiomatically introduced and then theKolmogorov�s �axioms� for probabilitymeasures are derived. R. Langevin sur-veys the history of integral geometry andits interactions with probability theory,measure theory, Riemannian geometryand topology, starting from 1777.

A philosophical article of M. Serfati, Ladialectique de l�indétermité, de Viète à Frege etRussell, is devoted to the representation of�datum� in symbolic mathematical writing.In P. Cegielski�s contribution, the historyof presentation, formalisation andaxiomatisation of mathematics fromancient times is described, leading to theZermelo-Fraenkel set theory as a base ofmathematics. O. Hudry�s article deals withthe four-colour problem, methods andresults of its study in historical order, andmeditates over the �correctness� of com-puter-aided proofs. The eighth article is aphilosophical survey on the conception ofscientific discovery, due to the scientist-philosophers Poincaré and Einstein: thecreative process is based on �free construc-tions of thought�; the author, M. Paty,compares it with other concepts and triesto connect them with the scientists� inven-tions. The ninth article presents the histo-ry of the negative answer to Hilbert�s tenthproblem on the solvability of diophantineequations, to which the author, Y.Matiasevitch, contributed. In the finalarticle, J. Bénabou studies the analogybetween the categories of �observable sets�(introduced in the text) and �ordinarysets�.

The book has appeared in the collectionL�écriture des mathématiques, directed by M.Serfati. It is aimed at a wide readership:mathematicians, philosophers, historians,teachers, and others interested in scienceand its development. (efa)

P. Taylor, Practical Foundations ofMathematics, Cambridge Studies in Advanced

RECENT BOOKS


Mathematics 59, Cambridge University Press,Cambridge, 199, 572 pp., £50, ISBN 0-521-63107-6�Our system conforms very closely to theway mathematical constructions have actu-ally been formulated in the twentieth cen-tury. The claim that set theory providesthe foundations of mathematics is only jus-tified via an encoding of this system, andnot directly�, says the author in paragraph2.2 after Chapter I on First order reason-ing� and after the first paragraph ofChapter II on �Constructing the numbersystem�. He does not use �the encoding�and he works with sets quite freely usingthe axiom of comprehension: if X is a set andϕ[x] is a predicate on X, then {x: X ϕ[x]} isalso a set. Beginning with Chapter IV, hisbasic mathematical tool is category theory.The use of categories as a basis for mathe-matics and the application of categoricalnotions and methods in logic and in com-puter science (more precisely, the devel-oping of logic and of computer science onthe basis of categorical notions and cate-gorical methods) has been widely andintensively examined during the last peri-od. The book presents a systematic expo-sition of this topic, explaining its ideas andsummarising the corresponding results.The author also aims to show how thesemodern ideas develop the classical onesand he presents many interesting histori-cal facts. The book is intended for pro-grammers and computer scientists, ratherthan for mathematicians and logicians, butit can be useful for both groups. (vt)

A. Terras, Fourier Analysis on FiniteGroups and Applications, LondonMathematical Society Student Texts 43,Cambridge University Press, Cambridge,1999, 442 pp., £18.95, ISBN 0-521-45108-6 and 0-521-45718-1The main goal of this book is to considerfinite analogues of symmetric spaces, suchas Rn and the Poincar upper half plane.The author describes finite analogues ofall the basic theorems in Fourier analysis,both commutative and non-commutative,including the Poisson summation formulaand the Selberg trace formula. One moti-vation for this study is to prepare theground for understanding the continuoustheory by developing its finite model. Thebook is written in such a way that it can beenjoyed by non-experts, such as advancedundergraduates, beginning graduate stu-dents, and scientists from outside mathe-matics. Several applications are included:the construction of graphs that are goodexpanders, reciprocity laws in number the-ory, the Ehrenfest model of diffusion ran-dom walks on graphs, and vibrating sys-tems and chemistry of molecules. (ae)

A. Tuganbaev, Distributive Modules andRelated Topics, Algebra, Logic andApplications Series 12, Gordon and Breach,Amsterdam, 1999, 258 pp., US$95, ISBN 90-5699-192-2 This book is not based on a few main the-orems, but is rather a collection of miscel-laneous results bearing on the centralnotion of a distributive module.

