NEWSLETTER No. 52June 2004

EMS Agenda ........................................................................................................... 2

Editorial by Ari Laptev ........................................................................................... 3

EMS Summer Schools .............................................................................................. 6

EC Meeting in Helsinki ........................................................................................... 6

On powers of 2 by Pawel Strzelecki ........................................................................ 7

A forgotten mathematician by Robert Fokkink ..................................................... 9

Quantum Cryptography by Nuno Crato ............................................................ 15

Household names in Swedish mathematics by Kjell-Ove Widman ..................... 17

"Sauvons la recherche" ......................................................................................... 21

Portugal in the 1st years of the IMU by E. Hernandez-Manfredini, A. Ramos ........ 23

Interview with Ari Laptev ...................................................................................... 25

ERCOM: EURANDOM (Eindhoven) ....................................................................28

Abel Prize, ICME-10 and ICMI awards ............................................................... 30

Forthcoming Conferences ..................................................................................... 36

Recent Books .......................................................................................................... 39

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Published by European Mathematical SocietyISSN 1027 - 488X

The views expressed in this Newsletter are those of the authors and do not necessari-ly represent those of the EMS or the Editorial team.

NOTICE FOR MATHEMATICAL SOCIETIESLabels for the next issue will be prepared during the second half of August 2004.Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department ofMathematics and Statistics, P.O. Box 68 (Gustav Hällströmintie 2B), FI-00014 University ofHelsinki, Finland; e-mail: tuulikki.makelainen@helsinki.fi

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EDITOR-IN-CHIEFMARTIN RAUSSENDepartment of Mathematical Sciences, Aalborg UniversityFredrik Bajers Vej 7GDK-9220 Aalborg, Denmarke-mail: raussen@math.aau.dkASSOCIATE EDITORSVASILE BERINDEDepartment of Mathematics,University of Baia Mare, Romania e-mail: vberinde@ubm.ro KRZYSZTOF CIESIELSKIMathematics InstituteJagiellonian UniversityReymonta 4, 30-059 Kraków, Polande-mail: ciesiels@im.uj.edu.plSTEEN MARKVORSENDepartment of Mathematics, TechnicalUniversity of Denmark, Building 303DK-2800 Kgs. Lyngby, Denmarke-mail: s.markvorsen@mat.dtu.dkROBIN WILSONDepartment of Pure MathematicsThe Open UniversityMilton Keynes MK7 6AA, UKe-mail: r.j.wilson@open.ac.ukCOPY EDITOR:KELLY NEILSchool of MathematicsUniversity of SouthamptonHighfield SO17 1BJ, UKe-mail: K.A.Neil@maths.soton.ac.ukSPECIALIST EDITORSINTERVIEWSSteen Markvorsen [address as above]SOCIETIESKrzysztof Ciesielski [address as above]EDUCATIONTony GardinerUniversity of BirminghamBirmingham B15 2TT, UKe-mail: a.d.gardiner@bham.ac.ukMATHEMATICAL PROBLEMSPaul JaintaWerkvolkstr. 10D-91126 Schwabach, Germanye-mail: PaulJainta@aol.com ANNIVERSARIESJune Barrow-Green and Jeremy GrayOpen University [address as above]e-mail: j.e.barrow-green@open.ac.ukand j.j.gray@open.ac.ukCONFERENCESVasile Berinde [address as above]RECENT BOOKSIvan Netuka and Vladimír SouèekMathematical InstituteCharles University, Sokolovská 8318600 Prague, Czech Republice-mail: netuka@karlin.mff.cuni.czand soucek@karlin.mff.cuni.czADVERTISING OFFICERVivette GiraultLaboratoire d’Analyse NumériqueBoite Courrier 187, Université Pierre et Marie Curie, 4 Place Jussieu75252 Paris Cedex 05, Francee-mail: girault@ann.jussieu.fr

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CONTENTS

EMS June 2004

EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

EXECUTIVE COMMITTEEPRESIDENTProf. Sir JOHN KINGMAN (2003-06)Isaac Newton Institute20 Clarkson RoadCambridge CB3 0EH, UKe-mail: emspresident@newton.cam.ac.uk

VICE-PRESIDENTSProf. LUC LEMAIRE (2003-06)Department of Mathematics Université Libre de BruxellesC.P. 218 – Campus PlaineBld du TriompheB-1050 Bruxelles, Belgiume-mail: llemaire@ulb.ac.beProf. BODIL BRANNER (2001–04)Department of MathematicsTechnical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: b.branner@mat.dtu.dk

SECRETARY Prof. HELGE HOLDEN (2003-06)Department of Mathematical SciencesNorwegian University of Science and TechnologyAlfred Getz vei 1NO-7491 Trondheim, Norwaye-mail: h.holden@math.ntnu.no

TREASURERProf. OLLI MARTIO (2003-06)Department of MathematicsP.O. Box 4, FIN-00014 University of Helsinki, Finlande-mail: olli.martio@helsinki.fi

ORDINARY MEMBERSProf. VICTOR BUCHSTABER (2001–04)Steklov Mathematical InstituteRussian Academy of SciencesGubkina St. 8, Moscow 117966, Russiae-mail: buchstab@mendeleevo.ru Prof. DOINA CIORANESCU (2003–06)Laboratoire d’Analyse NumériqueUniversité Paris VI4 Place Jussieu75252 Paris Cedex 05, Francee-mail: cioran@ann.jussieu.frProf. PAVEL EXNER (2003-06)Department of Theoretical Physics, NPIAcademy of Sciences25068 Rez – Prague, Czech Republice-mail: exner@ujf.cas.czProf. MARTA SANZ-SOLÉ (2001-04)Facultat de MatematiquesUniversitat de BarcelonaGran Via 585E-08007 Barcelona, Spaine-mail: sanz@cerber.mat.ub.esProf. MINA TEICHER (2001–04)Department of Mathematics and Computer ScienceBar-Ilan UniversityRamat-Gan 52900, Israele-mail: teicher@macs.biu.ac.il

EMS SECRETARIATMs. T. MÄKELÄINENDepartment of Mathematics and StatisticsP.O. Box 68 (Gustav Hällströmintie 2B)FI-00014 University of HelsinkiFinlandtel: (+358)-9-1912-2883fax: (+358)-9-1912-3213telex: 124690e-mail: tuulikki.makelainen@helsinki.fi website: http://www.emis.de

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EMS NEWS

EMS June 2004

200419-24 JuneEURESCO Conference.Symmetries and Integrability of Difference Equations: EuroConference on Analytic Linear and Nonlinear Difference Equations and Special FunctionsHelsinki (Finland).25 JuneEMS Executive Committee meeting at Uppsala (Sweden).Contact: Helge Holden, email: holden@math.ntnu.no26-27 JuneEMS Council meeting at Uppsala (Sweden).Contact: Helge Holden, email: holden@math.ntnu.no or Tuulikki Mäkeläinen email: tuulikki.makelainen@helsinki.fi27 June - 2 July 4th European Congress of Mathematics, Stockholm.Web site: http://www.math.kth.se/4ecm4-24 JulyEMS Summer School at Cortona (Italy).Evolution equations and applications.Contact: Graziano Gentili, email: gentili@unifi.it15-23 JulyEMS Conference at Bêdlewo (Poland).Analysis on metric measure spaces.Web site: http://impan.gov.pl/Bedlewo/15 AugustDeadline for submission of material for the September issue of the EMS Newsletter.Contact: Martin Raussen, email: raussen@math.aau.dk30 August - 3 SeptemberEMS Summer School at Universidad de Cantabria, Santander (Spain).Empirical processes: Theory and Statistical Applications.Web site: http://www.eio.uva.es/ems/3-5 SeptemberEMS-Czech Union of Mathematicians and Physicists (Mathematics Research Section).Mathematical Weekend, Prague (Czech Republic).Web site: http://mvs.jcmf.cz/emsweekend/5-6 SeptemberEMS Executive Committee Meeting in Prague.

20055-13 FebruaryEMS Summer School at Eilat (Israel).Applications of braid groups and braid monodromy.25 June - 2 JulyEMS Summer School at Pontignano (Italy).Subdivision schemes in geometric modelling, theory and applications.17-23 JulyEMS Summer School and European young statisticians' training camp at Oslo (Norway).13-23 SeptemberEMS Summer School at Barcelona (Catalunya, Spain).Recent trends of Combinatorics in the mathematical context.16-18 SeptemberEMS-SCM Joint Mathematical Weekend, Barcelona (Catalunya, Spain).25 September - 2 OctoberEMS Summer School and Séminaire Européen de Statistique at Warwick (UK).Statistics in Genetics and Molecular Biology.12-16 DecemberEMS-SIAM-UMALCA joint meeting in applied mathematics.Venue: the CMM (Centre for Mathematical Modelling), Santiago de Chile.

200716-20 JulyICIAM 2007, Zurich (Switzerland).

EMS AgendaEMS Committee

Cost of advertisements and inserts in the EMS Newsletter, 2004(all prices in British pounds)

AdvertisementsCommercial rates: Full page: £230; half-page: £120; quarter-page: £74 Academic rates: Full page: £120; half-page: £74; quarter-page: £44 Intermediate rates: Full page: £176; half-page: £98; quarter-page: £59

Inserts Postage cost: £14 per gram plus Insertion cost: £58 (e.g. a leaflet weighing 8.0 grams willcost 8x£14+£58 = £170) (weight to nearest 0.1 gram)

The Fourth European Congress ofMathematics will take place inStockholm between June 28 and July 2and is organized by the Royal Instituteof Technology (KTH) in collaborationwith Stockholm University (SU).

My possible involvement in the orga-nization of this event became apparentin July 2000 at a Conference in Bristol,when I received an e-mail from AnatolyVershik, who was a member of the EMSExecutive Committee (EC) and was par-ticipating in the 3ECM in Barcelona. Hetold me that there were some difficultiesin raising funds for organizing the next4ECM in 2004, which was planned to beheld Göteborg. On behalf of the EMSEC he asked me for some help since Iwas the Vice President of the SwedishMathematical Society (SMS) at thattime. Anatoly found it hard to believethat Sweden did not have the funds toorganize the 4ECM. Naturally, I did nothave much to say, but in the end ofAugust 2000 there was a meeting atKTH, where it was preliminarily agreedto try to raise funds for the 4ECM inStockholm.

At that meeting everybody assumedthat, as the future President of the SMS,I would take the major responsibility fororganizing 4ECM, although this wascertainly not my original intention.

In November 2000, Anders Lindquist,chairman of the MathematicsDepartment, KTH, Ulf Persson and thenPresident of the SMS, joined me inLondon to attend the EMC ExecutiveCommittee meeting. We were met withan element of friendly curiosity. Tryingto play a “hard game” our delegationwanted to have as much freedom inorganizing 4ECM as possible. To ourgreat surprise we found a lot of under-standing among the EMS EC members.In particular, when we suggestedLennart Carleson as a possible Presidentof the 4ECM Scientific Committee, theEC members said that they had the sameidea. So the meeting ended on a happynote and in mutual agreement.

Mostly due to Anders Lindquist’s con-tacts we were able to raise some sub-

stantial funds amazingly quickly.Among our major sponsors are the Knutand Alice Wallenberg Foundation, theSwedish Foundation for StrategicResearch, the Swedish ResearchCouncil, the Nobel Committees forPhysics and Chemistry of the RoyalSwedish Academy of Sciences and manyothers.

The Swedish Ministry of HigherEducation and the bank of SwedenTercentenary Foundation have also con-tributed to 4ECM funds by each provid-ing financial support of 1/2 million SEK.

We are extremely grateful to all thesponsors. Without their support the4ECM could not have been organized.

The 4ECM Committees worked verywell. Lennart Carleson as the Presidentof the Scientific Committee was mostefficient in organizing two meetings ofhis Committee at KTH. At the firstmeeting the decision regarding the struc-ture of the 4ECM was made and in June2003, at the Committee’s second meet-ing, all the speakers were chosen, withthe exception of those speakers forScience Lectures. The OrganizingCommittee chose the latter afterwards,in coordination with L. Carleson.

The task of finding a President of thePrize Committee was rather difficult andthe EMS Executive Committee was veryhappy when Nina Uraltseva finallyagreed to take on this important respon-sibility. The Prize Committee wasquickly formed and its work culminatedat the meeting in Stockholm at the end ofFebruary, when 10 EMS Prize Winnerswere chosen.

Being invited to the meeting of thisCommittee I can confirm that the maindecisions were made surprisingly quick-ly and in a very friendly way. It is noteasy to choose 10 prizewinners frommore than 50 young mathematiciansnominated for an EMS prize and whoalmost all deserve such a prize.

On behalf of the 4ECM OrganizingCommittee I would like to express mygratitude to the members of theScientific Committee for choosing anextremely interesting programme and to

the members of the Prize Committee fortheir hard work in selecting excellentcandidates for the EMS prizes.

By now the Organizing Committee hasconsidered two rounds of 4ECM posterapplications. There were two deadlines -15th of February and 20th of April.Altogether 358 posters were accepted.Unfortunately not all of them will bepresented at the 4ECM since some of theposter applicants depend on receivinggrants in order to attend.

There were more than 400 4ECMgrant applications, mostly from EasternEurope and the Former Soviet Union.Unfortunately our financial possibilitiesallow us to give only 209 grants, whichcover travel, a double hotel room, lunch-es and the Conference fee. We are verygrateful to the EMS EC for helping us toobtain $25,000 from UNESCO-ROSTE,which allowed us to support participantsfrom Central and Eastern Europeancountries. In total we have more than1,600,000 SEK for the 4ECM grants,which are approved mostly for youngparticipants.

Although organizing a Congress withmore than 1000 participants is hardwork, I have found it very rewardingtoo. It allowed me to start seeingMathematics much more as a whole sub-ject, with many people trying to createbeautiful new structures and find bril-liant methods with which to solve oldand new exciting problems.

Having only 1½ months remaining Iam beginning to feel slightly apprehen-sive about how it will all go. Almostfour years ago I could only guess howmuch work has to be done and even nowI am still not quite sure I understandfully!

EDITORIAL

EMS June 2004 3

EditorialEditorialAri Laptev (Stockholm)

Chairman, 4ECM Organizing Committee

Ari Laptev

EEvvoolluutt iioonn EEqquuaatt iioonnss aanndd

AAppppll iiccaatt iioonnss

SUMMER SCHOOL of the European Mathematical Society

Scuola Matematica Interuniversitaria

Supported by the Istituto Nazionale di Alta Matematica and the

European Commission

Cortona (Palazzone), Italy

4th to 24th July 2004

The SUMMER SCHOOL will be devoted to evolution problems, includ-

ing:

• semigroups and evolution equations in Banach spaces;

• linear and non-linear parbolic type problems using functional

analytical methods.

The School will provide an occasion to facilitate and stimulate discus-

sion between all participants, in particular, young researchers and

lecturers, for whom financial support is available.

Courses• Semigroups for linear evolution equations (Reiner Nagel,

University of Tubingen, Germany)

• Linear and non linear diffusion problems (Alessandra Lunardi,

University of Padova, Italy)

Website: http://www.matapp.unimib.it/smi

Some time ago, we wrote that theEuropean Mathematical Society hadbeen unsuccessful in its application forFramework 6 support of its summerschool programme. No sooner had thearticle appeared, than Vice-PresidentLuc Lemaire was contacted and toldthat the project had been promotedfrom the reserve list. The EMS wasworking in close collaboration withthe European Regional Committee ofthe Bernoulli Society, the Centre deRecerca Matematica (CRM) ofBarcelona, the Scuola MatematicaInteruniversitaria (SMI) of Firenze,the Mathematical Research andConference Centre in Bêdlewo(Poland), the Emmy Noether Institute(German-Israeli research institutelocated in Israel) and various groups ofmathematicians working in Europeanuniversities. We shall pass over thetraumatic hours of form filling and fur-ther negotiation, familiar to anyonewho gets involved in European grants,but in March the EMS obtained amajor grant of the EuropeanCommission (Marie Curie Series ofConferences) for the following eightSummer Schools. In the list below,attentive readers will perceive that notall the events happen in summer, butall bar the Bêdlewo meeting do countas EMS Summer Schools.

4-24 July 2004 at Cortona, Italy:Evolution equations and applications;

15-23 July 2004 at Bêdlewo, Poland:Analysis on metric measure spaces;

12-19 December 2004 at Munich,Germany: The Statistics of Spatio-Temporal Systems;

5-13 February 2005 at Eilat, Israel:Applications of braid groups andbraid monodromy;

25 June - 2 July 2005 at Pontignano,Italy: Subdivision schemes in geomet-ric modelling, theory and applica-tions;

17-23 July 2005 at Oslo, Norway:European young statisticians’ training camp;

13-23 September 2005 at Barcelona,Spain: Recent trends of Combinatoricsin the mathematical context;

25 September - 2 October at Warwick,UK: Statistics in Genetics andMolecular Biology.

We also obtained support fromUnesco-Roste for the SummerSchools:- Analysis on metric measure spaces,

Bêdlewo, Poland, 15 to 23 July2004;

- EMS Course on EmpiricalProcesses: Theory and StatisticalApplications in Laredo, from 29August to 3 September 2004.

For information on participation andconditions for financial support, pleasecheck on the Society’s web site:http://www.emis.de/etc/ems-summer-schools.html

Summer schools for the followingyearsOn June 24, the EU will have decidedon some modifications of the rules ofapplications for Summer Schools sup-port. We shall thus launch a new callfor proposals.

Call for proposalsThe EMS will pursue its programme ofSummer schools, aiming at runningsuch schools in pure and applied math-ematics. The guidelines for such eventsare that there must be a very strongcomponent of training of youngresearchers (in the first 10 years of theircareer) by means of integrated coursesand lectures at advanced level. This canbe supplemented by conference typeresearch lectures, but the training com-ponent is needed. The courses shouldaim at an international audience (nomore than 30% of participants shouldcome from a single state). The EMS

will help with advertisement andorganisation, as well as the applicationsfor financial support.

Proposals for future Summer Schoolscan be sent at any time by e-mail to LucLemaire (e-mail: llemaire @ulb.ac.be)

The deadline for proposals for EMSSummer Schools for 2006, 2007 and2008 is tentatively fixed on December31, 2004.

EMS mathematical weekendsThe EMS has launched a new format ofjoint meetings with its corporate mem-ber societies, following the model setout by the Portuguese MathematicalSociety in the meeting that took placein Lisbon from September 12 to 14,2003 (http:// www.math.ist.utl.pt/ems/).

These “EMS- joint mathematicalweek-ends” will start on a Friday, andfinish on the Sunday, both atlunchtime, so that they can be easilyattended during term-time. Each wouldcover around 4 subjects, chosen by thelocal organisers to fit the researchstrengths of the local mathematicians,or new subjects they would want todevelop. For each subject, a plenarylecture and two half-days of parallelsessions will be organised. Past experi-ence shows that such an internationali-sation of the meetings of national soci-eties helps to substantially increaseparticipation. The EMS will help withscientific organisation, publicity andfunding applications. With more thanfifty corporate members, the EMShopes to see regular meetings of thisformat. Note that mathematics depart-ments or individual members can alsoplan such meetings.

Two more Week-Ends are planned,one in Prague, Czech Republic,September 3-5, 2004 (http://mvs.jcmf.cz/emsweekend), and one in Barcelona,September 16-18, 2005.

Call for proposalsProposals for future JointMathematical Weekends can be sent atany time by e-mail to Luc Lemaire (e -mail: llemaire@ulb.ac.be).

EMS NEWS

EMS June 2004 5

EMS SummerEMS Summer SchoolsSchoolsLuc Lemaire (Brussels) and David Salinger (EMS Publicity Officer)

It wasn't cold for Helsinki at the end ofFebruary, just a few degrees belowzero. The streets were slippery and thesea was frozen, but the ExecutiveCommittee met indoors and got onwith its business.

The best news was that, after all, wewere likely to get funding for the sum-mer schools from the EU's sixth frame-work programme. Final confirmationarrived after the meeting and the fulllist is in this issue's agenda. This is theculmination of very hard work, innegotiation and form-filling, in partic-ular by Luc Lemaire and TuulikkiMäkeläinen. It means also that anadditional staff member, MikuKoskenoja, can be employed at theSociety's Helsinki office.

Other good news is that the Societyadmitted as Institutional members theMathematisches ForschungsinstitutOberwolfach, Institut Henri Poincaré,Centre Emile Borel, the InternationalCentre for Mathematics, and the AbdusSalam International Centre forMathematical Physics. The Committeerecommended to Council the followingsocieties for corporate membership:the Cyprus, Malta and RomanianMathematical Societies and theBelgian Statistical Society. It washoped that other statistical societieswould follow that lead; three of theseven forthcoming summer schoolswere in statistics.

The Society had been asked to sug-gest names to the European ScienceFoundation for a panel to considerapplications for young investigators'fellowships.

The Committee agreed to recom-mend to Council that the corporatemembership fee should be increasedslightly, but that the individual mem-bers' rate should remain the same.

The Committee had received no pro-posals for Diderot forums or EMS lec-tures. It was recognised that the for-mer were difficult to organize anddepended on local enthusiasm. Severaldifferent models were suggested torevive the EMS lectures: it was agreedto discuss this further at Council.

There was a strong field of candi-dates for the EMS prizes, but no nomi-

nations had been received for the FelixKlein prize.

It was agreed to support a mathe-matical weekend in Prague inSeptember 2004. The offer of a math-ematical weekend in Barcelona waswelcomed: this is now to take place on16-18 September 2005.

The President reported that GérardHuet, Terry Lyons, Simon Tavare andMartin Groetschel had agreed to speakat the European Science Open Forummeeting in Stockholm in August. LucLemaire had been invited to give a ple-nary lecture.

A meeting had been held in Brusselsabout the proposed European ResearchCouncil. If this went ahead, theSociety would have to be active toensure that funding through theResearch Council would be mathemat-ics-friendly.

The Committee discussed whetherthe Society's journal, JEMS, shouldhave a second series devoted to appliedmathematics. The chair of the AppliedMathematicss committee would beasked to investigate the viability ofsuch a journal. It was important thatJEMS should be recommended tolibraries by mathematicians, because,

now that it was being published by ourPublishing House, it was no longerincluded in the Springer package.

An additional 3000 euro wasapproved to be used for grants to sup-port 4ecm.

The French ministry had produced areport on Zentralblatt, which, whilecritical of a few aspects, gave overallsupport. It had recommended the set-ting up of an advisory board and theCommittee endorsed this approach.

Unfortunately, no one on theCommittee was able to carry forwardthe Digital Maths Library initiative.

The EMS publishing house hadstarted producing books and journals.It hoped to move into profit in 2005.

The Newsletter now had a copy edi-tor. Plans were afoot to have parts ofthe Newsletter in an electronic data-base.

The Committee had received a pro-posal asking it to support a centre ofmathematical excellence in Palestine(Bir-Zeit University). It was agreed toask the chair of the DevelopingCountries committee to look into this.

The President thanked Olli Martioand Tuulikki Mäkeläinen for organis-ing the meeting.

EMS NEWS

EMS June 20046

The Helsinki Meeting of the ExecutiveThe Helsinki Meeting of the ExecutiveCommitteeCommittee

David Salinger, EMS Publicity Officer

EC meeting

FEATURE

EMS June 2004 7

On powers of 2

Paweł Strzelecki (Warsaw)

Let’s have a look at the following interesting little problem:

Problem: Consider a sequence (an)n∈N consisting of the first digits of the successive powers of 2:

1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, ...

Will 7 ever appear in this sequence?

This is a translation from Polish of an article that appeared in the Polish journal Delta 7/1994.

This problem, in different versions,is well known in mathematical litera-ture. It can be found in A walking mathe-matician, by Zbigniew Marciniak (Delta7/1991), or in the famous book Ordi-nary differential equations, by V.I.Arnold.It is usually accompanied by auxiliaryfacts or suggestions, intended to makea solution more accessible. Neverthe-less, I do not know of any publicationin which the problem is presented to-gether with a complete solution, inclu-ding every detail. Let us try to re-medy this unfortunate situation. Westart with a rather desperate solutionrequiring just a piece of paper and apencil, or perhaps some more advan-ced versions of these tools (a good cal-culator or a computer?). An assiduousreader will easily verify that

246 = 70, 368, 744, 177, 664.

Going further with this experiment itcan be seen that 7 is the first digit of the56th, 66th, 76th, 86th and 96th power of 2(but the first digit of the 106th power of2 is 8, not 7). This method, however, isclearly far from being elegant.

It is time then for a better solution,which might also lead to new conclu-sions. First of all, we should try torealize what the statement, that 7 isthe first digit of a number 2n, actuallymeans.

The answer is simple: 7 is the first di-git of 2n if and only if for some natural

k we have

7 · 10k < 2n < 8 · 10k

We can get a simpler description of thiscondition if we take the decimal loga-rithms of both sides; this yields

k + log(7) < n log(2) < k + log(8).

Since decimal logarithms of 7 and 8 liebetween 0 and 1, we conclude that k issimply the integer part of the numbern log(2), which leads to the followinginequalitites:

log(7) < n log(2) − [n log(2)] < log(8).

And now it suffices to bring togethersome known facts.

Lemma 1. The number log(2) is irra-tional.

Lemma 2. If a number x is irrationaland c(n) := nx − [nx], then for any aand b in [0,1] infinitely many membersof the sequence c(n) lie within the in-terval (a, b).

Before we prove these two lemmaslet’s have a look at their consequences.First, by applying Lemma 2 so thatx = log(2), a = log(7), b = log(8),we get that 7 is the first digit of infi-nitely many powers of 2. If we applyLemma 2 again, to get the numbers x =

log(2), a = log(77)− 1, b = log(78)− 1,then because of the equalities 1 =[log 77] = [log 78], we conclude thatthe digit 7 can even appear twice, inthe first two places of the decimal no-tation of a power of 2. Using an ana-logous argument we readily discoverthat any finite sequence of digits canappear at the beginning of a decimalnotation of a power of 2, like 1995 or1234 or 567890 etc. At the end of thisarticle we exhibit a table of importanthistorical dates compared with the cor-responding powers of 2 - to satisfy thesceptics. We can even deduce more:

Corollary. If an integer p > 1 is not aninteger power of 10, then any sequenceof digits can appear at the beginningof the decimal notation of some nth po-wer of p for some n.

To prove the corollary, just observethat log(p) is irrational and then repeatthe same argument as above.

But then, why does 7 not appearamong the first members of the se-quence we introduced at the very be-ginning? Why does this deceitful se-quence pretend to be periodical? Thereason is simple. The number log(2) =0.3010299956... can be very well ap-proximated by the rational number 0.3,and for all rational x the sequencec(n) = nx − [nx] is periodical. This iswhy after seeing the first initial mem-

On powersOn powersof 2of 2

Pawe³ Strzelecki (Warsaw) Pawe³ Strzelecki (Photo "die arge lola", Stuttgart)

FEATURE

EMS June 20048

bers of the sequence a(n) we come toan erroneous conclusion that the se-quence has period 10 and 7 is not amember of it, while 8 appears quite of-ten.

In 1910 Waclaw Sierpinski, HermannWeyl and P. Bohl proved indepen-dently of each other that for an irra-tional x the sequence c(n) = nx − [nx]is homogeneously distributed over theinterval [0,1]. More precisely, if we takearbitrary a and b (with a < b) in [0,1],and let k(n, a, b) denote the number ofelements of the set

{c(i) : 1 ≤ i ≤ n, c(i) ∈ (a, b)},

then

limn→∞

k(n, a, b)n

= b − a

In more illustrative terms, the theorem

can be rephrased as follows: If we walkalong a circumference of perimeter 1for long enough and make steps of anirrational length, then we shall step oneach hole with a frequency which isdirectly proportional to the size of thehole.

