contents editorial team european mathematical society

NEWSLETTER No. 41

September 2001EMS Agenda ................................................................................................. 2

Editorial - Carles Casacuberta ..................................................................... 3

EMS Summer School at St Petersburg 2001 ................................................. 4

EAGER - ENI - EMS Summer School ........................................................... 5

Meeting of the EMS Council ......................................................................... 6

EMS Lecturer 2002 - Gianni Dal Maso ........................................................ 8

Interview with Marek Kordos ....................................................................... 9

Interview with Graham Higman ................................................................. 12

The Methodology of Mathematics, part 2 .................................................... 14

Mathematical Societies: Norwegian ........................................................... 17

Mathematical Societies: Estonian .............................................................. 18

Problem Corner ........................................................................................... 20

Mathematics Education on the Web ............................................................ 23

Forthcoming Conferences ............................................................................ 24

Recent Books ............................................................................................... 27

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

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

ISSN 1027 - 488X

The views expressed in this Newsletter are those of the authors and do not necessarilyrepresent those of the EMS or the Editorial team.

NOTICE FOR MATHEMATICAL SOCIETIESLabels for the next issue will be prepared during the second half of November 2001. Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department ofMathematics, P.O. Box 4, FIN-00014 University of Helsinki, Finland; e-mail:[email protected]

INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTERInstitutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat,Department of Mathematics, P. O. Box 4, FI-00014 University of Helsinki, Finland, or by e-mail: ([email protected]). Please include the name and full address (with postal code), tele-phone and fax number (with country code) and e-mail address. The annual subscription fee(including mailing) is 65 euros; an invoice will be sent with a sample copy of the Newsletter.

EDITOR-IN-CHIEFROBIN WILSONDepartment of Pure MathematicsThe Open UniversityMilton Keynes MK7 6AA, UKe-mail: [email protected]

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

SPECIALIST EDITORSINTERVIEWSSteen Markvorsen [address as above]SOCIETIESKrzysztof Ciesielski [address as above]EDUCATIONTony GardinerUniversity of BirminghamBirmingham B15 2TT, UKe-mail: [email protected] PROBLEMSPaul JaintaWerkvolkstr. 10D-91126 Schwabach, Germanye-mail: [email protected] ANNIVERSARIESJune Barrow-Green and Jeremy GrayOpen University [address as above]e-mail: [email protected] [email protected] andCONFERENCESKathleen Quinn [address as above]RECENT BOOKSIvan Netuka and Vladimir Sou³ekMathematical InstituteCharles UniversitySokolovská 8318600 Prague, Czech Republice-mail: [email protected] [email protected] OFFICERVivette GiraultLaboratoire dAnalyse NumériqueBoite Courrier 187, Université Pierre et Marie Curie, 4 Place Jussieu75252 Paris Cedex 05, Francee-mail: [email protected] UNIVERSITY PRODUCTION TEAMLiz Scarna, Kathleen Quinn

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CONTENTS

EMS September 2001

EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

EXECUTIVE COMMITTEEPRESIDENT (19992002)Prof. ROLF JELTSCHSeminar for Applied MathematicsETH, CH-8092 Zürich, Switzerlande-mail: [email protected]. LUC LEMAIRE (19992002)Department of Mathematics Université Libre de BruxellesC.P. 218 Campus PlaineBld du TriompheB-1050 Bruxelles, Belgiume-mail: [email protected]. BODIL BRANNER (20012004)Department of MathematicsTechnical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: [email protected] (19992002)Prof. DAVID BRANNANDepartment of Pure Mathematics The Open UniversityWalton HallMilton Keynes MK7 6AA, UKe-mail: [email protected] (19992002)Prof. OLLI MARTIODepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlande-mail: [email protected] ORDINARY MEMBERSProf. VICTOR BUCHSTABER (20012004)Department of Mathematics and MechanicsMoscow State University119899 Moscow, Russiae-mail: [email protected] Prof. DOINA CIORANESCU (19992002)Laboratoire dAnalyse NumériqueUniversité Paris VI4 Place Jussieu75252 Paris Cedex 05, Francee-mail: [email protected]. RENZO PICCININI (19992002)Dipartimento di Matematica e ApplicazioniUniversità di Milano-BicoccaVia Bicocca degli Arcimboldi, 820126 Milano, Italye-mail: [email protected]. MARTA SANZ-SOLÉ (19972000)Facultat de MatematiquesUniversitat de BarcelonaGran Via 585E-08007 Barcelona, Spaine-mail: [email protected]. MINA TEICHER (20012004)Department of Mathematics and ComputerScienceBar-Ilan UniversityRamat-Gan 52900, Israele-mail: [email protected] EMS SECRETARIATMs. T. MÄKELÄINENDepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlandtel: (+358)-9-1912-2883fax: (+358)-9-1912-3213telex: 124690e-mail: [email protected]: http://www.emis.de

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EMS NEWS

EMS September 2001

200115 NovemberDeadline for submission of material for the December issue of the EMSNewsletterContact: Robin Wilson, e-mail: [email protected]

19-21 NovemberEMS lectures at Università degli Studi, Tor Vergata, Rome (Italy)Lecturer: Michèle Vergne (Ecole Polytechnique, Palaiseau, France)Title: Convex PolytopesContact: Prof. Maria Welleda Baldoni, e-mail:[email protected]

22-23 NovemberFifth Diderot Mathematical ForumTitle: Mathematics and TelecommunicationsVenues: Eindhoven (Netherlands), Helsinki (Finland) and Lausanne(Switzerland)Contacts: Paul Urbach at Philips, Eindhoven ([email protected]); OlaviNevanlinna, Helskinki University of Technology ([email protected]);Gerard Ben Arous, Ecole Polytechnique Fédérale de Lausanne([email protected])

20029-10 FebruaryEMS Executive Committee Meeting in Brussels (Belgium), at the invitationof the Belgian Mathematical Society and the Université Libre de Bruxelles

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

24-28 February EMS Summer School in Eilat (Israel)Title: Computational Algebraic Geometry and ApplicationsContact: Mina Teicher, e-mail: [email protected]

1 MarchDeadline for Proposals for 2003 EMS Lectures. Contact: David Brannan, e-mail: [email protected]

31 May EMS Executive Committee meeting in Oslo (Norway)

1-2 June EMS Council Meeting, Oslo

3-8 JuneAbel Bicentennial Conference, Oslo

200425-27 June EMS Council Meeting, Stockholm (Sweden)

27 June-2 July4th European Congress of Mathematicians, Stockholm

EMS AgendaEMS Committee

Since the foundation of the EMS in 1990, theSocietys publications strategy has been asource of lengthy debates, often with dividedviews and energetic standpoints. Yet, a lullhas arrived when most of the former subjectsof discussion have become realities, whileexciting new ventures are about to start. Therecently appointed Managing Director of theEMS Publishing House, ThomasHintermann, starts his term of office inSeptember 2001. Before looking into thefuture, however, let me shortly review thepast.

The first stepsThe EMS Publications Committee started atthe same meeting that the Society was found-ed. Stewart Robertson was elected as Chair.His first task was to launch the Newsletter,which has appeared regularly every threemonths since September 1991. It was initial-ly edited by David Singerman (Southampton)and Ivan Netuka (Prague). An editorial teamfrom the Glasgow Caledonian University,represented by Roy Bradley and MartinSpeller, undertook the production in 1996,still in connection with the Prague team. Inearly 1999, Robin Wilson accepted the role ofEditor-in-Chief and built up the current edi-torial team. The inner format was designedby Jan Kosniowski, from Armstrong Press(Southampton), while the cover design stemsfrom Marie-Claude Vergne of IHÉS (Paris).It is unanimously recognised that, after theefforts and expertise of all the peopleinvolved, the EMS Newsletter has reachedmaturity in its content and layout.

The birth of JEMSThe desire to produce a learned journal wasalso as old as the Society itself, in spite ofmuch controversy about the manner in whichit should materialise. A letter of intent wassigned by EMS President FriedrichHirzebruch and representatives of Springer-Verlag in December 1994. At that stage, thenegotiations were conducted by StewartRobertson and David Wallace. The journalwas to have a distinct European flavour andcarry articles in as wide a variety of fields aspossible. Nevertheless, still a long way wasneeded until a contract was signed and thefirst issue of JEMS appeared in January 1999,after the impulse given by EMS PresidentJean Pierre Bourguignon. Thanks are due toSpringer for the friendly collaboration andjoint promotion of the journal. The board ofeditors of JEMS consists of Jürgen Jost(Editor-in-Chief), Luigi Ambrosio, GérardBen Arous, John Coates, Helmut Hofer,Alexander Merkurjev, and twenty-five associ-ate editors. As the Editor-in-Chief wrote in his1998 Newsletter editorial, JEMS aims at pre-serving the unity of mathematical thinking bypresenting profound and important advancesin both pure and applied mathematics.

Electronic versus traditionalArticles published in JEMS become availableon EMIS, the Societys website, two years afterthe printed version. By the middle of the pastdecade, debates about the impact and man-agement of electronic publications were ubiq-uitous. Peter Michor, former EMS Secretary,was strongly supportive of the new means ofstorage and diffusion. He created EMIS in1995, in collaboration with FIZ Karlsruhe.The website was implemented by Michael Jostand Bernd Wegner; it contains general infor-mation about EMS, currently compiled byVolker Mehrmann, and a large ElectronicLibrary giving free access to journals, mono-graphs and preprints. Bernd Wegner hasdedicated an enormous amount of time andability to this and other initiatives involvingEMS, such as the Mathematical PreprintsServer System, the EULER project, and theEMS Publishing House. Most notably, theEMS is an active partner of ZentralblattMATH since 1997. The process to makeZentralblatt a large European-based infra-structure started with a French-German coop-eration in 1995, highlighted by the creationof the Cellule MathDoc in Grenoble. It is cur-rently being cofunded by the EuropeanCommission under the LIMES project, whichis coordinated by FIZ Karlsruhe and in whichEMS is a major partner. The EMS has aDatabase Committee, chaired by LaurentGuillopé, after John Coates, who chaired itfrom its creation in 1995 until 2000 and anElectronic Publishing Committee, which hassuccessively been chaired by Peter Michor(since 1995) and Bernd Wegner (since 2001).

Other EMS publicationsOriginally, the text of EMS Lectures was to bepublished in JEMS. This idea was later dis-carded and a project came out, under thepresidency of Jean-Pierre Bourguignon, toproduce a series of volumes with materialfrom the EMS Lectures, EMS SummerSchools and Diderot Mathematical Forumevents. This initiative led to a letter of agree-ment with Springer in 1998, in which an EMSsubseries of Springer Lecture Notes inMathematics was originated. Two scientific

advisers were elected for this new series:Fabrizio Catanese and Ragnar Winther, andthe first volume of the series, authored byNigel Cutland, appeared as LNM 1751 dur-ing 2000. At present, two more volumes havebeen submitted, and four are in preparation.

The Publications CommitteeStewart Robertson retired and ended his termof office as Chairman of the PublicationsCommittee in April 1997. Marta Sanz-Soléwas elected Chair pro tempore and held thisposition until the end of that year. In October1997, terms of reference for the PublicationsCommittee were formally approved; amongother things, it was stated that ThePublications Committee has as its main taskto formulate a strategy of publications of theSociety, to suggest and discuss ideas for itsimplementation, and to supervise the devel-opment of EMS publications. The currentChair was appointed in 1998 and has been re-elected for 2001 and 2002. The other com-mittee members are the Publicity Officer(David Salinger), the Editor-in-Chief of JEMS(Jürgen Jost), the Newsletter Editor (RobinWilson), the Chair of the ElectronicPublishing Committee (Bernd Wegner), andthe Managing Director of the EMSph(Thomas Hintermann). This updated list ofmembers was agreed by the ExecutiveCommittee in March 2001, in accordancewith the terms of reference. The most imme-diate goal of the committee is to define its wayof action for the years to come, obviously incoordination with the activities of the EMSPublishing House.

The EMS Publishing HouseThe old dream of making EMS a publishingforce went through its decisive step soon afterRolf Jeltsch began his presidency in 1999.The creation of the EMS Publishing House(EMSph) was approved by the ExecutiveCommittee in November 2000, after adetailed proposal of an ad hoc committee.The EMSph is not financially dependent onthe EMS, but it belongs to a newly createdEuropean Mathematical Foundation. This isa non-profit organisation, subsidiary of EMS,with seat in Zurich; its statutes have recentlybeen registered under Swiss law. Accordingto these, the tasks of the Foundation includethe establishment and supervision of theEMSph, the furtherance of the activities of theEMS, and support of activities of corporatemember societies of the EMS.The EMSph aims to provide a frameworkwhere member societies can join efforts inprinting and distributing mathematical jour-nals. Two prospective meetings with editorsof learned journals were held in August 1998(Berlin) and July 2000 (Barcelona), resultingin clearly positive reactions and several pre-liminary intents of partnership. The EMSwishes however to preserve the current diver-sity of mathematical literature in Europe anddoes not want to damage any existing coop-eration schemes. As a non-profit body, theEMSph will fight to keep prices as low as pos-sible; on the other hand, the EMSph has toprove itself financially viable, and in the longrun it should enable further initiativestowards enhancing the visibility, strength andinfluence of the EMS.

EDITORIAL

EMS September 2001 3

EditorialEditorialCarles Casacuberta (Barcelona)

(EMS Publications Officer, Chair of the Publications Committee)

April, but there were a lot of applicationsafter the deadline. However, many whohad registered in time did not take part,for different reasons, including some mainlecturers. The support of the EMS allowedus to include more participants from EastEurope than was announced in the appli-cation.

The scientific Committee was A. Vershik(Chair), L. Faddeev, E. Brezin, P. Deift, V.Malyshev and O. Bohigos.

Scientific reportAsymptotic combinatorics with applications tomathematical physics was devoted to someareas of mathematics and mathematicalphysics that have been studied very inten-sively in recent years. The idea was to getspecialists in integrable systems, asymptot-ic combinatorics, representation theory,random matrix theory and quantum fieldtheory to give short courses of lectures onthe subject and attract young students tothose areas. The list of lecturers includedspecialists of the highest level (P. Deift, E.Brezin, L. Faddeev, etc.). The school wellconfirmed the organisers initial idea onthe fruitfulness of the interrelationsbetween asymptotic combinatorics andmathematical physics. Methods from thetheory of integrable systems and matrixproblems, as well as the theory of theRiemann-Hilbert problem together withasymptotic combinatorics and representa-tion theory, give very powerful tools forthe solution of many old problems (fluctu-ations of the eigenvalues of random matri-ces, counting the number of coverings ofalgebraic curves, universality of distribu-tion of spacing and other statistics charac-teristics of Young diagrams, etc.). Weemphasise some of the lectures on spectac-

The EMS Summer School 2001 andNATO Advanced Study Institute at StPetersburg, Asymptotic combinatorics withapplication to mathematical physics, attractedthe leading world-wide specialists in thetheory of integrable systems, randommatrix theory, the Riemann-Hilbert prob-lem and asymptotic combinatorics. Manyyoung mathematicians took part in thesummer school and the school was espe-cially important for them. One importantfeatures of the school was the presence ofmathematicians and physicists simultane-ously as lecturers and students. Theschool was successful and fruitful, andmany new contacts between participantswere established. More than half the par-ticipants were less than age 35.

The preparation was started in theautumn of 1999 after EMS ExecutiveCommittee meeting in Zurich. An appli-cation was sent to the NATO ScientificAffairs Division in July 2000, and a grantwas obtained in January 2001. A grantfrom the Russian Fund of BasicResearches was obtained in April 2001.Information about the school was pub-lished in the AMS Notices, the EMSNewsletter, the INTERNET sites of NATO,the AMS Mathematics Calendar, theInternational Euler Mathematical Instituteand the St Petersburg MathematicalSociety. A poster was prepared by theEMS and sent to many Universities.

The opening session was at the SteklovInstitute of Mathematics in St Petersburg,and all the lectures and seminars wereheld at the Euler InternationalMathematical Institute (part of the SteklovInstitute) and were supported by staff fromthe Euler and Steklov Institutes.

The deadline for the registration was 1

ular new results in mathematical physics(P. Deift), combinatorics (A. Borodin andR. Kenyon) and algebraic geometry(A. Okounkov).

The impressions of the participantswere very positive and most of them con-sidered the lectures to be very useful. Theprogramme was over-full, with additionaltalks. A round table on current problemswas organised, and the younger partici-pants took part in short discussions abouttheir studies. The lectures and some addi-tional matter will be published in the spe-cial volumes of the proceedings of theschool, to be edited by V. Malyshev and A.Vershik; Volume 1 will be published bySpringer-Verlag, and Volume 2 by theKluver Publishing House, in 2002.

Main lecturesP. Biane, Asymptotics of representations

of symmetric groups, random matricesand free cumulants

A. Borodin, Asymptotic representationtheory and Riemann-Hilbert problem

M. Bozejko, Positive definite functions onCoxeter groups and second quantizationof Yang-Baxter type

E. Brezin, An introduction to matrix mod-els

P. Deift, Random matrix theory and com-binatorics: a Riemann-Hilbert approach

L. Faddeev, 3-dimensional solitons andknots

J. L. Jacobsen, Enumerating coloured tan-gles

V. Kazakov, Matrix quantum mechanicsand statistical physics on planar graphs

R. Kenyon, Hyperbolic geometry and thelow-temperature expansion of the Wulffshape in the 3D Ising model

V. Korepin, Quantum spin chains andRiemann zeta function with odd argu-ments

I. Krichever, τ-functions of conformalmaps

V. Liskovets, Some asymptotic distributionpatterns for planar maps

V. Malyshev, Combinatorics and probabil-ity for maps on two-dimensional sur-faces

M. Nazarov, On the Frobenius rank of askew Young diagram

S. Novikov, On the weakly nonlocalPoisson and symplectic structures

A. Okounkov, Combinatorics and modulispaces of curves

G. Olshanski, Harmonic analysis on biggroups, and determinantal pointprocesses

L. Pastur, Eigenvalue distribution of uni-tary invariant ensembles of randommatrices of large order

S. Smirnov, Critical percolation is confor-mally invariant

R. Speicher, Free probability and randommatrices

A. Vershik, Introduction to asymptotictheory of representations

EMS NEWS

EMS September 20014

EMS Summer School at St PEMS Summer School at St Petersburetersburg 2001g 2001Asymptotic combinatorics with application to mathematical physics


The school is an introduction on how to usecomputer algebra systems such as SINGU-LAR and MACAULAY2 and packages suchas SCHUBERT (intersection theory) forresearch in algebraic geometry and itsapplications. The programme consists oflectures and practical exercise sessions witha computer.

The topics to be considered include:computer algebra systems, Gröbner basesand syzygies, ideal and radical member-ship, manipulating ideals and modules,Hilbert polynomials and Hilbert functions,elimination, computations in local ringsand Milnor numbers, homological algebra(constructive module theory, Ext and Tor,sheaf cohomology, Beilinson monads), pri-mary decomposition, normalisation, ringsof invariants, parametrisation, deforma-tions, intersection theory, applications tospecial varieties, computer vision and cod-ing theory.

The course will be directed by Prof.Wolfram Decker (Saarbrucken, Germany).There will be additional guest lectures byDr Jeremy Kaminski (Bar-Ilan, Israel).

ComputersPlease bring your laptop with you. Two stu-dents can use one computer, so if you coor-dinate with a fellow student you need bringjust one computer between you. Beforecoming you should download some soft-ware from the Internet this will be madeprecise in January 2002.

AccommodationThe site of the Conference, Eilat, is the

southernmost point of Israel, a resort cityhugging the shores of the Red Sea, sur-rounded by the magnificent EdomMountain Range and characterised by itscrystal clear waters and year-round sun-shine. Its unique undersea vista, flora andfauna can be admired by boat, from abreathtaking underwater observatory, or bysnorkelling or diving. The inland desertlandscapes are no less fascinating. Eilat iswell known for its mild winter climate. Theweather in Eilat during February is warmduring the day and cooler at night; tem-peratures range from 12C to 25C, and itseldom rains. Participants who extendtheir stay may like to join interesting day-trips from Eilat, inside Israel or over theborder to Jordan or Egypt (Moon Valley,Akaba, Petra, Wadi Rum, Nuweiba,Colored Canyon and Santa-Catarina).(Reservations can be made on the spotthrough the hotel concierge.)

The Summer School will take place inHotel Meridian. The hotel special confer-ence rates are $90 a day per person (FB) ina double occupancy; the registration fee is$80.

More details will appear in the secondannouncement.

ArrivalYou can get to Eilat by bus from the centralbus station in Tel Aviv (about 4.30 hours);this is very economical and is the way werecommend.

The standard way to arrive by air is totake a connecting flight to Eilat city airportfrom Ben-Gurion international airport.

EMS NEWS

There are also some direct flights fromEurope to Eilat: you may check this possi-bility with your own travel agency.Occasionally there are charter flights fromEurope to Ovda Airport (40 minutes fromEilat: no regular bus services, but taxis ortransportation provided by the chartercompany).

From Eilat airport or from the Eilat cen-tral bus station, take a taxi to the hotel, orwalk.

RegistrationThe number of places in the school is limit-ed. If you are interested in coming, fax theform below by 15 September to (972-3)-5353325, att. Boris Kunyavski, indicatingarrival/departure data and credit cardnumber.

Financial support for local expensesIf you belong to EAGER you can apply toyour node coordinator for support.

Otherwise, please contact BorisKunyavski at the Emmy Noether Institute:[email protected] indicating age, academicdegree and institution where awarded, PhDthesis title, name of advisor, nationality,place of residence and affiliation.

If you are getting financial support fromany of the above sources, please send yourhotel receipts to the responsible party afterthe conference for reimbursement.

OrganisationProfs. Mina Teicher and Boris Kunyavski ofthe Emmy Noether Institute (Conference Secretary: Ms Chen Fireman)

SponsorsEMS (European Mathematical Society),EAGER (European Algebraic GeometryEducation and Research) and ENI (EmmyNoether Research Institute forMathematics at Bar-Ilan University and theMinerva Foundation)

To be sent by fax to (972-3)-5353325, att. Boris Kunyavski Hotel registration formEMS/EAGER Summer SchoolComputational Algebraic GeometryEilat, Hotel Meridian, 25-28 February 2002

Surname:

First (and other) name(s):

Affiliation:

Tel.:

Fax:

e-mail:

Credit card number:

Arrival data (date and time):

Departure date:

Accompanying persons (with names, sex, ages, if under 16):

Accommodation: single room

double room with ...

Remarks:

EAGER - ENI - EMS Summer SchoolEAGER - ENI - EMS Summer SchoolComputational Algebraic Geometry and Applications

Eilat, Israel 24-28 February 2002First announcement

JEMS

Journal of the European

Mathematical Society

Volume 3, number 3 of JEMS con-tains:L. Birgé and P. Massart: Gaussianmodel selectionR. Meyer: Excision in Entire CyclicCohomology

Volume 3, number 4 of JEMS con-tains:A. de Carvalho and T. Hall:Pruning theory and Thurstons classi-fication of surface homeomorphismsO. Biquard and M. Jardim:Asymptotic behaviour and the modulispace of doubly-periodic instantons

The EMS Council meets every second year.The next meeting will be held in Oslo,Norway, on 1-2 June 2002, before the AbelCentennial Meeting in Oslo which beginson 3 June. The first session of the Councilmeeting will start at 10 a.m. on 1 June, andwill run all day with a break for lunch. Thesecond session will probably start at 9 or 10a.m. on 2 June, and may last most or all ofthe day with a break for lunch, dependingon the volume and complexity of the busi-ness on the agenda.

Invitation to suggest business for theCouncilThe Executive Committee is responsiblefor preparing the matters to be discussedat Council meetings. Items for the agendaof this Council meeting should be sent(preferably by e-mail) as soon as possible,and no later than 1 February 2002, to theEMS Secretariat in Helsinki. TheExecutive Committee will meet on 9-10February 2002 to put together the Councilagenda.

Delegates to the Council will be electedby the various categories of members, asper the Statutes.

Election to Council of representatives ofindividual EMS membersA person becomes an individual memberof EMS either through a corporate mem-ber, by paying an extra fee, or by directmembership. On 14 July 2001, there weresome 2247 individual members, and,according to our Statutes, these memberswill be represented on Council by 23 dele-gates. Nomination papers for these elec-tions appear in this Newsletter.

15 delegates were elected for the term 2000-03,so they will continue unless they inform theSecretariat to the contrary by 31 December2001. They are Giuseppe Anichini(Firenze, Italy), Vasile Berinde (Baia Mare,Romania), Giorgio Bolondi (Milano, Italy),Alberta Conte (Torino, Italy), ChrisDodson (Manchester, UK), Jean-PierreFrançoise (Paris, France), Salvador Gomis(Alicante, Spain), Laurent Guillopé(Nantes, France), Willi Jägger (Heidelberg,Germany), Klaus Habetha (Aachen,Germany), Tapani Kuusalo (Jyväskylä,Finland), Laslo Marki (Budapest,Hungary), Andrzej Pelczar (Krakow,Poland), Zeev Rudnick (Tel Aviv, Israel)and Gerard Tronel (Paris, France).

