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

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

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

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CONTENTS

EMS June 2000

EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

NEWSLETTER No. 36

June 2000

EMS News : Agenda, Editorial, 3ecm, Bedlewo Meeting, Limes Project ........... 2

Catalan Mathematical Society ........................................................................... 3

The Hilbert Problems ....................................................................................... 10

Interview with Peter Deuflhard ....................................................................... 14

Interview with Jaroslav Kurzweil ..................................................................... 16

A Major Challenge for Mathematicians ........................................................... 20

EMS Position Paper: Towards a European Research Area ............................. 24

Forthcoming Conferences ................................................................................. 28

Recent Books ..................................................................................................... 34

Personal Column .............................................................................................. 42

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

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

EXECUTIVE COMMITTEEPRESIDENT (1999�2002)Prof. ROLF JELTSCHSeminar for Applied MathematicsETH, CH-8092 Zürich, Switzerlande-mail: jeltsch@sam.math.ethz.chVICE-PRESIDENTSProf. ANDRZEJ PELCZAR (1997�2000)Institute of MathematicsJagellonian UniversityRaymonta 4PL-30-059 Krakow, Polande-mail: pelczar@im.uj.edu.plProf. LUC LEMAIRE (1999�2002)Department of Mathematics Université Libre de BruxellesC.P. 218 � Campus PlaineBld du TriompheB-1050 Bruxelles, Belgiume-mail: llemaire@ulb.ac.beSECRETARY (1999�2002)Prof. DAVID BRANNANDepartment of Pure Mathematics The Open UniversityWalton HallMilton Keynes MK7 6AA, UKe-mail: d.a.brannan@open.ac.ukTREASURER (1999�2002)Prof. OLLI MARTIODepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlande-mail: olli.martio@helsinki.fi ORDINARY MEMBERSProf. BODIL BRANNER (1997�2000)Department of MaathematicsTechnical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: branner@mat.dtu.dkProf. DOINA CIORANESCU (1999�2002)Laboratoire d�Analyse NumériqueUniversité Paris VI4 Place Jussieu75252 Paris Cedex 05, Francee-mail: cioran@ann.jussieu.frProf. RENZO PICCININI (1999�2002)Dipto di Matem. F. EnriquesUniversit à di MilanoVia C. Saldini 50I-20133 Milano, Italye-mail: renzo@vmimat.mat.unimi.itProf. MARTA SANZ-SOLÉ (1997�2000)Facultat de MatematiquesUniversitat de BarcelonaGran Via 585E-08007 Barcelona, Spaine-mail: sanz@cerber.mat.ub.esProf. ANATOLY VERSHIK (1997�2000)P.O.M.I., Fontanka 27191011 St Petersburg, Russiae-mail: vershik@pdmi.ras.ruEMS SECRETARIATMs. T. MÄKELÄINENDepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlandtel: (+358)-9-1912-2883fax: (+358)-9-1912-3213telex: 124690e-mail: makelain@cc.helsinki.fiwebsite: http://www.emis.de2

EMS NEWS

EMS June 2000

EMS Agenda2000

13-16 JuneEMS Lectures by Prof. Dr. George Papanicolaou (Stanford, USA).ETH, Zürich (Switzerland): Financial MathematicsContact: David Brannan, e-mail: d.a.brannan@open.ac.uk17-22 JuneEURESCO Conference in Mathematical Analysis at Castelvecchio Pascoli (Italy): Partial Differential Equations and their Applications to Geometry and PhysicsOrganiser: J. Eichhorn, Greifswald (Germany), e-mail: euresco@esf.org[This series of conferences is financed by the ESF.]3-7 JulyALHAMBRA 2000: a joint mathematical European-Arabic conference in Granada (Spain),promoted by the EMS and the Spanish Royal Mathematical SocietyContact: Ceferino Ruiz, e-mail: ruiz@ugr.es website: www.ugr.es/~alhambra20006 JulyExecutive Committee Meeting in Barcelona (Spain)7-8 JulyEMS Council Meeting at Institut d�Estudis Catalans, Carrer del Carme 47, E-08001 Barcelona(Spain), starting at 10 a.m. Contact: EMS Secretariat, e-mail: makelain@cc.helsinki.fi10-14 JulyThird European Congress of Mathematics (3ecm) in Barcelona (Spain)e-mail: 3ecm@iec.es website: www.iec.es/3ecm/24 July-3 AugustEMS Summer School, in Edinburgh (Scotland): New analytic and geometric methods in inverse problemsOrganiser: Erkki Somersalo, Otaniemi (Finland), e-mail: Erkki.Somersalo@hut.fi15 AugustDeadline for submission of material for the September issue of the EMS NewsletterContact: Robin Wilson, e-mail: r.j.wilson@open.ac.uk17 August-2 SeptemberEMS Summer School at Saint-Flour, Cantal (France): Probability theoryOrganiser: Pierre Bernard, Clermont-Ferrand (France), e-mail: bernard@ucfma.univ-bpclermont.fr28-30 AugustEMS Lectures by Prof. Dr. George Papanicolaou (Stanford, USA).University of Crete, Heraklion, Crete (Greece): Time Reversed AcousticsContact: David Brannan, e-mail: d.a.brannan@open.ac.ukAutumnFifth Diderot Mathematical Forum on Mathematics and TelecommunicationsDate and programme to be announced.Contact: Jean-Pierre Bourguignon, e-mail: jpb@ihes.fr22-27 SeptemberEURESCO Conference at Obernai, near Strasbourg (France):Number theory and Arithmetical Geometry: Motives and ArithmeticOrganiser: U. Jannsen, Regensburg (Germany), e-mail: euresco@esf.orgEURESCO Conference at San Feliu de Guixols (Spain):Geometry, Analysis and Mathematical Physics: Analysis and Spectral TheoryOrganiser: J. Sjöstrand, Palaiseau (France), e-mail: euresco@esf.org30 SeptemberDeadline for proposals for 2001 EMS LecturesContact: David Brannan, e-mail: d.a.brannan@open.ac.ukDeadline for proposals for 2002 EMS Summer SchoolsContact: Renzo Piccinini, e-mail: renzo@matapp.unimib.it11-12 NovemberExecutive Committee Meeting in London (UK), at the invitation of the London MathematicalSociety.15 NovemberDeadline for submission of material for the December issue of the EMS NewsletterContact: Robin Wilson, e-mail: r.j.wilson@open.ac.uk

200115 FebruaryDeadline for submission of material for the March issue of the EMS NewsletterContact: Robin Wilson, e-mail: r.j.wilson@open.ac.uk10-11 MarchExecutive Committee Meeting in Kaiserslautern (Germany), at the invitation of the Fraunhofer-Institut für Techno- und Wirtschafts Mathematik.9-21 JulyEMS Summer School at the Euler Institute, St Petersburg (Russia): Asymptotic combinatorics and applications to mathematical physics and representation theoryOrganiser: Anatoly Vershik, e-mail: vershik@pdmi.ras.ru3-6 September1st EMS-SIAM conference, BerlinOrganiser: Peter Deuflhard, e-mail: deuflhard@zib.de

EMS News: Committee and Agenda

One of the 25 societies of the Institute ofCatalan Studies (IEC), the CatalanMathematical Society (SCM) today facesimportant challenges. Some of these stemfrom its local contributions to WorldMathematical Year 2000, but the main onesprings from the EMS Council�s decision atBudapest in 1996 to select the SCM�s bid toorganise the Third European Congress ofMathematics (3ecm). This takes place inBarcelona from 10-14 July, in the Palau deCongressos (Congress Palace), with 1500participants expected.

That the SCM has such internationalresponsibilities should interest mathemati-cians and professionals who use mathemat-ics, and also most Catalonians, since one ofthe goals of the Rio de Janeiro declarationwas to promote mathematics as a basic ele-ment for world development. This fact, anatural consequence of the ingredients thathave contributed to the present level of sci-ence and technology, is today even morepronounced due to the progressive digitisa-tion of information and knowledge (includ-ing fundamental aspects of the financialworld), particularly with regard to the oper-ations that regulate its storing, transmissionand use.

According to its statutes, the SCM�s mis-sion includes: �to cultivate the mathematicalsciences; to spread its knowledge among theCatalan society; to foster its teaching andresearch, both pure and applied; and topublish whatsoever works that fit thesegoals�. What are the main activities in whichthe Society engages in order to fulfil theseaims?

The oldest is the organisation inCatalonia of the regional contribution to theMathematical Olympiad, including thepreparation sessions. Since 1996, it has alsoorganised the European �Kangaroo� orCangur, also called the Mathematical Feast.This year, on 16 March, there were over6000 participants from the top four levels ofsecondary education, with the collaboration(for the first time) of several city halls.

Last March the Trobada Matemàtica(Mathematical meeting) was held for thethird time. Taking place once a year, it isbased around five or six interdisciplinarylectures by mathematicians from Catalan-speaking lands.

The Society publishes two journals: theBulletí twice a year, and the SCM Notíciesfour times a year. It also publishes books,such as two classics recently translated intoCatalan: C. F. Gauss�s Disquisicions aritmè-

tiques and René Descartes� La Geometria.Among high school teachers, the RecullCangur has a growing popularity; this bookis a collection of Cangur problems, updatedevery year to prepare entrants for theMathematical Olympiad.

Another initiative is the Évariste Galoisprize for students, awarded each year. Inaddition, every two years the IEC awards theJosep Teixidor prize for the best PhD thesisand the prestigious Ferran Sunyer i Balaguerinternational prize for research memoirs.

The SCM carries out its activities in all thelands of the Catalan language and culture.Thus Cangur 1999 and 2000 were organizedjointly, for the first time in 1999, inCatalonia, the Balearic Islands and theValencia Country, while this year theTrobada was held in Valencia; the previoustwo meetings were held in Barcelona and itis hoped that the 2001 meeting will be inPalma de Mallorca. Since mathematics is auniversal language, the small linguistic vari-ations between the local tongues, whether inthe Cangur or the Trobada, are very pleasant,when perceptible, for the participants.

The Society also maintains close relationswith the institutions on which it depends forits existence. In particular, since most of thecourses that are offered need computerrooms and the Society has no such facilities,help from the mathematics schools has beenindispensable. Another example is the rela-tionship with the Institut Joan Lluís Vives (anetwork of the 18 universities in Catalan-speaking lands), which in 1999 housed the

Cangur and awarded special prizes to thetwelve best winners of the joint three com-munities; the award ceremony for this yearwill be in Barcelona in July, after the 3ecm.

The SCM is a member of the EMS and hasreciprocity agreements with several othersocieties, among them the Real SociedadMatemática Española (RSME) and theAmerican Mathematical Society (AMS).

The present SCM, with over 1000 mem-bers, was founded in 1986 as one of the foursocieties resulting from the splitting by theIEC of the Catalan Society for Physics,Chemistry and Mathematics (SCCFQM). TheSCCFQM was founded in 1931, inside theScience Section of the IEC (which itself datesfrom 1907), and was initially structured inthree branches: Physics and Chemistry;Mathematics and Engineering; andAstronomy, Meteorology and Geophysics.The 1986 SCCFQM resulted from a restruc-turing in 1968, in which the third branchdisappeared and the first was divided intotwo, Physics and Chemistry, and a secondrestructuring in 1973, in which the secondbranch was also split into two, Mathematicsand Engineering. In the late 1970s andearly 1980s, the Mathematics section expe-rienced considerable growth under the pres-idency of Manuel Castellet, who is currentlyPresident of the IEC and a member of theEMS Council: there was a substantialincrease in the membership and of the num-ber of organised activities, such as the 1980Spanish-Portuguese Meeting.

In the past twenty years, the SCM hasbeen a nexus between researchers in thethree mathematics schools (the University ofBarcelona, the Universitat Autònoma deBarcelona and the Universitat Politècnicade Catalunya), high school teachers, andmathematicians working in other organisa-tions. An institutional presentation to thegeneral public of World Mathematical Year2000 was held on 7 March at the Universityof Barcelona, jointly promoted by the SCM,the three mathematics schools and FEEM-CAT, an organisation of secondary levelteachers of mathematics. These same insti-tutions also organised a celebration ofWMY2000 at the Parliament of Catalonia inearly June.

To conclude, I hope that the Society cancontinue to play this integrating role, andthat more and more people value it as anattractive professional platform to join.

Sebastià Xambó-Descamps is President of theCatalan Mathematical Society

SOCIETIES

EMS June 2000 3

The Catalan Mathematical SocietyThe Catalan Mathematical Society

Sebastià Xambó-Descamps

In July Barcelona will host the ThirdEuropean Congress of Mathematics (3ecm).As the most visible manifestation of theEMS, the European Congresses ofMathematics aim to present the unity ofmathematics in an interdisciplinary, plur-al, and interactive framework with an inno-vative format. They provide a forum forthe diffusion of mathematical knowledgeand for cooperation between mathemati-cians.

The theme chosen for the 3ecm, Shapingthe 21st century, invites mathematicians tothink about the main streams that willguide the development of mathematics inthe near future. The scientific programmereflects this purpose. Some of the activitiesmay also encourage us to discuss the socialrole of mathematics and evaluate itsimpact on the development of new tech-nologies in industry, in education and,more generally, in culture. As in formerECMs there will also be an opportunity torecognise the talent of brilliant youngmathematicians through the award of EMSprizes.

By a happy coincidence, the 3ecm occursin World Mathematical Year 2000,declared by a resolution of UNESCO inNovember 1997. In Spain the Congress isconsidered the most important mathemat-ical activity of the year.

More than eight years ago, the leaders ofthe Catalan Mathematical Society andtheir collaborators made a determined bidfor the Congress. With effort and keen-ness, and with a background of self-confi-dence and hope, our colleagues proposedto host an ECM. Our bid was presented forthe first time in Paris in 1992 and for thesecond time, successfully, in Budapest in1996. The selection of Barcelona as the sitefor the 3ecm is a landmark in our history,the end of a peculiar political situation,and a contribution to the normalisation ofthe status of mathematics in Catalonia andin Spain compared with that of otherEuropean countries with a long traditionin this subject.

Spain, and Catalonia as a part of it, hasmissed out on the creative progress in sci-ence in the modern age. In the early 20thcentury there were many attempts to createand promote institutions that fostered sci-entific research � in particular, research inmathematics. These endeavours were sud-denly and sadly stopped by a long andcruel civil war. After April 1939 our coun-try was immersed in conditions of material

and cultural poverty in which scientificresearch was an unattainable luxury. Manyintellectual people decided, or wereforced, to choose exile as a way to be safe.Those who remained lived in an environ-ment of isolation compared with what washappening beyond the Pyrennees. Ourlack of freedom often led to introspectiveattitudes, largely incompatible with theprocess of scientific activity.

However, the internal dynamism of asociety oppressed by a political regime, but

with well-organised structures in clandes-tinity, made it possible for us to recover.From a timid attitude towards Europe weare now completely integrated in it. Thiscomplex social evolution has resulted inthe normalisation of our activities in allfields, and has led in particular to a rapiddevelopment in science. Mathematics hasbeen a part of this process. May I take thisopportunity to express our gratitude to ageneration of mathematicians who suc-ceeded in teaching mathematics to a highlevel and who trained research fellowsunder conditions that were not conduciveto enthusiasm.

A recent report by the Institute ofCatalan Studies makes an accurate quanti-tative and qualitative analysis of mathe-matical research in the period 1990-96; itsconclusion is that, in terms of production,the scientific level in our country is compa-rable with that of other European coun-tries with a long tradition in mathematics.Nevertheless, the presence of Catalan

mathematicians in an international scien-tific forum is very unusual; few of them aremembers of editorial committees of high-level mathematical journals or take part inscientific committees. Now, for the firsttime, the Catalan mathematical communi-ty is involved in the organisation of a large-scale scientific event. This clearly providesus with a wonderful opportunity toincrease the visibility of mathematics inour country and to collaborate to spreadthe aims of the EMS. This is also both a bigchallenge for us and a heavy responsibility.More than three years ago we undertookthe organisation of the 3ecm. We startedwith the Executive Committee as its kernel.Very soon we felt the need to enlarge theteam of mathematicians working for the3ecm. An organising committee wasappointed and support committees forspecific tasks were created. In addition wereceived the valuable contribution of col-leagues for special jobs. Many mathematicsstudents have volunteered to assist duringthe week of the congress.The organisation of the 3ecm would havebeen impossible without substantial back-ing from a wide variety of public and acad-emic institutions, private foundations andcorporations. Thanks to the funds provid-ed by a long list of sponsors we have beenable to implement the main objectives ofour organisational procedure:� to provide financial support for youngresearchers in mathematics, with specialattention to their professional situationand their country of origin.� to afford the scientific recognition pro-gramme designed by the EMS for excellentyoung European researchers in mathemat-ics.� to offer good facilities for an excellentscientific programme conceived by theScientific committee and the Round tablescommittee.

Attending a congress provides theopportunity to meet old friends, make newones and enjoy some free time. Barcelona,the venue of the 3ecm, is an attractive des-tination for cultural tourism. A privilegedgeographical situation, rich architectureand a large variety of museums focus theattention of a large number of visitors. Thediversity of cultural activities is one exam-ple of the dynamism of a society known forits open character, vitality and hospitality.We are convinced that the participants ofthe 3ecm will enjoy a high-quality scientificevent in a splendid environment.

EDITORIAL

EMS June 20004

BarBarcelona Editorialcelona Editorial3ecm: Shaping the 21st century3ecm: Shaping the 21st century

Marta Sanz-Solé (Barcelona)

EMS NEWS

EMS June 2000 5

EMS CouncilMeeting, 7-8 July

2000The EMS Council Meeting will

take place at the Institut d�Estudis Catalans, Carrerdel Carme 47, E-08001 Barcelona

(Spain), starting at 10 a.m. on 7 July

Contact: EMS Secretariat, e-mail:makelain@cc.helsinki.fi

Barcelona accommodationThe Council meeting room islocated at the CSIC Residence,where participants can behoused. You can see someinformation about it in:http://www.resa.es/home_eng.htmhttp://www.resa.es/csic/castellano/RESIDENCIA.HTMThe number of reservations islimited.The address is: Residence Hallfor Researchers, C/ Hospital,64, 08001 Barcelona, Spain.tel: (+34)-93-443-86-10; fax: (+34)-93-442-82-02e-mail: resa.inv@mx4.redestb.esJavier Martinez de Albeniz [albeniz@eco.ub.es], Treasurer of the SCM

3rd European Congress of MathematicsShaping the 21st Century

Barcelona, 10 � 14 July 2000Summary of main events(For further details, and information about satellite conferences, see EMS Newsletter 34.)

The plenary lectures will be given by:Robbert Dijkgraaf (Amsterdam), Hans Föllmer (Berlin), Hendrik Lenstra(Berkeley/Leiden), Yuri Manin (Bonn), Yves Meyer (Cachan), Carles Simó(Barcelona), Marie-France Vignéras (Paris), Oleg Viro (Uppsala/St Petersburg),Andrew Wiles (Princeton)Parallel lectures will be given by:Rudolf Ahlswede (Bielefeld), François Baccelli (Paris), Volker Bach (Mainz), VivianeBaladi (Paris), Joaquim Bruna (Barcelona), Xavier Cabré (Barcelona), Peter Cameron(London), Ciro Ciliberto (Rome), Zoé Chatzidakis (Paris), Gianni Dal Maso (Trieste),Jan Denef (Leuven), Barbara Fantechi (Udine), Alexander Givental (California),Alexander Goncharov (Providence), Alexander Grigor�yan (London), Michael Harris(Paris), Kurt Johansson (Stockholm), Konstantin Khanin (Edinburgh/Moscow), PekkaKoskela (Jyväskylä), Steffen Lauritzen (Aalborg), Gilles Lebeau (Palaiseau), NicholasManton (Cambridge), Ieke Moerdijk (Utrecht), Eric Opdam (Leiden), ThomasPeternell (Bayreuth), Alexander Reznikov (Durham), Henrik Schlichtkrull(Copenhagen), Bernhard Schmidt (Augsburg), Klaus Schmidt (Vienna), Bálint Tóth(Budapest)

Mini-Symposia:The speakers at each mini-symposium have been selected by the Chair (listedbelow). Quantum chaology [Sir Michael Berry, Bristol], Computer algebra [Wolfram Decker,Saarland], Mathematics in modern genetics [Peter Donnelly, Oxford], String theoryand M-theory [Michael Douglas, Rutgers], Mathematical finance: theory and practice[Hélyette Geman, Paris], Quantum computing [Sandu Popescu, Cambridge], Freeboundary problems [José Francisco Rodrigues, Lisbon], Symplectic and contact geome-try and Hamiltonian dynamics [Mikhail Sevryuk, Moscow], Curves over finite fields andcodes [Gerard van der Geer, Amsterdam], Wavelet applications in signal processing[Andrew Walden, London]

Round TablesMathematics teaching at the tertiary levelWhat is mathematics today?The impact of mathematical research on industry and vice versaThe impact of new technologies on mathematical researchBuilding networks of cooperation in mathematicsHow to increase public awareness of mathematicsShaping the 21st century

Contact AddressesCongress e-mail: Congress web site: http://www.iec.es/3ecm/or Mailing address: Societat Catalana de Matemàtiques, Institut d�Estudis CatalansCarrer del Carme, 47 E-08001 Barcelona, SpainPhone: (+34)-93-270-16-20 Fax: (+34)-93-270-11-80

EMS Secretariat

Tuulikki MäkeläinenSince the creation of the Society, the Secretariat of the EMS has been at theDepartment of Mathematics of the University of Helsinki and I have been in chargeof it. My main objective has been to try to help the Executive Committee and othersworking for the EMS by taking care of some of the day-to-day routines.My activities include preparations with the President, Secretary and Treasurer forthe Executive Committee and Council meetings; contacts with member societies;contacts with individual members; daily financial transactions, in consultation withthe Treasurer; keeping track of various bureaucratic handlings; replying to queries.Since receiving a M.Sc. degree in Mathematics at the University of Helsinki I havebeen working with mathematics and mathematicians, with the InternationalCongress of Mathematicians in Helsinki, the International Mathematical Union,with various mathematical journals, and now with the EMS. During these years Ihave met countless mathematicians. It was once suggested that I write a bookMathematicians in my life � I could, with a subtitle �with love�.Tuulikki Mäkeläinen, e-mail: tuulikki.makelainen@helsinki.fi Tuulikki Mäkeläinen

photo by Czes³aw Olech

Committee members present were: RolfJeltsch (President, in the Chair), DavidBrannan (Secretary), Bodil Branner, DoinaCioranescu, Luc Lemaire (Vice-President),Olli Martio (Treasurer), Andrzej Pelczar (Vice-President) and Marta Sanz-Solé. Apologieswere received from Renzo Piccinini andAnatoly Vershik. Also in attendance (by invita-tion) were Jean-Pierre Bourguignon, MireilleChaleyat-Maurel, Volker Mehrmann,Tuulikki Makelainen, David Salinger andBernd Wegner.

Officers� ReportsThe President announced that VolkerMehrmann had been elected Editor of theleft-hand side of EMIS, and David Salingerhad been elected EMS Publicity Officer. Hehad signed a sponsorship contract withMathDidaktik, made a presentation in Macao,and attended the AMU 2000 Pan-AfricanCongress in Cape Town. J.-P. Bourguignonhad attended a meeting in Denmark on digi-tising journals; it would be good for similarEuropean activities to work together in a co-ordinated way.

The Secretary was looking at remits of com-mittees in cases where existing ones wereavailable, and for possible remits for those thatwere lacking; he was directed to send the draftremits to Committee Chairs for comment.

The Treasurer reported briefly on thefinancial statements for the year 1999, whichwere accepted and signed. Waivers and reduc-tions of fees for certain corporate memberswho had difficulty in paying their fees wereagreed. A proposal for the Society budget forthe years 2001 and 2002 was approved andwill be submitted to the Society Council meet-ing in July 2000.

Council Meeting in Barcelona: 7-8 July 2000The delegates for the individual members forthe period 2000-03 will be: GiuseppeAnichini, Vasile Berinde, Giorgio Bolondi,Alberto Conte, C. T. J. Dodson, Jean-PierreFrancoise, Salvador S. Gomis, LaurentGuillope, Klaus Habetha, Willi Jager, TapaniKuusalo, Laszlo Marki, Andrzej Pelczar, ZeevRudnick and Gerard Tronel. The terms ofoffice of four existing members of theExecutive Committee end at the end of theyear 2000, but they are eligible for re-election.

The Executive Committee will propose toCouncil that Oslo be the site for the Councilmeeting in 2002, with dates 8-9 June 2002,and will make a recommendation as to thelocation of the European MathematicalCongress in Summer 2004.

Membership mattersIt was agreed to recommend to Council thatthe European Society for Mathematical andTheoretical Biology (ESMTB) and ITWM

Kaiserslautern be elected corporate members.The European Mathematical Trust hadreported its dissolution and thus its withdraw-al from EMS membership. It was noted that

there is a need for two separate classes forinstitutional membership, one for academicand one for commercial institutes, and thatthis should be taken care of whenever theStatutes are revised.

A reciprocity agreement between the EMSand the American Mathematical Society willbe recommended to the EMS Council forapproval. It will contain the sentence: �Thisexchange agreement shall in no way interferewith existing reciprocity agreements between

the AMS and corporate members of theEMS.� It will be proposed to Council that fur-ther reciprocity agreements can be approvedby the Executive Committee. The first such

agreements are likely to be between the EMSand the Australian Mathematical Society andthe Canadian Mathematical Society.

EMS Committee mattersThe procedure to elect members for EMScommittees was reviewed: The ExecutiveCommittee should elect the Chair, and theChair suggest committee members for theExecutive Committee to approve. When theacceptance of the committee member has

EMS NEWS

EMS June 20006

EMS Executive Committee meetingEMS Executive Committee meetingBedlewo 24-25 March 2000

Institute of Mathematics, Bedlewo

David Brannan, Rolf Jeltsch, Bernd Wegner and Olii Martio

been received, their name and address shouldbe put on EMIS.

The Summer School Committee will now beresponsible for the organisation of an annualseries of EMS lectures. A call for proposals will

be put on EMIS and in the Newsletter, with adeadline of 1 March in year N for year N+1;proposals should contain the name of the per-son responsible and location(s) for the lec-tures. Exceptionally, in the year 2000 thedeadline should be 30 September for lectures

to be given in 2001.It was agreed to draft an EMS position

paper by early May for the French EUPresidency which begins 1 July.

E. Mezzetti had accepted an invitation tochair the Women and MathematicsCommittee; her nomination of PolinaAgranovich, Nicole Berline, Bodil Branner,Kari Hag, Ina Kersten, Laura TedeschiniLalli, Irene Sciriha and Tsou Sheung-tsun asCommittee members was accepted.

Carles Broto and Rudolf Fritsch were addedto the Summer Schools Committee member-ship. It was agreed that EMS will pay for theproduction of posters for the EMS SummerSchools, as a standing arrangement. Progresswas reported on planning for summer schoolsin 2001 in St Petersburg and Prague.

I. Diaz, A. Quarteroni and Kjell-OveWidman have agreed to join the SpecialEvents Committee. The Diderot Forum on�Mathematics and Music� had been very suc-cessful.

It was agreed that information on the possi-bility of obtaining funds for East and CentralEuropean mathematicians from the Society�sCommittee for the Support of East EuropeanMathematicians should be more widely adver-tised.

Posters produced for the EMS PosterCompetition for World Mathematical Year2000 will be shown in Barcelona during theEMS Congress. An EU-supported �RaisingAwareness in Science Technology Week� willtake place throughout Europe from 6-12November 2000. It was decided to establish anEMS committee to plan the participation ofEMS in the �European Science andTechnology Week�, partly to replace theWMY2000 committee; and to support a pro-posal for a competition of papers for RaisingPublic Awareness in Mathematics.

Alberto Cabada was added to theCommittee for Developing Countries; VolkerMehrmann was added to the ElectronicPublishing Committee; and VincenzoCapasso, Jean-Paul Pier and Olli Martio hadjoined the Group on Relations with EuropeanInstitutions.

The Executive Committee decided to inves-tigate holding a meeting of editors of non-commercial European mathematics journals

during the Barcelona Congress.The Publicity Officer agreed to update

existing EMS publicity material prior to 3ecm,and revised arrangements for the productionof the annual EMS �Agenda� poster wereapproved.

Publication mattersP. Massart has joined the group of AssociateEditors of JEMS.

It was agreed that the EMS Newsletter will infuture provide a limited amount of free spacefor member societies, for their publications,meetings, etc., with equal rights for the EMSin their Newsletters, and to remind the EMSmember societies of the recommendation touse in their letterheads the phrase �memberof the European Mathematical Society�.Vivette Girault is now in charge of seekingadvertising for the EMS Newsletter.

The question of publishing a small bookleton �Early History of EMS� was discussed.

Relations with MathematicalInstitutions/OrganisationsThe 1st EMS-SIAM conference � on AppliedMathematics in a Changing World � has beenplanned to take place in Berlin from 3-6September 2001; the local organiser is PeterDeuflhard, and Vincenzo Capasso, HeinzEngl, Bjorn Engquist and Stefan Mueller werenominated to represent the EMS on theProgramme Committee.

The President is discussing several possibleforms of operation with the new President ofthe African Mathematical Union.

It was agreed that the EMS should apply formembership as a �large associated society� inICIAM.

Future Executive Committee meetings6 July 2000, in Barcelona11-12 November 2000, in LondonMarch 2001, at ITWN in Kaiserslautern

And finally ...The President thanked the Polish Academy ofSciences, Polish Mathematical Society andBogdan Bojarski for their assistance in organ-ising and supporting the meeting, and fortheir tremendous hospitality.

EMS and the International

Mathematical UnionThe EMS has been granted affiliate statusin the IMU.

According to the IMU Statutes, multi-national mathematical societies and pro-fessional societies can be affiliated with theIMU. The affiliated members have theright to participate in the IMU GeneralAssembly, but have no voting rights. Theyhave a right to submit proposals for jointactivities.

For information on the IMU, see, for instance,

http://elib.zib.de/IMU

European Research Conferences(EURESCO)

The European Science Foundation is invit-ing proposals for new conference series aswell as the continuation of establishedones. It aims to consolidate at around 50conferences per annum. The EURESCOprogramme supports series of top-levelmeetings devoted to the same general sub-ject. Normally the conferences in a seriestake place every other year.If a conference is already supported by theEuropean Commission, it can attract up toan extra 8000 euros. If not so supported,the sum can be up to 25,000 euros.Deadline: 15 September 2000Details from: www.esf.org/euresco or from

European Science Foundation,EURESCO, 1 quai Lezay-Marnèsia, 67080

Strasbourg Cedex, France

EMS NEWS

EMS June 2000 7

Prof Bojarski, host of the meeting

LIMES is a project that will be guided bythe EMS and funded within the FifthFramework Programme (FP5) of theEuropean Community. The acronymLIMES stands for Large Infrastructure inMathematics � Enhanced Services. It will dealwith the enhancement of ZentralblattMATH to a European database in mathe-matics. The project fits into the horizontalFP5 programme �Improving humanresearch potential and the socio-economicknowledge base�, providing and improvingaccess to research infrastructures. Theduration of the project is from April 2000to March 2004.

The partners of the LIMES project are:FIZ (Fachinformationszentrum Karlsruhe,Germany), MDC (Cellule de CoordinationDocumentaire Nationale pour lesMathématiques, France), Eidetica (TheNetherlands), SIBA (University of Lecce,Italy), DTV (Technical Knowledge Centreand Library of Denmark, Denmark),USDC (University of Santiago deCompostela, Spain), HMS (HellenicMathematical Society, Greece), TUB(Technical University of Berlin, Germany),with the EMS as the supervising society.The project coordinator will be FIZ. Thecontracts for the funding of the projectswere signed in March 2000 and the workstarted on 1 April.