Distributive modules are characterised inseveral different ways, and are studiedsimultaneously with uniserial and Bezoutmodules (each finitely generated submod-ule is cyclic). In some cases the notionscoincide (for example, over local rings) orare closely related (over semi-perfectrings). Right distributive rings aredescribed when they are semi-perfect rightGoldie rings, right perfect or semi-perfectright noetherian. It is proved that theKrull dimension of left or right regularmodules over noetherian right distributiverings is at most 1. Then semi-distributivemodules are defined as direct sums of dis-tributive ones. Rings over which all rightmodules are semi-distributive, serial rings,and rings such that each (finitely generat-ed) right module is serial, are charac-terised. The author continues with thestudy of tensor product and flat modules;the relation of this part to the topic is quitevague. Some strong results are obtainedfor modules over right invariant rings(rings whose right ideals coincide withtwo-sided ones). After that, the question ofthe left-right symmetry of distributivity istreated and distributive modules overcommutative rings are studied. Amongother things it is shown that an endomor-phism ring of a distributive module over acommutative ring is commutative. Finally,the question of preserving distributivitywhen passing from a ring to another usingcertain classical ring constructions istouched on. A remarkable fact is that amodule M is distributive if and only if itscharacter module Hom (M,E), where E isan injective cogenerator, is End (E) dis-tributive.

The author does not separate lemmas,propositions and theorems, and the textgives the impression of a homogeneousmass of claims; orientation is very difficult.Even though it is impossible to avoid allformal mistakes in such a large text, thenumber of inaccuracies in the book isgreater than one would expect. However,despite a few drawbacks, the book pro-vides ample material for anyone interestedin the topic. (pruz)

J. S. Wilson, Profinite Groups, LondonMathematical Society Monographs New Series19, Clarendon Press, Oxford, 1998, 284 pp.,ISBN 0-19-850082-3This book presents the study of a nice classof infinite groups that are built up fromtheir finite homomorphic images and,indeed, appear in the classical literature asGalois groups of algebraic field exten-sions. Although profinite groups attractthe attention of abstract group theorists wealso meet them as topological quotients ofcompact groups.

The book starts with topological prelim-inaries and introductory chapters givingthe basics of completions, Sylow theoryand Galois theory. These chapters are fol-lowed by the study of modules over groupalgebras for a profinite group, with coeffi-cients in a profinite commutative ring.The advanced part of the book explainsthe cohomology theory of profinitegroups. Among the main results is

Lazard�s description of cohomology alge-bras of uniform pro-p groups. The finalchapter is on finitely presented pro-pgroups.

The treatment is accessible to graduatestudents and includes exercises and histor-ical and bibliographical notes. The cleartopological character of the subject isexplored in the development of the theo-ry but is also well explained at elementarylevel. (rb)

M. C. K. Yang, Introduction to StatisticalMethods in Modern Genetics, AsianMathematics Series 3, Gordon and Breach,Singapore, 2000, 247 pp., US$75, ISBN 90-5699-134-5 The importance of genetics can be feltalmost daily. The topics the author choos-es are undoubtedly biased; as he explains:�these are the topics I wanted to knowmore about when I got into this field, andI hope that many beginners will share thesame interest in them�. The topics includequestions as how a gene is found, how sci-entists have separated the genetic andenvironmental aspects of a person�s intel-ligence, how genetics has been used inagriculture so that domestic animals andcrops are constantly improved, what aDNA fingerprint is and why there are con-troversies about it, and how genes wereused to rebuild evolutionary history?

The author believes he understandsthese questions and hopes that his readerswill not find gaps in how they areanswered. The book is written mainly forstatistics students and therefore some sta-tistical background beyond elementarystatistics is assumed. The author hopesthat a year of graduate study in most sta-tistical departments is sufficient back-ground. (jant)

