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

ASSOCIATE EDITORSSTEEN MARKVORSENDepartment of Mathematics Technical University of DenmarkBuilding 303DK2800 Lyngby, Denmarke-mail: [email protected] CIESIELSKIMathematics Institute Jagiellonian UniversityReymonta 4 30-059 Krakow, Polande-mail: [email protected] QUINNOpen University [address as above]e-mail: [email protected]

SPECIALIST EDITORSINTERVIEWSSteen Markvorsen [address as above]SOCIETIESKrzysztof Ciesielski [address as above]EDUCATIONVinicio VillaniDipartimento di MatematicaVia Bounarotti, 256127 Pisa, Italy e-mail: [email protected] PROBLEMSPaul JaintaWerkvolkstr. 10D-91126 Schwabach, Germanye-mail: [email protected] ANNIVERSARIESJune Barrow-Green and Jeremy GrayOpen University [address as above]e-mail: [email protected] [email protected] andCONFERENCESKathleen Quinn [address as above]RECENT BOOKSIvan Netuka and Vladimir SouèekMathematical InstituteCharles UniversitySokolovská 8318600 Prague, Czech Republice-mail: [email protected] [email protected] OFFICERMartin SpellerDepartment of MathematicsGlasgow Caledonian UniversityGlasgow G4 0BA, Scotlande-mail: [email protected] UNIVERSITY PRODUCTIONTEAMKathleen Quinn, Liz Scarna

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CONTENTS

EMS June 1999

EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

NEWSLETTER No. 33

September 1999

EMS News : Committee and Agenda .................................................................. 2

Editorial by EMS Vice-President Andrzej Pelczar .............................................. 3

Fourth Diderot Mathematical Forum : Mathematics and Music ...................... 4

Introducing the Editorial Team : Part 1 ........................................................... 5

EMS Council Meeting in Barcelona: First Announcement ................................ 6

Oxford Doctorate for Andrew Wiles ................................................................... 7

Interview with Tim Gowers ................................................................................. 8

ICIAM99 in Edinburgh ..................................................................................... 10

SIAM and EMS Joint Conference on Computational Science ........................... 11

A Universal Mathematical Resources Locator? ............................................... 12

1999 Anniversaries : Caspar Wessel ................................................................ 13

1999 Anniversaries : E. C. Titchmarsh .............................................................16

Societies Corner : Swiss Mathematical Society .................................................18

Societies Corner : Edinburgh Mathematical Society ........................................ 20

Forthcoming Conferences ................................................................................. 22

Recent Books ..................................................................................................... 26

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

ISSN 1027 - 488X

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

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

EXECUTIVE COMMITTEEPRESIDENT (19992002)Prof. ROLF JELTSCHSeminar for Applied MathematicsETH, CH-8092 Zürich, Switzerlande-mail: [email protected]. ANDRZEJ PELCZAR (19972000)Institute of MathematicsJagellonian UniversityRaymonta 4PL-30-059 Krakow, Polande-mail: [email protected]. LUC LEMAIRE (19992002)Department of Mathematics Université Libre de BruxellesC.P. 218 Campus PlaineBld du TriompheB-1050 Bruxelles, Belgiume-mail: [email protected] (19992002)Prof. DAVID BRANNANDepartment of Pure Mathematics The Open UniversityWalton HallMilton Keynes MK7 6AA, UKe-mail: [email protected] (19992002)Prof. OLLI MARTIODepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlande-mail: [email protected] ORDINARY MEMBERSProf. BODIL BRANNER (19972000)Mathematical InstituteTechnical University of DenmarkBuilding 303DK-2800 Lyngby, Denmarke-mail: [email protected]. DOINA CIORANESCU (1999-2002)Laboratoire dAnalyse NumériqueUniversité Paris VI4 Place Jussieu75252 Paris Cedex 05, Francee-mail: [email protected]. RENZO PICCININI (19992002)Dipto di Matem. F. EnriquesUniversit à di MilanoVia C. Saldini 50I-20133 Milano, Italye-mail: [email protected]. MARTA SANZ-SOLÉ (19972000)Facultat de MatematiquesUniversitat de BarcelonaGran Via 585E-08007 Barcelona, Spaine-mail: [email protected]. ANATOLY VERSHIK (19972000)P.O.M.I., Fontanka 27191011 St Petersburg, Russiae-mail: [email protected] SECRETARIATMs. T. MÄKELÄINENDepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlandtel: (+358)-9-1912-2883fax: (+358)-9-1912-3213telex: 124690e-mail: [email protected]: http://www.emis.de2

EMS NEWS

EMS September 1999

EMS Agenda1999

30 SeptemberDeadline for submission of proposals for the 2001 EMS Summer Schoolscontact: David Brannan, e-mail: [email protected]

9 - 10 OctoberExecutive Committee Meeting, hosted by the Swiss Mathematical Society and ETH,Zurich (Switzerland)

15 NovemberDeadline for submission of material for the December issue of the EMS Newslettercontact: Robin Wilson, e-mail: [email protected]

DecemberSecond announcement of the Third European Congress of Mathematics (3ecm), Barcelona(Spain)contact: S. Xambó-Descamps, e-mail: [email protected]: www.iec.es/3ecm/

3 - 4 DecemberFourth Diderot Mathematical Forum, on Mathematics and Music, in Lisbon (Portugal),Paris (France) and Vienna (Austria)contact: Mireille Chaleyat-Maurel, e-mail: [email protected]

200031 JanuaryNominations to the Secretariat for delegates of individual members (details to appear inthe December Newsletter)contact: EMS Secretariat, e-mail: [email protected]

February - MarchVoting for delegates of individual memberscontact: EMS Secretariat, e-mail: [email protected]

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

25 - 26 MarchExecutive Committee Meeting, hosted by the Polish Mathematical Society and theInstitute of Mathematics of the Polish Academy of Sciences, Bedlevo, near Pozñan(Poland)

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

3 7 JulyALHAMBRA 2000: a joint mathematical European-Arabic conference in Granada(Spain), promoted by the European Mathematical Society and the Spanish RoyalMathematical Society contact: Ceferino Ruiz, e-mail: [email protected]: www.ugr.es/~ruiz/

7 8 JulyCouncil Meeting in Barcelona (Spain)contact: EMS Secretariat, e-mail: [email protected]

10 14 JulyThird European Congress of Mathematics (3ecm) in Barcelona (Spain)contact: S. Xambó-Descamps, e-mail: [email protected]: www.iec.es/3ecm/

24 July 3 AugustEMS Summer School in Edinburgh (Scotland): New analytic and geometric methods in inverseproblemsorganiser and contact: Erkki Somersalo (Otaniemo, Finland), e-mail: [email protected]

17 August 2 SeptemberEMS Summer School at Saint-Flour, Cantal (France): Probability theoryorganiser and contact: Pierre Bernard (Clermont-Ferrand, France), e-mail: [email protected]

EMS News: Committee and Agenda

In October 2000 the EuropeanMathematical Society celebrates the tenthanniversary of its creation. Looking backon the past nine years we can try to evalu-ate the achievements of the EMS, countingits successes, pointing out possible mis-takes or unrealised plans while concentrat-ing on projects for the future. Such ananniversary provides a good occasion for adiscussion on some problems concerningthe general shape of the Society and of itsactivities, as for instance its role in thepreparation of European Congresses ofMathematics and the DiderotMathematical Forums, the patronageoffered to scientific conferences, and so on.Important aspects of the EMSs activity,and more generally of the presence of theSociety on the European scientific and cul-tural platform, concern the use ofEuropean financial sources; compare, forinstance, Luc Lemaires Editorial in theJune issue (EMS Newsletter 32).

It would be practically impossible andsuperfluous in view of Rolf JeltschsEditorial in the March issue (EMSNewsletter 31) to present here all theforms of EMS activity related to research inmathematics and its applications (with allits interdisciplinary aspects), internationalcooperation of mathematicians andnational societies (including, but notrestricted to, support for young mathe-maticians from countries facing financialdifficulties) and everything that has beendone, according to the general mission ofthe Society, for the popularisation of math-ematics, with special attention to the roleof mathematics in the evolution of culture,technology and civilisation. It would alsobe useless to try to describe all the presentand future problems of the Society and theEMSs role in the scientific and cultural lifeof Europe; such a task would be too ambi-tious and beyond the authors competence.I will thus limit myself to a few of them,selected in a quite arbitrary and subjectiveway.

The first group of problems concernsthe members of the EMS. Everyone knowsthat the membership is very heteroge-neous with its three categories of members.National societies, as corporate foundermembers, created the Society in M¹dralin(Poland) in October 1990. Expecting arapid growth in the number of individualmembers, these societies decided that the

EMS statutes should be rediscussed,revised and then stabilised, in some sense,as soon as the number of individual mem-bers became large enough; for details, seethe EMS statutes. Since the growth in theindividual membership has been slowerthan expected, this decision is still to beimplemented; one can say that at that timethe process of building up and shaping theSociety will be barely completed. Such an

opinion is not based on an official inter-pretation of the statutes and need not beshared by everyone; on the other hand, itis not totally unacceptable, and is formu-lated here as intellectual provocation inorder to point out that the most importantquestion concerning individual membersis: how do we increase their number?

On the quantitative development of theEMS we have to think about the possibleimpact of the growth in the individualmembership on the future shape of theSociety. The EMS can be briefly charac-terised as a scientific society that is alsoconcerned with the popularisation ofmathematics as well as its applications and(at least implicitly) with the teaching ofmathematics, stressing to the public therole played by mathematics in science, cul-ture, technology and civilisation in gener-al. It is natural that these tasks and goalswere first realised by researchers andteachers in higher education institutions;at present the individual members comealmost exclusively from these categories.However, since the membership of severalnational mathematical societies (corporatemembers of the EMS) also includes school-teachers and other mathematical alumninot working as scientists or academics, wemust expect that some of them will becomefuture members of the EMS we shouldwelcome them warmly. In the long runthis should change the future image of theSociety. I think that the main impactwould be a stronger interest by the EMS inthe teaching of mathematics (at all levels,including elementary and secondary).Even now the Society should extend its

interest in mathematical education in theEuropean dimension, especially withrespect to a general reform of educationalsystems (including fundamental reforms inmathematics teaching) in several Centraland Eastern European countries, in orderto be aware of what has already been doneand what we can expect in that area in thenear future.

Returning to the question of member-ship, let me add a historical remark aboutthe Polish Mathematical Society. Duringits foundation meeting in Kraków in April1919 (see the Societies section in the Juneissue), two founder members, Stanis³awZaremba and Stefan Banach, decided afterdiscussion that the Society should have apurely scientific character; all suggestionsproposing the popularisation of mathe-matics and other not strictly scientific activ-ities were officially rejected. Eventually,after some decades, the Society changed itsidentity and became an association thatincluded in its mission the popularisationof mathematics and (later) the teaching ofmathematics. The membership rules nowallow schoolteachers to be members of thePolish Mathematical Society.

The above remarks are of minor impor-tance with respect to membership of theEMS, but they raise a more importantproblem, that of understanding the wordmathematician. This seems to be importantfor numerous reasons; let me mention onlyone statistical data indicating the num-bers of mathematicians in European coun-tries. In Poland we have two so-called pro-fessional titles, licencjat and magister, equiv-alent to a B.Sc. and an M.Sc. in English orAmerican terminology. Licencjat is purelya professional title, while magister is thefirst step in an academic career: everyonewishing to get a Ph.D. must have the titlemagister. In official (administrative) termi-nology everybody who gets the profession-al title of licencjat in mathematics (or, ofcourse, the higher one: magister) is countedin the statistical data as a mathematician.In all databases concerning higher educa-tion only mathematicians with an M.Sc. arecounted as members of staff of a highereducation institution. It is probable thatfuture databases for higher education willindicate only those members of academicstaff with a Ph.D. Thus, in the future, froma statistical point of view, mathematiciansin higher education institutions in Polandwill all be doctors in mathematics. (I thinkthat the terminology varies from country tocountry.)