The mandates of 5 of the present 20 delegatesend on 31 December 2001, and so electionsmust be held for their positions. They areManuel Castellet (Barcelona, Spain),George Jaiani (Tbilisi, Georgia), MarinaMarchisio (Boves, Italy), Vitali Milman (TelAviv, Israel) and Jan Slovak (Brno, CzechRepublic). All of these can be re-elected, sincethey have served in this capacity for only 4 years.

Election to Council of delegates of othercategories of EMS membersFull EMS Members are national mathe-matical societies, which elect 1, 2 or 3 del-egates according to their size andresources. Each society is responsible forthe election of its delegates. Each societyshould notify the EMS Secretariat inHelsinki of the names and addresses of itsdelegate(s) no later than 1 February 2002.As of 1 July 2001, there were about 47 suchsocieties, which could designate a maxi-mum of about 69 delegates.

There is one associate EMS member: theGesellschaft für MathematischeForschung. According to the Statutes:delegates representing associate membersshall be elected by a ballot organized bythe Executive Committee from a list ofcandidates who have been nominated andseconded, and have agreed to serve.

There are three institutional EMS mem-bers: Institut Non-Linéaire de Nice, theMoldovian Academy of Sciences and theMathematical Institute of the SerbianAcademy of Sciences and Arts. Accordingto the Statutes: delegates representinginstitutional members shall be elected by aballot organized by the ExecutiveCommittee from a list of candidates whohave been nominated and seconded, andhave agreed to serve.

The EMS Secretariat will contact thesociety members in these three categoriesdirectly in connection with their delegates.

Membership of the EMS ExecutiveCommitteeThe Council is responsible for electing thePresident, Vice-Presidents, Secretary,Treasurer and other members of theExecutive Committee. The present mem-bership of the Executive Committee,together with their individual terms ofoffice, is as follows. President Professor

Rolf Jeltsch (1999-2002)Vice-Presidents

Professor Luc Lemaire (1999-2002)Professor Bodil Branner (2001-04)

SecretaryProfessor David Brannan (1999-2002)

TreasurerProfessor Olli Martio (1999-2002)

MembersProfessor Victor Buchstaber (2001-04)Professor Doina Cioranescu (1999-2002)Professor Renzo Piccinini (1999-2002)Professor Marta Sanz-Solé (2001-04)Professor Mina Teicher (2001-04)

The President may serve only one term ofoffice, so Rolf Jeltsch cannot be re-electedas President. David Brannan and RenzoPiccinini have indicated that they do notcurrently wish to be re-elected.

Under Article 7 of the Statutes, members

of the Executive Committee shall be elect-ed for a period of 4 years. Committeemembers may be re-elected, provided thatconsecutive service shall not exceed 8years. No current member has served onthe Executive Committee for 8 years, so allexisting Committee members are in prin-ciple available for re-election.

It would be convenient if potential nom-inations for office in the ExecutiveCommittee, duly signed and seconded,could reach the Secretariat by 1 February2002. It is strongly recommended that astatement of intention or policy is enclosedwith each nomination.

The Council may, at its meeting in Oslo,add to the nominations received and set upa Nominations Committee, disjoint fromthe Executive Committee, to consider allcandidates. After hearing the report bythe Chair of the Nominations Committee(if one has been set up), the Council willproceed to the elections to the ExecutiveCommittee posts.

If a nomination comes from the floorduring the Council meeting, there must bea written declaration of the willingness ofthe person to serve, or his/her oral state-ment must be secured by the chair of theNominating Committee (if there is such)or by the President. It is recommendedthat a statement of policy of the candidatesnominated from the floor should be avail-able.

Accommodation arrangementsDelegates to the Council meeting, who areplanning to attend the Abel CentennialMeeting, are advised that their accommo-dation arrangements should be madethrough the normal Abel CentennialMeeting organisation arrangements. Fordelegates to the Council who are notattending the Abel Centennial Meeting, anaddress for accommodation arrangementswill be provided later.

Secretariat: Ms. Tuulikki MäkeläinenDepartment of Mathematics P. O. Box 4 FIN-00014 University of Helsinki Finland e-mail: [email protected]

David BrannanSecretary of the EMSe-mail: [email protected]

Timetable for theCouncil Meeting

September 2001Information on the Council meeting isprinted in the EMS Newsletter. A nomina-tion form for delegates of the individualmembers to Council is given. Suggestionsfor Council business and for ExecutiveCommittee membership are invited.

Letters are sent to full, associate andinstitutional members, as well as continu-ing delegates, giving information on theCouncil meeting. (Delegates are kindlyrequested to keep the Secretariat informed

EMS NEWS

EMS September 20016

Meeting of the EMS CouncilMeeting of the EMS CouncilOslo: 1-2 June 2002First Announcement

of their correct and up-to-date addresses.)Specifically, points for the agenda and sug-gestions for future members of theExecutive Committee are invited.

2 NovemberDeadline for nominations for delegates toCouncil of individual members.

December A ballot paper for delegates of individualmembers to Council is sent to individualmembers in the EMS Newsletter. Thevenue and meeting times of the Councilmeeting are announced.

1 February 2002Close of voting for delegates to Council ofindividual members.

February Members elected as delegates to Council ofindividual members are contacted by theEMS Secretariat to inform them of theirelection, and to let them know the venue,meeting times and agenda of the Councilmeeting.

MarchThe results of the elections for delegates toCouncil of individual members areannounced in the EMS Newsletter. Thevenue, the meeting times, and the agendaof the Council meeting are given. A letter is sent to each delegate to Council,containing the venue, meeting times andagenda of the Council meeting.

AprilFinal material for the Council meeting issent to the delegates.

1-2 June 2002Council meeting in Oslo.

Nominations are required for Council del-egates representing individual members ofthe Society. On 14 July 2001, there weresome 2247 individual members and,according to our Statutes, these memberswill be represented on Council by 23 dele-gates.

15 delegates were elected for the term2000-03, so they will continue unless theyinform the Secretariat to the contrary by31 December 2001. These delegates are:

Giuseppe Anichini (Firenze, Italy)Vasile Berinde (Baia Mare, Romania)Giorgio Bolondi (Milano, Italy)Alberta Conte (Torino, Italy)Chris Dodson (Manchester, UK)Jean-Pierre Françoise (Paris, France)Salvador Gomis (Alicante, Spain)Laurent Guillopé (Nantes, France)Willi Jägger (Heidelberg, Germany)Klaus Habetha (Aachen, Germany)Tapani Kuusalo (Jyväskylä, Finland)Laslo Marki (Budapest, Hungary)Andrzej Pelczar (Krakow, Poland)Zeev Rudnick (Tel Aviv, Israel)Gerard Tronel (Paris, France)

The mandates of 5 of the present 20 dele-gates end on 31 December 2001, and soelections must be held for their positions.They are:

Manuel Castellet (Barcelona, Spain)George Jaiani (Tbilisi, Georgia)Marina Marchisio (Boves, Italy)Vitali Milman (Tel Aviv, Israel)Jan Slovak (Brno, Czech Republic)

All of these can be re-elected, since theyhave served in this capacity for only 4years.

Nominations are therefore now soughtfor 8 delegates to serve for the years 2002-05. With this notice in the Newsletter is anomination form. Completed nominationforms must arrive at the Societys office inHelsinki by 2 November 2001. (A photocopyof the nomination form is acceptable.) If thereare more nominations than the allowednumber of delegates, a postal election willbe held. Members will receive ballot formsin the December 2001 Newsletter, whichmust be returned by 1 February 2002.

Nominated individuals must be individ-ual members of the Society, and they mustbe proposed and seconded by individualmembers. The Society will pay subsistencecosts for them to attend the Council meet-ings, if necessary, but it is not able to covertravel costs except perhaps in cases of par-ticular hardship.

Candidates for election are invited tosubmit with their nomination form a shortbiography (not more than 200 words),together with a statement of not more than100 words in support of their candidature.These will be circulated to the Societymembers with the ballot forms. A copy ofthe biography and statement can be sent asa text file by e-mail to the Secretariat at thefollowing e-mail address: [email protected]

David BrannanEMS Secretary

EMS NEWS


Election of Council Delegatesrepresenting the Individual members of the Society

NOMINATION FORM FOR COUNCIL DELEGATE[A photocopy of this nomination form is acceptable.]

NAME OF CANDIDATE:................................................................TITLE OF CANDIDATE: ..............................................................(please print)ADDRESS OF CANDIDATE: .................................................................................................................................................................(please print)..................................................................................................................................................................................................................

E-MAIL ADDRESS OF CANDIDATE: ....................................................................................................................................................(please print)

NAME OF PROPOSER: .................................................................. SIGNATURE OF PROPOSER: ....................................................(please print)

NAME OF SECONDER: ................................................................ SIGNATURE OF SECONDER: ...................................................(please print)

I certify that I am an individual member of the EMS and that I am willing to stand for election as a delegate of individual mem-bers to the Council.

SIGNATURE OF CANDIDATE: .................................................... DATE: ...........................................................................................

Completed forms should be sent to: Ms T. Mäkeläinen, EMS Secretariat, Department of Mathematics, P.O. Box 4, FIN-00014University of Helsinki, Finland to arrive by 2 November 2001.

Note: While it is highly desirable that the Biography and Statement of candidates be received by e-mail (as text files), it is necessary that nominationforms are received in hard copy format to ensure the genuineness of signatures.

The EMS Lecturer for 2002 will beProfessor Gianni Dal Maso of theInternational School for Advanced Studies(SISSA) in Trieste, Italy.

It is planned that he will visit two differ-ent locations in Europe to give the sameseries of lectures to different audiences,affording as many interested mathemati-cians as possible the opportunity to attendand discuss the topics with him. EMSmembers interested in organising such avisit are invited to contact Professor DalMaso directly (e-mail: [email protected]) bymid-October, copying their e-mail toProfessor David Brannan (e-mail: [email protected]).

Brief biographyProfessor Dal Maso was born in Vicenza,Italy, in 1954; in 1955 his family moved toTrieste, where he received his basic educa-tion. He was a student of the ScuolaNormale of Pisa from 1973 to 1977, andgraduated in mathematics from theUniversity of Pisa in 1977, with Ennio DeGiorgi as his advisor. He was then a grad-uate student of the Scuola Normale di Pisafrom 1978 to 1981, working with Ennio DeGiorgi on many problems connected withthe theory of gamma-convergence that wasdeveloped in those years.

After serving as Assistant Professor ofMathematical Analysis in the Faculty ofEngineering of the University of Udinefrom 1982 to 1985, he moved to theInternational School for Advanced Studiesin Trieste. He worked there as AssociateProfessor of Mathematical Analysis from1985 to 1987, and has been Full Professorof Calculus of Variations since 1987. Hewas awarded the Caccioppoli Prize in 1991and the Medaglia dei XL per laMatematica of the Accademia Nazionaledelle Scienze detta dei XL in 1996.

At SISSA he has developed his researchinterests on gamma-convergence,homogenisation theory and free disconti-nuity problems, and has supervised 19PhD students working on these subjects.He is currently Head of the Sector ofFunctional Analysis and Applications ofSISSA.

Research interestsProfessor Dal Maso started his researchwork in Pisa while Ennio De Giorgi wasdeveloping the new notion of gamma-con-vergence to deal systematically with the fol-lowing phenomena: the solutions of varia-tional problems depending on a parame-ter may converge to the solution of a limitproblem even if the integrands of the func-tionals to be minimised do not converge inany reasonable sense, or converge to alimit integrand which is different from theintegrand of the functional minimised bythe limit of the solutions. Gamma-conver-gence is a very efficient tool to tackle thiskind of problems.

In his work in Pisa and Udine he studiedseveral problems related to gamma-con-

vergence. In particular he developed,together with Giuseppe Buttazzo, severaltechniques to prove under differenthypotheses that the gamma-limits of inte-gral functionals are still integral function-als, and he used gamma-convergence tech-niques to study the asymptotic behaviourof solutions to minimum problems withstrongly oscillating obstacles. Using thenotion of capacity, he also gave a completecharacterisation of the sequences of obsta-cle problems whose variational limit is stillan obstacle problem.

He later used these techniques to study,with Umberto Mosco, the asymptoticbehaviour of the solutions of Dirichletproblems for the Laplace equation in per-forated domains, and to determine thegeneral form of their variational limits, aswell as the fine properties of the solutionsof these limit problems. These results havebeen extended, with different collabora-tors, to the case of other linear and non-linear equations and systems.

At present his main research interestsare in free discontinuity problems. Theseare variational problems where the func-tional to be minimised depends on a func-tion and on its discontinuity set, whoseshape and location are not prescribed. Inmany cases the discontinuity set can beconsidered as the main unknown of theproblem. Examples are given by the min-imisation of the Mumford-Shah functionalin image segmentation, and by the mini-mum problems which appear in many vari-ational models for fracture mechanics,where the unknown crack is represented asthe discontinuity set of the displacementvector, and the functional to be minimisedis the sum of the elastic energy and of anintegral on the discontinuity set, whichrepresents the work done to produce thecrack.

Selected list of publicationsAn Introduction to Gamma-Convergence,

Birkhäuser, Boston, 1993.Asymptotic behaviour of minimum problems

with bilateral obstacles, Ann. Mat. Pura Appl.(4) 129 (1981), 327-366.

Some necessary and sufficient conditions for theconvergence of sequences of unilateral convexsets, J. Funct. Anal. 62 (1985), 119-159.

(with U. Mosco) Wiener criteria and energydecay for relaxed Dirichlet problems, Arch.Rational Mech. Anal. 95 (1986), 345-387.

(with G. Buttazzo) Shape optimization forDirichlet problems: relaxed formulation andoptimality conditions, Appl. Math. Optim. 23(1991), 17-49.

(with J. M. Morel and S. Solimini) A variationalmethod in image segmentation: existenceand approximation results, Acta Math. 168(1992), 89-151.

(with A. Garroni) New results on the asymptoticbehaviour of Dirichlet problems in perforateddomains, Math. Mod. Meth. Appl. Sci. 3 (1994),373-407.

(with I. Ambrosio and A. Coscia) Fine propertiesof functions with bounded deformation, Arch.

Rational Mech. Anal. 139 (1997), 201-238.(with F. Murat) Asymptotic behaviour and cor-

rectors for Dirichlet problems in perforateddomains with homogeneous monotone oper-ators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)24 (1997), 239-290.

(with A. Braides) Non-local approximation ofthe Mumford-Shah functional, Calc. Var.Partial Differential Equations 5 (1997), 293-322.

(with F. Murat, L. Orsina and A. Prignet)Renormalized solutions of elliptic equationswith general measure data, Ann. Scuola Norm.Sup. Pisa Cl. Sci. (4) 28 (1999), 741-808.

(with G. Alberti and G. Bouchitte) The calibra-tion method for the Mumford-Shah function-al, C.R. Acad. Sci. Paris I Math. 329 (1999),249-254.

(with R. Toader) A model for the quasi-staticgrowth of brittle fractures: existence andapproximation results, Arch. Rational Mech.Anal., to appear.

EMS NEWS

EMS September 20018

EMS LecturEMS Lecturer 2002 : Gianni Dal Masoer 2002 : Gianni Dal Maso

Bringing mathematicians

into biologyThe Human Frontier Science Program isan international funding agency, sup-ported by the G7 governments, theEuropean Union and Switzerland.The HFSP supports interdisciplinaryinternational collaborations in the lifesciences, with an increasing focus onbringing scientists from various fieldssuch as physics, mathematics, chem-istry, computer science and engineer-ing, together with biologists, to openup new approaches to understandingcomplex biological systems.

The HFSP promotes internationalcollaboration through collaborativeresearch grants and post-doctoral fel-lowships. Long-term and short-termfellowships are available for scientistsearly in their research careers. Long-term fellowships provide three years ofsupport to obtain research training inanother country in a new researcharea; the third year can be used in thehome country and under this condi-tion can be delayed for up to two years.The application deadline isSeptember each year. Short-term fellow-ships provide travel and subsistencesupport to visits from 2 weeks to 3months to acquire new techniques orestablish new collaborations; there isno deadline for this programme.Research grants support internationalcollaborative projects. Teams of scien-tists wishing to apply for a grant mustsubmit a letter of intent via the HFSPweb site. The next deadline for appli-cations for letters of intent to submitresearch grants is 30 March 2002.Further information can be obtainedfrom the HFSP web site athttp://www.hfsp.org

Delta is a popular mathematical, physical andastronomical monthly publication by the PolishMathematical Society, Polish Physics Societyand Polish Astronomical Society. The first issuewas published in January 1974. Since then,Professor Marek Kordos from the MathematicsInstitute of Warsaw University has been thejournals Editor-in-Chief.

In Poland, most mathematicians knowDelta, but it is less well known abroad.What is its purpose?The main purpose of Delta is to presentmathematics, physics and astronomy bypeople who work on these sciences. Webelieve that such scientists can presenttheir subjects in a clear way, and do notbelieve in the picture of science that is pre-sented by the professional journalists. Thetrue picture we obtain is the main advan-tage of our point of view. However, manyscientists do not believe that they can speakand write about their results for a generalaudience: this is the main disadvantage.

How was Delta created?The godfather of Delta was Professor LeonJesmanowicz, who convinced ProfessorRoman Sikorski, President of the PolishMathematical Society, of the idea. Theefforts of Sikorski and Tadeusz Iwianski,Secretary of the Society, who was veryclever at fighting administrative formali-ties, brought Delta into existence. If some-body wanted to discredit Delta he could saythat the event happened during the Gierekdecade, so Delta, together with small Fiat126p cars, Central Railway Stations,Lazienki Roads, etc., form the same com-pany. From the very beginning of Delta,the idea of presenting only mathematics inthe journal was given up, because theauthorities decided that mathematicsalone is not interesting enough to fill thewhole journal. In fact, I am very surprisedat such a point of view, but I think it ismuch better that Delta is not only a mathe-matical journal.

Where did the title come from?It was Jesmanowicz who invented the title.The idea was to put something mathemat-ical in the title and ∆ = b2 4ac is suppos-edly the best-known mathematical term.

How did you become Editor-in-Chief?The editor had to be somebody who wasinvolved in journalism in some way. At thattime there was precisely one man inPoland who fulfilled the required condi-tion a mathematician, who presentedquizzes and logical puzzles on TV.However, this candidate was rejected whenhe presented his idea of the journal letme not speak about the details, it is

enough to say that the authorities of thePolish Mathematical Society did not like it.

Then they were terrified, they did notknow what to do and desperately looked foran editor. Eventually, my candidature wasput forward by Andrzej Makowski, whothought that my previous work on the jour-nal Wiedza i Zycie (Knowledge and Life) wasa sufficient recommendation. They wereshort of time and nobody worried about anyprecise investigation. Sikorski phoned meand asked if I could visit him immediate-ly, if possible. Of course, I came to him atonce and then I was offered the job ofEditor-in-Chief of Delta. I asked for sometime to think the proposal over. Sikorski

said: Certainly. How much time do youwant?. I answered: One week. ThenSikorski told me: Of course, you have oneweek, but tomorrow you must take part ina special meeting with physicists and, asEditor-in-Chief, select the associate editor.So, in fact I never agreed to be the Editor-in-Chief, but I probably didnt have to, itwas not necessary.

I do not know how everything connectedwith Delta would have gone on if I hadntfound Tomasz Hofmokl. He was a wonder-ful man, a very good scientist and a verygood manager (this is rather rare), and anexcellent partner. Tomasz was capable ofeverything I couldnt do and of doingsome things I could do and this was reallyfine. My meeting with Sikorski took placeon 3 June 1973 and on 8 December 1973another meeting with the Boards of theMathematical and Physical Societies wasplanned. During this meeting we were topresent our plans concerning Delta.However, we did not present the plans, wepresented the first issue just printed. Thiswas a great surprise and perhaps for thatreason the Boards liked it very much. Inpractice, the official acceptance of theshape of Delta took place in September1974 in Torun, during the GeneralAssembly of the Polish MathematicalSociety, where there was a row. This rowwas provoked by the godfather of Delta,Leon Jesmanowicz himself, who said thatDelta was too effective as its covers were toocolourful: perhaps because of that, somefools might buy Delta. He regarded such asituation as bad. Such an opinion mayseem strange now, but in the 1970s Deltawas the most colourful journal sold inPoland, I think. Delta was defended by

INTERVIEW


Interview with MarInterview with Marek Kek Korordosdos(Editor-in-Chief of the Polish monthly Delta)

interviewers: Krzysztof Ciesielski and Zdzis³aw Pogoda

Marek Kordos

Marek Kordos (centre) with Krzysztof Ciesielski (right) and Zdzislaw Pogoda (left)

Professor Zofia Krygowska with the words:Mathematics is colourful. Such was thebeginning of Delta.

Who is Delta aimed at?We were given some suggestions about the

kind of readers. At the beginning, mostoutstanding Polish mathematicians had anambition to have an article published inDelta. On the basis of what they wrote, it isdifficult to point out the readers.Nevertheless, I do not think that the read-ers of Delta should be precisely charac-terised.

I always explain why I do not want topopularise science professionally.Somebody who does it professionally triesto take science in, to digest it and to showthe final result to the audience. It is dis-gusting, isnt it? I was always against such amethod of popularising science. Thespeaking science is my ideal. I mean sci-entists should be able to speak about theirwork in the way understandable for others.In my opinion, this is the only method ifwe want science to be accepted. Scientistsshould inform others what they do andwhat for. If the subject is interesting forthem, they should know how to interestothers in it. This is the reason why Deltawas always interested in particular authors,not in particular readers.

Some time ago, Delta went through a badperiod. After some years, elderly mathe-maticians stopped writing for it (usually forthe reasons such as illness or death) andso-called young doctors became themajority of our authors. These people (notall of them, of course) are probably theworst kind of authors: they want to write asquickly as possible about everythingtheyve managed to learn. They do notwant anyone thinking: Ah, they didntmention martingales? That means they donot know what martingales are! Then,Delta became really too difficult.

We still have discussions during editorialcommittee meetings. Some members ofthe committee want to reject the articleswhich seem too easy and trivial. I always

strongly recommend such articles, fre-quently somebody agrees with me, so wefight to accept the articles regarded as triv-ial by others. However, one must be verycareful here. In any article (easy or difficult it does not matter), there should beenough information. In my opinion, themost difficult task in our editorial job is tofind articles that can be read easily read say, as a newspaper not studied.

After the period of the authors from thePolish Mathematical School came anothergeneration came a rather strange one.They regarded mathematical techniques asmost important. They would prefer gath-ering different techniques (and perhapsnever apply them) to something converse,that is to fighting with problems withoutsuitable techniques being learnt. This gen-eration (not all of them, of course) saw noreason to inform a general audience abouttheir work. Fortunately, the next genera-tion turned out to be more normal, andyoung mathematicians are now very inter-ested in their connections with the non-mathematical world.

Somebody writes an article for Delta andsubmits it. It is read, judged...... and frequently accepted. In such a waywe get many articles by scientists from dif-ferent countries, and all continents! fromIndia, New Zealand and Vietnam, as wellas the USA, Great Britain and Germany.But it sometimes happens that theEditorial Board does not want to publish asubmitted article.

Sometimes you suggest corrections, dontyou?Ah, yes, of course, but I think that wealways do it in a reasonable way. Only oncedid I shorten an article, written by a pro-fessor of physics. It originally had elevenpages and after my rewriting had 2 pages.The author was very grateful to me, heliked it very much. I think it was the onlyarticle that was really changed. In fact, weinterfere only in articles we want to pub-lish, and the author has always the right toappeal.

Comparing the first Editorial Board withthe present board, one can see that only youhave remained. Of the present Committee,only Andrzej Maakowski was in the firstScientific CommitteeDelta cannot be the main point in some-bodys CV. People come to Delta and laterthey go away. Professor Hofmokl, forinstance, was the Assistant Editor of Delta,later he was Chairman of the Departmentof Experimental Physics at WarsawUniversity, and at the end of his life wasthe Head of Polish Internet. Besides scien-tific and academic careers, people leavingDelta often organise their own printingbusiness. Why I didnt I give up? I wantedto several times, but there was never anyfool who agreed to take over the job. Onthe other hand, it would be a pity to give itup. It seems that Sikorski sentenced me tolife imprisonment.

Delta had a stall during the 1983

International Congress of Mathematiciansin WarsawYes, it attracted people mainly because ofour many colourful covers. Every partici-pant of the Congress received a specialissue of Delta in English, containing select-ed articles from its first ten years. Somearticles into English can now be found onthe internet (www.mimuw.edu.pl/delta/).Also, a selection of our articles will be pub-lished in Catalonian, which will probablybe the beginning of an analogous journalthere. Articles from Delta are also fre-quently published in the Russian Empireof Mathematics.

Deltas sold together with newspapers. Howdoes this work?Ah, this is very interesting. At first, we pub-lished 30000 copies of the journal andeverything was sold. Later we jumped to50000! However, ten years ago economicchanges came, and several new journalsand newspapers started. We believed thatmany of our readers bought Delta becausethey could not get Penthouse or Daily Rag.Now, we publish 4500 copies, which seemsto be reasonable and is probably about thenumber of fans of mathematics and physicsin Poland. On the other hand, such num-bers of copies are not easy to distribute and

sell through a large number of news-stands.