Let’s translate this fact into our lan-guage of powers of 2. Let a(7, n)and a(8, n) be the number of sevensand eights, correspondingly, amongthe first n members of the sequencea(n). By the last formula, we have

limn→∞

a(7, n)n

= log(8) − log(7),

limn→∞

a(8, n)n

= log(9) − log(8),

so consequently

limn→∞a(7,n)a(8,n) = log(8)−log(7)

log(9)−log(8)= 1.1337... > 1.

This means that looking at suffi-ciently long initial fragments of the se-quence a(n), we will see slightly moresevens than eights.

The result of Bohl, Sierpinski andWeyl which we have previously men-tioned and our little fact on sevensand eights are actually simple conse-quences of a very general and deepergodic theorem due to G.D.Birkhoff(1931), but that’s another story.

A problem for the reader: For whatn does the number 2n have four conse-cutive sevens at the beginning? Whatabout five sevens? How do you esti-mate from above, the least n such thatthe decimal notation of 2n begins with1995 consecutive sevens?

A calendar involving powers of 2

EVENT Year Power of 2Foundation of the Polish State 966 2568 = 9.66... × 10170

The King of Poland Jan III beats the Turcs at Vienna 1683 2709 = 1.683... × 10212

The first edition of Newton’s Principia 1687 26143 = 1.687... × 101849

The Battle of Waterloo 1815 2931 = 1.815... × 10280

Foundation of the Warsaw University 1816 27339 = 1.816... × 102209

Birth of Stefan Banach 1892 28133 = 1.892... × 102448

Beginning of World War II 1939 25522 = 1.939... × 101662

End of World War II 1945 21931 = 1.945... × 10581

First published issue of ”Delta” 1974 21549 = 1.974... × 101081

AppendixProof of Lemma 1

If the equality log 2 = rm held true for

some natural numbers r and m, then bythe definition of logarithm, 10

rm = 2 and

consequently 10r = 2m. This, however, is acontradiction, since 5 is a factor of any po-wer of 10, whereas it is certainly not a factorof a power of 2.

Proof of Lemma 2First note that all the members of the se-

quence (cn)n∈N are different. Indeed, ifck = cm for some k, m ∈ N with k > m, then(k − m)x = �kx� − �mx�. However, the pro-duct of a positive natural number and anirrational number cannot be an integer.

Now, let n be a natural number such that1n < b − a. The numbers c1, c2, c3, ..., cn+1being pairwise different with ci ∈ [0, 1] fori = 1, 2, 3, ..., n + 1, we can use the pigeon-hole principle to infer that for some s and t,both s and s + t are included between 1 andn + 1 and satisfy the inequality

(1) 0 < ε := |cs − cs+t| ≤ 1n

< b − a.

From now on we will refer to a very use-

ful representation of points. Imagine thereal axis is wound around a circle T withradius 1 and a distinguished point 0. Foreach pair of numbers a, b ∈ [0, 1] the arc of Tthat corresponds to the interval [a, b] on thereal axis will be also denoted by (a, b). Letf : T −→ T represent the anticlockwise ro-tation by an angle of 2πx radians. Now, ins-tead of considering the numbers cn as ele-ments of the interval [0, 1], let’s think of theimages of the distinguished point 0 undersuccessive iterations of the function f on T.In fact, a moment of reflection should makeit clear that if

bn = f n(0) =

n︷ ︸︸ ︷

f ◦ . . . ◦ f (0),

then the length of the arc (0, bn) is equal tocn. Thus, due to equation (1), the length ofthe arc between the points bs and bs+t is lessthan b − a. This means that the tth iterationof f is a rotation by an angle of 2πε radians;the direction of this rotation has no signifi-cance for our purposes.

Now, this clearly implies that there areinfinitely many points bt, b2t, b3t, ... that be-long to the arc (a, b). Thus if you start at

the point 0 and follow along the circle T foran infinitely long time in the same directionmaking steps of length ε, then you step in-finitely many times over the arc (a, b). In-deed, its length b − a is bigger than thelength of your step. This completes theproof.

Paweł Strzelecki [P.Strzelecki@mimuw.edu.pl]studied mathematics in Warsaw and graduatedin 1987. He finished his PhD-thesis under thesupervision of Bogdan Bojarski in 1993. Sincethat time he has been working as an assistantprofessor at the Institute of Mathematics atWarsaw University where he now also servesas the deputy director for teaching affairs. In1994-1999 he served as the mathematics editorfor the Polish popular mathematical-physical-astronomical monthly ”Delta”. In 1993-1998he presented mathematics in a Polish TV pro-gram for young people. He was an Alexandervon Humboldt fellow in Bonn (1999 – 2001and 2002 – 2003). He likes nonlinear ellipticPDEs, clever hard-analytic proofs, compensatedcompactness and Sobolev spaces, bright moun-tain mornings, bright people and good food (nomatter from which corner of the world). He ishappily married to a non-mathematician.

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A forgotten mathematician

Robbert Fokkink (Delft, the Netherlands)

The original Dutch version of this article was published in Nieuw Archief for Wiskunde 5 (1), March 2004. The Newsletter’sthanks go to Nieuw Archief for the premission to reproduce it and to Reinie Erne (Leiden) for the translation into English.

Not so long ago, I was listening to alecture in which a fine result was ca-sually mentioned. It was a classical em-bedding theorem of Haefliger, whichroughly says that you can embed aspace X in Rn if the reflection (x, y) →(y, x) on X × X is a conjugate of the an-tipodal map x → −x on Sn−1. Someresearch into the literature on this areaquickly showed that E.R. van Kampenwas the first to come up with suchembedding theorems, sometime in theearly thirties. Very little about his workcan be found in mathematical historybooks and his collected works were notto be found in the library. This is re-markable, because E.R. van Kampen isa much cited mathematician, who evenhad a component of the AMS classifica-tion named after him (20F06). So whowas he?

Egbert van Kampen (l) with his sister Elizabeth and brother Laurens. Like Egbert, Eli-zabeth studied mathematics in Leiden. Laurens became a professor of economy at theVrije Universiteit in Amsterdam. Both lived to the age of 96.Photo: Archief Mieke van der Veen

The life ofE.R. van Kampen

Egbertus Rudolf van Kampen, knownas Egbert, was born on 28 May 1908in Berchem as the youngest in a familyof three children. Egbert’s parents hadmoved from the Netherlands to Bel-gium a couple of years before, when hisfather became an accountant at the Mi-nerva car factory in Antwerp.

The children spend the war years1914-18 with family in Amsterdam.When the war is over, the family moveto The Hague, where Egbert completeshigh school, Eerste Christelijke H.B.S.At his final examination in 1924, heis mentioned in the national press.The Telegraaf and the Amsterdammerreport of a young mathematician, a

unique talent in Europe, who has ob-tained his teaching qualifications formathematics, but nevertheless was notexempted from this subject. His studyof mathematics and physics continuesin Leiden, where he probably did getthe necessary exemptions as he com-pletes the studies very quickly. In 1927,at the age of only nineteen, he travelsto Gottingen. There he meets Alexan-droff and van der Waerden and is gi-ven the subject for his Ph.D. thesis: finda computable, topological definition ofthe notion “variety”. Egbert solves theproblem by showing that the local ho-mology group is a topological inva-riant. According to Van Kampen, a va-riety is a simplicial complex with theright local homology. Armed with thisdefinition, he generalizes the Alexan-der duality theorem from spheres to

general varieties and expands Veblen’stheory of intersections and intersectionmultiplicities. In 1929, he obtains hisPh.D. with the thesis Die kombinato-rische Topologie und die Dualitatssatze.The thesis, directed by Van der Woude,is missing from the University Libraryin Leiden.

In the summer of 1928, Van Kam-pen spends five months in Hamburg,where he meets Emil Artin. Aroundthat time, he is recruited by Johns Hop-kins University, but because he is stilltoo young to enter the United Stateswithout the accompaniment of his pa-rents, he first becomes an assistant toSchouten in Delft. He writes a num-ber of papers with Schouten on ten-sor analysis, which is Schouten’s spe-cialty. [4, 6, 9]

AA forgottenforgottenmathematicianmathematicianRobbert Fokkink (Delft, the Netherlands)

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In 1931, Van Kampen departs to JohnsHopkins. In his autobiography, MarkKac gives an anecdote about Van Kam-pen’s arrival in the USA: ”Universityprofessors, clergymen and those of se-veral other occupations could emigrateoutside the national quotas and VanKampen had one of these highly co-veted non-quota visas. There was astory that at the immigration desk hewas asked a routine question and ins-tead of answering ’professor’ or ’tea-cher’ or even ’mathematician’, he ans-wered ’topologist’. The immigration of-ficial could not find ’topologist’ on thelist of occupations qualifying for a non-quota visa and he naturally asked whata topologist does. As Van Kampen triedto explain, the immigration official gai-ned the impression that he was dealingwith a nut. He detained Van Kampenfor further examination and Johns Hop-kins authorities had to step in to extri-cate him. Van Kampen never confirmedor denied this story - but it was comple-tely in character”.

Van Kampen (l) with Ehrenfest, Rotterdam circa 1927

In his first year in the U.S., Van Kam-pen writes a paper on the fundamentalgroup, with which he truly establishesa name for himself. In 1933, he spendsa year at the Institute for AdvancedStudy, and in 1935 he gives a talk ontopological groups at the InternationalCongress at Moscow.

In his mid-twenties, Egbert vanKampen seems to be at the beginningof a brilliant career, but in reality hehas already done his most importantwork. He remains particularly produc-tive and, together with Aurel Wintner,fills the American Journal of Mathema-tics.

At the end of the thirties, in a let-ter home, Van Kampen begins to com-plain of headaches. He sees a specia-list who diagnoses a neck problem. Asa result he starts physiotherapy but itis later determined that the problem ismore serious, a melanoma that origina-ted from a birthmark near his left eye.In April 1941, it is operated on and inthe autumn Egbert takes up his coursesagain, full of good hope for his reco-very. This however is not to last andhe once more suffers from violent hea-daches and becomes deaf in his left ear.In December 1941 he is once more ad-mitted to the hospital where his healthdeteriorates quickly. In January 1942 asecond operation follows in vain. OnFebruary 10, Egbert falls into a coma.He dies the next day, only 33 years old.

The work ofE.R. van KampenEgbert van Kampen has worked onmany subjects: topology, group theory,tensor analysis, harmonic analysis andprobability theory. His most well-know articles are on topology andgroup theory. He was thought of assomewhat of an oracle, as part of hiswork was only discovered when othersstarted realizing it again.

The first publication in 1928Van Kampen’s first article [1] appearsin the Hamburger Abhandlungen in1928. It consists of ten sentences andone illustration and is exactly what onewould expect from a brilliant youngperson: original and illegible.

It concerns two curves α and β thatare attached to a surface V, of which itis questioned whether they can be se-parated from each other, without theendpoints leaving the surface.

In 1925, in an article with the sametitle, Emil Artin had claimed the follo-wing: if α and β are linked, then eitherα or β has a knot. The illustration inVan Kampen’s article shows that thisstatement is incorrect. It is clear thatα and β do not have a knot, but thatthey are linked anyway. Van Kampenshows this by computing that the fun-damental group of the complement ofα ∪ β in half-space is not free. He firstdraws an auxiliary line γ through the

endpoints of α and β, which, for thatmatter, is not strictly necessary. Withsome effort one can then see that thecomplement of α ∪ β ∪ γ in R3 has thesame fundamental group as the com-plement of α ∪ β in half-space. Thisgroup has three normal subgroups ofindex 2 and Van Kampen notes thatone of those has the group Z3 ⊕ Z/3Zas quotient. It follows from this thatthe fundamental group can not be free.The theory that Van Kampen is expan-ding is at that time brand new and notentirely worked out yet. The compu-tations that he needed to do were nosmall matter. Perhaps that is why heleft out all the details. Nowadays thecomputation is much simpler, not inthe least because of the work of VanKampen himself.

Van Kampen’s embedding obs-tructionIn 1932, Van Kampen publishes an ar-ticle concerning embeddings of com-plexes in Euclidean spaces [7]. Again,Artin’s article from 1925 providesthe inspiration. Van Kampen stu-dies an embedding problem that willonly be definitively solved much la-ter: under which conditions does ann-dimensional complex fit in R2n? In1930, Kuratowski had solved this forn = 1. Perhaps Van Kampen was notaware of that result, because he doesnot mention Kuratowski and he dis-poses of the case n = 1 as being simple.

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The first article (1928) of Van Kampen in the Hamburger Abhandlungen

Van Kampen’s proof only holds forn ≥ 3, but his criterion is best illus-trated using the “simple” case n = 1.In that case the complex is a graphG. For example, draw 5 points in theplane and join each of them to all theothers. The result is a depiction of thecomplete graph with 5 nodes in theplane. Some connecting lines may in-tersect each other, so the map is not anembedding. You might try to make itinto an embedding by trial and error,by removing intersecting edges and re-placing them by new edges. For thegraph with 5 nodes, any such attemptultimately ends unsuccessfully. In Fi-gure 1, for example, there is only oneintersection point left, which can not beremoved. Van Kampen shows that thistrial and error takes place within a co-homology class and that the graph canbe embedded precisely when that classis zero. At this time he uses other ter-minology, because in 1932, cohomologytheory has not really been developedyet.

Figure 1 The complete graph with 5nodes

For a map f : G �→ R2 with fini-tely many double points, Van Kam-pen defines a vector v with coordinatesin Z/2Z. The number of coordinatesof v is equal to the number of pairs{I, J} of disjunct edges of the graph.The {I, J}th coordinate of v determinesthe parity of f (I) ∩ f (J). Van Kam-pen then defines vectors w(p, E) in thesame space (Z/2Z)N as v, where p is avertex and E is an edge that does notcontain p. The {I, J}th coordinate ofw(p, E) equals 1 if and only if p ∈ I andE = J, or vice-versa. Let L be the linearspace generated by the w(p, E). VanKampen’s embedding criterion is thenthat G can be embedded if and only ifv ∈ L. For the map of the graph above,v has a single 1 and zeros everywhereelse. For all vectors w(p, E) the sum ofthe coordinates is even, so in this casev �∈ L.

In the higher dimensional case n ≥ 3,the edges become n-dimensional cellsand the vertices become (n − 1)-

dimensional cells. The definition of thevector v uses the “intersection num-ber”, the theory of which Van Kampenhad set up in his thesis. He first notesthat the obstruction against an embed-ding can only lie in the n-dimensionalcells. Next he deduces a criterion infour steps. Lemma 1 states that doublepoints can be removed pairwise, soonly the parity matters. Lemma 2states that double points on adjacentedges are no problem, so only disjunctsubgraphs matter. Lemma 3 states thatfor every vector in v + L there is a mapf : K �→ R2n giving that vector. Lemma4 conversely states that every f : K �→R2n gives a vector in v + L. The com-plete proof is barely one page long. Asan added bonus, Van Kampen showsthat every n-dimensional variety canbe embedded in R2n, a result that isnow known as Whitney’s embeddingtheorem.

It is an amazing result with only oneflaw: the proof of Lemma 1 is wrong.

Afterwards, Van Kampen sends a hastyrectification indicating a partial correc-tion. He writes that it is “unneces-sary to neglect orientation”, by whichhe probably means that the coefficientsmust be taken in Z instead of in Z/2Z.

After this Van Kampen no longer pu-blishes about embeddings, even whenWhitney develops a complete theorya couple of years later. Only in 1957does Arnold Shapiro show that VanKampen’s embedding criterion followsfrom the results of Whitney.

As noted before, Van Kampen’sproof only holds for dimension n ≥ 3.It is therefore surprising when in 1991,Sarkaria shows that the criterion is alsocorrect for n = 1. In 1994, Freedman,Krushkal, and Teichner show that thecriterion does not hold for n = 2. Kru-shkal and Teichner also introduce thename “Van Kampen’s embedding obs-truction”, some sixty years after the pu-blication of the original article.

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Group photo of staff and post docs at the Johns Hopkins campus in 1933, with f.l.t.r.: Egbert van Kampen, George Schweigert, BeatriceAitchison, William Morrill, Oscar Zariski, Charles Wheeler and John Williamson.Copyright: Johns Hopkins University Library

Three consecutive articles onthe fundamental group

At the beginning of the thirties, Os-car Zariski, arguably the most famousmathematician from Hopkins, works onthe fundamental group of the comple-ment of an algebraic curve. An alge-braic curve can be knotted in the pro-jective space, which, as in the case ofa usual curve in R3, can be read off thefundamental group of the complement.Zariski determined the generators ofthe fundamental group and found re-lations between them, but it was notclear to him then whether or not therewere more relations. Zariski presentedthe problem to Van Kampen who thensolved it and wrote three consecutivepapers [10, 11, 12] on the problem inthe American Journal of Mathematics,the internal publication of Johns Hop-kins. All three articles have becomestandard references, although they arerarely mentioned together. The alge-braic geometers refer to the first article,the topologists to the second, and thegroup theorists to the third. This is re-markable as in fact only the first articlestands alone, and even then only ba-rely.

The articles have been written inan uninvolved tone. The first articlestates: As the resulting proof seemed tooalgebraic for this simple and nearly topo-logical question, Dr. Zariski asked me topublish a topological proof which is contai-ned in this paper. Part two states: Theopportunity of simplifying the treatment ofa fundamental group by means of this theo-rem has been overlooked several times [. . . ]for this reason we do not think it super-fluous to devote a separate paper to it. Partthree states: It is this 2-dimensional me-thod of the proof and not any original re-sult which justifies the publication of thispaper.

The first part of the three articlesconcerns Zariski’s problem, which VanKampen solves by skillfully cutting thespace in two pieces U and V whicheach have a free fundamental group.He then divides every homotopy intosmall pieces that lie either completelyin U or completely in V. As the fun-damental group of both U and V isfree, the only freedom of choice isfor the pieces of homotopy that lie inthe intersection U ∩ V. This freedomturns out to exactly correspond withthe relations of Zariski. Therefore thereare no other relations. This result is

now known as the Zariski-van Kam-pen theorem.

The second article continues with thequestion of how one computes the fun-damental group of a reunion U ∪ Vfrom the groups of U and V. In its mo-dern form this result is formulated bymeans of coverings, but Van Kampenturns his attention to the cut that di-vides the space in two or more pieces.He is not entirely satisfied with his re-sult and writes: a path is shown to ageneral theorem, of which the general for-mulation would be more confusing thanhelpful, so that it is suppressed. Thissuppressed general theorem probablyconcerns a cut that is as general as pos-sible, which divides the space into anarbitrary number of pieces. It is Corol-lary 2 of this article, in which a sim-ply connected set cuts the space intotwo pieces, that will later be calledthe Seifert-van Kampen Theorem. Thistheorem was in fact first proved by Sei-fert, another Ph.D. student of Van derWaerden.

An annoying technical detail thatVan Kampen comes across is that whenhe divides a homotopy, he must conti-nue to link everything to the base

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point. All small pieces of homotopymust be linked to it. In the algebraictranslation, the pieces of homotopy be-come relations, and the links to thebase point become conjugations. Inthe third part of the trio of articles,Van Kampen makes a virtue of neces-sity by noting that the inverse transla-tion gives a graphic representation ofgroup relations. He translates words ingroups into bouquets of curves, whicheach represent the conjugacy class of arelation. With this manner of thinkinghe simplifies the proof of a result ofSchreier. This third article is only dis-covered in 1966, when Roger Lyndonand Carl Weinbaum use the diagramsto study relations with “small cancella-tion”. The bouquets of curves are nowcalled van Kampen diagrams.

The typical form of a Van Kampen diagram:relations on stalks. Rewriting a word cor-responds to gluing the bouquet together.

Pontrjagin-van Kampen dua-lity and later work

In 1933, Van Kampen spends a yearin Princeton, at which time Pontrjaginproves that compact Abelian groupsare dual to discrete Abelian groups.Pontrjagin assumes the groups to beseparable, because in 1933, the Haarmeasure has only been constructed forsuch groups. Van Neumann remarksthat duality also holds without separa-bility, if instead of the Haar measure,one uses Von Neumann’s theory foralmost-periodic functions. Van Kam-pen takes this up and writes a sur-vey article on the duality of groups[16].Since then this duality is also called thePontrjagin-van Kampen duality.

On this topic, Van Kampen is quitelavish and writes sixteen pages, begin-ning with the definition of a topolo-gical group and ending with the dua-lity of locally compact Abelian groupswithout the restriction of separability.The article is well readable and writ-

ten for the general public, in contrast toVan Kampen’s previous work, whichgave the impression of being merely areminder for the author himself.

This marks the beginning of a newperiod. After 1935, Van Kampen leavesthe area of algebraic topology for whatit is and mostly studies almost-periodicfunctions and differential equations. Itis no coincidence that this is the area ofinterest of Aurel Wintner, a colleagueat Johns Hopkins. Egbert van Kampenhas always allowed himself to be inspi-red by the people around him. At theend of his life, when the young MarkKac comes to Baltimore, Van Kampeneven writes a number of articles onprobability theory. His last article, ofconsiderable size, deals with productmeasures and convolutions. It appearsin the middle of 1940 [49].

Conclusion

At the age of only sixteen, Egbert VanKampen appeared in the newspaper asan unprecedented mathematical talent.He was a renowned mathematician butnevertheless, his work had to wait along time for recognition. Perhaps thiswas due to his inaccessible manner ofwriting, a proof is frequently over be-fore you notice it. Perhaps it wasdue to his discretion. In an Ameri-can newspaper a colleague noted, “Henever spoke very much about his ac-complishments”. While Van Kampen’sfame is based on the work of his begin-ning years, after 1933 he wrote a fur-ther fourty articles. Who knows whatbeautiful things still await us.

Egbert van Kampen, 1940.Photo: Archief Mieke van den Veen

Word of thanks

This overview would not have beenpossible without the photographs andnewspaper cut-outs of Mieke van derVeen, daughter of Elizabeth van Kam-pen. I would like to thank her for theauthorization to use this material andfor the care with which she saved it.Boudewijn van Kampen, also a familymember, provided the photograph ofEgbert van Kampen and Paul Ehren-fest. Egbert van Kampen’s great-great-grandfather was Jacobus van der Blij,and he was also the great-grandfatherof the retired professor of mathematicsvan der Blij from Utrecht.

The anecdote about the visum of VanKampen comes from the memoirs ofMark Kac, Enigmas of chance, London,Harper and Row, 1985. More anecdotescan be found in these memoirs, aroundpage 85. The autobiography of Kac wasbrought to my attention by Cor Kraai-kamp.

Slava Krushkal helped me with thedetails of the Van Kampen embeddingobstruction, Mark de Bock of the muni-cipality of Antwerp and Albert Schilt-meijer from Amsterdam sent me dataconcerning the Van Kampen family.Jim Stimpert of Johns Hopkins Librarysent information concerning Van Kam-pen’s publication.

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[3] Die kombinatorische Topologie und die Dua-litatssatze, Diss. Leiden, (Van Stockum, DenHaag), 1929.

[4] with J.A. Schouten, Zur Einbettung und Krum-mungstheorie nichtholonomer Gebilde, Math.Ann. 103, 752-783, 1930.

[5] Komplexen in Euclidische Ruimten, Handelin-gen XXIIIe Nederl. Natuur- en GeneeskundigCongres (1931), 123-124.

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[11] On the connection between the fundamentalgroups of some related spaces, Amer. J. Math.55 (1933), 261–267. Het artikel staat integraal in:Two decades of mathematics in the Netherlands,1920-1940, A retrospection on the occasion of thebicentennial of the Wiskundig Genootschap, E.M. J. Bertin, H. J. M. Bos and A. W. Grootendorst,Amsterdam, 1978.

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[17] The topological transformation of a simple clo-sed curve into itself, Amer. Journ. of Math. 57(1935), 142-152.

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[19] Almost periodic functions and compact groups,Ann. Math. (2) 37, 78–91.

[20] A unicity theorem for ordinary differential equa-tions and a postulate system for Lie groups, Bull.Amer. Math. Soc. 42 (1936), 330.

[21] Note on a theorem of Pontrjagin, Amer. J. Math.58 (1936), 177-180.

[22] The fundamental theorem for Riemann integrals,Amer. J. Math. 58 (1936), 847-850.

[23] with A. Wintner, On the canonical transforma-tions of Hamiltonian systems, Amer. J. Math. 58(1936), 851–863.

[24] On almost periodic functions of constant abso-lute value, J. London math. Soc. 12 (1937), 3-6.

[25] with A. Wintner, On bounded convolutions, Bull.Amer. Math. Soc. 43 (1937), 564-566.

[26] On the argument functions of simple closedcurves and simple arcs, Compositio math. 4(1937), 271-275.

[27] Remarks on systems of ordinary differentialequations, Amer. J. Math. 59 (1937), 144-152.

[28] with A. Wintner, On a symmetrical canonical re-duction of the problem of three bodies, Amer. J.of Math. 59 (1937), 153–166.

[29] with A. Wintner, Convolutions of distributionson convex curves and the Riemann zeta function,Amer. J. Math. 59 (1937), 175-204.

[30] with P. Hartman and A. Wintner, Mean motionsand distribution functions, Amer. J. Math. 59(1937), 261-269.

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[32] with A. Wintner, On divergent infinite convolu-tions, Amer. J. Math. 59 (1937), 635-654.

[33] On the addition of convex curves and the den-sities of certain infinite convolutions, Amer. J.Math. 59 (1937), 679-695.

[34] with A. Wintner, On a singular monotone func-tion, J. London math. Soc. 12 (1937), 243-244.

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[36] The theorems of Gauss-Bonnet and Stokes,Amer. J. Math 60 (1938), 129–138.

[37] with P. Hartman and A. Wintner, On the dis-tribution functions of almost periodic functions,Amer. J. Math. 60 (1938), 491-500.

[38] Invariants derived from looping coefficients,Amer. J. Math. 60 (1938), 595–610.

[39] with A. Wintner, On Liouville systems, NieuwArch. Wiskd. 19 (1938), 235–240.

[40] with P. Hartman and A. Wintner, Asympto-tic distributions and statistical independence,Amer. J. Math. 61 (1939) 477-486.

[41] with M. Kac and A. Wintner, On Buffon’s pro-blem and its generalizations. Amer. J. Math. 61(1939), 672–676.

[42] with M. Kac, Circular equidistributions and sta-tistical independence. Amer. J. Math. 61 (1939),677–682.

[43] On the asymptotic distribution of a uniformly al-most periodic function. Amer. J. Math. 61 (1939),729–732.

[44] with A. Wintner, A limit theorem for probabilitydistributions on lattices. Amer. J. Math. 61 (1939),965–973.

[45] with M. Kac and A. Wintner, On the distributionof the values of real almost periodic functions.Amer. J. Math. 61 (1939), 985–991.

[46] A remark on asymptotic curves, Amer. J. Math.61 (1939), 992–994.

[47] with M. Kac and A. Wintner, Ramanujan sumsand almost periodic functions, Amer. J. Math. 62,(1940), 107–114.

[48] with P. Erdos, M. Kac, and A. Wintner, Ramanu-jan sums and almost periodic functions. StudiaMath. 9, (1940), 43–53.

[49] Infinite product measures and infinite convolu-tions. Amer. J. Math. 62, (1940), 417–448.

[50] with A. Wintner, On the almost periodic beha-vior of multiplicative number-theoretical func-tions, Amer. J. Math. 62, (1940), 613–626.

[51] On uniformly almost periodic multiplicative andadditive functions, Amer. J. Math. 62, (1940),627–634.

[52] Elementary proof of a theorem on Lorentz ma-trices, Bull. Am. Math. Soc. 47 (1941), 288-290.

[53] Remark on the address of S. S. Cairns. Lecturesin Topology, pp. 311–313. University of MichiganPress, Ann Arbor, Mich., 1941.

[54] Notes on systems of ordinary differential equa-tions. Amer. J. Math. 63, (1941), 371–376.

[55] with A. Wintner, On the asymptotic distributionof geodesics on surfaces of revolution, Cas. Mat.Fys. 72 (1947), 1-6.