ObjectivesThe objective of the project is to upgradethe database Zentralblatt MATH into aEuropean-based world-class database formathematics and its applications, by aprocess of technical improvement andwide Europeanisation. Upgrading theexisting database, improving the presentsystem, and developing a new distributedsystem, both for the input and output ofthe data, are necessary to allowZentralblatt MATH to use the latest devel-opments and to anticipate future develop-ments in electronic technology. This willmake Zentralblatt MATH a world refer-ence database, offering full coverage of themathematics literature world-wide, includ-ing bibliographic data, reviews and/orabstracts, indexing, classification, excellentsearch facilities, and links to offers of arti-cles, with a European basis.

Improvements will be made in threeareas:1. Improvement of content and retrievalfacilities through sophisticated furtherdevelopment of the current data sets andretrieval programs. 2.Broader and improved access to thedatabase via national access nodes and newdata distribution methods. In particular,improvement of access for isolated univer-sities in regions in economic difficultiesand in associated states of Central andEastern Europe where a mathematical tra-

dition of excellence is under economicthreat. Stimulation of usage for all kinds ofresearch, as well as for funding organisa-tions before decision making by initial sup-port of two national test sides. 3. Improved coverage and evaluation ofresearch literature via nationally distrib-uted editorial units, development of tech-nologies for efficient database production,exemplified by two further Europeanmember states.

The LIMES project sets out to achievethese goals by stimulation of usage (nation-al access nodes and centres for dissemina-tion, including development of adequatelicensing models, and links with offers ofcomplete documents); better coverage andmore precise evaluation of literaturethrough improved update procedures (forthe data gathering, but also for data distri-bution to access points) via a network ofdistributed editorial units. Networkedaccess to the database these days is mainlyvia rapidly evolving WWW technologies,with high potential for further develop-ment of innovative access and retrievalmethods, and improvements in the qualityof data (such as author and source identifi-cation through data-mining techniques).

In contrast to the centralised Americanmodel, Zentralblatt MATH will be extend-ed to a European research infrastructurewith distributed sites (Editorial offices andAccess nodes) in the member states. TheEMS, representing the scientific communi-ty concerned, has promoted these ideasamong the national societies and individ-ual members.

Since the EMS is member ofZentralblatt MATH�s Editorial Board, it isguaranteed that the results of the projectwill be available for usage in furtherresearch infrastructure partnerships.Tools and methods developed during theproject will be usable by project partnerson a royalty-free license basis. Furtherresults, documentation and informationwill be made publicly available.Commercial exploitation of resultsbeyond European research infrastructureservices will be considered.

The role of participants FIZ operates the central editorial office forZentralblatt MATH (de/). Founded in1931, it is today the longest-running inter-national abstracting and reviewing servicein the field of pure, applied and industrialmathematics. Zentralblatt MATH coversthe entire spectrum of mathematics andtheoretical computer science, with specialemphasis on areas of applications, andprovides about 60,000 reviews per year. Asa consequence of this role, FIZ will beengaged in all work packages in the pro-ject.

MDC provides mathematics research

libraries and mathematics departmentswith technical assistance on computeriseddocumentation. In 1999, the MDCreleased edbm/w3, the first module of theEuropean Database Manager forMathematics software. In LIMES MDC willbe in charge of the extensive developmentof this software-like integration of newtools for search, identification, display, andupdate, the development and integrationof improved access control management,and it will help national access nodes withthe development and installation ofimproved update procedures.

Eidetica is the most recent spin-off com-pany of CWI. It was created by researchersin the Interactive Information Engineeringtheme, combining expertise in linguisticsand mathematical clustering of dependen-cy networks for textual domains. Eideticawill mainly develop new models and toolsfor identification of author and serialnames (two of the most requested featuresfrom the user community), based on data-mining techniques, and other improve-ments of the retrieval process.

SIBA is the structure of the University ofLecce, Italy, that coordinates, managesand develops the Telematics InformationSystem for Research. SIBA coordinates theSINM project, which aims at developing aNational Information System forMathematics which will allow the Italianmathematical community to have easyaccess to the bibliographic and full-textresources available in the libraries of thefield, on-line and on CD-ROM. In LIMESSIBA will establish an editorial unit forZentralblatt MATH in Italy, identifyingand covering mathematical publications ofthe region, according to high reviewingstandards and editorial policies.

DTV was established in 1942 as the cen-tral library of the Technical University ofDenmark (DTU) and the national libraryfor engineering and applied sciences inDenmark. The role of DTV will be twofold: 1.Setting up and running an editorial unit.The unit will be responsible for covering anumber of journals and other sources, tobe defined and agreed byDTU/MATH/DTV and the coordinator.The unit will be engaged in experimentingwith modern electronic-reviewing meth-ods, especially with the results of the SIBA-UNILE system development in this area. 2.Establishing a raw data system for theeditorial units. Following the agreement ofthe publishers, this data will be made avail-able for LIMES in two ways: � for end-user current awareness purpos-es, through a special LIMES gatewayadded to the Current Awareness System ofthe EMS and to the Zentralblatt MATHdatabase, � for cataloguing reuse by the editorialunits through the export of raw catalogu-ing data to the Zentralblatt editorial sys-tem.

The role of USDC will be to propagatethe use of the database in Spain and tofacilitate the access to it, first by installinga national site server and regularly updat-ing the database and new software releases,and second by setting up a task group for

EMS June 20008

EMS NEWS

Report on the LIMES-PReport on the LIMES-Prroject oject Michael Jost and Bernd Wegner

EMS June 2000 9

stimulating the usage of the database bySpanish universities, research centres andlibraries. A network structure for informa-tion in mathematics will be developed,integrating Zentralblatt MATH withnational information offers. This work willbe undertaken in cooperation mainly withCESGA (Centro de Supercomputacion deGalicia) and RSME (Real SociedadMatemática Española). HMS, founded in1918, aims to encourage and contribute tothe study and research of mathematics andits applications, as well as to improvemathematical education in Greece. HMSwill apply a methodology similar to that ofUSDC in a different environment.

TUB is one of the largest universities inGermany, offering curricula for the fullscale of engineering sciences. The mem-bers of its mathematics department havetraditionally cooperated with ZentralblattMATH by serving as consultants for theinput to the database. In particular, theEditor-in-chief of Zentralblatt MATH is

member of that department and can lookback on a 25-year period of experience inthis position. TUB will cooperate with allpartners, providing them with continuousadvice and caring about the evaluation ofresults. An important task for TUB will bethe development of editorial tools.

As a partner in the Editorial Board ofZentralblatt MATH, the EMS has set up anInnovations Committee whose main task isto define and enact innovative features andprocedures in Zentralblatt MATH. TheEMS has two roles to play in the project: itrepresents the end-user, and gives a pan-European framework to the project as awhole. In both roles, it will be part of theProject management.

Michael Jost, FachinformationszentrumKarlsruhe, Abteilung Mathematik,Franklinsträsse 11, D-10587 BerlinBernd Wegner, TU Berlin, FachbereichMathematik, Sekr. MA 8-1, Strässe des 17 Juni135, D-10623 Berlin

The European Mathematical Society hasbeen running a successful series of EMSLectures for some years. For example, inJune this year Professor GeorgePapanicolaou (Stanford, USA) will giveseries of lectures over a period of four daysat ETH in Zurich (Switzerland) onFinancial Mathematics and over three daysat The University of Crete, Heraklion,Crete (Greece) on Time reversed acoustics.

The EMS Lectures may be in Pure orApplied Mathematics or may span bothareas. With this activity, the Society aims toencourage European mathematicians(especially young mathematicians) to meetand study together current developmentsin Mathematics and its applications. Thelectures should be given over several daysin each of at least two locations, in order togive as many as possible the opportunity toattend.

Costs of participation should be keptlow, and (if possible) grants should beavailable to people from countries whichcannot afford any financial support. TheEMS will guarantee its moral support tothe selected lecture series, and will pay forposters advertising the lectures within theEuropean mathematical community; it willalso do its best to help the organisers toraise funds, and is likely to offer somefinancial support to organisers for partici-pants who are young or come fromEuropean countries with financial difficul-ties.

Topics (which may be single or compos-ite) for the lecture series, the sites, and theorganisers of the schools will vary fromyear to year to cover a wide range of thesubject.The Society is now inviting proposals for at leastone Lecture Series for 2001. Proposals shouldcontain at least: the topic (title and shortdescription), names of the likely lecturer, the site,the timing, conditions for participants and thename and address of the organiser submittingthe proposal.Please send proposals for series of EMS Lecturesin 2001 to:Professor D. A. Brannan, Faculty ofMathematics and Computing, The OpenUniversity, Walton Hall, Milton Keynes MK76AA, United Kingdome-mail: ; fax: (+44) 1908 652140if possible by 30th September 2000.

The Society would hope to decide onproposals within a month or so.

Correction to Newsletter 35On page 18, at the end of Tony Gardiner�sarticle on Mathematics in English schools,the final line was inadvertently omitted.The final sentence should have read: Inlater years (ages 12-15) Number and algebramay occupy 55% of the time, Shape, spaceand measures 30%, and Handling data 15%.

EMS NEWS

The European Mathematical Society hasbeen running a successful series ofSummer Schools for some years now.Readers of the Newsletter will recall, forexample, Newsletter reports on a 1996Summer School in Hungary on AlgebraicGeometry [issue 20] and a 1998 SummerSchool on Wavelets in Analysis andSimulation in France [issue 29].

The series is intended to include at leasttwo summer schools each year � preferablyat least one in Pure Mathematics and atleast one in Applied Mathematics. Withthis activity, the Society aims to encourageyoung European mathematicians to meetand study together current developmentsin Mathematics and its applications.

The Society�s Summer School Committeewill also consider sponsoring proposals forsummer schools fully organised by otherinstitutions. To meet the EMS expecta-tions, each school should be at pre-doctor-al level, should last from 2 to 3 weeks, andhave 100-200 participants � mainly gradu-ate students or young mathematicianscoming from several European countries.Costs of participation should be kept low,and (if possible) grants should be availableto people from countries which cannotafford any financial support. The EMS willguarantee its moral support to the selected

schools, and will pay for posters advertis-ing the summer schools within theEuropean mathematical community; it willalso do its best to help the organisers toraise funds.

Topics (which may be single or composite)for summer schools, the sites, and theorganisers of the schools are likely to varyfrom year to year to cover a wide range ofthe subject.

The Society is now inviting proposals for at leasttwo Summer Schools for 2002. Proposals shouldcontain at least: the topic (title and shortdescription), names of likely lecturers, the site,the timing, anticipated costs, conditions for par-ticipants, organising committee membership,and name and address of the organiser submit-ting the proposal.

Please send proposals for summer schools in2002 to: Professor Renzo Piccinini, Department ofMathematics and its Applications, Università diMilano, Bicocca, Via Bicocca degli Arcimboldi8, 20126 Milano, Italy; e-mail: , fax: (+39) 02 6448 7705, phone: (+39) 02 6448 7751,if possible by 30 September 2000.

The Committee would hope to decide onproposals within a month or so.

EMS SUMMER SCHOOLS � EMS SUMMER SCHOOLS �

CALL FOR PROPOSALSCALL FOR PROPOSALS

EMS 2001 EMS 2001 LECTURES -LECTURES -

CALL FOR PROPOSALSCALL FOR PROPOSALS

In 1900 David Hilbert went to the secondInternational Congress of Mathematicians

in Paris to give an invited paper. He spokeon The Problems of Mathematics, to sucheffect that Hermann Weyl later referred toanyone who solved one of the 23 problemsthat Hilbert presented as entering the hon-ours class of mathematicians. Throughoutthe 20th century the solution of a problemwas the occasion for praise and celebra-tion.

Hilbert in 1900By 1900 Hilbert had emerged as the lead-ing mathematician in Germany. He wasfamous for his solution of the major prob-lems of invariant theory, and for his greatZahlbericht, or Report on the theory of numbers,published in 1896. In 1899, at Klein�srequest, Hilbert published The foundationsof geometry as part of the commemorationsof Gauss and Weber in Göttingen. Hurwitzsaw clearly that the implications of that lit-tle book reached far beyond its immediatefield. As he put in a letter to Hilbert: You have opened up an immeasurable field ofmathematical investigation which can be calledthe �mathematics of axioms� and which goes farbeyond the domain of geometry.

Hilbert was therefore poised to lead theinternational community of mathemati-cians. He consulted with his friendsMinkowski and Hurwitz, and Minkowskiadvised him to seize the moment, writing: Most alluring would be the attempt to look intothe future, in other words, a characterisation of

the problems to which the mathematicians shouldturn in the future. With this, you might con-ceivably have people talking about your speecheven decades from now.

And if the Congress itself was something

of a shambles, Hilbert�s written textnonetheless made its impact, just asMinkowski had predicted.

One reason is undoubtedly that, forHilbert, problem solving and theory for-mation went hand in hand. Indeed, someof his problems are not problems at all, butwhole programmes of research: Hilbert�s6th problem, for example, calls for the

axiomatisation of physics. But he gavegood reasons for caring about his prob-lems, and bound it all up with his inspiringoptimism. In opposition to Emil du Bois-Reymond�s fashionable academic pes-

simism, Hilbert insisted that in mathemat-ics: We can know, and we shall know.

Hilbert presents his problems Hilbert�s problems came in four groups. Inthe first group were six foundational ones,starting with an analysis of the real num-bers using Cantorian set theory, andincluding a call for axioms for arithmetic,

EMS June 200010

FEATURE

The HilbertThe Hilbertprproblemsoblems

1900-20001900-2000Jeremy Gray

Hilbert�s ProblemsAsterisks denote the ten problems presented during the Paris lecture

1. * Cantor�s problem of the cardinal number of the continuum (continuum hypoth-esis)

2. * The compatibility of the arithmetical axioms3. The equality of two volumes of two tetrahedra with equal bases and equal alti-

tudes4. The problem of the straight line as the shortest distance between two points

(alternative geometries)5. Lie�s concept of a continuous group of transformations, without the assumption

of the differentiability of the functions defining the group6. * A mathematical treatment of the axioms of physics7. * The irrationality and transcendence of certain numbers8. * Problems of prime numbers (including the Riemann hypothesis)9. A proof of the most general law of reciprocity in any number field10. The determination of the solvability of a Diophantine equation11. Quadratic forms with any algebraic numerical coefficients12. The extension of Kronecker�s theorem on abelian fields to any algebraic realm

of rationality13. * A proof of the impossibility of solving any equation of the 7th degree by means

of functions of only two arguments14. A proof of the finiteness of certain complete systems of functions15. A rigorous foundation for Schubert�s enumerative calculus16. * The problem of the topology of algebraic curves and surfaces17. The expression of definite forms by squares18. The building-up of space from congruent polyhedra (n-dimensional crystallog-

raphy groups, etc.) 19. * Determining whether the solutions of regular problems in the calculus of vari-

ations are necessarily analytic20. The general problem of boundary values (variational problems)21. * A proof of the existence of linear differential equations with a prescribed mon-

odromy group22. * The uniformisation of analytic relations by means of automorphic functions23. Further development of the methods of the calculus of variations

David Hilbert, around 1900

Hermann Minkowski Adolf Hurwitz

EMS June 2000

and the challenge to axiomatise physics.The next six drew on his study of (algebra-ic) number theory, and culminated with hisrevival of Kronecker�s Jugendtraum, and thethird set of six were a mixed bag of alge-braic and geometric problems covering avariety of topics. In the last group were fiveproblems in analysis � the direction thatHilbert�s own interests were going. Heasked for a proof that suitably smoothelliptic partial differential equations havethe type of solutions that physical intuition(and many a German physics textbook)suggest, even though it had been knownsince the 1870s that the general problemof that kind does not. He made a specificproposal for advancing the general theoryof the calculus of variations.

The Hilbert problems very quickly suc-ceeded in getting young mathematicians tocreate the future that Hilbert had conjec-tured � not always accurately, however, aswe shall see. The Russian mathematicianSerge Bernstein travelled from Paris toGöttingen in 1904 to present his proofthat, under the conditions Hilbert had stat-ed, elliptic partial differential equations dohave analytic solutions. It would take morethan a single book to describe the resultsproduced in this mushrooming field in the20th century. The opportunistic andpower-seeking Bieberbach was anotherman drawn to the Hilbert problems for thefame that they could confer. In 1908 heshowed that there are only a finite numberof crystallographic groups in a Euclideanspace of any dimension, thus solving partof Hilbert�s 18th problem. His results con-firmed that there are 17 patterns of thiskind for the Euclidean plane, and 219 pat-terns (or crystal structures) for Euclidean3-dimensional space.

Hilbert and axiomatisation Hilbert himself did not work exclusively onthe problems, and nor did most of hismany students, who were instead oftendrawn in the early 1900s to the study of�Hilbert space�. After 1909, when his friendMinkowski died, Hilbert became more andmore interested in questions of axiomatisa-tion and the foundations of mathematics.His interest in axiomatising physics was tolead to several lecture courses but few pub-lications, and has accordingly been muchmisunderstood (see Leo Corry�s work forthe new picture). Hilbert lectured regular-ly and well, and he promoted the view thatan axiom was a fundamental idea fromwhich many others followed. So axiomsplayed a crucial role in organising theo-ries, as they had done in his Foundations ofgeometry.

As the years went by, the consistency ofarithmetic seemed more of a challenge. In1917 Hilbert wrote: Since the examination of the consistency is a taskthat cannot be avoided, it appears necessary toaxiomatise logic itself and to prove that numbertheory and set-theory are only parts of logic.

He imposed high standards on the task.What was needed, he said, were proofsthat: � every mathematical question be solvablein principle;

11

FEATURE

Title page of ICM Paris Proceeding

Hilbert�s lecture from the Paris Proceddings

� every result be checkable; � every mathematical question be decid-able in a finite number of steps: this is thedecision problem which asks: can a givenstatement be shown to beprovable/refutable in a given theory, or is itindependent?

By 1922 Hilbert became locked into adispute with Brouwer, who believed that

the human mind was strikingly limited inits ability to deal with infinite sets. Hilbert�shopes for the future of mathematics werefurther darkened by the fact that his beststudent, Hermann Weyl, also foundBrouwer�s ideas attractive, and for a timeseemed willing to forgo certain mathemat-

ical arguments on philosophical grounds.Hilbert attempted to get round their argu-ments in 1922 by defining mathematics asa system of signs and distinguishing care-fully between mathematics and meta-math-ematics. Mathematics was to be identifiedwith the stock of provable formulae, whileinference about the content was onlyadmissible at the level of a new meta-math-

ematics. On the basis of this distinctionbetween valid formulae and their interpre-tations he called for a proof theory � trulya remarkable idea.

In the event Brouwer withdrew from thecontest, Weyl found that he needed classi-cal analysis for his work on Lie groups, andthe crisis passed, although Hilbert�s ideasabout proof theory did not convince theexperts. But Hilbert continued, with hisassistants Ackermann and Bernays, and in1931 he published a forceful re-statementof his views on the occasion of his becom-ing an honorary citizen of his nativeKönigsberg. The lecture ends with a mov-ing affirmation of his deepest belief aboutmathematics: There are absolutely no unsolvable problems.Instead of the foolish ignorabimus, our answer ison the contrary: We must know, We shall know.

Ironically, the day before Hilbert lec-tured, the young Austrian logician KurtGödel also lectured in Königsberg on his

incompleteness theorem, the work that ispopularly said to have killed Hilbert�s pro-gramme, even though, as Gödel said in thefamous paper: I wish to note expressly that [this theorem does]not contradict Hilbert�s formalistic viewpoint.

The problem is that finding such proofshas proved elusive. No agreed place tostand has been found which compels uni-versal assent (such as the elementary rulesof logic) and which delivers all of set theo-ry. On the other hand, powerful negativeresults continued to accumulate. WhenAlan Turing showed in 1936 that the deci-sion problem is also unsolvable, the origi-nal hopes for Hilbert�s programme wereall in tatters.

The Hilbert problems between the WarsThe Hilbert problems themselves haveperhaps proved a more enduring part ofhis legacy than Hilbert�s own work onmathematical logic. Those on number the-ory have turned out, given Hilbert�s owninterests, to have been particularly wellput. Ironically, Hilbert�s own formulation

of Kronecker�s Jugendtraum was mislead-ing, but the Japanese mathematicianTakagi, who had studied under Hilbert inGöttingen in 1900 while writing his thesisfor the University of Tokyo, succeededwith the crucial generalisation to theabelian case in 1920. In 1923 Emil Artinbumped into Takagi�s work almost by acci-

dent and the work he did as a resultenabled Hasse to solve Hilbert�s 9thProblem (calling for a general reciprocitylaw) in 1927. Hasse had meanwhile alsosolved Hilbert�s 11th problem, on quadrat-ic forms, in 1923. In 1926 Artin solvedHilbert�s 17th problem in a restricted, butnonetheless very general, case. Hilbert�s7th Problem (to show that ab is an irra-tional transcendental number when a isalgebraic and not equal to 0 or 1, and b isirrational and algebraic) was solved by theSoviet mathematician A. O. Gelfond in1934, and by Th. Schneider independent-ly later the same year.

The Bourbaki connection After the Second World War the strugglefor the heart of mathematics was won bythe pure mathematicians. In this context avigorous contest developed for the mantleof Hilbert. Should it go to mathematicallogicians, or to applied mathematiciansworking in the tradition of Courant andHilbert, or to the number theorists andalgebraists? The most powerful advocate ofthe last of these views, both by word anddeed, was Bourbaki. André Weil and JeanDieudonné shared a view of mathematicsthat put Hilbert centre stage, although itwas a Hilbert created in their own image.They espoused the axiomatic method,which Dieudonné claimed has revealed unsuspected analogies and permit-ted extended generalizations; the origin of themodern developments of algebra, topology andgroup theory is to he found only in the employ-ment of axiomatic methods.

EMS June 200012

FEATURE

L E J Brouwer

Hermann Weyl

Kurt Gödel

Emil Artin

André Weil made the connection toHilbert even more forcefully. In 1947 hequoted Hilbert: A branch of science is full of life, as long as itoffers an abundance of problems; a lack of prob-lems is a sign of death.

Great problems, said Weil, furnish thedaily bread on which the mathematicianthrives. And turning to Hilbert�s famouslist of problems he singled out the 5thproblem (then still unsolved) on Liegroups, the Riemann hypothesis, and theproblem of generalising the theorems ofKronecker�s Jugendtraum which �stillescapes us, in spite of the conjectures ofHilbert himself and the efforts of hispupils�.

The Hilbert problems after the War Hilbert�s Paris address had brilliantly unit-ed problems with their theoretical contextand so, for a generation, did Bourbaki.The stock of the Hilbert problems rosewith theirs. The remaining ones acquiredan extra cachet for having held out, andthey too began to fall. The 5th problem, oncharacterising Lie groups, was solvedthrough the work of Gleason and ofMontgomery and Zippin in 1952. The14th problem on rings of invariants was

interpreted geometrically by Zariski andthen solved in the negative by Nagata in1959. In the 1960s and 1970s mathemati-cal logicians turned back to the Hilbertproblems. A high point was reached withthe award of a Fields Medal in 1966 to PaulCohen for showing that the axiom ofchoice and the continuum hypothesis areindependent of the other axioms of settheory. The axiom of choice is not one ofHilbert�s problems, but in calling for a con-sistent set of axioms for arithmetic, Hilberthad opened the way to similar analyses ofall of mathematics, and indeed by estab-lishing the independence of the continu-

um hypothesis, Cohen did indeed settleHilbert�s 1st problem.

In the early 1970s the Russian mathe-matician Yuri Matijasevich solved Hilbert�s10th problem (Is there a finite process

which determines if a polynomial equationis solvable in integers?) in the negative,using earlier papers by the Americanmathematician Martin Davis, HilaryPutnam, and Julia Robinson. JuliaRobinson had been close to solving thisproblem herself, and a powerful collabora-tion developed between her andMatijasevich, despite the Cold War. It fol-lows from their work that had Hilbert�s10th problem been answered positively,Goldbach�s conjecture (mentioned byHilbert in Paris) would have beenanswered in the negative � a connectionthat Hilbert surely had not suspected.

Few of Hilbert�s problems have dwin-dled with the years. Perhaps the completesolution of the 5th problem is a case inpoint. In 1986 Jean-Pierre Serre said: Still, it is true that sometimes a theory can bekilled. A well-known example is Hilbert�s 5th

problem. � When I was a young topologist, thatwas a problem I really wanted to solve � but Icould get nowhere. It was Gleason, andMontgomery-Zippin, who solved it, and theirsolution all but killed the problem. What else is

there to find in this direction? I can only thinkof one question: [but it] seems quite hard � but asolution would have no application whatsoever,as far as I can see.

Yet there are numerous exampleswhere the Hilbert touch has proved bene-ficial. The great development of algebraicnumber theory was surely animated byHilbert�s problems. The topic of partialdifferential equations was re-opened byHilbert with his 19th problem, and muchof that rich theory can be traced back tothe work it inspired. Not only are theimplications of the solutions and reformu-lations of the problems still to be workedout, some of the problems are still alive.Russian mathematicians have recentlyshown that in two cases the original �solu-tions� were flawed, and have given differ-ent and rigorous accounts. Aspects of bothhalves of the 16th problem (on real alge-braic curves and on vector fields) are stillopen questions. Hilbert�s deepest vision, ofthe intricate dance of theory and problemsin mathematics, is one that all mathemati-cians share, but few have articulated as wellas he.

Further reading F. Browder (ed.), Mathematical DevelopmentsArising from Hilbert�s Problems, Symposia in PureMathematics, American Mathematical Society28, Providence, Rhode Island, 1976. Leo Corry, �David Hilbert and theAxiomatization of Physics (1894�1905)�, Archivefor History of Exact Sciences 51 (1997), 83�198.Jeremy J. Gray, We must know, we shall know; aHistory of the Hilbert Problems, Oxford UniversityPress, 2000 (to appear).

Jeremy Gray is a Senior Lecturer in the Department ofPure Mathematics, The Open University, UK.

FEATURE

EMS June 2000 13

Martin Davis, Julia Robinson and Yuri Matijasevich

Andrew Gleason

Paul Cohen

You will be organising the first EMS-SIAMmeeting in Berlin from 2-6 September 2001.The main theme is �Applied Mathematics inour Changing World�. How would youdescribe what applied mathematics is?In principle, most substantial mathematics(including pure mathematics) is applicable.We now know that Hardy failed in aimingat mathematics that would surely never beapplicable � never say �never�. The ques-tion is, whether applicable mathematics,when applied, is already applied mathe-matics. It is my repeated experience thatsolving hard real-life problems usuallygoes far beyond just taking mathematicaltools off the shelf. Hence, �AppliedMathematics�, as I understand it, deals withquestions arising from quantitative prob-lems outside mathematics and the corre-sponding development of new mathemat-ics for their solution. This characterisationcertainly includes most parts of computa-tional mathematics (such as the design ofefficient algorithms in differential equa-tions or discrete mathematics), but alsomodelling (such as the regularisation ofinverse problems or the homogenisation ofmulti-scale problems), and visualisation, asfar as it requires the development of newalgorithmic tools (such as feature extrac-tion in fluid dynamics).

Where do you see the differences betweenapplied mathematics, industrial mathemat-ics and computational sciences and engi-neering?Even though I have worked quite a bit onreally tough industrial problems, I am nottoo fond of the definition of �industrialmathematics�: the only distinguishing fea-ture in my definition (above) of appliedmathematics might be that these problemsarise in industry. Computational sciencesand engineering, however, deserve somefurther thoughts: in my understanding,CSE does not say anything on the methodsthat are used to solve the problems, so thisfield necessarily includes good, bad andindifferent. I use the term �scientific com-puting� to describe the discipline that addi-tionally takes care of the efficiency ofmethods applied to computational prob-lems from sciences and engineering.There is, however, a close mutual depen-dence between scientific computing andcomputational science: only if computa-tional science brings up interesting prob-lems can scientific computing then workon the methods and, in turn, help to raisethe complexity barrier of tractable compu-tational problems in science and engineer-ing.

Do you think that applied mathematics haschanged in the last 20 years?Yes, indeed, but this is too long a story. Inshort: there has been a clear drive fromapplicable mathematics to applied mathe-matics. Today mathematics interferes withunexpectedly many facets of our life, part-ly visible to the public, but mostly invisible.

And even mathematicians recognise thatfact. The beauty of a theorem is more andmore compared with its use. Fortunatelyquite often beauty and use go nicelytogether (I still love beauty). But beautyalone has a short life, as we all know.

Do you expect further changes in thefuture? In what direction and with whatspeed?I hate to answer such questions, which typ-ically are asked at the end of a mathemati-cian�s active professional life. I would pre-fer to continue to be active for a while. I seea lot of changes, but would like to wait withthe answer.

You also chair the scientific committee.How are these changes reflected in the pro-gramme?The title �Applied Mathematics in ourChanging World� was deliberately chosenby the initiators from SIAM and the EMS.The key idea is to focus on some (not all!)aspects of mathematics where it interfereswith the typically fast changes in our realworld. Generally speaking, the fastestchanges arise in fields such as finance, net-work technology, environment and life sci-ences.

Where would you see the main emphasis inthe programme?Topics to be mentioned (among many oth-ers) are medicine (computationally assistedsurgery, immune system modelling),biotechnology (density functional theory,drug design � not to be confused withdesigner drugs!), material sciences (com-posite materials, crack modelling and sim-ulation), nanoscale technology (fast opticalnetworks, quantum electronics), climateresearch, communication (network simula-tion and optimisation), traffic (route guid-ance and planning), finance (option pric-ing, derivative trading � of course, not themathematical derivatives!), speech andimage recognition (filter analysis).

How many participants do you expect?We aim at an attendance of from 500 to1000 participants.

Do you have any particular message youwould like to give to potential participants?Whatever you have contributed in terms ofmathematics or want to contribute to thechanges of our world, come to the excitingcity of Berlin and contribute your part tothe conference community!

To change topic, you are President of theKonrad Zuse Zentrum in Berlin. Some ofour readers attended the ICM 1998 organ-ised by Martin Grötschel, also from thisinstitute, but might not know much about it.Could you describe it briefly?Here I am tempted to refer to a recent arti-cle in SIAM NEWS (March 1999) that Iwrote for just that purpose. Its title was ZIB� computational mathematics as key to keytechnology. There you can find a lot ofdetails about ZIB�s organisation, mission,funding, and scientific achievements.

You have been director since 1986 when itwas founded. What have been the mainachievements in this period?Yes, I founded ZIB in the scientific sense in1986, and Martin Grötschel joined ZIB in1991. As for the main achievements in thepast: there have been many extremely suc-cessful projects in Discrete Mathematicsand Indiscrete Mathematics (my subject!)that have been pursued within ZIB. TheZIB story has been a real success story sofar, I�m proud to say. Why don�t you justgive our web address below?

Perhaps the main project structure ofour research may be worth mentioningexplicitly: we distinguish between long-term projects (usually basic research, ori-ented towards the treatment of difficultproblem classes and funded by ZIB), medi-um-term projects (usually publicly fundedon the basis of competitive proposals), andshort-term projects (industrial problems).This span from basic research via algo-rithm development to real-life applicationshas turned out to be quite fruitful over theyears. Behind this concept stands my firmbelief that genuine progress in the solutionof hard real-world problems typicallyrequires thorough and creative basicresearch.