Has Delta ever published original researchpapers?One summer day in 1980 Professor KarolBorsuk called me and gave me a manu-script. In his article he wrote about his nowwell-known theorem that there exists anintrinsic isometry mapping n-dimensionalEuclidean space En onto a subset of thespace En+1 with arbitrarily small diameter.The article was written in a style suitablefor Delta. Borsuk wanted this theorem to befirst published in Delta, because it acted forimagination: in fact, if an alien fromAldebaran could move in the Euclideanspace of higher dimension, then the dis-tance from his feet to his nose might be the

INTERVIEW

EMS September 200110

A caricature of Marek Kordos drawn by LeonJesmanowicz

same as the distance from his feet to theEarth. Thus Delta was the first journalwhich published this result.

You havent published only this journal,have you?Delta has only 17 pages, so we had to pub-lish something else, as our writers wantedto write much more. We used to publishthree series of books: Deltas library, Deltapresents and Read! Perhaps you will under-stand: in this last series we published 26volumes. But first we published a book forschoolchildren Can you wonder? which sold200,000 copies! This is double the numberof copies of any most popular crime bookin Poland. We continued with a secondbook See it in another way, which was also abest-seller.

These two books were based on LittleDelta. For some years, each issue of Deltaincluded a special column for youngerreaders, called Little Delta. In 1981 we gaveup this column and start publishing anindependent journal Little Delta. After twoyears, its title changed to A glass and an eye,the editors changed, and the journal wasedited by the group from Bialystok. 99issues were published, and A glass and aneye existed for about ten years. Now wehave a Little Delta column again. We alsoproduced special TV programmes andeven some theatre and circus perfor-mances.

What about other columns in Delta?The oldest is Club 44, and is a challenge tothe readers. In each issue some mathemat-ical and physical problems are publishedand the readers are invited to send in solu-tions. These are marked, and a competitorwho gains 44 points joins the club. MarcinKuczma, who edits and organises the com-petition, was presented with the DavidHilbert Award by the World Federation ofNational Mathematical Competitions forthis column.

For many years we had a column calledLaboratory at home. Edited by Jan Gaj, thiswas very popular with the readers andslightly less so with their schoolteachers,who sometimes did not know how toexplain the physics of the experiments. In1979 the astronomers joined Delta, and westarted a new column Look at the sky.

We work in Warsaw, but have columnsedited by colleagues from other cities. In1991-97 we had a column named Epsilon.It appeared in 77 issues and was edited byour Krakow colleagues. They presentedmathematics in a light way, with a specialsense of humour. Now we haveGammalimatias written by a Wroclaw math-ematician.

In 1986, the International MathematicalOlympiad took place in Warsaw...Yes, it was really quite interesting. A groupselected from the members of Club 44 tookpart in the competition unofficially, butwith the acceptance of the Jury. The resultof our group was similar to that obtainedby the official Polish team.

Each year you organise a competition for a

mathematical paper by a secondary schoolstudentYes, we do this together with the PolishMathematical Society, and consider onlyoriginal results obtained by young people.The first competition took place in 1978.The winners are awarded Gold, Silver andBronze Medals and small amounts ofmoney, but the main prize is that the win-ners present their papers to the GeneralAssembly of the Polish MathematicalSociety. Also, the winning papers are pub-lished in Delta.

The level of that competition is reallyhigh. Some papers presented here arelater published in good professional math-ematical journals. The winner of the 1980Competition is now a professor of mathe-matics, while some others are on the way toprofessorships. Since 1995, Poland hasparticipated in the European Competitionfor Young Scientists, organised by theEuropean Community. There, papers arepresented by young students and school-children, aged 20 or less. All the mathe-matical papers from Poland in that compe-tition won in our Competition first, andgained one Silver Medal, two BronzeMedals and one special distinction in theEuropean Competition. These were all theprizes gained by mathematicians in theEuropean Competition.

What about the future of Delta?Our situation is complicated. It is similar toBialowieza National Park in Poland. Tworare species live in this National Park:large ones, European bisons and smallones, Laxmanns shrews. Bisons are at leastrepresentative, but what are shrews are?The existence of Delta depends on WarsawUniversity. As long as the Universityregards Delta as useful, or at least good,Delta will exist. This may be called ecologyoutside biology. In fact, there is not muchtrouble with Delta.

Any problems with censorship?Oh, yes, there were some problems at thevery beginning. In the first issue an articleabout the Department of Nuclear Physicswas going to be published. ProfessorHrynkiewicz, a Krakow physicist, talkedabout that Department and photographswere included. In the article it was notedhow large an area is taken by theDepartment. This piece of informationturned out to be top secret informationand we were strictly forbidden to publish it.It was so silly, that when we were orderedto cancel a sentence about this, we laughedabout it very much. I believe that welaughed too much, because as a result weforgot to cancel this piece of information.So, the issue was printed in 30000 copiesand this issue had no right to be distrib-uted. What to do? The copy editor of Deltavisited the Main Censorship Office, thenhe fainted and fell into the arms of the cen-sor-ladies in the office. When the ladiesbrought him back to consciousness hewhispered tragically about the extremelylarge amount of money he would have topay as the one who accepted the issue forpublication.

Then the ladies told him: Ah, its noth-ing, really. When, additionally, it turnedout that the copy editor brought with himtwo copies of the best-seller Cezars livesand wanted to leave these books in theoffice, everything cleared up very soon.

Speaking of censorship, they also can-celled a lot in the issue about ecology.

Look back at 27 years of Delta. What do youadmire mostly? Any articles...It is difficult to say which articles were thebest: I cannot judge it in any sensible way.I would rather mention whole issues. Iespecially liked three of them: the issueson ecology and the 17th century, and amini-monograph on complex numbers.Another valuable thing was a map of thesky. In 1985, we printed part of such a mapon the back cover of each issue: twelveparts put together formed a large umbrel-la with the sky pictured. I have never heardof anything similar to that.

Also, we made special glasses for stereo-graphic pictures, which was a swindle ... ah,you wanted a funny story about Delta, sonow you have one. In 1984 we wanted toprint special stereographic pictures. Thesepictures have to be printed in two colours,but when you look at them through specialcolour glasses you see three-dimensionalpictures. The glasses should have beenenclosed with the issue of Delta. We had toproduce glasses and we needed specialcolour transparents. We needed money forthat where from? No chance to getmoney. So, we started looking for some-body who would give us such transparents.In particular, we asked several institutionsengaged in international trade. In onesuch agency, we were told: It is possible.They wrote to the foreign producer of suchtransparents saying that they were consid-ering buying transparents, and asked forsample copies. These sample copies wereused for producing our glasses and theagency did not buy anything!

INTERVIEW


The cover of the book On differentgeometries, written by Marek Kordos and

published in the series Delta presents

Graham Higman was Waynflete Professor ofMathematics at Oxford University from 1960 to1984, following a period at the University ofManchester. His contributions to pure mathe-matics have been mainly in group theory andmathematical logic.

How did you become interested in mathe-matics?The chief reason was that my elder broth-er did chemistry! When I went to Oxford Icame up on a natural science scholarship but since hed read chemistry, I thought Idbetter read mathematics. My brother hadalready gone to Balliol College, so I wentthere too.

What sort of mathematics did you enjoy atOxford?I was a born pure mathematician I hadno respect for applied mathematics but Ihad to study it all the same to a certainextent.

Was there any pure mathematician whoparticularly inspired you?Well, Henry Whitehead, who was my tutor,of coursea very good tutor. I didntrealise how great he was undergraduatesdont. He certainly wasnt intimidating.But he had a sense of humour that wasrather different from mine, and his non-mathematical attitudes were also different.But apart from that, we got on very well.However, I dont think he took me serious-ly for the first couple of terms, because Idopted to do mathematics on a natural sci-ence scholarship, and he had no ideawhether I was any good.

While you were an undergraduate, youhelped to found the Invariant Society theundergraduate mathematical society?Yes, that was Whiteheads idea it was thefirst Oxford society for mathematicsundergraduates. I remember G.H. Hardygiving the opening address in 1936 onround numbers.

To what extent was algebra taught atOxford in the 1930s in the undergraduatesyllabus?Well, there was a good deal of old-fash-

ioned traditional algebra determinants,matrices, and that sort of thing. But therewas no group theory except that youcould do it as a special subject. In fact, youcould take one or two special subjects Itook two: the theory of groups and differ-ential geometry.

What happened after that?I did a D.Phil. straightaway; Whiteheadwas my supervisor. I suppose I was alreadyinterested in an academic career. Mytopic, Units in group rings, was designed tomake me into an algebraist. But I was alsointerested in combinatorial topology atthat time, as Whitehead was. He was atopologist but he still had an interest inalgebra of various kinds more in linearalgebra, I think, than in group theory.From his point of view, the point of study-ing units of group rings was because oftheir applications to K-theory.

When did you first get to know Philip Hall?I spent a year in Cambridge at the end ofmy D.Phil. course, and thats when I methim. However, Cambridge was a shockingplace to be at that time the beginning ofthe War. It was too near the East Coast! Ialso got to know Max Newman a little butmore so later, when I went to Manchester.He was interested in group theory and alsoin logic. He contributed greatly to mybecoming partly a logician and partly analgebraist.

What did you do during the War?I signed up as a conscientious objector,and then drifted in to the meteorologicaloffice partly because of that, I suppose. Ibegan in Lincolnshire, and spent part ofthe time in Northern Ireland, and also inGibraltar for a while. I didnt do any math-ematics I was just a straightforward fore-caster.

Did you enjoy your work there?Yes, as much as anyone enjoyed anythingin those days more than I would havedone in other jobs. It exercised the mind.

What happened after the War?Well, I can tell you why I became a mathe-matician after the War. By then, I wastotally disillusioned by my experiences inthe real world, and I decided that since Iknew nothing else Id better stay in the MetOffice. I was interviewed there, and at theend the interviewer asked me: If it hadntbeen for the War, what would you havedone? And I said, I suppose I would havetried to climb the academic ladder. Andhe said, Well, with qualifications likeyours, you still could. And I thought,Well, a nods as good as a wink to a blindhorse, so I left. They offered me a job, butit wasnt a particularly good one, so Idecided to apply for academic jobs.

INTERVIEW


Looking back: Graham HigmanLooking back: Graham Higmaninterviewers: ADRIAN RICE and ROBIN WILSON

G.H. Hardys notes on round numbers

Were academic jobs difficult to get at thattime? No, they were actively looking for people anybody with the right qualifications couldget one. In fact, I was offered a job atDurham. I was interviewed at Durham andManchester in the same week Durhamfirst and they offered me a job. Since Iknew that Max Newman was now atManchester, I asked Durham if I coulddelay my decision until Id heard fromManchester. But they said no, so I turnedDurham down and went to Manchester.

Who else was in Manchester at that time?Jack Good and David Rees BernhardNeumann and Walter Ledermann camelater.

Was Alan Turing still around?He must have been around when I gotthere, but I didnt meet him until I becameinterested in his stuff. I think he was in thedepartment of experimental physics, sincehe had to have a laboratory. He was rathera kind man, but rather up in the air,loose, one might say. He didnt give theimpression of being very organised. Iworked a little with him, when he solved theword problem or nearly solved it.

Were there any other colleagues atManchester with whom you worked, or wereyou a solitary figure?No, I was never very solitary. David Rees, asI said, and Trevor Evans and BernhardNeumann when he arrived. BernhardNeumann settled me down a great deal,though it was Max Newman who I chieflyremember with gratitude. In the first place,he took me at a time when Id not done anyserious mathematics for several years and Iwas rather doubtful of my capacity to do so he encouraged me a good deal. And theother thing about Max Newman was that hewas interested in various things he gave acourse of lectures which I remember verywell, on logic and word problems. My inter-est in that aspect of group theory springsvery much from him.

The two prominent figures in your career,Whitehead and Newman, were primarilytopologists. They were topologists primarily, but theywere both mathematicians with a wide range I hope people would describe me the sameway. They were interested in the funda-mental group as a topological tool, and thisled immediately to algebraic questions, ofcourse.

Have you noticed an increasing tendencytowards specialisation in recent years?Well, I think that analysts have always spe-cialised its in their nature theyre self-sufficient, arent they, whereas nobody elsecan afford to be self-sufficient. The D.Phil.,as an almost universal gateway to the pro-fession, wasnt there in my day. But afterthe War, it became obvious that we neededsomething like that.

With such a stimulating environment inManchester, why did you return to Oxford in

1955?In the first place I was very ambitious Istarted applying for professorships that Iwas perhaps not qualified for, and also forother jobs which rather unsettled me. ButI think Henry Whitehead was partly respon-sible for my returning to Oxford, becausehe thought I would be wasted in certainplaces that shall remain nameless. Hearranged a lectureship, which became areadership pretty quickly.

Then Whitehead died suddenly, in 1960.Yes, it was a shock for everybody. He diedin America a long way away at a meetingin Princeton. He was at a tragically youngage.

Whitehead had been the Waynflete Professorof Pure Mathematics at Oxford. Were youthe heir apparent the natural successor?I thought so, but whether anybody else did,I dont know! I applied, and I thought thatI had a reasonable chance and I was in factappointed.

During your time as Waynflete Professorfrom 1960-84, you built up a formidableresearch school in algebra in Oxford. Well, that is what I was there for, really.Perhaps the topologists had had it all theirown way for too long!

Your time in Oxford coincided with all theexciting work on the classification of finitesimple groups.Yes, but I was always on the periphery. I wasmore interested in infinite groups thanfinite groups, and I was never a very goodconjecturer I always thought there wouldbe far more simple groups than thereturned out to be. I tried to construct themrather to characterise them. In fact, Imissed a great opportunity to discover theConway groups. I should have known bet-ter than to go on footling along with finitegroups, but unfortunately I was really obsti-nate about it.

During the 1960s you became President ofthe London Mathematical Society. Had youalways been involved with the LMS?Id been a member since just after the War.I was on the Council for many years and wasassistant editor on the LMS Proceedings toJohn Todd. When I was on the Council Iattended all the meetings they had inLondon except on one occasion whensuch a fog descended that I arrived just asthe meeting was due to finish, and theredidnt seem much point in going.

In those days the outgoing president

looked for a new one. I succeeded A. G.Walker. On the whole I think I was a ratherconservative president. You have to remem-ber that the London Mathematical Societyis a very much larger institution than everbefore, and most of that increase took placeafter I was president. It was already begin-ning when I was president it was clear thatit had to come, but I was a bit frightened ofit.

Since you retired, have you kept up yourinterest in mathematics?Im interested in the kinds of mathematicsthat people do. Im interested in algebra.Im interested in logic. And Im particularlyinterested in seeing a greater number ofpeople working on the borderline betweenalgebra and logic. Im interested, forinstance, in model theory, and the work Ivedone on almost free groups; this leads youto consider the language in which you cantalk about these things. But above all, Iminterested in going on doing mathematics,even if it isnt very good mathematics,because theres nothing else that keeps mesane relatively sane, at least.

Do you still keep abreast of current mathe-matical developments?No, I dont claim to keep abreast of currentdevelopments I never did. I was never agood person for knowing the literature itwasnt my strong point. Others are far morewidely read than I am. What I do is fiddlearound with problems that are probably ofno interest to anybody else. On the wholeIve been interested in things that otherpeople havent been interested in. Ivenever felt that my lines were popular.

Is there a piece of your own work of whichyou are particularly proud?Oh yes theres no doubt whatever! Theone thing that I hope to be remembered foris the theorem on the necessary and suffi-cient condition on a finitely generatedgroup for it to be embeddable into a finite-ly generated group with finite presentation.In the first place, that is solely mine nobody even so much as conjectured itbefore. And in the second place it opens upvistas; for instance, an obvious corollary isthat there are universal finitely presentedgroups that is, finitely presented groupswhich contain every finitely presentedgroup as a subgroup. As far as I know,theres no proof of this which doesnt usemy theorem. There are other things thatIm proud of but thats the thing I think isreally important.

Is there any mathematician from the pastwho you wish you had met?I suppose, Kurt Gödel. I have an immenseadmiration for his work.

And finally, how would you like to be remem-bered?As a great mathematician, of course whatelse is there?

We thank the Mathematical Institute and Prof AKosinski for supplying the photographs in thisarticle

INTERVIEW


Henry Whitehead and friend

We return to the last three questions at thebeginning of Part 1 of this article:4.What is the methodology of mathematics, andwhat is the way it goes about its job?5. Is there research going on in mathematics? Ifso, how much? What are its broad aims or mainaims? What are its most important achieve-ments? How does one go about doing mathe-matical research?6.What is good mathematics? Is there a futurefor mathematics?

We re-emphasise that the tone of thisarticle is that of an address to students.We hope that its publication will encouragedebate about how, as Dantzig [1] put it, topresent mathematics with its cultural con-tent and not as a bare skeleton of techni-calities, with the danger of repellingmany a fine mind. Serious and difficultquestions remain on the balance betweenthis kind of teaching and the usual materi-al, and on assessment: students are rightlyinclined to believe that matters notassessed are thought to be unimportant. Ifmatters related to professionalism are notdiscussed, then there is a danger that naiveattitudes will prevail such as that goodmathematics is precisely the mathematicsdone by top mathematicians.

What is the methodology of mathematics?Here again is a subject that is rarely andnot widely studied. There is the commentof Paul Erdõs [4] that mathematics is ameans of turning coffee into theorems;perhaps, though, this does not help thebeginner too much. So let us look at someof the issues discussed in the books by P.Davis and R. Hersh, The mathematical expe-rience [2] and Descartes dream [3], particu-larly the section of the first book on Innerissues. This deals with a number ofthemes.SymbolsThe use of symbols and symbolic notationsis one of the characteristics of mathemat-ics, and one that puts off the general pub-lic. People will say they were able to domathematics till it got onto x and y.

The manipulation of symbols accordingto rules is still an important part of thecraft of mathematics. We find we have toteach people who wish to master (say) eco-nomics, but who are unable to deduce fromx + 2 = 4 that x = 2. This makes it verydifficult to understand the concepts of eco-nomics.

Very complicated relations can beexpressed symbolically in a way that canhardly be conveyed in words. This econo-my which symbols allow is improving con-tinually, as symbols are used in the denota-tion of advanced concepts and the rules ofsymbol manipulation are used to modelthe rules for the concepts.

It has been said, in an exaggerated way,

that the history of mathematics is the his-tory of improved notation. This reflectsthe finite nature of intelligence, whichrequires props and metaphors to help andguide it.

Some symbols are in themselvesmetaphors: examples are /, <, →, ∫.Others have acquired strong associations,so that we can use them as metaphors.Symbols are able to express with economyand precision, to use words of A.N.Whitehead. The use of particular symbolsis something that changes with time, asmathematicians become accustomed to,and appropriate, a new notation.

In some cases, a notation, brought aboutby the laziness of mathematicians, leads toa new theory. For example, expressions ofthe type (a11x1 + + a1nxn; ; am1x1 + + amnxn) get abbreviated over time to Ax,and to allow for the correct manipulationof this abbreviation the rules for matricesare worked out.

To give an example close to the heart ofsome of our research, the first author hasbeen concerned for many years as towhether the linear notation for mathemat-ics is a necessity or a historical result, basedon the needs of printing.

The analysis of this linguistic point hasled to a new kind of higher-dimensionalalgebra, in which symbols are related notjust to those to the left and to the right, butalso up and down or out of the page. Thisalgebra then becomes closer to, and moreable to model, some geometric situations,and this leads to the formulation andproofs of new theorems, and to new calcu-lations and insights.Abstraction This is an essential part of mathematics,and again is one part of what makes math-ematics incomprehensible to the generalpublic.

Mathematical structures are abstract.They are defined by the relations withinthem. They are thought of as non-sensual.The advantages of abstraction are at leastthree-fold:* An abstract theory codifies our knowl-

edge about a number of examples, andso makes it easier to learn their commonfeatures. Only one theory is needed, toreplace a multiplicity. This codificationexploits analogies, not between thingsthemselves, but between the behaviourand relations of things. Finding theseanalogies, the abstract theory thatreplaces a multiplicity, is an importantmethod in mathematics.

* Once the theory is available, it may befound to apply to new examples. Thisleads to the excitement and joy of: Thatreminds me of ...!. For this new exam-ple, a body of established theory is avail-able at the turn of a page.

* An abstract theory allows for simplerproofs. This is a surprise, but is com-monly found to be true. The abstracttheory allows for the distillation of essen-tials. It is of interest to know if a theo-rem or fact is true in the general situa-tion or only in the particular example.The abstract theory allows for theremoval of possibly irrelevant aspects.

Generalisation and extension This has some features in common withabstraction, but usually applies differently.Thus a generalisation of the (3, 4, 5) right-angled triangle is Pythagoras theorem,while an extension is Fermats last theo-rem, that the equation xn + yn = zn has nosolutions for positive integers x, y, z, if n >2. This has now been famously proved byAndrew Wiles [7].Proof The rigorousness of the notion of proof isa particular feature of mathematics. It iswhy mathematics is essential in engineer-ing, safety, physics, and so on.

The notion of proof, of validity, in math-ematics, is an aspect of the general ques-tion: what is the notion of validity in anarea of study? Each area, from social sci-ences, economics, chemistry, biology, edu-cation, law, literature, and so on, has itsnotion of validity, and the contrast anduses of this notion are of particular inter-est.

The question of what is acceptable as avalid argument in mathematics is still sub-ject to argument and discussion, particu-larly with the existence of very long proofs(for example, 15000 pages [5]), and withthe use of computers for visualisation,experimentation and calculation.Existence of mathematical objects A great mathematician has urged that themajor problem of mathematical educationis to teach the reality of mathematicalobjects. What is this reality? In what waydo these objects exist? For example,Eternity, by John Robinson, is a symbolic


FEATURE ARTICLE

The Methodology of MathematicsThe Methodology of Mathematics part 2

Ronald Brown and Timothy PorterThe first part of this article appeared in the June 2001 issue.

Eternity, by John Robinson

sculpture, but is also the construction of afibre bundle! (see [6]).

These questions have been a matter ofmajor interest to many philosophers ofmathematics, but their interest is perhapsin the process of being downgraded.Mathematics is often about processes. Thequestion of existence of a mathematicalstructure is maybe like asking whether thegame of chess exists. Clearly it does notexist in the way that tables and chairs exist,but nonetheless it influences many livesand, to put it crudely, passes the cash test.(Does it earn money? The answer is clear-ly: yes, for some for example, worldchampions and makers of chess equip-ment.)

The relation of mathematical conceptsand methods to processes is indicated bythe way that memory of muscular actionand rhythm are important aspects of math-ematical work. A lot of mathematics is con-cerned with the realisation and under-standing of the effect of repetitive process-es and methods.

Mathematicians are good at understand-ing and imagining moving things around,such as from one side of an equation toanother, or changing a pattern in space.They use movements of their hands andarms to convey what is happening. Theobjects and ideas of which mathematicianstalk are sometimes a kind of concatenationof a variety of such remembered processes.By contrast the representation of theseideas in writing is often bare and sparse,and this is part of the difficulty of learningthe use and application of these objectsand ideas. On the other hand, it alsoallows for each person to make the inter-pretation and internalisation most appro-priate to themselves.Infinity The taming of the infinite, or the enlarge-ment of the imagination to include infiniteoperations, is one of the joys of mathemat-

ics, and also one of its scandals. Are theseinfinite objects real? The surprise is thatthese infinite, possibly unreal, objects canbe used to prove finite real things, and thisagain is an aspect of the mystery of the sub-ject. Suppose for example that these infi-nite objects are used to prove the safety ofa nuclear installation, or of an aircraftlanding system? What credence should beplaced on such a proof? These are realissues.

Is there research going on in mathematics?Those who wish a practical test should lookat the change in Mathematical Reviews sinceit started in 1940. This monthly journalcontains abstracts of mathematical papers.Roughly speaking, a few paragraphs areenough for a five-page paper. The growthin numbers of pages over these years isabout eleven-fold: each month now about400 large pages of abstracts of mathemati-cal papers are published. This is indeedthe golden age of mathematics, both inquantity and quality.

The aims of this research are at variouslevels. One is the advancement in knowl-edge about particular types of structures,that are already well defined. Another isthe introduction of the study of new struc-tures, as they have appeared and beenshown to be relevant. There are new rela-tions between structures. There is the urgeto simplification, to find structures thatexplain and help us to understand the wayparticular structures behave in themselvesrelative to other structures.

What is difficult for the newcomer in thefield, and for the general public, to under-stand is how one goes about doing mathe-matical research. Here we give somepointers, by suggesting four ways of goingabout the job. There are certainly manymore, and individual researchers must inthe end devise their own strategy for suc-cess. It is also difficult to know how muchone must know before starting on mathe-matical research: a famous answer to thisparticular question was: Everything, ornothing.Method 1: Apply a standard method to a stan-dard type of problem This method is probably a part of everysuccessful research project, and has everyguarantee of success, provided that one issufficiently skilful in the standard method.Indeed, a common method of mathemati-cal research is to reduce a problem to onealready considered. If the original prob-lem is too difficult, then a standard strate-gy is to simplify the problem so that itbecomes of standard type, before addingthe complications that make it a new prob-lem. The general presumption might bethat one can do only easy things. So themethod is to reduce a problem to a typethat can be seen to be easy. If in doubt, dothe obvious thing first.