3 Robbert Fokkink [R.J.Fokkink@ewi.tudelft.nl] studied mathematics in Amsterdam and Delft. His PhD thesis from 1991 is ontopological dynamics. From 1992 to 1999 he worked at Delft Hydraulics. He is currently a member of the probability group at DelftUniversity.

"Quantum cryptography" is the final of athree-article sequence written by NunoCrato and published at the Portuguesenewspaper Expresso in September 2001.These three articles were submitted tothe Public Awareness of Mathematics2003 competition and won its FirstPrize.

Crato's articles were published in thewidely read magazine Revista, which isincluded in the weekly newspaperExpresso. Readership for this newspaperis about half a million people, which is alarge number for Portugal. In fact,Expresso is the most sold newspaper inthe country.

The first and second articles referredbriefly to the history of cryptography,discussed the asymmetric key solutionand the RSA method. They stressed theimportance of mathematical conceptsfor safe communication and showedtoday's relevance of mathematical cryp-tography. These two articles discussedconcepts that were novel for the generalreader, but that are sufficiently wellknown for most mathematicians. Thethird article discussed the emergentresearch on quantum cryptography,explaining one way by which the uncer-tainty of quantum world can be used toprovide safe communications. It is thisthird article we reproduce here.

In the next issue of this Newsletter,we will publish an interview with NunoCrato by José Francisco Rodrigues. Thisconversation between the twoPortuguese mathematicians addressessubjects such as the importance of math-ematics popularization, the difficultiesand caveats of this work, and its rele-vance for the public and the public statusof Mathematics.

QUANTUM CRYPTOGRAPHYIt sounds like science fiction, but it is areality: the most bizarre properties ofsubatomic particles allow us to createunbreakable ciphersThe security of bank transactions, elec-tronic commerce, and email communica-tion is based on the safest cryptographysystems created by man. But "safest"

does not mean "impossible to break".The safety of one of the most reliablemodern cryptography systems - the socalled RSA, which we have been report-ing on in this Revista - is based on thedifficulty of finding the prime factors ofvery large numbers. There are no algo-rithms known to date that enable theirfactorization in a reasonable amount oftime, even if we use the most powerfulcomputers available. However, if a math-ematician discovers a process thatenables a speedy factorization, or if anew generation of computers such asquantum computers are commercialized,the world of communication as we knowit can be easily jeopardized. In the eventthat one of these revolutionary techno-logic advances becomes widespread,electronic commerce would cease to besecure, the military would have to rein-vent its communication systems, andbank institutions would have to take astep back and make transactions at a veryslow pace. It would be the collapse ofinformational technology in our presentday society.

Not surprisingly, people are looking

for a new cryptography system that willbe absolutely secure. Before RSA canfail us, mathematicians, physicists, andcomputer scientists hope to develop anew procedure that will be virtuallyunbreakable. This might well be possi-ble, because the new system scientistsenvision to create is based on the mostprofound laws of matter, those that gov-ern the uncertainty of the quantumworld. The impossibility of getting toknow a priori the behavior of elementaryparticles is what will guarantee commu-nication security.

The idea has been brewing in theminds of scientists for some time now.Charles Bennet, a computer scientist atthe IBM Watson Research Center hasbeen one of those looking for a solutionto this problem. Finally, in the 80's heand his colleague Gilles Brassard man-aged to conceptualize a quantum cryp-tography system. For a long time theirideas were only dreams. During the lasttwo years, however, technological andscientific advances have made it possibleto build prototypes of quantum systemsthat seem to be absolutely unbreakable.

To discuss issues related to cipheredmessages, experts usually refer to threedifferent fictitious characters: Alice,Bob, and Eve. The first represents thesender, the second the receiver, and thethird the intruder who attempts to breakthe secret communication. At the core ofthe process proposed by Charles Bennetand his collaborators, there is a randomcode key as long as the message itself.This key is simply a binary number, asequence of zeros and ones led by a one.Alice begins by transforming the text shewants to send Bob by translating it intoanother binary number. Then, she addsthe key number to her message number.She sends the result to Bob, who has the

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A new book explaining how quantummechanics can be used to conceive aninnovative computing system

Quantum CrQuantum CryptographyyptographyNuno Crato (Lisbon)

key that Alice used. By subtracting thekey from the message received, he getsthe original message. To read it, he onlyhas to transform the sequence of zerosand ones into a sequence of letters, butthis is a routine procedure any computeris capable of doing.

For this system to be unbreakable, it isimportant that the key is a randomsequence of zeros and ones and that it isused only once. This usually means thatthose numbers have to be generated inadvance and that Alice has to send themto Bob. This is where problems usuallyarise. If Alice and Bob never meet face toface, as it happens in e-commerce, theyhave to exchange the key through somecommunication channel. How can theydo that? Well, they could agree on anoth-er key, but this again does not solve theproblem, as to do this Alice and Bobwould have to meet or trust a messen-ger... Since Eve is always peering overthem, absolute security seems impossi-ble.

This is where the quantum worldcomes into play by the hands of CharlesBennett and other computer scientists. Itis a strange world, with rules difficult tounderstand on the basis of our everydayexperiences. One of those rules is uncer-tainty. And this uncertainty is not basedon our insufficient knowledge; rather itis intrinsic to the very life of particles.How could this be used for safe commu-nication?

In order to create the random codekey, Alice begins by sending Bob asequence of light particles, that is, asequence of photons. She has two polar-izers in her system, one vertically orient-ed and another tilted at 45º, as we can seein the figure. To set up the key, she ran-domly alternates the polarizers and, forexample, makes zero correspond to a

vertically polarized photon and makesone correspond to a 45º polarized pho-ton. Bob has two other polarizers, one ofthem horizontally oriented and anotherone at –45º. When he receives Alice'sphotons, he makes them pass throughone of his polarizers, alternating the twoin a completely random fashion.

The photons that Alice sends and Bobreceives may or may not pass through hisdevice, depending on how the polarizersare oriented. If Alice sends a photonpolarized vertically and Bob attempts tomake it pass through his horizontalpolarizer, the particle is stopped and doesnot pass. If Alice sends Bob a photonpolarized at 45º and Bob receives it withthe polarizer at –45º, the particle is alsostopped and does not go through.Polarizers perpendicular to the photonpolarization do not let the particles pass.

Surprises arise when Alice sends avertically polarized photon and Bobreceives it with the diagonal polarizer orwhen Alice sends a diagonally polarizedphoton and Bob receives it with the hor-izontal polarizer - that is, when Alice andBob's polarizers make a 45º angle. In thiscase, the uncertainty principle of quan-tum mechanics is set in motion: half ofthe particles go through Bob's polarizerand the other half are stopped. And thishappens without any prior knowledge ofwhich particles are going to go throughand which particles are going to bestopped.

At this stage, only Alice knows thepolarization of the photons she sent. Andonly Bob knows which ones finishedtheir trip. This way, Bob discovers thepolarization established by Alice for thephotons that passed through. If a photonpassed through a diagonal filter, Alicemust have polarized it vertically and itcan be assigned the value zero. If a pho-

ton passed through a horizontal filter,Alice must have polarized it diagonallyand it can be assigned the value one.Those photons that were retained remaina mystery to Bob.

But now Alice needs to know whichphotons went through Bob's filters. Toknow this, Alice and Bob can communi-cate through a less secure system andthey can even be heard by Eve. Even ifEve finds out which particles reachedBob, she still cannot figure out which fil-ter they passed through. Thus, the key isestablished solely on the basis of thephotons that completed the voyage.Now, Alice and Bob can communicatewith complete security. The uncertaintyof the quantum world gives them the cer-tainty that their code cannot be broken.

Maybe we are close to implementingthis cryptography system. Only a fewyears ago it all sounded like science fic-tion, but recent technological break-throughs made it possible to create pro-totype computation models with viableapplications of these ideas. As we cananticipate, the technological challengesare huge. How do we transmit light aphoton at a time? How do we make surethese particles reach their destination?Nonetheless, these problems are beingsolved little by little. We are already ableto use quantum cryptography throughfiber cables and through air for a fewkilometers. Thus, we may not be far frombeing able to protect our secrets by mak-ing them travel a particle at a time - at thespeed of light.

Nuno Crato [ncrato@iseg.utl.pt] isAssociate Professor with habilitation atthe Mathematics Department of InstitutoSuperior de Economia e Gestão, LisbonTechnical University. He received hisPh.D. from the University of Delawareand worked previously in the U.S. for anumber of years. His research interestslie in Stochastic Processes and TimeSeries Analysis, with various applica-tions, from finance and economics to cli-mate models. He is a member of theboard of the Portuguese MathematicalSociety (SPM). In parallel with hisresearch and teaching activity, he is ascience writer. He has a column at thePortuguese newspaper Expresso and hasworked for a few TV science documen-taries. He is co-author of Trânsitos deVénus (2004), Eclipses! (1999) and otherscience books. The EuropeanMathematical Society awarded him theFirst Prize in mathematics populariza-tion writing in 2003.

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For natural reasons, this causerie onSwedish mathematicians – just in time forthe 4th European Congress ofMathematics in Stockholm - contains verylittle in the way of flesh on its bones. Thecanonical, meaty source is Gårding, 1998,from which I have used much material. Toavoid delicate questions concerning mycolleagues, I have decided not to mentionthe names of any mathematicians underthe traditional Swedish retirement age of65. This of course, for the most part,leaves contemporary mathematics out ofthe picture, but that is the way it has to be.

The Mittag-Leffler EraOn an international scale, Swedish mathe-matics begins with Gösta Mittag-Leffler(1846-1927). He learned analysis, includ-

ing complex analysis, in Uppsala in the1860’s, from his teachers H. Daug and G.Dillner. However, the knowledge gainedwould hardly have sufficed for a researchcareer had he not gone to Hermite, and inparticular to Weierstrass, to learn more. Infact a glance at Mittag-Leffler’s thesisdoes not convince one that he had mas-tered the definition of the derivative of acomplex-valued function of a complexvariable. “Die Strenge des Weierstrass”was not lost on him, however, and he start-ed producing mathematics papers which

were well up to the standard of the day.Not wanting to stay in Germany, and find-ing no suitable job in Sweden, he appliedfor, and was appointed to, a professorshipin Helsinki, where he had one student witha name that has remained in the books:Hjalmar Mellin (1854-1933), inventor ofthe transform bearing his name. As a pro-fessor in Stockholm from 1881 to 1911, hemade a successful career as a researcher,teacher, thesis adviser, and founder of theActa Mathematica, while at the sametime managing to amass a substantial for-tune as a businessman. He willed hishouse and fortune to the Royal SwedishAcademy of Sciences, making the futureInstitute Mittag-Leffler possible. A last-ing contribution to history was his suc-cessful effort to bring Sonya Kovalevskyto Sweden.

Several of Mittag-Leffler’s studentsbecame well known and respected mathe-maticians in their day. Ivar Fredholm(1866-1927, professor in Stockholm) isbest known for his work on integral equa-tions and spectral theory, with a theoremwhose generalization to abstract spacesgoes under the name of “Fredholm’s alter-native”. His main interest was mathemat-ical physics and partial differential equa-tions, where he constructed fundamentalsolutions for various classes of equationsin two and three variables. These studieswere continued by Fredholm’s student,Nils Zeilon (1886-1957, professor inLund), who managed to considerablyenlarge the classes of equations for whicha fundamental solution could be shown toexist. This line of investigation was latercarried to a logical end in the thesis ofHörmander.

Ivar Bendixson (1861-1935, professorin Stockholm), learned set theory from thepapers by Cantor, solicited by Mittag-Leffler for the Acta, and dynamical sys-tems from the papers of Poincaré. Theresult was the Cantor-Bendixson theoremin set theory and the Poincaré-Bendixsontheorem on the existence of closed cyclesof ODEs in two dimensions.

Edvard Phragmén (1863-1937, profes-sor in Stockholm), is of course best knownfor the Phragmén-Lindelöf theorem, but

also for his discovery of the famous errorin Poincaré’s first paper in the Acta.Having succeeded Sonya Kovalevsky as aprofessor, Phragmén left mathematics forinsurance and politics after about tenyears.

Decidedly less known is Helge vonKoch (1870-1924, professor inStockholm), a precocious mathematicianwhose work on infinite determinants andon continuous non-differentiable functionshave receded into the darkness of history.

Two of Mittag-Leffler’s students gaveup pure mathematics for other fields ofendeavour: Gustav Cassel (1866-1945)decided mathematics was too difficult andwent on to become the father of theStockholm school of economics, withNobel Prize winners Bertil Ohlin andGunnar Myrdal as prime exponents.

Number theory, more specifically ana-lytic number theory, had been taken upboth by Phragmén and von Koch in theearly years of the century, inspired by thesuccesses of Hadamard and de la ValléePoussin. Harald Cramér (1893-1985,professor in Stockholm), probablyinspired and tutored by Marcel Riesz,wrote a thesis on the subject in 1917, andcontinued to produce good papers for sev-eral years. According to legend however,he fell out with Mittag-Leffler, whopromised that as long as he, Mittag-Leffler, had any influence, Cramér wouldnever get a professorship. This canarguably be considered a stroke of luck,both for Swedish mathematics and forCramér himself, since Cramér then turnedto probability and mathematical statisticsand founded a thriving Swedish school inthe area. His magnificent monographfrom 1946, Methods of MathematicalStatistics, has rightly been considered astandard reference in the area.

Marcel Riesz (1886-1969, brother ofFredrik Riesz), came to Sweden to assistMittag-Leffler in the editing of the Acta,but stayed on at the University ofStockholm as a docent until 1927, when hebecame professor in Lund. Riesz intro-duced new areas of analysis in Sweden,and had a profound influence on Swedishmathematics. One of Riesz’s most famous

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Household names inHousehold names inSwedish mathematicsSwedish mathematicsSome who should be and some others

Kjell-Ove Widman (Institut Mittag-Leffler, Stockholm)

Gösta Mittag-Leffler

theorems, generalized by one of his stu-dents and known as the Riesz-Thorin inter-polation theorem, gave rise to a wholenew subject. Other claims to Riesz’s fameare his theorem on conjugate functions andhis work on Riesz potentials. The latterwork was used in potential theory and inconnection with the wave operator and thepropagation of singularities. Riesz’s per-sonality was decidedly un-Swedish. Heliked to talk at length, and was a continu-ous source of inspiration to his studentsthrough his indefatigable enthusiasm formathematics.

Björling and Holmgren descendantsBefore continuing with the students ofMarcel Riesz, we must go back to anothergerm of modern Swedish mathematics,namely to Carl Björling (1839-1910, pro-fessor in Lund). Inspired by the Danishmathematician Zeuthen, and in competi-tion with his colleague Anders Bäcklund,he took up algebraic geometry. This wasa subject much in vogue at the time duemainly to the work of Italian mathemati-cians11 Among them was Pasquale delPezzo, Duca di Cajanello and brother-in-law of Gösta Mittag-Leffler.

Björling had a student by the name ofAnders Wiman (1865-1957), who laterbecame professor in Uppsala and had twofamous students, Fritz Carlsson and ArneBeurling. Wiman had started out in thefootsteps of his teacher Björling, but inUppsala he also worked in analysis, in par-ticular in analytic functions.

Björling’s colleague, Anders Bäcklund(1845-1922), also dabbled in algebraicgeometry. By a quirk of history, Bäcklundfound a new field of interest when SophusLie was among the applicants for a mathe-matics chair in Lund, in 1871. A friend ofLie was supposed to hand in the applica-tion, but was negligent in his duties andthe application arrived one day late. Theact stayed in the office of the registrar,however, and Bäcklund was allowed tostudy Lie’s papers, among which he founda description of Lie’s contact transforma-tions. The Bäcklund transformation, ageneralization of Lie’s concept, ensued.Bäcklund later used his considerable pow-ers of analysis on problems in mechanicsand electricity. None of Bäcklund’s stu-dents continued with mathematics. Wewill now return to Wiman’s students,Beurling and Fritz Carlson.

Arne Beurling (1905-1986), was a stu-dent of Holmgren as well as of Wiman, buttook up Wiman’s interest in analytic func-tions. His thesis, in which he started thedevelopment of the concept of extremallength, was delayed for a year becauseBeurling’s father wanted him to join himfor an alligator safari in Panama. An addi-tional reason was that some of the resultsin the thesis had been anticipated by the

Finnish mathematician Lars Ahlfors,requiring Beurling to extend his investiga-tions. Beurling had a brilliant career, firstas professor in Uppsala, then as a memberof the Institute for Advanced Study. He isprobably unique in Swedish mathematicsin that he might have influenced the courseof history through his ingenious cryptana-lytic efforts during World War II. Themost dramatic event was the breaking ofthe German “Geheimschreiber”, T-52,which made it possible for the Swedishmilitary command to read top levelGerman communications for two or threeyears during the most critical part of thewar (the story is described in detail inBeckman, 2003). There were otherSwedish mathematicians involved as well:Gunnar Blom and C-O Segerdahl, bothstudents of Cramér, and Beurling’s studentBo Kjellberg.

In his own country, Beurling is consid-ered to be one of the handful of really bril-liant mathematicians that Sweden has pro-duced, but unfortunately, except for asmall group of cognoscienti, he is littleknown internationally. Part of the reasonmight be that his most productive timecoincided with a trend away from con-crete, down-to-earth analysis. Also, heseldom published, and had a tendency tokeep his ideas to himself, guarding themwith suspicion and jealousy. He had alasting influence on Swedish mathematicsthrough his many students, one of whichwas to join him on the Swedish mathemat-ical Olympus.

The second of Wiman’s students, FritzCarlson (1888-1952, professor inStockholm), also continued in the path ofhis teacher’s interest in analytic functions,and can indeed be seen as continuingMittag-Leffler’s program in the area.Carlson had two students, with an interna-tional reputation in different fields, HansRådström (1918-1969, professor inLinköping) and Tord Ganelius (1925-,professor in Göteborg). Ganelius is bestknown for his Tauberian theorems. He

had several students, the best known ofwhich was Björn Dahlberg (1950-1998,professor in Göteborg), an imaginativeanalyst who produced a result on themutual singularity of harmonic andboundary surface measure in Lipschitzdomains.

Anders Wiman had a colleague inUppsala, Erik Holmgren (1873-1943,professor in Uppsala), whose name wouldlive on because of an oft-cited theorem.His interest was partial differential equa-tions, and he proved a uniqueness theoremwhich plays an important part in the mod-ern theory of the subject.

Torsten Carleman (1892-1949, profes-sor in Stockholm), was easily Holmgren’s

most famous student. He wrote aboutintegral equations, partial differentialequations, analytic functions, quasi-ana-lytic functions, harmonic analysis, andkinetic gas theory, and made fundamentalcontributions in all of these areas. LikeBeurling however, he did not enjoy theinternational reputation that we theSwedes think he deserves. Again, part ofthe reason was the concurrent develop-ment of more abstract theories, whichattracted more attention. Other causesmay have been that he was actually aheadof his time, as for example with his gener-alization of Holmgren’s uniqueness theo-rem, which only drew attention throughlater developments in the area.Carleman’s personality was complicated.To quote Gårding: he was neither at easewith the world nor with himself.Carleman was director of the Mittag-Leffler Institute for 20 years, but there washardly any activity, due a great deal to thelack of funds.

Carleman did not have many students.One of them however, Åke Pleijel (1913-1989, professor at KTH, in Lund and inUppsala), continued and extendedCarleman’s work in spectral theory.Pleijel is perhaps best known for his

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Arne Beurling

Torsten Carleman

results on the asymptotic distribution ofeigenvalues for the Laplacian onRiemannian manifolds and for ordinarydifferential equations.

To finish the line of Holmgren’s descen-dants, we will mention Jaak Peetre(1935-, professor in Lund and inStockholm). Peetre started out in PDEs asa student of Pleijel, but then branched outinto interpolation theory and other fields,broadening his areas of interest substan-tially.

The Riesz LegacyThe first Riesz student to reach fame wasEinar Hille (1894-1980, professor atYale). A couple of years after his Ph.D. heleft for a postdoc position at Harvard andwas lost to Sweden for ever, although theauthor took a graduate course in algebrafrom him in Uppsala in the early 1960’s.Hille published widely, but is of coursebest known for his book on semigroup the-ory.

Otto Frostman (1907-1977, professorin Stockholm) wrote a beautiful thesis onpotential theory. Using modern measuretheory he could complete and considerablyextend Gauss’s ideas from the 1820’s tokernels of a very general type. Severalgood papers followed the thesis, but afterhis time as docent in Lund, Frostman hadto support himself as a high school teacheruntil a chair in Stockholm became avail-able. For a 15-year period, he was alsodirector of the Institute Mittag-Leffler.

Lars Gårding (1918-, professor inLund) soon developed into one of the pil-lars of Swedish mathematics in the lasthalf of the century. He worked initially ingroup representation theory - albeit withstrong connections to PDEs - but soonmade a name for himself in PDEs, wherehe published widely on elliptic and hyper-bolic equations and systems of equations.Presumably, his name will live on foreverin connection with the Gårding inequality.Later work on hyperbolic equationsincludes studies on lacunas, partly togeth-er with Atiyah and Bott. Gårding alsoused his considerable literary skills toadvantage; one example is the alreadycited book on Swedish mathematicsbefore 1950. Less known abroad are prob-ably his plays and essays, with mathemat-ical and philosophical themes. His colour-ful and outspoken personality, combinedwith a solid interest in music and culture,has never failed to make a lasting impres-sion on anybody who has come his way.

Sweden’s only Fields medallist so far,Lars Hörmander (1931-, professor inStockholm and Lund), received his PhD inLund in 1955, and was a full professor inStockholm in 1957, at the age of 25. Histhesis on linear partial differential equa-tions was a tour-de-force, and he has con-tinued in the same vain for decades, pub-

lishing widely on PDEs, several complexvariables, Fourier integral operators, andconvexity. His early monograph on linearPDEs grew with time into a formidablefour-volume oeuvre. After some politicalskirmishes, Hörmander left Stockholm forthe Institute of Advanced Study, causingthe Swedish parliament to institute a newlaw, Lex Hörmander, aiming at curtailingthe brain-drain. After a few years howev-er, Hörmander returned to his Alma Mater.Through his dozen or so Swedish PhD stu-dents, he has over 80 descendants (MGPdoes not list all of them). It would proba-bly not be too much of an exaggeration tosay that Hörmander is the Swedish mathe-matician who has had the largest impactever.

Two of his early students, GermundDahlquist and Vidar Thomée, movedaway from their master’s voice to contin-ue their careers in different parts of numer-ical mathematics, more about which willbe explained later. Other students includeJan Boman (1931-, professor inStockholm) and Christer Kiselman(1939-, professor in Uppsala). Kiselmanhas carried the torch of several - indeed,sometimes infinitely many - complex vari-ables, and has in turn had over a dozen stu-dents.

The Beurling LegacyBeurling’s students were given problemsin very different areas, attesting to theirmaster’s wide interests.

Carl-Gustav Esseen (1918-2001, pro-fessor at KTH and in Uppsala), wrote athesis on the best remainder in the law oflarge numbers, and continued in mathe-matical statistics.

Göran Borg (1913-1997, professor atKTH), was among the first to successfullystudy the inverse spectral problem forODEs. Due to the isolation caused by thewar, his results did perhaps not have animmediate following, receiving theirrecognition much later.

Bo Kjellberg (1920-1999, professor at

KTH), took up Beurling’s interest inNevanlinna theory, studying the relationsbetween maximum and minimum moduliof analytic functions in the plane.

The most famous of Beurling’s students,and the only one to reach - and surpass -his level, is Lennart Carleson (1928-,professor in Stockholm and Uppsala).Carleson started out doing bone-hardanalysis, studying exceptional sets of vari-

ous classes of harmonic and analytic func-tions. Interpolation problems for holo-morphic functions led to the solution ofthe corona problem. For the generalmathematical public, Carleson is bestknown for proving the i.e. convergence ofthe Fourier series of L2 functions, a prob-lem which had eluded analysts for morethan half a century. Outside harmonicanalysis he has, in particular in his lateryears, been very active in dynamical sys-tems. Carleson seems to be the only one tohave profited by Lex Hörmander, havingbeen awarded a personal professorship bythe Government in the 1960s. As a thesisadviser he has been prolific, with over twodozen students, and his descendants, closeto 100 in number, have populated themathematics departments of Swedish uni-versities and university colleges. He wasalso in practice the lone editor of the ActaMathematica for close to a 30-year period.In addition to traditional mathematicalactivities, Carleson contributed greatly byinfusing life into the slumbering InstitutMittag-Leffler. By finding funds torefurbish the Mittag-Leffler villa andbuilding apartments for guests, and byconceiving the format of the institute’s sci-entific programs, he managed to create acentre for mathematical research of aunique character.

Carleson’s most productive student hasbeen Hans Wallin (1936-, professor inUmeå), both when it comes to number ofpapers and number of students (MGP lists32 descendants, of which 24 were stu-dents). The rest of us, including GunnarAronsson (1939-, professor in

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Lars Hörmander

Lennart Carleson

Linköping), Lars Inge Hedberg (1935-,professor in Linköping), Ingemar Wik(1934-, Umeå), and the author, have beenmore modest in these respects.

The last one of Beurling’s students,Yngve Domar (1928-, professor inUppsala), wrote a beautiful thesis that didnot receive the attention it probablydeserved. Domar had a productive careerin Uppsala, doing harmonic analysis andsynthesis. He had 12 PhD students, one ofwhom, Matts Essén (1932-2003, profes-sor in Uppsala), came under the influenceof Bo Kjellberg and continued his careerin majorization problems for harmonic andanalytic functions.

Numerical MathematicsNumerical analysis took off as an indepen-dent subject in the 1960s, and three moreor less independent schools were created.My impression is that Sweden has had avery good reputation in this area. A resultof these activities was the creation of BIT,a Nordic journal for numerical mathemat-ics with Carl-Erik Fröberg (1918-, pro-fessor in Lund) as one of the drivingforces.

Although a student of Hörmander,Germund Dahlquist (1925-, professor atKTH) worked in numerical methods ofODEs and associated numerical algebraproblems from the beginning. Among hisstudents, in excess of 20, was Åke Björck(1934-, professor in Linköping).

Heinz-Otto Kreiss (1930-, professor inUppsala and at KTH), came fromGermany but took a PhD at the KTH andwent on to found a thriving school ofnumerical analysis of PDEs in Uppsala.He has also had long stints abroad.Among his students are Bertil Gustafsson(1939-, professor in Uppsala) and OlofWidlund (1939- ), who left Sweden earlyon for a career at the Courant Institute.

Another of Hörmander’s students wasVidar Thomée (1933-, professor inGöteborg), who started doing more tradi-tional Hörmander mathematics, but thenbranched out and founded a very success-ful school of PDE numerical analysis inGöteborg, with an international reputationand a number of good students to his cred-it.

New AreasIn the author’s youth, Swedish mathemat-ics was dominated by analysis. True, therewas Trygve Nagell (1895-198) in Uppsaladoing number theory, and Olof Hanner(1924?-) in Göteborg doing general topol-ogy, but both represented rather modest-scale activities.

This started to change when Jan-ErikRoos (1935-, professor in Stockholm)went away from Lund to study in Paris,and found out that there was somethingcalled algebra and algebraic geometry. He

came back and founded a school in thesubject in Stockholm, which is very activeto this day.

Bernt Lindström (1933-, professor atKTH) was formally a student of OttoFrostman, but wrote a thesis on combina-torial methods in number theory. He hadseveral students, thus creating a Swedishschool of combinatorics, representedtoday by at least three professors.