EMS June 200014

INTERVIEW

Interview with Interview with PPeter Deuflhareter Deuflhardd

(President, Konrad Zuse Zentrum, Berlin)interviewer: Rolf Jeltsch, EMS President (Zurich)

You have almost another decade of direc-torship ahead of you. In which directionshould ZIB evolve?We are always on our way and have lots ofnew ideas � enough for the years to come.

ZIB is one of the German supercomputercentres, and may even be THE centre. Willsupercomputing increase at ZIB and inGermany?First of all, supercomputing should bemore than just computing on supercom-puters. At least, this is the spirit of super-computing at ZIB, as opposed to othercomparable institutes. At ZIB, supercom-puting is part of our branch of ComputerScience. Secondly, as for the installation ofsupercomputers, Germany has a govern-mental road-map for the development ofsuch centres. The trouble is that only veryfew problems are especially well suited forsupercomputers, and very few expertsknow how to deal efficiently with them. Asa consequence, supercomputing does nothave a broad lobby, but must be regardedas a strategic issue for any industrial society� spoken in the direction of all the govern-ments of readers of this interview.

Should there be more supercomputer cen-tres in Europe?There is a natural decay time of supercom-puter technology, which more-or-less leadsto an optimal number of such centres foreach community. It depends on what youdefine as the community.

Currently most of these centres are fundednationally. Should the European Union gointo the business of maintaining such cen-tres?My personal opinion is that in the long runsupercomputers will be a commonEuropean strategic issue, but in the shortterm the organisational details would killthe performance if an EU option wererealised too early.

What is the relation between such centres inEurope and in North America, especiallythe USA?Like other such centres we have interna-tional connections both in research andnetworking, such as in metacomputing.For example, we did computations aboutthe collision of neutron stars and rotatingblack holes (the numerical solution ofEinstein�s equations of general relativity)simultaneously on four supercomputers inSan Diego, Urbana, Munich and Berlin.

Let me ask some questions related to yourpersonal research. In your early days inMunich and Heidelberg you worked onordinary differential equations. You notonly did theory but also software andapplied examples. How would you positionthe development of software at universities?There is no general answer to this kind ofquestion: some university groups take a lotof care in developing software, some don�t.However, the short-term restrictions ontypical university jobs are a severe obstaclefor any software developments. In addi-

tion, this kind of work is still insufficientlyvalued in terms of a university career. Thismay partly have to do with the fact that atsome places software development is donewithout any theoretical component � anattempt often prone to lead to rathershort-sighted software solutions. Back toyour question. In principle, universitiescould be good places for software design.

I would like to make an additionalremark in this context: without any realapplication work, which will usuallyinclude software implementation, universi-ty careers have increasingly been sloweddown in recent years, and speeded up oth-erwise. Just look at the many new universi-ty openings (and more to come in a fewyears when the present generation leaves)in the fields of scientific computing or indiscrete mathematics. It seems that a goodstrategy mix is necessary for young peopleto grade up successfully. By the way, we arealways looking for enthusiastic young grad-uates to join us.

How can they maintain the software if PhDstudents who did it have left the university?Attempts have been made to establish net-works to maintain software, but in myopinion with doubtful success. I do notknow a really good remedy for this unde-sirable effect. On the other hand, redesignof software, which is necessary anyway aftera certain period, is strongly supported bysome present languages like C++ or Javawhich partly reduces pressure from thisissue.

Have your style of working and the scientif-ic topics chosen changed when moving froma purely university environment inHeidelberg to the ZIB, where you are alsoattached to the Freie Universität Berlin?I still hold the Chair of Scientific

Computing at the Free University ofBerlin. And I still love teaching in closecontact with the next generation.Unfortunately, like other departmentsworld-wide, we do not have enough stu-dents. This is in extreme contradiction tothe excellent professional perspectives ofcomputational mathematicians that haveworked in real applications. As for my per-sonal research topics: with the ZIB asorganisational background, I have beenable to look at more complex computa-tional problems that play a role in real lifeand require a larger group. So my activitiesin Berlin cover all kinds of differentialequations, far beyond the Munich orHeidelberg topics of ODEs. To give oneexample, I worked on adaptive multi-levelfinite element methods for PDEs, especial-ly in cancer therapy and microwave tech-nology.

What are the research challenges you stillwant to attack in the future?

I still have a number of personal scientificdreams. Let me just mention the next stepsin the development of conformationaldynamics, which will play a decisive role inthe design of biomolecules, or the furtherspread of topics in quantitative medicinewhich opens a new door to computationalmathematics. It seems to be a coincidencethat some of my favourite topics appear onthe list of topics of the SIAM-EMS confer-ence!

I am sure that with you as pivotal person inthe organisation of the first EMS-SIAMconference, it will be a success.

For more information on the interviewee, seehttp://www.zib.de/deuflhard/for more information on the ZIB, seehttp://www.zib.de/

EMS June 2000 15

INTERVIEW

Design of biomolecules in a 3D virtual laboratory at ZIB

Jaroslav Kurzweil was born on 7 May 1926 inPrague. He graduated in mathematics in 1949at the Charles University in Prague and start-ed his scientific career as a student of ProfessorVojtech Jarník in the metric theory ofDiophantine approximations. Until 1951 hewas at the Czech Technical University and wasthen a research student at the MathematicalInstitute which became a part of the newlyfounded Czechoslovak Academy of Sciences inPrague. After a research visit to Poznan withW. Orlicz he studied analytic operationsdefined in a Banach space with values inanother Banach space. He has been fullyemployed at the Mathematical Institute of theAcademy since 1954. In 1955 he received hisCandidate of Sciences degree and was appoint-ed Head of the Department of OrdinaryDifferential Equations at the Institute. At thattime his work in ODEs started. His main inter-est was in stability theory, invariant manifoldsand problems concerning the continuousdependence of solutions on a parameter. In thislast field he created a new definition of an inte-gral based on Riemann-type integral sums thatturned out to be equivalent to the Perron ornarrow Denjoy integral. This was indepen-dently discovered by Ralph Henstock, and sincethe early 1960s has been studied intensively. In1958 Kurzweil received his Doctor of Sciencedegree, and in 1966 was appointed FullProfessor of Mathematics.

Soon after the revolution in 1989 Kurzweilwas elected Director of the MathematicalInstitute of the Academy, and he has held thisoffice till 1996. Since 1990 he has beenChairman of the Board for Accreditation,appointed by the Government of the CzechRepublic to give approval to educational activ-ities of all institutions of higher education.

Jaroslav Kurzweil has been Chairman of theUnion of Czech Mathematicians and Physicistssince 1996. In 1999 he was reelected for thenext three years.

The deep trace of Kurzweil�s work and per-sonality in Czech mathematics is evident andincontestable to everybody who has met him,either as a mathematician or simply as a man.

What made you choose mathematics as thesubject of your studies?Mathematics has attracted me since mychildhood. For example, I remember thatwhen I was a pupil at primary school Iasked adults questions of the type: �howcan we calculate the heights of a triangleif we know its sides?�. I wondered how itwas possible to construct the triangle andmeasure its heights, but I could not calcu-late them. I had no luck and nobodyhelped me. I was told not to rack my

brains, that all this was well known andthat it is impossible to find anything newin this direction. Later, as a student at sec-

ondary school (�gymnasium�) I used tosolve competition problems published bythe Union of CzechoslovakMathematicians and Physicists in the jour-nal Rozhledy matematicko-fyzikální(Horizons in Maths and Physics), intend-ed for secondary school pupils. I read arti-cles appearing in the Journal and alsosome popular books, such as On theApollonius problem, On Einstein�s theory ofrelativity and gravitation, and Introduction togroup theory. In 1945 I completed my sec-ondary education. I briefly considered thestudy of medicine, having been influencedby care I received in my childhood duringa long hospital stay after an accident.Nevertheless, I enrolled in the Faculty ofScience of Charles University in Prague tostudy mathematics and physics. My deci-sion was also affected by the fact that allsecondary school maths appeared quiteclear to me without learning it.

You became a university student in 1945,immediately after the end of World War II.How did your studies continue at thattime?The Nazi occupation in 1939-45 resultedin a drastic restriction of Czech schooleducation. All Czech institutions of higherlearning were closed from November1939 till the end of World War II. Thenumber of secondary schools wasreduced, the number of students allowed

to continue, and the teaching of somesubjects (such as history and literature)was prohibited. Students at higher gradeswere forced to work in Bohemia or inGermany, mostly in the war industry. Theinstitutions of higher education resumedtheir activities immediately after the endof the war, but the situation was not nor-mal. First, the number of professors haddecreased considerably: some had beenexecuted, some did not survive Nazi con-centration camps, others had died by nat-ural causes. Many younger teachers hadfound other jobs. This caused a genera-tion gap in the community of universityteachers, since people younger than 45years of age were quite an exception.Many former students deprived of thepossibility of studying after the abolitionof Czech universities in 1939 desired tocomplete their studies, and others appliedwho had graduated from secondaryschools during the war.

Thus the education in Mathematics Ireceived at the Faculty of Science from1945 to 1949 was rather unbalanced. Iattended excellent lectures and seminarsin topology and number theory, but Ilearnt nothing of differential equations,probability theory or numerical methods(no computers were then available). Igraduated with a thesis in metrical num-ber theory; it dealt with Hausdorff mea-sures of sets of real numbers that can bewell approximated by rational numbers. Iwas initiated in this field by VojtechJarník. Postgraduate studies had onlystarted in the early 1950s. After 1950some universities and research instituteswere allowed to enrol students in doctoralstudies and offer them scholarships. Thepreparation of the legislative frameworkonly started then, and so the first disserta-tions might not have been defended until1956.

After a short period of teaching at theFaculty of Mechanical Engineering of theCzech Technical University I arrived in1951 as a postgraduate student(�Aspirant�) to the institute, which laterbecame the Mathematical Institute of theAcademy of Sciences. The generation gapwas still strongly felt, there was a lot ofimprovisation and much was left to theinitiative and efforts of individual stu-dents. In the initial period of communistrule in Czechoslovakia it was difficult totravel even to politically allied countries,but I was lucky enough to spend severalmonths in 1953 studying in Poland. Mystay in Poznan with Professor Wladyslaw

EMS June 200016

INTERVIEW

Looking back: Looking back: JarJaroslav Koslav Kurzweilurzweil

(Prague)interviewers: Jirí Jarník and �tefan Schwabik

Orlicz was of extraordinary importancefor me. I succeeded, among other things,to solve the problem of whether it is pos-sible to uniformly approximate a functiondefined on a real Banach space by an ana-lytic function (defined by means of apower series). It appeared that in the classof uniformly convex spaces the affirmativeanswer holds if and only if there exists apolynomial q such that q(0) = 0 and infq(x); |x| = 1 > 0. In the special case of thespaces Lp and lp this occurs if and only if pis even.

What prospects did you have after com-pleting your doctoral studies?The institutions of higher learning werethen introducing new specialisations andthey also started to work with smallergroups of students; they therefore neededyounger experts for assistants� jobs. Alsoresearch institutes took on qualified spe-cialists, so there was no lack of job offers.In the Fifties our country did not havespecialists in some branches of mathemat-ics who would be able to communicateand cooperate with specialists in non-mathematical sciences or with peoplefrom �practical� professions. TheMathematical Institute was supposed toact to improve this situation. This was whythe following working groups were found-ed in it:� probability theory and mathematicalstatistics� theory of PDEs and mathematical theo-ry of elasticity� theory of ODEs� numerical methods� elementary mathematics.

When I finished my doctoral studies,the Director of the Mathematical InstituteVladimír Kníchal offered that I shouldremain in the Institute as a research work-er, allowing me to choose which group Iwanted to work in. I chose the group ofODEs, which was gradually increasing andin 1955 was established as a department.Two topics for the start of the researchwere given by Kníchal � the existence ofperiodic solutions, and Lyapunov func-tions and stability.

The work on the first problem was start-ed by a smaller group that ultimatelyformed an independent department deal-ing with evolution problems in all theirbreadth. I started to work on the othertopic, and soon succeeded in proving inthe case of asymptotic stability that thereexists a Lyapunov function with the corre-sponding properties and is of class C8,even if the right-hand side of the differ-ential equation is just continuous. Niceresults concerning Lyapunov characterisa-tion of stability under permanently actingperturbations and integral stability wereobtained by Ivo Vrkoc, who later startedresearch in stochastic perturbations andgradually formed a small group whichnow works in the theory of stochastic dif-ferential equations in infinite-dimension-al spaces.

Thus we have arrived at your researchactivities ...

I remained, together with several othercolleagues, closer to ODEs. We con-tributed to control theory in its begin-nings and to the theory of differentialinclusions. The research in the theory ofinvariant manifolds was based on a geo-metrical approach; its results wereapplied to ODEs, and in particular to dif-ferential equations with delayed argu-ment. Occasionally I was also engaged inother topics in mathematical analysis.There is not much sense in presenting thedetails of these results. On the otherhand, I would like to mention anotherbranch of research that stemmed fromour work in the ODEs.In classical theorems on the continuousdependence of the solutions of the differ-ential equation x� = f (x, t, e) on a para-meter e, it is assumed that f is a continu-ous function of the triple of variables (x, t,e). In the Fifties cases were considered inwhich the right-hand side of the differen-tial equation contains terms of the typecos t/e, f(x)/e cos t/e, so that with respect toe it is not continuous in a neighbourhoodof 0. Nevertheless, the primitive functionswith respect to t of such terms are e sin t/e,f(x) e2sin t/e, which are continuous. Thus anatural question arises under what addi-tional conditions the solutions of theequation depend continuously on theparameter e, provided that we assumeonly that the primitive function to f withrespect to t is continuous. When I waslooking for general principles governingthese problems I was working withLebesgue integrable functions andRiemann sums. This provided the inspira-tion for a modification of the Riemannintegral; the modification of the usual�epsilon-delta� definition consists inadmitting for d not only positive constantsbut positive functions (called gauges).

I published the modified definition forthe first time in 1957 and used it to intro-duce generalised differential equationsand establish results on the continuousdependence of solutions of generaliseddifferential equations on a parameter. Inthis way the concept of S(P)-integrablefunctions was obtained.

It was also shown there that a real func-tion g is S(P)-integrable if and only if it isPerron integrable, and that the integral inthe sense of S(P)-integrability is thePerron integral of the function g. (In thenotation S(P), the symbol S correspondsto the approach by Riemann sums, whileP refers to Perron integration.) I regardedthe definition as a contribution to the the-ory of the Perron integral; I did not dealwith it for about the next two decades, mymain field of interest having been ODEsand related problems.

The same definition was publishedindependently in 1961 by RalphHenstock. In 1969 McShane published ageneral theory of integration in abstractspaces, which in the special case of inte-gration on a one-dimensional intervalreduces to another modification of mydefinition. McShane proved that by thismodification we obtain exactly theLebesgue integrable functions.

The above works, including Henstock�s1963 monograph, aroused interest, whichled to the publication of several mono-graphs and a long series of papers.Numerous modifications of the originaldefinition were investigated, in particularfor integration in multi-dimensionalspaces, characterisation of primitives,convergence of sequences of functions. Inparticular, such modifications of the defi-nition for multi-dimensional integrationwere found which allow the usual trans-formation formula and can be used to for-mulate a general Stokes theorem (forexample, by W. Pfeffer in 1993). I joinedthis trend by a monograph in 1980 and bya series of papers that I published with co-authors in the years 1983-99.

Until recently, in connection with thesummation approach to integration, theconvergence of sequences of functions wasinvestigated, but no attention was paid tothe topologisation of the space of inte-grable functions. As we know, S(P) inte-gration is Perron integration and, as such,is an extension of Lebesgue integration.Therefore, if for f: [a, b] → R andg: [a, b] → R, f = g a.e., and if f is S(P)-integrable, then g is S(P)-integrable andthe two functions have a common primi-tive. It is natural to study both the conver-gence and the topology in the vectorspace Int S(P), which is the space of equiv-alence classes of S(P)-integrable functions.A convergence of sequences with reason-able properties can be introduced in thespace Int S(P), and we arrive at the fol-lowing problem: Does there exist a topol-ogy Cal U on Int S(P) such that (Int S(P),Cal U) is a complete topological vectorspace and every sequence convergent inthe above sense is convergent in (Int S(P),Cal U) as well?

In view of the first requirement thetopology may not be too coarse, by the lat-ter it may not be too fine. I have recentlyproved that the answer is affirmative.However, if we add the condition that (IntS(P), Cal U) is a locally convex space, thenthe answer is negative.

I am author or co-author of 57 paperswhich concern the above subjects, whichwere published between 1957 and 1991.

What brought you to work in mathemat-ics teaching?I got to mathematics teaching without anyeffort. In the so-called socialist regime thepublishing of textbooks for elementaryand secondary schools was directed cen-trally. There were publishing houses inboth Prague and Bratislava, whose taskwas to publish textbooks which were thenused in all schools throughout the coun-try. When the higher authorities decidedthat new textbooks for primary and sec-ondary schools were to be published, thetwo publishing houses divided the task ofpreparing mathematical textbooks forindividual grades. Then the publishinghouses started to seek for the authors andreferees. I do not know what had to beapproved �up there� before the publisherswere allowed to appoint the authors andreferees. Other institutions involved in

EMS June 2000 17

INTERVIEW

the process were the PedagogicalResearch Institutes, both in Prague andBratislava. They published, in small print-ings, the so-called experimental textbooksof various subjects. Those employed in theInstitutes were authors of most of theseexperimental textbooks and were oftenauthors or co-authors of the standardones also. In the Seventies I was asked bythe Director of the MathematicalInstitute, Jirí Fábera, to write a review ofone of these textbooks. The review wasunfavourable, sometimes even biting. Iwas therefore surprised when the head ofthe Mathematical Department of thePedagogical Research Institute, JanaMüllerová, thanked me and asked me togive my critical opinion of future manu-scripts. The result was that for severalyears I served as an advisor to thoseauthors of mathematical textbooks whoshowed interest.

In the Seventies a very formal approachto elementary mathematics, based onnaive set theory, was put through. First-grade pupils in primary schools (6-7 yearsold) learnt the union and the intersectionof sets, and set diagrams became idols.Maybe the authors actually believed thatthey had discovered how to make mathsattractive and accessible, but above all,they had enough influence to enforce thissystem all over the country. I was co-author of a letter addressed to thePedagogical Research Institute andsigned by about ten outstanding Czechmathematicians working at institutions ofhigher learning. We pointed out that theprocess of cognition of children proceedsalong other lines, that abstractions have tobe prepared by experience, and that theyare the top and not the start of the cogni-tive process. The letter was not published,but was circulated and had a positiveeffect. New authors were appointed forsome textbooks and the whole system wasmoderated.

The first director of the MathematicalInstitute (in the Fifties) was the famoustopologist Eduard Cech. He believed thatmaths could be taught at secondaryschools much better and more effectively.He had already organised meetings withteachers before World War II, lecturedand discussed with them, and even wroteseveral secondary school textbooks him-self. It was this tradition which led to thefoundation of a Department of Didacticsof Mathematics in the MathematicalInstitute. When the head of thisDepartment Josef Horálek died in 1984, Iwas put in charge of its activities; of courseI did not participate in the proper work ofthe Department, but I willingly discussedwith its members their experience, prob-lems and plans. In 1985 I becameChairman of the Committee forMathematical Terminology of the Unionof Czechoslovak Mathematicians andPhysicists. This Committee had earlierprepared for publication a list of mathe-matical terms and symbols, which wentmildly beyond the needs of primary andsecondary schools. The aim of theCommittee was to prepare a dictionary of

school mathematics with explanations ofthe terms. The first edition appeared in1981 and the Committee soon started toprepare an improved edition. It met reg-ularly on seminars, found authors forindividual chapters and in 1989 it hadprepared about 80 per cent of the manu-script; only two chapters were missing �unfortunately, the key ones concerninggeometry and computers. However, after1989 most of its members were chal-lenged with more urgent tasks and theCommittee ceased to work.

In 1987 the Union awarded me its high-est distinction and elected me its hon-orary member.

We now come to the year 1989...One consequence of the generation gapin the mathematical community, which Imentioned in connection with the Nazioccupation, was that I and some of mycontemporaries were entrusted very earlywith organisational and decision-makingtasks. I served as a member of numerousexpert committees, and committees foracademic and scientific degrees. From1956 to 1970 I was the Chief Editor of themathematical journal now publishedunder the name Mathematica Bohemica.

The year 1989 brought changes andhopes. The system of central directing wasto be abandoned and self-government atall levels was to be restored in all areas ofsociety, including education and science.Naturally I desired to participate in bring-ing about these changes. In 1990 I waselected Director of the Mathematical

Institute of the Academy of Sciences, andI remained in this office till 1996. In thesame year I was elected the Chairman ofthe Union of Czech Mathematicians andPhysicists.

As a consequence of the new law oninstitutions of higher studies, passed in1990, the government appointed theAccreditation Board that was entrustedwith important tasks regarding institu-tions of higher learning. I acted asChairman of this Board from 1990 to2000. I have strictly applied the opinionthat statistical and scientometrical datahave an auxiliary character and that everydecision of the Accreditation Boardrequires expert substantiation. In 1991 I

was elected a member of the ScientificBoard of the Charles University and ofthe Scientific Board of the Faculty ofMathematics and Physics. In 1994 I wasamong the founding members of theLearned Society of the Czech Republic,which continues a tradition going back tothe second half of the 18th century.

You lived during the periods when yourcountry was governed by Nazis and thenby Communists. How do you see the pre-sent?It is indeed a miracle that both regimes,Nazi entirely and Communist except forBelorussia, have disappeared fromEurope. Nonetheless, the introduction ofeach was made possible only due to a fail-ure of Western civilisation and education.This, in my opinion, is a warning for us aswell as for the future generations.

EMS June 200018

INTERVIEW

These stamp were issued in 1987 to commemorate the 125th anniversary of the Union ofCzechoslovak Mathematicians and Physicists. The middle one features Kurzweil�s teacher VojtechJarník (right).

The Belgian Mathematical Society and theDeutsche Mathematiker Vereinigung will holdtheir first joint meeting at the University ofLiege, 8-10 June 2001. It will consist of six50-minute plenary talks and 10-12 specialsessions.I. Daubechies (Princeton), C. Deninger(Münster) and P. Deuflhard (Berlin) havealready agreed to deliver plenary talks: The following six special sessions (andorganisers) have already been fixed:Arithmetic geometry (G. Cornelissen, AnnetteHuber, K. Künnemann, W. Veys);Functional analysis and functional analyticmethods in partial differential equations (K. D.Bierstedt, P. Laubin, R. Meise, J. Schmets);Global analysis (J. Brüning, L. Lemaire);Optimisation (M. Goemans, M. Grötschel,Ph. Toint, J. Zowe);Ordinary differential equations and dynamicsystems (F. Dumortier, B. Fiedler, J.Mawhin, J. Scheurle);Representation theory (D. Happel, C. Ringel,F. Van Oystaeyen, A. Verschoren).Information on the meeting can be foundon the homepage:http://math-www.uni-paderborn.de/Liege2001/Everybody interested in participating tothe meeting is asked to preregister bysending an e-mail to , mentioning theirname, institution, e-mail address and2001 BMS-DMV Meeting. All mathemati-cians who have pre-registered this way willautomatically receive the SecondAnnouncement of the meeting in October2000.

The Société Mathématique de France(SMF) maintains an electronic mailing listallowing each member to be periodicallyand freely informed of the SMF�s latestissues.

To sign up, send an e-mail message tosympa@ens.fr without subject or signatureand containing exclusively the line: subscribe publications-smf full name (forexample: subscribe publications-smf DupontJean)

To unsubscribe to the mailing list, sendan e-mail message to sympa@ens.fr withoutsubject or signature and containing exclu-sively the line:unsubscribe publications-smfBeware: the body of your message shouldcontain only raw text (neither html norWord attachment).

SociétéMathématique deFrance electronic

mailing list

Joint BMS-DMV meeting of the

Belgian and GermanMathematical Societies

8-10 June 2001This message is to announce a French ini-tiative on the occasion of WorldMathematical Year 2000. It is a web sys-tem for the collection of information on allconferences and seminar announcementsin all branches of mathematics that anyonewith a web access can query in a highlyuser-friendly way.It is called the Agenda des conferences mathé-matiques, ACM for short. Its web address isat http://acm.emath.fr. It was created byStephane Cordier (a maitre de conferencesat the University of Paris 6 and an activemember of the SMAI), has been function-ing in France since February 1998 andpresently indexes some 100 ongoing semi-nar series. Its set-up is financially support-ed by a small grant from the FrenchMinistry of Research, with the understand-ing that it should be made freely availableeverywhere, with the full support of theSMF (Société Mathématique de France)and the SMAI (Société de MathématiquesAppliquées et Industrielles). TheEuropean Mathematical Society also sup-ports it.ACM has recently been expanding to othercountries (Italy, Germany, Canada, etc.).Its user interface comes in several lan-guages and the aim is to include more, butthis will require the help of native speakersof other languages. National and regionalcorrespondents are also needed; some arealready available for France, Austria,Belgium, Germany, Italy and Canada.Mirrors will also be welcome everywherepossible, including on EMIS.At the core of ACM is a large database,but the interesting concept is that thedatabase gets created and maintainedautomatically, by querying the web pagesof the seminar series, conferences or col-loquia. Of course, it requires the activecooperation of their organisers, but this isstraightforward for them: the URL of theweb pages of the announcements have tobe indicated to the ACM-master and thesepages have to include some simple tagsthat the ACM engine can recognise. Thisprocedure allows for instantaneousupdates. Detailed explanations are avail-able on the ACM server, in English,German, Italian, Spanish and French).On the user�s side, the search engine canbe queried in these languages and can becustomised to each user�s profile or pro-files (which can be saved as Bookmarks orFavorites). Simple and complex searchesare possible, including geographicalregions and time periods. Setting up aprofile is a little lengthy the first time, butwell worth it because, once bookmarked,it performs a new search in one singlestep and keeps the date as relative, notabsolute. The result of a search is a list-ing of all the seminar talks, symposia talksand conferences in the database whichsatisfy the criteria, with their title and a

ACM (Agenda des

conferences

mathématiques)

link to their web sites. One even has theoption of asking for a timely e-mailreminder.As the geographical coverage increases,more numerous and focused regions willbe included. Apart from the present tengeneral subfields of mathematics that canbe selected for a focused search, each talkcan present a list of MSC2000 codes thatcan be used in the searches. The ACMdatabase also includes information onmathematical conferences, congresses,workshops and colloquia from the confer-ence calendar maintained by the AtlasMathematical Conference Abstracts(AMCA) at http://at.yorku.ca/amca/. It canalso include the information located on theEMIS conference board. For those loca-tions that do not have a web page, it will bepossible in France to post the announce-ments on a special site that will automati-cally be queried by the ACM robot. Thistype of procedure should be considered inall other countries at the country level, orby the national Societies of Mathematics.It already exists on the EMIS ConferenceBoard and can be used in European coun-tries.This an evolving project and the hope ofour Societies is that by being a tool for thewhole Mathematical community, Europe-wide and World-wide, it can evolve to sat-isfy the needs of all. The purpose of thismessage is to inform you of this new ser-vice, to suggest that you use it as often asyou need it and, most importantly, thatyou ask the organisers of meetings andseminars to contribute to ACM by usingthe procedure that allows it to extract therelevant information for its database.The purpose of this message is also toinform you that the whole concept, andsoftware, is a gift of the FrenchMathematical Societies and community totheir fellow mathematicians around theworld; but for ACM to succeed, the contri-bution of everyone is now needed, andmost notably our European fellowSocieties.One final note: ACM does not wish tocompete with other sites that proposesimilar information. On the contrary, it iswilling to incorporate all information thatresides on these sites, with direct links tothem. An example of such cooperation iswith ACMA, as noted above.We sincerely believe that the ACM con-cept can play an outstanding role in fos-tering the sense of a world-wide mathe-matical community. At the dawn of thethird millennium, mathematics can againbe an example to all other disciplines onthat front.Thank you for all the help you can give tothis idea, the care of which the SMF andSMAI now put into your hands.Rolf Jeltsch, President of the EMS, e-mail:jeltsch@math.ethz.chMireille Martin-Deschamps, President ofthe SMF, e-mail: mmd@math.uvsq.frPatrick Le Tallec, President of the SMAI, e-mail: letallec@ceremade.dauphine.frAlain Damlamian, Past-President of theSMAI, e-mail: damla@math.polytechnique.fr

EMS June 2000 19

NEWS

In today�s society, mathematics is presentmore widely than ever before, but this israrely acknowledged, even by mathemati-cians. As a result, citizens need to be com-fortable with mathematical situations, so asto be helped with the ever-complex choic-es they have to make.

This article arose from discussions withmany people inside and outside theresearch mathematical community (in par-ticular, teachers) and the perception Ideveloped from them. For the most part,mathematicians have not appreciated thelarge demand for mathematics in societyand the consequences they must draw fromthis. The starting point is to analyse whatmakes mathematics special among the sci-ences and how it is perceived by those whohave to deal with it in some way � parents,professionals, managers, and mathemati-cians themselves, as teachers or asresearchers.

Elaborating on the ordinary perceptionof mathematics, I present what I see as anew relationship of mathematics to society, andconsider detailed implications on its inter-nal organisation, together with needs to beaddressed in its development and teach-ing.

1.The ordinary perception of mathemat-ics Its role in basic teaching Around the world,one�s first encounters with mathematicshappen in elementary schools. Indeed, allcountries consider mathematics to be partof the basic training that children musthave. The form it takes varies from onecountry to the next, but basic operationson numbers are always taught, togetherwith fundamental geometric figures.Mathematics is also viewed everywhere as asubject of its own. From this perspective wecannot say that mathematics lies hidden.

When one digs deeper and considers theactual pedagogical practices, one realisesthat, although mathematics appears every-where, the intellectual process on whichmathematics is based is not always fullypresented. For example, the abstractionprocess that lies at the heart of mathemat-ics is often hidden behind routines thathave value but are only one side of thecoin. The same can be said of the researchprocess. Mathematics is sometimes pre-sented as an activity in which one has to

function automatically, when as mathe-maticians we know how much one has tostruggle to solve a problem or break it intosimpler pieces. In other words, if a mini-mum of technique is required to functionin a mathematical environment, the mean-

ing of what we manipulate cannot beignored if we are to ensure a learningprocess with long-lasting effects.

Some fallacies about mathematics Forsome people, mathematics is just the languageof the quantitative. They base their judge-ment on the fact that mathematics enter-tains a special relationship with language;this opinion is shared by some of our fellowscientists. We mathematicians know howwrong this is, and how much effort goesinto building concepts, establishing facts,and following avenues we once thoughtplausible but turn out to be dead ends.This widespread belief forces us to consid-er more carefully how mathematics inter-acts with other disciplines and what is theexact nature of the mathematical modelsthat appear ever more frequently.

Second, mathematics is seen as a dead sub-ject. One of the most frequent questionsthat professional mathematicians hearfrom laypersons and from media people,is: �What can you do in a subject whereeverything is known?� So many examplescan be used to prove the contrary that we

too often omit to make this point publicly� in particular, to our students. In my opin-ion, it is of fundamental importance that,as early as possible (probably even beforesecondary school), students meet the ideathat mathematics is still in the making, andthat thousands of mathematicians aroundthe world are working at solving outstand-ing and extremely varied problems.