Those who become skilful at applyingstandard methods may someday find thattheir skills apply to a problem no one elsehas considered, and that this leads to newand important results. Much of the educa-tion of a mathematician is concerned withacquiring the skills and knowledge appro-priate to work in a chosen area.Method 2: Attack a famous problem at the fron-tiers of knowledgeThis is the strategy of going for a famousproblem at a peak of knowledge. Theadvantage is that if you succeed, then youwill become famous. It is more difficult toassess your chances of success: you willprobably need some new ideas.

This seems to be the most ambitiousmethod for a young person. However, S.Ulam, in conversation with the first author

in 1964, suggested that while this methodmight appeal to young ambitious persons,concentration on this might also not allowthem to develop the kind of mathematicsmost personal and characteristic to them-selves, because they are solving other peo-ples problems.

Usually, though, one attacks smallerproblems at the frontiers of knowledge,problems to which less effort has beendevoted, and so where there is a greaterlikelihood of success. You will almost cer-tainly have to study to find what has beendone, what techniques are available, andwhich you need to master.

It is helpful to have problems whose cri-teria for success are clear: the answer is yesor no to some question. On the otherhand, failure to provide a solution is thenalso clear cut, as is finding the problem tooeasy. Mathematicians need to build intotheir strategy plans for dealing with bothtoo little and too much success on theproblem at hand.Method 3: Relate different areas of knowledgeIn this method you learn about the begin-nings of different areas, and find relationsbetween them so you fill in the gapsbetween the peaks, while often the toppeople are occupied with building up thepeaks. The advantages of this method arethat you learn something of differentareas, and in a useful way, since you haveto work to do the translations between thetwo areas. This is a good method for PhDtheses, since a supervisor can often see therelation without having worked out thedetails; it also advances the general unityof mathematics. Another advantage is thatit gets you used to the idea of provingsmall but useful results which help to fill inthe gaps and create the picture of what isgoing on.Method 4: Blue sky research Here you have some idea of a mathematicsthat ought to exist, and of its characteris-tics. You also have a few hints as to thekind of materials from which the mathe-matics ought to be made. The problem isthat proper mathematics requires defini-tions, examples, propositions, theorems,proofs, calculations, and in the beginningnone of these exist so they have to beassembled over a period of time. In whatorder should this be done, and how impor-tant will the work be? This can hardly bejudged until the theory is worked out, andsuch a theory does not emerge, like VenusAnadyamene, fully formed from the sea. Atheory accumulates in a journey over aperiod of years, and a gut feeling of theimportance of a line of investigation is nec-essary to motivate travel on a long road.

For decades we have both been workingon this kind of research, as well as on otherkinds. In the mid-1960s the first authorformulated the theme of higher-dimen-sional algebra, as mentioned earlier. Theaim was that of an algebra more closelyrelated to the geometry, and allowing amore general type of composition. Theexpectation was that this algebra wouldyield some formulations and proofs of newtheorems, which would automatically leadto new methods of calculation.


FEATURE ARTICLE

Eternity, by John Robinson

This in the end has proved correct, withmany people joining in the project. For along time (five years), though, all thatcould be said was that it was possible todraw pictures that suggested that the ideaswould have to work. The problem was alack of framework to express the algebracorresponding to the pictures, and to thegeometry. This framework was built upgradually, and it became ever more amaz-ing to see how natural and fitting a way itwas, once the ideas were thought about inthe correct manner. Thus, as suggestedby Wigner in the quotation given earlier,the aesthetic criteria for a proper theorywere nicely satisfied, and the theorybecame better than the vision that hadprompted it.

It has to be said that, paradoxically, thesecret of success in research is the success-ful management of failure. For if younever fail, then it is likely that the tasks youhave set yourself are simply too easy.Interesting research must have an elementof risk. You need strategies for dealingwith situations when things go wrong: theproblem may have proved too hard, or tooeasy. What comes next? Analysing thereasons for failure, and comparing thesereasons with the reasons for wanting to dothis problem in the first place, becomesinstructive for future work.

What is good mathematics?We would not like to attempt any finalanswer to this, but all of us should try andformulate some of the aspects we are look-ing for. Indeed, as editors of journals, wehave to make judgements on this questionon a daily basis. For a new mathematicalpaper we ask the questions: are the resultsnew? how far ahead of current literature dothey go? is the paper clear and well writ-ten? are the authors familiar with currentwork in the field and aware the relation oftheir results to the field? how surprisingare the results? how elegant are the meth-ods, and are any new methods introduced?

Some of the best mathematics is thatwhich introduces new ideas and conceptsthat make the previously difficult easy.This contradicts an impression you mayhave that mathematics is meant to be hard,and is good for you partly for that reason,like a cold bath. On the contrary, goodmathematics can (perhaps, should) beeasy. It is just that often we do not knowhow to do this. The combination of appar-ently simple arguments with a surprisingconclusion, perhaps with a surprising twist,is what we like best of all.

What is worrying is that many youngmathematicians go through their educa-tion without the notion of good mathe-matics even being debated. Yet for anyhuman activity, there is always the questionof its value, both for society and personal-ly. There is an argument that the teachingof a subject should reflect something of thevalues of the professionals in it; for exam-ple, for a professional, it is not enough justto produce an answer, but it is importantalso to produce (if possible) a satisfyingexplanation.

Thus we would argue for the advantages

of introducing pupils and students to thenotion of good exposition, and even to askthem to compete, not in problem solving,but in producing expositions and exhibi-tions of mathematical principles and appli-cations. We have found the work on pro-ducing a mathematical exhibition enor-mously instructive [8, 9].

Is there a future for mathematics?There is a view that there is no more basicmathematics to be found. This view iscomparable to the view of those who havesaid that physics was ended, the basic prob-lems having been solved. We feel, to thecontrary, that mathematics is undergoing arevolution a quiet one, but a revolutionnonetheless. This is occurring on twofronts.

There is first the computational revolution.For computation with numbers, or forgraphics presentation, this revolution iswell known. Less well known publicly is thecomputer software that can manipulatesymbols and axioms, and other softwarethat can carry out automated reasoning.In principle, these should give mathemati-cians power to calculate and reason a mil-lion-fold more than they can at present,and to deal with the complexities of sys-tems thought previously to be intractable.The prospective effect of these on theteaching of mathematics has yet to beproperly understood and assessed,although a lot of work is in progress. Theeffect on research has already been consid-erable and is likely to grow in its influence.

A more subtle revolution is the conceptu-al revolution. The emphasis on mathemat-ics as the study of structures is finding itsmathematicisation in category theory, themathematical and algebraic study of struc-tures. Category theory has revealed newapproaches to the basic concepts of math-ematics, such as logic and set theory, andindeed has made respectable the idea thatthe practice of mathematics needs not onefoundation, as traditionally sought, butalternative environments and a frameworkfor their comparison. These ideas are alsoimportant for the progress of computerscience, as (for example) in showing newapproaches to data structures.

Some dangers aheadOne of the pleasures of mathematics is theway it operates on various levels, whichthen interact. So the algebraic study ofmathematical structures has itself led tonew mathematical structures. Some ofthese structures have had notable applica-tions in mathematics and in physics.

Nevertheless, there are still many cur-rent dangers for mathematics. There is ageneral lack of appreciation of what math-ematicians have accomplished, and of theimportance of mathematics. Some of thishas come about through mathematiciansthemselves failing to define and explaintheir subject in a global sense to their stu-dents, to the public, and to governmentand industry. It is possible for a student toget a good degree in mathematics withoutany awareness that research is going on inthe subject.

Another danger is the growing relianceon computers as a black box to give theanswer, without any understanding of theprocesses involved, or of the concepts thatare intended to be manipulated. So boththe scope and the limitations of the com-puter fail to be understood, the mathemat-ical basis is neglected and perhaps fails tobe developed, and the computer may beused in ways that are inappropriate, orsimply limited by the software design. It issaid that some engineering firms are dis-pensing with their mathematical researchdepartments in favour of engineers manip-ulating software packages. Will this ensurethe safety or reliability of the product, andwill it allow the use of the most advancedmathematical concepts?

If these dangers are to be averted, thenan increased understanding and apprecia-tion of the questions with which we startedare essential. There may be ways of speed-ing up the process of transfer from theconceptual foresight of the mathematicianto the realisation in a scientific or techno-logical application. To find them, we needin society a real understanding of the workof mathematicians, and of the way mathe-matics has played a role in the society inwhich we live. It is our responsibility to thesubject we love to find ways of developingthis understanding.

Acknowledgements Many of the questionsraised in this article were discussed withstudents of the final-year Maths in Contextcourse we ran together at Bangor, and alsowith first-year honours mathematics stu-dents taking the course Ideas in Maths. Thecontributions of these students throughdiscussions and essays have strongly influ-enced our thinking. We also wish to thankRoger Bowers and Brian Denton who haverun a course on Mathematics in Society atLiverpool University.

References1 T. Dantzig, Number: the language of science,

1930, 2nd ed. 1954, Macmillan.2. P. Davis and R. Hersh, The mathematical expe-

rience, Penguin, 1981.3. P. Davis and R. Hersh, Descartes Dream,

Penguin, 1988.4. http://www-history.mcs.st-and.acuk/history/Math

ematicians/Erdos.html5. S. Gorenstein, The longest proof, Scientific

American.6. John Robinson sculpture: for a discussion of

this, see http://www.cpm.informatics.bangor.ac.uk/sculmath/wake.htm

7. http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Wiles.html

8. Bangor Maths exhibition group, Mathematicsand knots, Exhibition for the PopMaths Road-Show, 1989 (16 A2 boards); also a brochure,published by Mathematics and Knots, 1989, andthe website: http://www.cpm.informatics.bangor.ac.uk/

9. R. Brown and T. Porter, Why we made amathematical exhibition, The Popularisation ofMathematics (ed. G. Howson and P. Kahane),Cambridge, 1992.

School of Informatics, University of Wales,Dean St., Bangor, Gwynedd LL57 1UT, UK

FEATURE ARTICLE


The first attempt to create a mathematicalsociety in Norway was made in 1885 bySophus Lie, who was at that time professorin Oslo. This was a time when similar ini-tiatives took place in many European coun-tries. The Moscow Mathematical Societywas founded in 1864, London in 1865, andthe Finnish, French and Danish ones in1868, 1872 and 1873, respectively. InNorway, however, the mathematical com-munity at that time was too small, and theventure broke down when Lie moved toLeipzig in the following year. But a seriesof reforms in the high schools and at theuniversity in the second half of the 1800s(less Latin and Greek, more modern lan-guages, science and mathematics) led to amarked expansion of that community, anda formal organisation became necessary.In particular, the need for a Norwegianmathematical journal was felt. The diffi-culty was to find financial support for it,and to find persons able and willing to takeon the editorial work.

In 1918 the time had come, and prelim-inary discussions took place in the earlyautumn. Arnfinn Palmstrøm, who at thattime worked as an actuary and who from1919 until his untimely death in 1922 was

Norways first professor of insurance math-ematics, secured financial support fromthe major insurance companies.Government sources also responded posi-tively, and the Danish mathematician PoulHeegaard, who had just been appointed

The NorwegianMathematical Society

Bent Birkeland

professor of geometry in Oslo, was willingto edit the journal. He had valuable expe-rience from editing the DanishMathematical Journal for a couple of years.Finally, on 2 November 1918 (incidentally,Heegaards birthday), the NorwegianMathematical Society was born. Its pur-pose was stated broadly as connectingmathematically interested persons from allover the country, and the first more spe-cific task was to start a national mathemat-ical journal. Professor Carl Störmer waselected the Societys first president, whilePalmstrøm became its secretary and themore arduous task of editing the journalwas taken on by Heegaard for the mathe-matical side and Anton Alexander for thedidactic one.

The founding fathers were universitymathematicians, leading schoolteachers,actuaries, officers (mainly from the geodet-ic service) and students. At least two ofthem came to be closely involved with theSociety for more than sixty years: the num-ber-theorist Viggo Brun (1885-1978)became a university professor, whileFredrik Lange Nielsen (1891-1980)became a leader in the insurance world inNorway. The activities taken on were what onemight expect: meetings, publications, andsome lobbying for good mathematicalcauses. A few of these are described below.

PublicationsThe first issue of the Norsk MatematiskTidsskrift (Norwegian MathematicalJournal) appeared in 1919, opening, sadly,with the obituary of Ludvig Sylow, writtenby Thoralf Skolem. The first volume alsocontained contributions by the young andpromising number-theorists Viggo Brunand Trygve Nagel.

The journal was intended to serve twonot-quite-compatible purposes: to provide

interesting reading for the general mathe-matically interested public, and to giveyoung and aspiring mathematicians achance to have their work printed. Thatproblem found a temporary solution whenHeegaard succeeded in obtaining fundsfor a series of pamphlets, Norsk MatematiskForenings Skrifter (Publications of theNMF), where younger Norwegian mathe-maticians, including Øystein Ore, ThoralfSkolem, Trygve Nagel and Ragnar Frisch(Nobel laureate in Economics, 1969) hadsome of their early work published.Regrettably, this enterprise was discontin-ued in the 1930s for financial reasons.

The journal continued for 34 years.Finally, in 1952, it was amalgamated withthe corresponding journals in the otherScandinavian countries to form two newperiodicals, the Mathematica Scandinavicafor professional mathematics, and theNordisk Matematisk Tidskrift (Normat, forshort), which aims at a broader audienceand mainly prints work in theScandinavian languages. Both of these arestill active, under the joint auspices of thefive Nordic mathematical societies(Danish, Icelandic, Finnish, Swedish andNorwegian).Another publishing venture of the Societywas Sophus Lies collected works. The edi-tors, Friederich Engel and Poul Heegaard,published the first volume in 1922, but forfinancial and other reasons the seventhand last volume did not appear until 1960.

The NMF also published a series ofsmall popular booklets in the early 1990s,aimed mainly at college students, onthemes varying from number systems toNorwegian mathematicians.

Present activitiesMathematics competitions for college stu-

SOCIETIES


Societies CornerSocieties Corner

Arnfinn Palmstrøm (1867-1922)

Poul Heegaard (1871-1948)

Carl Störmer (1874-1957)

dents have been part of the Societys activ-ities nearly constantly from the beginning.For many years, starting in 1922, CrownPrince Olav awarded a prize for the bestsolutions to a series of problems posed inthe Journal. Later on, other sponsors tookover. For many years the Norwegiantelecommunications company Telenor hassponsored a mass contest in three stages,called the Abel Competition. The firststage involves several thousand students,and successive eliminations bring it downto about twenty participants in the finalround. Of these, the best six are selectedto take part in the InternationalMathematical Olympiad. (A note on thiscompetition appeared in EMS Newsletter 32in June 1999.)

Another activity is the winter seminarSki and Mathematics, early in January.This tradition was initiated in the 1960s byProfessor Karl Egil Aubert. The seminarwas arranged regularly for half a dozenyears, then more intermittently, until it wasresumed on a regular basis in 1997. Ittakes place at a hotel in the mountains, theprogramme being divided between out-door activities before lunch and mathemat-ics in the afternoon. (There is not muchdaylight after 3 p.m. at that time of theyear.)

World Mathematical Year 2000The international mathematical year wascelebrated by the NMF and by others, withactivities at the universities, in the schoolsand in the streets. Mathematically, themain event was a conference in Trondheimin January. Later in the year a large num-ber of events were aimed at the schools andthe general public. On one occasion thefirst 5000 digits of ð were written along themain street of Oslo!

The Abel Bicentennial ConferenceNiels Henrik Abel was born on 6 August1802. His bicentennial will be the occasionfor much celebration, aiming both atmathematicians from home and abroadand at society in general. The main event

will be the Abel Bicentennial Conference 2002,which takes place in Oslo from 3-8 June.The conference is arranged jointly by theNorwegian Mathematical Society, theNorwegian Academy of Science andLetters and the Norwegian MathematicalCouncil, with support from theInternational Mathematical Union and theEuropean Mathematical Society. The con-ference will present an overview of themathematical heritage of Niels HenrikAbel and, based on this heritage, will iden-tify new mathematical trends for the 21stcentury. There will be sessions on the his-tory of mathematics, algebraic geometry,complex analysis, differential equationsand non-commutative geometry. On thisoccasion the remaining unsold copies ofAbels works (edited by Sylow and Lie in1881) will be on sale, nicely bound inleather. This should be of interest to bothmathematicians and bibliophiles.

The Academic Mathematical Society, pre-decessor of the Estonian MathematicalSociety, was founded in Tartu 75 years ago.The statutes of the Society were registeredby the Council of the University of Tartuon 23 February 1926, although the open-ing meeting of the Society with 68 found-ing members present mostly teachersand students from the Universitys mathe-matics department took place in theFestive Hall of the University on 21 March1926. A lecture on Archimedean andEuclidean Sentences was delivered byProfessor Jaan Sarv. Professor GerhardRägo became the first President of theSociety, and his successors as Presidentwere Jüri Nuut (1927-32), Edgar Krahn(1932-36) and Hermann Jaakson (1936-40). In addition to mathematicians fromTartu, several members of the Societycame from other towns in Estonia.

The Academic Mathematical Societyplayed an essential role in the mathemati-cal life of pre-war Estonia. Interesting cur-rent results in the field of exact sciences,reports on research by members of theSociety, issues connected with the teaching

75 Years of the EstonianMathematical Society

Mati Abel

of mathematics, and topics from the histo-ry of mathematics were discussed at meet-ings of the Society.

After the annexation of Estonia by theSoviet Union in 1940, the activities of sci-entific societies were banned, and the lastmeeting of the Academic MathematicalSociety took place in November 1940. InJune 1941 the new puppet Soviet govern-ment in Tallinn ordered the University ofTartu to reorganise the activities of eigh-teen functioning academic societies at theUniversity (the Academic MathematicalSociety included). In reply to this regula-tion, Professor Hans Kruus, Rector of theUniversity, made a proposal to the govern-ment that the Academic MathematicalSociety be closed. The activities of othersocieties were reorganised in accordancewith the requirements presented to organ-isations in the Soviet society. Because ofthe German invasion of Estonia in July1941, the Soviet administration in Tallinncould not deal with the Rectors proposal:the Academic Mathematical Society wasnot closed down, but its activities werestopped. During the war HermannJaakson, the most recent President of theSociety, repeatedly tried to obtain permis-sion to continue the Societys activities, butthe new occupation system in Estoniaplaced demands on the statutes of academ-ic societies, with the result that theAcademic Mathematical Society changedits statutes several times during the war.After the war the activities of all formeracademic societies were forbidden by theSoviet regime in Estonia.

The idea of re-establishing theMathematical Society circulated amongEstonian mathematicians in the 1970s, butthe timing was not favourable. The firstattempt to re-establish the activities of theSociety in 1983 failed: the EstonianCommunist Party leadership did not givepermission to re-open it. In spite of thisrefusal, Estonian mathematicians contin-ued to seek ways to continue the Societysactivities.

The re-opening conference of theAcademic Mathematical Society, under the

SOCIETIES


Viggo Brun (1885 -1978)

Mati Abel and Imre Csiszar

new name of the Estonian MathematicalSociety, took place with 118 founding par-ticipants and 52 guests on 17 September1987, once again in the Festive Hall of theUniversity. Professor Ülo Lumiste, themain initiator of the re-establishment ofthe Society, opened the conference. Theparticipants elected the Board of theSociety, the President of the Society (ÜloLumiste) and the Honorary President(Academician Arnold Humal). ÜloLumiste led the activities of the Society fortwo three-year terms until 1993. Mati Abelhas occupied the Presidency since 1994.

Immediately after its re-establishment,the Society started its activities with therestored pre-war structure. The section onschool mathematics (from which the Unionof School Mathematics was formed in1989), and several working groups (oncomputer science, mathematical terminol-ogy, history of mathematics, etc.) began towork. It was decided to publish the year-books of the Society, to establish annualprizes for high-school final-grade students(for the best results in the Olympiads inmathematics) and for undergraduate stu-dents (for outstanding results in research).On a proposal from the heirs of

Academician Arnold Humal, the ArnoldHumal Annual Prize for a young mathe-matician was established. In 1995 theAssociation of School Teachers ofMathematics Teaching in Russian joinedthe Estonian Mathematical Society andstarted functioning as a section of theSociety.

Every other year the EstonianMathematical Society organises Days ofMathematics in Estonia, and the Union ofSchool Mathematics holds the annualSummer and Winter Days of the Unionwhich are very popular with teachers ofmathematics. In addition to workshopsand report meetings, the Society helps toorganise international and local confer-ences in mathematics, to publish mathe-matical books and a journal, to composereference books and dictionaries in mathe-matics, to organise open competitions andOlympiads in mathematics for high schoolstudents, and to train them for interna-tional competitions in mathematics. AnInternational Workshop on TopologicalAlgebras and an essay competition Thebeauty, magic and pain of mathematicswere held by the Society to celebrate WorldMathematical Year 2000 in Estonia.

With over 400 members, the EstonianMathematical Society was a foundingmember of the European MathematicalSociety, and since 2000 has been a memberof the International Mathematical Unionas an associated organisation.

On 17 February 2001, the EstonianMathematical Society celebrated its 75thanniversary with a conference in Tartu.Among the participants were Imre Csiszar(President of the Hungarian MathematicalSociety), Aleksanders ostaks (President ofthe Latvian Mathematical Society), UlfPerrson (President of the SwedishMathematical Society), Wies³aw Zelazko(President of the Polish MathematicalSociety in 1984-86 and a foreign memberof the Estonian Mathematical Society), andHans-Olav Johannes Tylli (a member ofthe Finnish Mathematical Society). JonasKubilius (President of the LithuanianMathematical Society) and Anatoly Vershik(President of the St PetersburgMathematical Society) sent anniversarycongratulations.

We cherish friendly ties and cooperationwith mathematicians from other countriesand are determined to expand our inter-national contacts.

SOCIETIES


Anniversary meeting of the society on 17 February 2001

ability is the number of submitted solu-tions to the problems.

Another selection rule for candidateswishing to become one of Romanian topyoung mathematicians invokes the resultsachieved in national or internationalevents during the previous year. But theyhave to get over two other hurdles to reachthe summit, the 'reasoning' round and astage that examines problem-solvingstrategies and a mastery of mathematicaltechniques. In August 1998, forty-eightstudents attended the final round, held inPitesti, undergoing two 4-hour papers.The top scorers are awarded prize money,books, medals or diplomas. In 2000, theeditorial department of Gazeta Matematicaconducted its twentieth annual contest. Inthe Western hemisphere no other publica-tion does as good a job as the Romanianperiodical, and in general the assortmentof mathematics events in Romania exceedsby far the range of contests elsewhere.

I wish to thank Professor VasileBerinde, of the Department ofMathematics and Computer Science in theNorth University of Baia Mare, for provid-ing me with information for use in thiscorner.

This leads to the next set of questions,from the Gazeta Matematica Contest inSeptember 1997.

The solution of problems is theladder by which the mind ascends into

the higher fields of original researchand investigation. Many dormant

minds have been aroused into activitythrough the mastery of a single prob-

lem.Benjamin Finkel and John Colaw,

American Math. Monthly, 1894, Vol. 1, no. 1, page 1

At the beginning of the twenty-first centu-ry, elementary mathematics is undergoingtwo major changes. The first is in teach-ing, where one moves away from routineexercises and memorised algorithmstowards creative solutions to unconven-tional problems. The second consists inspreading the culture of problem solving -especially in Eastern Europe. Romanianmathematicians have influenced bothtrends strongly, breaking new ground inestablishing competitions for youngsters.Following Paul Halmos, who said, 'prob-lems are the heart of mathematics', theyemphasised them in the classroom, inseminars, and in many publications of allkinds written to train their students to bebetter problem-solvers and problem-posers.

It has been claimed that mathematics inEast Europe flourished underCommunism because it is less susceptibleto political interference than other intel-lectual pursuits, and consequently attract-ed many talented minds to its study. Withungrudging admiration we observe thatRomania has produced more than its fairshare of top-quality mathematicians.What makes Romania unusual comparedwith other Western countries is the largeproportion of its distinguished mathemat-ical experts who have devoted time andenergy to the younger generation ofpromising scientists, through via theirproblem-solving competitions and associ-ated training units. These Academiciansknow the necessity for challenging thecleverest students with difficult problemsand teaching them the mathematical skillsrequired for their solution. This involve-ment by many mathematical experts has

also led to the creation of remarkably art-ful problems aimed at differing levels ofstudents. The variety of the questions,their suitability for a range of age groupsand the nature of the challenge has beenof a consistently high standard since thefirst competition in 1895.

The Gazeta Matematica CompetitionIn January 1910 the renowned mathemat-ical journal Gazeta Matematica rose to thefore by promoting mathematics competi-tions and created a separate maths eventnationwide. As a result the questions werecirculated to the remotest corners of thecountry overnight. In this way a competi-tion was launched that embraced thewhole Romanian area for the first time.

The Annual Gazeta Matematica Contesthad a life-span of about 40 years, and wasfinally replaced by the NationalMathematics Olympiad in 1950.However, the competition that was run bya mathematics journal had its eyes on par-ticular benchmarks. Priority was given tothose participants who had correspondedwell to the problems published in the jour-nal over a given period (perhaps a year).The columns of Gazeta Matematica repre-sent a hotbed of demanding questions thatcannot be solved in a short time, and thusthis periodical was an excellent vehicle foridentifying highly talented young persons.