It would be very useful to obtain moreprecise terminology on a European scale.It would be valuable in particular to con-tinue the extremely interesting panel dis-cussion on Demography of mathemati-cians at the Second European Congress ofMathematics (Budapest, 1996). Maybe theEMS should suggest something alongthose lines to stimulate some clarification.The first step would be to compare the ter-minology in various countries.

The next question I regard as importantsince it touches on one form of activity thatmakes the Society visible. Among severalinteresting problems discussed by mem-

EMS June 1999 3

EDITORIAL

EditorialEditorialby

EMS Vice-President

Andrzej Pelczar (Kraków)

Photo by Stefan Ciechanpublished with kind permission of Forum Akademickie

Fourth Diderot Mathematical ForumMathematics and Music

L. Mazliak (Paris)

On 3-4 December 1999 the Fourth Diderot Mathematical Forum, onMathematics and Music, will be held in Lisbon, Paris and Vienna. As for allevents in this series, each city has selected a specific subtheme on which themeeting will focus. These subthemes are: Lisbon: A historical study of the connection between the two domains; Paris: The problems around the formal systems for composition in the 20thcentury; Vienna: The mathematics of sound. A round table between the three sites is scheduled for Friday afternoon on thetheme: Is the link between mathematics and music a cultural or a natural one? Detailsof the programmes and local arrangements can be found on the web site:www.emis.de, with links to other sites.

The connections between mathematics and music are among the classicalthemes studied by philosophers. One of the main sources for this interest canbe found in the Pythagorean system that connects with elementary arithmeticthe fundamental components of sounds (such as those emitted by a vibratingstring) and all parts of the universe such as planets and stars. For centuries thisvision of the world, inherited from the ancient Greeks, was spread by theChurch-dominated Western scientific culture. It had the great advantage ofproviding a unified system of the World, ruled by simple laws that could beregarded as a proof of Gods rationality; all the teaching of medieval Christianscience could be summarised as Reason is Gods part in Man. With the helpof reasoning, Man could tear the hidden order of Creation from the appearingchaos of the world.

On the beautiful allegorical rose of Laon cathedral in the north of France,the artists of the Middle Ages have represented theology surrounded by itscourt: arithmetic, geometry and music are members of it. However, early inhistory (even in Greek times), serious fractures began to appear in a system thatwas too perfect and too rigid: slowly, but inevitably, mathematics and musichave followed separate ways. However, the weight of the cultural heritage wasso great that its effects were still present many centuries after musicians hadceased to have a strong connection with pure science. It is impressive that greatscientists, such as Kepler, Euler and Lagrange, have been interested in music ata scientific level and have looked for formal systems.

Attempts to find unified scientific explanations for music in the 19th centu-ry quickly miscarried: some of them were clever, but most were quite wretched,as described in Fichets Musical theories of music in the XIXth and XXth cen-turies. In a sense, the twelve-tone system of Schönberg and the Vienna schoolmay be seen as another attempt to obtain a unified system; when readingWeberns cycle of conferences in 1932 (Path to the new music), it is surprisinghow much energy the author spends in proving (through a careful choice ofexamples!) that the twelve-tone system embodies all the previous music.

The attempt was doomed to failure. However, the twelve-tone systemmarked a real turn in modern music: as an official ending of tonality, itopened the door to the idea that composers were free to choose their own com-positional systems. And here, mathematics returned to offer a platform onwhich artists may base their personal language. Also, new importance was givento the mathematical study of the particular physical phenomenon we call musi-cal sound, in order to obtain a better understanding of it and to develop newtechnology (such as new instruments) that can be used as a resource for com-posers.

At the turn of the 21st century, the Diderot Forum can be seen as an attemptto take stock of the millenial history of the lively connections between the twodomains.

bers of the EMS Council at the Budapestmeeting, one concerned a fundamentalquestion: is the organisation of large andnon-specialised conferences, such asInternational Congresses ofMathematicians or the EuropeanCongresses of Mathematics, really reason-able and fruitful? A weaker version of thisproblem questioned the need to organiseEuropean Congresses in view of the factthat an ICM is organised every four years or (almost equivalently) the need to organ-ise great and expensive internationalmeetings every two years. The arguments(presented here in simplified form) were oftwo kinds: (1) real scientific gain is now obtained

mostly by participating in specialisedconferences on specific topics, ratherthan in everything-touching big con-ferences;

(2) great meetings (congresses) are expen-sive, and the ratio of scientific gain tocost seems to be too small.

Radical options presented during this dis-cussion (and repeated a few times sincethen) described huge congresses as relicsof the past. Such radical opinions were(and are) not shared by many mathemati-cians, but they should be noted.

Presenting the opposite point of view,we notice that there is a growing trendtowards specialisation and a tendency toorganise specialised conferences and so-called workshops, and it is thus reasonableand necessary to bring together mathe-maticians working in distinct areas ofmathematics and give them opportunitiesto exchange ideas and enable interdiscipli-nary discussions. It is also important tobuild interpersonal relations, so that math-ematicians can meet each other as friendsand not only as scientific partners.

There are other important, and proba-bly deeper, reasons that make large andregular congresses very fruitful. Each suchcongress summarises (in some sense) whathas been achieved in mathematics duringthe last few years and identifies the mostimportant results obtained since the pre-ceding congress. This is realised by theprestigious prizes awarded to the authorsof the most spectacular achievements andby the selection of the invited talks. Listsof invited lectures (both plenary and sec-tional) indicate the most important fields not necessarily those with the most impor-tant and spectacular results! Thus, themajor international and prestigious con-gresses both summarise the past and stim-ulate the future, at least until the next con-gress.

Accepting the above arguments, onemight say that the InternationalCongresses of Mathematicians are suffi-cient and that the European Congressesshould be abolished. However, analysingthe results and opinions from the first twoEuropean Congresses, we can see that theywere well placed between the ICMs, andthat their format ensures that the aboveaspirations are realised and help to moti-vate the next congress. We also note thatthe European Congresses do not duplicatethe ICMs; in particular, the European

Congresses present prizes to young mathe-maticians, and there are round table dis-cussions on special topics. We note finallythat progress in mathematics, and thegrowth in numbers of researchers andmathematical scientific centres densityin the time scale of mathematically impor-tant events make it reasonable to organ-ise large and prestigious congresses inde-pendently of the ICMs. EuropeanCongresses play a proper role in that

respect. We add that the acceptance of theidea of the European Congresses has beenemphasised in spectacular form by thestrong competition for permission toorganise the Third European Congress ofMathematics; Barcelona was the winner ofthat competition. I am sure that the ThirdCongress will be fully successful and willdismiss all doubts concerning the organisa-tion of European Congresses.

EMS September 19994

EDITORIAL

EMS NEWS

EMS September 1999 5

Robin Wilson (Editor-in-Chief) is a Senior Lecturer in Mathematics at the OpenUniversity, UK, where he has been since 1972. He is also a Fellow of Keble College,Oxford University, and a frequent Visiting Professor at Colorado College, USA. Hereceived his Ph.D. from the University of Pennsylvania in 1968 for a thesis in number the-ory.His mathematical interests lie mainly in graph theory and the history of mathematics. Inthe former he has been particularly involved with graph colourings and in the latter hisconcerns are mainly British mathematics, particularly of the late 19th and early 20th cen-turies, and the history of combinatorics. He has written and edited over twenty books inthese areas, and won the Mathematical Association of Americas Lester Ford award foroutstanding expository writing. He has served on the British Combinatorial Committeeand the committee of the British Society for the History of Mathematics. He is actively involved with the popularisation of mathematics and enjoys lecturing toschool and college students and to adults interested in mathematics.

Krzysztof Ciesielski (Associate editor and Societies editor) works in the Mathematics Institute of theJagiellonian University, Kraków, from which he obtained his Ph.D. in 1986. His mathematicalinterests include dynamical systems, topology and analysis. He has been a Correspondent of TheMathematical Intelligencer since 1987. His wife Danuta is also a mathematician, who is interested inclassical geometry and complex analytic geometry.

He and his friend Zdzislaw Pogoda have co-authored about 100 articles presenting mathemat-ics to a general audience. They have been given several awards, including the Dickstein Prize(1995) which is given once every few years by the Polish Mathematical Society for an outstandingcontribution to mathematical culture; they are the youngest ever recipients of this prize. Theyhave written two popular books: Boundlessness of Mathematical Imagination (1995) and MathematicalDiamonds (1997). Both of these books were best-sellers in Poland.

Steen Markvorsen (Associate editor and Interviews editor) received his Ph.D. from theTechnical University of Denmark in 1983. His main mathematical interest is in differ-ential (and distance-) geometry. In particular he is concerned with the geometric syn-thesis between curvature, form and function, including its applications to a spectrum oftopics ranging from general relativity to biology. Together with a group of researchershe is currently exploring the role and potential of computer experimentation in mathe-matical research, in particular within the area of curvature geometry.He has been a member of the board of the Danish Mathematical Society and is activelypromoting and disseminating mathematics inside and outside university circles.

Introducing the Editorial Team : part 1

Kathleen Quinn (Associate editorand Conferences editor) is a Lecturerin Mathematics at the OpenUniversity, UK. She obtained herPh.D. in 1991 from LondonUniversity. Her research interestslie mainly in design theory andthe applications of combinatoricsto cryptography. She was a lectur-er in London (at the RoehamptonInstitute) from 1991 to 1995, andthen a research fellow at the OpenUniversity for four years.

June Barrow-Green (Anniversaries editor) is a Research Fellow in the History ofMathematics at the Open University, UK. She graduated from Kings College, London,and received her doctorate from the Open University. Her research interests include19th- and 20th-century British and European mathematics (in particular, the work ofHenri Poincaré). She is also working on the use of databases and the use of the WorldWide Web as research tools in the history of mathematics. A former secretary of the British Society for the History of Mathematics (BSHM), she iscurrently a member of the BSHM Council.

The EMS Council meets every second year.The next meeting will be held in Barcelonaon 7-8 July 2000, before the 3rd EuropeanCongress of Mathematics. The exact loca-tion will be announced later.

Delegates to the Council will be electedby the following categories of members, asper the Statutes.(a) Full Members: Full Members are nation-al mathematical societies, which elect 1, 2or 3 delegates according to their size andresources. Each society is responsible forthe election of its delegates. Each societyshould notify the Secretariat of the EMS inHelsinki of the names and addresses of itsdelegate(s) no later than 10 March 2000.As of 1 July 1999, there were 47 such soci-eties which could designate a maximumof 69 delegates.(b) Associate Members: There are two associ-

ate members, namely the Gesellschaft fürMathematische Forschung and theEuropean Mathematical Trust. Their cur-rent common delegate is elected until1999, so their delegate has to be elected in2000. According to the Statutes, delegatesrepresenting associate members shall beelected by a ballot organised by theExecutive Committee from a list of candi-dates who have been nominated and sec-onded, and have agreed to serve.(c) Institutional Members: There are three

institutional members, Institut Non-Lineare de Nice, the Moldovian Academyof Sciences and the Mathematical Instituteof the Serbian Academy of Sciences andArts. Their common delegate is elected till1999, so their delegate has to be elected in2000. According to the Statutes, delegatesrepresenting institutional members shallbe elected by a ballot organised by theExecutive Committee from a list of candi-dates who have been nominated and sec-onded, and have agreed to serve.(d) Individual Members: A person becomes

an individual member either through acorporate member, by paying an extra fee,or by direct membership. On 30 June1999, there were some 1900 individualmembers and, according to our statutes,these members will be represented by 19-20 delegates. The final count of individualmembers for these elections will be madeon 1 November 1999.