These misconceptions are encapsulatedin the unfortunate reputation of the word�abstract�, when it is precisely in abstractingfrom a situation what makes a phenome-non general that mathematics is at its best.Henri Poincaré gave a great definition ofmathematics. For him, �faire des mathé-matiques, c�est donner le même nom à deschoses différentes�. From that point ofview, opposing �concrete� and �abstract� canbe nonsense. Indeed, extracting from acomplex situation what constitutes its coreis what makes us really understand howthings work. We must therefore spend timein making this process explicit, emphasisethe role it plays in the foundation of math-ematics. In my opinion, here lies one of themain roots of the quid pro quo that current-ly affects the relationship of mathematicsto society.

Some other sources of misunderstandingin connection with education A fact thatrelates the study of mathematics in schoolto that of languages is that mathematicalknowledge is accumulated, so that a lack ofunderstanding at one stage affects the abil-ity to understand and progress later. Thisdependence is not as strong in other disci-plines such as physics, history or geogra-phy.

Another aspect that is largely countrydependent is the role that mathematicalperformance plays in the selection processat school. From this point of view, Franceused to be an extreme case because of itsdual system of higher education, splitbetween the so-called �Grandes Écoles� andthe universities, and the weight attributedto mathematics in the entrance competi-tions to the �Grandes Écoles�.

A partial and brief conclusion: It is clearto me that the image mathematicians give to thegeneral public (an image that is amplified by themedia) needs to be worked at to make it more inline with what they actually do. This requires aconsiderable and sustained effort on the part ofmathematicians.

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EMS June 200020

A major challenge for A major challenge for mathematiciansmathematicians

The undervaluing of the role of mathematics in today�s society

Jean-Pierre Bourguignon

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EMS June 2000 21

2.What is the nature of mathematics? In this section I shall concentrate on a fewtopics that make mathematics special as a science. It is my opinion that mathemati-cians have not worked enough on this frontand are not always in a good position to�make a case� for their needs. This impliesthat more thought needs to be given to theobjective basis for the misconceptions dis-cussed in the previous section. Last but notleast, we must take a historical perspectivewhen discussing these issues, becausemathematics has a long history (maybe thelongest of all sciences), and because the sit-uation is changing very rapidly. So what isso special about mathematics?

A special relationship to language Toaddress this question we must realise thatthe way one expresses mathematics evolveswith time. The now well-accepted idea thatmathematics can be totally formalised wasstated as a general paradigm only at theend of the 19th century. It is largely aphilosophical stand and has little in com-mon with the actual work done by mostmathematicians or teachers.

We can view this evolution as the culmi-nation of the need to use a precise language.Since mathematicians like to borrow theirvocabulary from ordinary language, wemust realise that there is a price to pay �namely, that students may resent it as aform of robbery, while we would ratherview it as a form of poetry. More attentionshould be devoted to explaining why we soappropriate the common language, andwhy we would be in trouble without it. Weshould behave as responsible citizens andexplain what we make of commonresources. After all, are we not forcing ourstudents to live double lives, depending onwhether we are talking in a mathematicalcontext or not? This is not easy and shouldbe identified as such.

The constraint of expressing mathemat-ics by a well-controlled language is certain-ly one reason for seeing mathematics onlyas a language. We require so much atten-tion to this matter, and for good reasons,that we should carefully explain why,describe those mathematical activities thatare not affected by this practice, and iden-tify times when forcing oneself to stickstrictly to a formalised language is a hand-icap. This particular difficulty appearsboth for advanced students and for begin-ners, and may be one of the major hurdlesthat young researchers must pass.

A special link with truth Confronted witha mathematical statement, a mathemati-cian�s goal is to prove that it is �true�. Whatdoes this mean? Of course, now that math-ematicians agree to work in the context ofa potentially completely formalised theory,its meaning can be none other than �thestatement can be deduced from the basicaxioms agreed upon�. In fact, if we are toaddress this question in the context of therelations of mathematics to society, we areforced to take it in a broader, more philo-sophical, perspective because we have toconfront it with reality. This amounts todeciding whether there is a mathematicalreality of which this statement is a part,

and of the status of this reality in relationwith ordinary �sensible� reality. On thisbroader issue, mathematicians have differ-ent views: from the �Platonists� at one endof the spectrum who believe that they �dis-cover� new territories and new facts of themathematical world, to the �intuitionists� atthe other end who view mathematical con-structions as purely human scaffoldingsbased on consensus opinions among arestricted community � in other words,that mathematics is �invented�. I have nodoubt that the vast majority of mathemati-cians are closer to the Platonist viewpointthan to the other one, or at least that �theyspend a good portion of their professionaltime behaving as if they were�, as AndréWeil once put it.

The possibility of using today resultsestablished centuries ago is another aspectof the special relation that mathematicshas with truth. It gives it the possibility oftranscending civilisations, and makes itstransmission much easier since it is a prioriless ideology dependent. This founds itsclaim for universality. Of course, this is allbased on the role that proofs play in math-ematics, in spite of their changing appear-ance with time. Making this dimensionaccessible to students is of course the majorchallenge that teachers face at all levels.This gives us mathematicians a specialobligation towards the history of our disci-pline, and requires that it be properlyincorporated in our teaching, both as doc-umented proof of its evolution, and of theprogress it has made. Facts accessible inthe past to a very restrictive group of greatminds can be, and are, correctly assimilat-ed today by large crowds of students,because new concepts and new techniqueshave been developed in the meantime.Here lies for me the best antidote againstthe spreading of the devastating belief thatmathematics is a dead subject. Let me notein passing that efforts in the direction justmentioned will also give us stronger argu-ments to support our worldwide strugglefor keeping good libraries. To do present-day mathematics, we need access to docu-ments that may be a century old or more,and this need will stay with us in the future.

Another consequence of the relationshipof mathematics to truth, also worth point-ing out in connection with education, is thefollowing. A person who has understood amathematical fact becomes able to chal-lenge his or her teacher, and his or herclassmates. This experience of winning byexercising the mind is of the utmostimportance in training critical minds andgiving them access to autonomous think-ing. This of course needs to be thoughtthrough in relation to the very diversesocial environments in which schools areoperating.

The unbelievable success story of mathemat-ics throughout history Mathematicians tendto be shy about their field, even though thehistory of mathematics is one of the mostextraordinary success stories of the humanadventure. Think that at the end of the17th century precise mechanisms weredesigned in order to say something suffi-

ciently pertinent about infinity to be ableto handle major questions connected withit. This was the birth of modern analysis.Think that, in spite of formidable evidenceimposed on mathematicians by everydayexperience, the early part of the 19th cen-tury saw the birth of new geometries, gen-eralising the well-established Euclideangeometry, and opening the way to a moreradical broadening of the scope thatenabled general relativity to take ground.Think that major problems concerningprime numbers and Diophantine equa-tions have been solved one after another,thanks to numerous technical and moreconceptual achievements. Think that ran-dom quantities can now be properly esti-mated on the basis of a few preciselyanalysed assumptions. Many of these revo-lutions were the basis for major changes inthe organisation of society: we come backto this later. Why are we so shy?

3. A new relationship to societyThis part is the most controversial, since Idare to offer some perspectives for thefuture. It may be good to start by taking alast look at the evolution of our disciplinein the 20th century, in order to identify thechanges that I claim force us to continue.

Mathematics and mathematicians in the20th century In the 20th century, mathe-matics, as many other sciences, enjoyedunsurpassed development. Many newresults were obtained � some very spectac-ular, such as the solution of the Fermatproblem � but the real change was in thefiner web of results and the extraordinarybroadening and ramification of areas thattransformed our discipline. At the end ofthe 20th century, the yearly production ofresearch articles had reached the 60,000mark, when in the 1950s there had beenonly 5000. This is undoubtedly themechanical effect of the growth of themathematical community throughout thepast century. Whereas some 250 partici-pants took part in the 1900 Paris Congress,the century ended with the BerlinCongress that hosted 4000. Today, thenumber of mathematicians activelyengaged in research is estimated at about50,000. At the same time we should keepin mind that the number of biologists isnow estimated at about 1 million.

In a more qualitative vein, the 20th cen-tury witnessed the generalisation of the useof the axiomatic approach. The distance itallows to take from the immediate realitysurrounding us has also some drawbacks �namely, that of providing arguments forpeople who say that mathematics hasdeserted the real world. Symmetrically, italso frees mathematicians from relatingwhat they do to real-life problems.Nevertheless, there is no doubt in my mindthat this newly gained freedom is a fantas-tic advantage, provided that we do not iso-late ourselves and pretend that we havenothing to say and nothing to learn fromother scientists, from engineers, and fromsociety in general.

Before leaving this question of theaxiomatic method, it is important to men-tion that it also had an extraordinary

impact on the teaching of mathematics atan advanced level, allowing us to teachmuch more quickly to a wider group of stu-dents. In that sense, it has greatly con-tributed to the development of the mathe-matical enterprise. The impact it has hadon the teaching at lower levels has beendiversely appreciated but, if some experi-ences were really negative, the status quowas no more acceptable, for the very rea-sons that we are presently examining.

Coupling mathematics with powerfulcomputing tools Already by the 1970s, newmeans of computation had appeared inthe form of powerful computers, exhibit-ing first a fantastic number-crunching abil-ity but later also unexpected wonderfulvisualisation capacities. These develop-ments have continued at an unabatedpace. Can one measure the types of impactthat the advent of computers has had onmathematics? I propose to organise themin four families:

� with computers, we can immediatelydeal with huge data sets, with the result thatsome problems have now become accessi-ble, where earlier mathematical methodswere not feasible; this is typical with statis-tical data. Here, the key question is tolearn how to extract relevant informationfrom huge data sets, and this is a mathe-matical problem.

� these new machines, which handle onlyfinite data, force us to reconsider the relationsbetween the finite and the infinite, about con-vergence of processes. We can teach a com-puter how to handle very large numbers,thereby making some areas of mathemat-ics, such as number theory or dynamicalsystems, amenable to experimentation.Here too, the net result is a considerablebroadening of what is accessible to mathe-maticians; as an example, we can namecryptography, again an activity whosenature is completely mathematical.

� machines have the capacity to repeat a sim-ple algorithm a very large number of times. Theexample of Julia sets and fractals may havebeen overused, but we have to learn how toexploit their attractiveness and beautyproperly; this has changed the view wehave on some structures, transformingtheir pathological appearance into a realrichness that has provided natural scien-tists with more models relevant to somepractical situations.

� computers open new horizons to the numer-ical treatment of sophisticated equations, a typ-ical example being that of weather fore-casting.

By opening new domains to mathemat-ics, changing our views on complexity, andgiving new tools to mathematicians to solvetheir problems, the impact of new comput-ing tools is therefore considerable, evenwithout mentioning the very deep mathe-matical problems posed by machines andby networks connecting them. This isundoubtedly one of the new frontiers ofmathematics. What was once considered bymathematicians as a threat provides agreat opportunity to revitalise the disci-pline and make it relevant to many moresituations.

The omnipresence of mathematical

objects in everyday life We now come towhat seems the most important aspect ofthe change occurring at the end of the lastcentury � the omnipresence of mathematicalobjects in everyday life.

This phenomenon is also connected withthe new possibilities offered by computers,but other aspects of modern society thathave no relation whatsoever with mathe-matics play a major role in it. Every day weuse many objects that incorporate somepart of mathematics, and often recentones. Most of the time even we mathemati-cians do not realise it, because the mathe-matical component is not immediately vis-ible and pinning it down requires somethinking.

The most obvious case is that of thepocket calculator, but computers alsoexhibit impressive mathematical ability;their use is widespread, and we are con-fronted with them everywhere. Less evi-dent examples are CD-players, TV sets,cars, etc. This proliferation has been madepossible thanks to the widespread use ofhigh-technology products producedcheaply.

The notion of industrial product Today�ssociety is dominated by the notion of industrialproduct. This means in particular that manyobjects and structures that surround ushave been designed in a well-defined man-ner. Objects have to fulfil a definite task,and their mere existence is due to the factthat a market for them has been identified.Usually this requires optimising the objector the structure in various ways, a stepinvolving a mathematical process that ear-lier would have been a purely hardwareproblem.

This goes further. Even if one is interest-ed in keeping an industrial product longerthan was planned by its inceptor, one soonencounters problems because (and this isespecially true for high-tech products) theyquickly become incompatible with theirenvironment � they cannot be used anymore. This compatibility problem poses ata large scale the question of keeping thememory of the past in proper form and,after what we said earlier, this is no minorconcern for mathematicians. The notion ofan industrial product incorporates its lim-ited life span.

Towards a communications societyClaiming that we are entering a societydominated by communications is not orig-inal. Here, we consider the consequencesof this change from a mathematical pointof view.

First, more information is made avail-able to a broader section of the public,leading usually to too much information.To learn how to navigate in such a world isnot easy, since one needs in particular toknow whether a piece of information isreliable. In view of the quantity it is clearthat most of the available information can-not have been properly checked. If onewants citizens to have control of the factson which to base their judgements, oneneeds to train them already at school to dothat. This makes exposure to critical rea-soning more important, and training todistinguish bias a necessity.

If this reliance on communications hasthe positive aspect of abolishing distance,it also makes us dependent on the qualityof information collected by various kindsof sensors that can be taught more easily tosend falsified information than our eyes,our fingers, etc.

Where is mathematics around us? Wenow give some explicit examples of themathematics around us. We make no claimto be exhaustive, and there is certainlysome personal bias.

In modern societies complex systems areeverywhere, to the extent that this featurecan even be seen as one of their character-istics. One needs to regulate these mecha-nisms, and this requires a new type ofknowledge that is basically mathematical.The management of such systems hasimportant consequences for the develop-ment of society.

Biological systems provide many examplesof complex systems that we must come toterms with. This domain became one of thegreat adventures of the second part of the20th century since the discovery of somefundamental mechanisms of living organ-isms such as DNA structure. Genomics,which comes out of this, requires handlinghuge amounts of information, some of itdiscrete (the digits of the 4-letter alphabetcoding of DNA) and some of it geometrical(being able to act on the DNA requires oneto know very precisely the geometric acces-sibility to the proper sequence).Mathematical tools to deal with such prob-lems are not obviously available, and arelikely to require new mathematical devel-opments around the question of how tostructure large data sets. Telecommunication systems have already hada wide impact on the way that the wholesociety functions. Again, when we considerobjects through which we communicates,besides the fact that one punches numberson a keyboard, there is no hint that thereis some mathematics behind them. It looksas though the chips, the laser (as in CD-Roms) and the wires are the whole story,when in reality the design of the networkplays an equally important role, and this iswhere mathematics enters. The perfor-mance of a telecommunication network ismeasured through its capacity for notaltering the information it carries. This canoften be done through error-correctingcodes, which encapsulate sophisticatedalgebraic properties � that is, purely math-ematical knowledge.

Another side of telecommunications hasto do with the quantity of data to be trans-mitted. Given a given physical systemdesigned to carry data, it may be necessaryto compress the data (typically, images) soas not to exceed the capacity of the net-work. Designing compression and decom-pression algorithms is another challengingproblem, in which (for example) wavelettheory, a truly mathematical modernadventure, has had an important impact.

The Global Positioning System (GPS) isanother example of a modern tool thatmust incorporate very advanced mathe-matics in order to reach the accuracy thatwidens its use. Indeed, in order to improve

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EMS June 200022

precision from 10 metres to 10 centime-tres, corrections that take care of effectsaccounted for by the theory of relativitymust be considered. For GPS again, opti-mising the choice of satellites from whichone estimates the position on the earthrequires the use of sophisticated mathe-matics, such as discrete group theory.

Finally, there is still another side toproblems posed by transmission of data �their security. In many cases, for one rea-son or another, one needs to be sure thatthe information is kept confidential all theway through. Encoding it is the solution,provided that the code cannot be brokenby a person different from the one that issupposed to receive it. Here again, one hasto deal with discrete data, and discretemathematics is a part of mathematics that,in many countries, has not yet been giventhe same priority as the a priori more noblepart using real numbers.

Data collection and analysis is yet anotherarea where we get used to new ways offunctioning and may miss the underlyingmathematics. Indeed, at the supermarket,we have got used to no longer seeing thecashier type the price of goods we buy,thanks to the systematic use of barcodes.The net gain for the shop is not only thetime saved in typing but the automaticmanagement of the reserves. Here againthe recognition process on which the bar-code system is based is purely mathemati-cal.

Statistics and polls We read statistics innewspapers every day, and to be able todecipher what they say and what they omithas become of the greatest importance intoday�s world. The fact that we can collectand deal with large data gives us moreopportunities, but citizens need a basictraining in this field in order to allow themto detect when they are being manipulat-ed.

Other information that we now find reg-ularly in the news media are polls. We areexposed to many of them, and in thisrespect we regularly have the feeling thatinformation drawn from them is definitelyimproper. Since they have become a nec-essary part of the process by which peopleform their opinions, they are very impor-tant to democracy. Knowing how to esti-mate the error margin and how polls canbe falsified is therefore essential.

Using appropriate rigorous mathemat-ics, it is possible to prove whether a statis-tical claim is correct or not. It is probablyin this area that a contribution towardscritical thinking through mathematics isreally possible, and at the same time indis-pensable.

Automatics and robotics Many complex sys-tems are run using control parameters.Automatic pilot systems that run planes areexamples of systems where control theoryplays an important role.

In fact, we are confronted with manyother automated systems in our everydaylife: lifts, TV and telephone satellites, cars,etc. Scheduling, the theory that ensuresthat a given network can provide the ser-vice it is designed for, also relies on appro-priate mathematical theory. Most trans-

port systems rely on such networks.Indeed, when we are waiting for a bus, atrain or a plane, we focus our attention onthe material carrier, but the important (buthidden) fact is that this object is part of awhole network. All this relies on sophisti-cated graph theory.

Optimisation of forms hides a lot of mathe-matics. We often use objects whose shapeshave been designed using advanced math-ematical tools, most of the time by extrem-ising some functional � just think of aero-dynamic forms of cars or planes to min-imise air resistance, and thus to improvefuel efficiency. Many other examples canbe given that involve purely design criteria,lower production costs, or improve a mate-rial strength.

An interesting example is given by opti-mising the places where the body of a caris linked to its frame. In choosing themproperly (for example, at nodes of thebasic harmonic vibrations of the body), onecan minimise the transmission to theinside of the car of vibrations caused bybumps on the road.

Yet another example, again taken fromthe car industry, has to do with the shapingof windshields, which are usually curvedsurfaces. A difficult question is to deter-mine the shape of the windshield when it ismade flat, before one �forms� it by heatingit again to make it deformable.

Banking products and insurance Many ofthe above examples come from differentareas, and point to one fact that makesmathematics different from other sciences� namely, that there is no industrial sectorthat can easily identify itself with the disci-pline.

Recently, this may have changed withthe new trends taken by the banking andinsurance industries. There are severalreasons for this change. We have alreadyencountered one of them � the capacity tocollect and quickly analyse large data sets.Another is linked to the new telecommuni-cation possibilities, and the fact that todaythere is really one world financial market.Yet another is deeper, and has to do withthe evolution of the banking activity whichactually functions more and more like aninsurance. A typical example of thischange is given by derivatives and options,the promise to buy a certain good at aprize determined by a well-defined mecha-nism, based on information that will beavailable later. The key point is to estimatethe risks that one takes by making such adeal. One must use sophisticated proba-bilistic models in order to evaluate theindices on which the price is based andwhich depend on random parameters. Thekind of stochastic processes that areinvolved turn out not to belong to any ofthe standard categories that have beenextensively studied in connection withprocesses met in the natural sciences, butrather with the general theory of stochasticprocesses, a body of knowledge long con-sidered highly theoretical if not esoteric.These developments have led banks andinsurance companies to hire mathemati-cians, and have given birth to a new branchof our discipline, financial mathematics, in

which special training is now offered at anumber of institutions of higher education.

4. Final wordsFrom all the above challenges concerning

the image of mathematics, the most seriousseems to me the need to make it evident thatmathematics is a science and alive. The restshould follow.

In my opinion, we mathematicians mustagree to put more consideration into, and toshow more curiosity for, what is happeningaround us. We also have to agree to saymore about our activities. This requirestaking more initiatives towards the generalpublic. We should be helped in that thereare many mathematical products aroundus. It is our duty to exhibit what is mathe-matical about them.

Secondly, we have many opportunities tomake people dream. Mathematics stillremains a great creative adventure. New chal-lenges are ahead of us. The need in ourdiscipline for great minds remains aspressing as it has been throughout history. To achieve such goals, we need to findappropriate ways to connect ourselves withteachers in schools. In the long run, this isthe only channel through which these newtrends will be passed on to the new gener-ation. The future of mathematics lies in thehands of those younger people that our genera-tion will prove able to attract to mathematics.

A version of this article was presented at a�Symposium on the Mathematics of the XXthcentury� held in September 1998 inLuxembourg.

The previous issue of the Newsletter had anarticle about the oldest national mathe-matical society in existence, the WiskundigGenootschap. As with British stamps, thereis no need for an explicit reference to itscountry of origin.

An editorial error crept into the expla-nation of wiskunde, the Dutch word formathematics. Many foreigners relatewiskunde to the German word Wissenschaft,and incorrectly infer that wiskunde is Dutchfor science, which has the Latin scire for toknow or, in German, wissen as its root. TheDutch have wetenschap for Wissenschaft, anduse weten for to know. Weten and wissenderive from the same root, which also pro-duced wits in English and the Dutch wordwis, meaning sure or certain. This is theword that Simon Stevin used in coining theword wiskunde. Thus, wiskunde literallytranslates as surology � a non-existent wordcoined by the American mathematicianBarry Mazur from Harvard.

As the article pointed out, a �new tomb-stone� for Ludolph van Ceulen containing�his� 35 decimals of π will be unveiled in thePieterskerk in Leiden. For ceremonial rea-sons, the date of the event has now beenadvanced to 5 July 2000.

More on Dutch mathematics and π

Peter Stevenhagen

FEATURE

EMS June 2000 23

The following is the EMS�s response to theEuropean Commission Communication�Towards a European research area�, issued inJanuary 2000 (see the websitehttp://www.europa.eu.int/comm/research/area.html). The text was sent to CommissionnerBusquin, who received an EMS delegation (R.Jeltsch, J.-P. Bourguignon and L. Lemaire) on23 May, for a constructive exchange of ideason the EMS document.

The European Mathematical Society hasanalysed the communication and relateddocuments, and welcomes the new orien-tations that it describes proposing a newstage of development for EuropeanResearch. It wishes to contribute to theopen debate initiated by CommissionerBusquin, bearing in mind its ownEuropean mission and its direct connec-tion with research and training activitiesthroughout the countries of the Union,the Associated States and all other coun-tries of the European continent.

ContentsI. Some special features of mathematicsII. Fundamental research and its role in

industrial developmentIII. Human capitalIV. Centres of excellenceV. Infrastructures and electronic databasesVI. Central and Eastern Europe and Third

WorldVII. E.U. programmes and proceduresVIII. BenchmarkingIX. Summary of recommendations

I. Some special features of mathematicsAlthough the E.M.S. wishes to raise somepoints pertaining to aspects of research inall fields, it feels the need to point tosome particular features of mathematicalresearch, notably in Europe.

Some of these characteristics shouldhave some bearing on the organisation ofthe E.U. programmes. Let it be clear thatwe do not call for a blind increase inmathematics funding, or for adoptingrules that would favour mathematics, butfor an increased flexibility in the defini-tion and the running of the programmesthat would leave all opportunities forexcellent proposals to be considered in allfields, with best possible efficiency.Different research traditions indeedrequire different frameworks, and webelieve that our comments do not applyonly to mathematics.

Mathematical research in Europe isrecognised as among the best in theworld: several indicators place it ahead ofthat of North America and of Asia. Othercharacteristics that distinguish it from sis-ter sciences, and in particular from theexperimental sciences, are the following:

(1) A mathematical research team at agiven location is generally smaller, andresearch on a given subject is geographi-cally more dispersed. That the quality of

mathematical research is not necessarilydirectly dependent on a concentration ofheavy resources is certainly an advantage� but requires that this fact be taken intoaccount when setting up programmes.This doesn�t mean that no support at all isneeded for mathematics, and computa-tional mathematics and scientific compu-tations will request investments � e.g. aslarge infrastructures.(2) The unity of mathematics has always

made itself felt with considerable force,and at the end of the 20th century theinterplay of mathematical ideas betweenthe different sub-domains had probablyreached an all-time high in intensity.There is no doubt that this phenomenonsets mathematics apart from many otherbranches of science which have split upinto multiple specialities, each with itsspecific culture and sociology. A directconsequence is that the task of a panel ofmathematicians having to evaluateresearch is much easier, the existence offactions and of schools in conflict with oneanother remaining exceptional.(3) The distance between fundamentalmathematics and applications is relativelyshort and is getting shorter both in timeand in content. In some cases, mathemat-ical subjects developed for reasons purelyinternal to mathematics have suddenlyled to concrete applications, in sharp con-trast with the traditional pattern of neigh-bourhood relationships: pure science �applied science � applications. By way ofexample, we might mention the use ofstochastic processes in finance, or of num-ber theory in cryptography and systemsecurity.(4) In mathematics there is an extremelyclose relationship between research andtraining. In fact, most researchers inmathematics are also teachers and

European support for research has animmediate effect on the improvement ofmathematical qualifications in the man-power of the future. It is the appropriatemoment to point out that the use of peo-ple mastering an advanced mathematicaltraining has broadened enormously.Besides finance, we can mention telecom-munication as a whole, the drugs indus-try, material design and so on.(5) In Central and Eastern Europe and inthe Third World the state of research inmathematics is often better than in theother sciences, for relatively low cost ini-tiatives have often allowed the creation ofcentres of excellence; without beingexhaustive, we could name some centresthat are at the highest world level: theIMPA in Rio de Janeiro, the BanachCentre in Warsaw, the Tata Institute inBombay. Some institutes in Vietnam andAfrica can also be mentioned.Cooperation in mathematics with suchcountries should not be seen as a one-wayaid to development, but as a trueexchange between active centres, fromwhich EU groups can benefit directly.

II. Fundamental research and its role inindustrial developmentThe European Mathematical Society ishappy to see the importance of funda-mental research underlined at variousplaces of the Communication.

The private sector mostly invests inshort-term applied research, and thisimplies that the public bodies � States andEuropean Union � must bear the majorresponsibility of long-term investment infundamental research.

All opinions converge today on the factthat the time-lag before a discovery can beput to practical use is getting shorter andshorter. But the time before commerciali-sation is getting short because the numer-ous (and often cheap) innovations thatcome up everyday from the world ofresearch are more quickly sorted out, andnot because the analysis of needs hasmotivated the discovery. We are in an erawhere the precise demand is not shapingall the markets.

A very significant example will illustratethis. The necessary scientific basis thatgave rise to the explosion of the commu-nication society (the internet, ...) is nottied to discoveries that were motivated bythat development, but by very theoreticalresults, obtained outside the pursuit ofcommercial aim and following the initia-tive of the scientists (bottom-upapproach).

Indeed, the laser was built as a physicalcuriosity in the sixties (the production ofcoherent light). It has now become a nec-essary ingredient of fast transmission

EMS NEWS

EMS June 200024

EMS Position paper on the Commission�s communication:Towards a European research area

Luc LEMAIRE (Bruxelles)

along optical fibres � as well as being usedevery day when we listen to compact discs,use code bars and have surgery per-formed.

Likewise, number theory was studiedfor centuries as an ultimate amusement ofmathematicians � but it led to cryptogra-phy and error correcting codes, withoutwhich safe transmission on the internetand e-commerce would not exist.

Also, the developments in non-linearanalysis, like wavelets in image process-ing, have preceded the use of sophisticat-ed non-linear models in high technologyindustry.

In many commercial developments ofthe future, we should expect to observethe same pattern: scientific discoveries ofdifferent fundamental fields being quiteunexpectedly transformed into innovativenew tools, creating a new market.

What can the E.U. do in this context?Naturally, it must support the develop-ment of industrial technology in an inte-grated way, to avoid negative effects ofinternal competition.

But it must mostly create a Europeanarea of scientific freedom, where the sci-entists will create new science followingtheir own approach. It must then encour-age or produce risk capital sufficientlyquickly when a discovery can lead tounexpected applications. In short,Europe has to give a chance to the unex-pected. This goes against the long-estab-lished idea of very close control on pro-grammes and choices made a priori.

III. Human capitalIn this age of science and technology, it isobvious that the future of Europe willdepend crucially on the level of itsresearchers, and in fact of all its work-force. Energetic measures are thusrequired to encourage the pursuit of sci-ence, improve the training of theresearchers and fight the brain drain toother continents.

III.1 In mathematics, the crux of the mat-ter is probably the question of availabilityof postdoctoral positions, and the E.M.S.wants to underline the very positiveresults already achieved by the E.U.through its Network and Fellowship pro-grammes. It calls for a substantialincrease of means in these activities(together with other changes described insection III 2 below), which would preventhaving to reject excellent proposals, as isnow the case in the network category.

Why are postdoctoral positions soimportant for the future of science inEurope? We all observe that scientists inEurope are often better trained than inthe U.S. � up to the doctoral level. Thisconsiderable investment in educationhaving been done, Europe often offers nogood opportunity for further employ-ment, leaving to the U.S. the benefit ofhiring highly competent scientists. As aresult, the doctoral and postdoctoral U.S.programmes work with a majority of non-U.S. citizens.

A number of undisputed centres of

excellence do exist in Europe but theyhave less success, for the simple and basicreason that they do not have enough posi-tions available!

The E.U. can have a strong effect onthis problem provided it continues itsefforts in the appropriate form. It is inparticular extremely important thatyoung researchers after their PhD havethe opportunity to enlarge their perspec-tive, by gaining access to centres of highlevel outside of narrowly defined scientif-ic projects.

Practically, it would first be importantto create and impose a tax-free status forthe European postdoctoral researchers inmobility paid by the budget of theCommission, with the hope that Stateswould follow that example with their ownfellowships. For the moment, the situa-tion varies from country to country and itis ridiculous to see that the sameresearcher is twice as expensive inBelgium as in the U.K. (in fact, inBelgium the E.U. will have to pay 180.000Bef per month, the researcher receiving60.000!). A European tax-free statuswould overnight double the number ofpositions offered by the Commission.

Secondly, we propose to define a new(optional) format for a postdoctoral fellowin mobility � namely, two years in anotherE.U. country, then one year in their coun-try of origin (for the moment, this existsonly for researchers from regions in eco-nomic difficulties).

Such a move would greatly improve theattractiveness of Europe for the followingreasons.

For the moment, the time gap betweencompletion of the PhD and the possibilityof getting a permanent position is noteasily covered by postdoctoral positions inEurope � and as a result many have to goto the U.S., where they often remain.

The 2+1 positions would be muchmore attractive: the researcher couldcompletely concentrate on his/her workwhile abroad, without having to preparemultiple applications for the followingyear. Back home typically during year 3,he/she would be in a better position toseek permanent employment.

Finally, it happens from time to timethat a bright researcher is offered simul-taneously a 2-year E.U. fellowship and apermanent job at home, to take effectimmediately. The choice is obvious � butnot optimal for the development ofEuropean science. The employer (e.g.,University) will not usually agree to givean immediate leave of absence. If on theother hand, a third year were offered towork in the same place, it could be anoth-er matter.

Altogether, these 2+1 positions wouldnot be seen as an obstacle to a localcareer, but on the contrary could becomehighly attractive positions, well recog-nised in a Curriculum.

III.2 The networks are also importantcomponents of the postdoctoral pro-gramme � well integrated with the effec-tive development of research.