At the start of the global economicalrace in the 1980s, Romanian authoritiesagain became aware of the significance ofGazeta Matematica in drafting new mathe-matical blood. The old 'tool' has beenrevived, and the Gazeta MatematicaCompetition now runs annually in sum-mer. The competitors can choose betweenproblems for the Gazeta's contest andtricky questions designed to prepareyoung people for mathematics competi-tions. Both types of problems are pub-lished in Romanian and English. In 1998about 2100 teasers of the first type wereamassed, while in the second some 280problems were devised for junior contes-tants and roughly 900 challenging ques-tions for senior contributors. A usefulbarometer of the increase in mathematical

PROBLEM CORNER


PPrroblem Corneroblem CornerContests from Romania, part 4

Paul Jainta (Schwabach, Germany)

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With the transition of the traditional jour-nal Zentralblatt für Didaktik der Mathematik(ZDM) to a web service, a more systematicEuropean extension of this originallyGerman-based enterprise took place. Themost prominent sign of this is the involve-ment of the EMS in the editing of the webversions, accompanied by the installationof some European backbones, helping withthe management of the input (local andnational), for the ZDM-database MATHDI(MATHematics DIdactics). Indeed, thecombined printed service ZDM, providingsurvey articles and a documentation part,has been split between the databaseMATHDI and an electronic journal whichpublishes the ZDM-surveys. Both offersare easily accessible through EMIS, with itsmore than 40 mirrors.

MATHDI on the webAs mentioned above, MATHDI developedfrom the literature documentation activi-ties in ZDM; its general aim is thus to pro-vide an information service and referenc-ing tool for education in mathematics andcomputer science, which was the tradition-al role of this part of ZDM. At present,MATHDI is the largest and most up-to-date world-wide database service in thisfield. Even before it was available on theweb, this part of ZDM had been availableelectronically via STN for quite a longperiod, under the English subtitleInternational Reviews on MathematicalEducation. Searches in MATHDI can thusbe extended back over a period of 25years.

The input for MATHDI has been takenfrom as many relevant documents as possi-ble, and currently journal articles frommore than 400 world-wide journals arecovered. These are complemented byinformation on textbooks and other mono-graphs, dissertations, conference papers,curricula, software and teaching aids in thetheory and practice of education in mathe-matics and computer science. As is com-mon practice with literature informationdatabases, all publications are announcedby bibliographic data, reviews or abstracts,and some additional information. Thegeneral texts and abstracts are usually inEnglish, although exceptionally they mayappear in French, German or Spanish.

At present the contents of MATHDIcomprise more than 100,000 entries, withan annual increase of 6000 entries; thisincludes about 21,000 reviews from rele-vant publications in the US. Current cov-erage of European publications is improv-ing, by means of a more reliable acquisi-tion and handling of national offersthrough co-operational units; here, in par-ticular, Portugal, Spain, Italy, Greece,Hungary and Yugoslavia should be men-tioned; hopefully, other European coun-

tries will catch up. World-wide coverage isalso becoming more complete, as can beseen for Mexico and Argentina. Moreover,cooperation with ERIC in the US providesan excellent representation of US-American publications in MATHDI.

As a consequence of the involvement ofthe EMS in MATHDI, it has been possibleto apply the same search software to thisservice as for its big brother ZentralblattMATH. This software was made availablefrom the partner MDC of ZentralblattMATH in Grenoble, and it is continuallyimproved there. As a result, a good inter-face and a selection of search menus of dif-ferent levels of expertise are available.Search results are obtained in TeX-sourcecode, but additional views such as displaysin Postscript can be chosen. Linking facil-ities, such as hypertext links to authors, aremade available and such facilities are grow-ing. Classification codes are assigned reg-ularly, according to an extended schemefor didactics and its reduced version inMSC 2000, and the free trial access forZentralblatt MATH has been copied forMATHDI. Anyone can do searches withouthaving a subscription, but in the latter caseonly the first three answers are displayed:just go to the URL http://www.emis.de andclick on the MATHDI-box.

The subjects covered by MATHDI arewider than a non-experienced user mightimagine. Information is provided onresearch in mathematical education,methodology of didactics of mathematics,mathematical instruction from elementaryschool to university teaching and teachertraining, elementary mathematics and itsapplications, education-relevant popularmathematics, education in computer sci-ence, basic pedagogical and psychologicalissues for mathematics and general educa-tion in science, and administrative issuessuch as curricula and teaching pro-grammes.

There is thus a variety of users for whichMATHDI should be an important tool.Almost everyone thinks of professionals indidactics for mathematics and computerscience, but it is also directed at all inter-ested in research and education in theseareas, at instructors and lecturers, and atteachers in all types of schools where math-ematics is relevant. The information avail-able from MATHDI is a basic infrastructurefor the professional work of educationaltechnologists, curriculum experts, and pol-icy-makers in education and educationaladministrators. On the formal side, it alsooffers an important tool for librarians andinformation specialists.

As an additional attractive offer, MATH-DI is distributed on CD-ROM. The mostrecent update is MATHDI 2001, and inspite of missing functionality for some link-ing facilities, it is a good solution for a

stand-alone installation. Since the data-base is comparatively small and spe-cialised, the price is kept low so that indi-viduals may consider buying the CD-ROMfor their private installation.

The electronic version of ZDMEfforts to keep the pricing of ZDM at a lowlevel led to the decision to separate thepart dealing with information on publica-tions from the journal-like part that pro-vides complete articles. Moreover,acknowledging the growing importance ofelectronic media and taking into accountthe reduced production costs for electron-ic versions, it was simultaneously decidedthat the full articles should be offered in anelectronic journal. Since traditional sub-scribers should not be ignored, this part isoffered as a dual journal as long as there issufficient interest in the print version.

As mentioned above, survey articles oneducation in mathematics and computerscience will mainly appear here, and it hasbeen no problem to convince authors invit-ed to write an article that they should nor-mally deliver their article in some not-too-sophisticated TeX-dialect. The journal isoffered through FIZ Karlsruhe, but there isan agreement with ElibM in EMIS thatback volumes will be made freely accessi-ble. This is already the case, and thosewho are interested are invited to look atEMIS and click on the journal at the end ofthe journals box in EMIS under the elec-tronic library. Frequent users may thenenter a bookmark to their directories.

This change of ZDM is a consequence ofthe transition of publications to newmedia, and profits a lot from the supportof the EMS. It furthermore underlines theengagement of the EMS in the develop-ment of education in mathematics.

Addresses: Gerhard König, ZDM, FIZ-Karlsruhe, 76344 Eggenstein-Leopoldshafen,Germany [[email protected]]; Bernd Wegner, TU Berlin, Fakultät II, Institutfür Mathematik, 10623 Berlin, Germany [[email protected]].

MATHDI


MAMATHDI - mathematics education on the WTHDI - mathematics education on the WebebGerhard König and Bernd Wegner

Bernt Wegner with Einstein

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

27-30: Acoustics and Music: Theory andApplications 2001 (AMTA 2001), SkiathosIsland, GreeceInformation:Web site: http://www.worldses.org/wses/calen-dar.htm27-30: Mathematics and Computers inBiology and Chemistry 2001 (MCBC 2001),Skiathos Island, GreeceInformation:Web site: http://www.worldses.org/wses/calendar.htm27-30: Mathematics and Computers inBusiness and Economics 2001 (MCBE 2001),Skiathos Island, GreeceInformation:Web site: http://www.worldses.org/wses/calendar.htm27-30: Automation and Information: Theoryand Applications 2001 (AITA 2001), SkiathosIsland, GreeceInformation:Web site: http://www.worldses.org/wses/calendar.htm

1-5: Aspects of Hyperbolic Geometry,Fribourg, SwitzerlandInvited speakers: include J. Anderson(Southampton), M. Boileau (Toulouse), M.Bourdon (Lille), B. Bowditch (Southampton), S.Buyalo (St. Petersburg), J.-L. Cathelineau(Nice), C. Drutu (Lille), J. Dupont (Aarhus), U.Hamenstädt (Bonn), G. Knieper (Bochum), U.Lang (ETZ Zürich), G. Martin (Auckland), J.-P.Otal (ENS Lyon/Orléans), J. Parker (Durham),F. Paulin (ENS Paris), N. Peyerimhoff(Bochum), A. Reid (Austin), J.-M. Schlenker(Toulouse), B. Stratmann (St. Andrews), E.Vinberg (Moscow)Scientific Board: Christophe Bavard (Uni.Bordeaux I), Gérard Besson (Uni. Grenoble I),Ruth Kellerhals (Uni. Fribourg), ViktorSchroeder (Uni. Zürich)Site: Department of Mathematics, University ofFribourg, SwitzerlandInformation:e-mail: [email protected] site: http://www.unifr.ch/math/conference2-5: SYM-OP-IS 2001 XXVIII YugoslavSymposium on Operations Research (SYM-OP-IS), BelgradeTopics: applications of operations researchmethods in organisational, technical, techno-logical, economic and other systemsProgramme: will be available by 21 SeptemberLanguages: Serbian and English

October 2001

September 2001

Programme committee: Sinisa Borovic(Belgrade), chairmanOrganizing committee: Svetomir Minic(Belgrade), chairman; Obrad Cabarkapa(Belgrade), secretary. The organisation thisyear will be coordinated by the MilitaryAcademy of the Yugoslav ArmySponsors: Yugoslav Army, YugoslavOperational Research Society, Faculty ofTransport and Traffic Engineering, MihajloPupin Institute, Economics Institute, Faculty ofEconomics, Faculty of Mining and Geology,Faculty of Organizational Sciences, Faculty ofMechanical Engineering and Faculty ofMathematicsProceedings: refereed proceedings of selectedpapers will be published prior to the confer-ence and available to all participantsSite: Military Academy, Neznanog junaka 28,Belgrade, YugoslaviaDeadlines: already expired Information: contact SYM-OP-IS 2001Organising Committee, Neznanog junaka 28,VP 2102-4, 11002 Belgradee-mail: [email protected] site: http://www.vj.yu/symopis/6: 1st Annual Arf Lecture, Ankara, Turkey[in memory of Cahit Arf]Speaker: Prof. Dr. Gerhard Frey (Essen)Title: Bauer Groups and Data SecurityOrganisers: Department of Mathematics atMiddle East Technical University, Ankara, andTurkish Mathematics FoundationAdvisory board: M. G. I. Keda, R. Langlandsand P. Roquette

2-8: International Centre for MathematicalSciences Workshop on Classical andQuantum Integrable Systems and theirSymmetries, Edinburgh, UK[satellite workshop of the Isaac NewtonInstitute for Mathematical Sciences programmeon Integrable Systems (July to December2001)]Theme: the emphasis will be on quantum inte-grable systems and the symmetry approach,where recent progress has been rapid andintenseAim: to analyse current developments and toencourage the exchange of ideas betweenresearchers in the field of integrable systemsand those working in representation theory,string theory and other areas of modern math-ematics and physicsMain speakers: expected to include JohnCardy (UK), Ivan Cherednik (USA), PatrickDorey (UK), Jonathan Mark Evans (UK),Kentaro Hori (USA), Michio Jimbo (Japan),Hitoshi Konno (Japan), André Leclair (USA),Jean-Michel Maillet (France), Barry McCoy(USA), Eugene Mukhin (USA), Maxim Nazarov(UK), Nikita Nekrasov (France), FrancescoRavanini (Italy), N. Reshetikhin (USA), RyuSasaki (Japan), Junichi Shiraishi (Japan),Evgueni Sklyanin (UK), Feodor Smirnov(France), Kanehisa Takasaki (Japan), GerardWatts (UK), Alexei Zamolodchikov (France)Organising committee: Ed Corrigan (York),Chris Eilbeck (Heriot-Watt), Tetsuji Miwa(Kyoto), Robert Weston (Heriot-Watt)Sponsors: Engineering and Physical Sciences

December 2001

Research Council and Isaac Newton Institutefor Mathematical SciencesGrants: limited funds (worth 250 pounds perperson) to support the attendance of mathe-maticians or research students in UK universi-tiesSite: Heriot-Watt UniversityDeadlines: for registration, 1 OctoberInformation:e-mail: [email protected] site: www.ma.hw.ac.uk/icms/current/cqis

1 February-30 April: Special ResearchTrimester on Dynamical Systems, PisaTopics: non-uniformly and partially hyperbolicsystems, quasi-periodic orbits, holomorphicdynamics and foliations, interaction betweendynamical systems and biology, interactionbetween dynamical systems and physics (includ-ing celestial mechanics)Main speakers: M. Brunella (CNRS Dijon), A.Chenciner (Universit de Paris VII), D.Dolgopyat (Pennsylvania State University), C.Favre (CNRS, Paris), R. Ferriere (EcoleNormale Paris), G. Forni (PrincetonUniversity), A. Giorgilli (Milano Bicocca), V.Yu. Kaloshin (Princeton University), R.Krikorian, J. Laskar (CNRS-IMC Paris), P. LeCalvez (Paris XIII), C. Liverani (Roma TorVergata), S. Luzzatto (Imperial CollegeLondon), S. Marmi (Udine and SNS Pisa), J.Mather (Princeton University), G. Moreira(IMPA), J. Palis (IMPA), J. Rivera (SUNY), D.Sauzin (CNRS, Paris), S. Smirnov (KTHStockholm), L. Stolovich (Université PaulSabatier,Toulouse), M. Viana (IMPA), J.-C.Yoccoz (College de France), L. S. Young(Courant Institute)Programme committee: S. Marmi (Italy), J.Mather (USA), J. Milnor (USA), J. Palis (Brasil),J.-C. Yoccoz (France)Organising committee: C. Carminati (Italy), G.Da Prato (Italy), M. Giaquinta (Italy), S. Marmi(Italy)Sponsors: Istituto Nazionale di AltaMatematica, Scuola Normale SuperioreSite: Scuola Normale Superiore (Piazza deiCavalieri 7, Pisa, Italy)Grants: a limited number of grants supportingpreferably long-term visits (at least one month)will be available for PhD students and PostDocs. Applications should be made as soon aspossibleInformation:e-mail: [email protected] site: http://www.math.sns.it/degiorgi/dynsys/11-15: Neural Networks and Applications(NNA 02), Interlaken, SwitzerlandInformation:Web site: http://www.worldses.org/wses/calendar.htm11-15: Fuzzy Sets and Fuzzy Systems (FSFS02), Interlaken, SwitzerlandInformation:Web site: http://www.worldses.org/wses/calendar.htm11-15: Evolutionary Computations (EC 02)Interlaken, SwitzerlandInformation:Web site: http://www.worldses.org/wses/calendar.htm

18-19: Workshop on Under- and Over-deter-mined Systems of Algebraic and DifferentialEquations, Karlsruhe, GermanyTheme: algebraic or differential equations,

March 2002

February 2002

CONFERENCES


FForthcoming conferorthcoming conferencesencesCompiled by Kathleen Quinn

computer algebra, numerical analysisScope: computational approaches to under-and over-determined systems. Submissions areexpected on theory and applications, algo-rithms and software. The workshop will be ofan interdisciplinary nature; the intention is tobring together researchers from many differentfields in order to foster communicationbetween different communitiesTopics: all aspects of under- and over-deter-mined systems: completion, exact or approxi-mate solutions, structure analysis, symbolicand/or numerical treatment, Gröbner or invo-lutive bases for polynomial or differential sys-tems, differential algebraic equations (DAEs),symmetry analysis, applications in all fields ofmathematics or sciencesCall for papers: downloadable from the con-ference web siteProgramme committee: J. Calmet (Karlsruhe,Workshop Chair), V. P. Gerdt (Dubna, Chair),W. M. Seiler (Mannheim, Chair), J. Apel(Leipzig), G. Carra Ferro (Catania), G.Czichowski (Greifswald), L. Lambe(Bangor/Rutgers), E. L. Mansfield(Canterbury), B. Mourrain (Sophia-Antipolis),P. J. Olver (Minneapolis), E. Pankratiev(Moscow), V. P. Shapeev (Novosibirsk), J.Tuomela (Joensuu)Proceedings: submissions are not formally ref-ereed and may be submitted later elsewhere.Informal proceedings will be distributed to par-ticipants. Authors of outstanding submissionswill be invited to contribute to a special issue ofthe AAECC journalSite: University of KarlsruheDeadlines: for submission, 1 December; forregistration, 15 FebruaryInformation:Web site: http://iaks-www.ira.uka.de/iaks-calmet/ADE21-22: Eighth Rhine Workshop on ComputerAlgebra, Mannheim, GermanyAims: to serve as a regional forum forresearchers in the field, and in particular tooffer an opportunity to young researchers andnewcomers to present their workTopics: all aspects of computer algebra, fromtheory to applications and systemsCall for papers: downloadable from the con-ference web siteProgramme committee: H. Kredel(Mannheim, Workshop Chair), W. M. Seiler(Mannheim, Chair), M. Bronstein (SophiaAntipolis), R. Buendgen (Boeblingen), J.Calmet (Karlsruhe), J. Della Dora (Grenoble),A. Cohen (Eindhoven), J. C. Faugere (Paris), V.P. Gerdt (Dubna), M. MacCallum (London), D.Mall (Zurich), E. L. Mansfield (Canterbury), T.Recio (Santander), M. Schlichenmaier(Mannheim), W. K. Seiler (Mannheim), T.Sturm (Passau), C. Traverso (Pisa), W. Werner(Heilbronn), F. Winkler (Linz), E. Zerz(Kaiserslautern)Proceedings: submissions are not formally ref-ereed and may be submitted later elsewhere.Informal proceedings will be distributed to par-ticipantsSite: University of MannheimDeadlines: for submissions, 1 December; forearly registration, 15 February; late registrationat higher fee until start of conferenceInformation:Web site: http://www.uni-mannheim.de/RWCA26-4 April: International Centre forMathematical Sciences EuroSummer Schooland Instructional Conference onCombinatorial Aspects of MathematicalAnalysis, Edinburgh, UKAim: to instruct young mathematicians in top-ics involving combinatorial ideas in mathemati-cal analysis (10-day course)

Topics: combinatorial number theory, combi-natorial methods in convexity, combinatorialmethods in harmonic analysis, concentration ofmeasure, geometric inequalities, Ramsey meth-ods in Banach spacesMain speakers: Keith Ball (UC London),Franck Barthe (Marne la Valle), WilliamBeckner (U Texas, Austin), Béla Bollobás(Cambridge and Memphis), Anthony Carbery(Edinburgh), Michael Christ (U of California,Berkeley), Apostolos Giannopoulos (U of Crete,Heraklion), Gil Kalai (Hebrew University ofJerusalem), Michel Ledoux (Toulouse), TedOdell (Texas A&M), Imre Ruzsa (HungarianAcademy of Science), Gideon Schechtman(Weizmann Institute), Terence Tao (UCLA),Christoph Thiele (UCLA)Programme: the conference will take the formof a series of courses, each comprising 2, 3 or 4one-hour lectures. The courses are structuredso that the more basic material is presentedduring the first week, leading to moreadvanced lectures in the second weekOrganising committee: Tony Carbery(Edinburgh), Mike Christ (UC Berkeley), TimGowers (Chair, Cambridge), Vitali Milman (TelAviv), Terry Tao (UCLA)Sponsors: EC Framework V and theEngineering and Physical Sciences ResearchCouncil (EPSRC) of the UKSite: the James Clerk Maxwell Building of theUniversity of EdinburghGrants: some funding available to assist EUnationals who are under the age of 35. ICMSalso has funds from the LMS to assist UKmathematicians. Please refer to the website forfull detailsDeadlines: for registration, 16 DecemberInformation:e-mail: [email protected] site: www.ma.hw.ac.uk/icms/meetings/2002/cama February 2002

13-17: 34th Journées de Statistique, FrenchStatistical Society, Brussels and Louvain-la-Neuve, BelgiumThemes: statistical analysis of functional data,actuarial and financial econometrics, resam-pling methods, non-parametric inference andmodelling, voice and writing recognition, epi-demics, genomics as well as mathematical sta-tisticsTopics: sequential analysis, multivariate robustanalysis, classification, applications of copulasin insurance and finance, change point detec-tion, censored and missing data, Gibbs sam-pling, frontier estimation, extreme value theo-ry, stochastic finance, data and files merging,the 150th anniversary of the first InternationalCongress of Statistics, statistical software fortext analysis, algebraic methods in time seriesanalyses, non-standard mathematical methodsin statistics and regression analysis, latent vari-ables models, hierarchical Bayes models, sto-chastic models in telecommunication, largepanels of time series data, deconvolution prob-lems, stochastic processes in functional spaces,data depth, wavelets and time series, chemo-metrics, environmetrics, statistics in astronomy,statistics and language, continuous time sto-chastic processes, spatial and directional statis-tics, risk theory, image processing and neuralnetworksLanguages: French and EnglishProgramme committee: Marc Hallin (ULB),Chairman Lucien Birgé (Paris 6), HenriCaussinus (Toulouse), Christian Genest (Laval),Irène Gijbels (UCL), Ludovic Lebart (ENST-CNRS), Bernard Ycart (Paris 5)

May 2002

Organizing committee: Léopold Simar (UCL),Chair, Jean-Jacques Droesbeke (ULB), ClaudeCheruy (INS), Michel Denuit (UCL), CatherineVermandele (ULB), Bernadette Govaerts(UCL), Thérèse Lekeux (ULB), ClaudiaLemoine (UCL), Sophie Malali (UCL)Grants: for students, statisticians from develop-ing countriesDeadlines: for registration at lower fee, 15March; for abstracts, 15 January Information:e-mail: [email protected] site: www.stat.ucl.ac.be/JSBL200225-3: XXII International Seminar on StabilityProblems for Stochastic Models (SPSM) andSeminar on Statistical Data Analysis (SDA),Varna, BulgariaTopics of SPSM: limit theorems of probabilitytheory, theory of probability metrics, asymptot-ic statistics, limit theorems for stochasticprocesses, queueing theory, applications ofprobability theory, insurance and financialmathematicsTopics of SDA: robust statistics, statistical algo-rithms, application of stochastic models in theindustryOrganizing Committee: Vladimir Zolotarev(Russia, Chair of SPSM), Dimitr Vandev(Bulgaria, Chair of SDA)Site: International Home of Scientists FredericJoliot CurieInformation:Web site: http://stabil.fmi.uni-sofia.bg27-29: Spring School on Frobenius Manifoldsin Mathematical Physics, Enschede, TheNetherlandsInformation:e-mail: [email protected]: 6th Congress of SIMAI, Chia Laguna,SardiniaTopics: applied mathematics and the applica-tions of mathematics in industry, technology,environment and societyInformation:Web site:http://www.iac.rm.cnr.it/simai/simai2002

4-13: 3rd Linear Algebra Workshop BLED2002, Bled, SloveniaMain theme: the interplay between operatortheory and algebraProgramme: talks in morning sessions, work insmaller groups in the afternoonsProgramme committee: R. Drnovek (SLO), L.Grunenfelder (CAN), T. Koir (SLO), M.Omladiè (SLO), H. Radjavi (CAN) Organizing committee: R. Drnovek (SLO), T.Koir (SLO), M. Omladiè (SLO)Site: Hotel Golf, BledDeadlines: for registration, 1 DecemberInformation:e-mail: [email protected],Web site: http://www.ijp.si/ftp/pub/STOp/law/5-9: Conference in Honour of Hans Wallin,Umea, SwedenThemes: 1) potential theory, function spaces,approximation theory and related topics; 2)mathematics educationAim: to bring together researchers in some ofthe areas in which Hans Wallin has been work-ing, in order to get a picture of some currentresearch on these areas. We hope that the con-ference will be useful for a comparatively broadaudience, and we especially encourage Ph.D.students to participateMain speakers: A. Ambroladze (Sweden), D.Broomhead (Great Britain), L. Carleson(Sweden), Z. Cieselski (Poland), S. Janson(Sweden), D. S. Lubinsky (South Africa), P.