The mandates of 11 of the present 17delegates end on 31 December 1999, andso elections must be held for their posi-

tions. They are: G. Anichini, G. Bolondi,B. Branner, J.-M. Deshouillers, K.Habetha, M. Karoubi, T. Kuusalo, A.Lahtinen, L. Màrki, R. Piccinini, and D.Puppe. Of the eleven, B. Branner, J.-M.Deshouillers and M. Karoubi cannot be re-elected because they have served in thiscapacity for eight years.

Nomination papers for these electionswill appear in the December issue of theNewsletter. Six delegates were elected forthe term 1998-2001, so they will continueunless they inform the Secretariat to thecontrary by 31 December 1999.

The Executive Committee is responsi-ble for preparing the matters to be dis-cussed at Council meetings. Items for theagenda of this meeting of the Councilshould be sent as soon as possible and nolater than 10 March 2000 to the EMSSecretariat in Helsinki.

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

Professor Rolf Jeltsch (1999-2002)Vice-Presidents

Professor Andrzej Pelczar (1997-2000)Professor Luc Lemaire (1999-2002)

SecretaryProfessor David Brannan (1999-2002)

TreasurerProfessor Olli Martio (1999-2002)

MembersProfessor Bodil Branner (1997-2000)Professor Marta Sanz-Solé (1997-2000)Professor Anatoly Vershik (1997-2000)Professor Doina Cioranescu (1999-2002)Professor Renzo Piccinini (1999-2002)

Under Article 7 of the Statutes, membersof the Executive Committee shall be elect-ed for a period of four years. Committeemembers may be re-elected, provided thatconsecutive service shall not exceed eightyears. Andrzej Pelczar has served on theExecutive Committee for eight years, so hecannot be re-elected.

It would be convenient if potential nom-inations for office in the ExecutiveCommittee, duly signed and seconded,could reach the Secretariat by 10 March2000. It is strongly recommended that astatement of intention or policy is enclosedwith each nomination. If the nominationcomes from the floor during the Councilmeeting there must be a written declara-tion of the willingness of the person toserve, or his/her oral statement must besecured by the Chair of the NominatingCommittee (if there is such) or by thePresident. It is recommended that a state-ment of policy of the candidates nominat-ed from the floor should be available.

The Council may, at its meeting, add tothe nominations received and set up aNominations Committee, disjoint from theExecutive Committee, to consider all can-didates. After hearing the report by theChair of the Nominations Committee (ifone has been set up), the Council will pro-ceed to the elections to the ExecutiveCommittee posts.

Delegates to the Council meeting, whoare planning to attend the EuropeanCongress of Mathematics, are advised thattheir accommodation arrangementsshould be made through the ECM. Fordelegates to the Council who are notattending the ECM, an address for accom-modation arrangements will be providedlater.

Secretariat: Ms. Tuulikki Mäkeläinen Department of Mathematics P.O. Box 4, FIN-00014 University of Helsinki, Finland

David BrannanSecretary of the EMS

Annex : Timetable for theCouncil Meeting

September 1999: Letters are sent to full,associate and institutional members as wellas delegates giving information on theCouncil meeting. Specifically, points forthe agenda and suggestions for futuremembers of the Executive Committee areinvited. (Delegates are kindly requested tokeep the Secretariat informed of their cor-rect and up-to-date addresses.) 1 November 1999: Following the by-laws,the number of individual members isrecorded to determine the number of theirdelegates. December 1999 Newsletter: Informationon the Council meeting is printed again. Anominating slip for the delegates of theindividual members is given and sugges-tions for Executive Committee membersare invited. 31 January 2000: Deadline for nomina-tions for delegates of individual members.February 2000: The ballots for delegatesof individual members are sent to individ-ual members. March 2000 Newsletter: Candidates fordelegates of individual members areannounced. The venue and meeting timesof the Council meeting are repeated. April 2000: A letter is sent to each dele-gate, containing the agenda of the Councilmeeting. June 2000 Newsletter: The results of theelections for delegates of individual mem-bers are announced. The venue, the meet-ing times, and the agenda of the Councilmeeting are given. June 2000: Material for the Council meet-ing is sent to the delegates.

EMS NEWS

EMS September 19996

Meeting Meeting of the of the

EMS CouncilEMS Council7-8 July 2000, Barcelona

First announcement

In June Andrew Wiles received an honorarydoctorate from his alma mater, OxfordUniversity. For those readers who wish to tryout their Latin, the citation by the PublicOrator Jasper Griffin was as follows; a trans-lation appears below. We thank Jasper Griffinfor permission to reprint his citation.

Professor ANDREW WILES, FRS, EugeneHiggins Professor of Mathematics, PrincetonUniversity.Neminem, credo, in hac hominum litteratissimo-rum frequentia reppereris quin mecum hanc for-mulam conceptis verbis concinere possit: Nullam ininfinitum ultra quadratum potestatem in duaseiusdem nominis fas esse dividere. Haud ita pri-dem res aliter se habebat, nos plerique Ciceroni ipsiadsensi eos dixissemus qui mathematici vocenturnon solum recondita in arte et multiplici subtiliquesed etiam in magna rerum obscuritate versari; sedhodie et Pythagorae arcana vulgi aures titillant, ethomines devia illa mathematicorum latibula visi-tant ita indocti ut nihilominus curiositate ducan-tur. itaque quisquis famam sibi adpetit huius quemproduco vitam contemplatus ne eruditum geome-trarum pulverum aspernetur. hic enim cum diu inintimis rei algebraicae medullis habitarit, tam pri-mum nobiscum quam postea apud Cantabrigienseset postremo apud Princetonienses, tam insignemdenique consecutus est gloriam ut etiam ab insciisneque isti studiorum generi adscriptis agnoscatur.qui usque a primis aetatis suae annis austeranumerorum scientia delectatus notissimum illudPetri Fermati theorema, a tot tamque nobilibusmathematicis CCC ferme annos frustra temptatum,sibi proposuit probandum; quod quidem firmataiam aetate post alia egregia facinora rationibusexquisitissimis summoque acuminis firmamentousus tandem firmissime elaboravit. longum sit sispinosissimas istius incepti difficultates, ingeniosis-sima huius artificia, singillatim percensere coner,praesertim cum L, tantum homines esse dicanturqui quantum hic perfecerit animo recte aestimarepossint, quorum in numero me non esse confiteoratque concedo. erant homines quibus hic nimisaudax videbatur, qui tantae claritudinis problemasolus aggrederetur; erat tempus quo ipse paene des-perarat, cum theorema illud, devictum iam, utvidebatur, atque superatum, tamen tanquamHydra illa Lernaea insperatas difficultates subitoprotulit atque produxit. sed res bene vortit: vicittandem vivida vis animi, invenit SphinxOedipodem suum, cui ita plauserunt universimathematici ut dolerent quidam sibi ademptas essehaud ingratas frustra ratiocinandi molestias, solu-tum denique esse venerabile istud aenigma.

Preasento temporum nostrorum Archimeden,numerorum magistrum singularem, theorematisultimi enodatorem incomparabilem, AndreamWiles, Societatis Regiae Sodalem, CollegiMertonensis Socium honoris causa ad scitum, utadmittatur honoris causa ad gradum Doctoris inScientia.

Admission by the ChancellorMathematicorum princeps ingeniosissime, quiquaestionem perdifficilem deficientibus ceteris vicogitationis devicisti, ego auctoritate mea et totiusUniversitatis admitto te ad gradum Doctoris inScientia honoris causa.

Paraphrase in EnglishI do not imagine that there is anybody in thislearned company who could not recite in uni-son with me the formula: There is no whole-number solution to the equation xn + yn = zn,where n is greater than 2. That is a very newstate of things. Until recently most of uswould have agreed with Cicero, who said thatmathematicians concern themselves with asubject matter which is not only various andrarified but also obscure; but now discoveriesin mathematics appeal to the ears of theunlearned, and quite ordinary people feel aninterest, even if not a well informed one, in itsmost abstruse areas. Anyone who is interest-ed in becoming famous should consider thecareer of Professor Andrew Wiles and thinktwice about passing up a mathematicalcareer. After spending many years at numbertheory, first here, then at Cambridge, andmost recently in Princeton, he has attainedsuch celebrity that he has become recognis-able to laymen and to those with no profes-sional interest in the subject. At a very earlyage he was attracted by algebraic number the-ory and decided that he would try to producea proof of the last theorem of Pierre Fermat,that classic problem which over the last threehundred years had been attempted withoutsuccess by so many eminent mathematicians.In his maturity he crowned his many otherachievements by producing a definitiveproof, by means of procedures of extraordi-nary subtlety and intellectual range. It wouldtake far too long if I were to try to explain hisachievement in detail, with its perplexing dif-ficulties and Professor Wiless most ingenious

solutions; especially as there are said to beonly fifty people in the world who fullyunderstand it, and I freely confess that I amnot one of them. There was a time when peo-ple were inclined to criticise him for over-confidence, in taking on such a problem sin-gle-handed; there was a time when he cameclose to despair, as the theorem, which hadappeared to be defeated, suddenly put forthnew and unexpected difficulties. But all waswell: his intellectual power prevailed, like anOedipus he solved the riddle of the Sphinx.His achievement has been greeted with uni-versal applause by the mathematical commu-nity, although some of them view with regretthe disappearance of a venerable puzzle, andthe loss of the mingled pleasure and pain ofinconclusive mathematical endeavour.

I present the Archimedes of our time, theoutstanding master of numbers, the incom-parable unriddler of the Last Theorem,Professor Andrew Wiles, FRS, HonoraryFellow of Merton College, for admission tothe honorary degree of Doctor of Science.

Admission by the ChancellorYou are a prince among mathematicians: byyour intellectual power you have solved amost intractable problem where others havefailed. Acting on my own authority and thatof the whole University, I admit you to thehonorary degree of Doctor of Science.

NEWS


OxforOxford doctorate d doctorate for for

AndrAndrew Wew Wilesiles

EurEuropean Wopean Women in Mathematicsomen in MathematicsThe video Women and Mathematics across Cultures, is available from the EWM office inHelsinki. It was shown at the ICM98 in Berlin after the panel on women and math-ematics.The video explores the impact of cultural differences of the female condition allow-ing four women mathematicians who have studied and worked in Europe and Northand South America to tell their stories. Following a five-minute introduction to EWM,including some surprising statistics about women mathematicians in Europe, the fourwomen recount their personal experiences. The length of the video is 25 minutes,and was directed by Marjatta Näätänen in collaboration with Bodil Branner, Kari Hagand Caroline Series in 1996.The cassettes are in VHS and the prices are as follows:

PAL SECAM/NTSCin Europe 200 FIM 250 FIMoutside Europe 220 FIM 270 FIM(6 FIM is approximately 1 euro)SECAM and NTSC cassettes are not equipped with English subtitles, but the video isin English and a written text of the interviews is provided.Please send your order via e-mail to [email protected] stating which systemyou want. After we receive your payment, the cassette will be mailed to you. Please,pay to Euroopan naismatemaatikot, account number 800017-702454141 withLeonia Bank plc, Helsinki, Finland, with swift code PSPBFIHH via Euro NettingSystem or via Eurogiro. The telex of Leonia Bank is 121 698. Personal checks causeus a lot of work and expense, so they cannot be used; neither can we accept creditcards.Mailing address: Riitta Ulmanen, Department of Mathematics, P.O.Box 4, FIN-00014University of Helsinki, Finland.