With their present rules, the rate of suc-cess has become too small to make choic-es optimal, and the lack of flexibilitymake them less adapted to sciences likemathematics, where at each node groupsare usually small. Thus, a more flexiblerule should allow the inclusion of moresmall groups, without necessarily increas-ing the global budget of a network.Moreover, the panel of experts should begiven much more freedom, including thepossibility to award less money thanrequested or intervening to suggest themerger of two applications.

The freedom of action of the panel hasbeen abruptly suppressed at the begin-ning of the Fourth FrameworkProgramme, and in mathematics the ruleslead to a dramatic decrease of the cost-efficiency of the programme.

III.3 The demography of researchers andprofessors in European universitiesshould be studied in detail for every field,to get a fairly precise estimate year byyear of future needs. For mathematics,this was done a few years ago by theE.M.S. This study should be updated, andwould show some strong variations in thenumber of retirements over differentperiods (still following the effect of WorldWar 2, of the growth in the sixties, the cri-sis in the seventies, and so on).

An ambitious policy is badly needed. Itshould insure, by an adequate number ofdoctoral and postdoctoral positions, thatthe future professorial positions in uni-versities will be awarded to high-classresearchers.

III.4 A serious technical problem arises inthe definition of Marie Curie trainingsites. Indeed, they are open only to doc-toral students and not postdoctoral ones.In certain fields, this corresponds to well-identified needs, but not in others (likemathematics). We propose to open thesesites to both doctoral and postdoctoralresearchers, thus allowing for optimalchoices by the institutes in each field.

III.5 Finally, anyone who has invited toEurope a scientist from outside the E.U.knows that the visa problem is a majorheadache and a huge loss of time. Insome cases, a foreign scientist invited andpaid by a government will not be able tocome, because the same government willnot provide him/her with a visa.

IV. Centres of excellenceThe E.M.S. supports strongly theCommission�s view that centres of excel-lence are an important component ofEuropean research, well positioned forefficient and justified E.U. support.

However, those centres cannot be iden-tified (as in the Communication of theCommission) by their capacity to produceindustrially applicable discoveries. Wenote however that the accompanying doc-ument: Action for �centres of excellence�with a European dimension gives bettercriteria, and a list of existing centresshowing a broader vision.

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As mentioned above, centres of excel-lence in mathematics exist in Europe atthe highest level, and are by no meansinferior to U.S. centres like the I.A.S. inPrinceton or the M.S.R.I. in Berkeley.

Seven of these centres have in factalready constituted a network ofEuropean dimension, pre-tailored for thescope of a European Research Area. TheEuropean Post-Doctoral Institute isindeed constituted by the Institut desHautes Etudes Scientifiques (Bures-sur-Yvette, France), the Isaac NewtonInstitute for Mathematical Sciences(Cambridge, UK), the Max-PlanckInstitut für Mathematik (Bonn,Germany), the Max-Planck Institut fürMathematik in den Naturwissenschaften(Leipzig, Germany), the ErwinSchrödinger Institut (Vienna, Austria),the Institut Mittag-Leffler (Djursholm,Sweden) and the Banach Center (Warsaw,Poland).

The aim and function of this E.P.D.I. isso much in line with the E.U. programmethat its case should be examined by theappropriate scientific panels. In the past,strict rules on different E.U. programmesmade the EPDI not eligible � usually tothe regret of scientific panels who rated itas excellent.

Other networks in mathematics havealso established themselves at the highestlevel, either in the E.U. programmes oroutside, and we wish to mention here theNetwork MACSI ( Mathematics, comput-ing, simulation in Industry), initiated byECMI and ECCOMAS.

In order to allow for a choice of optimalcentres and networks, the E.M.S. insistsonce more on the importance of flexibili-ty in the rules: a network of centres ofexcellence need not be very large, neednot be founded on a very specialisedtopic, and need not consist of large labo-ratories, although of course it can.

Finally, all calls for proposals should bewidely open to competition � science inEurope is not limited to a list of centresfixed once and for all.

V. Infrastructures and electronic data-basesThe E.M.S. gratefully acknowledges thefact that the definition of a large infra-structure was enlarged in the FifthProgramme, thereby opening the way to asuccessful application of support of thecomprehensive European database ofmathematical literature, Zentralblatt-MATH.

Further support for electronic databaseand libraries will be essential for thedevelopment of mathematics, and Europemust reach the level of an unavoidablepartner in the world competition in thisdomain.

A special feature of that branch is thatdocuments are never obsolete: a brilliantproof by Bernhard Riemann or ElieCartan can be of extreme contemporaryimportance, because ideas remain validand the tradition of mathematics publica-tion does not include repetition of pastresults. Thus a present day researcher will

effectively need access to the whole of theliterature for his/her work.

On the other hand, the price of sub-scriptions to journals, in particular pub-lished by commercial companies, is risingunreasonably: increases of 10 to 20% peryear are the norm, 60% often happens.Thus, the average university librarybecomes unable to acquire the basic nec-essary tool for its mathematicsresearchers.

As a consequence, some U.S. enterpris-es. have embarked on an ambitious pro-gramme of digitising the existing scientif-ic literature, in order to distribute it elec-tronically. This effort is funded by privatefoundations, but access will be reserved toinstitutes paying a heavy annual fee, like-ly to remain out of reach for mostEuropean universities.

For these reasons, the E.M.S. pressesthe European Commission to take strongaction aiming at developing electronicdatabases and libraries, including thesteps of developing the data and its distri-bution. A more precise analysis shouldshow whether Europe should catch upwith the digitising initiated in the U.S. bycompetition or by partnership. In the lat-ter case, a sufficiently strong positionmust be reached before hoping for a realpartnership. By strong action, we mean inparticular that the Commission supportbe granted not only for access to infra-structure, or for research related to thedevelopment of the infrastructure, but fordirect funding of the installation itself.

This does not contradict any principleof subsidiarity, once it is acknowledgedthat the infrastructure is a truly Europeanone and could just not be conceived at thescale of one state.

We note also that the E.M.S. has theproject of establishing itself as aPublishing House, which would work as anon-profit organisation. The EuropeanCommission might think of supportingthis kind of endeavour at the initial stage,for the long-term benefit of the scientificcommunity.

VI. Central and Eastern Europe andThird WorldAs indicated above, very high-level math-ematics research is pursued both inCentral and Eastern Europe and in vari-ous developing countries. In Central andEastern Europe, a high mathematicallevel has been maintained as a tradition,and using the fact that restricted access toexpensive material over a short period oftime did not bring irreversible damage.

This tradition is under threat in somecountries, simply because economic prob-lems have brought down the salaries ofuniversity teachers below a minimal level� thus producing a massive exodus to thewest.

It is of the interest of the whole ofEurope to preserve its high and long-last-ing tradition of mathematical research inall countries. To disrupt it in the Eastwould irreversibly damage a remarkableschool of talent, which Europe will needin the near future.

Thus we advocate strong relations withCentral and Eastern Europe, on the basisof scientific partnership at high level. Thevery low cost of living there can be used asan advantage, to organise activities morecost-efficiently.

Concerning the countries of LatinAmerica, Asia and Africa, Europe could atthis stage develop strong scientific rela-tions, which would be beneficial in thelong run. Indeed, these countries do notusually cherish the idea of being absorbedunder the umbrella of powerful neigh-bours (U.S.A. for Brazil, Japan for Korea,...). Mathematically, some centres in thesecountries are at the highest level, and aprogramme of cooperation would be veryefficient. Currently, cooperation is onlyfunded on specific themes defined by theCommission (pollution, forest, health, ...),missing the necessity for these countriesto be integrated in the future society ofknowledge. Various European states havestrong historical and cultural links withdeveloping countries, and these linkscould form the basis of future coopera-tion.

VII. E.U. programmes and proceduresSome rules currently enforced by theCommission do not allow for maximalefficiency of the programmes. It seemsessential to identify and modify them.

VII.1 The rule of subsidiarity, pushed tosome extreme, has sometimes been inter-preted as saying that the Commission willnot fund research itself, but onlyEuropean added value. This is a self-defeating interpretation which we strong-ly regret. The European Space ofResearch must be in itself a Europeanendeavour, and thus must be funded with-out administrative restrictions.

In the present programme, examplescan be shown in which proposers, panelsand administrators of the programmeagree that a realisation is worthy of sup-port, yet the rules prevent it.

For infrastructure, such restrictionswere described in Section V.

For the programme �Raising PublicAwareness�, the description of the pro-gramme gives the impression that it willbe easier to fund a round table of discus-sion than an actual concrete realisation(the latter being pushed in the category ofaccompanying measures).

VII.2 The E.U. procedures, applicationsand contracts are absurdly complicated,notably because they aim at covering inextreme detail many different situationsin the same format. It seems that all doc-uments were drawn up by wary lawyers,and not by scientists. The smallest appli-cation is the object of 50 to 80 pages ofinstructions on the Cordis website. A sim-ple shift of a few months in the executionof a project requires a full amendment tothe contract.

Quite often, the questions asked in theapplication form constitute a barragebetween the proposer and the scientificpanel. This is in part due to a restrictive

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EMS June 200026

view of subsidiarity: instead of simply say-ing what research he plans to do, for thepanel to read, the proposer will have toanswer a series of questions, distinguish-ing training, research, added Europeanvalue, externalities, ... It has gone so farthat some high-level scientists do notapply any more, whereas other hire the(paid) service of consulting agencies, whospecialise in knowing how to transform ascientific project into a �viable� E.U. appli-cation.

VII.3 The scientific panels play, and mustplay, an essential role in the peer review-ing of proposals. They must examine theproposals and have an actual meeting toexchange information and opinions. Toask for separate marks out of 100 andthen to make a simple average (a processunfortunately used by INTAS) transformsthe choice into a lottery, different expertshaving different scales for their marks.

The composition of panels must behandled with great care. At the beginningof the Fifth Programme, a short sentencesuddenly appeared in the texts, restrict-ing the choice of panel members by theE.U. staff to scientists who had filled in anapplication form.

The result could have been expected: inmany disciplines, the top-class scientistsdid not agree to fill in a long form inorder to be included in a database ofpotential experts; on the other hand,26,000 people applied, producing a listimpossible to handle.

We propose to go back to the precedingprocedure: the panel secretaries togetherwith the present panel members makeproposals, taking into account the level ofthe experts and their fields. Proposals arealso made from the outside, and the panelsecretary and the director of the pro-gramme make a final choice. They canalso call personally some well-known sci-entists and ask them to take part in thepanel.

Anomalies in the way a panel memberbehaves are monitored by the E.U. staff,and in case a problem arises the panelmember is not appointed again.

Our experience of the mathematics andinformation science panel is that it hasevolved well, with competent guidance ofpanel secretaries at high level.

In case a panel were to concentratearound a specific subfield, it is up to theE.U. administration (directors and panelsecretaries) to make the corrections.

VII.4 As already mentioned, panelsshould be given more freedom by allow-ing added flexibility to the rules.

VII.5 In some E.U. programmes, part ofthe first application must be anonymous,in the sense that the scientific programmemust be described but the name of scien-tists in charge must be deleted. This pro-cedure may give an impression of impar-tiality, but in fact it is absurd. The level ofexpertise of the researchers is the mainfactor in the success of a programme, andto delete that information is to deprive

the assessor from a major element ofjudgement. Moreover, if an assessor issuspected of helping his or her friends, itis clear that these friends will tell him theyare behind the application. So nothing isgained but very much is lost in thatprocess.

VII.6 To summarise this section, wereduce our recommendations to threesimple mottoes: more flexibility, fewerrules, less paperwork.

VIII. BenchmarkingThe E.M.S. certainly approves of bench-marking exercises, but not to the extentthat it uses up a substantial part of E.U.funding and of scientists� time.

Moreover, the benchmarking describedcompares best practice in various E.U.countries, whereas the initial aim of theprogramme seems to be to catch up onU.S. efficiency, where organisation ofresearch is concerned.

Maybe a benchmarking of practice withthe U.S. would simply confirm that theirefficiency is related to more freedom, lesscontrol, less planning, more facilities topatent a discovery, and simplified proce-dures to start a company, and these couldbe major guidelines for Europe.

IX. Summary of recommendationsThe European Mathematical Society rec-ommends that1. the Commission take strong initiatives

to develop in Europe a large area offundamental research, preserving a�bottom-up� approach leaving the dooropen for the unexpected discoveriesthat could shape tomorrow�s society.Concurrently, administrative simplifi-cations and encouragement of risk cap-ital should allow for quick exploitationof these discoveries.

2. more flexibility be introduced in thedefinition of E.U. programmes, so thatsciences with different traditions andformat can be efficiently supported.

3. the Improving Human Capital pro-gramme be maintained and amplified,with the following specific suggestions:

� the creation of a tax-free status for doc-toral and post-doctoral Europeanresearchers;

� the creation of �2+1 fellowships�, withtwo years in another country and aone-year return fellowship;

� a study of the demography of universi-ty professors in each field, and ade-quate measures to insure a supply ofhigh-level researchers when necessary;

� the development of the idea of centresof excellence and networks of such cen-tres, with flexible rules and definitions,allowing for the different ways of dif-ferent fields.

4. the Commission firmly support thedevelopment of European electronicdatabases and libraries, to allow accessto the necessary scientific informationall over the continent.

5. the Central and Eastern Europeanmathematicians be integrated rapidlyinto all the Commission programmes,

as equal scientific partners,6. E.U. programmes include mathemati-

cal collaboration with researchers ofdeveloping countries, where a goodmathematical level has been reached.

7. all Commission procedures be enor-mously simplified, to avoid possibleneglect of programmes by the best sci-entists; this applies to all steps of theprocess, from the initial applications tothe final reports.

8. the scientific panels be given morefreedom to judge, amend or make sug-gestions about the proposals.

Moreover, it would be extremely usefulto:� allow both doctoral and postdoctoral

researchers in the Marie Curie trainingsites;

� go back to the previous system ofappointments for panel members, tobe chosen jointly by the present mem-bers and the Commission staff;

� suppress the idea of evaluating anony-mous proposals;

� stress the importance of easy visa pro-cedures for recognised researchers.

In EMS Newsletter 35 we featured theBelgian stamp issued to commemorateWorld Mathematical Year 2000. Furtherstamps have since been issued byLuxembourg, Slovakia and Argentina.Two of these are shown below: theLuxembourg stamp features the WMY2000 logo, Fermat�s last theorem, Stokes�theorem, the Riemann zeta-function andthe decimal expansion of p, while theArgentinian stamp depicts the infinitysign. If any reader can send the Slovakianstamp to the Editor, it will appear in thenext issue.

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EMS June 2000 27

World Mathematical YearStamps

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

2�7: 6th International Conference on p-AdicAnalysis, Ioannina, GreeceInformation: contact A. K. Katsaras, Dept. ofMath., Univ. of Ioannina, 45110, Ioannina,Greece,tel: (+30)-651-98289, fax: (+30)-651-46361 e-mail: akatsar@cc.u0i.gr Web site: http://www.uoi.gr/conf_sem/p-adic [For details, see EMS Newsletter 34]

2�15: NATO Advanced Study Institute 20thCentury Harmonic Analysis � a Celebration,Tuscany, ItalyInformation:Web site: http://www.cs.umb.edu/~asi/analy-sis2000

3�5: International Conference on MonteCarlo and Probabilistic Methods for PartialDifferential Equations, Monte Carlo, MonacoInformation:e-mail: Monique.Simonetti@sophia.inria.fr

3�5: Rencontre de Probabilistes, Lectures onProbabilistic Topics in 3D Fluids, Barcelona,SpainSpeaker: Franco Flandoli (Università di Pisa) Aim: the meeting will be devoted to someadvanced lectures on probabilistic methods for3D fluids. These will present the state of the artin a subject that lies at the interface betweenfluid dynamics and probability theory. Twopoints of view will be considered: the study ofstochastic Navier-Stokes equation and the mod-elling of vortex structures by probabilisticmethods Organisers: the probability teams of theUniversity Paris 13 and the University ofBarcelonaInformation: Web site:http://orfeu.mat.ub.es/~gaesto/fluid.htm

3�7: ALHAMBRA 2000, Granada, SpainInformation: contact ALHAMBRA 2000Conference eurocongres Avda. Constitución, 18- Blq.4 E-18012 - Granada, Spaintel: (+34)-958-209-361, fax: (+34)-958-209-400 e-mail: alhambra2000@ugr.es, eurocongres@mx3.redestb.esWeb site: http://www.ugr.es/local/alhambra 2000[For details, see EMS Newsletter 34]

July 2000

3�7: ANTS IV Algorithmic Number TheorySymposium, Leiden, the NetherlandsInformation:e-mail: ants4@wins.uva.nl Web site: http://www.math.leidenuniv.nl/ants4/[For details, see EMS Newsletter 34]

3�7: Functional Analysis Valencia 2000,Valencia, SpainInformation: contact: K. D. Bierstedt or J.Bonet, Univ. Paderborn, FB 17, Math., D-33095 Paderborn, Germany or UniversidadPolitècnica de Valencia, Departamento deMatemática Aplicada, E-46071 Valencia, Spain e-mail: VLC2000@uni-paderborn.deWeb site: http://math-www.uni-paderborn.de/VLC2000[For details, see EMS Newsletter 32]

3�9: Euro-Summer School on MathematicalAspects of Evolving Interfaces, Madeira,Portugal Information:e-mail: maei2000@lmc.fc.ul.pt Web site: http://maei.lmc.fc.ul.pt [For details, see EMS Newsletter 34]

4�6: Catop 2000, Fribourg, SwitzerlandScope: categorical topological methodsInformation:Web site: http://www.unifr.ch/math/catop2000/ [For details, see EMS Newsletter 34]

4�7: 2nd International Conference onMathematical Methods in Reliability,Bordeaux, FranceInformation: contact Dr. Valentina Nikoulina,Université Victor Segalen - Bordeaux 2,Statistique Mathematique, UFR MI2S, B.P. 6933076 Bordeaux Cedex, FRANCE,tel: (+33) 5 57 57 10 70 & 5 57 57 14 25, ax: (+33) 5 56 98 57 36 & (+33) 5 57 57 12 63e-mail: vnikou@mi2s.u-bordeaux2.fr,Nikolaos.Limnios@utc.frWeb site: http://www.mass.u-bordeaux2.fr/MI2S/MMR2000/[For details, see EMS Newsletter 34]

5�7: Scandinavian Workshop on AlgorithmTheory, Bergen, NorwayInformation:e-mail: telle@ii.uib.noWeb site: http://www.ii.uib.no/swat20005�8: Ordinal and Symbolic Data Analysis(OSDA 2000), Brussels, BelgiumInformation: Web site: http://www.ulb.ac.be/sciences/ulbmath/osda2000 [For details, see EMS Newsletter 35]

6�8: 6th Barcelona Logic Meeting, Barcelona,SpainInformation:e-mail: 6blm@crm.es Web site:

http://www.mat.ub.es/~logica/news.html orhttp://www.crm.es/[For details, see EMS Newsletter 34]

9�15: Mathematical Methods for ProteinStructure Analysis and Design, Taranto, Italy

10�14: 3rd European Congress ofMathematics (3ecm), Barcelona, SpainInformation: contact Societat Catalana deMatemátiques, Carrer del Carme 47, E-08001Barcelona, Spaintel: (+34)-270-16-20, fax (+34)-93-270-11-80 e-mail: 3ecm@iec.esWeb site: http://www.iec.es/3ecm/[For details, including satellite conferences, seeSecond Announcement in EMS Newsletter 34]

10�14: ICMS Workshop: Dynamical Systems,Edinburgh, UK[satellite meeting of the International Congressin Mathematical Physics, 17-22 July, London,UK]Information:e-mail: icms@maths.ed.ac.uk

10�14: International Conference onNoncommutativity � Geometry andProbability, Nottingham, UK[satellite meeting of the International Congressin Mathematical Physics, 17-22 July, London,UK]Information:e-mail: alison.wilde@ntu.ac.uk

10�14: IUTAM Symposium on Free SurfaceFlows, Birmingham, UKInformation:Web site:http://www.mat.bham.ac.uk/research/iutam. htm[For details, see EMS Newsletter 33]

10�15: 2nd World Conference onMathematics and Computers in MechanicalEngineering, Vouliagmeni, GreeceInformation:Web site: http://www.softlab.ntua.gr/~mastor/mcme2000.htm

10�15: 2nd World Conference onMathematics and Computers in Physics,Vouliagmeni, GreeceInformation:Web site: http://www.softlab.ntua.gr/~mastor/mcp2000.htm

10�15: 4th World Conference on Circuits,Systems, Communications and Computers,Vouliagmeni, GreeceInformation:Web site: http://www.softlab.ntua.gr/~mastor/cscc2000.htm

13�14: Computational Challenges for theMillennium, Cambridge, UKInformation: Web site: http://www.ima.org.uk/mathematics/confmillennium.htm[For details, see EMS Newsletter 35]

16�21: Advanced Research Workshop onFinite Geometries, UKInformation:Web site:http://www.maths.susx.ac.uk/Staff/JWPH/

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EMS June 200028

FForthcoming conferorthcoming conferencesencescompiled by Kathleen Quinn

17�20: IUTAM Symposium 2000/10Diffraction and Scattering in Fluid Mechanicsand Elasticity, Manchester, UKInformation: contact Professor DavidAbrahams, Department of Mathematics,University of Manchester, Oxford Road,Manchester M13 9PL, UK, tel: (+44)-161-275-5901, fax: (+44)-161-275-5819 e-mail: i.d.abrahams@ma.man.ac.ukWeb site: http://www.keele.ac.uk/depts/ma/iutam/[For details, see EMS Newsletter 33]

17�21: Colloquium on Semigroups, Szeged,HungaryInformation:e-mail: algebra@math.u-szeged.hu

17�21: 9th International Conference onFibonacci Numbers and their Applications,Luxembourg-City, LuxembourgInformation:e-mail: howard@mthcsc.wfu.edu

17�22: Colloquium on Lie Theory andApplications, Vigo, SpainInformation: contact I Colloquium on LieTheory and Applications, E. T. S. I.Telecomunicación, Universidad de Vigo, 36280 Vigo, Spain tel: (+86) 81 21 52 // (+86) 81 24 45, fax: (+86) 81 21 16 // (+86) 81 24 01 e-mail: clieta@dma.uvigo.es Web site: http://www.dma.uvigo.es/~clieta/[For details, see EMS Newsletter 33]

17�22: International Congress onMathematical Physics, London, UKInformation:Web site: http://icmp2000.ma.ic.ac.uk/

19�26: 3rd World Congress of Non-linearAnalysts (WCNA-2000), Catania, Italy

22�28: New Mathematical Methods inContinuum Mechanics, Anogia, CreteInformation: e-mail: ball@maths.ox.ac.uk [For details, see EMS Newsletter 35]

23�31: ASL European Summer Meeting(Logic Colloquium 2000), Paris, FranceInformation: e-mail: asl@math.uiuc.eduWeb site: http://lc2000.logique.jussieu.fr[For details, see EMS Newsletter 33]24�29: SIAG OP-SF Summer School 2000,Laredo, SpainTheme: orthogonal polynomials and specialfunctions Information: e-mail: ran@cica.es or pacomarc@ing.uc3m.es

24�3 August: EMS Summer School, NewAnalytic and Geometric Methods in InverseProblems, Edinburgh, UKInformation: contact Erkki Somersalo, HelsinkiUniversity of Technology, Finland e-mail: esomersa@dopey.hut.fi

25�30: Colloquium on Differential Geometryand its Applications, Debrecen, HungaryInformation: Web site: http://pc121.math.klte.hu/diffgeo

29�4 August: Curves and Abelian Varietiesover Finite Fields and their Applications,Anogia, Crete [For details, see EMS Newsletter 35]

30�5 August: ICOR 2000, Innsbruck, Austria

31�3 August: 3rd Conference of BalkanSociety of Geometers, Bucharest, RomaniaInformation: contact V. Balan, UniversityPolitehnica of Bucharest, Department ofMathematics I, Splaiul Independentei 313, RO-77206, Bucharest, Romania, fax: (+401) 411.53.65e-mail: vbalan@mathem.pub.ro [For details, see EMS Newsletter 33]

31�4 August: Numerical Modelling InContinuum Mechanics (Theory, Algorithms,Applications), Prague, Czech RepublicInformation:e-mail: nmicm@karlin.mff.cuni.czWeb site: http://www.karlin.mff.cuni.cz/katedry/knm/nmicm2000 [For details, see EMS Newsletter 35]

31�4 August: Workshop on PartialDifferential Equations: Thermo, Visco andElasticity, Konstanz, Germany Information: local organiser:reinhard.racke@uni-konstanz.deWeb site: http://www.mathe.uni-konstanz.de/~racke/announ/ws2000.html [For details, see EMS Newsletter 35]

2�9: Summer School on MathematicalPhysics (emphasis on Quantum FieldTheory), Sandbjerg Manor, DenmarkInformation:Web site: http://www.maphysto.dk/events/

2�18: Rings, Modules and Representations �Constanta 2000, Constanta, RomaniaInformation:Address: University of Stuttgart, MathematischesInstitut B/3, Pfaffenwaldring 57, 70550Stuttgart, Germanyfax: (+49)-711-685-5322e-mail: buro@poolb.mathematik.uni-stuttgart.deWeb site: http://web.mathematik.uni-stuttgart.de/~ovid[For details, see EMS Newsletter 35]

3�5: Recent Development in the Wave Fieldand Diffuse Tomographic Inverse Problems,Edinburgh, UKInformation:e-mail: icms@maths.ed.ac.uk

4�9: Stokes� Millennium Summer School,Skreen, County Sligo, Ireland[Third in a series of conferences organised tohonour the life and work of Sir George GabrielStokes at his birthplace �within the sound of theAtlantic breakers�]Theme: Navier-Stokes equations Topics: computational methods and applica-tions, including asymptotic and perturbationmethods, as well as numerical analysis. Therewill also be talks on history and applications tometeorology and engineering

August 2000

Programme and invited speakers: the Schoolis in two parts, and participants may attendeither or both. 4�5 August: instructional course on new numer-ical methods for Navier-Stokes equations,organised by J. Miller (Dublin) and G. Shishkin(Ekaterinburg) with A. Hegarty (Limerick) andE. O�Riordan (Dublin City). 6�9 August: work-shop with invited lectures from J. R. King(Nottingham), V. Entov (Moscow), A. R. Davies(Aberystwyth), P. G. Drazin (Bristol), P.Clarkson (Kent), B. Straughan (Glasgow), E.Mansfield (Kent), A. D. Craik (St. Andrews), P.Lynch (Met Eireann), L. Crane (Dublin).Lectures are scheduled for mornings and earlyevenings, leaving afternoons free for discus-sions and interactionsLanguage: English Organisers: Alastair Wood (Dublin CityUniversity), Sir Michael Berry (Bristol) Sponsors: the Institute of NumericalComputation and Analysis. Support from theEU under the Euro Summer SchoolProgramme has been applied forSite: lectures will be held in the formerParochial School, in the very room whereStokes received his primary educationNotes: because of the capacity of the scenic andhistoric venue, attendance is restricted to 50.Accommodation will be in farmhouse bed-and-breakfast and self-catering house in the parish.Meals will be taken communally in theParochial Hall.Deadlines: for application forms and payment,7 July 2000Information: contact Carmel Reid or AlastairWood, School of Mathematical Sciences, DublinCity University, Dublin 9, Irelandtel: (+353) 1 7045293, fax: (+353) 1 7045786

Web site:http://webpages.dcu.ie/~wooda/stokes/stokes2000.html e-mail: Carmel.Reid@dcu.ie orAlastair.Wood@dcu.ie

6�9: ISSAC 2000 International Symposium onSymbolic and Algebraic Computation, StAndrews, ScotlandInformation: Web site: http://gap.dcs.st-and.ac.uk/issac 2000/

8�12: XVIII Nevanlinna Colloquium,Helsinki, FinlandInformation:e-mail: pekka.tukia@helsinki.fiWeb site: http://www.math.helsinki.fi/~analysis/NevanlinnaColloquium/[For details, see EMS Newsletter 33]

12�17: Ninth International Colloquium onNumerical Analysis and Computer Scienceswith Applications, Plovdiv, Bulgaria Sessions: Acceleration of convergence,Numerical simulation, Numerical approxima-tion, Numerical methods in complex analysis,Numerical methods in linear algebra, Intervalarithmetic, Numerical algebraic or transcen-dental equations, Mathematical programming,optimisation and variational techniques,Numerical analysis for ordinary differentialequations, Numerical analysis for partial differ-ential equations, Computer arithmetic andnumerical analysis, Computer aspects ofnumerical algorithms, Parallel and distributed

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EMS June 2000 29

algorithms, Concurrent and parallel computa-tions, Computer networks, Discrete mathemat-ics in relation to computer science, Computeraided design, Theory of data, Programming,Image processing, Pattern recognition,Communication systems, Information systems,Manufacturing systems, Data base, Softwaretechnologies, Software engineering,Applications in mechanics, physics, chemistry,biology, technology and economics Chairman of the organising committee:Professor Drumi Bainov, P.O. Box 45, Sofia1504, Bulgariatel: (+359) 2437343, fax: (+359) 29879874 e-mail: dbainov@mbox.pharmfac.acad.bg

17�3 September: EMS Summer School inProbability Theory, Saint-Flour, Cantal,FranceInformation: contact: P. Bernard, Laboratoirede Mathematiques Appliquées, Univ. BlaisePascal, F-63177 Aubiere, tel/fax: (+33) 4 73 40 70 64e-mail: bernard@ucfma.univ-bpclermint.fr [For details, see EMS Newsletter 34]

19�25: Discrete and Algorithmic Geometry,Anogia, CreteInformation:e-mail: ziegler@math.tu-berlin.de[For details, see EMS Newsletter 35]

20�23: 3rd International Workshop onScientific Computing in ElectricalEngineering SCEE�2000, Warnemünde,GermanyInformation:Web site: http://www.SCEE-2000.uni- rostock.de

21�25: International Association forMathematics and Computers World Congress(IMACS 2000), Lausanne, SwitzerlandInformation: contact Prof. Robert Owens,IMACS Congress 2000, DGM-IMHEF-LMF,Swiss Federal Institute of Technology, CH-1015 Lausanne, Switzerlandtel: (+41)-21-693.35.89, fax: (+41)-21-693.36.46 e-mail: robert.owens@epfl.chWeb site: http://imacs2000.epfl.ch[For details, see EMS Newsletter 32]

27�1 September: 9th Summer St PetersburgMeeting in Mathematical Analysis, StPetersburg, RussiaInformation:Web site: www.pdmi.ras.ru/EIMI/2000/analysis9/index.html

30�2 September: Innovations in HigherEducation 2000, Helsinki, FinlandInformation:e-mail: sari.lindblom-ylanne@helsinki.fiWeb site: http://www.helsinki.fi/inno2000

1�4: Constantin Caratheodory Congress,Evros, GreeceScope: measure theory, function theory, partialdifferential equations and their applicationsInformation:e-mail: vougiou@edu.duth.gr