June 2002

CONFERENCES


Mattila (Finland), E. B. Saff (USA), J.-O.Stromberg (Sweden), A. Teplayev (USA), O.Bjorkqvist (Finland), G. Gjone (Norway), M.Niss (Denmark)Programme committee: Alf Jonsson, JohanLithner, Kaj Nyström, Tord Sjödin, PeterWingrenOrganising committee: Margareta Brinkstam,Jan Gelfgren, Tord Sjödin, Britt-Marie StockeSite: Umea university, Umea, North of SwedenDeadlines: for registration, 31 OctoberInformation: e-mail: [email protected] site: http://www.math.umu.se/aktuellt/HWkonferens.htm10-16: Aarhus Topology 2002, Aarhus,DenmarkTheme: algebraic topologyMain speakers: include Raoul Bott (USA),Ralph Cohen (USA), Yakov Eliashberg (USA),Jesper Grodal (USA), Karsten Grove (USA),Lars Hesselholt (USA), Mike Hopkins (USA),Wolfgang Lück (Germany), Mike Mandell(USA), Fabien Morel (France), Bob Oliver(France), Erik K. Pedersen (USA), Zoltan Szabo(USA), Ulrike Tillmann (UK), Vladimir Turaev(France)Organising committee: Johan Dupont (Chair),University of Aarhus; Hans Jørgen Munkholm,SDU, Odense University; Lars Hesselholt, MIT,USA; Lisbeth Fajstrup, Aalborg UniversitySite: University of AarhusDeadline: for registration, to be announced atour web siteInformation:Web site: http://www.imf.au.dk/AT2002/17-19: BEM 24, 24th InternationalConference on Boundary Element Methodsand Meshless Solutions Seminar, Sintra,PortugalOrganiser: Wessex Institute of Technology,UK and the University of Coimbra, PortugalSponsors: International Society of BoundaryElements (ISBE) and the International Journalof Engineering Analysis with BoundaryElementsDeadline: for papers, 15 JanuaryInformation: contact Conference Secretariat ,BEM02, Wessex Institute of Technology,Ashurst Lodge, Ashurst Southampton, SO407AA, UK, tel: 44 (0) 238 029 3223, fax: 44 (0)238 029 2853e-mail: [email protected] site: http://www.wessex.ac.uk/conferences/2002/be02/index.html27- 3 July: Fifth International Conference onCurves and Surfaces, Saint-Malo, FranceInformation:e-mail: [email protected] site: &http://www-lmc.imag.fr/saint-malo/

16-22: 7th International Spring School:Nonlinear Analysis, Function Spaces andApplications (NAFSA 7), Prague, CzechRepublicInformation:e-mail: [email protected] site: http://www.math.cas.cz/~nafsa7[For details, see EMS Newsletter 40]

3-10: Logic Colloquium 2002 (ASL EuropeanSummer Meeting), Münster, GermanyMain speakers: Jeremy Avigad (Pittsburgh,PA), Arnold Beckmann (Münster), TimCarlson (Columbus, OH), Robert Constable(Ithaca, NY), Kosta Dosen (Toulouse), Moti

August 2002

July 2002

Gitik (Tel Aviv), Volker Halbach (Konstanz),Bakhadyr Khoussainov (Auckland), SteffenLempp (Madison, WI), Toniann Pitassi(Tucson, AZ), Thomas Scanlon (Berkeley, CA),Ralf Schindler (Wien), Patrick Speissegger(Madison, WI), Katrin Tent (Würzburg), LevBeklemishev (Moscow/Utrecht), Steven Cook(Toronto, ON), Olivier Lessmann (Chicago,IL), Simon Thomas (Piscataway, NJ)Sessions: Computability theory [organisers:Steffen Lempp (Madison, WI), ManuelLerman (Storrs CT), Andrea Sorbi (Siena)],Frank Stephan (Heidelberg), IskanderKalimullin (Kazan), Sebastiaan Terwijn(Amsterdam), Charles McCoy (Madison, WI),Vasco Brattka (Hagen), Ivan Soskov (Sofia),Elias Fernandez-Combarro Alvarez (Oviedo);Non-monotonic logic [organisers: KarlSchlechta (Marseille), Krister Segerberg(Uppsala)], Nick Asher (Austin, TX),Alexander Bochman (Holon), Dov Gabbay(London), Daniel Lehmann (Jerusalem), DavidMakinson (Paris), Rohit Parikh (New York,NY), Renata Wassermann (São Paulo); Set the-ory [organisers: Alessandro Andretta (Torino),Sy Friedman (Vienna)], David Aspero(Barcelona), Doug Burke (Las Vegas, NV),James Hirschorn (Helsinki), Ilijas Farah (NewYork, NY), Rene Schipperus (Beer-Sheva),Paul Larson (Toronto, ON), Su Gao(Pasadena, CA)Programme committee: Klaus Ambos-Spies(Heidelberg), Sam Buss (San Diego CA), ZoéChatzidakis (Paris), Alekos Kechris (Pasadena,CA, Chair), Peter Koepke (Bonn), PeterKomjath (Budapest), Manuel Lerman (Storrs,CT), Vann McGee (Cambridge, MA), WolframPohlers (Münster), Michael Rathjen (Leeds),Krister Segerberg (Uppsala), Boris Zilber(Oxford)Organising committee: Manfred Burghardt(Bonn), Justus Diller (Münster), Peter Koepke(Bonn), Benedikt Löwe (Bonn), MichaelMöllerfeld (Münster), Wolfram Pohlers(Münster, Chair), Andreas Weiermann(Münster)Sponsors: Association of Symbolic LogicInformation:Web site: http://www.math.uni-muenster.de/LC2002/10-11: Colloquium Logicum 2002, Münster,Germany[satellite conference of Logic Colloquium 2002]Main speakers: Toshiyasu Arai (Hiroshima),Joan Bagaria (Barcelona), Andre Nies(Chicago, IL), Martin Otto (Swansea), CharlesParsons (Harvard), Anand Pillay (Urbana-Champaign, IL), Michael Rathjen (Leeds),Johan van Benthem (Amsterdam/Stanford CA)Organizing & scientific committee: JustusDiller (Münster), Peter Koepke (Bonn),Benedikt Loewe (Bonn), Wolfram Pohlers(Münster, Chair), Christian Thiel (Erlangen),Wolfgang Thomas (Aachen), AndreasWeiermann (Münster)Organizer: DVMLGSponsors: Deutsche Vereinigung fürMathematische Logik und für Grundlagen derExakten Wissenschaften (DVMLG)Information:Web site: http://wwwmath.uni-muenster.de/LC2002/25-30: Wireless and Optical Communications(WOC 02), Miedzyzdroje, PolandInformation:Web site: http://www.worldses.org/wses/calendar.htm25-30: Nanoelectronics, Nanotechnologies(NN02), Miedzyzdroje, PolandInformation:Web site: http://www.worldses.org/wses/calendar.htm

4-7: International Conference on DynamicalMethods for Differential Equations,Valladolid, SpainTheme: the focus is on those recent advances intopological methods and ergodic theory whichare relevant to the analysis of ordinary differen-tial equations, partial differential equations andfunctional equations, as well as on their applica-tions to science and technologyScientific committee: Amadeu Delshams(Spain), Russell Johnson (Italy), Rafael Obaya(Spain), Rafael Ortega (Spain)Organising committee: Ana I. Alonso, SylviaNovo, Carmen Núñez, Rafael Obaya, Jesús Rojo(all Universidad de Valladolid, Spain)Main speakers: L. Diaz (Brazil), A. Jorba(Spain), U. Kirchgraber* (Switzerland ), P.Kloeden (Germany), R. Krikorian (France), Y.Latushkin (USA), R. de la Llave (USA), R.Markarian (Uruguay), W. de Melo* (Brazil ), J.A. Rodríguez (Spain), G. R. Sell (USA), Y. Yi(USA) (* to be confirmed)Proceedings: selected papers from the confer-ence will be published in a special issue ofJournal of Dynamics and Differential EquationsSite: Castillo de la Mota (Medina del Campo),Valladolid, SpainGrants: a limited number of financial grants areavailable for graduate and doctoral students Deadline: for pre-registration and submissionof abstracts, 28 FebruaryInformation:e-mail: [email protected] site: http://wmatem.eis.uva.es/~dmde02/February 20035-7: 4th IMACS Symposium on MathematicalModelling, Vienna, AustriaAim: to give scientists and engineers using ordeveloping models or interested in the develop-ment or application of various modelling toolsan opportunity to present ideas, methods andresults and discuss their experiences or prob-lems with experts of various areas of specialisa-tionScope: theoretic and applied aspects of the var-ious types of mathematical modelling (equa-tions of various types, automata, Petri nets,bond graphs, qualitative and fuzzy models, etc.)for systems of dynamic nature (deterministic,stochastic, continuous, discrete or hybrid withrespect to time, etc.). Comparison of modellingapproaches, model simplification, modellinguncertainties, port-based modelling, and theimpact of items such as these on problem solu-tion, numerical techniques, validation, automa-tion of modelling and software support formodelling, co-simulation, etc. will be discussedin special sessions as well as applications ofmodelling in control, design or analysis of sys-tems in engineering and other fields of applica-tion. Presentations of modelling and simulationsoftware and a book exhibition will be organ-isedOrganiser: Division for Mathematics of Controland Simulation (E114/3) at Vienna University ofTechnology.Chair: of IPC, Univ. Prof. Dr. Inge TrochSite: Vienna University of TechnologyDeadlines: for submission of abstracts, 15 May2002; for notification of authors, 15 October2002, for full paper, 1 December 2002Information:Contact Univ. Prof. Dr. Inge Troch, ViennaUniversity of Technology, WiednerHauptstrasse 8 - 10 A-1040 Wien, Austria, tel:+431-58801-11451, fax: +431-58801-11499e-mail: [email protected] site: http://simtech.tuwien.ac.at/ MATHMOD

September 2002

CONFERENCES


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

H. Bass and A. Lubotzky, Tree Lattices,Progress in Mathematics 176, Birkhäuser, Boston,2001, 233 pp., DM 108, ISBN 0-8176-4120-3and 3-7643-4120-3 This is an advanced book on geometricalmethods in group theory and combinatorialgroup theory. The authors study groups ofautormorphisms of locally finite trees stress-ing parallels with the theory of Lie groups.Applications to combinatorics and numbertheory are given. In a sense, this is a continu-ation of the classical monograph of J. P. Serre,Trees (Springer, 1980), and indeed the book isdedicated to Serre in admiring tribute. (jnes)

V. I. Bernik and M. M. Dodson, MetricDiophantine Approximation on Manifolds,Cambridge Tracts in Mathematics 137, CambridgeUniversity Press, Cambridge, 1999, 172 pp.,£27.50, ISBN 0-521-43275-8 This book deals with metric Diophantineapproximations on smooth manifolds embed-dable in a Euclidean space. The text startswith an overview of basic problems and resultson Diophantine approximation in one dimen-sion, and the necessary analytic backgroundon such manifolds needed for transition tometric aspects of the Diophantine approxima-tion on manifolds. In the second chapter,they extend Khintchines and Groshevs theo-rem on simultaneous approximation to cer-tain manifolds, and prove a conjecture of A.Baker related to his previous extension ofSprinduks theorem. The next three chap-ters are devoted to Hausdorff dimension,especially to different techniques developedfor upper and lower bounds of the associatednull sets. Chapter 6 contains an account of p-adic Diophantine approximation on mani-folds. The final chapter deals with variousapplications, such as the wave equation andthe rotation number. Each chapter ends withnotes containing material for further readingand historical comments. Especially valuableare those connected with results spread injournals within the former Soviet Union.

This book can be recommended not only tothose interested in number-theoretic aspects,but also to those whose interests lie in topicsrelated to dynamical systems. The book isessentially self-contained and very readable.(p)

E. D. Bloch, Proofs and Fundamentals. A FirstCourse in Abstract Mathematics, Birkhäuser,Boston, 2000, 424 pp., DM 108, ISBN 0-8176-4111-4 and 3-7643-4111-4 This book presents an elementary abstractbasis of mathematics in three natural parts:logic, basic set notions and methods, and basicmathematical structures; the names of theseparts are: Proofs, Fundamentals, Extras.

Part 1 is an explanation of elementarynotions of first-order logic and an expositionof how to use them to produce logically cor-rect arguments and proofs. Part 2 is a pre-

sentation of elementary informal set theory;the chapters are: Sets, Functions, Relations,Infinite and Finite Sets. Part 3 introducessuch basic structures as groups, partiallyordered sets, lattices, positive integers, ratio-nal and real numbers, and presents computa-tions with finite sets. The text has over 400exercises, with hints for selected ones.

The presentation of the subject is sufficient-ly precise, and motivations of introducednotions are discussed. The book can be seenas a solid base of general mathematicalnotions and methods. (jml)

D. Bao, S.-S. Chern and Z. Shen, AnIntroduction to Riemann-Finsler Geometry,Graduate Texts in Mathematics 200, Springer,New York, 2000, 431 pp., DM 98, ISBN 0-387-98948-XThis book offers the most modern treatmentof the topic and will attract both graduate stu-dents and a broad community of mathemati-cians from various related fields. The authorsstart with a short but informative historicalexposition and with a guide to the contents ofthe book. They show how the original moti-vation of the topic came from physics andmention other fields of applications, such asecology and biology.

Whereas a Riemann structure is a smoothfamily of inner products, a Finsler structurecan be viewed as a smooth family of generalMinkowski norms along a manifold. One ofthe main goals of this book seems to be toanswer the following question: to what extentcan the basic notions and theorems from glob-al Riemann geometry be generalised toFinsler structures, either general ones, or spe-cialised ones?

The exposition is based mainly on the so-called Chern connection and its curvature,which are studied in detail in the first part ofthe book. Through this approach, one learnsthat the following items make sense in a gen-eral Finsler geometry: Schurs lemma, thegeneralised Gauss-Bonnet theorem, Jacobifields, the Hopf-Rinow theorem, index formand the Bonnet-Myers theorem, cut and con-jugate loci and Synges theorem, the Cartan-Hadamard theorem and Rauchs first theo-rem. These topics are studied in the secondpart of the book. The final part of the book isdevoted to special Finsler spaces, and is con-nected with the names of H. Akbar-Zadeh, F.Brickel, A. Deicke and Z. Szabó. One chapterillustrates the general concepts in pureRiemannian geometry.

Full credit is given to the traditionalJapanese, Romanian and other schools ofFinslerian geometry, as well as to the mathe-maticians who made isolated contributions tothe topic. The authors seem anxious to be fairas concerns citations. The book includes animpressive 393 exercises and some examplesusing Maple. (ok)

J. F. Bonnans and A. Shapiro, PerturbationAnalysis of Optimization Problems, SpringerSeries in Operations Research, Springer, New York,2000, 601 pp., DM 139, ISBN 0-387-98705-3 This monograph presents a compact overview

of perturbation analysis of continuous optimi-sation problems. The framework of the bookis abstract: the optimisation problem consid-ered is parametrised by a parameter varyingin a Banach space, theoretical results are for-mulated for Banach spaces, and the duality ofBanach spaces is the main tool for proofs.The book investigates the continuity and dif-ferentiability of the optimal value and the setof all optimal solutions with respect to theparameter.

The book has seven chapters. The intro-ductory chapter describes the topics relationto other fields, such as non-linear program-ming, optimal control and variational inequal-ities. Chapter 2 contains the backgroundmaterial needed for a full understanding ofthe text: basic functional analysis, duality inBanach spaces, recession cones, directionaldifferentiability of a function, tangent cones,the basic elements of multi-function theory,properties of convex functions, and conjugate(Fenchel) and Lagrangian duality. The thirdchapter discusses first- and second-order opti-mality conditions for the optimisation prob-lems; this is based on Lagrangian dualityextended to a generalized Lagrangian.Chapter 4 is the main part of the book, givinga comprehensive study of stability and sensi-tivity analysis of an optimisation problemparametrised by a parameter varying in aBanach space. Subsections present a first-order differentiability analysis of the optimalvalue and a discussion of the quantitative sta-bility of optimal solutions and Lagrange mul-tipliers, followed by a second-order analysis inLipschitz and Hölder stable cases. A specialsubsection treats second-order analysis infunction spaces. The fifth chapter bringsadditional materials and applications, includ-ing a discussion of variational inequalities,non-linear programming, semi-definite pro-gramming and semi-infinite programming.Chapter 6 is relatively self-contained, and con-siders optimisation problems based on partialdifferential equations, such as the Dirichletproblem, optimal control of a semi-linearelliptic equation, the state-constrained opti-mal control problem and the obstacle prob-lem. The final chapter gives bibliographicalnotes and references for additional reading.

The monograph contains the present stateof knowledge of the topic. The text is writtenin a clear and correct manner and is arrangedin order to help the reader. The book can berecommended for graduate students,researchers and practitioners in optimisationtheory. (pl)

X. Buff, J. Fehrenbach, P. Lochak, L.Schneps and P. Vogel, Espaces de modules descourbes, groupes modulaires et théorie deschamps, Panoramas et Synthéses 7, SociétéMathématique de France, Paris, 1999, 143 pp.,FRF 120, ISBN 2-85629-073-6This book consists of lecture notes from athree-day workshop. The first two parts (Élé-ments de géométrie des espaces de modulesdes courbes, and Groupoïdes fondamentauxdes espaces de modules en genre 0 et cate-gories tensorielles tressées) are devoted to thestructure of Teichmüller space of Riemannsurfaces, while the third, more or less inde-pendent, part (Invariants de Witten-Reshetikhin-Turaev et théories quantiquesdes champs) is devoted to quantum groupinvariants of links and 3-dimensional mani-folds. Topics of these sections intersect in thenotion of braided or strict monoidal category.

In the first part we find a definition of

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Recent booksRecent booksedited by Ivan Netuka and Vladimír Sou³ek

Teichmüller space Tg,n of Riemann surfaces ofgenus g with n distinct labelled points andalso two alternative descriptions of this space in terms of hyperbolic metrics and by sub-groups of PSL(2, R). The space Tg,n is topolo-gised by the Fenchel-Nielsen coordinateswhich are then used to describe a compactifi-cation CMg,n of the corresponding modulispace Mg,n by adding degenerate points. Thedecorated Teichmüller space Tg,n, intimatelyrelated to graph complexes of Kontsevich, isalso studied.

The central object of the second part is thecompleted Teichmüller groupoid T0,n, that is,the profinite completion of the fundamentalgroupoid of the moduli space CM0,n based at aneighborhood of most degenerate configura-tions of CMg,n. This groupoid admits a natur-al action of the Galois group Gal(Cl(Q)/Q).The Grothendieck-Teichmüller group GT,isomorphic for n ≥ 4 to the automorphismgroup of T0,n, is also introduced. The cele-brated conjecture that Gal(Cl(Q)/Q) is iso-morphic to GT is then formulated.

The third part explicates how a ribbonbraided monoidal category induces a linkinvariant or, more generally, an invariant ofpairs (M, L) of a 3-manifold M and a link Lembedded in M. Particular examples of thesebraided monoidal categories are provided byrepresentations of quantum groups. The restof this part indicates how (in some cases) theseinvariants are equivalent to topological quan-tum field theories.

The concise and self-contained exposition isprimarily aimed at non-specialists and gradu-ate students, but a specialist might also finduseful information and proofs that are diffi-cult to locate in the existing literature. (mm)

D. Bump, Algebraic Geometry, World Scientific,Singapore, 1998, 218 pp., £26, ISBN 9-810-23561-5 This book is designed as a text for a one-yearcourse in basic algebraic geometry at the grad-uate level. It is divided into two parts.

The first part is devoted to the general the-ory of affine varieties over an algebraicallyclosed field. The theory of affine varieties issystematically and clearly explained in thefirst part of the book, and some prerequisitiesfrom algebra are included. The dimensionand products of affine varieties are studiedand a section is devoted to theory of projectivevarieties. The second part is devoted to thetheory of algebraic curves, and the main topicis the theory of complete non-singular curves.The questions studied are the ramificationproblem, extensions and completions of afield with respect to a valuation, differentials,residues and the Riemann-Roch theorem.Special attention is paid to the theory of ellip-tic curves and their properties and to the zetafunction of a curve. As mentioned by theauthor, the most significant omission is inter-section theory. Each chapter has a set of exer-cises illustrating its content and helping a bet-ter understanding of the theory. As a whole,the textbook offers a good introduction toalgebraic geometry. (jbu)

E. B. Burger, Exploring the Number Jungle: AJourney into Diophantine Analysis, StudentMathematical Library 8, American MathematicalSociety, Providence, 2000, 151 pp., US$20,ISBN 0-8218-2640-9 This book provides an excellent survey ofDiophantine analysis. It is surprising how theauthor is able to explain many interesting and

famous facts in the topic in just 150 pages.Twenty modules (chapters) are followed bymany exercises, formulated as lemmas or the-orems with many hints, and remarks andquestions.

The book contains the Dirichlet andHurwitz theorems, continued fractions, theMarkoff spectrum, the Pell equation, Liouvilleand Roths results, elliptic curves, the geome-try of numbers and its application to simulta-neous diophantine approximations, uniformdistribution, and p-adic analysis. Many proofsare omitted (the Mordell and Mazur theo-rems), some are only sketched (the Roth theo-rem), while some are presented with fullproofs (the Lagrange theorem, the Minkowskiconvex body theorem, the linear forms theo-rem and Kroneckers theorem). The bookincludes an index and references for furtherreading, and can be warmly recommended tostudents and university teachers. (bn)

M. Capiñski and T. Zastawniak, ProbabilityThrough Problems, Problem Books inMathematics, Springer, New York, 2001, 257 pp.,DM 109, ISBN 0-387-95063-X This book is intended to accompany anundergraduate course in probability; the onlyprerequisite are basic algebra (including ele-ments of set theory) and calculus. A brief sur-vey of the terminology and notation of set the-ory and calculus is provided at the beginningof the book.

The body of the book is divided into twelvechapters. Chapter 1 concerns elements ofmodelling of random experiments. Chapters2-6 are devoted to classical probability spacesand related combinatorial problems, fieldsand σ-fields of sets, finitely and countablyadditive probability. Chapter 7 focuses onconditional probability and independence,Chapter 8 concerns random variables andtheir distribution functions, while Chapter 9offers problems on their expectations andvariances and is devoted to conditional expec-tations. Charateristic functions are the subjectof Chapter 11, while Chapter 12 containsproblems connected with the laws of largenumbers and the central limit theorem, withan emphasis on consequences and applica-tions. Each chapter is divided into theory(including basic notions and theorems), prob-lems, hints and solutions. Hints are given forall problems and fully worked solutions for themajority of them. All problem sectionsinclude expository material. (mahu)

S. Choi, The Convex and ConcaveDecomposition of Manifolds with RealProjective Structures, Mémoires de la SociétéMathématique de France 78, Société Mathématiquede France, Paris, 1999, 102 pp., FF 150, ISBN 2-85629-079-5 This small book is devoted to a study of prop-erties of real manifolds with a flat projectivestructure. The author introduces a notion ofi-convexity, generalising the ordinary notionof convexity, and he proves a decompositiontheorem showing that a real projective mani-fold of dimension n is either (n1)-convex orcan be decomposed into simpler real projec-tive manifolds. Special attention is paid tothe three-dimensional situation, as motivationfor such a study came from the study ofhyperbolic 3-manifolds. The methods devel-oped are applied to a classification of radiantaffine three-dimensional manifolds. Thebook brings a systematic and ordered treat-ment of this new field, which was not previ-ously available in book form. (vs)

C. J. Colbourn and A. Rosa, Triple Systems,Oxford Mathematical Monographs, ClarendonPress, Oxford, 1999, 560 pp., £80, ISBN 0-19-853576-7This book presents the current knowledgeabout triple systems, collecting together com-mon themes and providing an accurate por-trait of an incredible variety of problems andresults. Representative samples of majorstyles of proof techniques are provided. Thebook is intended primarily for readers with abasic knowledge of combinatorial design the-ory.

After a historical background, the first threechapters explain in detail the basic materialon constructions and existence of triple sys-tems. The next five chapters describe topicsconnected with triple systems: isomorphism,enumeration, subsystems and automorphisms.Chapters 9-23 treat a number of challengingproblems on triple systems in detail. Chapters24 and 25 provide a guide to two related class-es of triple systems in which the triples containordered pairs. A comprehensive bibliographyon triple systems is provided. (jj)

J. B. Conway, A Course in Operator Theory,Graduate Studies in Mathematics 21, AmericanMathematical Society, Providence, 1999, 372 pp.,US$49, ISBN 0-8218-2065-6This book is a continuation of the authorsprevious text A Course in Functional Analysis(Springer, 2nd ed., 1990). Its aim is to covercentral topics of operator theory in a formthat is accessible to graduate students.

The book is divided into eight chapters.The first two chapters, devoted to C*-algebrasand normal operators, overlap Chapters 8and 9 from the previous edition, but theymake the book more independent. The theo-ry of C*-algebras is continued in Chapter 5,where irreducible representations and posi-tive maps are examined. As an application,the dilation theorem is proved. Chapter 3 dis-cusses ideals of bounded operators in aHilbert space, especially the ideal of compactoperators. Chapter 6 deals with a study ofFredholm operators and compact perturba-tions. The Weyl-von Neumann-Berg theoremon almost diagonalisation of normal operatorsis proved here. There are two central chap-ters: Chapter 4 examines non-normal opera-tors like isometries in particular shifts, andsome deep connections between operator the-ory and analytic functions are shown; Chapter7 treats von Neumann algebras and their clas-sifications. The final chapter explores rela-tions between sets of operators and their com-mon invariant subspaces (reflexive andhyper-reflexive operators).

This book is written in a very readable style.In spite of the many recent results included, areader is not lost in technical explanations,since the main ideas and comments are simul-taneously given. Many exercises make thisbook convenient for independent study, withcross-references throughout. The whole textis well organised, rendering the book suitablefor anybody interested in the above topics.(jmil)

A. Croft, R. Davison and M. Hargreaves,Engineering Mathematics, with CD, PearsonEducation, London, 2001, 969 pp., £39.99,ISBN 0-130-26858-5This book was written to serve the mathemat-ical needs of students of a first course in engi-neering, primarily for students of electronic,electrical, communication and system engi-

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neering. The book has two main aims. The first is to

provide an accessible and readable introduc-tion to engineering mathematics at the degreelevel; the second is to encourage the integra-tion of engineering and mathematics.