You come from a distinguished intellectualfamily with Cambridge connections?Well, I would not put it that strongly, but mygreat-great-grandfather was a famous neuro-physiologist in his day, and his name is stillvery much known to some of my colleagues atTrinity College in that area. My great-grand-father was a civil servant who is best knownfor editing Fowlers Modern English Usage,and for a book he wrote called Plain Words.This was originally commissioned as a guideto good writing style for civil servants, but itturned out to have a much broader appealand is still in print today. My father is a com-poser, which is a highly cerebral occupationeven if it is not usually classified as an intel-lectual one. He and I are often struck by sim-ilarities between his working methods whenwriting a piece of music and mine when tack-ling a complicated mathematics problem.My father, grandfather and great-grandfa-ther all studied at Cambridge, and so did sev-eral relatives on my mothers side. Althoughmy parents lived in London, I boarded atKings College School, Cambridge, where Iwas a chorister.

I believe that Andrew Wiles also went to thesame school?Yes, ten years before me so we had teachersin common. Mary Briggs, the wife of theheadmaster at the time, had graduated inmathematics from Girton College, where shewas taught by Mary Cartwright, and duringmy last year or so she gave me some individ-ual teaching. I was very lucky to be taught byher, and by my later teachers as well.

Thanks to Mrs Briggs I got a scholarshipto Eton and there I had another inspirationalteacher, Norman Routledge, who had been afellow of Kings. He did not allow himself tobe limited to the syllabus but ranged far morewidely. In my last two years at Eton, themathematics specialists were given a weeklysheet of challenging problems which wereonly loosely based on the syllabus, if at all. Ofcourse, boys being boys, we tended to donothing for five days and then rush at themfor two days, but even so it was a very valuableexperience. Such a thing was rare then and Iam afraid it is even rarer now in the days ofschool league tables and the like.

Then I got into Trinity and fell under thespell of Dr Bollobás another believer inhard questions off the syllabus. So through-out my education I always had strong andgood influences.

You had an excellent but not an outstandingundergraduate career?That would be a fair summary.

How did you choose your subject for research?In my third year I did pure mathematics andI gradually narrowed down my preferences toanalysis rather than algebra or geometry.The course in Part III (the Cambridge fourthyear) which I enjoyed most was in the geom-etry of Banach spaces, given by Dr Bollobás,and the thought of being supervised by himappealed to me very much anyway. Lookingback it is amusing to remember how little

idea I had of what research in different areaswould be like when I made such an importantchoice. But I was lucky and found myself inan area that suited me very well, and with anexcellent supervisor.

Just as I was finishing my Ph.D., PeteCasazza visited Cambridge for a year. I hada bit of a lean patch for eighteen months orso around that time, but had a boost to mymorale in the middle of it when, encouragedby him, I solved a problem in the finite-dimensional theory. He used to carry arounda list of unsolved problems in Banach spacetheory, mostly infinite-dimensional, workingup enthusiasm for them. At the time, thereceived wisdom amongst Banach space the-orists was that the infinite-dimensional theo-ry had stagnated, with the interesting prob-lems being inaccessible. He very definitelyfelt otherwise, and had a surprisingly pre-scient view of how things might work out.

The next year I made my main infinite-dimensional breakthrough, solving the so-called unconditional basic sequence problem,which was also solved independently byBernard Maurey, and for some time myresearch programme was obvious manyquestions in the infinite-dimensional theory(including several on Casazzas list) suddenlylooked possible with the new techniques andideas that had been introduced, and over thenext couple of years or so, many of them fell,to me and to others. Towards the end Iobtained my dichotomy result. This neededdifferent techniques, based on results inRamsey theory, though the problem itself wasclosely related to my other work. I was par-ticularly pleased because it was a positiveresult, and I had felt that I was getting a rep-utation as a counter-examples mathemati-cian. Incidentally, many of my counter-examples have resulted from trying to provepositive results its just that theres notmuch you can do about it when one of theseturns out to be false.

I think Maurey is the only mathematician withwhom you have collaborated to the extent ofwriting a joint paper?Yes he is, and even that collaboration aroseout of the accident of our solving the sameproblem at the same time. Although it wasinitially a shock (for both of us) to discoverthat we had to share the result, which wasmuch more important than anything I hadpreviously done, I am now very glad to havecollaborated with Maurey, and proud of ourtwo joint papers. The second of these grewout of the first, and I would definitely nothave been able to do it on my own.

In general, although I have nothingagainst joint work and sometimes resolve todo more of it, I think I work more naturallyon my own. When I discuss mathematics withother people, I often find that either I dontunderstand what they are saying, or they pro-voke some idea in me which I want to thinkabout alone and free of distractions. This isprobably just a sign of inexperience on mypart.

How did you decide to change subject?

Despite appearances, I havent exactlychanged subject, because my Banach spacesresults and what I have done more recentlyboth fall under a general heading of whatmight be called applied combinatorics. Thatsaid, I did make a fairly conscious decision toapply what combinatorial skill I had in areasother than Banach space theory, for severalreasons. First, given that I have only one life,I dont want to spend twenty years of it in asingle area of mathematics, and I cantunderstand those who do. Secondly, I feltunder a certain amount of external pressureto show that I could do more than churn outresults on Banach spaces (not that I myselfregarded the process as churning out).Thirdly, and most importantly, I always had aside interest in combinatorics (not surprising-ly, given my research supervisor) and wantedto solve one or more of the beautiful prob-lems that I had known about for years.

My general approach to research is to tryproblems with a reputation for being diffi-cult, but, in order to avoid the danger of wast-ing years getting nowhere, I like to have sev-eral on the go at once, spending a month ortwo here, a month or two there, until one ofthem reveals a soft underbelly. This proce-dure increases the chances of happy acci-dents. For example, I built up my under-standing of infinite-dimensional Banachspaces by trying and failing to solve the so-called distortion problem, later solved byOdell and Schlumprecht. Without that, Iwould not have seen how to do the otherproblems.

More recently, I decided to think aboutthe Kakeya problem, starting completelyfrom scratch, because I had been told that itwas very important. I hadnt been going longwhen I realised (what I now know is anabsolutely standard observation) that it wasbasically a combinatorial problem aboutarithmetic progressions, which caused me tothink, by no means for the first time, aboutSzemerédis theorem. I then had an idea forhow to tackle the special case of arithmeticprogressions of length 3, and found that Ihad reinvented Roths original proof for thiscase. However, this spurred me on to thinkabout progressions of length 4, and a monthor two later I came up with a complete prooffor the general case. I wrote it out as a fin-ished paper, with all the details. I then dis-covered that a seemingly unimportant lemmawas false, and the hole, like a hole in a pieceof knitting, expanded to the point where Irealised that the main lemma on which my

INTERVIEW

EMS September 19998

Interview with TInterview with Tim Gowers (Cambridge)im Gowers (Cambridge)Fields Medallist 1998interviewer: Tom Körner

proof was based was also false. This was acrushing blow, but by now I was hooked onthe problem. Bollobás was very encouragingat this point and said there must surely besomething in the ideas I had had. In the endI managed, after a big struggle, to get a newproof of the general result. I dont think Iwould have had the stamina for it if it had notbeen for the earlier disappointment. Thewhole thing occupied me for two years andhas led on to another clear research pro-gramme which should keep me going forsome time. Meanwhile, I still have no ideahow to solve the Kakeya problem.

Does it bother you that the things you work onare unfashionable?Not from a personal point of view, because ithas not impeded my career, but I have beenlucky. There are others in my sort of areawho are bothered, and with some justifica-tion. I do think it is good to have to workhard to interest other mathematicians thesituation becomes unhealthy when it is eitherimpossible to do so because ones area is com-pletely out of favour, or too easy because it isthe latest craze.

The unfashionability of combinatorics ispartly a result of the familiar contrastbetween elementary methods and bigmachinery. Twentieth-century mathematicshas seen many triumphs of the latter, and thishas naturally influenced peoples opinions. Ithink also that if you get used to big machin-ery, then problems which can be stated to aschoolchild begin to seem babyish. The stan-dard answer of a combinatorialist to such anattitude would be Why not have a go at solv-ing one?, but this does not convince every-body.

Combinatorics has a reputation for beingrather isolated, with few applications to therest of mathematics, or at least importantmathematics. I think this view rests on a mis-apprehension. Combinatorics does not con-tain many powerful and difficult generalresults like the Atiyah-Singer index theorem,which can be directly applied over and overagain. However, over the years, combinatori-alists have built up a considerable expertisein solving certain types of problems whichwould have been hopeless a generation agoand which can be used in many externalfields. Although progress in mathematics canbe a result of truly understanding difficultconcepts until the proofs write themselves, Ibelieve that it is not sufficiently recognisedthat many problems are at heart combinator-ial, and in the end if you want to solve themyou simply have to get your hands dirty.

What is your attitude to teaching?I enjoy teaching, not all the time of course,but I find it sufficiently rewarding to wish togo on doing it. I enjoy the process of work-ing out why something that I understandcauses difficulty to a student. (Unfortunately,the answer is often boring they just haventlearnt the relevant definitions or somethinglike that.) I am fortunate enough to supervisea number of the very best undergraduates inthe country, who are sometimes better than Iam at what I am supposed to be teachingthem.

I find that lecturing can be directly benefi-cial to my research, because I have to under-stand even quite elementary material muchbetter if I am going to stand in front of twohundred intelligent people and explain it to

them. Bollobás brings up his students not touse notes, which I think improves lecturesimmeasurably. As he puts it, how can weexpect our audience to learn several coursesfor an examination if we, who are supposedto be experts, cannot even learn a small frac-tion of one? It is hard work, but there isnothing quite like the feeling of having suc-cessfully given a complicated lecture frommemory.

Teaching is of course very important,since our future audience depends on it. Itirritates me that many writers seem to treatbooks and papers more as an opportunity todisplay their own knowledge and achieve-ment than as a genuine attempt to convey itto their readers that is, to teach it. In par-ticular, it is very common for proofs to bepresented over-neatly, so that, although onecan see that the steps are correct, one has towork much harder to understand how any-body could have thought of them. Just a fewwell-placed remarks can make an enormousdifference (Actually, this is a natural thing todo because ...) but they are surprisingly rare.Many papers, or chapters in books, could dowith much longer introductions, setting thescene, explaining why the results are interest-ing and explaining the difficulties to be over-come. But I am not the first to air these com-plaints. I firmly believe that one should aimnot just at a specialist audience, but also atthose who would prefer to skim a paper andget some idea of what is happening withoutworrying about the technical details.

We have tended to talk about Cambridge.What about your time elsewhere?Although my heart has always been inCambridge, I had a very happy and produc-tive four years at University College London,and am glad to have experienced life else-where (though never a life without cloisters).The department suited me well, as there wereseveral people there David Preiss, Keith

Ball, David Larman, Ambrose Rogers withinterests similar to mine. I used to commutefrom Cambridge, and found the train a con-genial place to work, making at least one gen-uine breakthrough on it.

When did you first think that you might win aFields medal?Initially, I assumed that it was impossible toget one for work in Banach spaces, which atleast saved me the bother of thinking aboutit. However, about two years before the lastInternational Congress I started getting mys-terious e-mail messages asking for lists ofpublications, copies of papers, and so on.The messages were usually labelled urgent,but did not explain why they were urgent.Even then, it was a long time before I daredto wonder whether I was being considered.