Foreign InfluxWith the fall of the iron curtain, a numberof Russian mathematicians immigrated toSweden, changing the face of mathematicshere inalterably. Also, professors havebeen recruited from Germany, Holland,Italy, Norway, Poland, Romania, the UK,and the United States (I have probably for-gotten one or two). Due to my self-imposed age limit, the names will not bementioned here, with the exception ofVladimir Maz’ya (1937- ), who wasdrafted to Linköping by Lars IngeHedberg. Maz’ya, working mostly inPDEs, and enormously productive bySwedish standards (several books by hishand have appeared in the last few years),has given analysis in Sweden and inLinköping new impetus and new direc-tions.

Women MathematiciansAs mentioned above, Mittag-Leffler man-aged to convince the board of the then pri-vate Stockholm University (known asStockholm’s högskola in those days) toappoint Sonya Kovalevsky (1850-1892)

to a chair of mathematics. Unfortunately,she died at a relatively young age, beforereally making an imprint here.

With Kovalevsky as a model, one wouldhave expected locally groomed womenmathematicians to appear soon after.

These however, failed to materialize, andthe female gender has been sadly under-represented in the professorial ranks. Afew have received PhDs though, and somehave become lecturers. One can mentionSonja Lyttkens (student of Carleson),Anna-Lisa Wold-Arrhenius (Domar),and Kersti Haliste (Carleson). (There aresome unmentionables also.)

When in the 1990s, there were finallyone or two women appointed to mathe-matics and applied mathematics chairs,they were imports from Germany andFinland.

Swedish mathematics of lateFor several decades after 1950, the num-ber of professors of mathematics inSweden was very small, on the order of10-15, and most of them worked in analy-sis. Life was dominated, and the standardset, by the three giants, Carleson, Gårding,and Hörmander. This meant that you sel-dom published, and preferably only yourbest and polished results. In retrospect,this time seems idyllic, although I do notthink we, the lesser souls, thought so at thetime, the worries of finding a bread-win-ning position occupied our minds toomuch. Since then, Swedish mathematicshas taken on a completely different, inter-national flavour, for good and for bad (onbalance, surely for good). The number ofprofessors is more than 75, although dueto a strange law, some of these are teach-ing positions, and professorships in nameonly. The breadth of areas of mathematicsbeing studied has increased enormously,while some of the traditional subjects havereceded almost into oblivion. The tradi-tion of publishing has altered, the pressureto publish has increased, and so has thecompetition for positions. It is not uncom-mon that a lectureship draws 50 appli-cants. In short, we have joined the world.

ReferencesGårding, Lars: Mathematics and

Mathematicians. Mathematics inSweden before 1950. AmericanMathematical Society and LondonMathematical Society 1998.

Beckman, Bengt: Codebreakers: ArneBeurling and the Swedish Crypto pro-gram during World War II. AmericanMathematical Society 2003.

Kjell-Ove Widman [widman@ml.kva.se]received his PhD under Lennart Carlesonin Uppsala in 1966. Professorships inUppsala and Linköping were followed bya longish stint abroad. He then returnedto Sweden to become Director of InstitutMittag-Leffler in 1995. His scientificinterests include partial differential equa-tions, calculus of variations, and cryptog-raphy.

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EMS June 200420

Sonya Kovalevsky

SOCIETIES CORNERThe Societies Corner is a column concern-ing mathematical societies in Europeancountries. Over the last five years, this col-umn has presented numerous articles thatdescribed the history of a particular societyor the society's present day work. ManyEuropean Mathematical Societies havebeen portrayed, and we wait for the contri-butions from those societies that have yet topresent.

However, until now some importantaspects of the societies' lives have notappeared in the column.

The life of a society does not only consistof its historical development and its currentactive work. A number of events happen inthe every day life of a society.

Do you remember the general assemblyof your society three years ago and a fasci-nating discussion that took place there?How about a farewell party and an excitingtoast by the President, or a controversialdecision about something undertaken bythe council last year? Several anecdotesconcerning famous mathematicians or for-mer presidents of the society, connectedwith their work in the society? And whatabout your society competition for thepaper written by school pupils, and somepapers presented there? What about anoth-er activity? A host of funny or importantstories, so many interesting events hap-pened...

Don't keep them to yourself! Please,write about them and submit to the Societycolumn. Let others enjoy your stories! Letothers know about the everyday life of thesociety! If it is not written down now, aftersome years it may be forgotten.

Of course, we will not stop publishingarticles of the same style as before, we justwant to have something more.

If you feel that anything like that wouldinterest others, please do not hesitate and

contact the column editor, KrzysztofCiesielski (e-mail: ciesiels@im.uj.edu.pl)in the first instance.

During the first semester of 2004, a very sig-nificant initiative entitled 'Sauvons laRecherche' ('Let's save scientific research') hastaken place in France, inducing an unexpectedand powerful social movement of popular sup-port for scientists' requests. Several dozenthousand scientists, as well as several hundredthousand non-scientists, have signed a peti-tion, with opinion polls indicating that morethan 80% of the population are in support ofthe scientists' requests. These requests wereextremely simple. The first one was that thegovernment cancels its recent funding restric-tions and gives the funding that was initiallypromised back to the research institutions. Thesecond one was that the 550 research positionsproposed by the government as non perma-nent positions are turned into civil servantspositions as was the case in pre-ceding years. Finally, the thirdwas that a national discussion isorganized under the name'Etats généraux de la recherche'.In the French political andsocial tradition, Etats générauxis thought of as a large bottom-up discussion converging tonational proposals for further actions andchanges, following the model of the famousinitial Etats Generaux that was prelude to theFrench revolution. When the influence of theinitiative grew bigger, the opening requests,initiated in research institutions such as INRAand CNRS, were rapidly added to by a requestfor the creation of 1000 new positions for hir-

ing young scientists in Universities.On March 9, given the lack of satisfactory

response from the government, over 2000 sci-entists in charge of research responsibilities(heads of research laboratories or research pro-grams) collectively resigned from theirresponsibilities. This was considered as amajor event by the media. A few weeks later,the government lost the regional elections in aspectacular way. A change of policy concern-ing research was one of the three main changesrecommended by President Chirac to take intoaccount the electoral regional failure (the twoothers being the funding of young artists, andthe support of unemployed people, two othertopics for which important demonstrations andactions had taken place in the precedingmonths). The ministry in charge of research

was changed and the requestsof the scientists accepted with-in a few days.

The current situation,beyond all initial hopes, is thatall the requests motivating the'Let's save scientific research'initiative have been accepted.

The Etats Généraux de laRecherche are currently being prepared inlocal research groups, universities andresearch institutions and should define propo-sitions for the fall. A 'programmation law'making scientific research a national priorityshould be voted before the end of the year andvarious changes to the organization of scientif-ic research in France should follow.

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EMS June 2004 21

Krzysztof Ciesielski

France: "Sauvons la rFrance: "Sauvons la recherecherche"che"Colette Moeglin (Paris, Jussieu)

and Marie-Francoise Roy (Rennes, France)

Colette Moeglin Marie-Francoise Roy

The French mathematical community isnaturally active in this context. The Frenchmathematical societies SMF (SociétéMathématique de France) and SMAI (Sociétéde Mathématiques Appliquées &Industrielles) have supported the 'Let's savescientific research' initiative from its beginningand encouraged their members to be active inthe various demonstrations, contacts with thepopulation and preparation of the 'Etatsgénéraux'.

The organization of research and of scien-tific curricula has some very specific featuresin France. Firstly in many areas of science, likephysics, biology and chemistry, a majority ofscientists are not university professors butrather full time researchers, and moreover,civil servants paid by the state. The mainresearch institution is CNRS, which covers allaspects of scientific research from archaeologyand linguistics to biology, computer scienceand mathematics. However smaller and morespecialized research institutes like INRA forbiology and medicine, and INRIA for com-puter science, are also present, and the relativeroles of these institutes and CNRS is not clear.Important research programs are defined anddirected by the Ministry of Research, indepen-dently of research institutes. The role ofUniversities, as well as regional institutions, inthe orientation of research has to be redefined,and the duality between full time permanentresearch positions and university positions isbeing questioned.

Another specific French feature is the exis-tence of 'classes prépatoires'. Rather thangoing to universities, the best scientific stu-dents prepare competitive exams in specificclasses, taught by very competent full timeprofessors in the secondary school administra-tive system, having no contact with research.The engineering schools they enter after thispreparation are mostly independent of the uni-versities. The very best of these students, ifthey enter into major 'grandes écoles' such asEcoles Normales Supérieures, EcolePolytechnique and a few others, will be in con-tact with scientific research in the remainder oftheir studies. However, an important propor-tion of these good students will stay apart fromthe university and scientific research systemduring their whole curriculum. On the otherhand, students deciding to choose universitiesare often poorly motivated and not adequatelyprepared and do not meet the expectations ofthe university professors who are active scien-tists.

Status and PerspectivesThe present context is moreover characterizedby two important facts: the number of scien-tists retiring from research or teaching posi-tions is growing, and the number of students

choosing scientific studies in universities isdropping.

In this general context, French mathemati-cal data can quickly be summarized. The com-munity of professional mathematicians (uni-versity teachers and full time researchers) hasabout 3500 members, only 10% of them beingfull time CNRS researchers, and about half ofthese are concentrated in Paris or the Ile deFrance region. In spite of the small number offull time researchers in mathematics, CNRSplays a very important role in the organizationof French mathematical research: about 85%of the mathematicians belong to one of about60 joint CNRS-Universities research laborato-ries, and the important equipments of theFrench mathematical community are support-ed by CNRS (CIRM, IHP, the network ofmath libraries, the project Numdam fornumeration of old mathematical publication,the network of computer engineers for mathe-matics). 25% of INRIA (around 100 people)are mathematicians. The loss of students,although less spectacular than in physics forexample, is high in mathematics, reachingabout 25% in the last four years. The numberof university positions in mathematics startedto decrease in 2003 and about 100 positionshave disappeared. The mechanism is the fol-lowing: mathematicians retire or are promotedand their position is given to another depart-ment where student numbers are greater (sportand theatre are currently popular among stu-dents).

A vicious circle is thus currently at work:less students, less positions for mathemati-cians, less motivation to study mathematics....and in a few years the French mathematicalatmosphere, recently active and full ofpromise, could switch to long-term depres-sion.

This situation is paradoxical since theimportance of mathematics in a growing num-

ber of scientific and technological researchareas is widely acknowledged. Industry andservices need more and more people with agood mathematical background and an abilityto adapt to the rapid changes in science andtechnology. French mathematicians have toopen their eyes and ears and redefine new cur-ricula for students and new scientific collabo-rations with the outer world.

A forum discussion 'Let's save scientificresearch' has been proposed by SMF andapproved by SMAI, cf. http://smf.emath.fr/Forum/. Concrete suggestions following thisdiscussion will be proposed before the sum-mer. On the European scale, a petition 'For aEuropean Research' in support of a EuropeanResearch Council (ERC) for basic researchhas been launched at http://fer.apinc.org/ .European mathematical societies and individ-ual European mathematicians are very wel-come to participate in our French discussionsand bring us some fresh air and new ideas!Please visit and send your contribution (by e-mail to smf-debats@smf.ihp. jussieu.fr).

Colette Moeglin [moeglin@math.jussieu. fr]graduated at Univ. Paris 6 under JacquesDixmier with a thesis on primitive ideals ofenveloping algebras. Her current research isconcerned with automorphic forms and repre-sentations of p-adic groups. She is employedby CNRS as Directrice de Recherche at theInstitute of Mathematics at Paris-Jussieu,where she also acts as Directrice Adjointeamd as editor-in-chief of JIMJ, the Journal ofthe institute.

Marie-Francoise Roy [marie.francoise.roy@univ-rennes1.fr] works on algorithmicswithin real algebraic geometry. She is a pro-fessor of mathematics at the University ofRennes 1 and a member of the council of theSMF.

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EMS June 200422

Monsieur François FillonMinistre de l’Education Nationale

de l’Enseignement Supérieur et de la Recherche110 rue de Grenelle

75357 Paris 07 SP

Paris, le 13 avril 2004

Monsieur le Ministre,

L'organisation future de la recherche scientifique en France va faire l'objet de discussions et dedécisions dans les semaines qui viennent. Nous nous félicitons de ces nouvelles perspectives.

Par la présente, nous avons l’honneur de solliciter un entretien avec vous pour vous présenterquelques particularités de l'organisation actuelle des sciences mathématiques qu'il nous semble importantde voir prises en considération dans le cadre des adaptations nécessaires. Il nous revient aussi de vous

In [5], Graciano de Oliveira describes theessentials of the life, aims and activities ofthe Portuguese Mathematical Society,together with the adverse historical contextthat seriously hindered its normal develop-ment and obstructed Portuguese mathemat-ics during the dictatorship years. DeOliveira pays special attention to the histo-ry of mathematics in Portugal in the1940’s. Digging a little further back intime, and following Costa & Malonek [2],and Ramos & Malonek [6], [7], these notesaim to attract more attention to the effortsmade by Portugal to establish contacts withthe international scientific community inthe post-Great War period; an aspect ofPortuguese scientific history which hasoften not received due attention, if not beencompletely forgotten.

We also intend these notes to be a contri-bution to recover-ing the memory ofFrancisco Mirandada Costa Lobo,( 1 8 6 4 - 1 9 4 5 ) ,whose dedicationand efforts to makethese contacts pos-sible and to stimu-late mathematicallife in Portugal inthe 1920's seem tohave faded frommemory as well.

Costa Lobo wasnot only a first-class scientist, but also animportant ambassador for Portugal in theinternational scientific field, without whoman important chapter in the history ofPortuguese academic life would, no doubt,be missing. Costa Lobo, an astronomer,was Director of the ObservatórioAstronómico de Coimbra, and was alsopresident of the Instituto de Coimbra from1915. The Instituto, recognised by theInternational Association of ScientificAcademies as an academy of sciences, hadan interest in international intellectual life.Under the influence of its director, theInstituto focused its interests in the area ofmathematics.

At the time, the Instituto made a consid-erable contribution to Portuguese mathe-matical life, and opened the country to theinternational mathematical world: In its1920 General Assembly, the Institutonamed Emile Picard, Charles de la ValléPoussin, and Gabriel Koenigs, as Honoraryforeign members.

Later, after the 1924 International

Congress of Mathematicians (ICM) inToronto, the Instituto’s Assembly namedas Corresponding Members the followingrenowned mathematicians:Élie Joseph Cartan, from France;John Charles Fields, from Canada;Karl Rudolf Fueter, from Switzerland;Tullio Levi-Civita, from Italy;Salvatore Pincherle, also from Italy;John Lighton Synge, from Ireland;Grace Chisolm Young, from England;William Henry Young, from England, andClément Servais, from Belgium.

Within the next couple of months, theAssembly would also add to the list, thenames of Nicolau Kryloff, from KievUniversity, who gave a two-month courseon Advanced Analysis at the University ofCoimbra, and Jacques Hadamard, fromFrance. Finally, in 1925, the file was com-pleted with the name of the mathematicalphysicist Sir John Larmor, from England.

Some of the above were to becomePresidents of different ICMs: Pincherle in1928 in Bologna; Fueter in 1932, in Zurich;and Hadamard, Honorary President of the1950 Congress in Cambridge, USA. Thepresence of these personalities in Portugalallowed a much needed interchange ofinformation and widened Portugal’s inter-national scientific horizon. In particular,one can see works by de la Vallé, G. C.Young, Kryloff, and Servais, published bythe Instituto’s magazine. This impressivelist of collaborators is a token of CostaLobo’s dedication and commitment todeveloping mathematics in Portugal, aswell as to forging relations between hiscountry and the mathematical world.

It was Costa Lobo’s connection with the

International Mathematical Union (IMU)and his involvement in several Congressesof the International Research Council(IRC), which provided him with the oppor-tunity to become acquainted with severalforeign mathematicians, with whom hedeveloped strong ties of friendship. This isespecially true of his participation in theToronto Congress in 1924. The long trip bysea provided an environment that was pro-pitious for such developments.

Costa Lobo’s activities and contactsenabled him to make the state of mathe-matical research abroad known in his coun-try, as well as to report about the wayMathematical Education was conducted inforeign universities. He was quiteimpressed with several aspects of theUniversity of Toronto, (its organization,academic life, interaction with other uni-versities, the subjects offered in the differ-ent courses, in particular those in mathe-matics), and describes them in detail inLobo [4]. This can also be seen in Costa &Malonek [2].

Mathematical activities in Portugal in the1920’s, and especially, Portugal’s earlyassociation with the IMU are rather forgot-ten matters. Yet, it is to be noted thatPortugal, was involved in the very creationof the IMU. First, taking part in theConstitutive Assembly of the InternationalResearch Council in Brussels, in 1919,where preliminary moves towards the cre-ation of the IMU were made, and later, in1920, as a participant in the ICM inStrasbourg, where the IMU was definitive-ly founded. On this occasion, thePortuguese representative was Costa Lobo,who presented a communication Sur lacourve déscrite par le pôle sur la surface

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EMS June 2004 23

PORPORTUGALTUGAL IN IN THE FIRSTTHE FIRST YEARS OFYEARS OF THE THE INTERNAINTERNATIONALTIONAL MAMATHEMATHEMATICALTICAL UNIONUNION

(Enrique Hernández-Manfredini and Anabela Ramos, Universidade de Aveiro, Portugal)

Francisco Mirandada Costa Lobo.

(1864-1945)

Enrique Hernández-Manfredini Anabela Ramos

de la terre. Later, in 1924, Portugal attend-ed the Congress that took place in Toronto,represented by Costa Lobo and Fernandode Vasconcelos. The former presented acommunication on Nouvelles théoriesphysiques: application à l’astronomie; thelatter, one entitled Sur quelques points del’histoire des mathématiques des Egipcienset aussi sur le Sidhantas des Indiens.

It will be remembered that politicalissues and sensitivities proper of the post-war period pervaded the life of the IMUand the IRC’s meetings. They were haunt-ed by animosities towards the countriesthat had fought against the allied powers.As a consequence, these countries werebanned from membership of the IMU andfrom participation in the 1924 TorontoCongress. Regarding these political issueswithin the context of the IMU and the IRC,it may be of interest to note that CostaLobo did not directly address or commenton them. However, his views about warrelated topics were the ones that could beexpected from a citizen of one of the alliedcountries. (cf. Lobo [3]).

In the year that followed the TorontoCongress, it has to be mentioned the rôleplayed by Costa Lobo in the realization inCoimbra, in 1925, of the Second JointCongress of the Associação Portuguesapara o Avanço das Ciências1, together withits Spanish counterpart, the AssociaciónEspañola para el Avance de las Ciencias.2.

The congress took place from 14th.to19th of July and embraced a wide spectrumof topics: mathematics, astronomy andphysics of the Earth, physico-chemistry,natural sciences, philological, philosophi-cal and historical sciences, and medical sci-ences and applications. Apart from thepeninsular representatives, the congresswas attended, amongst others, by represen-tatives of The French Institute, The RoyalAcademy of Belgium and the French andEnglish Associations for the Advancementof Science.

On this occasion, Costa Lobo sharedresponsibilities with an important andinternationally known Portuguese mathe-matician: Francisco Gomes Teixeira. Thelatter took part as President of theAssociação Portuguesa, while Costa Loboacted as President of the Congress’Executive Committee.

Although we lose track of Portugueseparticipation in the 1928, 1932 and 1936ICMs, we note the presence of Portuguesemathematicians at the 1950 Cambridge-USA Congress. Communications weresubmitted by Hugo Ribeiro, A. Gião, A. Monteiro, M. Peixito, and R. L. Gomes.However, these scientists took part in theCongress as individuals and not as repre-sentatives of Portugal. Moreover, onlyHugo Ribeiro was personally present, as he

was living in the USA. The others, exiledin Brazil, limited themselves to sendingtheir contributions. This was an indicationthat political pressure exerted on thePortuguese universities by the Salazarregime had began to show its effects. It iswell known that during the Salazar regime,intellectual life and scientific researchactivities in Portugal were severely ham-pered. For more details of how this situa-tion arose and subsequently evolved, see deOliveira [5], Bebiano [1], and Ramos &Malonek [6].

In these notes we have focussed ourattention on a specific part of the history ofmathematics in Portugal in the 1920’s.Details of ensuing events, especially thefoundation and life of the PortugueseMathematical Society can, again, be foundin de Oliveira [5].

References.

[1] Bebiano, Natália. Bento Caraça(1901-48): A beautiful mind.. News-letter of the EMS 44, June 2002.

[2] Costa, Teresa & Malonek, Helmuth.Uma iniciativa notável de Costa Lobo.(A remarkable initiative of CostaLobo). Helios, http/www.mat.uc.pt/~obsv/obsv.

[3] Lobo, Francisco Miranda da Costa.Congresso Internacional de Matem-áticas. (International MathematicalCongress). O Instituto, JornalScientífico e Literário, No. 67,Coimbra, 1920.

[4] Lobo, Francisco Miranda da Costa.Congresso da União MatemáticaInternacional. (Congress of theInternational Mathematical Union). OInstituto, Jornal Scientífico e Literário,No. 71, Coimbra, 1924.

[5] de Oliveira, Graciano. The PortugueseMathematical Society. Newsletter of

the EMS 46, December 2002.[6] Ramos, Anabela & Malonek,

Helmuth. Portugal e a fundação daUnião Matemática Internacional.(Portugal and the foundation of theInternational Mathematical Union).To appear.

[7] Ramos, Anabela & Malonek,Helmuth. Um capítulo esquecido –Sócios Matemáticos extrangeiros.doInstituto de Coimbra na terceira déca-da século XX. (A forgotten chapter -foreign Mathematicians, as membersof the Coimbra Institute in the thirddecade of the 20th century.) Boletimde la Sociedade Portuguesa deMatemática, No. 49, 2003.

Enrique Hernández Manfredini lectures inMathematical Logic and Foundations ofMathematics at the University of Aveiro-Portugal. He graduated in mathematicseducation at Chile University and obtainedhis Ph. D. in mathematics from (the for-mer) Bedford College, London University.He has worked at the Chile University, the(former) Technical University of Santiago-Chile, the Cuban Academy of Sciences, theUniversity of Havana, British Telecom,and at a variety of secondary and furthereducation colleges. His main interest liesin the realm of the foundations of mathe-matics. Anabela Ramos is an assistant at theMathematics Department of AveiroUniversity, where she tutors and lectures inmathematical analysis. She completed herfirst degree in mathematics, and later hermaster’s degree, at the University ofCoimbra. Her MSc thesis dealt with one ofPedro Nunes’ works: O De Erratis OrontiiFinaei. She has previously worked as asecondary school mathematics teacher.Her main interest focuses on Portuguesemathematical history.

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EMS June 200424

University of Coimbra

1 Portuguese Association for the Advancement of Science.2 Spanish Association for the Advancement of Science

Ari Laptev (Stockholm) is theChairman of the 4ECMOrganizing Committee.

Role of the European Congressof Mathematics (ECM)

You are chairing the OrganizingCommittee of the 4ECM, one ofthe major mathematical events inEurope in 2004. What is in youropinion the role and the necessi-ty of big, interdisciplinary meet-ings like ECMs?I used to be quite sceptical aboutlarge-scale meetings in generaland EMS Congresses in particular.However, my involvement inorganizing the 4ECM has radical-ly changed my attitude. I see nowhow useful such meetings can be.Compared with specialized scien-tific conferences, where one meetsclose colleagues, big meetingshave quite a different function.They give an overview of newtendencies in Mathematics andplay an important role in develop-ing its different areas. There is anopinion that Mathematics hasbecome too large and that it is dif-ficult for the majority of partici-pants to follow most of the talks atthese meetings. During the lastcentury many new fields havebeen created, and establishedbranches of Mathematics havesplit into sub fields. However,development during the last twen-ty years has to a large extent beenthe opposite. There is a clear ten-dency towards unification of dif-ferent branches of mathematicalresearch. Many new break-

throughs were achieved by com-bining techniques from differentareas of Mathematics. Therefore Ithink that IMU and EMSCongresses will play an evenmore important role than before,allowing mathematicians to focuson problems of general interest.

Since 1897, mathematicians havemet periodically at InternationalCongresses (ICMs), with pro-grammes including all disci-plines. Mathematics is the onlyscientific field keeping this tradi-tion. The much younger ECMsmight be seen as ICMs on asmaller scale. What makes ECMsso special?At the time when it became clearthat Stockholm might be the nexthost for the European Congress ofMathematics, the opinion of someof my colleagues was that EMSCongresses are not needed, pre-cisely for this reason. Theyclaimed that such meetings imi-tate IMU Congresses on a smallerscale, are not of the highest quali-ty and therefore unnecessary. Idisagree. Looking back at thethree EMS Congresses organizedin Paris, Budapest and Barcelona,it is noticeable that they were alldifferent from the IMUCongresses in character as well.When organizing the 4ECM wealso hoped to introduce new fea-tures. For example, already at thevery early organizational stage we

decided to involve the EuropeanNetworks by giving members anopportunity to speak at theCongress.

In order to receive a grant fromBrussels or ESF, one has to gathera very representative bunch ofteams from different Europeancountries working in the samearea of Mathematics. The compe-tition is extremely high but oncethe project is awarded, theNetwork obtains quite substantialfunding for developing a specificarea of research. However, theyhave very little interaction withmathematicians who do notbelong to their Network.Therefore, European Congressescan play a coordinating role forthese programmes.By inviting the speakers from dif-ferent European Networks to the4ECM, we want to give EuropeanNetworks the possibility of com-municating with each other. Wealso hoped that by inviting suchspeakers we could make the4ECM more attractive to themembers of the correspondingresearch groups. Indeed, as we seenow, many members of thesegroups will be participating in the4ECM and will have their ownsatellite meetings before or afterthe Congress.

This time we only have half aday for the Network presenta-tions. I personally think that itwould have been even better if we

INTERVIEW

EMS June 2004 25

An interview An interview with with

Ari LaptevAri LaptevInterviewer:

Marta Sanz-Solé (Universitat de Barcelona)

Ari Laptev Marta Sanz-Solé

could have devoted a whole day tothe European Network activities.For example, half a day for gener-al Invited Lectures and anotherhalf for networks mini-symposia,where the members of the corre-sponding Network could choosethe speakers themselves. One ofthe advantages of such mini-sym-posia compared with “usual”Network meetings could be thatsuch a meeting would gather abroader audience, allowing peopleto both invite speakers from paral-lel groups and participate them-selves in other Network mini-symposia.

We also kept in mind thatincluding European Network pre-sentations might be important forfuture EMS relations with theauthorities from Brussels and withthe ESF in Strasbourg. We areexpecting some representativesfrom Brussels to attend the 4ECMOpening Ceremony.

Another new feature of the4ECM is the involvement of sci-entists from areas outsideMathematics. Nobel Institutes inPhysics and Chemistry of theSwedish Royal Academy ofSciences are supporting theCongress. This enables us toinvite prominent specialists inPhysics, Chemistry and Biology,who will share their views on thesignificance of Mathematics intheir subject in a lecture of theirown choice.

Finally, I would also like tomention that there are 10 prizesawarded to young mathematiciansat EMS Congresses. They are dif-ferent in character compared withFields Prizes and allow us to drawattention to a group of youngEuropean mathematicians who areat the beginning of their scientificcareer and have already made sub-stantial contributions in theirareas of research.

I am sure that the organizers ofthe 5ECM in Amsterdam will alsohave new ideas, which will con-tribute to the future style of theEMS Congresses.

Mathematics in Science andTechnology

The theme of the 4ECM -Mathematics in Science andTechnology - might be a state-ment addressed to those whothink that Mathematics candevelop without any cross con-nections and interactions withscientists and engineers. At thesame time, it is carrying a clearmessage about the new opportu-nities and developments forMathematics in the 21st century.Could you comment on this?In his article written for theBarcelona Intelligencer, Jean-Pierre Bourguignon pointed outthe importance of interactionsbetween different areas withinMathematics and also the interac-tion of Mathematics with othersubjects. He writes:“Mathematicians are already andas I see it, will be more and more,confronted with the need to broad-en the conception they developedof their discipline and how theypractice it.”