The first three chapters include a review ofsome important functions and techniquesfrom previous courses. These chapters con-tain a review of algebraic techniques, engi-neering functions and trigonometric func-tions. The next chapters include descriptionsof many topics: coordinate systems, sequencesand series, vectors, matrix algebra, complexnumbers, differentiation, integration, Taylorpolynomials, Taylor and Maclaurin series,ordinary differential equations, the Laplacetransform, difference equations and the z-transform, Fourier series and Fourier trans-form, functions of several variables, vector cal-culus, line and multiple integrals, statisticsand probability. There are four appendicesand an index. As a supplement, an interactiveCD testing and assessment package is includ-ed. (jkof)

P. R. Cromwell, Polyhedra, CambridgeUniversity Press, Cambridge, 1999, 451 pp.,£32.50, ISBN 0-521-55432-2 and 0-521-66405-5 This book comprehensively documents manyways that polyhedra have appeared in the his-tory of mathematics and the sciences. It is anunusual book as it combines the style of a his-torical essay with a description of scientificachievements and goals, and it aims to provebasic results and theorems. The figures arenice, ranging from Kepler and Dürer to mod-ern times, and are complemented by a schol-arly exposition. To quote one of the reviews:it is a labor of love and successfully comple-ments earlier books, most notably those ofGrünbaum. In some places the writing is a bitvague: for example, while disscussing theproof of the 4-colour theorem, the authordescribes Appel and Hakens attempt withoutmentioning the recent work of Robertson,Seymour and Thomas. This is a book for abroad mathematical and scientific audience.(jnes)

J. W. Dauben (ed.), The History ofMathematics from Antiquity to the Present: ASelective Annotated Bibliography, Revised edi-tion on CD-ROM, American Mathematical Society,Providence, 2000, US$ 49, ISBN 0-8218-0844-3This is not a book, but a CD-ROM. This sec-ond CD-ROM edition is the revised andupdated first edition, which was published in1985 by Garland Publishing in New York. Itis an unusual and comprehensive guide to thehistory of mathematics, containing some 4800bibliographical entries with annotations; ofthese 2800 are entirely new and many of theremaining ones have been updated.

Thirty-eight historians of mathematics fromten countries have participated as contribut-ing editors. They present the best databasewith the best introductions for all topics. Thesections are devoted to Egyptian, Babylonianand Greek mathematics, the Arabic, Latin andHebrew traditions, European mathematics inthe 15th, 16th, 17th and 18th centuries, mod-ern mathematics in the 19th and 20th cen-turies, mathematics in Africa and the Orient,and women in mathematics. Each sectioncontains major works on any given topic orperiod, accompanied with critical descriptionsof these works. The authors emphasise the

most useful and authoritative secondarysources and other types of primary source(texts, manuscripts, correspondence, etc.).Anyone who wishes to become acquinted withthe history of mathematics can begin with thisimpressive bibliographical database. (mnem)

M. Dimassi and J. Sjöstrand, SpectralAsymptotics in the Semi-Classical Limit,London Mathematical Society Lecture Note Series268, Cambridge University Press, Cambridge,1999, 227 pp., £ 24.95, ISBN 0-521-66544-2 This book is based on a course given by theauthors at various universities in France. Itsmain theme is an application of methods ofmicrolocal analysis to spectral problems insemi-classical limit. Main notions are brieflyreviewed (local symplectic geometry, self-adjoint operators), and the authors thendevelop the basic theory of h-pseudodifferen-tial operators and a functional calculus forthem). WKB methods are used for the con-struction of local asymptotic solutions of theSchrödinger operator and the method of sta-tionary phase is explained. There are discus-sions of tunnel effects, asymptotic expansionsfor the trace and of spectral results in variousspecial situations. Each chapter ends with his-torical remarks, useful comments and recom-mendations for further reading. The bookcan be used for a one-semester course on thetopic. (vs)

S. Dineen, Complex Analysis on InfiniteDimensional Spaces, Springer Monographs inMathematics, Springer, London, 1999, 543 pp.,DM 179, ISBN 1-85233-158-5This book contains a comprehensive study ofproperties of holomorphic functions on opensubsets of infinite-dimensional complex topo-logical vector spaces. The first two chaptersare devoted to a study of polynomials in aninfinite-dimensional setting. They are intro-duced using multi-linear maps and tensorproducts, their relation to geometric conceptsof Banach space theory is discussed and theduality theory for polynomials is developed.Chapter 3 introduces basic definitions of holo-morphic maps between infinite-dimensionalspaces, studies their Taylor and monomialexpansions and introduces main topologieson the space of holomorphic maps on an openset. The next two chapters contain the centraltheme of the book a comparison of the mainthree topologies on the space of holomorphicmaps. To understand conditions under whichsome of them coincide is a complicated ques-tion whose answer needs a surprising varietyof tools. T hfinal chapter is devoted to extension proper-ties of holomorphic maps in the infinite-dimensional situation. Each chapter endswith exercises containing additional material,and many of them are commented on in theAppendix.

A special feature of the book is a compre-hensive and detailed description of the histo-ry of the subject and of its individual results,contained in the notes ending each chapterand in the appendix. The authors efforts inthis respect add a special value to the book.The reader is supposed to know basic complexfunction theory, topology and Banach spacetheory; a knowledge of several complex vari-able theory is helpful, but not necessary. Thebook is very well written and can be recom-mended to mathematicians working in thefield as well as those from other fields inter-ested in the subject. (vs)

M. Dummett, Elements of Intuitionism, OxfordLogic Guides 39, Clarendon Press, Oxford, 2000,331 pp., £60, ISBN 0-19-850524-8 The first edition of this book appeared in1977. The authors intention is to give basicinformation about the fundamental ideas ofintuitionism, and especially to clarify two suchideas underlying intuitionistic mathematics.The first is a general theory of meaning for amathematical language, according to whichthe only thing that can make a statement trueis an intuitively acceptable proof, representinga certain kind of mental construction. Thesecond is the concept of infinite effectivesequences, which are developed in Chapters 3(Choice sequences and spreads), 4 (The for-malism of intuitionistic logic) and 5 (Thesemantics of intuitionistic logic).Philosophical remarks form Chapter 7. Someparts from the first edition are revised forexample, the account of Brouwers proof ofthe Bar theorem, as well the treatment of gen-eralised Beth trees.

This book provides a comprehensive pre-sentation of the subject and can be read with-out special preliminary knowledge. (jml)

J. Fauvel, R. Flood and R. Wilson (eds.),Oxford Figures: 800 Years of the MathematicalSciences, Oxford University Press, Oxford, 2000,296 pp., £35, ISBN 0-19-852309-2This book chronicles the development ofmathematical research and studies at OxfordUniversity from its foundation to the 20thcentury. It is a story of the intellectual andsocial life of the mathematical community (thecommunity of professors and students and thewider scientific community in Britain andthroughout the world).

The authors describe those parts of mathe-matics that were covered in lectures and howthey were treated at Oxford, the content ofexaminations and how they were realised,what and how was changed during more than800 years (the transformation of the mathe-matical curriculum and the role of mathemat-ics in British society). The aspects of the his-tory of mathematics in the periods of medievalOxford, renaissance Oxford, the mid-17thcentury, the Newtonian school, GeorgianOxford, the mid-19th century and the 20thcentury are described so that the reader cansee how mathematics was developed centuryby century. There are stories about manywell-known and sometimes surprising figures,such as Robert Boyle, Christopher Wren,Edmond Halley, Percy Bysshe Shelley,Charles Dodgson (Lewis Carroll), John Wallis,Isaac Newton, Thomas Hornsby, Henry Smithand James Joseph Sylvester.

Appendices containing lists of the holdersof the Savilian, Sedleian, Waynflete, RouseBall and Wallis Chairs since their foundations,and an index of names, are included at theend of the book. The book is very well writtenand beautifully illustrated. It can be recom-mended to anyone interested in the history ofmathematics and the history of teaching.(mnem)

D. Gardy and A. Mokkadem (eds.),Mathematics and Computer Science.Algorithms, Trees, Combinatorics andProbabilities, Trends in Mathematics, Birkhäuser,Basel, 2000, 340 pp., DM 148, ISBN 3-7643-6430-0 These proceedings of a colloquium held inVersailles in September 2000 consist of 28 ref-ereed research papers on diverse mathemati-cal problems, mainly motivated by computer

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science. With a few exceptions, the questionsbelong to probability theory or combinatorialenumeration, and the topics include estimat-ing various statistics of random trees, randomgeneration of words, probabilistic analysis ofalgorithms (QUICKSORT, genetic algo-rithms, routing in a network, an algorithm forrendezvous in graphs, calculating the station-ary vector for discrete-time stochastic automa-ta, universal prediction algorithm for mixingsources), Markov chains, random walks,branching processes, large deviations forpolling systems, general enumeration tech-niques, enumerating certain paths in lattices,asymptotic analysis of coefficients of multivari-ate generating functions, and 0-1 laws. A good3-page overview of the contents is given in thepreface. (jmat)

J. R. Giles, Introduction to the Analysis ofNormed Linear Spaces, Australian MathematicalSociety Lecture Series 13, Cambridge UniversityPress, Cambridge, 2000, 280 pp., £19.95, ISBN0-521-65375-4This book presents a basic course in function-al analysis. The text is very readable andoffers a detailed explanation of the subject.Some recent results are also included.

The book starts with basic properties ofnormed linear spaces (including the Schauderbasis), classes of examples and the theory ofHilbert spaces. The next chapters deal withspaces of continuous linear mappings (includ-ing Banach algebras), the analytic form of theHahn-Banach theorem, reflexivity and subre-flexivity (including a complete proof of theBishop-Phelps theorem), the open mappingand closed graph theorems and the uniformboundedness principle. Further chapters aredevoted to various types of continuous linearmappings (conjugate mappings, adjoint oper-ators, projection and compact operators),main properties of spectra of linear and com-pact operators (a proof of Lomonosovs theo-rem on invariant subspaces) and spectral the-ory (culminating with spectral theories fornormal, compact and hermitian operators).

The text includes many exercises, both ele-mentary and advanced. An appendix containsthe set theory results used in the text (Zornslemma, the Schröder-Bernstein theorem andthe Hamel basis). At the end of the book thereare historical notes showing the developmentof functional analysis from the late 19th cen-tury. Two late significant advances concern-ing (Ekelands) variational principles andAsplund spaces are treated.

The book is addressed mainly at seniorundergraduate and beginning postgraduatestudents, who are assumed to be familiar withelementary real and complex analysis, linearalgebra and the theory of metric spaces. (jl)

J. Hadamard (J. J. Gray and A. Shenitzer(eds.)), Non-Euclidean Geometry in the Theoryof Automorphic Functions, History ofMathematics 17, American Mathematical Society,Providence, 1999, 95 pp., US$19, ISBN 0-8218-2030-3In the 1920s J. Hadamard wrote a survey onautomorphic functions, in connection with thepreparation of an edition of the collectedworks of N. I. Lobachevski. It was translatedby A. V. Vasiliev into Russian and edited by B.A. Fuks and published in 1951. The text isnow available in a very readable English edi-tion, prepared by Abe Shenitzer. Most of theresults in the booklet are stated without proof,and are not intended for novices. However, itcan be recommended as supplementary read-

ing to anybody interested in the subject. The text is divided into six chapters and

enlarged by two introductory prefaces writtenby J. Gray. The first preface (just 1 page)describes the historical background andHadamards connections with Russian mathe-matics. The second one gives a very usefulbrief account of the history of the theory ofautomorphic functions from 1880 to 1930.The Hadamard chapters range from realisa-tions of the Lobatchevskian plane, the mostimportant properties of properly discontinu-ous subgroups, and Fuchsian and Kleinianfunctions, to the uniformisation of algebraiccurves and solutions of ordinary differentialequations with algebraic coefficients.

This booklet provides interesting readingfor anybody interested in the theory of auto-morphic functions, and this English editionwill certainly be of interest not only to thisgroup of working mathematicians. (p)

M. Hazewinkel (ed.), Handbook of Algebra,Vol. 2, Elsevier, Amsterdam 2000, 878 pp.,US$177.50, ISBN 0-444-50396-X This handbook is the second part of a com-prehensive guide through modern algebra.The whole project is divided into nine sec-tions, and the second volume partially coverssix of them. Section 2 contains a treatment ofcategory theory, some parts of homologicaland homotopical algebras and model-theoret-ic algebras. Section 3 presents the theory ofcommutative rings and algebras, associativerings and algebras and the deformation theo-ry of rings and algebras. In Section 4 varietiesof algebras and Lie algebras are explained.Section 5A is devoted to groups and semi-groups and Section 6C deals with the repre-sentation theory of continuous groups. Part Eis devoted to abstract and functorial represen-tation theory. (lbi)

C. Hillermeier, Nonlinear MultiobjectiveOptimization: A Generalized HomotopyApproach, International Series of NumericalMathematics 135, Birkhäuser, Basel, 2001, 135pp., DM 88, ISBN 3-7643-6498-XThis book motivates and surveys the princi-ples and classical methods of multi-objectiveoptimisation, including a recent stochasticapproach and concentrating on the presenta-tion of a new generalised homotopy approach.

The homotopy methods require that allfunctions in the optimisation problem aretwice continuously differentiable. They havebeen analysed mostly in the one-dimensionalparameter case, whereas the proposedapproach allows a multi-dimensional para-metrisation useful for the solution of multi-objective optimisation problems. The set ofefficient points is examined from the view-point of differential topology, and the sug-gested numerical algorithm is applied both toan academic example and to optimisationproblems dealing with the design and opera-tion of industrial systems. The book will inter-est applied mathematicians and engineers.(jd)

S. Kaufmann, A Crash Course inMathematica, Birkhäuser, Basel, 1999, 200 pp.(CD-ROM included), DM 48, ISBN 3-7643-6127-1 and 0-8176-6127-1This book, and the accompanyingMathematica notebooks on the CD-ROM, givethe reader the basics of Mathematica in a com-pact form. The book itself is basically a print-out of the Mathematica notebooks containedon the CD-ROM. The author discusses the

following points: user interface (front end),actual calculator (kernel) and additionalmacros (packages).Everything is complemented by examples thatare kept at a simple mathematical level andare largely independent of special technical orscientific applications. Emphasis is placed onsolving such standard problems as equationsand integrals, and on graphics. After workingthrough this course, readers will be able tosolve problems independently and to findadditional help in the on-line documentation.Depending on their interests and needs, com-pleting the first two parts of this course may besufficient, as they include the most importantcalculations and graphics functions. The thirdpart is more technical, and the fourth partintroduces programming in Mathematica.

The CD-ROM can be used with MacOS,Windows 95/98/NT or Unix. Up-to-dateinformation and any corrections to the bookcan be accessed at http://www.ifm.ethz.ch/\~kaufman. (mbr)

A. G. Kulikovskii, N. V. Pogorelov and A.Yu. Semenov, Mathematical Aspects ofNumerical Solution of Hyperbolic Systems,Monographs and Surveys in Pure and AppliedMathematics 118, Chapman & Hall/CRC, BocaRaton, 2001, 540 pp., £63.99, ISBN 0-8493-0608-6This book presents various methods and tech-niques for the numerical solution of hyperbol-ic systems of partial differential equations, andtreats a number of problems with importantapplications.

The book consists of seven chapters. InChapter 1 the basic concepts and notationsare introduced. Chapter 2 is concerned withthe formulation of basic approaches to thenumerical solution of quasi-linear hyperbolicsystems, both in the conservative and non-conservative forms. The methods of Godunov,Courant-Isaacson-Rees, Roe and Osher aretreated, and attention is paid to higher-orderschemes with reconstruction and limiting pro-cedures. The next chapters are devoted toparticular mechanical problems. Chapter 3deals with gas dynamics equations and thesolution of Euler equations equipped with var-ious state equations. In Chapter 4, shallowwater equations are considered, and Chapter5 is devoted to numerical solution of MHDproblems. Chapter 6 is an attempt to outlineproblems of solid dynamics that are governedby hyperbolic systems. Finally, Chapter 7introduces the notion of non-classical discon-tinuity, discusses its various aspects and treatsseveral applications.

This book is a substantial addition to theexisting literature, particularly because it con-tains a number of applications. It will be ofinterest to students and researchers in fluiddynamics and continuum mechanics and invarious fields of physics. It contains a numberof figures and examples and will be useful forspecialists dealing with practical computation.Although the word Mathematical appears inthe title, it is not written in a mathematicalstyle. The results are not formulated as theo-rems and the mathematical theory of numeri-cal schemes as convergence and error esti-mates is not mentioned. (mf) S. Lang, Fundamentals of DifferentialGeometry, Graduate Texts in Mathematics 191,Springer, New York, 1999, 535 pp., DM 109,ISBN 0-387-98539-XThere are many books on the fundamentals ofdifferential geometry, but this one is quiteexceptional; this is not surprising for those

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who know Serge Langs books. The aim is topresent the fundamentals shared by differen-tial topology, differential geometry, and dif-ferential equations.

The various notions of the smooth calculuson manifolds form the core of the book asusual, but there are two distinct guidelines vis-ible; the coordinate-free treatment of allproofs and main theorems, and the emphasison the developments in global analysis andgeometry after the 1960s. These features fitwell together, since much of the recent inter-est heads towards various aspects of an infi-nite-dimensional character, while the coordi-nate-free approach leads naturally to differen-tial geometry modelled on Banach spaces(manifolds without further structure), self-dual Banach spaces (pseudo-Riemannianstructures), and Hilbert spaces (Riemannianstructures). Although these models are notgeneral enough to include many standardinfinite-dimensional objects, such as thespaces of all smooth mappings between finite-dimensional manifolds, Langs treatment pro-vides the most conceptual way that complete-ly recovers and generalises (but also simpli-fies) the more usual expositions with x1,,xn,dx1,,dxn , Γi

jk , etc. To indicate the wide area covered in about

500 pages, we list the chapter titles: differen-tial calculus; manifolds; vector bundles; vectorfields and differential equations; operationson vector fields and differential forms; thetheorem of Frobenius; metrics; covariantderivatives and geodesics; curvature; Jacobilifts and tensorial splitting of the double tan-gent bundle; curvature and the variation for-mula; an example of seminegative curvature;automorphisms and symmetries; immersionsand submersions; volume forms; integrationof differential forms; the Stokes theorem;applications of the Stokes theorem; the spec-tral theorem. The book is designed rather forgraduates, although it is explicitly based onlyon elementary calculus, topology and linearalgebra. It can be warmly recommended to awide audience. (jslo)

E. H. Lieb and M. Loss, Analysis, 2nd edition,Graduate Studies in Mathematics 14, AmericanMathematical Society, Providence, 2001, 346 pp.,US$39, ISBN 0-8218-2783-9 The first edition of the book appeared in 1997(for its review, see EMS Newsletter 25,September 1997). In this second edition,besides correcting misprints, the authors haveadded a new Chapter 12 containing a discus-sion of a semi-classical approximation for theSchrödinger equation, using the Glaubercoherent states and various bounds on eigen-values and their sums. In addition, Chapter 8has been extended (compactness criterion,and Poincaré, Nash and logarithmic Sobolovinequalities) and there are further additions inChapter 1 (integration using simple functions)and Chapter 6 (Yukawa potential). The num-ber of exercises has been increased. Thisattractive book can be highly recommendedfor its style, clarity and interesting choice ofmaterial. (vs)

M. Mesterton-Gibbons, An Introduction toGame-Theoretic Modelling, Second edn., StudentMathematical Library 11, American MathematicalSociety, Providence, 2000, 368 pp., US$39, ISBN0-8218-1929-1 This is the second updated edition of a suc-cessful textbook. It is an introduction to gametheory and applications from the perspectiveof a mathematical modeller. Unlike theoreti-

cally oriented textbooks, the emphasis is onconcrete examples, and the authors explana-tion proceeds from specific to general, so thatstudents can follow the motivation of generalmathematical concepts and constructions.The author avoids rigorous statements andproofs of theorems, referring instead to stan-dard mathematical texts. Thus the readerneeds only basic knowledge of calculus andlinear algebra and some experience withstudying mathematical texts.

By tradition, games are classified as eithercooperative or non-cooperative, although theauthor considers this dichotomy imperfect,since almost every conflict has an element ofcooperation and almost all cooperations havean element of conflict.

The book is divided into seven chapters.The explanation begins with the concept ofNash equilibrium in non-cooperative games.In Chapter 1 the author shows that a non-cooperative game can have several Nash equi-libria. The next chapter gives three criteriaenabling us in such situations to distinguishqualitative properties of different Nash equi-libria; in particular, various concepts of equi-librium stability are introduced and com-pared. Chapters 3 and 4 are devoted to coop-erative games; the author introduces coopera-tive games in strategic form and the corre-sponding Nash bargaining solution, charac-teristic function games, Shapley value and theShapley-Shubik index. In Chapter 5, the con-cept of the well-known Prisoners Dilemma isdescribed and a closely related concept, theso-called Cooperators Dilemma, is intro-duced. Using the latter concept, the authorinvestigates cooperation within the context ofa noncooperative game. Chapter 6 is devotedto population games; this chapter shows thatgames are valuable tools in a study of both ani-mal and human behaviour in some non-coop-erative conflict situations. Chapter 7 containssome concluding remarks, whose aim is togive an objective evaluation of the usefulnessof games for studying real conflict situations.

Each chapter concludes with exercises,whose solutions are presented in an appendix.Another appendix contains an explanation ofthe tracing procedure suggested by Harsanyi.(kzim)

T. Miwa, M. Jimbo and E. Date, Solitons,Cambridge Tracts in Mathematics 135, CambridgeUniversity Press, Cambridge, 2000, 108 pp., £ 25,ISBN 0-521-56161-2 The simplicity of the structure of the set ofsolutions of linear equations is due to the factthat it is a linear vector space. Nothing com-parable applies to non-linear equations.Nevertheless, for certain classes of non-linearPDEs, such as KP-hierarchy of the famous KdVequations, there is an extraordinary substitutefor a missing linear structure: the set of solu-tions is an orbit of an infinite-dimensional Liegroup, and this symmetry makes a descriptionof the set of solutions possible. The book con-tains a discussion of these equations frommany complementary point of views (symme-tries of KdV equations, Lax form of equations,integrable systems, Hirota equations and ver-tex operators, bosonic and fermionic Fockspaces, the Boson-Fermion correspondence,transformation groups of equations and taufunction, infinite-dimensionalGrassmannians, Young diagrams and charac-ters, Hirota equation as Plücker relations).The treatment is based on ideas of M. Sato,developed by M. Kashiwara and the authors.

The book has the pleasant spirit of informal

lectures on the subject, but basic facts areproved. Reading of the book requires only abasic background and there are a lot of exam-ples illustrating the theory. The authors suc-ceed in explaining the essentials in just 100pages, and this charming book can be recom-mended to anybody interested in the moderndevelopment in the mathematics arising frommathematical physics. (vs)

K. Peeva, H.-J. Vogel, R. Lozanov and P.Peeva, Elseviers Dictionary of Mathematics,Elsevier, Amsterdam, 2000, 997 pp., US$209.50,ISBN 0-444-82953-9This is a useful guide for readers, writers andtranslators and all specialists exploring themultilingual scientific terminology in English,German, French and Russian. The dictionarycontains nearly 12000 entries with almost5000 cross-references. The terminology cov-ers arithmetic, algebra, geometry, set theory,discrete mathematics, logic, linear algebra,calculus, differential equations, vector alge-bra, field theory, probability and statistics,optimisation, numerical methods, mathemati-cal programming, modern algebra, computeralgebra, category theory, applied mathemat-ics, the theory of automata and formal lan-guages, the theory of games and commonlyused entries in computer architecture. (in)

G. M. Phillips, Two Millenia of Mathematics.From Archimedes to Gauss, CMS Books inMathematics 6, Springer, New York, 2000, 223pp., US$49.95, ISBN 0-387-95022-2This book is an extended collection of inter-esting mathematical topics from number the-ory and analysis such as Archimedes and π,the discovery of exponential and logarithmicfunctions, Napier and Briggss logarithms,Newtons interpolation polynomial, finite andother differences, the Euclidean algorithm,Fibonacci numbers, prime numbers, Gaussscongruences, and Diophantine equations) ranging over two millennia. It does not pre-tend to be a comprehensive history of mathe-matics of this period.

In five chapters (From Archimedes toGauss, Logarithms, Interpolation, Continuedfractions, More number theory), the authorshows that many interesting and importantresults in mathematics have been discoveredby ordinary people and not only by greatgeniuses. Each chapter includes the history ofits topic with an interpretation of the mathe-matical problems. The book shows how andwhy some results in mathematics have beendiscovered or obtained, by following in thesteps of well-known mathematicians who dis-covered them. It is a useful source of mathe-matical material for teachers, undergraduatestudents, students and the vast numbers ofamateurs who love mathematics. (mnem)

L. Polterovich, The Geometry of the Group ofSymplectic Diffeomorphisms, Lectures inMathematics, Birkhäuser, Basel, 2001, 132 pp.,DM 44, ISBN 3-7643-6432-7The main topic of the book is a discussion of arole of the group Ham(M, Ω) of Hamiltoniandiffeomorphisms of a symplectic manifold(M, Ω) in geometry and classical mechanics.Under suitable assumptions on M, the groupHam(M, Ω) is the connected component of theidentity of the group of all symplectic diffeo-morphisms. In mechanics it is just the groupof admissible motions. A basic question relat-ed to elements f∈ Ham(M, Ω) is how to evalu-ate the minimal amount of energy required togenerate the given diffeomorphism f. The

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answer can be obtained using a distancedefined on the group Ham(M, Ω) by means ofthe canonical biinvariant metric on Ham(M,Ω) which is a solution of a certain naturalvariational problem on (M, Ω). The theory isbased on geometrical properties of the groupHam(M, Ω), developed first by Hofer in 1990.In recent years these questions have beenintensively studied in the framework of sym-plectic topology, using the Gromov theory ofpseudo-holomorphic curves, Floer homologyand the theory of symplectic connections.