When did you realise that you wanted to be amathematician?It was a gradual process. It was the subject Ienjoyed most at school, partly because Ifound it easiest. The system laid out a seriesof hurdles in ones path and to some extent Ijust found that I had jumped the hurdlesrather than being forced to find another job.I became certain that I would like to be a pro-fessional mathematician some time when Iwas an undergraduate though, even then, Ihad little idea of what this meant.

And when did you feel that you had become amathematician?Several times. At almost every hurdle, in fact.When I became a research student. When Igot a research fellowship. But when I solvedthe unconditional basic sequence problem,which was a problem with a name, which oth-ers had tried, then I felt for the first time thatI had truly fulfilled a boyhood ambition.

We thank the London Mathematical Society for permission toreproduce these photographs

INTERVIEW


Seven Fields Medallists at the opening of De Morgan House, the new headquarters of the LondonMathematical Society (see EMS Newsletter 31). From left to right: Tim Gowers, Klaus Roth, SimonDonaldson, Sir Michael Atiyah, Alan Baker, Richard Borcherds and Daniel Quillen.

ICIAM 99

in Edinburgh5 9 July 1999

Rolf Jeltsch (President of EMS)

More than 1500 applied mathematiciansgathered from all over the world for a weekin Edinburgh. They came to attend the 4thInternational Congress on Industrial andApplied Mathematics. ICIAM99 followed thetraditions of its successful predecessors inParis (1987), Washington (1995) andHamburg (1995).

The opening ceremony was held in thebeautiful McEwan Hall, a semicirculararena with fantastic paintings on the ceil-ing of the half dome. Unfortunately,H.R.H. Prince Philip, Duke of Edinburgh,a Joint Patron of the Congress, couldntattend the opening as he was attending aceremony to honour the late King Husseinof Jordan. Instead, a message from himwas read and he was able to join the con-gress dinner in the Playfair Library at theOld College of the University ofEdinburgh. Sir Michael Atiyah, Chair ofICIAM99, responded to the openingspeech and Lord Sainsbury, UK Ministerof State responsible for Science, welcomedthe delegates. The Celtic Brass opened theceremony which announced the four newCICIAM prizes, created since ICIAM95 inHamburg. The president of CICIAM,Reinhard Mennicken, presented the prizesto the winners.

Jacques-Louis Lions was awarded theLagrange prize in recognition of hisexceptional contributions to applied andindustrial mathematics throughout hiscareer. He was cited as one of the mostdistinguished and influential scientists ofthis century in the domain of applied andindustrial mathematics. A few of his out-standing contributions to our science andhis famous books were mentioned. Lionsfounded and developed an importantschool of applied mathematics in Francewhich has had a strong influence in manyother countries. He has participated inmany industrial programmes. With thechoice of Jacques-Louis Lions the prizecommittee has set an extremely high stan-dard for future winners. The Lagrangeprize is a gift of the Société desMathématiques Appliquées et Industrielles

(SMAI), the Sociedad Espanola deMathematica Aplicada (SEMA) and theSocieta Italiana di Mathematica Applicatae Industriale (SIAMAI).

The Collatz prize 1999, awarded to ascientist under 42 years of age, went toStefan Mueller for his highly original andprofound contributions to applied mathe-matics, the calculus of variations and non-linear partial differential equations, themechanics of continua, and mathematicalmaterial sciences. Mueller, born in 1962,studied mathematics and physics in Bonn,Edinburgh and Paris. He became full pro-fessor at the University of Freiburg at theage of 32, and shortly after, Vice-Directorof the famous mathematical ResearchCentre at Oberwolfach. After a briefappointment at ETH Zurich, he becameone of the three directors of the MaxPlanck Institute for Mathematics in theSciences in Leipzig in 1996. StefanMueller is one of the very few young math-ematicians in the world who combine high-quality mathematical skills with a feelingfor real world problems. The Collatz prizewas sponsored by the Gesellschaft fürAngewandte Mathematik und Mechanik(GAMM).

The CICIAM Pioneer prize was awardedto Ronald R. Coifman of Yale University andHelmut Neunzert of the University ofKaiserslautern for very different contribu-tions to applied mathematics. Coifmanwas honoured for his pioneering work inexploiting harmonics, and especiallywavelet analysis, to provide computationalmethods and algorithms in a wide varietyof important contexts involving signal andimage processing. Applications haveincluded FBI data files for fingerprints andmany other problems involving compres-sion and/or restoration of images andsound. Who has not seen, when clickingon a webpage, how the images on the pageare built up from large scale wavelets tofine scales in seconds? Neunzert was hon-oured for his work over the last twentyyears in developing technomathematics,both as a scientific discipline and as a cur-riculum now offered at more than twenty-five universities, and in developing thespecialisation of industrial mathematicsthrough active consulting and modelling,playing a leading role in the EuropeanConsortium for Mathematics in Industry,and for founding and directing theFrauenhofer Institute for Techno- andEcono-mathematics at Kaiserslautern.[You can read more about HelmutNeunzert in an interview he gave to HeinzEngl in the June 1999 issue of the EMSNewsletter.] The Pioneer prize is funded bythe Society for Industrial and AppliedMathematics (SIAM), and is given for pio-neering work introducing applied mathe-matical methods and scientific computingtechniques to an industrial problem areaor a new scientific field of applications.

Grigory Issakovic Barenblatt was awardedthe CICIAM Maxwell prize in recognitionof outstanding originality in his work inapplied mathematics. He is one of themost distinguished Russian applied math-ematicians, and is well known for his

important contributions to the mathemati-cal theory of fluid motion, solid structure,non-linear waves, scaling and asymptotics.He constructed a deep connection betweennon-linear waves and general scaling argu-ments. Among the many applications ofthis highly original and amazing theoryare the scaling of turbulence, the analysisof failure in solids, the dynamics of reser-voirs and the analysis of stratification ingeophysical fluid mechanics. The Maxwellprize is sponsored by the Institute ofMathematics and its Applications (IMA)and the James Clerk Maxwell Foundation,to provide international recognition to amathematician who has demonstratedoriginality in applied mathematics.

These prizes will be presented everyfour years at the ICIAM. Clearly, morethan through their official descriptions,the world standing of their first winnerswill set the level of achievement for futurewinners. The presentation of these prizesconcluded the opening ceremony.

The conference was one of superlatives and not only with respect to these prizesand their winners. In one week there were33 plenary speakers and about 210 mini-symposia (usually consisting of 3-4 lec-tures, but some running up to 26 lectures),and about 1100 contributed papers werepresented. Everything ran in parallel,even the plenary sessions, and sometimes28 mini-symposia were running simultane-ously, so you cannot expect your corre-spondent to tell you all the highlights; itwas simply impossible to attend and absorball that was presented. The few lectures Idid attend were selected by personal inter-est, curiosity, or just by chance. But theproportion of excellent lectures wasextremely high. The two Schlumbergerlectures I attended by F. P. Kelly onMathematical modelling of the internet and S.Popescu on What is quantum computation?were real eye-openers for me. MargaretWrights lecture on What, if anything, is newin optimization? and James A. Sethians pre-sentation on Fast marching methods and levelset methods: evolving interfaces in fluidmechanics, computational geometry and materi-al sciences really surpassed their usual(already fascinating) performances.Among the mini-symposia, I would like tosingle out the one on Computational scienceand engineering: How to organize? How toteach?, because this new curriculum willhave to be discussed in the near future bymany mathematics departments.

For those with insatiable appetites whofound the presented material insufficient,there were additional scientific events likethe Symposium on mathematics and the lawand a meeting on Maxwellian themes withtopics like Testing Einstein in space: amarriage of physics and technology or AMaxwellian approach to modern cosmolo-gy. ICIAM99 really covered all the fieldsone could imagine and even ones onecouldnt.

Three additional prizes were awardedin the McEwan Hall, between two morningplenary lectures.

The Dahlquist prize, established bySIAM in 1995, was awarded to Linda

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EMS September 199910

Petzold for her important contribution toeffective numerical methodology for dif-ferential equations, especially the analysisof methods for differential algebraic equa-tions, the construction of effective tech-niques for their solution, and the integra-tion of these and other techniques intorobust software, thus making possible thereliable solution of large classes of ordi-nary and partial differential equations aris-ing from engineering and science applica-tions. This prize is awarded on a biannualbasis.

Germund Dahlquist received the newlyestablished Henrici prize. He was cited forhis outstanding research and leadership innumerical analysis. Dahlquist created thefundamental concepts of stability, A-stabil-ity and the non-linear G-stability for thenumerical solution of ordinary differentialequations. He succeeded, in an extraordi-nary way, in relating stability concepts toaccuracy and proved the deep results thatare nowadays called the first and secondDahlquist barriers. His interests, likeHenricis, are broad, and he has con-tributed significantly to many parts ofnumerical analysis. As a human being andscientist, he gives freely of his talent andknowledge to others and will remain amodel for many generations to come. Thiswas the official citation; I wish to add that Imyself benefited a lot from his generosity.The Henrici prize is jointly sponsored bySIAM and The Swiss Federal Institute ofTechnology, ETH Zurich.

Matteo Frigo and Steven Johnson of theMassachusetts Institute of Technologyreceived the Wilkinson Prize for FFTW,the fastest Fourier transform in the West,a C library for the computation of the dis-crete Fourier transform that automaticallytunes the computation for any particularhardware in order to produce efficientcode. The Wilkinson prize for numericalsoftware was established in honour of theoutstanding contributions of James HardyWilkinson to the field of numerical soft-ware. This prize is jointly sponsored byArgonne National Laboratory, the

ICIAM99


National Physical Laboratory, and theNumerical Algorithms Group (NAG).

You can find more information on allprizes, the exact citations, the prize com-

mittees and the whole scientific pro-gramme of ICIAM99 on the webpage:http://www.ma.hw.ac.uk/iciam99/.Finally, I should add that the EMS was pre-sent at ICIAM99 with a booth, jointly withZentralblatt MATH. It gave the Chair of theApplications Committee (Heinz Engl) andmyself a good opportunity to meet mem-ber societies and their representatives aswell as other societies; you will be readingabout some of the results from these talksin future Newsletters. One effect can benoted in this issue: SIAM and EMS areplanning a cooperation agreement andyou will find an article by the SIAM presi-dent, Gil Strang below. A correspondingarticle by myself will appear in theSeptember issue of SIAM News.

On Saturday the Committee forInternational Conferences on Industrialand Applied Mathematics (CICAM) had abusiness meeting. It prepared the electionof the new president, to be done by mailthis autumn. It also established newstatutes. We shall report on these newdevelopments in the next Newsletter.

We look forward to the next ICIAM, tobe held from 7-11 July 2003, in Sydney,Australia.

SIAM and EMS joint conference onComputational Science

Gil Strang, SIAM PresidentI am writing to introduce SIAM, the Society for Industrial and Applied Mathematics.This is an especially good time to add new connections between SIAM and the EMS,because our two societies are planning a joint conference. It will be the first majorconference for SIAM in Europe and the first big conference on Computational sci-ence for the EMS. Rolf Jeltsch, EMS President, is preparing a similar column forSIAM News, to introduce the EMS to members of SIAM.

I hope you have seen SIAM News, which goes to all members and is very widelyread and above all the SIAM journals which are at the centre of our work. Thereare now eleven journals covering a wide range, from classical problems in analysisto discrete mathematics and optimisation. Maybe this is an important point aboutSIAM it has a very broad view of what applied mathematics really is.