With the title “Mathematics inScience and Technology” wewanted to emphasize the impor-tance of developing Mathematicsfor other Sciences andEngineering, and try to identifythe place of Mathematics withinthe modern scientific community.The attitude that Mathematics canbe good only for mathematiciansshould be changed and we all haveto try to be more open mindedtowards a large number of newareas in Science and Engineeringdeveloped during the last 20years. There is a danger that math-ematicians will lose respect fromother scientists if they are not pre-pared to collaborate.

There is also a danger thatMathematics as a subject could beput aside and become insignifi-cant in the main stream of thedevelopment of Science. Weshould not take the importance ofMathematics for granted; we haveto continue to prove it.

As I already mentioned, when

organizing the 4ECM we had theidea of using the fact that Swedenis the country of Nobel Prizes andinvited representatives from otherSciences to give Plenary talks. Webelieve that a partnership withspecialists from other scientificareas could be extremely fruitful.

We have made an effort to estab-lish better contact with AppliedMathematics by inviting Plenaryand Invited Speakers covering avariety of applications.

Visibility of Mathematics

The celebration of a large scien-tific meeting like the 4ECM is anexcellent opportunity to increasethe visibility of Mathematics.What aspects of the scientificprogramme help the most to showwhy Mathematics is useful andwhy a knowledge-based societyneeds mathematically well-edu-cated people?It is a very good opportunityindeed. Looking at the list of reg-istered participants and at some ofthe applications to Poster ses-sions, I can see many scientistswho cannot be classified as PureMathematicians. Obviously, thismeans that people from areas ofresearch related to Mathematicsare interested in the 4ECM and indeveloping contacts with mathe-maticians. This is a promising ten-dency and once more shows theimportance of big general meet-ings of Mathematics where peoplefrom different areas can meet eachother.

When trying to the predictfuture, we sometimes have discus-sions in our Department abouthow much one will needMathematics in perhaps 50 yearstime. Will future engineers needto know the multiplication table?For example, this skill is hardly a

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EMS June 200426

requirement for a supermarketcashier nowadays, but it definitelywas very important in the past.Will the teaching of Mathematicssurvive or will it die out in a sim-ilar way to the teaching of Latin?It is difficult to say. However, it isclear that the training of logicalthinking will always be necessary.

Moreover, progress of Scienceand Engineering relies more andmore on complicated computa-tions, which always have to beevaluated. It is difficult to imaginethat this can be done without deepknowledge of Mathematics.

Young mathematicians

One cannot think of the future ofscience without the incomingflow of the younger generations.Why do young people feel attract-ed to do research inMathematics? What is the impor-tance of Congresses showingboth the unity and the diversity ofMathematics and its most recentbreakthroughs?It is a mystery how society contin-uously produces a number of peo-ple who are gifted in Mathematicsand who have the necessary pas-sion to pursue it. At the beginningof May we had a meeting at ourDepartment at KTH evaluatingapplications for PhD positions.There were 30 applications.Although our financial possibili-ties allowed us to accept only fourapplications, we were very happyto see that Mathematics is anattractive subject for young peo-ple in Sweden. However, such anattraction is based on many fac-tors, which are often insignificantwhen taking separately, butbecome very encouraging whenviewed together. I believe that thevisibility of the EMS Congresseswith their awards to youngresearchers plays an importantpart in making Mathematicsattractive to young people.

Needless to say that more than200 grants, which were distrib-uted to mostly young participantsfrom Central and Eastern

European countries, will supportthe popularity of Mathematics incountries where in general, condi-tions for mathematicians are stillvery poor.

You live in a country with inten-sive activity and influence in thedevelopment of Science andTechnology. Do education policymakers put enough effort intoensuring good mathematicaleducation at Swedish schools?Are scholars aware of the attrac-tive, exciting side ofMathematics?It is clear that we have a big prob-lem with mathematical educationin Swedish schools and it lookslike it is a common phenomenonfor many other European coun-tries too. One of the main reasonsbehind such development is thatnowadays the status of teachers israther low. As a result, this profes-sion is not very attractive to tal-ented people. Our governmenttries to improve this situation butit is not an easy problem, especial-ly if one is not prepared to investa substantial amount of money init. A relatively “cheap” solution -investing into research in mathe-matical education - has not hadmuch of an effect.

Being a good teacher in any sub-ject, in particular Mathematics, isa talent that is based on a solidknowledge of the subject. Theresource of talented teachers islimited and it is vital for any soci-ety not to lose such people and tomake the teaching professionmore attractive. At Universitylevel, we try to do our best to putfuture Engineers on the “righttrack”, but it becomes more andmore difficult because studentshave not been sufficiently pre-pared in Mathematics at school.

Stockholm is known as a “City ofScience”. Do you plan to presentthe 4ECM to journalists, to letpeople know that it will be the“City of Mathematics” aroundthe dates of the congress?

We shall try our best and hope

that we shall be able to involve themedia. The possibility of gettingextra publicity is also an addedadvantage of inviting Nobel Prizewinners to give lectures at theCongress. The Swedish Academyof Science has already shown itsinterest in publicizing the 4ECM.However, it is difficult to competewith the European Football Cup inPortugal, the last week of whichcoincides with the 4ECM. You canguess which event will be themost attractive to journalists.

And the outcome?

You and your colleagues on theOrganizing Committee are devot-ing enormous efforts into thepreparation of this importantevent, and many of the country’sInstitutions are sponsoring theCongress. What kind of tangiblebenefits could you expect fromall this? (For instance, increas-ing student’s interest forMathematics, gaining govern-ment support to improve mathe-matical education and research,strengthening scientific coopera-tion between different depart-ments across the country, etc.)I do not think that one should tryto measure how profitable the4ECM will be for Mathematics inEurope, Sweden or for ourDepartment at KTH. However, Ibelieve that a meeting of thisdimension, as well as everythingimportant which is happening inMathematics, is definitely helpfulfor its visibility. I believe that ulti-mately it will play an importantrole in the future funding ofMathematics at every level.

We expect that the 4ECM will bevery good for Swedish PhD stu-dents. Many of them are partici-pating in the Congress, and meet-ing prominent mathematicianspersonally will definitely inspiretheir commitment to Mathematics.

Thank you very much for thisinterview. I am sure that the4ECM will be a success.

INTERVIEW

EMS June 2004 27

History and peopleThe stochastic sciences have been develop-ing so fast in the past decades that anincreasing need was felt for internationalcooperation and bundling of researchresources. This led to the establishment ofEURANDOM, in July 1997. The instituteopened its doors in September 1998. Thefirst scientific director was Willem vanZwet. Since October 2000, Frank denHollander has been the scientific director,with Henry Wynn (London, UnitedKingdom) as scientific co-director.

Frank den Hollander, scientific director

Peter Bickel (Berkeley, USA) chairs thescientific council, which consists ofrenowned European scientists in the areasof probability, statistics and stochasticoperations research. In July 2004, DonDawson (Ottawa, Canada) will take overthis position.

Financial resourcesEURANDOM has two main financiers: theNetherlands Organisation for ScientificResearch (NWO) and Eindhoven

University of Technology (TU/e). Together

they provide 60% of the budget. Both par-ties have a representative on the Board ofEURANDOM and they jointly appoint athird external member. Other fundingsources are: European science organisa-tions, industrial contracts, EU-funds, ESF-networks, etc.

MissionThe mission of EURANDOM is to fosterresearch in the stochastic sciences andtheir applications. It achieves this missionby helping talented young researchers findtheir way to tenured positions in academiaand industry. It does this by carrying outand facilitating research through post-doc-toral and graduate appointments, visitorexchange and workshops, and by takinginitiatives for collaborative research at theEuropean level.

FocusFrom the summer of 2004 onwards, theresearch at EURANDOM will be re-aligned into three parallel programmes,each with three themes:

Random Spatial Structures (RSS):- Critical Phenomena

- Disordered Systems- Combinatorial Probability

Queuing and Performance Analysis (QPA):- Performance Analysis of Production

Systems- Performance Analysis of

Communication Systems- Queuing Theory

Statistical Information and Modelling(SIM):

- Statistical Learning- Biomedical and Bio-molecular

Statistics- Industrial Statistics.

Harry Kesten (Cornell University, USA),EURANDOM Chair in 2002

RSS focuses on systems consisting of alarge number of interacting random com-ponents, possibly with disorder. It aims tocapture their large space-time behaviourthrough a combination of probabilistic,combinatorial and ergodic techniques, withspecial emphasis on critical phenomenaand universality.

ERCOM: European Research Centres on MathematicsA lot of mathematical activity is hosted at the 24 European research centres - quite different in size and scope- represented in the EMS committee ERCOM. The Newsletter wishes to give these institutions the opportu-nity to present themselves in the coming issues and years. This will hopefully give the readers an idea of howthey might use the possibilities offered by these centres: as organiser or participant of a workshop or a sem-inar or as a visitor. Quick information about the ERCOM institutions and about the ERCOM committee oftheir directors is found on the web site http://www.crm.es/ERCOM.

EURANDOM, a European research institute for the stochastic sciences“The feeling of sparkling Champagne”

EURANDOM

EMS June 200428

QPA is concerned with the performanceanalysis of stochastic networks, in particu-lar, production and communication sys-tems. Queuing theory and stochastic analy-sis provide the means to deal with controland optimization issues.

SIM combines analytic and algebraicmethods to address a variety of statisticalquestions, including statistical classifica-tion, bootstrap, micro-arrays, gene expres-sion, design of experiments, predictivemaintenance and quantum statistics.

Each programme hosts six to eight post-doctorates and graduate students, is super-vised by senior scientific advisors, and isguided by an international steering com-mittee. Through the choice of themes, spe-cial attention is given to the interfaces withphysics, biology, telecommunication andindustry.

In addition to the above three research pro-grammes, EURANDOM runs two researchprojects, one on reinsurance and one onbattery management.

Staff and researchEURANDOM offers an attractive environ-ment for carrying out research on randomphenomena. Since its start, some 60 post-doctorates and graduate students have beenworking at EURANDOM. About 80% ofthe junior staff leaves EURANDOM fortenured positions in academia or industry(11 so far in The Netherlands). At anytime, about 25 junior researchers are work-ing at EURANDOM.

Vacancies at EURANDOM may occur atany time during the year. The opening ofpositions is not restricted by deadlines, anddepends on the number of junior staff perprogramme at a given moment of time.Candidates who wish to apply for an exter-nal research grant are welcome. The insti-tute offers assistance in the applicationprocedure. Post-doctoral appointments aretypically for two years, though shorterperiods are possible as well. Graduate stu-dents are appointed for three to four years.

The EURANDOM visitor programme isanother important element that makes theinstitute into a lively place. Senior andjunior researchers come from all over theworld, either on invitation or upon request,and participate in the various activities.Workshop topics are chosen by the scien-tific advisors and by the steering commit-tees of the research programmes. Ideas fortopics are collected from post-doctorates

and senior visitors, and through variousEuropean networks and programmes inwhich EURANDOM participates.

Links to research in The NetherlandsEURANDOM is located on the TU/e cam-pus in Eindhoven. EURANDOM is anindependent foundation. The location onthe university campus allows for specialties with the university, in particular withthe Department of Mathematics andComputer Science. Through the appoint-ment of scientific advisors, the institutemaintains intensive relations with othermathematics departments in TheNetherlands.

Links to research in EuropeEuropean cooperation runs via the scientif-ic council and the steering committees ofthe research programmes, via the appoint-ment of post-doctorates, graduate students,and through the workshop and visitors pro-gramme. Each year EURANDOM organis-es some 10 workshops on a range of topics,with on average 40-45 participants fromaround the world. Each year some 40junior and senior researchers from outsideThe Netherlands visit EURANDOM forperiods of one week up to several months(on average 60-70 visitor weeks per year).EURANDOM thus provides a stimulatingenvironment to enhance collaborationwithin Europe.

EURANDOM participates in an ECThematic Network (Pro-Enbis), two ECNetworks of Excellence (PASCAL andEURO-NGI), a Dutch-German BilateralResearch Group (BRG), and chairs an ESFNetwork (RDSES). EURANDOM has

close ties with MaPhySto (Aarhus,Denmark), the Stochastic Centre(Göteborg, Sweden), and the NorwegianComputing Centre (Oslo), which share astrong focus on the stochastic sciences.

Several national research organizations(e.g. DFG, FWO, NSF) have supportedEURANDOM through grants for post-doc-torates and visitors to carry out theirresearch at EURANDOM.

Location and facilitiesEURANDOM is situated in a spaciousbuilding, full of light, glass and colours, onthe TU/e campus in Eindhoven.

Post-doctorates get together on a dailybasis, run a bi-weekly post-doctorate semi-nar, and participate in the busy schedule ofseminars and mini-courses the instituteoffers. Social events are organised at regu-lar intervals, ranging from hiking on bike(with or without umbrella!), to admiringtulip exhibitions, ice-skating, canoeing,etc.

Junior staff and long-term visitors have anoffice with computer facilities and areassisted at various levels, such as findingaccommodation and arranging a visa andwork permit. There is a small in-houselibrary, carrying most international sto-chastic science journals. Extensive libraryfacilities are available on campus, much ofit on-line.

The TU/e acts formally as employer ofEURANDOM personnel; therefore all staffcan use the facilities on campus (library,sporting, restaurant, etc.).

EURANDOM looks forward to welcomingyou at the institute. Come and share withus the “feeling of sparkling Champagne”.For more detailed information on the insti-tute and its current activities, visit ourwebsite: http://www.euran dom.tue.nl.

ERCOM

EMS June 2004 29

I.Singer, M. Atiyah and King Haraldduring the Prize Ceremony

(Photo: Knut Falch/Scanpix)

The second Abel Prize has beenawarded jointly to Michael FrancisAtiyah and Isadore M. Singer. TheAtiyah-Singer index theorem is one ofthe great landmarks of twentieth cen-tury mathematics, profoundly influ-encing many of the most importantlater developments in topology, differ-ential geometry and quantum field the-ory. Its authors, both jointly and indi-vidually, have been instrumental inrepairing a rift between the worlds ofpure mathematics and theoretical par-ticle physics, initiating a cross-fertil-ization that has been one of the mostexciting developments of the lastdecades.

We describe the world by measuringquantities and forces that vary overtime and space. The rules of nature areoften expressed by formulas involvingtheir rates of change, so-called differ-ential equations. Such formulas mayhave an "index", the number of solu-tions of the formulas minus the num-ber of restrictions that they impose onthe values of the quantities being com-puted. The index theorem calculatedthis number in terms of the geometry

of the surrounding space. A simple case is illustrated by a

famous paradoxical etching of M.C.Escher, "Ascending and Descending",where the people, going uphill all thetime, still manage to circle the castlecourtyard. The index theorem wouldhave told them this was impossible!

The Atiyah-Singer index theoremwas the culmination and crowningachievement of a development of ideasmore than 100-years old, fromStokes's theorem, which students learnin calculus classes, to sophisticatedmodern theories like Hodge's theory ofharmonic integrals and Hirzebruch'ssignature theorem.

The problem solved by the Atiyah-Singer theorem is truly ubiquitous. Inthe 40 years since its discovery, the

theorem has had innumerable applica-tions, first within mathematics andthen, beginning in the late 70's, in the-oretical physics: gauge theory, instan-tons, monopoles, string theory, the the-ory of anomalies, etc.

At first, the applications in physicscame as a complete surprise to boththe mathematics and physics commu-nities. Now the index theorem hasbecome an integral part of their cul-tures. Atiyah and Singer, together andindividually, have been tireless in theirattempts to explain the insights ofphysicists to mathematicians. At thesame time, they brought modern dif-ferential geometry and analysis as itapplies to quantum field theory to theattention of physicists and suggestednew directions within physics itself.This cross-fertilization continues to befruitful for both sciences.

Michael Francis Atiyah and IsadoreM. Singer are among the most influen-tial mathematicians of the last century,and are still working. Over a period of20 years they worked together on theindex theorem and its ramifications,and with their theorem they changed

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EMS June 200430

Abel Prize 2004 to Abel Prize 2004 to Atiyah and SingerAtiyah and SingerThe Norwegian Academy of Science and Letters has decided to award theAbel Prize for 2004 jointly to

Sir Michael Francis Atiyah, University of Edinburghand

Isadore M. Singer, Massachusetts Institute of Technology"for their discovery and proof of the index theorem, bringing together topol-ogy, geometry and analysis, and their outstanding role in building newbridges between mathematics and theoretical physics."

Wreath ceremony at theAbel monument, Oslo

the landscape of mathematics. Atiyah and Singer came originally

from different fields of mathematics -Atiyah from algebraic geometry andtopology, Singer from analysis. Theirmain contributions in their respectiveareas are also highly recognized.Atiyah's early work on meromorphicforms on algebraic varieties, and hisimportant 1961 paper on Thom com-plexes, are such examples. Atiyah'spioneering work together withFriedrich Hirzebruch on the develop-ment of the topological analogue ofGrothendieck's K-theory, had numer-ous applications to classical problemsof topology and turned out later to bedeeply connected with the index theo-rem.

Singer, jointly with Richard V.Kadison, initiated the subject of trian-gular operator algebras. Singer's nameis also associated with the Ambrose-Singer holonomy theorem and theRay-Singer torsion invariant. Singer,together with Henry P. McKean, point-ed out the deep geometrical informa-tion hidden in heat kernels, a discov-ery that had great impact.

Isadore M. Singer was born in Detroitin 1924, and received his undergradu-ate degree from the University of

Michigan in 1944. After obtaining hisPh.D. from the University of Chicagoin 1950, he joined the faculty at theMassachusetts Institute of Technology(MIT). Singer has spent most of hisprofessional life at MIT, where he iscurrently an Institute Professor.

Singer is a member of the American

Academy of Art and Sciences, theAmerican Philosophical Society andthe National Academy of Sciences(NAS). He served on the Council ofNAS, the Governing Board of theNational Research Council, and theWhite House Science Council. Singerwas vice president of the AmericanMathematical Society from 1970 to1972.

In 1992 Singer received theAmerican Mathematical Society'sAward for Distinguished PublicService. The citation recognized his"outstanding contribution to his pro-fession, to science more broadly and tothe public good."

Among the other awards he hasreceived are the Bôcher Prize (1969)and the Steele Prize for LifetimeAchievement (2000), both from theAmerican Mathematical Society, theEugene Wigner Medal (1988), and theNational Medal of Science (1983). When Singer was awarded the SteelePrize, his response, published in theNotices of the AMS, was: "For me theclassroom is an important counterpartto research. I enjoy teaching under-graduates at all levels, and I have ahost of graduate students, many ofwhom have ended up teaching memore than I have taught them." Singerhas also written influential textbooksthat have inspired generations ofmathematicians.

Michael Francis Atiyah was born inLondon in 1929. Atiyah got his B.A.and his doctorate from Trinity College,

Cambridge. He spent the greatest partof his academic career in Cambridgeand Oxford. He has held many promi-

nent positions, among them the highlyprestigious Savilian Chair ofGeometry at Oxford and that of Masterof Trinity College, Cambridge. Atiyahhas also been professor of mathemat-ics at the Institute for Advanced Studyin Princeton.

Atiyah rejuvenated British mathe-matics during his years at Oxford andCambridge. He was also the drivingforce behind the creation of the IsaacNewton Institute for MathematicalSciences in Cambridge and became itsfirst director. Atiyah is now retiredand an honorary professor at theUniversity of Edinburgh.

Michael Francis Atiyah hasreceived many honours during hiscareer, including the Fields Medal(1966). He was elected a Fellow of theRoyal Society in 1962, at the age of32. He was awarded the Royal Medalof the Society in 1968 and its CopleyMedal in 1988. Atiyah was presidentof the Royal Society from 1990 to1995. Atiyah has also served as presi-dent of the London MathematicalSociety (1974 - 1976) and has playedan important role in the shaping oftoday's European MathematicalSociety (EMS).

Atiyah was responsible for thefounding of the Inter-Academy Panelthat brought together many of theworlds academies of science. TheInter-Academy Panel has now beenpermanently established and will playa major role in the integration of sci-entific policy throughout the world.Atiyah also instigated the formation ofthe Association of EuropeanAcademies (ALLEA). Atiyah hasbeen president of PugwashConferences on Science and WorldAffairs.

Among the prizes he has receivedare the Feltrinelli Prize from theAccademia Nazionale dei Lincei(1981) and the King FaisalInternational Prize for Science (1987).Michael Francis Atiyah was knightedin 1983 and made a member of theOrder of Merit in 1992.

The Newsletter plans to publish aninterview with the laureates in its nextissue.

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EMS June 2004 31

The 10th International Congresson Mathematics Education,ICME-10, in Denmark this sum-mer, has got the following visions:

• A well organised congress cele-brating the ICME tradition.

• A congress with a strong scien-tific programme representingthe state-of-the-art in bothmathematics education researchand teaching practices.

• A strong Nordic representationamongst participants.

• A large representation of math-ematics teachers from all edu-cational levels.

• A congress that makes a differ-ence to the practice of teachingmathematics.

• A gender balanced congress.• A congress that gives space and

time for informal discussionsamong the participants.

• A financially balanced con-gress.

The last vision might need somefurther development. Two monthsbefore the congress the number ofregistrations is a little below the

budget, but not that bad: 484 fromthe Nordic Countries, 590 fromthe rest of Europe and 1032 fromthe rest of the world including 355from the USA. If you want theexact number of participants fromyour country check at www.icme-10.dk.

The programme will be asplanned in the second announce-ment.

The ICMI Felix Klein and HansFreudenthal medals for 2003The International Commission onMathematical Instruction (ICMI),founded in Rome in 1908, has, forthe first time in its history, estab-lished prizes recognizing out-standing achievement in mathe-matics education research. TheFelix Klein Medal, named for thefirst president of ICMI (1908-1920), honours a lifetime achieve-ment. The Hans FreudenthalMedal, named for the eighth pres-ident of ICMI (1967-1970), recog-nizes a major cumulative programof research. These awards are tobe made in each odd numberedyear, with presentation of themedals, and invited addresses bythe medallists at the followingInternational Congress onMathematical Education (ICME).

These awards, which pay tributeto outstanding scholarship inmathematics education, serve notonly to encourage the efforts ofothers, but also to contribute to thedevelopment, through the publicrecognition of exemplars, of highstandards for the field. Theawards represent the judgment ofan (anonymous) jury of distin-guished scholars of internationalstature, chaired by Prof. Michèle

Artigue of the University Paris 7.ICMI is proud to announce the

first awardees of the Klein andFreudenthal Medals.

The Felix Klein Medal for 2003is awarded to Guy Brousseau,Professor Emeritus of theUniversity Institute for TeacherEducation of Aquitaine inBordeaux, for his lifetime devel-opment of the theory of didacticsituations, and its applications tothe teaching and learning of math-ematics.

The Hans Freudenthal Medalfor 2003 is awarded to CeliaHoyles, Professor at the Instituteof Education of the University ofLondon, for her seminal researchon instructional uses of technologyin mathematics education.

Citations of the work of thesemedalists can be found below.Presentation of the medals, andinvited addresses of the medalists,will occur at ICME-10 inCopenhagen, July 4-11, 2004.

Citation for the 2003 ICMIFelix Klein Medal to Guy

BrousseauThe first Felix Klein Award of theInternal Commission onMathematical Instruction (ICMI)is awarded to Professor GuyBrousseau. This distinction recog-nises the essential contributionGuy Brousseau has given to thedevelopment of mathematics edu-cation as a scientific field ofresearch, through his theoreticaland experimental work over fourdecades, and to the sustainedeffort he has made throughout hisprofessional life to apply the fruitsof his research to the mathematicseducation of both students and

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EMS June 200432

ICME-10 and ICMI awardsICME-10 and ICMI awards

teachers. Born in 1933, Guy Brousseau

began his career as an elementaryteacher in 1953. In the late sixties,after graduating in mathematics,he entered the University ofBordeaux. In 1986 he earned a‘doctorat d’état,’ and in 1991became a full professor at thenewly created University Institutefor Teacher Education (IUFM) inBordeaux, where he worked until1998. He is now ProfessorEmeritus at the IUFM ofAquitaine. He is also DoctorHonoris Causa of the University ofMontréal.

Guy Brousseau

From the early seventies, GuyBrousseau emerged as one of theleading and most originalresearchers in the new field ofmathematics education, convincedon the one hand that this field mustbe developed as a genuine field ofresearch, with both fundamentaland applied dimensions, and onthe other hand that it must remainclose to the discipline of mathe-matics. His notable theoreticalachievement was the elaborationof the theory of didactic situa-tions, a theory he initiated in theearly seventies, and which he hascontinued to develop with unfail-ing energy and creativity. At a time

when the dominant vision wascognitive, strongly influenced bythe Piagetian epistemology, hestressed that what the field neededfor its development was not apurely cognitive theory but oneallowing us also to understand thesocial interactions between stu-dents, teachers and knowledge thattake place in the classroom andcondition what is learned by stu-dents and how it can be learned.This is the aim of the theory ofdidactic situations, which has pro-gressively matured, becoming theimpressive and complex theorythat it is today. To be sure, this wasa collective work, but each timethere were substantial advances,the critical source was GuyBrousseau.

This theory, visionary in its inte-gration of epistemological, cogni-tive and social dimensions, hasbeen a constant source of inspira-tion for many researchers through-out the world. Its main constructs,such as the concepts of adidacticand didactic situations, of didacticcontract, of devolution and institu-tionalization have been madewidely accessible through thetranslation of Guy Brousseau’sprincipal texts into many differentlanguages and, more recently, thepublication by Kluwer in 1997 ofthe book, ‘Theory of didactical sit-uations in mathematics - 1970-1990’.

Although the research GuyBrousseau has inspired currentlyembraces the entire range of math-ematics education from elemen-tary to post-secondary, his majorcontributions deal with the ele-mentary level, where they coverall mathematical domains fromnumbers and geometry to proba-bility. Their production owesmuch to a specific structure – theCOREM (Center for Observationand Research in MathematicsEducation) – that he created in1972 and directed until 1997.

COREM provided an originalorganisation of the relationshipsbetween theoretical and experi-mental work.

Guy Brousseau is not only anexceptional and inspiredresearcher in the field, he is also ascholar who has dedicated his lifeto mathematics education, tireless-ly supporting the development ofthe field, not only in France but inmany countries, supporting newdoctoral programs, helping andsupervising young internationalresearchers (he supervised morethan 50 doctoral theses), contribut-ing in a vital way to the develop-ment of mathematical and didacticknowledge of students and teach-ers. He has been until the ninetiesintensely involved in the activitiesof the CIEAEM (CommissionInternationale pour l’Etude etl’Amélioration de l’Enseignementdes Mathématiques) and he was itssecretary from 1981 to 1984. At anational level, he was deeplyinvolved in the experience of theIREMs (Research Institutes inMathematics Education), fromtheir foundation in the late sixties.He had a decisive influence on theactivities and resources these insti-tutes have developed for promot-ing high quality mathematicstraining of elementary teachers formore than 30 years.

Citation for the 2003 ICMIFreudenthal Medal to Celia

HoylesThe first Hans Freudenthal Awardof the International Commissionon Mathematical Instruction(ICMI) is awarded to ProfessorCelia Hoyles. This distinctionrecognises the outstanding contri-bution that Celia Hoyles has madeto research in the domain of tech-nology and mathematics educa-tion, both in terms of theoreticaladvances and through the develop-ment and piloting of national andinternational projects in this field,

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EMS June 2004 33

aimed at improving through tech-nology the mathematics educationof the general population, fromyoung children to adults in theworkplace.

Celia Hoyles

Celia Hoyles studied mathemat-ics at the University ofManchester, winning the Daltonprize for the best first-class degreein Mathematics. She began hercareer as a secondary teacher, andthen became a lecturer at thePolytechnic of North London. Sheentered the field of mathematicseducation research, earning aMasters and Doctorate, andbecame Professor of MathematicsEducation at the Institute ofEducation, University of Londonin 1984.