List of reviewers for 2000The Editor would like to thank the following fortheir reviews this year: J. Andìl, J. Antoch,R. Bashir, J. Beèváø, M. Nìmcová-Beèváøová, V. Beneš, L. Beran, J. Bureš,E. Calda, K. Èuda, A. Drápal, V. Dupaè,J. Dupaèová, J. Eisner, A. Elashvilli, M.Engliš, E. Fašangová, M. Feistauer, E.Fuchs, S. Hencl, J. Hurt, M. Hušek, M.Hušková, T. Kepka, M. Klazar, J.Kopáèek, O. Kowalski, J. Kratochvíl, M.Kruík, M. Kríek, M. Kulich, P. Lachout,M. Loebl, J. Lukeš, J. Málek, J. Malý, P.Mandl, M. Markl, J. Matoušek, D.Medková, J. Milota, J. Mlèek, K. Najzar,J. Nekovár, J. Nešetril, I. Netuka,Z.Pluhar, P. Pyrih, Š. Porubský, J. Rohn,T. Roubíèek, P. Ruièka, P. Simon, J.Slovák, V. Souèek, J. Trlifaj, V. Trnková,J. Tuma, J. Vanura, J. Veselý, L. Zajíèek.All of these are on the staff of the CharlesUniversity, Faculty of Mathematics andPhysics, Prague, except: J. Eisner, M.Engliš, M. Kruík, M. Kríek, M. Markl,D. Medková and J. Vanura (MathematicalInstitute, Czech Academy of Sciences), M.Nìmcová-Beèvárová and Š. Porubský(Technical University, Prague), J. Nekovár(Cambridge University, UK), E. Fuchs and J.Slovák (Masaryk University, Faculty ofNatural Sciences, Brno).

RECENT BOOKS

EMS December 200042

The numbers 35�38 refer to the issue numbers,in March, June, September and December,respectively; the second number is the page num-ber.

EditorialsVagn Lundsgaard Hansen (WMY2000President) 35-4Marta Sanz-Solé (3ecm organiser) 36-3Olli Martio (EMS Treasurer) 37-3Anatoly Vershik (EC retiring member)

38-3

Introducing . . .WMY2000 team 35-5New members of the ExecutiveCommittee 38-5

EMS NewsMessage from the EMS President (RolfJeltsch) 35-3Executive Committee meetings(Bedlewo, Barcelona and London)

36-6, 37-5, 38-63rd European Congress of Mathematics(3ecm) 36-5, 37-8, 37-10, 38-11Prizes awarded at 3ecm 37-12World Mathematical Year 2000

35-4, 38-12EMS Council 35-3, 37-6Report on the LIMES-Project 36-8EMS Summer Schools and Conferences

36-9, 37-14

EMS Lectures 36-9EMS Position Paper: Towards a Europeanresearch area (Luc Lemaire) 36-24EMS Committee for Women andMathematics 37-15, 38-4EMS Poster Competition 37-19

Feature articlesJeremy Gray: The Hilbert problems 1900-2000 36-10Jean-Pierre Bourguignon: A major chal-lenge for mathematicians 36-20Philip Maini: Mathematical modelling inthe biosciences 37-16Aatos Lahtinen: The pre-history of the EMS

38-14

Short articlesA. D. Gardiner: Mathematics in Englishschools 35-16Paul Jainta: Problem Corner 35-20, 37-30Ulf Rehmann: The price spiral of mathe-matics journals 38-29Hans J. Munkholm: Joint AMS-Scandinavia meeting 38-11M. Joswig & K. Polthier: Digital modelsand computer assisted proofs 38-30 Agenda des conferences mathématiques 36-19

InterviewsLars Gårding 35-6Peter Deuflhard 36-14Jaroslav Kurzweil 36-16Martin Grötschel 37-20Bernt Wegner 37-24Sir Roger Penrose 38-17Vadim G. Vizing 38-22

SocietiesDutch Mathematical Society [WiskundigGenootschap] 35-12, 36-23Danish Mathematical Society (BodilBranner) 35-14Catalan Mathematical Society (SebastiàXambó-Descamps) 36-3London Mathematical Society (AdrianRice) 37-28L�Unione Matematica Italiana (GiuseppeAnichini) 38-26

2000 anniversariesSonya Kovalevskaya (June Barrow-Green) 35-9Eugenio Beltrami (Jeremy Gray) 35-11The Hilbert Problems (Jeremy Gray)