Another important point about SIAM is that it is truly international, as mathe-matics is. More than a third of our members live and work outside North America.I am personally very happy about that, and am working in my two years as presidentto do everything possible for applied mathematics worldwide. So much of the bestresearch is joint work we are already working together! I have had total encour-agement in this cooperation from inside and outside SIAM.

Let me mention some new directions for the society? One is to establish an activ-ity group in the life sciences. Biology, medicine and pharmacology are areas oftremendous scientific growth, and the mathematical and computational parts areincreasingly important. The activity groups in SIAM are responsible for focussedconferences every two years and I believe that this new group will grow quickly. Theenvironment is also a part of their interest. The linear algebra group has been par-ticularly active in Europe, and I anticipate a new group in 2000 on imaging science.

May I call your attention to our web page: www.siam.org. It has much more infor-mation about the society than I can give here. You can see the calendar of confer-ences, and how to join SIAM, and the editorial boards of all the journals (which nowcome electronically before the printed editions). I am happy to give my own email([email protected]) for anything I can do.

One more word about our joint conference. The first discussion was with RolfJeltsch and Heinz Engl. The whole topic of computational science and engineeringis coming into prominence for students too. The key is to know about the scienceas well as the mathematics (and the computation). I am convinced that the confer-ence will be a success.

Jacques-Louis Lions, winner of the Lagrange prize, at the 1994 ICM.

Late last June, an electronic message witha similar title was sent to the presidents ofall the member societies of the EMS. It wassigned by Rolf Jeltsch, president of theEMS, as well as by the presidents of the twoFrench Mathematical Societies, the SMFand the SMAI. Its purpose was to presenta new French initiative for WorldMathematical Year 2000, and to urge allmathematicians in Europe and outside touse this new service. Insofar as such a toolis useful only if it is used by many, and asits usefulness increases exponentially withthe number of sites it indexes, we presentit in the columns of the EMS Newsletter andurge all EMS members to take a look at it.

What is ACM/UMRL? The original name of this service isAgenda des Conférences Matématiques(ACM for short, not to be confused withother ACMs on the web); its web address is<http://acm.emath.fr>. It was createdeighteen months ago at the initiative ofStephane Cordier, a Maitre deConferences at the University of Paris 6and an active member of the SMAI, tocover France. ACM has been supportedfrom its onset by a small grant from theFrench Ministry of Research, with theunderstanding that it should be madefreely available everywhere. Its success inFrance has been such that the idea ofextending it to the whole of Europe andelsewhere has become realistic, and this iswhat the EMS, the SMAI and the SMF aresupporting for the year 2000. Since theacronym ACM has many meanings (amongothers, the Association for ComputingMachinery), the following web name isproposed: ACM/UMRL, or even UMRL,which could stand for UniversalMathematics Resources Locator.

At the core of ACM/UMRL is a databaseof seminar series, conferences and collo-quia in mathematics, including dates andtimes, locations, titles, speakers, and

(optionally) key words, MSC2000 codes,etc. We present it here from the end-userspoint of view, and from that of the organ-iser of a series of lectures or a conference.

ACM/UMRL for the end-userOn the users side, UMRL appears as theinterface for a search engine. It can bequeried in English and French (this can beextended to other languages with helpfrom native speakers). It can be customisedto each users profile(s), which can be savedas Bookmarks or Favorites and re-used atwill. Simple and complex searches arepossible, including geographical regionsand time periods. Although setting up aprofile can be lengthy the first time (*), itis worth while because once bookmarked itperforms new searches in a single step(and keeps the date as relative, notabsolute). The result of a search is a list ofall the seminar talks, symposia talks andconferences in the database which satisfythe search criteria, with their titles andlinks to their web sites (abstracts, direc-tions, etc.). One can even ask for a timelye-mail reminder.

At present, some sixteen geographicalareas are used, as well as ten general sub-fields of mathematics that can be selectedfor a focused search. Each talk can alsopresent a list of key words and MSC2000codes that can be used in the searches.

The database includes information onmathematical conferences, congresses,workshops and colloquia from the confer-ence calendar maintained by the Atlas ofMathematical Conference Abstracts(AMCA) at http://at.yorku.ca/amca/. It alsoincludes the information located on theEMIS conference board. We suggest thatyou give it a try at http://acm.emath.fr.ACM/UMRL has been expanding regularlyin Europe (Austria, Germany, Italy, theNetherlands) and North America, and asimple query can certainly be instructive.One can easily imagine the many uses that

such a large database can have. But how isthe ACM/UMRL database maintained?

ACM/UMRL for the organiser of a confer-ence, colloquium or seminar seriesHere is the key concept: the ACM/UMRLdatabase gets created and maintainedautomatically, using a web-engine thatqueries the web pages of all seminar series,conferences or colloquia registered withACM/UMRL. This requires the activecooperation of the organisers of theseevents, but that is straightforward: theURLs of the announcement web pageshave to be indicated to the ACM/UMRL-master and these pages must include somevery simple tags that the ACM/UMRLengine can recognise and retrieve.Another advantage of this procedure isthat the database always remains up-to-date. Detailed explanations are availableon the ACM/UMRL server <http://acm.emath.fr> in English, German, Italian,Spanish and French (so far!).

How can ACM/UMRL grow? ACM has recently been expanding to coun-tries other than France. Its user interfacehas been translated into several languagesand it is hoped that more will be included.National and regional correspondents arealso needed; some are already in place,from France, Austria, Belgium, Germany,Italy, the Netherlands and Canada.Mirrors will also be most welcome fromeverywhere possible; a mirror is plannedon the EMIS servers.

For those locations that have no webpage, it will be possible in France to postannouncements on a special web-site thatwill automatically be queried by theACM/UMRL robot. This type of procedureshould be considered at country level, pos-sibly by the national mathematical Society.As the geographical coverage increases,more numerous and focused regions willbe included. If the need arises, more sub-fields of Mathematics can be defined, butthe searches can always be made via key-words and MSC2000 codes.

Obviously, ACM/UMRL is an evolvingproject and we hope that by being a com-mon tool for the mathematical communityit can evolve to satisfy the needs of all andto help foster the sense of a global mathe-matical community.

One final note: ACM/UMRL does notintend to compete with other sites thatpropose similar information. On the con-trary, it is willing to incorporate all infor-mation that resides on these sites, withdirect reciprocal links with them. Anexample of such an existing cooperation iswith ACMA, as indicated above.

(*) At present, a bug in the way Netscape v.4.6handles long URLs (and the bookmarks createdby ACM/UMRL are long indeed) appears to cor-rupt bookmarks when one tries to rename them.If this happens to you, try not to rename thebookmark. This was only noted on the Macplatform.

RESOURCES LOCATOR


A universalA universalmathematicalmathematicalrresouresources ces locator?locator?Alain Damlamian Past-President Société de Mathématiques Appliquées et Industrielles (SMAI)

The Norwegian surveyor Caspar Wessel isnow recognised as the first to have given ageometrical interpretation of complexnumbers and their rules of composition.This year we celebrate the bicentenary ofhis paper in Det Kongelige DanskeVidenskabernes Selskabs Skrifter, Nye Samling,V, Kiøbenhavn, 1799, pp. 469-518 [the col-lected papers of The Royal DanishAcademy of Sciences and Letters].Although his work was ignored and had noinfluence on later mathematical develop-ments, his story is worth telling.

In 1796 Wessel completed the triangu-lation of Denmark and Schleswig that,together with astronomical observations,formed the basis of the first real cartogra-phy of Denmark; this work was done underthe auspices of the Academy. In the sameyear Wessel wrote a mathematical treatiseentitled Om Directionens analytiskeBetegning, et Forsøg anvendt fornemmelig tilplane og sphæriske Polygoners Opløsning (Onthe analytical representation of directions;an attempt applied chiefly to solving planeand spherical polygons). Its first part dealswith directions in a plane, and the second,

Caspar Wesselon representing complex

numbers (1799)Bodil Branner

less well-known, part deals with directionsin space. The treatise was presented at ameeting on 10 March 1797 to the Academyand accepted for publication. From thestart (1742) the Academy had decided topublish articles written by members inDanish, not Latin. The statute of theAcademy was relaxed in 1796 to allow non-members to submit articles and Wesselstreatise became the first of this kind to bepublished.

At the end of the 18th century there wasnobody around who could understand orappreciate the scope of Wessels work. Ifthere had been, the paper would probablyhave been translated into Latin or Germanas other important publications were.There were many people of German originin Copenhagen at the time, such as J. N.Tetens, who had come to Copenhagenfrom a professorship at Kiel University inphilosophy and mathematics. He encour-aged Wessel to write his results down and,as the leader of the mathematical class ofthe Academy, he presented Wessels workat the meeting. Wessel himself was not pre-sent.

Soon after, Wessels treatise seems tohave been completely forgotten. It wasrediscovered about a century later whenSophus A. Christensen mentioned it in hisdoctoral thesis (On the development of mathe-matics in Denmark and Norway in the 18thcentury). The three opponents at his

defence were professors at CopenhagenUniversity, two official (H.-G. Zeuthen andJulius Petersen) and one from the audi-ence (T.-N. Thiele). Immediately after,Christian Juel, docent at the TechnicalUniversity, called attention to Wesselsachievement by publishing a short paper(in Danish) about it and Sophus Liereprinted Wessels paper in the Archiv forMathematik og Naturvidenskab in 1895. TheAcademy honoured Wessel by publishing aFrench translation of his treatise by H.-G.Zeuthen in 1897. Last year the Academyheld a Wessel Symposium organised byJesper Lützen, and this year a completeEnglish translation of Wessels treatise willappear (for the first time) in the proceed-ings [1], together with a Wessel biographyand a paper on the history of complexnumbers.

Caspar Wessel was born in 1745 inVestby in Norway, south-east of Christiania(now Oslo). He was the sixth child (of 14)of the curate Jonas Wessel and his wifeHelene Marie Schumacher. When he was12 years old he was sent to the CathedralSchool in Christiania with two of his elderbrothers. There was no university inNorway, so the three brothers moved toCopenhagen, the capital of the dualmonarchy Denmark-Norway, to study atthe university there, the elder brothersJohan Herman and Ole Christopher in1761 and Caspar in 1763. They all hadsome financial support from home to startout with, but it soon became necessary tofind extra income. Both Ole Christopherand Caspar became involved with survey-ing for the Academy. Ole Christopherearned his living this way while he was astudent; after taking a degree in law in1770 he embarked on a successful legalcareer, attaining one of the highest posi-tions in Norway. For Caspar surveyingbecame an engagement for life although,like his brother, he completed a degree inlaw. Their brother Johan Herman Wesselbecame a poet. Although he was not veryproductive and died rather young, many ofhis poems are still remembered in Norwayand Denmark; they have a special humourand are hard to translate.

The first plan for a modern topograph-ical measurement of Denmark was pro-posed to the King in 1757 by a student,Peder de Koefoed. The plan was approvedand Koefoed started to work, but unfortu-nately he died three years later havingaccomplished little. But the time was ripeand the way prepared for a more ambitiousplan put to the Academy by Christen Hee,professor in mathematics, and ThomasBugge (1740-1815), Koefoeds assistant.The plan involved both topographical sur-veying and the method of triangulation todetermine geographical coordinates. Thislong-term project of great national (andinternational) interest became essential forthe development and strengthening of theyoung Academy.


ANNIVERSARIES

1999 Anniversaries

Title page of Wessels 1799 treatise.