Her early research in the area oftechnology and mathematics edu-cation, like that of manyresearchers, began by exploringthe potential offered by Logo, andshe soon became an internationalleader in this area. Two books pub-lished in 1986 and 1992 (edited)attested to the productivity of herresearch with Logo. This was fol-lowed, in 1996, by the publicationof Windows on MathematicalMeanings: Learning Cultures andComputers, co-authored withRichard Noss, which inspired

major theoretical advances in thefield, such as the notions of web-bing and situated abstraction,ideas that are well known toresearchers irrespective of the spe-cific technologies they are study-ing.

From the mid nineties, herresearch on technology integratedthe new possibilities offered byinformation and communicationtechnologies as well as the newrelationships children developwith technology. She has recentlyco-directed successively two pro-jects funded by the EuropeanUnion: the Playground project inwhich children from differentcountries designed, built andshared their own video games, andthe current WebLabs project,which aims at designing and eval-uating virtual laboratories wherechildren in different countriesbuild and explore mathematicaland scientific ideas collaborativelyat a distance. As an internationalleader in the area of technologyand mathematics education, shewas recently appointed by theICMI Executive Committee as co-chair of a new ICMI Study on thistheme.

However, Celia Hoyles’ contri-bution to research in mathematicseducation is considerably broaderthan this focus on technology.Since the mid nineties, she hasbeen involved in two further majorareas of research. The first, aseries of studies on children’sunderstanding of proof, has pio-neered some novel methodologicalstrategies linking quantitative andqualitative approaches that includelongitudinal analyses of develop-ment. The second area hasinvolved researching the mathe-matics used at work and she nowco-directs a new project, Techno-Mathematical Literacies in theWorkplace, which aims to developthis research by implementing andevaluating some theoretically-

designed workplace training usinga range of new media.

In recent years Celia Hoyles hasbecome increasingly involved inworking alongside mathematiciansand teachers in policy-making.She was elected Chair of the JointMathematical Council of the U.K.in October 1999 and she is a mem-ber of the Advisory Committee onMathematics Education (ACME)that speaks for the whole of themathematics community to theGovernment on policy mattersrelated to mathematics, from pri-mary to higher education. In2002, she played a major role inACME’s first report to theGovernment on the ContinuingProfessional Development ofTeachers of Mathematics, and con-tributed to the comprehensivereview of 14-19 mathematics inthe UK. In recognition of her con-tributions, Celia has recently beenawarded the Order of the BritishEmpire for “Services toMathematics Education”.

Celia Hoyles belongs to thatspecial breed of mathematics edu-cators who, even while engagingwith theoretical questions, do notlose sight of practice; and recipro-cally, while engaged in advanc-ing practice, do not forget thelessons they have learned fromtheory and from empiricalresearch. Celia Hoyles’ commit-ment to the improvement of math-ematics education, in her countryand beyond, can be felt in everydetail of her multi-faceted, diverseprofessional activity. Her enthusi-asm and vision are universallyadmired by those who have beenin direct contact with her. It isthanks to people like Celia Hoyles,with a clear sense of mission andthe ability to build bridgesbetween research and practicewhile contributing to both, that thecommunity of mathematics educa-tion has acquired, over the years, abetter-defined identity.

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In November 2003, the African Mathematics Union (AMU) andthe International Conference of Mathematical Sciences presentedmedals to a group of promising young African Mathematicians.The winners are:Olatunde Akiulade (Nigeria; mathematical physics)Bamidele Awojoyogbe (Nigeria; mathematical physics)Abba Gumel (Nigeria, now Canada; applied mathematics)Oluwole D. Makinde (Nigeria, now South Africa; applied math-ematics)Guy M. Kniet (Gabon; statistics)James A. Oguntuase (Nigeria, now Italy; pure mathematics)

The French Academy of Sciences has awarded several prizes inmathematics for 2003: Claire Voisin (Paris) has been awarded thePrix Sophie Germain; Louis Boutet de Monvel (Paris) receivedthe Prix Fondée par l’État; The Prix Jacques Hadamard was award-ed to Wendelin Werner (Paris-Sud); Claude Bardos was award-ed the Prix Marcel Dassault; Gilles Lebeau (Nice) received thePrix Ampère de l’Électricité de France; The Prix Gabrielle Sand etM. Guido Triossi was awarded to Damien Gaboriau (CNRS) andJean-Marc Delort (Paris-Nord) received the Prix Langevin.

The 2003 Fermat Prize for Mathematical Research has beenawarded to Luigi Ambrosio (Scuola Normale Superiore) for hiscontributions to the calculus of variations and geometric measuretheory and their relations with partial differential equations.

Two scientists were awarded the Max Planck Research Award2003 in Mathematics from the Max Planck Society:Stephan Luckhaus (Leipzig, Germany) for his trail blazing modelshowing clear cut solutions for the transport of water and othersubstances in soil. These solutions can also describe the growth oftumour cells in healthy tissue, which, up until now, was onlydescribed theoretically; andWolfgang Lück (Münster, Germany) for his landmark research onalgebraic topology.

In 2003 the first John von Neumann Awards were presented toMarina von Neumann Whitman, Charles Simony and toGeorge Dyson.

Mikhael L.Gromov (IHES, Bures-sur-Yvettes, France) and JayGould (Courant Institute, NYU, NY, USA) have been namedrecipients of the 2003-2004 Frederic Esser Nemmers Prize inMathematics.

The Institut de Recherche Mathématique Avancée (IRMA) hascreated an annual prize in memory of the mathematician PaulAndré Meyer (1934-2003). The prize honours an outstandingyoung probabilist, working in the field of stochastic processes.The first IRMA prize in Memory of Paul André Meyer was award-ed in February 2004 to Thomas Duquesne (Paris-Sud).

The 2004 Emil Artin Junior Prize in Mathematics has been award-ed to Gurgin R. Asatryan (Yerevan, Armenia).

The Norwegian Academy of Science and Letters has awarded thesecond Abel Prize (2004) jointly to Sir Michael Atiyah

AWARDS(Edinburgh, UK) and to Isadore M. Singer (MIT, USA) for theirdiscovery and proof of the index theorem. More information canbe found in this issue of the Newsletter.

The 2004 Gödel Prize for outstanding papers in the area of theo-retical computer science, sponsored jointly by the EuropeanAssociation for Theoretical Computer Science (EATCS) and theSpecial Interest Group on Algorithms and Computing Theory ofthe Association of Computing Machinery (ACM-SIGACT), hasbeen awarded to Maurice Herlihy (Brown Univ., RI, USA), NirShavit (Sun Labs, USA), Michael Saks (Rutgers, NJ, USA) andFotios Zaharoglou (Univ. CA, San Diego, USA) for importantpapers on the topological structure of asynchronous computability.

Former Newsletter Editor Robin Wilson (Open University, UK)has been appointed Gresham Professor of Geometry for the years2004-07. Congratulations!

Erratum: The editor wishes to apologize for an incorrect statement in issue50 about awards given in Poland in 2003. The correctedannouncements are:Micha³ Szurek (Warsaw) has received the Dickstein Great Prizeof the Polish Mathematical Society for his achievements in popu-larizing mathematics.Pawe³ Wilczyñski (Kraków) has been awarded the firstMarcinkiewicz Prize of the Polish Mathematical Society for stu-dents’ research papers.

We regret to announce the deaths of:

Martine Babillot (July 2003)

Andrey Andreevich Bolibruch (11 November 2003)

Armand Borel (11 August 2003)

David Fowler (13 April 2004)

Miguel de Guzman (14 April 2004)

John Hammersley (2 May 2004)

Hans Hermes (10 November 2003)

Eckehart Hotzel (28 September 2003)

Martin Kneser (16 February 2004)

John Lewis (21 January 2004)

Hans-Joachim Nastold (26 January 2004)

Peter Owens (30 December 2003)

Paul Wauters (26 October 2003)

Heiner Zieschang (5 April 2004)

DEATHS

Personal columnPersonal columnPlease send information on mathematical awards and deaths to the editor.

PERSONAL COLUMN

EMS June 2004 35

Please e-mail announcements of Europeanconferences, workshops and mathematicalmeetings of interest to EMS members, toone of the following addressesvberinde@ubm.ro or vasile_berinde@yahoo.com. Announcements should bewritten in a style similar to those here, andsent as Microsoft Word files or as text files(but not as TeX input files). Space permit-ting, each announcement will appear indetail in the next issue of the Newsletter togo to press, and thereafter will be brieflynoted in each new issue until the meetingtakes place, with a reference to the issue inwhich the detailed announcementappeared.

June 26 - July 1: 7th InternationalConference of The MathematicsEducation into the 21st Century Project,Ciechocinek, Poland.Information:e-mail: arogerson@vsg.edu.au Web site (for previous conferences only):http://math.unipa.it/~grim/21project.htm[For details, see EMS Newsletter 50]

27 June - 2 July: 4th EuropeanCongress of Mathematics, Stockholm,Sweden.Information:Web site: http://www.math.kth.se/4ecm[For details, see also EMS Newsletter 50,51]

June 30-July 2: Convexity and DiscreteGeometry. A conference on the occasionof the sixtieth birthday of TudorZamfirescuTheme: This conference will reflect TudorZamfirescu’s lifelong passion for geome-try, his broad range of mathematical inter-ests, and his commitment to students andyoung mathematiciansLocation: Eötvös Loránd University,Faculty of Science, Pázmány Péter sétány1/a H-1117 Budapest, HungaryOrganizers: K. Bezdek (Univ. of Calgary& Eötvös Univ.); K. Böröczky (EötvösUniv.); A. Heppes (Eötvös Univ.); ÉvaVásárhelyi (Eötvös Univ.)Confirmed Speakers: K. Bezdek (Univ.of Calgary, Canada); P. Gritzmann (Tech.Univ. of Munich, Germany); P.M. Gruber(Tech. Univ. of Vienna, Austria); A. Heppes (Eötvös Univ., Hungary);

July 2004

J.-I. Itoh (Kumamoto Univ., Japan); S. Marcus (Univ. of Bucuresti, Romania);J. Myjak (Univ. dell’Aquila , Italy)Deadline: for preliminary registration(express of interest) is April 10, 2004. Information: Eva Vasarhelyi, e-mail:vasar@ludens.elte.hu

1-15: Moonshine - the First QuarterCentury and Beyond. A Workshop onthe Moonshine Conjectures and VertexAlgebras, International Centre forMathematical Sciences, Edinburgh UKInformation:e-mail: icms@maths.ed.ac.ukWeb site: http://www.ma.hw.ac.uk/icms/meetings/2004/moonshine[For details, see EMS Newsletter 51]

3-10: Conference on SymplecticTopology ECM Satellite ConferenceStare Jab³onki, Poland Information:e-mail: symp@univ.szczecin.pl Web site: http://symp.univ.szczecin.pl[For details, see EMS Newsletter 51]

15-23: EMS Conference at Bêdlewo(Poland). Analysis on metric measurespaces.Information:Web site: http://impan.gov.pl/Bedlewo/

26-31: 6th World Congress of theBernoulli Society and the 67th AnnualMeeting of the Institute of MathematicalStatistics, Barcelona (Spain)Information:e-mail: wc2004@imub.ub.esWeb site: http://www.imub.ub.es/events/wc2004/[For details, see EMS Newsletter 49]

16-20 Workshop “Automorphisms ofCurves”, Lorentz Center, Leiden, The Netherlands.Organizers: Gunther Cornelissen andFrans Oort (Universiteit Utrecht)Aim: We will discuss the latest develop-ments in the theory of group actions onalgebraic curves, a field that has been lift-ed to a new level of sophistication in recentyearsTopics: deformation theory, lifting to char-acteristic zero, relation to moduli ofcurves, patching method, techniques from

August 2004

fundamental groups, equivariant families,non-archimedean uniformization, andgroup theoretic aspectsKeynote participants include: A. Abbes;P. Bradley; T. Chinburg; R. Guralnick; D. Harbater; F. Kato; S. Maugeais; R. Pries; M. Raynaud; N. Stalder; S. Wewers; M. ZieveInformation: e-mail: cornelis@math.uu.nlor oort@math.uu.nlWeb site: http://pascal.lc.leidenuniv.nl/

16-21: Bunyakovsky InternationalConference, Kyiv, Ukraine Theme: Honouring the bicentenary ofB.Ya.Bunyakovsky (1804-1889), eminentmathematician of Tsarist Russia, born inUkraine. The scope of the conference willinclude the range of the problemsBunyakovsky was interested in and madehis own contribution Topics: (1) mathematical analysis; (2)number theory; (3) theory of probabilityand its applications, actuarial mathematics;(4) mathematics education and mathemati-cal terminologyLanguages: English, Ukrainian, RussianOrganizers: Institute of Mathematics ofthe National Academy of Sciences ofUkraine, Ministry of Science andEducation of Ukraine, National TechnicalUniversity of Ukraine “KPI”, VinnytsyaNational Technical University, T. Shevchenko Kyiv National University,Ukrainian Mathematical Society Programme committee: Y. Berezansky;O. Boholyubov; V. Vyshensky; O. Hannyushkin; M. Gorbachuk; M. Zgurovsky; V. Klochko; V. Korolyuk;V. Melnyk; B. Mokin; M. Portenko; A. Yurachkivsky; M. Yadrenko (all fromUkraine); L. Brylevskaya (St-Petersburg,Russia); E. Seneta (Sydney, Australia), R. Andrushkiw (New York, USA)Organising committee: A. Samoilenko(Chairman); M. Gorbachuk and M. Swirzhevsky (Vice-Chairmen); H. Syta(Secretary); V. Boyko; Y. Vynnyshyn; I. Yegorchenko; T. Karataeva; O. Magda;V. Ostrovskyi; N. Ryabova; S. Spichak -Institute of Mathematics of the NASU; N. Pankratova - National TechnicalUniversity of Ukraine “KPI”; V. Grabko -Vinnytsya National Technical UniversityLocation: Kyiv, Institute of Mathematicsof National Academy of Sciences ofUkraine Tereshchenkivska Str.,3 01601,Kyiv-4, UkraineNote: From August 20 to 21, conferenceparticipants have the opportunity to takepart in a two days’ after-conference excur-sion to Bar, the native place of ViktorBunyakovsky, and to Vinnytsya, stayingthe night in the City Hotel in VinnytsyaDeadline: for abstracts May 15, 2004Information:Web site: http://www.imath.kiev.ua/~syta/bunyak

18-21: The Thirteenth International

CONFERENCES

EMS June 200436

Forthcoming conferencesForthcoming conferencescompiled by Vasile Berinde (Baia Mare, Romania)

Workshop on Matrices and Statistics, inCelebration of Ingram Olkin’s 80thBirthday, Bêdlewo, near Poznañ, PolandInformation:e-mail: matrix04@main.amu.edu.plWeb site: http://matrix04.amu.edu.pl/[For details, see EMS Newsletter 50]

23-September 2: InternationalConference-School on Geometry andAnalysis dedicated to the 75th anniver-sary of Academician Yu. G. Reshetnyak,Novosibirsk, RussiaInformation: contact Sergei Vodopyanov(Chairman of the Organizing Committee), e-mail: angeom@math.nsc.ru Web site: http://www.math.nsc.ru/conference/ag 2004/indengl.htm[For details, see EMS Newsletter 51]

24-27: International Conference onNonlinear Operators, DifferentialEquations and Applications (ICN-ODEA-2004), Cluj-Napoca, RomaniaInformation: e-mail: nodeacj@math.ubb-cluj.roWeb site: http://www.math.ubbcluj.ro/~mserban/confan.htm[For details, see EMS Newsletter 49]

30 - September 3: 7th French-RomanianColloquium in Applied Mathematics,Craiova (Romania)The previous editions of this Colloquiumwere organized in Iasi (1992), ENS-Paris(1994), Cluj (1996), Metz (1998),Constantza (2000), and Perpignan (2002).The present edition is organized by theDepartment of Mathematics of the Facultyof Mathematics and Informatics,University of Craiova (Romania), underthe patronage of the Institute of Statisticsof the Romanian Academy, UniversitéPierre et Marie Curie (Paris 6), and theSociété de Mathématiques Appliquées etIndustrielles (SMAI).Scientific Committee: D. Cioranescu(chairman, Paris 6), M. Iosifescu (chair-man, Romanian Academy), V. Radulescu(chairman, Craiova), V. Barbu (Iasi), S. Basarab (Bucharest), F. Bethuel (Paris6), H. Brezis (Paris 6), P. Deheuvels (Paris6), J.-M. Deshouillers (Bordeaux), C. Duhamel (Paris), D. Gaspar(Timisoara), U. Herkenrath (Duisburg), I. Ionescu (Chambéry), Y. Maday (Paris 6),C. Niculescu (Craiova), A. Rascanu (Iasi),I.A. Rus (Cluj), G. Salagean (Cluj)Invited Speakers: V. Bally (Paris), C. Bandle (Basel), J.-Y. Chemin (Paris),Ph. G. Ciarlet (Paris and Hong Kong), A. Damlamian (Paris), G. Dinca(Bucharest), C. Faciu (Bucharest), O. Goubet (Amiens), G. Haiman (Lille), D. Iftimie (Lyon), P. Jebelean (Timisoara),C. Le Bris (Paris), B. Miara (Noisy-le-Grand), S. Micu (Craiova), G. Nenciu(Bucharest), D. Polisevschi (Bucharest), R. Precup (Cluj), T. Ratiu (Lausanne), M. Sofonea (Perpignan), M. Théra

(Limoges)Local Organizing Committee:V. Radulescu (chairman), D. Busneag, N. Constantinescu, M. Cosulschi, D. Ebanca, M. Gabroveanu, M. Ghergu, A. Matei, M. Preda, N. Tandareanu, C. Vladimirescu, I. Vladimirescu.Invited Sessions:1) Contrôle des systemes gouvernés pardes équations aux dérivées partielles, orga-nized by Marius Tucsnak (Nancy I)2) Homogénéisation et applications auxsciences des matériaux, organized by HoriaEne (Bucharest)3) Mathématiques financieres, organizedby Radu Tunaru (London)4) Biomécanique, organized by MarcThiriet (Paris)5) Inégalités et applications, organized byConstantin Niculescu (Craiova)Information: To participate in theColloquium it is necessary to fill out theRegistration Form and send it to the LocalOrganizing Committee.Abstracts of conference lectures and com-munications will have been published bythe start of the Colloquium. TheOrganizing Committee plans to publish theProceedings of the Colloquium in the localmathematical journal, Annals of theUniversity of Craiova, Series Mathematicsand Informatics. Additional informationabout the Colloquium may be obtained at:http://www.inf.ucv.ro/colloque2004Contacts: M. Basarab, LaboratoireJacques-Louis Lions, Université Paris 6,175 rue du Chevaleret, 75013 Paris,France. e-mail: matei@ann.jussieu.frV. Radulescu, Department of Mathematics,University of Craiova, 200 585 Craiova,Romania, e-mail: colloque@inf.ucv.ro

3-5: EMS-Czech Union of mathemati-cians and Physicists (MathematicsResearch Section). MathematicalWeekend, Prague (Czech Republic).Information: Web site:http://mvs.jcmf.cz/emsweekend/

8-11: Dixiemes journees montoises d’in-formatique theorique, a Liege (TenthMons theoretical computer science days,in Liege)Information: e-mail: M.Rigo@ulg.ac.be; Web site: www.jm2004.ulg.ac.be[For details, see EMS Newsletter 50]

13 – 18: 3º CIME Course - StochasticGeometry, Martina Franca, Taranto,ItalyInformation: e-mail: cime@math.unifi.itWeb site: http: //www.math.unifi.it/CIME Address: Fondazione C.I.M.E. c/oDipart.di Matematica “U. Dini”Viale Morgagni, 67/A - 50134 FIRENZE(ITALY)

September 2004

Tel. +39-55-434975 / +39-55-4237111 FAX +39-55-434975 / +39-55-4222695[For details, see EMS Newsletter 51]

20-24: 12th French-German-SpanishConference on Optimization Avignon,FranceInformation: e-mail: alberto.seeger@univ-avignon.fr; Web site: http://www.fgs2004.univ-avignon.fr [For details, see EMS Newsletter 50]

23-26: 4th International Conference onApplied Mathematics (ICAM-4), BaiaMare, Romania (previous editions in1998, 2000 and 2002)Information: e-mail: marietag@ubm.ro;icam4@ubm.roWeb site: http://www.ubm.ro/site-ro/facultati/departament/manifestari/icam4/index.html[For details, see EMS Newsletter 49]

26-October 1: Potential theory andrelated topics, Hejnice, Czech RepublicAim: to introduce progress, prospects andperspectives of the fieldScope: various aspects of potential theoryand related fieldsMain speakers: L. Beznea (Romania);A. O’Farrell (Northern Ireland); S. Gardiner (Northern Ireland); W. Hansen(Germany); O. Martio Finland); M. Rockner (Germany)Note: During the conference there will bea chance to congratulate our colleague IvanNetuka on the occasion of his birthday. Hisworks will briefly be described by JurgenBliedtner (Germany)Format: invited and contributed talksSessions: only plenary sectionOrganizers: Technical University Liberecand other institutionsOrganizing committee: M. Brzezina(Technical Univ. Liberec);M. Dont (Czech Technical Univ., Prague);K. Janssen (Heinrich-Heine-Univ.,Dusseldorf); J. Lukes (Charles Univ.,Praha); D. Medkova (Czech Academy ofSciences, Prague); Karl-Theodor Sturm(Rheinische Friedrich-Wilhelms-Univ.,Bonn); Jiri Vesely (Charles Univ., Praha)Proceedings: will not be publishedLocation: Hejnice, Czech RepublicInformation: e-mail: jvesely@karlin.mff.cuni.czWeb site: http://www.karlin.mff.cuni.cz/PTRT04

29-October 2: XII Annual Congress ofthe Portuguese Statistical Society,Evora, PortugalTheme: all subjects in Probability andStatistics and their applicationsAim: to promote the exchange of ideas andthe spread of recent scientific develop-mentsMain speakers: L. Centeno (Portugal); D. Cox (UK); B. Efron (USA); J. Mexia

EMS June 2004 37

(Portugal); D. Peña (Spain); J. Branco(Portugal; tutorial on Cluster Analysis)Format: invited conferences, contributedtalks, tutorialSessions: plenary conferences, 3 or 4 par-allel sessions, poster sessions, tutorialLanguages: Portuguese and EnglishCall for papers: if you wish to present acontributed presentation, please send anabstract before 15 May 2004 to the e-mail:spe2004@uevora.pt, indicating keywordsand the preferred form of presentation (talkor poster). Maximum one A4 page in Wordformat with a 12 font and 1.5 line spacingOrganizers: Portuguese Statistical Society(SPE), University of EvoraExecutive and Scientific Committee:F. Rosado (Lisboa, Congress President); C. Braumann (Evora); M. E. Graça Martins(Lisboa); P. Oliveira (Coimbra); C. D. Paulino (Lisboa)Local Organising Committee:C. Braumann; R. Alpizar-Jara; F. Mendes;M. OliveiraProceedings: refereed, to be publishedafter the Congress (assuming, as it hap-pened in all the previous 11 Congresses,that adequate financial support will beavailable)Location: Evora Hotel (Evora), specialaccommodation fares at the Hotel for par-ticipants and accompanying personsSocial program: dinner, excursion, otherevents Reduced registration fees: for studentsand members of the SocietyNotes: registration and accommodationforms can be downloaded from the websiteDeadlines: for abstracts, 15 May 2004;registration with reduced fee if sent before15 May 2004Information: e-mail: spe2004@uevora.ptWeb site: www.eventos.uevora.pt/spe2004Address: Comissao Organizadora XIICongresso SPE, Departamento deMatematica, Universidade de Evora, RuaRomao Ramalho 59, 7000-671 EVORA,PORTUGAL Fax: +351-266745393;Phone: +351-266745370

6-9: HYKE Conference on ComplexFlows, Centre de Recerca Matematica,Bellaterra, SpainAim: The main objective of the conferenceis to highlight new developments of eithernumerical or analytical nature in kineticand hydrodynamic equations. We wouldlike to foster the interaction with applica-tions, making an emphasis on two applica-tions: granular media and astrophysicalflows, with special sessions devoted tothem.Co-ordinators: J.A. Carrillo (ICREA-UAB); A. Marquina (Univ. de Valencia) Programme Committee: A.V. Bolylev(Karlstadt Univ.); F. Bouchut (École

October 2004

Normale Supérieure, Paris); D. Kroener(Univ. Freiburg); S. Noelle (RWTHAachen); L. Pareschi (Univ. di Ferrara); M. Pulvirenti (Univ. di Roma); J. Soler(Univ. de Granada); C. Villani (ÉcoleNormale Supérieure, Lyon) Speakers: E. Caglioti (Univ. di Roma I,Italy); B. Despres (Univ. Paris VI, France);L. Desvilletes (ENS Cachan, France); F. Filbet (Univ. d'Orléans, France); J.A. Font (Univ. de Valencia, Spain); A. Goldshtein (Technion Haifa, Israel); L. Gosse (IAC Bari, Italy); T. Goudon(Univ. des Sciences et Technologies Lille1, France); C. Helzel (IAM Bonn,Germany); J.M. Ibánez (Univ. de Valencia,Spain); P.E. Jabin (ENS Paris, France);K.H. Karlsen (Univ. of Bergen, Norway);D. Levermore (Univ. of Maryland, USA);A. Mangeney (Institute de Physique duGlobe de Paris, France); J.M. Marti (Univ.de Valencia, Spain); C. Mouhot (ENSLyons, France); S. Osher (UCLA, USA);T. Poeschel (Humboldt-Universität -Charité, Germany); S. Rjasanow (Univ.Saarland, Germany); G. Russo (Univ. di

Catania, Italy); O. Sánchez (Univ. deGranada, Spain); A. Santos (Universidadde Extremadura, Spain); H.J. Schroll(Lund Univ., Sweden); S. Serna (Univ. deValencia, Spain); B. Sjogreen (KTHStockholm, Sweden); M. Torrillhon(ETHZ, Switzerland); G. Toscani (Univ. diPavia, Italy); J.J.L. Velázquez (Univ.Complutense de Madrid, Spain) Oral Presentations and Posters: Thereare 10 free time slots of 25' for oral pre-sentations. Also, there will be a poster ses-sion throughout the whole conference. Wewould appreciate your registration as soonas possible and that a title of your presen-tation is sent no later than June 6th. Wewill send notification of acceptance and thetype of your contribution, before June15th. Deadlines: For application for financialsupport, June 5, 2004. For registration andpayment, June 30, 2004Information: e-mail: ComplexFlows@crm.es Web site: http://www.crm.es/ComplexFlows

EMS June 200438

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

Y. A. Abramovich, C. D. Aliprantis: AnInvitation to Operator Theory, GraduateStudies in Mathematics, vol. 50, AmericanMathematical Society, Providence, 2002, 530pp., US$69, ISBN 0-8218-2146-6 The book is an immense graduate text devot-ed to the basic notions of linear operator theo-ry in Banach spaces and Banach lattices. Basicconcepts of the theory (Banach spaces, opera-tors and functionals, Banach lattices, bases,fundamentals of measure theory and basicoperator theory) are contained in preliminarychapters. The chapters that follow these are onoperators on AL- and AM-spaces and specialclasses of operators (finite-rank operators,multiplication operators, the Fredholm andstrictly singular operators) and on integraloperators. The next part of the book is devot-ed to spectral theory (including holomorphicfunctional calculus and the Riesz-Schaudertheory), positive matrices and irreducibleoperators. The last two chapters also containsome recent results, published for the firsttime in the book. The first of these chaptersdeals with the famous invariant subspaceproblem, while the last one presents a com-prehensive study of operators satisfying theDaugavet equation. Every chapter ends with alarge number of exercises and problems ofvarious levels of difficulty. There are morethan 600 exercises offering useful practise tothe reader, supplementing examples and coun-terexamples, or extending the theory present-ed in the text. Complete solutions of all exer-cises are given in a separate companion vol-ume “Problems in operator theory” (cf. thenext review). The book is written in an under-standable way and it requires only a basicknowledge of functional analysis, measuretheory and topology, together with a suitablebackground in real analysis. It can be warmlyrecommended to graduate students and every-body who wishes to become acquainted withbasic elements and deeper properties of linearoperators on Banach spaces and Banach lat-tices. (jl)