September 2000

2�9: International Conference on Topologyand its Applications, Ohrid, Macedonia Invited speakers: P. J.Collins (University ofOxford), J. Dydak (University of Tennessee,Knoxville), A. Koyama (Osaka KyoikuUniversity), Y. Lisica (Russian University ofPeoples Friendship, Moscow), L. Rubin(University of Oklahoma, Norman), J. M. R.Sanjurjo (Universidad Complutense, Madrid),M. Scheepers (Boise State University, Idaho)Programme committee: Prof. Yuri Lisica(Moscow), Prof. Jose Sanjurjo (Madrid), Prof.Nikita Shekutkovski Local organising committee: NikitaShekutkovski (president), Biljana Krsteska,Liljana Babinkostova Call for papers: abstracts of 300-500 wordsshould be sent no later than 1 July to the mail-ing address of the conference: ICTA2000,Institute of Mathematics, Faculty of NaturalSciences and Mathematics St.Cyril andMethodius University, Skopje, Macedonia, orby e-mail to:icta2000@iunona.pmf.ukim.edu.mk. If theabstract is sent electronically in TeX format,then it should also be sent in compiled form (asa .jep file ready for printing) or printedSponsor: Faculty of Natural Sciences andMathematics at St. Cyril and MethodiusUniversity, Skopje, Macedonia Information:Web site: http://www.pmf.ukim.edu.mk/mathe-matics/ icta2000.html

3�6: 31st European Mathematical PsychologyGroup Meeting (EMPG 2000), Graz, AustriaInformation: e-mail: empg2000@psyserver.kfunigraz.ac.atWeb site:http://psyserver.kfunigraz.ac.at/empg2000/[For details, see EMS Newsletter 35]

3�10: Noncommutative Geometry, Taranto,Italy

4�6: BEM 22 � 22nd InternationalConference on the Boundary ElementMethod, Cambridge, UKAim: major new advances are made in theboundary element field each year and this con-ference provides a way of disseminating thecurrent research within the international scien-tific and engineering community. The worldconference on boundary element methodsoriginated in 1978 and continues to act as themain focus for major developments in the tech-nique. The proceedings are always published inbook form and constitute a permanent recordof the development of the method over the lasttwo decadesTopics: dynamics and vibrations, fracturemechanics and fatigue, inelastic problems,composite materials, plates and shells, contactmechanics, geomechanics, material processingand metal forming, soil and soil structure prob-lems, electrostatics and electromagnetics, bio-mechanics, fundamental principles, computa-tional techniques, advanced formulations,refinement and adaptive techniques, sensitivityanalysis and shape optimisation, inverse prob-lems, applications in optimisation, industrialapplications, thermal problems, fluid dynamics,fluid flow, groundwater flow, interfacial andfree surface flow, transport problems, wavepropagation problems, acoustics, high perfor-

mance computing, dual reciprocity methodand basis functions Programme: see the web siteOrganizer: Wessex Institute of Technology,Southampton, UK Sponsors: International Society of BoundaryElements (ISBE) Deadlines: submit papers as soon as possibleNote: Wessex Institute of Technology is alsoorganising BETECH 2001 � 14th InternationalConference on Boundary Element Technology,12�14 March 2001 in Orlando, Florida, USA. Abstracts should be submitted as soon as possi-ble. See the web site at http://www.wessex.ac.uk/conferences/2001/Information: contact Susan Hanley,Conference Secretariat, BEM 22, WessexInstitute of Technology, Ashurst Lodge,Ashurst, Southampton SO40 7AA, UKtel: (+44) 238 029 3223, fax: (+44) 238 029 2853 e-mail: shanley@wessex.ac.uk Web-site: http:/www.wessex.ac.uk/conferences/2000/bem22/

4�6: Mathematics of Surfaces, Cambridge,UKInformation:Web site: http://www.ima.org.uk

4�8: FGI2000 French-German-ItalianConference on Optimisation, Montpellier,FranceInformation: contact: Bernard Lemaire,Mathématiques, Université de Montpellier II,Place Eugène Bataillon, 34095 Montpelliercedex 05e-mail: fgi2000@math.univ-montp2.fr Web site: http://www.math.univ-montp2.fr/ [For details, see EMS Newsletter 34]

4�15: Spatial Structures in Biology andEcology: Models and Methods, ABiomathematics Summer School, Taranto,ItalyInformation:Web site:http://www.mat.unimi.it/~miriam/ESMTB/MARTINA-SS/summer_school.html

5�7: Quantitative Modelling in theManagement of Health Care, Salford, UK Information: Web site: http://www.ima.org.uk/mathematics/conferences.htm[For details, see EMS Newsletter 34]

5�8: Brno Colloquium on Differential andDifference Equations, Brno, Czech Republic Topics: qualitative theory of differential anddifference equations and applications Aim: to bring together people working in theabove mentioned areasMain speakers: O. Dosly (Czech Rep.), I. Gyori(Hungary), L. Gorniewitz (Poland), T. Kusano(Japan), I. Kiguradze (Georgia), M. Marini(Italy), S. Schwabik (Czech Rep.) Programme: invited survey lectures, short com-munications, posters and extended abstracts Language: English Programme committee � advisory board: P.Drabek, J. Jaros, I. Kiguradze, F. Neuman, S.Schwabik Organizing committee: M. Bartusek (chair-man), Z. Dosla, O. Dosly, A. Lomtatidze, J.

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Vosmansky Proceedings: to be published in 2001 Abstracts: will be distributed at the beginningof the Colloquium Site: Santon hotel in the beautiful area of theBrno dam lake, available by city transport Conference fee: USD 170 (before 31 July),USD 190 (later); this covers the Registration fee(usual conference expenses, proceedings andsocial programme) and the package of boardand lodging for the whole period of the confer-ence Deadline: for abstracts and registration, 15June 2000 (later applications may be accepted) Information:e-mail: cdde@math.muni.cz Web site: http://www.math.muni.cz/cdde

5�16: Advanced Course on AlgebraicQuantum Groups, Bellaterra, SpainInformation:e-mail: quantum@crm.esWeb site: http://crm.es/quantum

10�17: Summer School on Geometry ofQuiver-Representations and PreprojectiveAlgebras, Isle of Thorn, UKInformation: contact Karin Erdmann,Mathematical Institute, University of Oxford,Oxford OX1 3LB, UK e-mail: erdmann@maths.ox.ac.uk Web site: http://www.mathematik.uni-bielefeld.de/~sek/summerseries.html [For details, see EMS Newsletter 34]

11�13: Optimal Discrete Structures andAlgorithms � ODSA 2000, Rostock, Germany Scope: the emphasis will be on the interactionsbetween several aspects of discrete mathemat-ics, mathematical optimisation, and theoreticalcomputer scienceTopics: combinatorial optimisation and algo-rithms on discrete structures, extremal prob-lems in posets, design theory, coding theory,cryptography, graph theory, complexity theoryInvited speakers: C. J. Colbourn (Burlington),P. Frankl (Tokyo), J. R. Griggs (Columbia), C.T. Hoang (Thunder Bay), D. Kreher(Houghton), A. Pott (Magdeburg), J. P. Spinrad(Nashville), I. Wegener (Dortmund), G. Ziegler(Berlin)Programme: nine invited plenary lectures (55minutes) and contributed talks (25 minutes) inparallel sessionsOrganisers: A. Brandstädt, K. Engel, H.-D.Gronau, R. Labahn (Rostock) Proceedings: a special issue of Discrete AppliedMathematics with papers related to the confer-ence and refereed according to the usual pro-ceduresInformation: e-mail: odsa@mathematik.uni-rostock.de Web site: http://www.math.uni-rostock.de/odsa

11�14: International Colloquium in Honourof Professor Michel Mendès, Bordeaux,France Information:Web site: http://www.math.u-bordeaux.fr/~stan/Colloque/MMF.html [For details, see EMS Newsletter 35]

11�15: Boundary Integral Methods: Theoryand Applications, Bath, UKInformation:

Web site: http://www.ima.org.uk/mathematics/conferences.htm[For details, see EMS Newsletter 34]

12�15: Imaging and Digital ImageProcessing: Mathematical Methods,Algorithms and Applications, Leicester, UKInformation:Web site: http://www.ima.org.uk/mathematics/conferences.htm[For details, see EMS Newsletter 34]

12�15: IWOTA � Portugal 2000 InternationalWorkshop on Operator Theory andApplications, Faro, PortugalInformation: contact F.-O. Speck,Departamento de Matematica, InstitutoSuperior Tecnico, U.T.L., 1049-001 Lisboa,Portugaltel: +351-21-8417095, fax: +351-21-8417598, e-mail: fspeck@math.ist.utl.ptWeb site: http://www.ualg.pt/cma/iwota[For details, see EMS Newsletter 35]

13�15: International Conference of the RoyalStatistical Society, Reading, UKInformation:e-mail: rss2000@reading.ac.uk

15�18: Physical Interpretations of RelativityTheory, London, UKInformation:e-mail: michael.duffy@sunderland.ac.uk

18�22: International Data AnalysisConference, Innsbruck, AustriaInformation:e-mail: viertl@tuwien.ac.at Web site:http://www.statistik.tuwien.ac.at/ida2000/

18�23: International Congress on DifferentialGeometry, Bilbao, Spain[in memory of Alfred Gray (1939-98)][some details have changed since the announcementin EMS Newsletter 35]Theme: differential geometryTopics: special Riemannian manifolds, homo-geneous spaces, complex structures, symplecticmanifolds, geometry of geodesic spheres andtubes and related problems, geometry of sur-faces, computer graphics in differential geome-try and MathematicaMain speakers: T. Banchoff (USA), R. Bryant(USA), H. Ferguson (USA), T. Friedrich(Germany), K. Grove (USA), S. Gindikin (USA),A. Huckleberry (Germany), D. Joyce (England),M. Mezzino (USA), V. Miquel (Spain), E. Musso(Italy), R. Palais (USA), M. Pinsky (USA), A. Ros(Spain), D. Sullivan (USA), I. Taimanov(Russia), J. Wolf (USA)Programme: plenary lectures, a video confer-ence, oral communications, a poster session,and a round table discussion �Where does thegeometry go? A research and educational per-spective�Language: EnglishCall for papers: we are using the facilities ofAtlas Mathematical Conference Abstracts. Ifyou wish to present an oral communication or aposter, please submit an extended (up to twopages) abstract (either plain ASCII or TeX) viahttp://at.yorku.ca/cgi-bin/amca/ submit/cadq-01.Abstracts accepted by the organising committee

will become available at http://at.yorku.ca/cgi-bin/amca/cadq-01 Programme committee: T. Banchoff (USA), J.P. Bourguignon (France), S. Donaldson(England), J. Eells (England), S. Gindikin(USA), M. Gromov (France), O. Kowalski(Czech Republic), M. Mezzino (USA), S.Novikov (USA), M. Pinsky (USA), A. Ros(Spain), S. Salamon (England), L. Vanhecke(Belgium), J. Wolf (USA) Organising committee: M. Fernández (chair-man, Spain), L. C. de Andrés (Spain), L. A.Cordero (Spain), A. Ferrández (Spain), R.Ibáñez (Spain), M. de León (Spain), M. Macho-Stadler (Spain), A. Martinez Naveira (Spain), L.Ugarte (Spain)Sponsors (provisional): Universidad del Paìs,Vasco-Euskal Herriko Unibertsitatea, AulaDocumental de Investigaciòn, Ayuntamiento deBilbao-Bilboko Udala, Bilbao IniciativasTuristicas, Burdinola, Canon, Casco Viejo,Codorniù, Colegio Mayor Miguel Unamuno,Deia, European Mathematical Society,Euskaltel, La Fortaleza, Gobierno Vasco-EuskoJaurlaritza, Iberia, Iparlat-Kaiku, Jagoleak,Kodak, Kukuma, Metro Bilbao, El Mundo,Norbega Coca-Cola, Panda Software, RadioNerviòn-Telebilbao, Radio Popular, RealSociedad Bascongada de Amigos del Paìs, RealSociedad Matemàtica Espaõla, Seur, SociedadEstatal España Nuevo Milenio, Staedtler,Urrestarazu, Wolfram Research Inc.Proceedings: to be publishedSite: the buildings of Facultad de CienciasEconomicas y Empresariales (AvenidaLehendakari Agirre 83, Bilbao) Grants: probably support for participants fromcountries in a difficult economic situation andyoung mathematicians Deadlines: for registration, 30 June; forabstracts, 31 May Information:e-mail: gray@lg.ehu.esWeb site: www.ehu.es/Gray

18�27: 8th Workshop on Stochastic andRelated Fields, Famagusta, North CyprusInformation:Web site: http://mozart.emu.edu.tr/workshop

19�22: Fractal Geometry: MathematicalTechniques, Algorithms and Applications,Leicester, UKInformation: Web site: http://www.ima.org.uk/mathematics/confractalgeometry.htm[For details, see EMS Newsletter 34]

19�22 SCAN 2000: 9th GAMM-IMACSInternational Symposium on ScientificComputing, Computer Arithmetic andValidated Numerics, Karlsruhe, GermanyInformation:Web site: http://www.scan2000.de/

22�23: International Colloquium onMathematics and Art, Maubeuge, FranceAims: (1) to allow to artists to present worksinspired by mathematics and/or using mathe-matics; (2) to permit mathematicians to presentand discuss works that can inspire mathematics(tilings, all kinds of surfaces and topologicalobjects, trajectories); (3) to debate the use of artas a tool to help the teaching of mathematics,and to compare experiences and projects

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Advisory committee: R. Brown (Bangor), J.Bochnak (Amsterdam), C.-P. Bruter (Paris 11),M. Chaves (Porto), M. Emmer (Roma 1), T. C.Kuo (Sydney), Richard Palais (Brandeis), V.Poenaru (Paris 11)Proceedings: to be publishedInformation: contact C. P. Bruter, fax: 01 60 92 51 43 e-mail: bruter@univ-paris12.fr

22�27: EURESCO Conference on Geometry,Analysis and Mathematical Physics: Analysisand Spectral Theory, San Feliu de Guixols,Spain Information:http://www.esf.org/euresco/00/pc00127a.htm[For details, see EMS Newsletter 35]

22�27: EURESCO Conference on NumberTheory and Arithmetical Geometry: Motivesand Arithmetic, Obernai (near Strasbourg),FranceOrganiser: U. Jannsen (Regensburg)Information: available from Web site: http://www.esf.org/euresco

22�28: 4th International Conference onFunctional Analysis and ApproximationTheory (4th FAAT), Acquafredda di Maratea ,Potenza, ItalyInformation: contact any of the organisingcommittee or visit the website Web site:http://www.dm.uniba.it/maratea/FAAT2000.htm[For details, see EMS Newsletter 35]

25�27: International Workshop on AutomatedDeduction in Geometry, Zurich, SwitzerlandInformation:e-mail: wang@calfor.lip6.fr

26�30: XI Biannual Conference of ECMI TheEuropean Consortium for Mathematics forIndustry, Palermo, ItalyInformation:Web site: http://www.ecmi.dk/

29�2 Oct: IXth Oporto Meeting OnGeometry, Topology And Physics, Oporto,Portugal [series formerly known as the Oporto Meetingson Knot Theory and Physics]Aim: to bring together mathematicians andphysicists interested in the inter-relationbetween geometry, topology and physics and toprovide them with a pleasant and informalenvironment for scientific interchangeThemes: this year the focus will be on differen-tial geometry and algebraic geometry Programme: mainly short courses, of approxi-mately three lectures each, given by the mainspeakers, and a limited number of seminars(more details later). The talks are at theadvanced graduate or postdoctoral level, andshould be of interest to all researchers wishingto learn about recent developments in the over-lap between geometry, topology and physicsMain speakers: Maks. A. Akivis (Jerusalem)and Vladislav V. Goldberg (Newark), TheGeometry of Submanifolds with Degenerate GaussMappings (joint course); Joseph Landsberg(Toulouse), Representation Theory and theGeometry of Projective Varieties; Dietmar Salamon(Zurich), Floer Homology; Simon Salamon

(Oxford); Rick W. Sharpe (Toronto), A Survey ofInfinite Dimensional Lie Groups Information:Web site:http://fisica.ist.utl.pt/~jmourao/om/omix/om00b.html

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

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

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

20�22: One Hundred Years of the JournalL�Enseignement Mathématique, Geneva,SwitzerlandTopics: on the occasion of the centennial ofthe journal L�Enseignement Mathématique (a lead-ing journal in the internationalization of math-ematics education at the beginning of the 20thcentury), it was felt appropriate to hold a sym-posium with the aims of looking at the evolu-tion of mathematics education over the lastcentury and identifying some guidelines andtrends for the future, taking into account,among other sources, the documents, debatesand related papers that appeared inL�Enseignement Mathématique. The emphasis ofthe symposium is on secondary education (stu-dents in the age range of about 12 to 18 or 19years). In addition to proposing a reflection onthe history of mathematics education and theevolution of mathematics and its teaching andlearning in the 20th century, the symposiumalso gives the opportunity of a gathering ofsome of the main actors, during the lastdecades, in mathematics education as consid-ered from an international perspectiveOrganizers: ICMI, University of GenevaProgramme committee: Daniel Coray(Switzerland), Fulvia Furinghetti (Italy), HélèneGispert (France), Bernard R. Hodgson(Canada), Gert Schubring (Germany)Information:e-mail: EnsMath@math.unige.chWeb site: http:/www.unige.ch/math/EnsMath

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

30�4 November: Evolution Equations 2000:Applications to Physics, Industry, Life

October 2000

Sciences and Economics, Levico Terme(Trento), Italy Aim: this conference is the seventh of a seriesdevoted to evolution equations. The purpose isto bring together experts in the field of evolu-tion equations in order to discuss the state ofthe art of the theory and the specific applica-tions to different classes of problems arisingfrom applicationsTopics: ranging from functional analytic meth-ods for partial differential equations to specificproblems arising in applications, with particu-lar emphasis on non-linear evolution equations,semigroup methods and partial differentialequations, stochastic evolution equations, appli-cations to mathematical physics and industry,applications to mathematical biologySpeakers: S. Albeverio , H. Amann, S. Anita, J.Antoniou, W. Arendt , J. P. Aubin , A. V.Balakrishnan, V. Barbu, P. W. Bates, C. Batty,Ph. Benilan , P. Cannarsa , J. Van Casteren, Ph.Clement, T. Coulhon , M. Crandall, G. DaPrato, M. Demuth , O. Diekmann, G. Dore, J.Escher, A. Favini, G. Goldstein, J. Goldstein, M.Gyllenberg, K. Hadeler , M. Iannelli, K. Kunish, M. Langlais , A. Lasota, Y. Latushkin, S. O.Londen, G. Lumer, V. P. Maslov, J. Milota, E.Mitidieri, R. Nagel, J. van Neerven, F.Neubrander, E. Obrecht , S. Oharu, B. dePagter, I. Prigogine, J. Pruss, A. Pugliese, S.Romanelli, M. Rockner, W. Schappacher, B. W.Shultze, G. Simonett, J. Sprekels, Th. Sturm, K.Taira, L. Tubaro, A. Valli, V. Vespri , G. Webb,L. Weis, A. Visintin, Y. ZabczykProgramme: mainly morning and afternoonsessions; also a poster session and opportunityfor workshopsScientific committee: S. Albeverio (Bonn), H.Amann (Zurich), W. Arendt (Ulm), P. Benilan(Besancon), M. Crandall (Santa Barbara), G. DaPrato (Pisa), J. Goldstein (Memphis), G. Lumer(Mons and Brussels), F. Neubrander (BatonRouge), S. Oharu (Hiroshima), J. Prüss (Halle),G. Webb (Nashville), L. Weis (Karlsrue)Local organizing committee: M. Iannelli, A.Pugliese, L. Tubaro, A. Valli, A. VisintinSponsors: European Commission, DGXII,Human Potential Programme, High levelScientific Conferences; C.I.R.M. (CentroInternazionale per la Ricerca Matematica,Trento-Italy); I.N.D.A.M. (Istituto Nazionale diAlta Matematica, Roma�Italy); University ofTrento, Mathematics DepartmentInformation: contact Mr. A. Micheletti �Secretary of CIRM, Centro Internazionale perla Ricerca Matematica Istituto Trentino diCultura 38050 Povo (Trento)tel: (+39)-0461-881628, fax: (+39)-0461-810629 e-mail: michelet@science.unitn.it Web site: http://alpha.science.unitn.it/cirm/

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

November 2000

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EMS June 200032

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

27�1 December: Foundations of Probabilityand Physics, Vaxjo, SwedenAim: to reconsider foundations of probabilitytheory in connection with foundations ofphysics (quantum as well as statistical)Topics: the following problems will be dis-cussed: (A) Creation of the conventional probabilitytheory: Kolmogorov�s axiomatics, 1933 (Whyand How?). Is probability a measure or a fre-quency? A role of ergodicity. Why didKolmogorov�s model become dominating inmathematics and physics? (B) Creation of probabilistic foundations ofquantum mechanics: Is quantum probability ameasure, frequency, potentiality or degree ofmy belief? The great diversity of opinions. Whydoes everybody use finally Kolmogorov�s mea-sure-theoretical approach? (C) Probability in statistical physics, The role ofergodicity, Individual and statistical ergodicity. (D) Einstein-Podolsky-Rosen paradox, Bell�sinequality, locality, reality, probability.Conventional probabilities and Bell�s inequality(via Bell, Clauser, Horne, Shimony, Holt,Aspect, Home, Selleri, d�Espagnat, Fine, Rastal,Greenberger, Zeilinger,�). Thermodynamicalapproach (via de Broglie, Lochak, de Muynck).Quantum fluctuations, stochastic dynamics.Von Mises frequency approach and Bell�sinequality. Fluctuating probabilities, modifiedBell�s inequality. Quantum probabilities.Negative probabilities. p-adic probabilities. p-adic model of fluctuating reality. (E) Quantumcomputers and information and probability. (F) Paradoxes of probability theory (in particu-lar, violations of the law of large numbers orergodicity). Randomness, complexity and foun-dations of probability theory and physicsInvited Speakers: S. Albeverio (Bonn Univ.),H. Atmanschpaher (Inst. of Psychology,Freiburg), L. Ballentine (Simon Fraser), J.Bricmont (FYMA), A. Holevo (Steklov Math.Inst.), S. Gudder (Denver Univ.), P. Lahti(Turku Univ.), T. Maudlin (Rutgers), A.Shimony (Boston Univ.), J. Summhammer(Atominst. Wien)Organisers: L. Accardi (Rome), W. De Muynck(Eindhoven), T. Hida (Meijo University,Japan), A. Khrennikov (Sweden), V. Maximov(Belostok Univ., Poland), L. Accardi (Univ.Rome-2, Italy), W. De Muynck (Univ. ofEindhoven, the Netherlands), T. Hida (MeijoUniversity, Japan), A. Khrennikov (VäxjöUniversity, Sweden), V. Maximov (BelostokUniv., Poland)Organising institution: International Centerfor Mathematical Modeling [in physics, tech-nique and cognitive sciences], Vaxjo University,SwedenDeadline: 1 October 2000Information:e-mail: send abstracts and application forms to

Karl-Olof.Lindahl�@msi.vxu.se Web site: http://www.msi.vxu.se/aktuellt/konferenser.html

18�20: 5th International Conference onMathematics in Signal Processing, Coventry,UKInformation: contact Pamela Bye, The Instituteof Mathematics and its Applications, CatherineRichards House, 16 Nelson Street, Southend-on-Sea, Essex SS1 1EF, Englandfax: (+44) 1702 354111 e-mail: conferences@ima.org.ac.uk [For details, see EMS Newsletter 34]

28�3 February: 2001 XXI InternationalSeminar on Stability Problems for StochasticModels, Eger, Hungary[Dedicated to the 70th jubilees of M. Arató(Debrecen, Hungary) and V. M. Zolotarev(Moscow, Russia), Honorary Chairmen of theSeminar]Topics: limit theorems in probability theory;characterisations of probability distributionsand their stability; theory of probability metrics;stochastic processes and queueing theory; actu-arial and financial mathematics; applications ininformatics and computer sciences; applied sta-tisticsLanguage: English Programme committee: Chairmen: Z. Daróczyand G. Pap (Debrecen, Hungary), V. Yu.Korolev (Moscow, Russia). Members: A.Balkema (Amsterdam), A. Benczúr (Moscow),V. E. Bening (Moscow), A. V. Bulinski(Moscow), L. Csôke (Eger), S. Csörgô (Szeged),I. Fazekas (Debrecen), J. Fritz (Budapest), V. V.Kalashnikov (Moscow), J. Kormos (Debrecen),V. M. Kruglov (Moscow), P. Major (Debrecen),G. Michaletzky (Budapest), J. Mijnheer(Leiden), T. Móri (Budapest), E. Omey(Brussels), A. Plucinska (Warsaw), L. Szeidl(Pécs), M. van Zuijlen (Nijmegen)Secretaries of the organising committee: A.V.Kolchin (Moscow), S. Baran (Debrecen) Sponsors (provisional): Steklov MathematicalInstitue of the Russian Academy of Sciences(Moscow), Institute of Mathematics andInformatics of the Universtiy of Debrecen(Hungary), Károly Eszterházy College ofEducation (Eger)Proceedings: to be published in the Journal ofMathematical SciencesDeadline: for registration and for abstracts, 15October 2000Information:e-mail: stabil@math.klte.hu orkolchin@mi.ras.ru Web site:http://neumann.math.klte.hu/ ~stabilhttp://bernoulli.mi.ras.ru

1�6: Eighteenth British CombinatorialConference, Brighton, United KingdomAim: the promotion of all branches of combi-natorics within the UK and around the worldMain speakers: M. Aigner (Germany), ThePenrose polynomial of graphs and matroids; I.Anderson (UK), Cyclic designs; A. R. Calderbank

July 2001

January 2001

December 2000

(USA), Orthogonal designs and space-time codes forwireless communication; L. A. Goldberg (UK),Sampling and counting unlabelled structures; B.Mohar (Slovenia), Graphs on surfaces and graphminors; M. S. O. Molloy (Canada), Graph colour-ing with the probabilistic method; J. G. Oxley(USA), The interplay between graphs and matroids;J. A. Thas (Belgium), Ovoids; spreads and m-sys-tems of finite classical polar spaces; D. R. Woodall(UK), List colourings of graphsProgramme: hour-long talks by the mainspeakers and parallel sessions with 20-minutetalks on any relevant topic Call for papers: see the web site � full versionup in July 2000 Organizers: J. W. P. Hirschfeld, R. P. Lewis Sponsor: British Combinatorial Committee Proceedings: papers based on the main talkswill be published in a book available at the startof the conference; the publisher is probablyCambridge University Press. Papers based onother talks can be submitted to the further pro-ceedings, probably to be published in a volumeof Discrete MathematicsSite: University of Sussex, http://www.sussex.ac.ukDeadline: for submission of abstracts, 30 April2001 Information:e-mail: bcc2001@susx.ac.uk Web sites:http://www.maths.susx.ac.uk/Staff/JWPH/http://hnadel.maps.susx.ac.uk/TAGG/Confs/BCC/index.html

5�18: Groups St Andrews 2001, Oxford,England[Sixth in the series of Groups St AndrewsConferences]Speakers: M. D. E. Conder (Auckland), P. W.Diaconis (Stanford), P. P. Palfy (Budapest), M.P. F. du Sautoy (Cambridge), M. R. Vaughan-Lee (Oxford) Programme: short courses of lectures given bythe speakers above, and a full programme ofone hour invited lectures and short researchpresentations Topics: all aspects of group theory. The shortlecture courses are intended to be accessible topostgraduate students, postdoctoral fellows,and researchers in all areas of group theory Accommodation: is booked at St Anne�s andSomerville Colleges, Oxford, England. Lectureswill be held in the Mathematical Institute,Oxford Scientific organising committee: ColinCampbell (St Andrews), Patrick Martineau(Oxford), Peter Neumann (Oxford), EdmundRobertson (St Andrews), Geoff Smith (Bath),Brian Stewart (Oxford), Gabrielle Stoy (Oxford) Site: University of OxfordInformation: Groups St Andrews 2001,Mathematical Institute, North Haugh, StAndrews, Fife KY16 9SS, Scotland e-mail: gps2001@mcs.st-and.ac.uk Web site: http://www.bath.ac.uk/~masgcs/gps01/

27�31: Equadiff 10, CzechoslovakInternational Conference on DifferentialEquations and their Applications, Prague,Czech RepublicInformation: Web site: http://www.math.cas.cz/

August 2001

CONFERENCES

EMS June 2000 33

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

A. Balog, G. O. H. Katona, A. Recski andD. Szász (eds.), European Congress ofMathematics, Budapest, 1996, Vols. I, II,Progress in Mathematics 168, 169,Birkhäuser, Basel, 1998, 334 and 402 pp.,each DM159, ISBN 3-7643-5497-6 and 3-7643-5496-8The tradition of European Congresses ofMathematics started in 1992 in Paris andfrom the beginning they have been majorevents in European mathematical life.These volumes present the second one inBudapest in 1996. The main lectures weregiven by N. Alon, Randomness in discretemathematics; D. McDuff, Symplectic topology;J. Kollár, Low degree polynomial equations; A.S. Merkurjev, K-theory and algebraic groups;V. Milman, High-dimensional convexity theo-ry; and St. Müller, Microstructures, geometryand the calculus of variations. There were tenprize winners announced at the Congress,some of whose lectures can be found inthese volumes: W. T. Gowers, Geometry ofBanach spaces; A. Huber, Extensions ofmotives; D. Kramkov, Statistics and mathe-matics of finance; J. Matou�ek, Geometric setsystems; and L. Polterovich, Symplectic geom-etry. The volumes conclude with records offive round tables during the Congress(electronic literature; mathematicalgames; women and mathematics; publicimage of mathematics; education). Writtenversions of invited lectures in both volumesprovide a very good overview of recentsubstantial progress in a broad variety ofmathematical fields and should be of inter-est to any working mathematician. (vs)

R. Becker, Cônes convexes en analyse,Travaux en cours 59, Hermann, Paris, 1999,245 pp., ISBN 2-7056-6384-3At the end of the 1950s, Gustav Choquetdeveloped his theory of integral represen-tations for compact convex subsets in local-ly convex spaces, and also for locally com-pact convex cones. Later he published sev-eral papers generalising his theory to thecase of weakly complete convex cones. Thebook contains a postlude where the masterChoquet himself explains the history andthe value of the theory he introduced.

The author starts with basic propertiesof weakly complete convex cones andintroduces the main objects (extremalpoints, hats, topologies on certain cones,conic measures, maximal measures, conesbien-coiffé, the Cartier-Fell-Meyer theo-rem and its generalisation). He then intro-duces several applications of the theory(proofs of the Bochner-Weil theorem andof Bernstein�s theorem based on theChoquet theory) and presents a short out-

line of the Brelot and Bauer axiomaticpotential theory. Further chapters aredevoted to a deeper study of conic mea-sures, zonoforms, adapted spaces andother subjects.

It would have been useful if the authorhad also mentioned the notable results ofN. Boboc and A. Cornea from the theoryof lower semicontinuous cones, or theresults of J. Bliedtner and W. Hansen con-cerning the simpliciality of certain cones ofsuperharmonic functions in potential the-ory.

This book is readable by anybody with abasic knowledge of functional analysis. (jl)

M. Berger, Riemannian Geometry Duringthe Second Half of the Twentieth Century,University Lecture Series 17, AmericanMathematical Society, Providence, 2000, 182pp., US$34, ISBN 0-8218-2052-4This is a new edition of the survey whichappeared for the first time as Jahresberichtder Deutschen Mathematiker-Vereinigung,Band 100, Heft 2, 1998. It starts with a his-torical review devoted to Riemanniangeometry up to 1950 �from Gauss andRiemann to Chern�. The next six chapterscover the following basic topics: pinchingproblems, curvature (sectional, Ricci,scalar) of a given sign, finiteness, compact-ness, collapsing, space forms, Einsteinmanifolds, the Yamabe problem, spectraand eigenfunctions, period geodesics, geo-desic flow, volumes, systols, holonomygroups, Kähler manifolds, spinors, har-monic maps, generalisations ofRiemannian geometry, submanifolds andgeometric measure theory.