The book is a good introduction to the topicand offers a description of recent develop-ments in Hofer geometry and its applicationsin dynamics and ergodic theory. It can be rec-ommended for mathematicians and physicistsinterested in problems related to symplecticgeometry and mechanics. (jbu)

S. Rigot, Ensembles quasi-minimaux avec con-trainte de volume et rectifiabilité uniforme,Mémoires de la Société Mathématique de France82, Société Mathématique de France, Paris, 2000,104 pp.,FRF 150, ISBN 2-85629-086-0 In this volume, interesting results areachieved concerning a useful generalisation ofthe classical isoperimetric problems. In par-ticular, the regularity of quasi-minimising setsfor a variational problem related to isoperi-metric inequality is investigated.

The object of study is a measurable set Gsatisfying

∫ |grad χG | ≤ ∫| grad χH | + g(|G ∆ H|),for each measurable test set H with relativelycompact symmetric difference G ∆ H and |H|finite and equal to |G| (integration over thewhole of Rn). Here g: (0, ∞) → (0, ∞) is fixedso that g(t) = o(t(n-1)/n) as t → 0. It is shownthat a suitable representative of the set G hasthe Ahlfors regular boundary, and that thenumbers of connected components of G andRn\G are estimated by an universal constant.Moreover, when g(t) is of the form Ctp, thenC1,α-partial regularity of the boundary isobtained, and for n ≤ 7, full C1,α regularity fol-lows. In the final chapter, the above resultsare applied to the (physically motivated) prob-lem of the minimisation of

E(G) = Hn-1(∂G) + ∫GxG K(x y) dx dy,where Hn-1 is Hausdorff measure and K is anintegrable kernel with compact support.(jama)

A. Rubinov, Abstract Convexity and GlobalOptimization, Nonconvex Optimization and itsApplications 44, Kluwer Academic Publishers,Dordrecht, 2000, 490 pp., £135, ISBN 0-7923-6323-X The contents of this book can be divided intotwo parts. The first part presents the notionof abstract convexity and the basic andadvanced theorems relating to this topic.These results are then applied to the study ofsome classes of functions and sets. Thisapproach includes elements of monotonicanalysis and elements of star-shaped analysis,as well as a study of quasi-convex functions.The results of the first part are then used inthe study of global optimisation, which is thesubject of the second part of the book. Theclassical notions of Lagrange and penaltyfunctions are presented and extended. Newmethods of global optimisation based mainlyon solvability results for inequality systemsand optimisation of the difference of abstractconvex functions are then developed. Thefinal chapter is devoted to the numericalaspects of global optimisation.

The book is recommended to specialists inglobal optimisation, mathematical program-ming and convex analysis, as well as to engi-neers and specialists in mathematical model-ling. (jvy)

D. G. Saari, Chaotic Elections! AMathematician Looks at Voting, AmericanMathematical Society, Providence, 2001, 159 pp.,US$23, ISBN 0-8218-2847-9 Two events motivated the author to write thisbook: the 2000 U.S. Presidential election andthe election debate about its outcome inFlorida, and the final resolution, in some ofthe authors recent papers, of a 200-year-oldmathematical problem concerning the sourceand explanation of the paradoxes and prob-lems of voting procedures. The purpose ofthe book is to explain what can go wrong inelections, and why.

The book is divided into six chapters.Chapter 1 has an introductory character, andcontains a critical analysis of the voting proce-dure employed in the 2000 U.S. Presidentialelection; in this chapter Arrows ImpossibilityTheorem is presented. In Chapter 2, theauthor explains the importance of the choiceof election procedure. Although little interestis usually devoted to this problem, the choiceof election procedure can substantially influ-ence the outcome of the elections. This isdemonstrated on examples both from therecent past (U.S. Presidential elections in the1990s) and from earlier (such as the 1860election in which Lincoln won the Presidencywith a majority vote in the Electoral College).Some general results concerning chaotic elec-tion outcomes and their consequences arepresented in Chapter 3, while Chapter 4analyses the so-called strategic behaviour ofvoters. This behaviour means that the votersdo not vote sincerely according to their realpreferences, but try by a different choice toachieve a better total election result indirectlyfrom their subjective point of view. InChapter 5, the author discusses a provokingquestion concerning the conditions underwhich the outcome of an election with a cho-sen voting procedure reflects the real will andpreferences of the voters. The final chaptergeneralises some experience from the analysisof elections and investigates a more generalconcept of aggregation, which is involved bothwith various election procedures, and also inother areas far beyond voting, such as thechoice of power indices, non-parametric sta-tistics and various probability assertions.

The book is not a research monograph; itsreading requires little more than high-schoolmathematics, since the author wanted toinform practitioners involved with designingpractical voting or aggregation rules (political,economic or technical) what may happenwhenever any group makes a choice undergiven rules. (kzim)

A. Semmes, Some Novel Types of FractalGeometry, Oxford Mathematical Monographs,Clarendon Press, Oxford, 2001, 164 pp., £49.95,ISBN 0-19-850806-9Let (M, d) be a metric space with a doublingmeasure µ. If A is a subset of M and f: A → Ris a function, then the quantity Dε f(x) = ε -1

sup|f(y) f(x)|: y ε A, d(x, y) ≤ εwith a small ε> 0 measures oscillation of f near the pointx, just as the ordinary gradient does in theclassical Euclidean setting. A decent calculusis possible on a space, if one can estimatelarge-scale quantities by integrating micro-scopic measurement oscillations.

One exact statement of this type consists ofinequalities of Poincaré type. The validity ofPoincaré inequalities is a property of the space(M,µ), which holds if there exist positive con-stants C and k such that

∫B |f(x) a|dµ(x) ≤ Cr ∫kB Dε f(x) dµ(x), for any ball B = B(x, r), any function f definedon kB (with mean value a over B) and any pos-itive ε < r. There are many recent treatmentsthat consider the Poincaré inequalities as astarting axiom. It is then important to under-stand in which situations this postulate mayhold. As fractal-like structures, Ahlfors regu-lar spaces of dimension s are considered: theseare spaces on which the s-dimensionalHausdorff measure can be taken as the refer-ence measure µ. Among them BPI spaces arestudied as an appropriate generalisation ofself-similarity. An important class of spaces isbased on Laaksos construction, where the keystep consists in taking the product of anAhlfors regular space of dimension s1 withthe line interval. Another source of spacescomes from geometric structures such asHeisenberg groups (or, more generally,Carnot spaces). Besides the validity ofPoincarés inequalities, various other mappingproperties are investigated, such as measure-mapping properties, rigidity of the structureunder mappings, and continuous families ofmappings. Tangential properties of Lipschitzand regular mappings are also studied,including rectifiability and its generalisations.

This book provides a thorough discussionon the current state and perspectives of thetopic. The visionary aspect is emphasised.There are many questions and conjectures inthe text, but fewer theorems. Proofs, whereincluded, are rather indicated than elaboratedin detail. This volume is a great source ofinspiration and a guide for students andresearchers looking for new areas to study.(jama)

R. Siegmund-Schultze, Rockefeller and theInternationalization of Mathematics Betweenthe Two World Wars, Science Networks/HistoricalStudies 25, Birkhäuser, Basel, 2001, 341 pp., DM170, ISBN 4-7643-6468-8This book is the first detailed study of the briefbut very influential role played by theRockefeller family in the field of mathematics.It is based on extensive research by theRockefeller Archive Center in the archives ofHarvard University and the Universities ofNew Hampshire, Göttingen and New YorkCity.

In the first chapter the author comments onthe process of internationalisation and mod-ernisation of mathematics and the conditionsfor international scientific collaborationbetween the First World War and the Nazi dic-tatorship after 1933. The second chapterdescribes the beginning of the InternationalEducation Board, a central role of theFellowship Program in the careers of the mod-ern generation of mathematicians (S. Banach,A. Weil, B. L. van der Waerden, etc.) and thesupport of the Rockefeller Foundation for newmathematical publications and for the foun-dation of new mathematical journals. Thethird chapter analyses the comparative devel-opments of mathematics in Europe and USAin the 1920s and 1930s. The fourth chapterexplains the practice of the FellowshipPrograms of the International EducationBoard between 1923 and 1928 and of theRockefeller Foundation after 1928. Here thereader can find the criteria for selections ofFellows, the Fellowship programs, some

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Fellowship lists, etc. The rise of Soviet(Russian) mathematics and problems of itscollaboration with the Rockefeller Foundationand its political self-isolation are commentedon. The fifth chapter shows the role ofRockefellers help in the foundation of newscientific institutes, mainly in Europe. Itincludes the foundation of the newMathematical Institute in Göttingen, theInstitute of Henri Poincaré in Paris, theMathematical Institute in Djursholm inSweden and the School of Mathematics of theInstitute for Advanced Study in Princeton,and their role in promoting cooperation inscience. The sixth chapter shows the changesof programmes of the Rockefeller Foundationafter 1933 and during the first years of theSecond World War. The seventh chaptergives a very short review of Rockefeller sup-port for mathematics between 1945 and 1950.

The book concludes with seventeen appen-dices. The first fourteen are letters andreports of distinguished mathematicians,workers or clerks of the RockefellerFoundation, illustrating Rockefellers contri-bution to science, and mathematics in particu-lar. The next three appendices contain lists ofFellows in mathematics up to 1945,Guggenheim Fellows in mathematics up to1945 and mathematicians from Europe whowere supported by the RockefellerFoundation Emergency Fund. This book canbe recommended to everybody who wants toknow more about the birth of modern andinternational mathematics during the firsthalf of the 20th century. (mnem)

K. A. Sikorski, Optimal Solution of NonlinearEquations, Oxford University Press, Oxford,2001, 238 pp., £39.50, ISBN 0-19-510690-3The aim of this book is to review the state ofthe art in methods for optimal (or nearly opti-mal) solution of non-linear problems. In par-ticular, the following problems are consid-ered: finding roots of non-linear equations,approximation of fixed points (of both con-tractive and non-contractive functions) andcomputation of the topological degree. Themethods are required to be robust, and guar-antee that the computed and exact solutionsdiffer by a prescribed tolerance for a specifiedclass of problems. The notion of optimality isdeveloped by means of information-basedcomplexity theory, which creates the leastoverhead for the particular method.

This book is self-contained. Each chapter isprovided with exercises and detailed annota-tions that encourage further reading. Theprospective audience of the book ranges fromresearches in computational complexity topractitioners in numerical computation. Forthe latter group, the book represents an alter-native view to the classical local convergenceanalysis of non-linear iterative techniques. (vj)

S. M. Stigler, Statistics on the Table. TheHistory of Statistical Concepts and Methods,Harvard University Press, London, 1999, 488 pp.,£30.95, ISBN 0-674-83601-4 This book is a collection of twenty-two essays,divided into five parts: Statistics and social sci-ence, Galtonian ideas, Some seventeenth-cen-tury explorers, Questions of discovery, andQuestions of standards. The author is wellknown for his papers dealing with the historyof statistics; these papers were published inprominent journals and now serve as basicmaterial for most of the essays. The title ofthe book is borrowed from a letter of KarlPearson: I am too familiar with the manner in

which actual data are met with the suggestion thatother data, if they were collected, might show some-thing else to believe it to have any value as an argu-ment. Statistics on the table, please, can be my solereply.

To give a flavour of the essays, I quote a fewsentences from the chapter Stiglers Law ofEponymy: For Stiglers Law of Eponymy in itssimplest form is this: No scientific discovery isnamed after its original discoverer. Thus in thefield of mathematical statistics it can be found thatLaplace employed Fourier Transforms in printbefore Fourier published on the topic, that Lagrangepresented Laplace Transforms before Laplace beganhis scientific career, that Poisson published theCauchy distribution in 1824, and that Bienaymstated and proved the Chebychev Inequality adecade before and in greater generality thanChebychevs first work on the topic. One mightalso expect that Gaussian distributions wereknown before 1809 when Gauss associated itwith the least squares method. In fact, Gausshimself cites Laplace who dealt with this dis-tribution in 1774; moreover, both Gauss andLaplace knew a 1733 publication by A. deMoivre, which is now considered as the originof the Gaussian (or normal) distribution. Bythe way, the title of the next essay is Who dis-covered Bayess Theorem?

All the essays are very interesting, and someof them should be a compulsory part of thegeneral education of every statistician inparticular, the chapters devoted to the historyof the maximum likelihood method and theinvention of the least squares method. Thisbook can be warmly recommended to anyoneinterested in probability and statistics and intheir history. (ja)

O. Stormark, Lies Structural Approach toPDE Systems, Encyclopaedia of Mathematics andits Applications 80, Cambridge University Press,Cambridge, 2000, 572 pp., £70, ISBN 0-521-78088-8 This monograph is a comprehensive introduc-tion to geometric methods for the study of sys-tems of partial differential equations. Theresults presented here concern local solutionsof systems. The geometric approach is basedon ideas developed by Sophus Lie, Elie Cartanand Ernest Vessiot. Any system of PDEs canbe considered as a submanifold in the corre-sponding jet bundle, which is equipped with acanonical contact pfaffian system or duallywith a canonical system of vector fields. Theproblem of solving the system is transportedinto the problem of finding integral manifoldsof the pfaffian or vector field system.

In the book, the duality between these con-cepts is described, and the Frobenius andCartan local existence theorems are proved.The notions of involutivity and prolongationsof a given system of PDEs are introduced andthe results obtained are applied to specialfirst- and second- order PDEs. Anotherobject related to a system of PDEs on M is thecontact transformation, a local diffeomor-phism of M that preserves the system. Thefamily of all contact transformations form a(Lie)-pseudogroup of local diffeomorphismsof M. For the special first-order PDE systemthere is a special Lie pseudogroup,called alocal Lie group, whose structure plays aninteresting role in the study of solutions of thesystem. The Cartan theory of Liepseudogroups is also explained, together withthe equivalence problem. Using the Drachclassification, an arbitrary PDE system can bereduced to a first- or second-order in oneunknown, and both cases are studied in detail

here. Many special and interesting examplesof PDEs are discussed, and their solutions aredescribed at the end of the book. This book isa good source for anybody interested in PDEs,differential geometry, Lie group theory andrelated fields. (jbu)

Tian-Xiao He, Dimensionality ReducingExpansion of Multivariate Integration,Birkhäuser, Boston, 2001, 225 pp., DM 156,ISBN 0-8176-4170-X and 3-7643-4170-X This book discusses a technique for numericalintegration by using dimensionality-reducingexpansions (DRE) to reduce a higher-dimen-sional integral to lower-dimensional integrals,with or without a remainder. Some importantand common aplications of DRE include theconstruction of boundary-type quadrature for-mulas (BTQF) and asymptotic formulas foroscillatory integrals.

The book is organised as follows. Chapters1 and 2 discuss the construction of DREs andBTQFs with various degrees of algebraic pre-cision. In Chapters 3 and 4, the author usesDREs to approximate oscillatory integrals,and establishes their corresponding numericalquadrature formulas. Chapter 5 demon-strates how to construct DREs over a complexregion, by using the Schwarz function and theBergman kernel. The final chapter examineshow the solutions of certain differential equa-tions can be used to construct exact DREs, andhow, conversely, some DREs can be utilised toderive a scheme of the boundary elementmethod, used for evaluating the numericalsolution of a boundary-value problem of apartial differential equation.

This book will be a useful guide for a widerange of readers in pure and applied mathe-matics, statistics and physics, and can also beused as a textbook for graduate and advancedundergraduate students. (knaj)

J. L. Walker, Codes and Curves, StudentMathematical Library 7, American MathematicalSociety, Providence, 2000, 66 pp., US$15, ISBN0-8218-2628-X This booklet arose from a series of lecturesheld at the Institute for Advanced Study inPrinceton. It consists of seven chapters andthree appendices, written in a clear and con-versational style and oriented towards a gen-eral audience. The reader obtains an intro-duction to classical coding theory, particularlyits algebraic geometric branch.

The first two chapters include the basicnotions and results of coding theory:Hammings distance and weight, dimension,etc.; all introduced notions are illustratedusing basic types of codes as error detecting orcorrecting, linear, cyclic, Reed-Solomon orISBN. The illustrations are carefully selectedand are naturally embedded into the exposi-tion. There follow basic results on the dimen-sion and on absolute and asymptotic bounds(Plotkin or Gilbert-Varshanov) on the maxi-mum number of codewords. In the introduc-tory geometric part, the reader learns thebasics of algebraic curves (Chapter 3), theirsingularities and genus (Chapter 4) and func-tions and divisor theory on curves (Chapter 5);these are necessary for the last two chapters,which are devoted to the description ofGoppas construction of algebraic geometrycodes (Chapter 6) and the Tsfasman, Vladutand Zink bound for asymptotically good codes(Chapter 7). Appendices review basic algebra-ic notions, such as groups, rings, ideals andfinite fields, and collect some important topicsin coding theory that are not covered in the

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text.As already mentioned, the book is written in

a lucid style. It contains worthwhile exercisesscattered through the text, and can be warmlyrecommended to all interested to learn aboutthe basics of classical coding theory. (p)

Jianhong Wu, Introduction to NeuralDynamics and Signal Transmissions Delay, DeGruyter Series in Nonlinear Analysis andApplications 6, Walter de Gruyter, Berlin, 2001,182 pp., DM 99, ISBN 3-11-016988-6 This book is a textbook for graduate andsenior undergraduate students. The mathe-matics used is elementary, in the sense that itrequires only basic knowledge of the theory ofmatrices and of ODEs. It is not fully self-con-tained, as it also makes use of theorems fromliterature, quoted without proof.

The contents of the book are as follows.Chapter 1 outlines the basics of neuroscience in particular, the structure of a single neu-ron and of a network of neurons, and themechanism of neural signal transmission.Chapter 2 introduces a general mathematicalmodel describing the dynamics of neural net-works. In Chapter 3, models of simple net-works that can perform some elementaryfunctions (storing, recalling and patternrecognition) are analysed. Chapter 4 aims ata global analysis of neural networks from theviewpoint of the existence and stability ofequilibria: two important approaches, viaLiapunov functions and monotone dynamicalsystems, are developed. The final, andlongest, chapter focuses on how the dynamicsof a neural network is altered when allowingfor time-delayed terms; the problemsaddressed here include delay-independentstability, delay-induced periodic/chaotic oscil-lations and delay-induced change of thedomain of attraction. (dp)

M. Zinsmeister, Thermodynamic Formalismamd Holomorphic Dynamical Systems,SMF/AMS Texts and Monographs 2, AmericanMathematical Society, Providence, 1999, 82 pp.,US$19, ISBN 0-8218-1948-8 This book links two different parts of mathe-matics and physics: the formalism of statisticalphysics and the theory of holomorphicdynamical systems. It is intended forresearchers from other fields, and is inspired

by classical treatises on these subjects, such asD. Ruelles book Thermodynamic formalism: themathematical structures of classical equilibrium sta-tistical mechanics (Addison-Wesley, 1978). Thefirst four chapters of this book deal with theconcepts of ergodicity, entropy (of a dynami-cal system) and the Perron-Frobenius theo-rem. Chapters 5 and 6 deal with conformalrepellers and iteration of quadratic polynomi-als; in particular, the Hausdorff dimension ofJulia sets of these mappings is disscused.Finally, Chapter 7 introduces the notion of aphase transition and applies some tools of thistheory (in particular that of Legendre trans-form) to a further study of the above prob-lems.

There are few texts on the relationshipsbetween statistical mechanics and the theoryof iterated quadratic maps in a plane. Thisshort book successfully helps to fill this gap. Itis clearly written and gives an excellent intro-duction to the subject. (mzah)

D. Zwillinger and S. Kokoska, StandardProbability and Statistic Tables and Formulae,

Chapman & Hall/CRC, Boca Raton, 2000, 554pp., ISBN 1-58488-059-7This book is in the form of a handbook, pro-viding tables and comprehensive lists of defin-tions, concepts, theorems and formulae inprobability and statistics. Its emphasis is onbasic statistics, as taught in most statisticalcourses, but also covers many advanced topicsfrom statistics and probability theory, such asnon-parametric statistics, quality control,experimental design, Markov chains, martin-gales, resampling, queueing theory, self-simi-lar processes and the elements of stochasticcalculus. Almost every table is accompaniedby a textual description and at least one exam-ple that uses a value from the table. Most con-cepts are illustrated with examples and step-by-step solutions. One section is devoted toclassic and interesting problems, where thereader finds formulations and solutions of anumber of well-known probability problems,such as Buffons needle problem, Bertrandsbox paradox, Simpsons paradox and the sec-retary problem. The book also contains infor-mation on electronic resources. (mah)

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Ferran Sunyer i Balaguer (1912-1967) wasa self-taught Catalan mathematician who,in spite of a serious physical disability, wasvery active in research in classicalMathematical Analysis, an area in which heacquired international recognition. Eachyear in honour of the memory of FerranSunyer i Balaguer, the Institut dEstudisCatalans awards an international mathe-matical research prize bearing his name,open to all mathematicians. This prize wasawarded for the first time in April 1993.

CONDITIONS OF THE PRIZE§ The prize will be awarded for a mathe-

matical monograph of an expositorynature, presenting the latest develop-ments in an active area of research inMathematics in which the applicant hasmade important contributions.

§ The monograph must be original, writ-ten in English, and of at least 150 pages.The monograph must not be subject toany previous copyright agreement. Inexceptional cases, manuscripts in otherlanguages may be considered.

§ The prize, amounting to 10,000 euros,is provided by the Ferran Sunyer iBalaguer Foundation. The winningmonograph will be published inBirkhäuser Verlags series Progress inMathematics, subject to the usual regula-tions concerning copyright and authorsrights.

§ The submission of a monograph impliesthe acceptance of all of the above condi-tions.

§ The name of the prize-winner will beannounced in Barcelona in April 2002.

SCIENTIFIC COMMITTEEThe winner of the prize will be proposed

by a Scientific Committee consisting of:§ H. Bass (University of Michigan)§ A. Córdoba (Universidad Autónoma de

Madrid)§ W. Dicks (Universitat Autónoma de

Barcelona)§ P. Malliavin (Université de Paris VI)§ A. Weinstein (University of California at

Berkeley)

SUBMISSION OF THE MONOGRAPHSMonographs should preferably be typesetin TEX. Authors should send a hard copyof the manuscript and two disks, one withthe DVI file and one with the PS file(PostScript), and enclosing an accompany-ing letter to the Ferran Sunyer i BalaguerFoundation. Submissions should be sentbefore 1 December 2001 to the followingaddress:Centre de Recerca Matemàtica (IEC)Fundació Ferran Sunyer i BalaguerApartat 50E-08193 Bellaterrae-mail: [email protected]

PREVIOUS WINNERS§ Alexander Lubotzky (Hebrew University of

Jerusalem), Discrete groups, expanding graphsand invariant measures, Progress inMathematics 125.

§ Klaus Schmidt (University of Warwick),Dynamical Systems of Algebraic Origin, Progressin Mathematics 128.

§ V. Kumar Murty and M. Ram Murty(University of Toronto), Non-vanishing of L-Functions and Applications, Progress inMathematics 157.

§ A. Böttcher and Y. I. Karlovich (TUChemnitz-Zwickau) and (Ukrainian Academyof Sciences), Carleson Curves, MuckenhouptWeights, and Toeplitz Operators, Progress inMathematics 154.

§ Juan J. Morales-Ruiz (Universitat Politècnicade Catalunya), Differential Galois Theory andNon-integrability of Hamiltonian Systems,Progress in Mathematics 179.

§ Patrick Dehornoy (Université de Caen),Braids and Self-Distributivity , Progress inMathematics 192.

§ Juan-Pablo Ortega and Tudor Ratiu (ÉcolePolytechnique Fedérale de Lausanne),Hamiltonian Singular Reduction (to be pub-lished).

§ Martin Golubitsky (University of Houston)and Ian Stewart (University of Warwick), TheSymmetry Perspective (to be published).

For further information on the FerranSunyer i Balaguer Foundation, see Web: http://www.crm.es/info/ffsb.htm


The 2002 FThe 2002 Ferran Sunyer i Balaguer Perran Sunyer i Balaguer Prizerize

Société Française de Statistique French Statistical Society

34th Journées de StatistiqueMay 13-17, 2002

The 34th Journées de Statistique of the Société Française de Statistique will be held in Brussels and Louvain-la-Neuve in Belgium. Thisevent is jointly organized by the Institutes of Statistics of the Université libre de Bruxelles and of the Université catholique deLouvain.

The main topics include statistical analysis of functional data, actuarial and financial econometrics, resampling methods, non-parametric inference and modelling, voice and writing recognition, epidemics, genomics as well as mathematical statistics. In addi-tion to these main topics, a number of invited sessions will focus on various subjects.

The conference is open to all persons interested in statistics, affiliated to universities, national institutes of statistics, business orindustry.

For further information, please consult : JSBL 2002

Institute of Statistics, Université catholique de Louvain20 Voie du Roman Pays, B-1348 Louvain-la-Neuve, Belgium

Telephone : +32 10 47 43 54 -Fax : +32 10 47 30 32

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