Y. A. Abramovich, C. D. Aliprantis:Problems in Operator Theory, GraduateStudies in Mathematics, vol. 51, AmericanMathematical Society, Providence, 2002, 386pp., US$49, ISBN 0-8218-2147-4 This book contains complete solutions tomore than 600 exercises, given in the com-panion volume “An invitation to operator the-ory” (cf. previous review) by the sameauthors. It can also be viewed as a supplemen-tary text to courses in operator theory and theirrelatives. (jl)

I. Agricola, T. Friedrich: Global Analysis:Differential Forms in Analysis, Geometry

and Physics, Graduate Studies inMathematics, vol. 52, American MathematicalSociety, Providence, 2002, 343 pp., US$59,ISBN 0-8218-2951-3

The book is based on courses taught at theHumboldt University for students of mathe-matics and physics. Topics treated in the bookcover analysis, differential geometry, Liegroups and Lie algebras and main applicationsin mathematical physics. The main tools usedthroughout are differential forms and their cal-culus. They are introduced first (together withthe Stokes theorem) on open subsets in Rn,then on submanifolds in Rn. The next essentialchapter addresses Pfaffian systems.Fundamental theorems on curves and surfacesin R3 are discussed carefully using all theadvantages offered by the language of differ-ential forms. To allow the reader to deal withsymmetry questions, a brief introduction toLie groups and homogeneous spaces is given.The last three chapters use material preparedearlier in the book for a discussion of basicapplications in mathematical physics – sym-plectic geometry and its relations to mechan-ics, statistical mechanics and thermodynam-ics, and finally electrodynamics. The book isnice to read. It is very clearly written, under-standable and contains a lot of pictures. Eachchapter ends with many carefully chosen exer-cises. The book can be useful for students aswell as for teachers preparing courses on thetopics covered. (vs)

M. Aguilar, S. Gitler, C. Prieto: AlgebraicTopology from a Homotopical Viewpoint,Universitext, Springer, New York, 2002, 478pp., €79,95, ISBN 0-387-95450-3This is a basic course on algebraic topology,and I would like to stress at the beginning thatit is excellently written and composed and canbe strongly recommended to anybody wishingto learn the field. It is already clear from thetitle that the authors adopted the homotopicalapproach to algebraic topology. This meansthat the book is more topological and lessalgebraic. Important technical tools here areinfinite symmetric products and Moorespaces. For example, homology groups aredefined as homotopy groups of infinite sym-metric product of the space under considera-tion. Eilenberg-Mac Lane spaces are infinitesymmetric products of Moore spaces, and viaEilenberg-Mac Lane spaces, cohomologygroups are defined. Taking the approach cho-sen in the book into account, the authors werewell aware of the fact that they must pay spe-cial attention to computations. The reader canfind many examples, calculations, and also anumber of exercises. (There is a whole chap-ter devoted to cellular homology.) However,we do not want to create an impression thatthe book covers only homology and cohomol-ogy. The book deals also with K-theory and itsmany applications, and later on, generalizedcohomology theories are treated. The authors

have also included the theory of characteristicclasses of vector bundles. The book has twoappendices (A: Proof of the Dold-Thom theo-rem, B: Proof of the Bott periodicity theorem),references consisting of 83 items and a longlist of symbols. There are many remarks andcomments making the orientation of the read-er in the field of algebraic topology easier, andsuggesting directions for further studies. Thereading requires rather modest prerequisites;some basic point set topology and some basicinformation about groups. Let us mention alsothat the book had a predecessor, namely a pre-liminary version in Spanish. (jiva)

J.P. Allouche, J. Shallit: AutomaticSequences: Theory, Applications,Generalizations, Cambridge UniversityPress, Cambridge, 2003, 571 pp., £37,50ISBN 0-521-82332-3The book can serve both as an introductioninto the study of automatic sequences, and asa systematic survey of known results andapplications. The topic mentioned in the titleis far from being the only interesting theme ofthe volume. Related topics range from combi-natorics on words and formal languagesthrough number theory and formal powerseries to physics. The basic concept is definedin the fifth chapter using finite automata for-malism. In the following chapter an equivalentcharacterization in terms of fixed points ofmorphisms is given. Properties of automaticsequences bring together in a natural wayresults from number theory and theoreticalcomputer science, for example, questions oftranscendence of numbers and power series.Relations between these areas are oftenneglected due to different conventions in lan-guage and notation. The book presents a lot ofresults from both fields for the first time in aunified framework. Material is presented in aclear and attractive way with motivation, exer-cises, open questions, and a very rich bibliog-raphy. The presentation is self-contained in aremarkable degree. The book contains areview of various areas needed such as peri-odicity in words, subword complexity, alge-braic and transcendental numbers, numerationsystems, finite automata, Turing machines orcontinued fractions. Famous sequences likeThue-Morse, Fibonacci or Rudin-Shapiro arealso studied. The book will be of use forbeginners as well as for advanced students andresearchers. (šh)

F. J. Almgren, Jr.: Plateau’s Problem: AnInvitation to Varifold Geometry, revised edi-tion, Student Mathematical Library, vol. 13,American Mathematical Society, Providence,2001, 77 pp., US$19, ISBN 0-8218-2747-2This is a small booklet describing germs ofgeometric measure theory and their applica-tions to the solution of the Plateau problem incalculus of variation. It means that we arelooking for a “surface” of smallest area havinga given closed curve as its boundary. Thebook starts with many examples of soap filmsillustrating an unexpected behaviour of solu-tions. The author then introduces differentialforms and their integration over rectifiablesets. It leads in a natural way to a notion of acurrent (i.e., an element of the dual to thespace of k-forms in a suitable topology) and it

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Recent booksRecent booksedited by Ivan Netuka and Vladimír Souèek (Prague)

is an inspiration for the definition of a vari-fold. The last part describes a solution of thePlateau problem in the context of varifolds.The first version of the book appeared in the60’s. The new revised edition contains anexpanded bibliography, reflecting the evolu-tion of the topic. After all these years, thebook is still a beautiful introduction to thePlateau problem and it is a pleasure to read.(vs)

K.T. Arasu, A. Seress, Eds.: Codes andDesigns, Ohio State University MathematicalResearch Institute Publications 10, Walter deGruyter, Berlin, New York, 2002, 322 pp.,€128, ISBN 3-11-017396-4This volume contains the proceedings of aconference honouring Dijen K. Ray-Chaudhuri on the occasion of his 65th birth-day held at Ohio State University in 2002.Dijen Ray-Chaudhuri is a well-known combi-natorist who is most renowned for BCH-codes, which appear in every textbook on cod-ing theory. He is also an influential teacher.The volume contains 22 contributions, writtenmostly by his former students, and relates tovarious aspects of algebraic combinatorics,coding theory, tilings, block designs andgraph theory. Highlights of Dijen Ray-Chaudhuri’s research are reviewed in the firstcontribution by A. Seress, which also includesa list of his papers. It is a nice volume hon-ouring a very fine mathematician. (jneš)

S. Ariki: Representations of QuantumAlgebras and Combinatorics of YoungTableaux, University Lecture Series, vol. 26,American Mathematical Society, Providence,2002, 158 pp., US$31, ISBN 0-8218-3232-8The main part of the book is devoted to astudy of representations of quantum algebras.The author uses the quantum deformation ofthe Kac-Moody Lie algebra Ak(1) as the mainworking example. The first three chapterscontain a brief review of concepts needed fora description of Ak(1) by generators and rela-tions. The next three chapters contain a dis-cussion of crystal bases for integrable repre-sentations of Ak(1) and a description of behav-ior of crystal bases under tensor products. Theexistence of crystal bases is proved in chapters7, 8 and 9. The proof is based on relationsbetween crystal bases and the Lusztig canoni-cal bases. The next two chapters contain a dis-cussion of the third possible approach to crys-tal bases, which uses a combinatorial con-struction based on (multi)-Young diagrams,due to Misra and Miwa. The final part of thebook contains the proof of a conjecture (due toLeclerc, Lascoux and Thibon) concerning therepresentation theory of cyclotomic Heckealgebras of classical type. The book offers anice introduction to combinatorics related rep-resentation theories and their applications.(vs)

I. Bauer, F. Catanese, Y. Kawamata, T.Peternell, Y.-T. Siu, (Eds.): ComplexGeometry, Springer, Berlin, 2002, 340 pp.,€79,95, ISBN 3-540-43259-0The book contains a collection of papers ded-icated to Hans Grauert on the occasion of his70th birthday. The reader can also find here a

list of his publications, a long list of his doc-toral students (40 altogether), together withthe program of the conference organized inGöttingen on this occasion in April 2000.There are 17 research contributions on manytopics from complex geometry (often relatedto work of H. Grauert) by W. Barth; T. Bauerand his coworkers; I. Bauer, F. Catanese andR. Pignatelli; A. Bonifant and J. Fornaess; C.Ciliberto and K. Hulek; J.P. Demailly; H.Flenner and M. Lübke; A. Huckleberry and J.Wolf; Y. Kawamata; S. Kebekus; T. Peternelland A. Sommese; K. Oguiso and De-QiZhang; T. Oshawa, S. Schröer and B. Siebert;Y.-T. Siu; E. Viehweg and K. Zuo, and by J.Wisniewski. (vs)

M. C. Beltrametti, F. Catanese, C.Ciliberto, A. Lanteri, C. Pedrini (Eds.):Algebraic Geometry, Walter de Gruyter,Berlin, 2002, 355 pp., €148, ISBN 3-11-017180-5The book is based on talks given at the con-ference in memory of Paolo Francia, held inGenova in September 2001, and it containsinvited papers dedicated to Paolo Francia. Thecontents cover a wide spectrum of algebraicgeometry. The life and work of Paolo Franciais briefly described in the introduction, togeth-er with the list of his publications. The bookcontains 18 contributions, by L. Badescu andM. Schneider (Formal functions, connectivityand homogeneous spaces), L. Barbieri-Viale(On algebraic 1-motives related to Hodgecycles), A. Beauville (The Szapiro inequalityfor higher genus fibrations), G. Borelli (Onregular surfaces of general type with pg = 2and non-birational bicanonical map), F.Catanese and F.O. Schreyer (Canonical pro-jections on irregular algebraic surfaces),C.Ciliberto and M. M.Lopes (On surfaces withpg =2, q=1 and non-birational bicanonicalmap), A. Conte, M. Marchisio and J. Murre(On unirrationality of double covers of fixeddegree and large dimension), A. Corti and M.Reid (Weighted Grasmanians), T. de Fernex,L. Ein (Resolution of indeterminacy of pairs)V. Guletskii and C. Pedrini (The Chow motiveand the Gedeaux surface), Y. Kawamata(Francia’s flip and derived categories), K.Konno (On the quadric hull of a canonical sur-face) A. Langer (A note on Bogomolov’sinstability and Higgs sheaves), A. Lanteri andR. Mallavibarrena (Jets of antimulticanonicalbundle on Del Pezzo surfaces of degree < 2),M. M. Lopes and R. Pardini (A survey on thebicanonical map of surfaces with pg = 0 and K2

>2), F. Russo (The antibirational involutionsof the plane and the classifications of real DelPezzo surfaces), V. V. Shokurov (Letters of abi-rationalist: IV.Geometry of log flips), A. J.Sommese, J. Verschelde and Ch. Wampler (Amethod for tracking singular paths with appli-cation to the numerical irreducible decompo-sition). In addition the book contains informa-tion on the conference consisting of a list oflectures and a list of participants. (jbu)

N. Bourbaki: Elements of Mathematics. LieGroups and Lie Algebras, Chapters 4-6,Springer, Berlin, 2002, 300 pp., €99,95, ISBN3-540-42650-7 The book is one of the Bourbaki volumes,translated into English. The first three chap-

ters of the book contain part of a theory need-ed in a study of Lie groups and Lie algebras.Chapter 4 is devoted to Coxeter groups andTits systems. Chapter 5 contains a discussionof groups generated by reflections. Finally,Chapter 6 treats root systems, ending withtheir classification. (vs)

O. Bratteli, P. Jorgensen: Wavelets Througha Looking Glass, Applied and NumericalHarmonic Analysis, Birkhäuser, Boston,2002, 398 pp., €85,98, ISBN 0-8176-4280-3The book gives a general presentation of somerecent developments in wavelet theory, withan emphasis on techniques that are both fun-damental and relatively timeless, having ageometric and spectral-theoretic flavour. Theexposition is clearly motivated and unfoldssystematically, aided by numerous graphics.Excellent graphics show how wavelets dependon the spectra of the transfer operators. Somenew results are presented for the first time(e.g., results on homotopy of multiresolutions,on approximation theory, and the spectrum ofassociated transfer and subdivision operators).The book is divided into six chapters. Eachchapter, and some sections within chapters,open with tutorials or primer of varyinglength. The tutorials are written in a style thatis much more informal. They are in fact meantas friendly invitations to the topics to follow,with the emphasis on friendliness. Key topicsof wavelet theory are examined: connectedcomponents in the variety of wavelets, thegeometry of winding numbers, the Galerkinprojection method, classical functions ofWeierstrass and Hurwitz and their role indescribing the eigenvalue-spectrum of thetransfer operator, Perron-Frobenius theory,and quadrature mirror filters. Concise back-ground material for each chapter, open prob-lems, exercises, bibliography, and comprehen-sive index, make this work a fine pedagogicaland reference source. The book also describesimportant applications to signal processing,communications engineering, computergraphics algorithms, qubit algorithms andchaos theory, and is aimed at a broad reader-ship of graduate students, practitioners, andresearchers in applied mathematics and engi-neering. The book is also useful for othermathematicians with an interest in the inter-face between mathematics and communica-tion theory. (knaj)

E. Bujalance, F.-J. Cirre, J.-M. Gamboa, G.Gromadzki: Symmetry Types ofHyperelliptic Riemann Surfaces, Mémoiresde la Sociéte Mathématique de France, Vol91, Sociéte Mathématique de France, Paris,2001, 122 pp., FRF 150, ISBN 2-85629-112-0The aim of the publication is to present classi-fication of all real structures of hyper-ellipticRiemann surfaces, together with their fullgroup of analytic and antianalytic automor-phisms, their topological invariants, theirdescription in terms of polynomials equationsand explicit formulae for the correspondingreal structures. A real structure τ on aRiemann surface X is an antianalytic involu-tion on X. A real form on X is a class of equiv-alence of τmodulo conjugation by elements ofthe full group of analytic and antianalyticauthomorphisms. For example, if the curve X

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is defined by a set of polynomials with realcoefficients, then the complex conjugationgives real structure on X. Most complexcurves have no real forms but some have morethan one. A number of connected componentsof the fixed point set of the involution τ andconnectivity of its complement in X togetherform an invariant, which characterizes theconjugacy class of τ. The symmetry type of Xis the (finite) set of these invariants for all realforms of X. In the first chapter, the authorsintroduce combinatorial methods used in theclassification. The second chapter is devotedto a description of the full group of automor-phisms and computation of the number of realforms. The third chapter contains a computa-tion of symmetry types of hyper-ellipticRiemann surfaces, divided into ten differentsubcases. (vs)

C.H. Clemens: A Scrapbook of ComplexCurve Theory, Graduate Studies inMathematics, vol. 55, American MathematicalSociety, Providence, 2003, 188 pp., US$39,ISBN 0-8218-3307-3This is very nice book by C.H. Clemens andwould make an indispensable tool for anyonewilling to enter the realm of algebraic geome-try. Although not written in the tight mathe-matical form Definition-Theorem-Proof, itcovers and explains motivation for the intro-duction of algebraic language to problems ingeometry using many suitable examples. Thefirst chapter recalls all notions of classicalgeometry related to conic curves - projectivespace, linear system of conics, cross ratio,hyperbolic space and rational points onquadrics. The second chapter describes cubic(i.e., elliptic) curves. A unifying pictureemerges by comparison of elliptic curves overcomplex numbers (i.e., the variation of Hodgestructure) and elliptic curves over finite fields(i.e., generating function for the number ofrational points over successive extensions offinite fields). The third and fourth chaptersoffer an introduction to elliptic, modular andtheta function theory, moduli spaces and basicfacts of cohomology theory - Jacobians, dual-ity in sheaf cohomology, Abel-Jacobi map,etc. The last two chapters contain topics fromgeometry of higher genus Riemannian sur-faces, e.g. Prym varieties, Schottky groupsand hyperelliptic curves of higher genus. (pso)

J. Cnops: An Introduction to DiracOperators on Manifolds, Progress inMathematical Physics, vol. 24, Birkhäuser,Boston, 2002, 211 pp., €72,90, ISBN 0-8176-4298-6A Dirac operator on a Riemannian manifold Macts on sections of a bundle S of left modulesover the canonical Clifford bundle Cl(M).This construction allows for many specialcases, a choice of the Dirac operator dependson a choice of the bundle S. The book is devot-ed to such cases, where the manifold M inher-its its Riemannian structure from an embed-ding to a (pseudo)-Euclidean space Rp,q. Thespecial case of embedded manifolds makes itpossible to study various special constructionsof the bundle S. The embedding of M into Rp,q

also allows the use of the Clifford algebraClp,q of Rp,q for various computations related tothe normal bundle of M. The first chapter of

the book treats Clifford algebras of a givenvector space with a non-degenerate scalarproduct. Embedded manifolds, correspondingcovariant derivatives and spinor fields areintroduced in the second chapter. Various ver-sions of Dirac operators are introduced andtheir properties are discussed in the third chap-ter. A description of conformal mappingsusing Clifford algebra 2 x 2 matrices and theirrelation to Dirac operators are given inChapter 4. The next two chapters contain adiscussion of the unique continuation propertyof solutions of a Dirac equation and theirboundary values. (vs)

P. M. Cohn: Basic Algebra. Groups, Ringsand Fields, Springer, London, 2003, 465 pp.,€64,95, ISBN 1-85233-587-4This book, together with its sequel, “FurtherAlgebra and Applications” (to appear), is anew and revised version of the author’sfamous text “Algebra (Vols. 2 and 3)”, whichis now out of print. Besides basics on groups,rings, modules and fields, the book containsclassical Galois theory of equations, an intro-duction to quadratic forms and ordered fields,valuation theory, and classical commutativering theory (including primary decompositionand Hilbert Nullstellensatz). The final chapterdeals with infinite field extensions. The bookcontains numerous exercises, and moreover,motivation and many illuminating commentson the subject. There is no doubt that the bookwill take the position of its predecessor inbeing one of the most outstanding introducto-ry algebra textbooks. (jtrl).

J. S. Cohen: Computer Algebra andSymbolic Computation: MathematicalMethods, A. K. Peters, Natick, 2003, 448 pp.,US$59, ISBN 1-56881-159-4The book is a readable introduction to com-puter algebra. It presents a theoretical basis ofrecently published Computer Algebra andSymbolic Computation: ElementaryAlgorithms, of the same author. The bookexplores applications of algorithms to auto-matic simplification, greatest common divisorcalculation, resultant computation, polynomi-al decomposition, and factorisation. After areview of basic background concepts, algo-rithms for manipulation and evaluation ofnumerical objects are given. Automatic alge-braic and trigonometric simplification isworked out thoroughly and it gives the readerrather detailed information about one of theessential algorithms implemented in computeralgebra software. Another of the importantnumerical objects - single variable polynomi-als - is discussed in two chapters. Attention ispaid in particular to the algorithm for polyno-mial division, which is the basis to the algo-rithm for polynomial expansion, Euclideanalgorithm, algorithm for partial fractionexpansion, and algorithm for performingarithmetic operations for expressions in sim-ple algebraic number fields. This is followedby a generalization of these concepts and algo-rithms to multivariate polynomials. Twoimportant concepts of computer algebra, theresultant and Gröbner basis, are introduced.The book culminates in the last chapter,devoted to polynomial factorisation and amodern algorithm for it. The book will be use-

ful for undergraduate students of mathematicsand computer science. The text is also acces-sible to a more general audience interested incomputer algebra and its applications. (mer)

E. von Collani (Ed.): Defining the Scienceof Stochastics, Sigma Series in Stochastics,Vol. 1, Heldermann Verlag, Lemgo, 2004, 260pp., €30, ISBN 3-88538-301-2Although the words “stochastic” and “sto-chastics” are frequently used, their exactmeaning may not be generally known. TheGreek noun stochos means target, aim, orguess. Stochastikos is a person who is skilfulin predicting. From this point of view, the title‘Ars conjectandi’ of the famous book by JacobBernoulli has the Greek equivalent ‘στοχστικητεχνγ’ (stochastic techniques). Formerly, theword “stochastic” was used mainly in thephrase “stochastic process”. Today one feelsthat “stochastic” means more than only “ran-dom”. The book contains contributions devot-ed to the problem of how to define the scienceof stochastics. The articles are revised andupdated versions of lectures delivered at thesymposium “Defining the Science ofStochastics”, which took place in Germany inOctober 2000. The publication is divided intothree parts, each one having three papers. Thefirst part describes the present state of statis-tics and probability theory in comparison tothe science of stochastics. The second partdeals with randomness, contingency, andwhite noise. The third part is devoted to sto-chastics in public education. The authors ofindividual contributions are S. Kotz, U.Herkenrath, V. Kalashnikov, J. Kohlas, C. S.Calude, T. Hida, M. Dumitrescu, and E. vonCollani (2×). The authors emphasize historyand nature of statistical science and propose toestablish “Stochastics” as an independent,mathematically based exact science, whichinvestigates the inherent variability exhibitedby any real phenomenon. A great variety oftopics in the published papers, reflects the fastgrowing diversification of statistical sciences.It might be that the reader finds some of themto be too far from his/her specialization, buteverybody will find some contributions inter-esting and stimulating. In Biographical Notes,on p. 215, one reads that Thomas Bayes lived1702-1761. According to the inscription onhis tomb in Bunhill Field cemetery in London,he died on April 7, 1761, at age of 59. Someauthors introduce that he was probably born in1701. (ja)

J.-M. Delort: Global Solutions for SmallNonlinear Long Range Perturbations of TwoDimensional Schrödinger Equations,Mémoires de la Sociéte Mathématique deFrance 91, Sociéte Mathématique de France,Paris, 2002, 94 pp., €23, ISBN 2-85629-125-2In the book, a global existence of solutions tothe two dimensional Schrödinger equations isproved. The equation allows for a quadraticnonlinearity containing the space derivatives.The initial data are small. The setting is ‘criti-cal’ in the sense that the nonlinearity becomesnon-integrable when applied to the fundamen-tal solution of the Schrödinger operator. Toovercome this difficulty, a suitably chosenfunction is subtracted from the solution to can-cel out the ‘worst’ terms. The main technique

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of the book is classical harmonic analysis.Symbolic calculus, dyadic decomposition,carefully developed linear theory, togetherwith some nonlinear estimates (e.g., for prod-ucts) are the keystones of the final success,and the solution is found in a suitable weight-ed space. The book is rather technical but ismostly self-contained. (dpr)

Y. Eliashberg, N. Mishachev: Introductionto the h-Principle, Graduate Studies inMathematics, vol. 48, American MathematicalSociety, Providence, 2002, 206 pp., US$30,ISBN 0-8218-3227-1The Gromov “h-principle” was introduced inthe 80’s, in Gromov’s book on partial differ-ential relations. This is really a principle, not atheorem, and it should be adapted to everynew application. It describes the behaviour ofa system of partial differential equations orinequalities having a big space of solutions(e.g., dense in the space of functions or fieldsconsidered). It was inspired by work on theC1-isometric embedding theory (Nash andKuiper) and the immersion theory in differen-tial topology (Smale and Hirsch). TheGromov book is not easy to understand. Thebook under review offers the reader a nice andunderstandable treatment of two methodsbased on the “h-principle” – the holonomicapproximation methods and convex integra-tion theory (which was treated in a more gen-eral situation in the book by D. Spring, pub-lished in 1998). These two methods covermany interesting applications. The book con-tains a discussion of applications of the holo-nomic approximation method in symplecticand contact geometry, including a review ofbasic notions needed for the applications. Thebook is very interesting, readable and shouldbe of interest to more than just geometers. (vs)

F. Fauvet, C. Mitschi, Eds.: FromCombinatorics to Dynamical Systems, IRMALectures in Mathematics and TheoreticalPhysics 3, Walter de Gruyter, Berlin, 2003,241 pp., €36,95, ISBN 3-11-017875-3This is the proceedings of the conference“Journées de calcul formel en l’honneur deJean Thomann”, which took place at IRMA(Institut de Recherche MathématiqueAvancée) in Strasbourg in March 2003. Themeeting was devoted to dynamical systems,computer algebra, and theoretical physics.The first paper, by M. Espie, J.-C. Novelli andG. Racinet, contains computations of certaingraded Lie algebras related to multiple zetavalues. The article by E. Corel gives an alge-braic and algorithmic treatment of formalexponents at an irregular singularity. The arti-cle by J. L. Martins studies irregular mero-morphic connections in higher dimensions.The paper by M. A. Barkatou, F. Chyzak andM. Loday-Richaud describes various algo-rithms for the rank reduction of differentialsystems. The article by M. Canalis-Durandpresents a “pattern recognition algorithm” forGevrey power series. The article by D.Boucher and J.-A. Weil gives a new proof ofthe non-integrability of the planar three-bodyproblem. The paper by L. Brenig uses theQuasi Polynomial formalism to obtain effec-tive integrability for some dynamical systems.The article by R. Conte, M. Musete and T.-L.