ANNIVERSARIES

Page 50 of Wessels 1787 surveying report, Trigonometriske Beregninger; in Section 5 he expresses the direction of the nth tangent vector in Fig. 3 by Tn.(cos wn + √-1. sin wn).

Hee and Bugge educated the two firstsurveyors, one of whom was OleChristopher Wessel, and the work startedin 1762. Two years later Ole Christopherneeded an assistant and chose Caspar. Atfirst Bugge was responsible for everythingincluding the trigonometrical measure-ments (the triangulation), the most theo-retical part of the surveying, but OleChristopher Wessel soon took over thisresponsibility as well. However, his salaryas an assistant was so low that Caspar couldnot survive on it alone, and he asked theAcademy for a rise if he took on the draw-ing of maps in addition to his surveying.He was granted increased pay and maderesponsible for the construction, reductionand drawing of maps based on the geo-graphical and trigonometrical surveying,then presenting the model from which thecoppersmith would make plates and pro-duce the final prints. A test map hadalready appeared, representing theCounty of Copenhagen; officially it hadbeen drawn by Ole Christopher Wessel,but from the surveying diary we know thatCaspar had drawn part of it.

From May to September or October, thesurveying took place in the countrysidefrom early morning to late evening (weath-er permitting) every day except Sundays. Afair copy of the surveying diary had to besent to the Academy, together with thetopographical maps measured that sum-mer, marking the locations of towns,churches, castles, mills and woods, thecourses of roads and streams, and the posi-tions of coastlines and islands. During thewinter there should then be time for study,but for Caspar Wessel this proved difficultsince he was then working on the reductionof maps. By 1778 he had became so des-perate that he asked for a sabbatical yearwith full salary. This was not popular sincehe had just accepted responsibility for the

trigonometrical surveying, to succeed thesuccessor of Ole Christopher. Neverthe-less, Hee wrote a supporting letter: None ofthe surveyors has been more useful to us than hehas, during the summers he has been surveyingand in the winter time he has been working as adesignator, which in the fourteen years he hasstayed with the surveying has ruined his healthand been an obstacle to his studies in such away, that if he once again has to interrupt hisstudies he is perdu and will never pick them upagain. Last winter when he nolens volens had todraw the general map of Zealand he was oncemore distracted in his studies, and then Ipromised him never again to disturb his circles.The sabbatical was granted and Wessel fin-ished his degree in law. No trigonometri-cal surveying took place that year.

From 1779 to 1796 he worked as atrigonometrical surveyor, still spending hissummers in the countryside measuring.During the winters he was occupied withtrigonometrical calculations based on thecollected data, judging the validity of themeasurements, and constructing anddrawing the resulting triangular maps. Hisreports also contained shorter articlesdescribing the methods behind the mea-surements and cartography. His workrequired both practical and theoreticalskills, as well as accuracy, patience and abreadth of outlook.

The purpose of the trigonometrical sur-veying was to establish a network oftrigonometrical points, to determine theirtriangular net, and to supplement this withan astronomical determination of the lati-tude and longitude of some of the pointsand of the direction of the meridianthrough such points. One distance, a baseline, between two neighbouring points inthe triangular net had to be measured withgreat care; all angles of all triangles in thenet were measured several times and fromthese data the rest of the distances were

calculated. Angles were measured by theso-called geographical instrument or Ekströmscircle. For trigonometrical surveying theinstrument was used with the circle in hor-izontal position and placed above onetrigonometrical point of a triangle, withone telescope pointing towards one of theother points in the chosen triangle and theother telescope in the direction of the thirdpoint in the triangle. The instrument wasalso used for astronomical measurementswith the circle in vertical position, measur-ing the height of the sun or certain stars.

The origin of the triangular net hadbeen chosen as the Royal Observatory inCopenhagen placed in the Round Tower.The building of the Round Tower wasstarted in 1637 and the observatory wasestablished by Longomontanus, a pupiland former assistant of Tycho Brahe(1546-1601). Brahe had been the first touse a triangular net in surveying, to deter-mine the position of his observatoryUraniborg on the island Ven relative to thecoastlines of Sweden and Denmark. Themethod was later pioneered by the FrenchAbbé Jean Picard who visited Denmark in1671-72 in order to compare the longi-tudes of the new Paris Observatory andUraniborg and the Round Tower. Whenthe trigonometrical surveying of Denmarkstarted in 1762, that of France had alreadybeen completed, forming the basis of thefamous Cassini-maps.

From May 1782 to the summer of 1785Wessel was again on leave from theAcademy, but this time with its recommen-dation. He was made responsible for thecomplete trigonometrical surveying of theduchy Oldenburg West of Bremen. In a let-ter of recommendation, Bugge wrote in1781, He possess a lot of theoretical knowledgeof algebra, trigonometry and mathematicalgeometry, and as far as the last point is con-cerned, he has come up with some new andbeautiful solutions to the most difficult problemsin geographical surveying. The Oldenburgarea was later (around 1824) re-triangulat-ed by Gauss; his instrument and part of histriangular net can be seen on German 10DM notes.

In the 1779 surveying report Wesselexplained how a map of an ellipsoid can beobtained by projecting points of the ellip-soid onto a cone and unfolding. The conehe used for Denmark was formed by thetangents to the meridians through pointswith the same latitude as the Observatory.When he returned from Oldenburg heelaborated on this method and in his 1787report he described how he came closer tothe actual measurements by using severalcones, each one over points of a fixed lati-tude; in the unfolding he pieced the dif-ferent cones together along tangents to acommon meridian. In his description ofhow to calculate the coordinates of a pointin the unfolding, he expressed a directionin the plane by T(cos w + √-1.sin w). So atleast as early as 1787 he had the geometricinterpretation of complex numbers asdirections in a plane. There is no trace ofthe product rule in this report. But know-ing that he was a master in handlingtrigonometrical formulas and noticing how


ANNIVERSARIES

The geographical instrument shown in vertical position, from Thomas Bugges Beskrivelse overden Opmaalings Maade som er brugt ved de Danske geografiske Karter, Kiøbenhavn, 1779.

he wished to change from a direction givenby (cos wn + √-1.sin wn) to one given by(cos wn+1 + √-1.sin wn+1) through a turn ofthe angle of (wn+1 - wn), it is possible thathe obtained the geometrical interpretationof the product rule by pursuing this fur-ther.

As already explained, he wrote his oneand only mathematical treatise in 1796. Itwas clearly inspired by his work as survey-or, but there is no reference to this, andthe results he obtained did not simplify thetrigonometrical calculations in the survey-ing. In part two of his treatise he expresseddirections in space by refering to twoorthogonal complex planes with the realaxis in common. Although his notationmay look rather clumsy and the resultsseem to drown in pages of symbols, there isan underlying simple and elegant geomet-rical idea which enabled him to obtain auniversal formula from which he derivedall the spherical trigonometrical formulas.

In 1805 Wessel resigned. But he stillworked for the Academy when they neededhis help; for instance, in 1808 he drew atriangular map of the duchies Schleswig-Holstein and added some explanations inFrench, since the maps were requested bythe French Emperor Napoleon and sent toDépôt général de la Guerre in Paris. In1815 he was made a knight of theDannebrog in recognition of his excep-tional contribution to surveying. He diedin 1818.Acknowledgement: This article is based onjoint work with Nils Voje Johansen; ourWessel biography in [1] contains a list ofthe material used.

Bibliography1. Caspar Wessel, On the analytical represen-tation of direction. An attempt applied chiefly tosolving plane and spherical polygons (translat-ed by Flemming Damhus). With introductorychapters by Bodil Branner, Nils Voje Johansenand Kirsti Andersen (edited by BodilBranner and Jesper Lützen, C.A. Reitzel,1999, ISBN 87-7304-298-6.

2. Olaf Pedersen, Lovers of Learning. AHistory of the Royal Danish Academy ofSciences and Letters 1742-1992.Munksgaard, Copenhagen, 1992, pp. 89-104 & 119-132.

Bodil Branner teaches in the Department ofMathematics, Technical University of Denmark.

On 1 June, around 100 people came to theMathematical Institute in Oxford to com-memorate the 100th anniversary of thebirth of Edward (Ted) CharlesTitchmarsh (1899-1963). This gathering,organised by David Edwards, includedmany of his former colleagues and fourgenerations of his family, including two ofhis three daughters. Talks were given onhis life (Robin Wilson) and his work in theareas of Fourier analysis (David Edwards),the Riemann zeta-function (Roger Heath-

E.C. Titchmarsh (b. 1899) Robin Wilson

Brown) and eigenfunctions (NorrieEveritt).

Early YearsThe Titchmarsh family can be traced

back many centuries, in the area to thesouth-west of Cambridge. There is a villagecalled Titchmarsh in Northamptonshire but it was around Royston that E. C.Titchmarshs forebears lived as localtraders. Titchmarshs grandfather ran agrocery shop, but Titchmarshs fatherentered the church and became aCongregational Minister first inNewbury, Berkshire, and later in Sheffield(Yorkshire).

Ted Titchmarsh was born in Newbury,the second of four children. In Sheffield heattended King Edward VII School, wherehe performed well in most subjects. Helater wrote: The first occasion on which Idistinguished myself was when I was in oneof the fourth forms. The headmaster forsome unknown reason made the wholeupper school do an arithmetic paper, thesame for all forms. The mathematical spe-cialists in the sixth form came out top, andI came next . . . At this point one had tochoose either classical or modern subjects:I was put on the classical side. I learntenough Latin to pass and enough Greek tofail. It had become clear that mathematicswas my real subject, and I began to spe-cialise in it.

In December 1916 Titchmarsh won theOpen Mathematical Scholarship to BalliolCollege, Oxford, going up in October1917 for a term. However, he then wentaway for almost two years on war service, asSecond Lieutenant in the Royal Engineers(Signals), going to France from August1918. He became a dispatch rider, onhorse and then motorcycle. He returned toOxford in October 1919, being tutored byJ. W. Russell. Mary Cartwright has written:At Russells first lecture the room waspacked to the doors, and Russell said: Ah,theres my clever pupil Mr Titchmarsh he knows it all, he can go away. Russelldictated his lectures word for word andexamples were handed out and then, ifnecessary, solution to examples. Some ofTitchmarshs solutions replaced the officialones. Certainly his student career was verysuccessful, and he gained a First ClassHonours Degree and won the Junior andSenior mathematical scholarships. Later,

Ted Titchmarsh was to write: I was howev-er principally influenced by G. H. Hardy.From him I learned what mathematicalanalysis is, and at his suggestion I devotedmyself to research in pure mathematics.

Oxford in the 1920sHis most important contemporary in

the 1920s was indeed Hardy, who had leftCambridge in 1920 to succeed WilliamEsson (former deputy to J. J. Sylvester) asSavilian Professor of Geometry in Oxford.Hardy was the founder of the Oxfordresearch school in analysis, a tradition thatTitchmarsh followed ten years later whenhe succeeded Hardy as Savilian professor.Other Oxford contemporaries includedEdwin Elliott, the first holder of theWaynflete Chair in pure mathematics, whowrote important books on invariant theorybut had no sympathy with foreign modernmethods; W. L. Ferrar later describedElliott: a man who has written books whichhave put the works of his rivals on thebookshelves is the worst lecturer who everpicked up chalk, Elliott retired in 1921,being replaced by Alfred Lee Dixon whomade few mathematical innovations butknew some very pretty things in 19th-cen-tury geometry; Dixon, a keen sportsman,was so strong that he could break a walnutin the crook of his arm. On the appliedside, the Sedleian Chair of NaturalPhilosophy was held for over forty years byAugustus Love, who worked in continuummechanics and electrodynamics and wrotea classic book on elasticity. In addition, thenew Rouse Ball Chair of Mathematics wasfounded in 1928 and the first holder was E.A. Milne.