Because the survey is devoted almostexclusively to global aspects ofRiemannian geometry, the Riemannianmanifolds are usually supposed to be com-pact, or at least complete. The expositionis systematic and careful, yet also elegantand charming, written with a pedagogicalmastery that is characteristic of the author.The book includes a short introduction,comments on the main topics, 33 pages ofbibliography, and subject and authorindices. (ok)

E. Berlekamp and T. Rodgers (eds.), TheMathemagician and Pied Puzzler, A. K.Peters, Natick, 1999, 266 pp., £22, ISBN 1-56881-075-XThis book is a collection of nearly fortyarticles from recreational mathematics,written by foremost puzzlers, magiciansand mathematicians. It contains manyinteresting puzzles, games, and mathemat-ical problems (for example, is it possible tofit unit squares on an n × n sheet of graphpaper labelled with 0s and 1s so that everyneighbourhood is odd). What most ofthese articles have in common is that theyshow the �magic face� of mathematics, in

contrast to the mathematics that the read-er may have learnt at school.

The book is dedicated to MartinGardner, the famous author of theMathematical Games column in ScientificAmerican for many years. The name of thismathemagician and pied puzzler guaran-tees that the book will bring enjoyment toa wide range of readers. (ec)

C. Cercignani and D. H. Sattinger,Scaling Limits and Models in PhysicalProcesses, DMW Seminar 28, Birkhäuser,Basel, 1998, 190 pp., DM58, ISBN 3-7643-5985-4 and 0-8176-5985-4This book, written by experts on theirresearch areas, is in two parts. In the firstpart, Scaling and Mathematical Models inKinetic Theory, Carlo Cercignani provides awell-written quick introduction to pertur-bation and scaling methods in the mathe-matical theory of the Boltzmann equation.The second part, Scaling, MathematialModelling, and Integrable Systems, was writ-ten by David Sattinger, and presents anintroduction to dispersive waves and aderivation of the associated completelyintegrable systems, as the Korteweg deVries and non-linear Schrödinger equa-tions, via a careful treatment of the inversescattering theory for the Schrödingeroperator.

Both parts contain interesting historicalcomments and a huge bibliography. Thebook can be recommended to non-special-ists in mathematics, physics and engineer-ing who desire a quick insight into theseareas. (jmal)

S. B. Cooper and J. K. Truss (eds.), Setsand Proofs and Models and Computability,London Mathematical Society Lecture NotesSeries 258, 259, Cambridge University Press,Cambridge, 1999, 436 and 419 pp., each£29.95, ISBN 0-521-63549-7 and 0-521-63550-0

These two volumes contain articles byinvited speakers at Logic Colloqium �97 (themajor international meeting of theAssociation of Symbolic Logic); all authorsare specialists in their respective fields. Inthese volumes, the reader can obtain com-prehensive information about the currentstate of research in almost all branches ofmathematical logic.

The first volume contains papers from alarge variety of problems concerning setsand proofs. We mention here ordinalanalysis, forcing large cardinals, coveringproperties concerning sets and construc-tive logic, intuitionistic propositional logic,complexity of the propositional calculus,mathematical truth, and infinite-timeTuring degrees concerning proofs.

As well as classical problems of modeltheory (the topological stability conjectureor relative categoricity in abelian groups)and computability (computability andcomplexity), the reader can find in the sec-ond volume effective model theory, non-standard models of arithmetic or logic anddecision making. (k³)

R. G. Douglas, Banach Algebra Techniquesin Operator Theory, Graduate Texts in

RECENT BOOKS

EMS June 200034

Recent booksRecent booksedited by Ivan Netuka and Vladimír Sou³ek

Mathematics 179, Springer, New York, 1998,194 pp., DM98, ISBN 0-387-98377-5This book is the second edition of the 1972original version and remains unchanged,except that several mistakes and typo-graphical errors have been corrected.There is also a brief report with referenceson the current state of the double-starredopen problems.

The book begins with basic results inBanach space theory (main examples, lin-ear functionals, weak topologies, theHahn-Banach and open mapping theo-rems, Lebesgue and Hardy spaces). Thesecond chapter is devoted to Banach alge-bras (also the Gelfand-Mazur theorem andthe Gelfand transform). Further chaptersexplain topics in the geometry of Hilbertspaces and operators on Hilbert spacesand C*-algebras (normal and self-adjointoperators, the Gelfand-Naimark theorem,the spectral theorem and functional calcu-lus, weak and strong operator topologies,W*-algebras, the Fuglede theorem). Onechapter is devoted to compact andFredholm operators and index theory (theCalkin algebra, the Fredholm alternative,representations of the C*-algebra of com-pact operators). The last two chapters dealwith Hardy spaces (Beurling�s theorem,the F. and M. Riesz theorem, the maximalideal space of H8, the Gleason-Whitneytheorem, abstract harmonic extensions,maximal ideal spaces) and with Toeplitzoperators (the spectrum and invertibilityof Toeplitz operators, localisation,Widom�s result on the connectedness ofthe spectrum). At the end of each chapterthere are notes suggesting additional read-ing and a large number of problems of var-ious difficulty.

The book is accessible to anybody with abasic knowledge of analysis, general topol-ogy, measure theory and algebra. (jl)

P. Drábek, P. Krej³í and P. Taká³,Nonlinear differential equations, ResearchNotes in Mathematics 404, Chapman &Hall/CRC, Boca Raton, 1999, 196 pp., £30,ISBN 1-58488-036-8

The three papers in this volume areadapted versions of a series of lecturesgiven by the editors at the Seminar inDifferential Equations, Chvalatice, CzechRepublic, in 1998. They present theauthors� recent research in a way that isalso comprehensible to PhD studentsworking in differential equations. The pre-sentations are based on recent papers andpreprints and also contain new resultsThe first contribution, on non-lineareigenvalue problems and the Fredholmalternative (P. Drábek), studies the eigen-values of the p-Laplacian, solvability of theelliptic boundary value problem with right-hand side, and points to the connectionwith the optimal Poincaré inequality. Mostof this paper deals with the one-dimen-sional case.

The second text, on evolution variation-al inequalities and multi-dimensional hys-teresis operators (P. Krej³í), is devoted to adiscussion on the influence of the geome-try of the convex constraint on analyticalproperties of the solution operator of a

variational inequality. The physical moti-vation in plasticity is explained in the text.The last contribution, on degenerate ellip-tic equations in ordered Banach spacesand applications (P. Taká³), is concernedwith quasi-linear elliptic boundary valueproblems with Dirichlet boundary condi-tions in bounded domains in Rn, includingthe p-Laplacian. The questions of exis-tence and uniqueness of solutions,Fredholm alternatives, and comparisonprinciples are studied and compared withthe linear (p = 2) case. Systems of equa-tions are also studied. In the last sectionthe abstract part-metric method in anordered topological cone is introducedand illustrated on an elliptic system. (efas)

J. von zur Gathen and J. Gerhard, ModernComputer Algebra, Cambridge UniversityPress, Cambridge, 1999, 753 pp., £29.95,ISBN 0-521-64176-4This monograph is devoted to modernalgorithms for solving classical algebraicproblems. These algorithms form the basisof various contemporary computer algebrasystems, and are presented, together withtheir complexity analysis, with the neces-sary theoretical background and accompa-nied by numerous applications. Thisapproach makes it possible for the readerto comprehend even the rather sophisti-cated recent methods. Readers can alsoimprove their understanding by solvingsome of the numerous exercises andresearch problems, following each chapter.The text is divided into five parts, namedafter the founders of the respective areas.In Part I (Euclid), the Euclidean algorithm,its modular forms, and the subresultanttheory for polynomials are examined.Applications include the ChineseRemainder Algorithm, Diophantineapproximations, and decoding of BCHcodes. Part II (Newton) deals mainly withthe fast arithmetic based on the FastFourier Transform (FFT). The Karatsubaand Strassen algorithms, as well as fastChinese remaindering and the fastEuclidean algorithm, are presented. Thereis a special chapter on the applications ofFFT to image compression. Part III(Gauss) deals with polynomial factorisa-tion. Various algorithms are presented infull detail (such as the Zassenhaus one anda polynomial-time algorithm for factorisa-tion in Q[x]). The main ingredients areHensel lifting and short vectors in lattices.Part IV (Fermat) presents the two basictasks of algorithmic number theory: test-ing for primality and the factoring of inte-gers. There is a special chapter on applica-tions to cryptography, such as the RSAcryptosystem. Part V (Hilbert) deals withthree more advanced topics: the Gröbnerbases of ideals in polynomial algebras(including complexity analysis of theBuchberger algorithm), symbolic integra-tion of rational functions, and symbolicsummation. As applications, Gröbnerproof systems and Petri nets are consid-ered, as well as the determination of allspatial conformations of cyclohexane.

The monograph has an appendix sum-marising the necessary background mater-

ial from algebra, linear algebra and com-plexity theory. There is a 30-page list ofreferences, a list of notation and an index.The monograph is a welcome survey of afast-growing and topical area of modernmathematics and computer science. (jtrl)

E. A. Gavosto, S. G. Krantz and W.McCallum (eds.), Contemporary Issues inMathematics Education, MathematicalSciences Research Institute Publications 36,Cambridge University Press, Cambridge, 1999,174 pp., £12.95, ISBN 0-521-65471-8 and0-521-65255-3The problem of how to teach mathematicshas become extremely important recently.Many things have changed. Undoubtedly,computers have had a serious influence onthe teaching process. Further, most of ourstudents major in non-mathematical fieldssuch as biology, medicine, econometrics,technology, etc. To understand their mainsubject, students must sometimes knowsome advanced parts of mathematics, suchas partial differential equations or the fastFourier transform. The usual courses ofmathematics are limited and the teachercannot present the necessary backgroundwith complete proofs. Even in calculuscourses for students of mathematics it maynot be possible to include all proofs at theusual level of rigour.

This booklet contains writings from theConference on the Future of MathematicsEducation at Research Universities held inBerkeley in 1996, and is divided into fourparts: Mathematics education at the university,Case studies in mathematics education, Thedebate over school mathematics education, andReports from the working groups. The papersstress the importance of motivation inteaching mathematics (including its histor-ical background) and in the presentationof surveys of advanced topics. It is alsoimportant to consult teachers of other sub-jects; for example, it is quite surprising tolearn of the level of interest in mathemat-ics among some biology professors, whosay that they never met a differential equa-tion in their research that could be solvedanalytically. The mathematical communitypolarises on what is the appropriate bal-ance between theory, technique and appli-cations. This book presents the views ofseveral mathematics instructors and pro-fessors on effective mathematics teaching.Some authors include useful examples todemonstrate their ideas.

This book is valuable and can be recom-mended to mathematicians who are willingto use the experience and advice of theircolleagues for improving the teachingprocess. (ja)

R. E. Gompf and A. I. Stipsicz, 4-Manifolds and Kirby Calculus, GraduateStudies in Mathematics 20, AmericanMathematical Society, Providence, 1999, 558pp., US$65.00, ISBN 0-8218-0994-6This book gives an excellent introductioninto the theory of 4-manifolds and can bestrongly recommended to beginners inthis field. It is carefully and clearly written;the authors have evidently paid greatattention to the presentation of the mater-

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ial, and the prerequisites are not too high.A postgraduate student will probably nothave to look very often to other sources.The exposition contains many really pret-ty and interesting examples and a greatnumber of exercises; the final chapter isthen devoted to solutions of some of these.I think that this type of presentation makesthe subject more attractive and its studyeasier. We add for young mathematiciansthat the theory of 4-manifolds has devel-oped very quickly in the last two decades,and can currently be considered as a verypromising field of study. (jiva)

A. Hajnal and P. Hamburger, Set Theory,London Mathematical Society Student Texts48, Cambridge University Press, Cambridge,1999, 316 pp., £16.95, ISBN 0-521-59344-1 and 0-521-59667-XThis textbook on set theory is based oncourses given by András Hajnal in EötvösLoránd University in Budapest. It dividesinto two parts.

Part I, Introduction to set theory, explainsthe basics in an almost naive way, but withenough rigour to give a proof, not only aclaim, of (say) the result that there is no setof all sets. All standard material for anundergraduate course is here. This partends with cardinal exponentiation, König�stheorem and the Hausdorff formula. A 40-page appendix follows, where an axiomat-ic treatment is developed, with emphasison the logical background (interpretationof the language, absoluteness of formulasand relative consistency).

Part II, the major part of the book, con-tains Hajnál�s beloved subject, infinitarycombinatorics. Stationary sets, ∆-systems,Ramsey�s theorem, partition calculus, setmappings, square-bracket arrows, largecardinals, saturated ideals � all these arepresented in considerable detail. The cul-mination is Shelah�s theorem on the power2ℵ

ω. Although this is no longer a text forundergraduates, the clarity of presentationand the choice of the easiest proofs contin-ues. A very fine feature of the book is itsexercises. Each chapter contains about tenof them, usually demanding more than amechanical application of theoremsproved in the text. Give this book fifteenminutes� attention � you will find that youmust buy it! (ps)

F. Hirsch and G. Lacombe, Elements ofFunctional Analysis, Graduate Texts inMathematics 192, Springer, New York, 1999,393 pp., DM98.00, ISBN 0-387-98254-7This book concerns that part of functionalanalysis that does not require advancedtopological tools and does not cover topicslike locally convex spaces or cornerstonesof functional analysis (the open mappingand closed graph theorems and the Hahn-Banach theorem). The first part of thebook is devoted to function spaces andtheir duals. It includes chapters on thespace of continuous functions on a com-pact set (the Stone-Weierstrass and Ascolitheorems), on locally compact spaces andRadon measures (Daniell�s approach tointegration theory, Radon measures), rudi-ments from the theory of Hilbert spaces

(the Riesz representation theorem, dualsof Hilbert spaces, weak convergence andHilbert bases) and a brief theory of Lp-spaces (also duals, the Radon-Nikodýmtheorem, weak convergence and convolu-tion). The second chapter is dedicated tooperator theory (operators on Banach andHilbert spaces and their spectra, opera-tional calculus on Hermitian operatorsand compact self-adjoint operators,Fredholm theorems, kernel operators). Acomprehensive final part of the book is anintroduction to the theory of distributions.Multiplication, differentiation and convo-lution of distributions are studied in a fairamount of detail. There are chapters onthe fundamental solutions of a differentialoperator, with classical examples of theLaplacian, the heat and wave operatorsand the Cauchy-Riemann operator. Asapplications, several regularity results � inparticular, those concerning the Sobolevspaces W1,p(Rn) � are proved. In the finalpart the Laplacian on an open subset of Rn

is studied (the Dirichlet problem, classicaland variational formulations, eigenvaluesof the Dirichlet Laplacian, and the heatand the wave problems).

The book is equipped with a rich collec-tion of exercises, with notes containingexamples and counter-examples, applica-tions to test the reader�s understanding ofthe text, and digressions to introduce newconcepts. Answers are presented at theend. It is readable by anybody with a basicknowledge of analysis (differential calculusin several variables, integration theorywith respect to a measure, the basics ofBanach space theory and the topology ofmetric spaces). It can be recommended tograduate students and will help moreadvanced students and researchersimprove their knowledge of some parts offunctional analysis. (jl)

F. den Hollander, Large Deviations, FieldsInstitute Monographs 14, AmericanMathematical Society, Providence, 2000, 143pp., US$49, ISBN 0-8218-1989-5The subject of this book is large deviationtheory. It corresponds to a graduate coursetaught by the author at the Fields Institutefor Research in Mathematical Sciences inToronto in 1998. The book divides intotwo parts.

Part A (Chapters I�V) describes the the-ory, while Part B (Chapters VI�X) presentsapplications. Chapters I and II contain theclassical basic large deviation theory fori.i.d. random variables. Chapter III pre-sents general definitions and theorems in amore abstract context. Chapter IV containsa discussion of large deviations for Markovchains, while Chapter V deals with largedeviations for random sequences withmoderate dependence.

A number of interesting applications ofthe above general theory are presented inPart B; for example, there are applicationsto statistical hypothesis testing (ChapterVI), random walk environment (ChapterVII), heat conduction with random sourcesand sinks (Chapter VIII), polymer chains(Chapter IX) and interacting diffusions(Chapter X). There are many exercises,

with solutions at the end of the book.This is a useful book on large deviations. Itcan be used as a text for advanced PhD stu-dents with a really good background inmathematical analysis and probability the-ory. (mhusk)

D. H. Hyers, G. Isac and T. M. Rassias,Stability of Functional Equations inSeveral Variables, Progress in NonlinearEquations and Their Applications 34,Birkhäuser, Boston, 1998, 313 pp., DM178,ISBN 0-8176-4024-X and 3-7643-4024-XThis book presents the current state ofsolutions of one of Ulam�s problems: is anε-homomorphism of metric groups closeto some homomorphism? The affirmativeanswer for the Cauchy functional equationon a Banach space has been well known fora long time (Hyers, 1941). This questioncan be extended in various ways � forexample, how does the answer depend onthe structure of the group (without com-mutativity the result is not true)?, underwhat conditions is an ε-homomorphism ahomomorphism (so-called superstability)or a homomorphism plus something�small� (stability modulo ...)? Similar ques-tions can be asked for multiplicative func-tionals on a Banach algebra, for polynomi-al functionals on a Banach space (∆h

n f =0), for inequalities instead of equalities(e.g. ε-convex functions), for set-valuedmaps, etc. Recent results on Ulam�s otherproblems concerning the stability of sta-tionary points or local minimum points arealso described.

The authors present both classical andrecent results clearly and in a self-con-tained fashion. Since only a basic knowl-edge of algebra and functional analysis isrequired for understanding much of thistext, it is accessible for graduate students.(jmil)

V. Ivrii, Microlocal Analysis and PreciseSpectral Asymptotics, Springer Monographsin Mathematics, Springer, Berlin, 1998, 731pp., DM178, ISBN 3-540-62780-4The main topic of this comprehensivemonograph (more than 700 pages) con-cerns asymptotic formulae for the numberN(λ) of eigenvalues of an operator A, whichdo not exceed λ, and corresponding pre-cise remainder estimates.

The first part of the monograph devel-ops the necessary tools of micro-localanalysis for h pseudo-differential and h-Fourier integral operators, and containsthe main theorems on propagation of sin-gularities. The second part contains semi-classical spectral asymptotics (as h → 0+)for various types of operators of their fam-ilies. In particular, there are chaptersdevoted to the Schrödinger and Diracoperators with strong magnetic field (thecoupling constant µ being another para-meter). The third part contains theoremsdescribing estimates of the spectrum whichare then applied in many special and inter-esting cases in the fourth part of the bookto get asymptotic formulae for spectrum incorresponding cases.

The book is well organised, is written ina careful and systematic way, and contains

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a huge amount of unpublished new mater-ial. The reader is expected to have a goodbackground in the theory of pseudo-differ-ential and Fourier integral operators. (vs)

J. Jost, Riemannian Geometry andGeometric Analysis, Universitext, Springer,Berlin, 1998, 455 pp., DM 78, ISBN 3-540-63654-4This is the second edition of a textbookbased on the material of a three-semestercourse given by the author at theUniversity of Bochum. It includes materialneeded for the study of geometrical andanalytic properties of Riemannian mani-folds.

At the beginning is a nice introductionto Riemannian geometry. In the first fourchapters, all basic material for the subjectis collected and presented in an under-standable way. The fifth chapter developsMorse theory and the sixth chapter con-tains a description of symmetric spaces,offering a rich source of important exam-ples in Riemannian geometry. In the sec-ond part of the book, the author returns tothe structure of geodesics on Riemannianmanifolds using analytical tools (geodesicsare treated as critical points of the energyfunctional). In addition, existence and fur-ther properties of harmonic maps betweenRiemannian manifolds are intensivelystudied, with special attention to the two-dimensional case.

In this second edition, some new andvery interesting material is added. Seiberg-Witten theory has recently found manyspectacular applications in geometry andthe topology of manifolds. The author firstcollects basic material for this theory (spingeometry, the Dirac operator), and thentreats Seiberg-Witten equations as an ana-logue of the Ginsberg-Landau equations.Each chapter ends with appendices called�Perspectives�, containing further resultsand directions for possible research. Thebook will be very useful for readers frommany areas of mathematics. (jbu)

R. Knobel, An Introduction to theMathematical Theory of Waves, StudentMathematical Library 3, AmericanMathematical Society, Providence, 1999, 196pp., US$23, ISBN 0-8218-2039-7This book provides an introduction to thebasic terminology and methods for solvingand analysing partial differential equationsthat describe wave phenomena � travellingand standing waves, the Fourier method,the method of characteristics, shock andrarefaction waves, and the viscositymethod for conservation laws. Deepermathematical concepts are mentionedwithout proof, but are well illustrated withsimple models (such as the superpositionprinciple on the one-dimensional waveequation, and shock waves and viscositysolution on the traffic-flow problem). Thederivation of models from physical laws isalso given in some cases (the sine-Gordonequation, vibrating string, conservationlaw). The book also contains elementarycomputer experiments (such as graphingthe analytically obtained solution) andexercises. To understand the text only a

knowledge of calculus is required. Thisbook is based on an undergraduate sum-mer course. It may provide an interestingfirst reading on high analysis at an ele-mentary level. (efas)

V. Lakshmikantham and S. G. Deo:Method of Variation of Parameters forDynamic Systems, Series in MathematicalAnalysis and Applications 1, Gordon andBreach Publishers, New Delhi, 1998, 317 pp.,£54, ISBN 90-5699-160-4Many results on differential equationswhose linear parts are perturbed by (small)non-linearities are based on so-called vari-ation of parameters formulae. These for-mulae can also be extended to certaincases where there are no adequate linearparts. In this way, the method of variationof parameters is closely related to modifi-cations of the Lyapunov second method(Lyapunov-like functions), comparisonprinciples and monotone iteration tech-niques. This book brings together variousaspects of these procedures for ordinarydifferential, integro-differential, delay, dif-ference, impulse differential, and stochas-tic differential equations. Applications aregiven to stability, controllability, periodicsolutions, and more generally to boundaryvalue problems. There are also brief sec-tions devoted to abstract differential equa-tions and more special equations. Thisbook partly overlaps other books writtenby the first author, but has the advantageof collecting similar ideas together. (jmil)

J. D. Lamb and D. A. Preece (eds.),Surveys in Combinatorics, 1999, LondonMathematical Society Lecture Note Series 267,Cambridge University Press, Cambridge, 1999,298 pp., £24.95, ISBN 0-521-65376-2This volume contains the nine invited talkspresented at the 17th BritishCombinatorial Conference, held at theUniversity of Kent in 1999. A special fea-ture is W. T. Tutte�s article, The coming ofmatroids, which covers the development ofthe theory of matroids from the point ofview of one of the masters.

The remaining eight survey articlesdeal with graph theory (4), finite geome-tries (2), probability and complexity (1)and combinatorial designs (1). C.Thomassen surveys parity arguments andproblems in graph theory and their con-nection to the cycle space, and proves anew result on odd K4-subdivisions in 4-connected graphs. R. Thomas discussesrecent and not-so-recent excluded minortheorems for graphs, connected to linklessembeddings, Hadwiger�s conjecture,Pfaffian orientations, and other topics. N.C. Wormald surveys results and methodsfor random graphs with restricted vertexdegrees, stressing results for regulargraphs. J. Pach describes some resultsfrom geometrical graph theory whereedges are represented in the plane bystraight-line segments, and considers somegeneralisations to hypergraphs (here sim-plices replace segments). S. Ball gives anoverview of the recent method in finitegeometry where objects are represented bypolynomials (problems concern nuclei,

affine blocking sets, maximal arcs, andunitals). K. Metsch�s article is on Bose-Burton type theorems for finite projective,affine and polar spaces. M. Dyer and C.Greenhill focus on approximate samplingfrom combinatorially defined sets andapproximate counting using Markov chainMonte-Carlo methods, viewed from theperspective of combinatorial algorithms.C. J. Colbourn, J. H. Dinitz and D. R.Stinson are concerned with applications ofcombinatorial designs to communication,cryptography and networking; their exten-sive bibliography has 159 items. (jnes)

G. F. Lawler and C. N. Lester, Lectures onContemporary Probability, StudentMathematical Library 2, AmericanMathematical Society, Providence, 1999, 95pp., US$19, ISBN 0-8218-2029-XThis well-written booklet is based on lec-tures and computer labs for undergradu-ates, held by the authors at the IAS/parkCity Mathematics Institute. The subjectsinclude random walks (simple and self-avoiding), Brownian motion, card shuf-fling, Markov chains, Markov chainMonte-Carlo, spanning trees, computersimulation of random walks and models infinance. The booklet is divided into 13chapters (lectures) and includes a numberof exercises. The reader is assumed to befamiliar with the basic elements of proba-bility. The authors try to present topicsthat are accessible to advanced undergrad-uates and to show the appeal of parts ofmodern probability, and make the lecturesvery attractive. To avoid long calculationsthey rely either on intuition or on results ofsimulations. The emphasis is on a proba-bilistic style of thinking. (mhusk)

Tan Lei (ed.), The Mandelbrot Set, Themeand Variations, London Mathematical SocietyLecture Notes Series 274, CambridgeUniversity Press, Cambridge, 2000, 365 pp.,ISBN 0-521-77476-4This volume provides a systematic exposi-tion of current knowledge concerning theMandelbrot set, and presents the latestresearch in complex dynamics. Topics dis-cussed include the universality and localconnectivity of the Mandelbrot set, para-bolic bifurcations, critical circle homeo-morphisms, absolutely continuous invari-ant measures, and matings of polynomials,along with the geometry, dimension andlocal connectivity of Julia sets. In additionto presenting new work, this collectionpresents important results hitherto unpub-lished or difficult to find in the literature.

The book contains a collection of six-teen papers (four in French). The prefacepresents an excellent historical back-ground of the topic. The papers aregrouped according to the properties dis-cussed (universality of the Mandelbrot set,quadratic Julia sets and the Mandelbrotset, Julia set of rational maps, foundation-al results). The book will be of interest tograduate students in mathematics, physicsand mathematical biology, as well as toresearchers in dynamical systems andKleinian groups. (pp)

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H. van Maldeghem: Generalized Polygons,Monographs in Mathematics 93,Birkhäuser, Basel, 1998, 502 pp., DM218,ISBN 3-7643-5864-5 and 3-8176-5864-5This book gives a contemporary survey ofresults in the theory of generalised poly-gons. The approach used is mostly geo-metrical in nature, and several algebraicresults are also proved in a geometric way.A generalised polygon (the notion is due toTits) is a geometry (an incidence structure)that generalises the notion of an ordinarypolygon. This notion is then related toanother combinatorial structure called abuilding and also to more general geome-tries, such as partial geometries, partialquadrangles, near polygons, etc. Anothertype of structure used to describe gener-alised polygons is a finite group of a spe-cial type.

Each chapter starts with an introduc-tion, containing motivation and historyrelated to its content. We find a classifica-tion and description of the classical andmixed quadrangles, classical hexagons,and non-classical hexagons and octagons.There is a chapter in which the coordinati-sation of polygons is studied, and anotherchapter contains a description of homo-morphisms of polygons and their auto-morphisms groups. Moufang polygons areinvestigated in great detail in several chap-ters. The final chapter is devoted to topo-logical polygons and contains results fromalgebraic topology. The book contains alot of material and is complements themonograph by Payne and Thas, FiniteGeneralized Quadrangles, Pitman ResearchNotes in Mathematics 110, 1984. (jbu)

J. T. McClave and T. Sincich, Statistics8e, Prentice Hall, New Jersey, 2000, 848 pp.,£25.99, ISBN 0-13-022329-8 and 0-13-022574-6

This is an undergraduate introductorytext in probability and statistics. The chap-ters devoted to probability contain suchclassical elementary topics as events, sam-ple spaces, conditional probability, anddiscrete and continuous random variables.Statistical chapters describe confidenceintervals, tests of hypotheses, analysis ofvariance, linear regression and modelbuilding, contingency tables, and non-parametric statistics. The book containsplenty of examples and exercises based onvery rich collections of real data.Numerical processing of data is illustratedon the TI-83 Graphic Calculator with ref-erences to the well-known statistical soft-ware packages SPSS, Minitab and SAS.

This textbook has become very popular.Its eighth edition includes a new chapteron multiple regression, and two otherchapters have been rewritten. The authorsinclude relevant recent results, such as theWilson adjusted confidence interval for apopulation proportion, which is based on apaper published in 1998. On the otherhand, the authors could have mentionedthat the Tukey method for multiple com-parisons was generalised for unequal sam-ple sizes, since the generalisation is avail-able in many statistical packages andbelongs to the most sensitive methods.

This book contains no mathematicalderivations of statistical methods, but ithelps the reader to understand statisticalmethodology and carry out necessary cal-culations. The data used in the book isavailable on a diskette in ASCII format.There also exist supplements for theinstructor and supplements available forpurchase by students. (ja)

A. N. Parshin and I. R. Shafarevich (eds.),Algebraic Geometry V, Encyclopaedia ofMathematical Sciences 47, Springer, Berlin,1999, 247 pp., DM159, ISBN 3-540-61468-0The main topic of this volume is the struc-ture theory of Fano varieties. A smoothprojective variety is a Fano variety if its anti-canonical divisor is ample. Examples ofFano varieties in dimension 2 are the so-called del Pezzo surfaces, and many otherexamples of Fano varieties can be foundhere.

In the first part of the book, the basicproperties of Fano varieties and resultsconcerning defining equations of varietiesrelated with Fano varieties are presented.The classification problem for algebraicvarieties was recently developed thanks tothe Mori theory of minimal models and itsextension from surfaces to higher-dimen-sional varieties. A minimal model isdefined as a normal projective variety witha numerically effective canonical divisor.The Mori conjecture says that any irre-ducible algebraic variety over an algebraicclosed field of characteristic 0 is birational-ly equivalent either to a minimal model orto a fibration over a variety of smallerdimension with the general fibre being aFano variety. This book gives a very goodsurvey of known results on Fano varieties,and also includes contemporary resultsand possible generalisations. There is a listof open questions and problems. (jbu)

M. S. Petkovi³ and L. D. Petkovi³,Complex Interval Arithmetic and itsApplications, Mathematical Research 105,Wiley-VCH, Berlin, 1999, 284 pp., DM 198,ISBN 3-527-40134-2Real interval arithmetic was introduced inthe 1960s as a tool for handling uncertain-ties in the data, round-off errors and com-putations in finite precision arithmetic.Complex interval arithmetic, introducedten years later, is still much less known.The present book is the first monographdevoted entirely to this subject and its var-ious applications.

The two basic types of complex intervalarithmetic (rectangular and circular) areintroduced in Chapter 1. In Chapter 2,methods for estimating range of complexfunctions are studied with help of so-calledcircular centred forms. Chapter 3 is devot-ed to best approximations of the range ofcomplex functions by discs; in particular,the functions eZ, Zn and Z1/k are studied indetail. A typical application of complexarithmetic is the inclusion of polynomialzeros. This problem is first handled inChapter 4 for single zeros, using intervalslope methods, and its exposition contin-ues in Chapter 5 (simultaneous inclusion

of complex zeros) and Chapter 6(improved inclusion methods for polyno-mial complex zeros). Various possibilitiesfor parallel implementations of inclusionmethods are presented in Chapter 7.Numerical stability of iterative processes isdiscussed in Chapter 8, and the book con-cludes in Chapter 9 with numerical com-putation of curvilinear integrals. The bibli-ography has 183 items.