Yee, presents an example of an analytical tra-jectory in a chaotic system. The last article, byJ. Della Dora and M. Mirica-Ruse, presentsformalization for the concept of a hybrid sys-tem. (pkur)

T. Forster: Logic, Induction and Sets,London Mathematical Society Student Texts56, Cambridge University Press, Cambridge,2003, 234 pp., £18,95, ISBN 0-521-53361-9,ISBN 0-521-82621-7The book is an introduction to mathematicallogic and set theory. It contains everythingthat is expected from such a book - language,syntax, semantics, model, deducibility, com-pleteness theorem, normal form theorem,recursion and axioms of ZFC. The style of thebook differs from similar textbooks. The cen-tral notion in the book is a recursive data type,which means a set defined by a recursion. Thisnotion is introduced early, hence some knowl-edge of set theory is tacitly assumed at thevery beginning (and it also induces some cir-cular arguments at the end). The whole pre-sentation is not as rigorous as could be expect-ed in textbooks of this type. On the other hand,the book contains a lot of philosophical expla-nations pointing to aspects usually ignored.Another interesting point is that the set ofexercises contains some, which are rare tomeet, for example: “Dress up the traditionalproof that √2 is irrational into a proof by well-founded induction on N x N.” Once you arefamiliar with a classical textbook on the top-ics, reading this one as a second choice is fun.(psi)

R. Garnier, J. Taylor: Discrete Mathematicsfor New Technology, second edition, Instituteof Physics Publishing, Bristol, 2002, 749 pp.,£25, ISBN 0-7503-0652-1The book is designed for students of comput-er science. It contains main mathematical top-ics needed in their undergraduate study. Themain core of mathematics contained in thebook belongs to standard material needed forany student of mathematics. The choice oftopics was inspired by a first-year mathemat-ics course for undergraduate students of com-puter science at the University of Nottingham.The reader can find here sections on logic,mathematical proofs, sets, relations, relationaldatabases, functions and functional depen-dence, matrix algebra and systems of linearequations, basic algebraic structures, Booleanalgebra and graph theory and its applications.The book also contains applications (relation-al databases, normal form of databases). In thesecond edition, the authors added a lot of newexercises and examples, illustrating discussedconcepts. The book contains a lot of well-ordered and nicely illustrated material. (vs)

J. C. Gower, G. B. Dijksterhuis: ProcrustesProblems, Oxford Statistical Science Series30, Oxford University Press, Oxford, 2004,233 pp., £55, ISBN 0-19-851058-6,Let X1 and X2 be given matrices. The simplestProcrustes problem is to find a matrix T thatminimizes X1T-X2 . In analogy with thestory of Procrustes, son of Poseidon, we canimagine that X1 is the unfortunate traveller, X2is the Procrustean bed, and T is the treatment

(racking, hammering, or amputation). Underclassical conditions, minimization of X1T-X2is the multivariate multiple regression prob-lem with the solution T=(X1′X1)-1X1′X2. Thetwo-sided form of the Procrustes problem is tominimize X1T1-X2T2 under some condi-tions. A further variant consists of the doubleProcrustes problem concerning minimizationof T2X1T1-X2 . The solutions of the men-tioned problems depend on the set of matricesT, which are allowed to be used. In somecases T is constrained to be an orthogonalmatrix, which represents a generalized rota-tion, in other problems T must be a permuta-tion matrix and so on. Generally we can saythat Procrustean methods are used to trans-form one set of data to represent another set ofdata as closely as possible. The book also con-tains oblique Procrustes problems, weighting,scaling, and missing values. Further topics areaccuracy and stability, and some applications.The authors inform that the process of writingthe book was similar to the machinations ofProcrustes, the material seemed to be continu-ously stretching and so it was necessary tochop things. However, this is probably the fateof many mathematical papers and books. (ja)

F. von Haeseler: Automatic Sequences, deGruyter Expositions in Mathematics 36,Walter de Gruyter, Berlin, 2003, 191 pp.,€78,50, ISBN 3-11-015629-6Automatic sequences can be equivalentlydefined in terms of finite automata (they areproduced by them), substitutions (they aretheir fixed points), or by their kernel (which isfinite). The book studies a generalized conceptof automatic sequences, where the sequence isindexed by elements of a finitely generatedgroup, instead of natural numbers.Accordingly, ordinary substitutions arereplaced by mappings based on expandinggroup endomorphisms, and a correspondingnotion analogous to finite automata is devel-oped. These concepts are introduced and dis-cussed in the first three chapters. Chapter 4studies automatic subsets and automatic maps.The final chapter combines automaticsequences with some additional algebraicstructures, and deals especially with Mahlerequations over the ring of formal power series.The book is written quite succinctly, and is notvery easy to read. A certain degree of mathe-matical maturity is required just to grasp thebasic definition of the automatic sequence.The volume will be useful mainly foradvanced researchers in the field, interested inbroadening their view of the subject. (šh)

I. Chajda, G. Eigenthaler, H. Länger:Congruence Classes in Universal Algebra,Research and Exposition in Mathematics, vol.26, Heldermann, Lemgo, 2003, 217 pp., €28,ISBN 3-88538-226-1The book explores a number of different fea-tures of congruence on universal algebras. Itcovers the subject from the early works ofMaltsev to recent results. One can find a goodamount of material concerning properties ofblocks (in general and in varieties), relation-ship to quotients and subalgebras or localproperties of congruence. The first two chap-ters explain basic notions and give quite a few

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examples of algebras. The study of congru-ence starts with the classical notions of per-mutability, distributivity, modularity anddirect decomposability. Further chapters dealwith the role of subalgebras, single blocks,constant terms or ideals. Special chapters aredevoted to extension properties of congru-ence, regular and coherent algebras, local con-ditions and one-block congruence. (eb)

P. Imkeller, J.-S. von Storch, Eds.:Stochastic Climate Models, Progress inProbability, vol. 149, Birkhäuser, Basel,2001, 398 pp., DM 196, ISBN 3-7643-6520-XThe proceedings of the workshop onStochastic Climate Models, held in Chorin in1999, present an exiting and stimulating sum-mary of probabilistic developments in climatephysics. Stochastic processes, stochastic andpartial differential equations, random dynami-cal systems, local and large deviations asymp-totic procedures, have comprised the mostactive mathematical areas in the field over thepast 30 years, since Klaus Haselmann sug-gested a climate stochastic model whereweather fluctuations randomly force the cli-mate in the same way that fluid moleculesforce Brownian pollen particles. Hence theLangevin stochastic equation and stochasticcalculus, enter the model in a very naturalway. The structure of the book reflects its aimto review and explain recent mathematicaladditions to the list of tools for climate model-ling, to help the climate physicists to under-stand what these tools are about. Chapter 1 byD. Olbers, K. Fraedrich, J. S. von Storch andR. Temam exhibits the most important climatemodels, from those simple ones to more com-plex ones. Chapter 2 by L. Arnols, M. Denker,M. Kesseböhmer, Y. Kiefer, Ch. Rödenbeck,Ch. Beck and H. Kantz, searches for possiblesources of stochasticity in the models from themathematical as well as physical point ofview. Chapter 3 by P. Imkeller, J. Duan, P. E.Kloeden, B. Schmalfuss, J. Zabzyk and P.Müller, is a very informative compendium ofprobabilistic tools mentioned above that arealready at work in the climate dynamics.Chapter 4 by J. Egger, J.A. Freund, A.Neiman, L. Schimanski-Geier, A.H.Monahan, L. Pandolfo, P. Imkeller, P.Sardeshmukh, P. Penland, M. Newman andW.A. Woyczynski, completes the presentationof climate mathematics with a discussion ofsome of the more special models for the fluc-tuating eastward flows or planetary waves.The book is strongly recommended as anexcellent source of information and inspira-tion, both to mathematicians and physicistsinterested in the field. (jste)

P.-Y. Jeanne: Optique géométrique pour dessystemes semi-linéaires avec invariance dejauge, Mémoires de la Sociéte Mathématiquede France, Sociéte Mathématique de France,Paris, 2002, 160 pp., €23, ISBN 2-85629-123-6The main topic of the book is a study of theproperties of solutions of field equations for ascalar field or a spinor field coupled to aYang-Mills field, and construction of approx-imate solutions of a special shape. First, in the60’s, solutions that rapidly oscillated in highfrequencies were found by Y. Choquet-

Bruhat, using a modification of WKB meth-ods for solutions of linear partial differentialequations. Meanwhile, justification for ideascoming from geometric optics was developedfor solutions of certain semi-linear or quasi-linear equations. The first part of the bookcontains an exposition of a structure of theequations. The second part describes familiesof approximate solutions (to any order), in theshape of single-phase expansions, rapidlyoscillating at high frequency. The third partcontains a description of exact solutions,asymptotic to previous expansions. (vs)

J.-M. Kantor (Ed.): Ou en sont les mathé-matiques?, Société Mathématique de France,Paris, 2002, 440 pp., ISBN 2-7117-8994-2The Société Mathématique de France has pub-lished the journal Gazette des mathématicienssince the 70’s. Its aim is a diffusion of mathe-matical knowledge. The book under reviewbrings a sample of papers from the journalsince its foundation. Papers are complementedby comments written by a contemporary spe-cialist, describing recent evolution in the field.As a whole, it is a fascinating read. Papers(written in French) describe eighteen differenttopics of enormous interest, reviewed in areadable and understandable way, anddesigned for a general audience. The topicsinclude classical themes (the third and the fif-teenth Hilbert problems, theory of invariants,arithmetic algebraic geometry), importantmathematical topics (stochastic calculus,deformation of various mathematical struc-tures, symplectic geometry and topology, thework of M. Gromov, convex polytopes, singu-larities of maps, error correcting codes), verymodern themes (mathematical ideas comingfrom quantum field theory, quantum groupsand invariants of nodes, Wiles’ proof of theTaniyama-Weil conjecture), as well as someapplications (image recognition, mathematicsin meteorology, topology in molecular biolo-gy). It is a very nice collection, which can con-tribute in an essential way to the general math-ematical education of any reader. (vs)

A. Katok: Combinatorial Constructions inErgodic Theory and Dynamics, UniversityLecture Series, vol. 30, AmericanMathematical Society, Providence, 2003, 121pp., US$29, SBN 0-8218-3496-7The book is an updated version of the unpub-

lished notes “Constructions in ergodic theo-ry”, written by the author in collaboration withE.A.Robinson Jr., in 1982-83. The first partentitled “Approximations and genericness inergodic theory”, deals with the approxima-tions of measure-preserving transformationsby periodic processes, which are permutationsof partitions of the space. Several types ofsuch periodic processes are considered and thespeed of approximation is evaluated. Themain theorem says that the set of measure-pre-serving transformations of a Lebesgue space,which admit a periodic approximation of agiven type and speed, is residual. The secondpart entitled “Cocycles, cohomology and com-binatorial constructions”, deals with algebraicconstructions in ergodic theory. The conceptsof rigidity, stability and effectiveness areinvestigated. Main applications includeDiophantine translations of the torus, Anosov

diffeomorphisms, horocycle flows and inter-val exchange transformations. (pkur)

B. Lapeyre, É. Pardoux, R. Sentis:Introduction to Monte-Carlo Methods forTransport and Diffusion Equations, OxfordTexts in Applied and EngineeringMathematics, Oxford University Press,Oxford, 2003, 163 pp., £60, ISBN 0-19-852592-3, ISBN 0-19-852593-1The book is devoted to the applications ofMonte Carlo methods (i.e., simulation tech-niques based on random numbers), for solu-tions of partial differential equations. A prob-abilistic representation of solutions of theseequations is given in the following way: thetransport or diffusion type equations are inter-preted as the Fokker-Planck equations associ-ated with Markov processes. Transport equa-tions in particle physics, the nonlinearBoltzmann transport equation, and the linksbetween second order partial differential dif-fusion equations and Brownian motion, are allinvestigated. For each type of problem, limitsof methods are discussed and specific tech-niques used in practice are described. Anessential point is that even if Monte-Carlomethods need not converge, it is always possi-ble to control the reliability of the result usingan inexpensive additional calculation. Thereader should have a good background inmathematical analysis. (jste)

W. K. Li W.: Diagnostic Checks in TimeSeries, Monographs on Statistics and AppliedProbability 102, Chapman & Hall/CRC, BocaRaton, London, 2004, 216 pp., US$69,95,ISBN 1-58488-337-5There are many books on time series analysisbut this is the first monograph specialized todiagnostic checking. Construction of a modelfor time series data usually consists of threesteps. At the beginning, a preliminary model ischosen. Then the parameters are estimated.The third stage is called model diagnosticchecking. It involves techniques like residualplots and procedures for testing if the residu-als are approximately uncorrelated. If it isfound that the model is not adequate, theprocess starts with the first step again, thistime with some new information from the pre-vious analysis. The book describes a rich vari-ety of diagnostic checks. It starts with the uni-variate and multivariate linear models.Although attention is paid mainly to ARMAmodels, the author also writes about periodicautoregression and Granger causality tests. Achapter is devoted to robust modelling anddiagnostic checking, from a robust portman-teau test to the trimmed portmanteau statistic.An important part of the book describesresults on nonlinear models. It contains good-ness-of-fit tests, tests for general nonlinearstructure, tests for linear vs. specific nonlineartime series, and methods for choosing two dif-ferent families of nonlinear models. Then themodels for conditional heteroscedasticity arepresented. Finally, fractionally differencedprocesses are described and a few special top-ics are mentioned like non-Gaussian timeseries and power transformations. Unit rootand co-integration tests are not included. Theauthor is a known specialist in time seriesmodelling. His approach is a practical one and

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each topic is presented from a model builder’spoint of view. A long list of references is alsoappreciated as an important part of the publi-cation. The monograph is very useful for sta-tisticians working in time series analysis. (ja)

S. Markvorsen, M. Min-Oo: GlobalRiemannian Geometry: Curvature andTopology, Advanced Courses in MathematicsCRM Barcelona, Birkhäuser, Basel, 2003, 87pp., €24, ISBN 3-7643-2170-9The small booklet contains lecture notes fromtwo advanced courses. The first one was writ-ten by S. Markvorsen (Distance geometricanalysis on manifolds). It contains a compari-son theory for distance functions on immersedsubmanifolds in Riemannian manifolds and itsrelations to seemingly unrelated topics. Theseinclude isoperimetric inequalities, diffusionprocesses (mean exit time comparison), tran-sience, warped products, and all that related tothe Laplacian. The second lecture notes arewritten by M. Min-Oo (The Dirac operator ingeometry and physics). The first partdescribes traditional results on the Dirac oper-ator, its index formula and the Lichnerowiczformula. The second part treats the Gromovnotion of K-area and the corresponding funda-mental K-area inequality, while the third partdiscusses various aspects of the famous posi-tive mass theorem. Both reviews are very use-ful for the orientation of the reader in the field,details should be found in the correspondingliterature. (vs)

P. Monk: Finite Element Methods forMaxwell’s Equations, NumericalMathematics and Scientific Computation,Clarendon Press, Oxford, 2003, 450 pp.,£59,95, ISBN 0-19-850888-3The aim of this book is to provide an up-to-date and sound theoretical foundation forfinite element methods in computational elec-tromagnetism. The emphasis is on finite ele-ment methods for scattering problems thatinvolve the solution of Maxwell’s equationson infinite domains. Suitable variational for-mulations are developed and justified mathe-matically. The main focus of the book is anerror analysis of edge finite element methodsthat are particularly well suited to Maxwell’sequations. The methods are justified forLipschitz polyhedral domains that can causestrong singularities in the solution. The bookis divided into 14 chapters. After the prepara-tory work in Chapters 2 and 3 (functionalanalysis and abstract error estimates, Sobolevspaces and vector function spaces), Chapter 4contains a discussion of a simple model prob-lem for Maxwell’s equations. Chapters 5 and6 form a central part of the book. They presenta detailed description of the original Nédélecfinite element spaces. In Chapter 7, the authorgives the finite element discretization of thecavity problem and presents two convergenceproofs for this method. Chapter 8 presentsbasic topics concerning finite elements andgives a brief introduction to hp finite elementmethods for Maxwell’s equations. A centraltask of computational electromagnetism is theapproximation of scattering problems. Thepresentation of these problems starts inChapter 9, with classical scattering by asphere, where the author derives the famous

integral representation of the solution toMaxwell’s equations called the Stratton-Chuformula. In addition, the author derives classi-cal series representations of the solution ofMaxwell’s equations. These are used inChapter 10 to derive a semi-discrete methodfor the scattering problem utilizing the elec-tromagnetic equivalent of the Dirichlet toNeumann map. A fully discrete domain-decomposed version of this algorithm is pro-posed and analysed in Chapter 11. In Chapter12, the author studies a coupled integral equa-tion and finite element method due to Hazard,Lenoir and Cutzach. Chapter 13 is devoted toa discussion of some issues related to practicalaspects of solving Maxwell’s equations. Thefinal Chapter gives a short introduction toinverse problems in electromagnetism. Thebook will be of interest to mathematicians,physicists and researchers concerned withMaxwell’s equations. The three chapters, 4, 5and 7, could be useful in a graduate course onfinite element methods. (knaj)

S. Montgomery, H.-J. Schneider: NewDirections in Hopf Algebras, MathematicalSciences Research Institute Publications 43,Cambridge University Press, Cambridge,2002, 485 pp., £55, ISBN 0-521-81512-6Hopf algebras are abstractions of algebras offunctions on a group. As such, they wereintroduced to topology in the first half of thelast century. Roughly speaking, Hopf algebrais an associative algebra that also carries a“comultiplication”, and these two structuresare compatible in an appropriate sense. Thesecond advent of Hopf algebras was markedby a seminal talk on quantum groups, present-ed by V. Drinfel’d in Berkeley in 1986. Sincethen, Hopf algebras have found a great num-ber of applications, not only in the theory ofquantum groups (which are particular types ofHopf algebras) but also in low-dimensionaltopology, algebra and geometry. The bookunder review is a collection of expository arti-cles that attempt to summarize the recentdevelopments and offer a new understandingof classical topics. The contributions werewritten by N. Andruskiewitsch, S. Gelaki, G.Letzler, A. Masuoka, D. Nikshych, F. vanOystaeyen, D. Radford, P. Schauenburg, H.-J.Schneider, M. Takeuchi, L. Vainerman and Y.Zhang. Most of the authors were participantsof the Hopf Algebras Workshop held at MSRIas a part of the 1999-2000 Year onNoncommutative Algebra. (mm)

P.T. Nagy, K. Strambach: Loops in GroupTheory and Lie Theory, de GruyterExpositions in Mathematics 35, Walter deGruyter, Berlin, 2002, 361 pp., US$148, ISBN3-11-017010-8The book is a comprehensive treatment onvarious topological and geometrical aspects ofthe theory of loops (i.e., generally non-asso-ciative binary structures with neutral elementsand unique division). The loops are represent-ed as stable transversals (alias sections) intheir one-sided multiplication groups, andthese are equipped with additional structure.Thus topological, differentiable, algebraicloops are treated via Lie groups, algebraicgroups, symmetric spaces, etc. The book isdivided into two parts. The first one presents

basic material, with special attention paid toBol loops and Bruck loops. The second part isdevoted to smooth loops on low dimensionalmanifolds. (tk)

V. Paulsen: Completely Bounded Maps andOperator Algebras, Cambridge Studies inAdvanced Mathematics 78, CambridgeUniversity Press, Cambridge, 2003, 300 pp.,£47,50, ISBN 0-521-81669-6This book is intended as an (advanced) intro-ductory text on the main results and ideas insome of the major topics of modern operatortheory. The first half of the book is devoted tothe study of completely positive and com-pletely bounded maps between C*-algebrasand their connections with the dilation theory.The adverb completely is related to a collec-tion of matrix norms on the algebra. The sec-ond part explains connections with varioustypes of similarity problems (e.g. the Kadisonconjecture and Pisier’s theory), and propertiesof operator systems (e.g., characterization ofexistence of isometric representations of oper-ator algebras - Blecher, Ruan, Sinclair theory,which is applied to new proofs of classicalresults on interpolation of analytic functions).Another application is Pisier’s theory of theuniversal operator algebra of an operatorspace and his results on similarity and factor-ization degree. The book is carefully written,proofs are often accompanied with notes help-ing to explain the situation. Several theoremsare proved by various methods, e.g., there arefive different proofs of the von Neumanninequality. The text can be used either for agraduate course in operator theory or for anindependent study, since it contains 205 exer-cises. The author recommends that the readerbe acquainted with R. G. Douglas’ book“Banach Algebra Techniques in OperatorTheory”. (jmil)

R. Plato: Concise Numerical Mathematics,Graduate Studies in Mathematics, vol. 57,American Mathematical Society, Providence,2003, 453 pp., US$85, ISBN 0-8218-2953-XThis excellent textbook is a concise introduc-tion to the fundamental concepts and methodsof numerical mathematics. The author coversmany important topics using well-chosenexamples and exercises. Many of the present-ed methods are illustrated by figures. Fornumerous algorithms, practical and relevantwork estimates are given and pseudo codes areprovided that can be used for implementa-tions. The author presents approximately 120exercises with different levels of difficulty.Topics covered include interpolation, the fastFourier transform, iterative methods for solv-ing systems of linear and nonlinear equations,numerical methods for solving ODE’s, numer-ical methods for matrix eigenvalue problems,approximation theory, and computer arith-metic. The book is suitable as a text for a firstcourse in numerical methods, for mathematicsstudents or students in neighbouring fields,such as engineering, physics, and computerscience. In general, the author assumes knowl-edge of calculus and linear algebra only.(knaj)

I. Reiner: Maximal Orders, LondonMathematical Society Monographs 28,

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Clarendon Press, Oxford, 2003, 395 pp., £70,ISBN 0-19-852675-3Let R be a noetherian integral domain withquotient field K, and let A be a finite dimen-sional K-algebra. Recall that an R-order in theK-algebra A, is a sub-ring Λ of A, having thesame unit element as A, and such that K . Λ ={ Σαi mi (finite sum)| αi ∈ K, mi ∈ Λ}= A. Amaximal R-order in A is defined in the usualway as such an R-order that is not properlycontained in any other R-order in A. Afteralgebraic preliminaries, basic properties oforders are investigated in Chapter 2. Maximalorders in skew-fields, over discrete valuationrings and over Dedekind domains, are studiedin Chapters 3, 5 and 6, respectively. Chapter 3is devoted to Morita equivalence and Chapter7 deals with crossed-product algebras. Thelast two parts investigate simple algebras overglobal fields and a local and global theory ofhereditary orders. (lbi)

A. Shen, N.K. Vereshchagin: ComputableFunctions, Student Mathematics Library, vol.19, American Mathematical Society,Providence, 2002, 166 pp., US$29, ISBN 0-8218-2732-4The book is based on lectures for undergradu-ate students at the Moscow State UniversityMathematics Department and the content cov-ers a majority of basic notions of general the-ory of computation, excluding computationalcomplexity. It begins with a definition of com-putable functions, based on an informal defin-ition of algorithms. Then, the classic notionsof topics such as decidability, universal func-tions, fixed point theorem, enumerable sets,oracle computations and arithmetical hierar-chy, are presented, clarified and developed. Aprecise definition of an algorithm is given inthe ninth chapter (Turing machines); the wordproblem is treated here as well. Chapter 10(Arithmeticity of computable function), devel-ops the subject on the mentioned precise base;Tarski’s theorem and Gödels’ undecidabilitytheorem are presented. The last chapter isdevoted to recursive functions. The book canappropriately provide knowledge of the themeto programmers and to students of mathemat-ics and computer science. (jml?)

W. Sieg, R. Sommer, C. Talcott, Eds.:Reflections on the Foundations ofMathematics. Essays in Honour of SolomonFeferman, Lecture Notes in Logic 15, A.K.Peters, Natick, 2002, 444 pp., US$45, ISBN 1-56881-170-5The book consists of contributions submittedmostly by participants of the symposium onthe foundations of mathematics, organized tohonour Solomon Feferman, which was held atStanford University, in December 1998. Thevolume is organized into four parts: Proof-the-oretic analysis, Logic and computation,Applicative and self-applicative theories, andPhilosophy of modern mathematical and logi-cal thought. For example, in the first part wecan find contributions concerning ordinalanalysis (such as Avigad’s and Buchholz’scontributions), Simpson’s paper that treatspredicative analysis, where Feferman’s sys-tem IR is compared with Friedman’s ATR0,and Friedman’s article on combinatorial finitetrees. To demonstrate the variety of themes,

let us mention the Fenstad contribution“Computability theory: structure and algo-rithms”, Rathjen’s “Explicite mathematicswith monotone inductive definitions. A sur-vey”, and Mancosu’s “On the constructivity ofproofs. A debate among Behmann, Bernays,Gödel, and Kaufmann.” (jmlc)

Y. Talpaert: Differential Geometry withApplications to Mechanics and Physics,Monographs and Textbooks in Pure andApplied Mathematics, vol. 237, MarcelDekker, Inc., New York, 2001, 454 pp.,US$185, ISBN 0-8247-0385-5The book contains a careful exposition ofbasic facts concerning manifolds, differentialforms and their integration, Stokes theorem,exterior differential systems, Lie derivativesand basic facts on Lie groups and Lie alge-bras, Riemannian geometry and their use inphysics. Comparing the new English versionwith the original French edition, the last threechapters were substantially reworked andextended (these are chapters on Riemanniangeometry, Lagrange and Hamilton mechanics,and symplectic geometry). The book is writtenin a very understandable and systematic way,with a lot of figures. A very good feature ofthe book is a collection of more than 130 exer-cises and problems, which are completelysolved, resp. answered. The book can be rec-ommended for a wide range of students as afirst book to read on the subject. It can be alsouseful for the preparation of courses on thetopic. (vs)

L. Vainerman, Ed.: Locally CompactQuantum Groups and Groupoids, IRMALectures in Mathematics and TheoreticalPhysics 2, Walter de Gruyter, Berlin, 2003,247 pp., €34,53, ISBN 3-11-017690-4These are the proceedings of the workshop"Quantum Groups, Hopf Algebras and theirApplications", which was held in Strasbourgin February 2002. They contain an"Introduction of the editor" and 7 original arti-cles. All of them represent substantial contri-butions to the theory of quantum groups andgroupoids, and every specialist in the fieldshould be familiar with them. The introduc-tion describes the development of the theoryof quantum groups and groupoids in a veryconcise and deep way, so that even a mathe-matician who is not a specialist on this topiccan adequately understand why the theory hasproceeded in this way or that. For the non-spe-cialist, let us also mention that the paper by S.Vaes and L. Vainerman (On low dimensionallocally compact quantum groups) containsPreliminaries, where we can find basic infor-mation about these groups. This paper repre-sents the continuation of the research of bothauthors on extensions of locally compactquantum groups. J. Kustermans and E.Koelink (Quantum SUq~(1,1) and itsPontryagin dual), present an overview of thequantum group SUq~(1,1) and study itsPontryagin dual. A. Van Daele, in his paper(Multiplier Hopf*-algebras with positive inte-grals: A laboratory for locally compact quan-tum groups), gives a survey of the theory ofalgebraic quantum groups and their relationswith locally compact quantum groups. Theremaining four papers deal with quantum

groupoids (M. Enock: Quantum groupoidsand pseudo-multiplicative unitaries; P.Schauenberg: Morita base change in quantumgroupoids; K. Szlachányi: Galois actions byfinite quantum groupoids; J.-M. Vallin:Multiplicative partial isometries and finitequantum groupoids). (jiva)

S. H. Weintraub: Representation Theory ofFinite Groups: Algebra and Arithmetic,Graduate Studies in Mathematics, vol. 59,American Mathematical Society, Providence,2003, 212 pp., US$45, ISBN 0-8218-3222-0The book is an introduction to the group rep-resentation theory with emphasis on finitegroups. Since group representations are justmodules over group algebras, the classicalresults on semisimple group representationsare naturally obtained in Chapter 3, as partic-ular cases of more general results concerningthe structure of modules over semisimplerings. The latter results are presented inChapter 2. Chapter 4 deals with induced rep-resentations and culminates in the proof ofBrauer's theorem (concerning irreduciblecomplex representations of G of exponent nbeing defined over Q (n√1)). The remainingthree chapters provide an introduction to mod-ular representation theory, again by applyingthe powerful module theoretic approach. Thebook is well written, and contains many exam-ples worked out in detail (for example, deter-mining all irreducible complex and modularrepresentations of A5). It is a suitable text fora year long graduate course on the subject.(jtrl)

V. I. Zhukovskiy: Lyapunov Functions inDifferential Games, Stability and Control:Theory, Methods and Applications, vol. 19,Taylor & Francis, London, 2003, 281 pp.,£70, ISBN 0-415-27341-2The book is devoted to the linear theory of dif-ferential games under uncertainty. It is sup-posed that nothing is known on their nature(i.e., the stochastic approach is ruled out). Thepresence of uncertainties moves the problemof the choice of strategy to the field of "multi-criteria" dynamical problems. The results aremainly given for quadratic payoff functions,two players and non-cooperative games. Thebook is divided into two parts. The first onedescribes the foundations of differentialgames under uncertainties. The central role isplayed by the notion of a vector guarantee.Two approaches, based on the analogue of thevector saddle point and the vector maximin,are used for solving the multicriteria dynami-cal problems. The second part is devoted tothe concept of equilibrium of objections andcounter-objections, as well on the active equi-librium. The reader should have a basicknowledge of ordinary differential equationsand optimization. For example, the dynamicprogramming approach (i.e., the Belmanequation) and the method of Lyapunov func-tions are frequently used. Both parts end withresults, comments on the history of ideas, andreferences (131 items). These are well orient-ed in works of the Russian and Ukrainianschools. Each part is also accompanied byexercises, with their solutions given at the endof the book. (jmil)

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