36-10John Napier (John Fauvel) 38-24

Summer Schools and Conferences Oberwolfach programme 2001 35-19Forthcoming conferences (KathleenQuinn) 35-25, 36-28, 37-32, 38-31EMS Summer School in Edinburgh(Erkki Somersalo) 38-10EMS-SIAM Joint Conference on AppliedMathematics 37-14

Book reviewsRecent books (Ivan Netuka & VladimírSoucek) 35-32, 36-34, 37-36, 38-35Book review by Sir Michael Atiyah 36-40

MiscellaneousWMY2000 stamps

35-18, 36-27, 37-4, 37-27, 38-13

INDEX


MATHEMATICAL REVIEWSAssociate Editor

Applications and recommendations are invited for a full-time position as an Associate Editor of Mathematical Reviews (MR), tocommence as soon as possible after 1 April 2001, and no later than 1 July 2001.

The Mathematical Reviews division of the American Mathematical Society (AMS) is located in Ann Arbor, Michigan, not farfrom the campus of the University of Michigan. The editors are employees of the AMS; they also enjoy many privileges atthe University. At present, MR employs fourteen mathematical editors, about six consultants and a further sixty non-mathe-maticians. MR�s mission is to develop and maintain the AMS databases of secondary sources covering the published mathe-matical literature. The chief responsibility is the development and maintenance of the MR Database, from which all MR-relat-ed products are produced: MathSciNet, the journals Mathematical Reviews and Current Mathematical Publications, MathSciDisc,and various other derived products. The responsibilities of an Associate Editor fall primarily in the day-to-day operations ofselecting articles and books suitable for coverage in the MR database, classifying these items, determining the type of cover-age, assigning those selected for review to reviewers, editing the reviews when they are returned and correcting the galleyproofs. An individual with considerable breadth in both pure and applied mathematics is sought; preference will be given tothose applicants with expertise in the broad area of applied mathematics, and in particular in one or more of the followingareas: numerical analysis (Section 65) or mathematical economics and life sciences (91, 92). The ability to write good Englishis essential and the ability to read mathematics in major foreign languages is important. It is desirable that the applicant haveseveral years� relevant academic (or equivalent) experience beyond the Ph.D.

The twelve-month salary will be commensurate with the experience the applicant brings to the position. Interested appli-cants are encouraged to write (or telephone) for further information. Persons interested in taking extended leave from an aca-demic appointment to accept the position are encouraged to apply.

Applications (including curriculum vitae, bibliography, and name, address and phone number of at least three references)and recommendations should be sent to Dr Jane E. Kister (Executive Editor), Mathematical Reviews, P.O. Box 8604, AnnArbor, MI 48107-8604, USA (e-mail: tel: (+1)-734-996-5257; fax: (+1)-734-996-2916. The closing date for applications is 1February 2001.

The American Mathematical Society is an equal opportunity employer.

EMS NewsletterIndex for 2000

We list below information about some appoint-ments, awards and deaths that have occurred inthe past few months. Since this list is inevitablyincomplete we invite you to send appropriateinformation to the Editor [[email protected]] or to your Country representative (see Issue34) for inclusion in the next issue. Please alsosend any items you feel should be included infuture Personal Columns.

Semyon Alesker (Israel), Raphael Cerf,Emmanuel Grenier, Vincent Lafforgue,Paul Seidel and Wendelin Werner(France), Dominic Joyce and MichaelMcQuillen (UK) and Stefan Nemirovski(Russia) were awarded EMS prizes at theThird European Congress in Barcelona;details of their work can be found in EMSNewsletter 37.

Pierre Auger (Lyon), Gérard Bricogne(Orsay) and Thibauld D�Amour (IHES)have been elected to membership of theAcadémie des Sciences (Paris).

John Ball (Oxford) has been elected as aforeign member of the Académie desSciences de Paris.

Grigory Barenblatt (Russia) has beenawarded the Maxwell Prize by ICIAM (TheInternational Council for Industrial andApplied Mathematics).

Richard Borcherds (Cambridge) has beenawarded an Honorary Doctorate by theUniversity of Birmingham.

Elisabeth Busser and Gilles Cohen havebeen jointly awarded the d�Alembert Prizefor 2000 by the Société Mathématique deFrance.