This book demonstrates the usefulnessof complex interval arithmetic for solvingnumerical problems arising in complexanalysis. It may be of interest both to spe-cialists and graduate students. In particu-lar, the part of the book that discussesinclusion of polynomial zeros (Chapters 4-6) may be found useful by specialists incontrol theory. (jroh)

M. Picardello and W. Woess (eds.),Random Walks and Discrete PotentialTheory, Symposia Mathematica 39, CambridgeUniversity Press, Cambridge, 1999, 361 pp.,£16.95, ISBN 0-521-77312-1This book contains the proceedings of theconference on �Random walks and discretepotential theory�, held in Crotona (Italy) inJune 1997. It covers the interplay betweenthe behaviour of a class of stochasticprocesses (random walks) and structuretheory. Written by leading researchers, thiscollection of invited papers presents linkswith spectral theory and discrete potentialtheory, and includes probabilistic andstructure theoretic aspects. Its interdisci-plinary approach spans several areas ofmathematics, including geometrical grouptheory, discrete geometry and harmonicanalysis. The book will be of interest toresearchers and postgraduate students,both in mathematics and statistical physics.(pp)

R. C. Read and R. J. Wilson, An Atlas ofGraphs, Oxford Science Publications,Clarendon Press, Oxford, 1998, 454 pp., £75,ISBN 0-19-853289-XThis is an unusual book, somewhat in thespirit of the Atlas of finite simple groups,although on a simpler and less technicallevel (and in a smaller format). It containssome 10000 pictures of graphs, tables ofthe number of graphs with given proper-ties, and extensive tables of properties ofdepicted graphs. Apart from its artisticvalue it will be of use to graph theory spe-cialists, chemists and combinatorialists ingeneral. It is certainly the most massiveamount of graph data in existence.

A small remark: for such an exhaustiveatlas perhaps the list of �special graphs�(listed on pp. 263-265) is somewhat tooshort. (jnes)

J. Robertson and W. Webb, Cake-CuttingAlgorithms: Be Fair If You Can, A. K.Peters, Natick, 1998, 181 pp., £26, ISBN 1-56881-076-8Since the 1950s the fair division of a cakehas been the popular subject of many eso-teric papers, as well as of serious mathe-matical research: we seek ways to divide acake �fairly� among several people in sucha way that each person is �satisfied� with the

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piece received. We can trace the roots of the problem to

the Polish group (Banach, Knaster), andthe problem was presented by H.Steinhaus in 1947. Since then the cake-cut-ting problem has blossomed and has beenwidely popularised. The book also includesnotes on other similar problems (dividingbeach-front property, the trimming algo-rithm, dividing an estate, a risk-free bet)while other well-known ones (the problemof the Nil, the problem of similar regions,the bisection problem, the ham sandwichproblem, the border problem) are left out.

The purpose of the book is to discussthe cake-cutting problem, to explain themeaning of a �fair share�, and to offer algo-rithms for its solution. The first few chap-ters provide a survey of the problem whilelater ones contain proofs of previouslyintroduced problems. Topics in the bookinclude fairly dividing a cake (two andmore persons), pieces or crumbs, howmany cuts are needed?, unequal shares,moving knife algorithms, the Stromquistalgorithm, disagreement, strong fair divi-sion, envy-free division, exact division,applying graph theory, impossibility theo-rems, Ramsey partitions and fair division.Exercises are included at the end of eachchapter. The book is readable by anybodywith a basic mathematical knowledge andcan be enjoyed by both students andresearchers. (jl)

T. Roubí³ek: Relaxation in OptimizationTheory and Variational Calculus, deGruyter Series in Nonlinear Analysis andApplications 4, Walter de Gruyter, Berlin,1997, 474 pp., DM298, ISBN 3-11-014542-1Having an integral functional as the argu-ment of �direct methods� that prove exis-tence of the minimum relies on lowersemi-continuity of this functional, which isclosely related to convexity properties ofthe integrand. In many cases, a lack of con-vexity results in oscillating minimisingsequences (or their gradients), whose weaklimits are not minimisers of the problem.In the late 1930s, L. C. Young showed thatthe non-existence of minima of some func-tionals can be overcome by enlarging theset of competing curves. These �gener-alised curves� can be described by means ofprobability measures, now widely known as�Young measures� and the procedure(enlarging) known as �relaxation�.

In Chapter 2 the author deals with anabstract and original theory of suitableconvex and (σ)-compact envelopes oftopological spaces. This is made specific inChapter 3 where the main emphasis is onconvex σ-compactifications of Lebesguespaces. It is shown that Young measuresand their generalisations can be viewed asspecial cases of these compactifications,and various topological properties of par-ticular compactifications are studied indetail. The rest of the chapter is mainlydevoted to approximation theory andnumerical analysis of general convex com-pactifications, enabling efficient computerimplementation of relaxed problems.

Optimal control theory is treated in

Chapter 4, starting with an abstract frame-work and continuing with optimisationproblems with controls on Lebesguespaces. Much attention is paid to first-order optimality conditions for the relaxedproblem and to integral and localised(Pontryagin-type) maximum principles.Four examples are analysed: the optimalcontrol of non-linear dynamical systems,elliptic and parabolic partial differentialequations and the optimal control of theHammerstein integral equation. The treat-ment of the first example (dynamical sys-tems) also includes its numerical approxi-mation and computational results.The programme of Chapters 5 and 6 is therelaxation in the variational calculus in thescalar (Chapter 5) and the vectorial case(Chapter 6). The author begins with con-vex σ-compactifications of Sobolev spacesW1,p and constructs relaxation schemes forthe reflexive case (e.g., problems ofmartensitic microstructure in elasticity, p> 1), as well as the non-reflexive case (e.g.,the non-parametric minimal surface prob-lem, p = 1). Weierstrass-type optimalityconditions are derived for the relaxedproblem, and its stability, correctness andwell-posedness are investigated. The theo-retical results for p > 1 are accompaniedwith a numerical approximation theory,including optimality conditions forapproximate solutions and sample calcula-tions.

The full strength of convex compactifi-cations is explored in Chapter 7, dealingwith the relaxation in game theory, whereboth convexity and compactness are need-ed to ensure the existence of solutions.The chapter contains two examples: gameswith dynamical systems and elliptic games,the former also with some numericalresults.

The book, containing many originalresults, has a large and bibliography andcommentary, as well as reviews of classicaland standard facts from topology, optimi-sation, functional and mathematical analy-sis (Chapter 1) which makes it fairly self-contained. It is a comprehensive and use-ful source of information for researchersand for graduate students working in opti-mal control theory, the calculus of varia-tions or the optimisation of distributedparameter systems. (mkr)

R. Roussarie, Bifurcations of PlanarVector Fields and Hilbert�s SixteenthProblem, Progress in Mathematics 164,Birkhäuser, Basel, 1998, 204 pp., DM118,ISBN 3-7643-5900-5

In this book the author studies bifurca-tions of limit periodic sets in families X? ofplanar vector fields depending on a para-meter λ in Rk, where the phase space is R2

or in general a surface of genus 0. As anexample, polynomial vector fields Xλ

n (x, y) = P(λ, x, y) δ/δx + Q(λ, x, y) δ/δywith polynomials P, Q of degree ≤ n areconsidered. The problem under consider-ation is the change of behaviour of a vectorfield when λ varies Xλ � more precisely, thechange of singular elements (singularpoints and isolated periodic orbits � limitcycles), when λ crosses a bifurcation set

Σ ⊂ Rk. The number of limit cycles thatbifurcate from a periodic limit set Γ, calledthe cyclicity of Γ in Xλ, plays a crucial role.The cyclicity depends only on the germ(unfolding) of Xλ along Γ and a generalconjecture is that it is finite for any analyt-ic unfolding. Hilbert�s 16th problem isassociated with the question: �Prove thatfor any n ≥ 2, there exists a finite numberH(n) such that any polynomial vector fieldof degree ≤ n has less than H(n) limitcycles�. There is a positive answer to thisproblem in Chapter 2, but � as the authoradds � an algorithm for finding the boundH(n) is unknown.

The author recalls in Chapter 1 thePoincaré-Bendixson theory, and inChapter 2 studies limit periodic sets andtheir cyclicity, concentrating on quadraticvector fields. The single vector fields arestudied in Chapter 3, using the desingu-larisation theory. In Chapter 4 regularlimit cycles, in Chapter 5 elementarygraphics, and in Chapter 6 non-elemen-tary graphics, are considered and studiedby methods of differential and analyticgeometry, using the Bautin ideal andasymptotic methods. (jeis)

I. P. Stavroulakis and S. A. Tersian,Partial Differential Equations. AnIntroduction with Mathematica andMaple, World Scientific, Singapore, 1999,297 pp., £23, ISBN 9-8102-3891-6This book provides an elementary intro-duction to the classical theory of first andsecond-order partial differential equa-tions. Linear problems are studied and theexplicit formulae for problems with homo-geneous and non-homogeneous boundaryconditions are provided. The contents ofthe book coincide, for example, with thefirst part of the monograph by L. Ewans,Partial Differential Equations, AMS, 1998(where, however, a more detailed discus-sion and comments are given); the sepa-rate chapters of the book can also be foundin other (cited) textbooks. Here, perhaps,more convenient examples for students isgiven.

Although the book has a promising sub-title, the usage of Mathematica consists onlyof drawing graphs (of insufficient quality)of solutions, and there are very few (onlytwo?) applications of Maple throughout thewhole book. (jmal)

D. W. Stroock, Probability Theory, anAnalytic View, Cambridge University Press,Cambridge, 2000, 536 pp., £18.95, ISBN 0-521-66349-0This is a new (paperback) edition of thebook first published in 1993 and reviewedin EMS Newsletter 14 (1994), p.36. The newedition is described as �revised�, but thealterations are minimal � splitting somesections into two and adding a couple ofnew exercises. (vd)

K. Taira, Brownian Motion and IndexFormulas for the de Rham Complex,Mathematical Research 106, Wiley-VCH,Berlin, 1998, 215 pp., DM198, ISBN 3-527-40139-3It is a well-known fact that cohomology

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groups of a manifold (with values in R) canbe represented by means of the de Rhamcomplex and harmonic forms. Similar factsfor relative cohomology groups are lessstandard. This book offers a nice systemat-ic treatment of these questions from ananalytic point of view. The main tools usedare some PDE techniques, with an addi-tional insight given by interpretation interms of Brownian motion and probabilitytheory. The first four chapters arepreparatory and contain basic facts neededfrom differential geometry, functionalanalysis, Markov processes, Fourier inte-gral operators and spaces of currents. Thecalculus of pseudo-differential operators isused in the main part of the book for atreatment of interior elliptic boundaryvalue problems and for proofs of the mainindex theorems. The book is well organ-ised and is written in an understandableway with many intuitive explanations andremarks. (vs)

K. Thomsen, Limits of CertainSubhomogeneous C*-Algebras, Mémoires dela Société Mathématique de France 71, SociétéMathématique de France, Paris, 1997, 125pp., ISBN 2-85629-064-7The main topic of this booklet is the struc-ture of a certain type of simple unital C*-algebras. Typical C*-algebras consideredhere are algebras of matrix-valued contin-uous maps defined on the interval [0, 1] oron the unit circle T, with special propertiesin a finite sequence of distinguished pointsas well as inductive limits of a sequence offinite direct sums of such algebras. Themain result is a classification theorem for aspecial type of such algebras and its exten-sion to the non-unital case. Proofs of themain theorems use several different previ-ously developed methods and are quitetechnical. The book is suitable for special-ists in the field. (vs)

J. Winkelmann, Complex AnalyticGeometry of Complex ParallelizableManifolds, Mémoires de la SociétéMathématique de France 72-73, SociétéMathématique de France, Paris, 1998, 219pp., ISBN 2-85629-070-1The main subject of this book is the studyof complex parallelisable manifolds � com-plex manifolds that are quotients of com-plex Lie groups by discrete subgroups.Special attention is paid to quotients bydiscrete subgroups that are cocompact orat least of finite covolume. As a specialcase, arithmetic groups are studied and acriterion for cocompactness of arithmeticsubgroups is presented.

The corresponding lattices in complexLie groups are discussed and their basicproperties are presented. The special casesstudied here are complex tori, and it isproved that they are the only quotientsadmitting a Kähler metric. Complex ana-lytic subspaces of quotients and theirKodaira dimension are discussed.Properties of holomorphic mappings andautomorphism groups of parallelisablemanifolds are investigated; it seems thatholomorphic mappings are often automat-ically equivariant.

An important part of the book is devotedto the study of homogeneous vector bun-dles. Some conditions for their flatness areproved, and the classification of flat bun-dles over parallelisable manifold is given.The author also treats deformations ofcompact parallelisable manifolds and thestructure of the related cohomologygroup. As an important example, complexnilmanifolds are studied and it is shownthat they are always built from complextori admitting a certain kind of complexmultiplication. The author also presentsalgebraic groups for which any invariantholomorphic function invariant withrespect to Zariski dense discrete subgroupsis constant.

This book can be recommended to allreaders interested in the topics covered.(jbu)

W. Woess, Random Walks on InfiniteGraphs and Groups, Cambridge Tracts inMathematics and its Applications 78,Cambridge University Press, Cambridge, 2000,334 pp., £40, ISBN 0-521-55292-3The topic of random walks is situatedsomewhere between probability, potentialtheory, harmonic analysis, geometry,graph theory and algebra. The beauty ofthe subject stems from linking, both in theway of thinking and in the methodsemployed, of different fields. The maintheme of this book is the interplay betweenthe behaviour of a class of stochasticprocesses (random walks) and discretestructure theory. The author considersMarkov chains whose state space isequipped with the structure of an infinitelocally finite graph, or as a particular case,of a finitely generated group. The transi-tion probabilities are assumed to be adapt-ed to the underlying structure in some waythat must be specified precisely in eachcase.

From a probabilistic viewpoint, thequestion is what impact the particular typeof structure has on various aspects of thebehaviour of the random walk. Conversely,random walks may also be seen as a usefultool for classifying, or at least describing,the structure of graphs and groups. Linkswith spectral theory and discrete potentialtheory are also discussed.

This book will be essential reading forall researchers in stochastic processes andrelated topics. According to the author,this book is not self-contained, by thenature of its topic. Another question maybe whether it is usable by graduate stu-dents. Here let us quote the author: �It hasbeen my experience that usually, this mustbe taken with caution, and is mostly trueonly in the presence of a guiding hand thatis acquainted with the topic. Proofs aresometimes a bit condensed, and it may bethat even students above the student levelwill need pen and paper when they want towork through them seriously � in particu-lar because of the variety of different meth-ods and techniques that I have tried tounite in this text. This does not mean,however, that parts of this book could notbe used for graduate or even undergradu-ate courses.� (jant)

Book Review by Sir Michael Atiyah

Conversation on Mind, Matter andMathematics by Jean-Pierre Changeux andAlain Connes edited and translated by M B DeBevoise Princeton University Press, 1995 260pp., £19.95, ISBN: 0 691 08759 8

Does mathematics have an existence inde-pendent of our physical world? Do mathe-maticians discover theorems, rather thaninvent them? Such questions have exer-cised the minds of philosophers and math-ematicians since the time of Plato, andmany books have addressed the issues.What is unusual and topical about thisbook is that it records a dialogue betweena mathematician, Alain Connes, and a biol-ogist, Jean-Pierre Changeux. Each is aninternational authority in his field andboth are Professors at the Collège deFrance, so that they speak from first-handexperience and the intellectual level ishigh. As a neuro-physiologist Changeux isparticularly interested in the structure andorganisation of the brain. He is intriguedby the capacity of the brain to handle thekind of sophisticated intellectual abstrac-tions represented by mathematics, and hisdialogue with Connes is an attempt to getto grips with this problem.

Most mathematicians would agree thattheir subject has evolved from the tradi-tional disciplines of arithmetic and geome-try. We have built an elaborate superstruc-ture on these twin foundations and so thenature of �mathematical reality� can beadequately dealt with by posing basic ques-tions about the integers 1, 2, 3 ... and aboutthe Euclidean geometry of triangles, cir-cles, etc. Connes is an unabashed Platonist.For him there is an �archaic mathematicalreality� represented by objects such as cir-cles or integers, which has an existence,independent of experience or of thehuman mind. One practical consequenceof such a belief, already adopted by NASA,is that mathematics provides the best formof potential communication with anyextra-terrestrial intelligence.

Any mathematician must sympathizewith Connes. We all feel that integers, orcircles, really exist in some abstract senseand the Platonist view is extremely seduc-tive. But can we really defend it? Had theuniverse been one-dimensional or evendiscrete it is difficult to see how geometrycould have evolved. It might seem thatwith the integers we are on firmer ground,and that counting is really a primordialnotion. But let us imagine that intelligencehad resided, not in mankind, but in some

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vast solitary and isolated jelly-fish, burieddeep in the depths of the Pacific Ocean. Itwould have no experience of individualobjects, only with the surrounding water.Motion, temperature and pressure wouldprovide its basic sensory data. In such apure continuum the discrete would notarise and there would be nothing to count.

Even more fundamentally, in a purelystatic universe without the notion of time,causality would disappear and with it thatof logical implication and of mathematicalproof. Connes actually alludes to thisphilosophical dilemma in the context ofrelativistic cosmology.

It may be argued that such �gedankenuniverses�� are not to be taken seriously.Our actual universe is a given datum andthe inevitable background of all intelligentdiscussion. But this is tantamount to con-ceding that mathematics has evolved fromthe human experience. Man has createdmathematics by idealising or abstractingelements of the physical world. The num-ber 2 for example represents the commonattribute of all pairs of objects that we haveencountered, in the same way as the word�chair� represents what is common to allthe individual items of furniture that we siton. Admittedly chairs are not so preciselyand unambiguously defined as numbers: isa three-legged stool a chair? Language isinherently more fuzzy or open-ended thanmathematics, but mathematics can proper-ly be viewed as a special kind of language.This applies not only to the nouns orobjects but also to the verbs or processessuch as addition and subtraction.

Connes will protest that there are thou-sands of different languages reflecting thehistory and culture of a particular people,whereas mathematics is universal andunique. Surely this gives it a special status?This can be debated at length, but a shortanswer is that the diversity of languagesdisguises a fundamental structural similar-ity, which is why dictionaries help us totranslate. Moreover different mathemati-cal notations, such as the Roman numeralsystem, do exist but western civilisation hasproduced a uniformity which was notinevitable.

In describing mathematics as a lan-guage it is important to emphasise that alanguage is not merely a set of words andgrammatical rules for producing coherentsentences. Words mean something andthey relate to our experience. In a similarway a mathematical statement has a mean-ing and one which, I would argue, restsultimately on our experience.

For a Platonist like Connes mathematicslives in some ideal world. I find this a diffi-cult notion to grasp and prefer to say morepragmatically that mathematics lives in thecollective consciousness of mankind. I amnot here embracing any obscure metaphys-ical notion, but merely observing thatthere are two essential components ofmathematics. In the first place it deals withconcepts and abstract processes which livein the conscious mind of the individualmathematician. Secondly, it must be com-municable to other mathematicians. Thefamous Indian genius Ramanujan fre-

quently produced marvellous formulae bysome unknown mental process which couldneither be described nor repeated. I thinkConnes would accept that such formulaedo not become an accepted part of mathe-matics until others have understood themand verified their validity.

Our collective consciousness may not besufficiently ideal for Connes but it is theworld where all ideas live, not just mathe-matical ones.

Where does this point of view leave thedichotomy between creation or discoveryin mathematics? By resisting the embraceof the Platonic world have we lost the pos-sibility of making �discoveries��? Is everytheorem man-made? Not at all. Since ourconcepts are based on the physical worldwe can discover facts about these conceptsexperimentally. For example the formula

3 × 5 = 5 × 3can be discovered by setting out 15 objectsin a rectangular array. One can then moveon to the more general algebraic formula

a × b = b × a,where a and b represent arbitrary integers.In a similar way the famous formula ofPythagoras

a2 + b2 = c2

relating the length of the sides of anyright-angled triangle was no doubt discov-ered experimentally. So man creates theconcepts of mathematics but he discoversthe subsequent connections between them.The reason he can have it both ways is thatmathematics is firmly rooted in the realworld.

The legally minded reader may objectthat I have slurred the difference betweenan empirical discovery, always concernedwith the specific, and a proof, concernedwith the general. I plead guilty, but only ona technicality and I hope the judge will belenient. Sometimes a discovery will carrywith it the seeds of a formal proof, as withthe example that 3 × 5 = 5 × 3.Sometimes one has to work harder asPythagoras had to do. But if possession isnine-tenths of the law, discovery is nine-tenths of the proof.

In his dialogue with Connes, Changeuxkeeps hitting the Platonist rock. As a hard-headed experimental scientist Changeuxwants to identify mathematics with whatactually goes on in the brain. For him thisis the only reality and the only place wheremathematics exists. Connes disputes thisextreme attitude and prefers to say thatmathematical reality (which exists else-where) is reflected in the neurologicalprocesses of the brain. To confuse the twois like identifying a piece of literature ormusic with the ink and paper on which it isrecorded. It is hard to disagree, but fasci-nating questions remain.

The brain is frequently compared to anelectronic computer and artificial intelli-gence is now a branch of computer science.But, as the dialogue brings out, this analo-gy has serious limitations and may in factbe obscuring some of the essential features.As Connes notes, the speed of transmissionof signals in the brain is vastly less than thecorresponding speeds in modern comput-ers. As a result computers are much better

and faster than humans at certain kinds ofcalculation, but in most important respectsthey are still a long way behind. For exam-ple, computers are not yet serious contrib-utors to mathematical theory. Perhaps byanalysing the structure of mathematics wemay learn something about the way thebrain operates. In particular a key featureof mathematics is its hierarchical nature.Examples of patients with specific braindamage show that particular levels of rea-soning are associated with definite parts ofthe brain.

For Changeux, the comparison betweenthe structure of mathematics and the struc-ture of the brain can be looked at bothfrom the evolutionary aspect and in termsof function. In evolutionary terms, thebrain has had to create a hierarchy of lev-els that adequately reflect the physicalenvironment and the challenges it poses.Man has been the ultimate winner of theevolutionary process and his brain has thestructure needed to produce mathematics.Would a different neurological solutionhave led to a different kind of mathemat-ics, or does mathematics depend only onthe functional capacity of the brain, not onits biological mechanism?

If one views the brain in its evolutionarycontext then the mysterious success ofmathematics in the physical sciences is atleast partially explained. The brainevolved in order to deal with the physicalworld, so it should not be too surprisingthat it has developed a language, mathe-matics, that is well suited for the purpose.The sceptic can point out that the strugglefor survival only requires us to cope withphysical phenomena at the human scale,yet mathematical theory appears to dealsuccessfully with all scales from the atomicto the galactic. Perhaps the explanationlies in the abstract hierarchical nature ofmathematics which enables is to move upand down the world scale with comparativeease.

The internal workings of the mathemat-ical mind were well described by JaquesHadamard who distinguished three stagesin attacking a problem: preparation, incu-bation and illumination. What these corre-spond to neurologically is a challengingquestion which generates a lively discus-sion between Connes and Changeux. Isthere for example a Darwinian element inthe search for successful ideas? HenriPoincaré argued that the subconsciousmind generates random ideas and �illumi-nation� occurs when one of these is select-ed.

The original Socratic dialogues wereartificially constructed to present a coher-ent view. The dialogue between Connesand Changeux is quite different. It is therecording of real-life arguments where thespeakers are frequently at cross-purposesand operate in different planes. For thereader this can be irritating but it alsoencourages him to become involved andframe his own answers, as I have endeav-oured to do!Sir Michael Atiyah, OM, was formerly Presidentof the Royal Society and Master of TrinityCollege Cambridge.

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We list below information about someappointments, awards and deaths thathave occurred during late 1999 andearly 2000. Since this list is inevitablyincomplete, we invite you to send appro-priate information from other countriesto Barbara Maenhaut [b.m.maen-haut@open.ac.uk] or to your Countryrepresentative (see Issue 34) for inclu-sion in the next issue. Please also sendany items you feel should be included infuture Personal Columns.

AppointmentsPaul Garthwaite (Aberdeen) andChris Jones have been appointedProfessors in the StatisticsDepartment at the Open University,and Pia Larsen, Karen Vines andCatriona Queen have been appoint-ed Lecturers.

Robert Hunt (Cambridge) has beenappointed Deputy Director of theIsaac Newton Institute forMathematical Sciences.

Robert MacKay and Andrew Stuarthave been appointed Professors atWarwick University; John Rawnsleyand Colin Rourke have been pro-moted to Personal Chairs; ClaudeBaesens, Peter Topping and BalaszSzendroi have been appointedLecturers.

AwardsLuigi Ambrosio (Scuola NormaleSuperiore of Pisa) was awarded theRenato Caccioppoli Prize (1998) bythe Italian Mathematical Union forhis contributions to the Calculus ofVariations and Partial DifferentialEquations.

F. Bethuel and F. Helein have beenawarded the 1999 Fermat Prize inMathematics for �several importantcontributions to the theory of varia-tional calculus which have conse-quences in physics and geometry.

The French Académie des Scienceshas awarded special mathematicalprizes to Laurent Clozel (Orsay),Pierre Colmez (Paris), SylvestreGallot (Grenoble), Laurent Manivel(Saint Martin d�Hères) andWandelin Werner (Orsay).

Yves Colin de Verdière (Grenoble)has been awarded the Prix Ampèrede l�Electricité de France by theFrench Académie des Sciences, forfundamental contributions to spec-tral theory.

Joachim Cuntz (Münster) wasawarded the 1999 Gottfried LeibnizPrize, given by the DeutscheForschungsgemeinschaft, for workin functional analysis.

Riccardo De Arcangelis (Napoli)was awarded the Calogero Vinti

Prize (1998) by the ItalianMathematical Union in recognitionof his theoretical research in theCalculus of Variations.

Persi Diaconis (Stanford) was elect-ed to hold the first UK HardyFellowship from May-August 2001,at Queen Mary and WestfieldCollege, London.

Matthias R. Gaberdiel has beenawarded a Royal Society UniversityResearch Fellowship to work in theDepartment of Applied Mathematicsand Theoretical Physics,Cambridge.

Mikhail Gromov (Paris and NewYork) was awarded the 1999 BalzanPrize for Mathematics, awarded bythe Fondazione InternazionalePremio E. Balzan.

Alice Guionnet (Orsay) has receivedthe 1999 Oberwolfach Prize for out-standing research in stochastics.

Nicholas J. Higham (Manchester)has been awarded a Royal SocietySenior Research Fellowship.

Tom Hoeholdt (TechnicalUniversity of Denmark) has beenelected an IEEE Fellow, from 1January 2000, for �fundamentalcontributions to the theory, analysisand decoding algorithms of algebra-ic geometry codes�.

Peter Hydon (Guildford, Surrey)has been promoted to Reader inmathematics.

Kurt Johansson (Royal Institute ofTechnology, Stockholm) and DavidWilson (Microsoft Research) havebeen awarded the Rollo DavidsonPrizes for 2000.

Marco Manetti (University of Roma�La Sapienza�) has been awardedthe Giuseppe Bartolozzi Prize (1999)by the Italian Mathematical Unionfor his contributions to the theory ofalgebraic surfaces.

Bernard Maurey (Paris) has beenawarded a State Prize by the FrenchAcadémie des Sciences for funda-mental work on Banach spaces.

Alun Morris (Aberystwyth) receivedan OBE and Andrew Wiles FRS(Princeton) received a knighthoodin the British New Years HonoursList. Sir Roger Penrose FRS(Oxford) has been appointed to theOrder of Merit.

Stefan Müller (Leipzig) and DieterLüst (Potsdam) have been awardedthe 1999 Gottfried Leibniz Prize,awarded by the DeutscheForschungsgemeinschaft, for workin functional analysis.

Erich Novak (Erlangen-Nürnberg)is the first winner of the Prize forAchievement in Information-BasedComplexity.

Juan-Pablo Ortega and TudorRatiu (Lausanne) have been award-ed the 1999 Ferran Sunyer i BalaguerPrize by the Institut d�EstudisCatalans for their monograph enti-tled Hamiltonian SingularReduction. The awards will be pre-sented on 10 July during the 3ecm inBarcelona.

James C. Robinson has been award-ed a Royal Society UniversityResearch Fellowship to work in theMathematics Institute, Warwick.

Guido Sanguinetti (Genova) wasawarded the Franco Tricerri Prize(1999) by the Italian MathematicalUnion for his doctoral work in dif-ferential geometry.

Jean-Pierre Serre (Paris) has beenawarded the 2000 Wolf Prize inMathematics, for his many funda-mental contributions to topology,algebraic geometry, algebra andnumber theory; he shares this prizewith Raoul Bott.

Volker Strassen (Konstanz) hasbeen awarded the 1999 GeorgCantor Medal by the DeutscheMathematiker Vereinigung.

Alain-Sol Sznitman (Zürich) hasbeen awarded the 1999 Line andMichel Loève International Prize inProbability.

John Toland, FRS (Bath) has beenawarded an honorary degree fromQueen�s University, Belfast.

Jack van Lint (Eindhoven) receivedthe honorary degree of Doctor ofMathematics on 24 March 2000,from the University of Ghent.

Don Zagier (Bonn) has been award-ed the Chauvenet Prize by theMathematical Association ofAmerica for his outstanding exposi-tory article �Newmans short proof ofthe prime number theorem� in theAmerican Mathematical Monthly.

DeathsWe regret to announce the deathsof:George K. Batchelor, FRS (30March 2000)Mary Bradburn (31 January 2000)Donna M. Carr (February 2000)David G. Crighton, FRS (12 April2000)Lord Halsbury (14 January 2000)Timothy Swan Harris (22 July1999)Nathan Jacobson (5 December1999)Bertha Jeffreys (18 December 1999)Jürgen Kurt Moser (17 December1999)Peeter Muursepp (3 November1999)John A. Nohel (1 November 1999)Edwin J. Redfern (8 January 2000)Robert F. Riley (4 March 2000)Gottfried T. Ruttimann (11 October1999)Ken Stroud (3 February 2000)Jeffrey D. Weston (10 March 2000)

EMS June 200042

PERSONAL COLUMN

PPersonal Columnersonal Column

EMS June 2000 43

POSTDOCTORAL POSITIONS IN MATHEMATICS IN THE UNIVERSITY OF AVEIRO

The Research unit Mathematics and its Applications of the University of Aveiroaccepts applications for post-doctoral research positions in the area ofFunctional Analysis until June 30th, 2000. Each position has a duration of atmost one year and shall start no sooner than September 1st, 2000, at a date con-venient for both parties.

The areas of research are- Non-Standard Methods in Analysis and Topology;- Function Spaces on Irregular Domains;- Convolution Type Operators and Diffraction Theory.

The successful applicant is supposed to work as a member of one of the teamsof the Group of Functional Analysis in one or more of these topics.

Candidates should have a PhD � preferably obtained within the last 5 years � inan area relevant to at least one of the subjects mentioned above and shall beselected on the basis of the following documents:

1. a letter of intent,2. curriculum vitae,3. official proof of PhD award,4. copy of two research works

(preferably in Portuguese, English, French or Spanish) 5. copy of passport.

The above documents must be sent, by registered mail, toSecretaria do Departamento de Matemática(c/o Prof. António Caetano)Universidade de Aveiro3810-193 AVEIROPortugal

For more information, please contact acaetano@mat.ua.pt