Mark Chaplain (Dundee), GwynethStallard (Milton Keynes), Andrew Stuart(Warwick) and Burt Totaro (Cambridge)have been awarded Whitehead Prizes for2000 by the London Mathematical Society.

Michele Conforti (Padua) has shared the2000 Delbert Ray Fulkerson Prize for apaper on the decomposition of balancedmatrices.

Alain Connes (Paris) has been awarded aClay Research Award by the ClayMathematics Institute for revolutionisingthe field of operator algebras and invent-ing modern non-commutative geometry.

Simon Donaldson (London) has beenelected as a foreign associate of the USNational Academy of Sciences.

Awards

Ludwig Elsner (Bielefeld) has been award-ed the Hans Schneider Prize in LinearAlgebra by the International LinearAlgebra Society (ILAS).

Athanassios Fokas (London) has beenawarded the Naylor Prize for 2000 by theLondon Mathematical Society for substan-tial contributions to the theory of inte-grable systems.

Nigel Hitchin (Oxford) has been awardedthe Sylvester Medal of the Royal Society ofLondon for contributions to geometry.

Sir Tony Hoare (Cambridge) has beenawarded an Honorary Doctorate byOxford Brookes University.

John Howie (St Andrews) has been award-ed an Honorary Doctorate by the OpenUniversity, UK.

Laurent Lafforgue (Paris) has been award-ed a Clay Research Award by the ClayMathematics Institute for work on theLanglands programme

Jacques-Louis Lions (Paris) has beenawarded the Lagrange Prize by ICIAM.

Terry Lyons (Oxford) has been awardedthe Pólya Prize for 2000 by the LondonMathematical Society for fundamentalcontributions to analysis and probability.

Robert MacKay (Warwick) and PaulTownsend (Cambridge) have been electedFellows of the Royal Society of London.

Stefan Müller (Leipzig) has been awardedthe Collatz Prize by ICIAM.

Helmut Neunzert (Kaiserslautern) hasbeen awarded a SIAM Pioneer Prize byICIAM.

Hilary Ockendon (Oxford) has beenawarded an Honorary Doctorate by theUniversity of Southampton.

Sir Roger Penrose (Oxford) has beenawarded the Order of Merit. Membershipof this order is limited to 24 people; anoth-er current holder is Sir Michael Atiyah.

Sergei Pereversev (Ukraine) has beenawarded the 2000 Prize for Achievement inInformation-based Complexity for manyoutstanding contributions to the area.

Istvan Reiman and János Suranyi(Hungary) and Francisco Bellot Rosado(Valladolid) have been awarded Paul ErdosNational Awards of the World Federation

of National Mathematics Competitions(WFNMC).

Jean-Pierre Serre (Paris) has been award-ed an Honorary Doctorate by theUniversity of Durham.

Ian Stewart (Warwick) has been awardedthe Institute of Mathematics and itsApplications Gold Medal for 2000 forexceptional service to mathematics andresearch

Vera Sós (Budapest) has been elected as anHonorary Fellow of the Institute ofCombinatorics and its Applications.

John Toland (Bath) has been awarded theSenior Berwick Prize for 2000 by theLondon Mathematical Society for an out-standing piece of research.

Hendrik Van Maldeghem (Ghent) hasbeen awarded a Hall Medal by the Instituteof Combinatorics and its Applications.

Benjamin Weiss (Jerusalem) has beenelected as a foreign honorary member ofthe American Academy of Arts andSciences.

PERSONAL COLUMN

EMS December 200044

PPersonal Columnersonal Column

We regret to announce the deaths of:

Slawomir Biel (1 September 2000)

Florent J. Bureau (28 June 1999)

V. N. Fomin (23 February 2000)

Rainer Hettich (23 July 2000)

Aubrey Ingleton (28 June 2000)

Frank Leslie (15 June 2000)

Thomas Lippold (10 June 2000)

Cyril Offord (4 June 2000)

Jean-Marie Painvin (July 2000)

Ian Sneddon (4 November 2000)

Terence Stanley (15 October 2000)

Dirk Struik (21 October 2000)

Ion Suliciu (24 November 1999)

Andreas Tamanas (15 August 2000)

Lyndon Woodward (12 June 2000)

Deaths

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