But in pure mathematics, Hardy was themain influence. He invited many foreignmathematicians to Oxford, such as thenumber-theorist Edmund Landau, theRussian emigré Abram Besicovitch, andGeorg Pólya, the first Rockefeller Fellow.Hardys weekly evening mathematicalgatherings were of great interest toTitchmarsh, Mary Cartwright and others,as were the meetings of the OxfordMathematical and Physical Society, found-


ANNIVERSARIES

E.C. Titchmarsh as a young lecturer

One of Titchmarshs favourite occupations was playing criket

ed by Sylvester in 1888 and celebrating its200th meeting in 1925. Hardys influencealso extended beyond mathematics; hisregular cricket matches frequentlyinvolved Titchmarsh, who had a passionfor the game indeed, his uncle was a pro-fessional cricketer.

The 1920s were important years forTitchmarsh. In 1923 he was appointed to aSenior Lectureship at University College,London, giving undergraduate lectures,supervising postgraduates, and starting topublish his research in major journals. Alsoin 1923 he became a Prize Fellow by exam-ination at Magdalen College, Oxford,awarded to graduates of outstandingmerit; he held this position for seven years,and was thereby able to keep in touch withOxford mathematics. Meanwhile, hisfather had become a church minister inEssex and Titchmarsh fell in love withKathleen, the Church Secretarys daugh-ter. Kathleen called him Oxfords most B.M. [brilliant mathematician]. They mar-ried in 1925 and had three daughters.

In 1929 he was appointed Professor ofPure Mathematics at Liverpool University,succeeding Charles Burkill. While there,

he was elected a Fellow of the RoyalSociety, became President of the LiverpoolMathematical Society, and wrote his firstbook, a Cambridge tract on the Riemannzeta-function.

In 1931 Hardy returned to Cambridgeto take up the Sadleirian Chair vacated onthe resignation of E. W. Hobson. Bychance, Titchmarsh was visiting Oxford toexamine a doctorate and bumped intoFerrar who asked him whether hedapplied for Hardys vacant Oxford Chair.Titchmarsh said no, but (encouraged byFerrar) thought that he might. He sent inan application on a single sheet saying thathe wished to apply for the geometry Chairbut could not undertake to lecture ongeometry as Hardy had done. Two dayslater he was appointed and the statuteswere soon changed to say that the SavilianProfessor of Geometry no longer had tolecture on geometry. Writing to OswaldVeblen about the appointment, Hardysaid: I fancy Littlewood [one of the elec-tors] would have preferred Besicovitch; butI expected the electors, with the opportu-nity of taking a genuinely first rate Oxfordproduct, to do so. The man I am unhappyabout is Mordell: first cut out fromCambridge by my decision to stand there,and then here by the Oxford candidate.The Chair was associated with NewCollege, where he held a number of col-lege positions.

Savilian Professor of GeometryDuring his remaining thirty years in

Oxford his research publications contin-ued to appear at a great rate, graduallyshifting from number theory and trigono-metrical series to Fourier transforms,eigenfunction expansions, analysis forphysicists, and the theoretical backgroundto relativistic quantum mechanics. It wasalso during this time that he wrote his best-known textbooks. His tract on theRiemann zeta-function had been written inLiverpool, but was much developed andrewritten twenty years later in Oxford. Hisfirst major book, The Theory of Functionswas later described as the rebellion of ayoung, widely read professor against thenarrow range to which mathematicalanalysis was then so often confined. Laterwritings included classic texts on Fourierintegrals and eigenfunction expansions

associated with second-order differentialequations. His method of writing them wasinteresting; after researching on a topic fora couple of years he would sign off with abook representing the synthesis of his owndiscoveries. He also wrote a popularMathematics for the General Reader.

His achievements became widely recog-nised. He was President of the LondonMathematical Society from 1945 to 1947and received its highest honours the DeMorgan Medal and the Berwick Prize.Sheffield University awarded him an hon-orary doctorate in 1953 and he was a ple-nary speaker at the AmsterdamInternational Congress the next year. TheRoyal Society awarded him their SylvesterMedal in 1955.

On a personal basis, he was a man of fewwords, but his silences were benevolentand never oppressive. While in Oxford hebecame Curator of the MathematicalInstitute, issuing keys to new graduate stu-dents. Sir Michael Atiyah recalls that onarrival in Oxford as a graduate student Iwas duly ushered into his big room, wherehe was sitting at his desk. I sat down and hehanded over the key, and I then expecteda speech of welcome or some words ofadvice, but we just sat in silence. After fiveminutes I left.

In January 1963, suddenly and com-pletely unexpectedly, he died in his arm-chair at home. There were many tributes,as everyone was very fond of him. His col-league Charles Coulson wrote: There weremany things about Ted that I have alwaysmuch admired his utter humility, whichnever betrayed anything but the greatestsimplicity; his complete integrity ... and hiskindness to me when I arrived first; to hisstudents (who worshipped him) and toeveryone. But perhaps the final tribute

can be given in the words of a young visi-tor to his house who, on being told laterthat he had been playing dominoes with agreat mathematician, remarked: Well, hedidnt seem like it. ReferenceMary Cartwright, Obituary of E. C. Titchmarsh, Journalof the London Mathematical Society 39 (1964), 544-565.

We thank Jennifer Andrews for supplying these photographs.


ANNIVERSARIES

Hardy leads a criket team of Oxford mathematicians during a British Association meeting in 1926;Titchmarsh is fourth from the right. Hardy dubbed this photograph Mathematicians v The Rest of the World.

Titchmarsh receiving an honorary degree fromSheffield University in 1953

E.C. Titchmarsh in later life

Societies Corner is a column concerning themathematical societies in European countries.The articles in this column could describe thehistory of a particular society or discuss someevent connected with the society. If you feel thatyour society would interest others, please contactthe column editor, Krzysztof Ciesielski (e-mail:[email protected]) in the first instance.

Urs StammbachThe Swiss Mathematical Society wasfounded in 1910, fairly late in compari-son with similar societies in otherEuropean countries. There were, how-ever, earlier organisations that, at leastpartially, served the needs of Swiss math-ematicians. We first look briefly at theseforerunners.

Forerunner Organisations Before 1800 there were few academicinstitutions in Switzerland, but two hadalready achieved European renown inmathematics by the 18th century. Oneof these was the University of Baslewhich was founded in 1460 and which,mainly due to the various members ofthe Bernoulli family, became one of themathematical centres of Europe in the18th century. The other one was theAcadémie de Genève which was foundedin 1559. It too achieved Europeanrenown in mathematics, due to peoplelike G. Cramer (1704-52) and J.-L.Calandini (1703-58).

When the other universities inSwitzerland were founded in the 19thcentury, mathematics did not, apparent-ly, take high priority. All that wasexpected of the discipline at that time

Swiss Mathematical

Society

was the education of specialist teachersin the subject. This is somewhat surpris-ing in view of the fact that, at the time,there were Swiss mathematicians ofinternational importance. Ch. F. Sturm(1803-55) from Geneva was professor atthe Ecole Polytechnique and theSorbonne in Paris and Jakob Steiner(1796-1863) from Berne was at theUniversity of Berlin. Mention must alsobe made of Steiners friend, LudwigSchläfli (1814-95); although he wasemployed as a lecturer at the Universityof Berne, his pay was so low that hecould not make a living out of his work.Eventually, in 1853, he was made a pro-fessor, albeit with a small salary.

Only at the Polytechnicum in Zurich,founded in 1855, was the situation formathematicians somewhat better. As abasic scientific discipline within the cur-riculum of a technical education, mathe-matics enjoyed a high priority statusright from the beginning. In addition,in the very early years, and mainly dueto the activity of E. B. Christoffel, aschool for specialist teachers was estab-lished, where scientific mathematicscould also be pursued within the frame-work of a mathematical seminar. ThePolytechnicum quickly made a name foritself as a technical school throughoutGerman-speaking Europe, and thisespecially applied to the area of mathe-matics. It attracted excellent youngmathematicians from Germany whospent the first years of their scientificcareers in Zurich: R. Dedekind, E. B.Christoffel, H. A. Schwarz, H. Weber, G.F. Frobenius, A. Hurwitz and H.Minkowski all stayed for shorter orlonger periods. One of the great draw-backs for the Polytechnicum in thosedays was the fact that it did not have theright to confer doctoral degrees. Thisbecame possible in 1909, and in 1911the name was changed to the

Eidgenössische Technische Hochschule(Federal Institute of Technology) or, togive it its German acronym, ETH.

In keeping with the spirit of the times,scientific societies opened their doors inmany university towns. In accordancewith the spirit of the enlightenment,their aim was to bring new knowledge tointerested parties and circles, by organ-ising gatherings and producing regularpublications. The NaturforschendeGesellschaft (Scientific Research Society)of Zurich was founded in 1746, and wasone of the first such organisations inEurope. Its quarterly publication was ofgreat importance to the mathematicalcommunity; until well into the 20th cen-tury, numerous mathematical essayswere published in it by, for example, E.B. Christoffel, R. Dedekind, H. A.Schwarz, G. F. Frobenius and L. Schläfli,and also by H. Weyl and H. Hopf. Anational Naturforschende Gesellschaft didnot come into being until 1815. Withinthis body, an informal mathematical sec-tion was set up and H. A. Schwarz, whowas at the ETH, became its first presi-dent. The activities of the sectionpetered out, however, after Schwarz wascalled to Göttingen in 1875.

First International Congress ofMathematicians, 1897

The first International Congress ofMathematicians took place in Zurich in1897. One might be tempted to assumethat this event would have led to thefounding of a national mathematicalsociety, but strangely this was not thecase. Under the supervision of C. F.Geiser, the congress was jointly organ-ised by mathematicians from the ETHand the University of Zurich. OtherSwiss universities were not involved inthe organisation, nor were many of theparticipants from other Swiss universi-ties. This was undoubtedly due to the


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Societies Corner

The three founding members of the Swiss Mathematical Society. From left to right, R. Fueter (1880-1950), H. Fehr (1870-1954) and M. Grossmann (1878-1936).

splintered situation and lack of cohesionin the area of mathematics inSwitzerland at the time.

Eulers Complete Works During the congress, F. Rudio broughtup the idea of publishing the completeworks of Leonard Euler. The decision todo so led indirectly a few years later to the founding of the SwissMathematical Society. It was clear toeveryone involved in this project that thepublication of Eulers complete workswould take many years and require con-siderable sums of money, and it wasclearly necessary that the project be sup-ported by a wide scientific community.At the celebrations for the 200thanniversary of Eulers birth in Basle, F.Rudio and C. F. Geiser, amongst others,submitted a corresponding proposal toSchweizerische Naturforschende Gesellschaft.Both Rudio and Geiser were at the ETH;the former was director of the libraryand had wide-ranging historical inter-

ests, while the latter was a talentedorganiser with good connections inpolitical circles. The so-called EulerCommission, charged with preparingthe publication of Eulers completeworks, was founded in the very sameyear, in 1907. Under the presidency ofRudio, the commission worked swiftlyand had soon clarified the financial andscientific aspects of the project.Following the commissions positivereport, the project of bringing outEulers complete works in the originalwas started. The first volume appearedin 1911, and today more than seventyvolumes have been published. It isexpected that the project will be com-pleted within the next few years.

Founding of the Swiss MathematicalSociety

During the preparations for the pub-lication of Eulers works, the peopleinvolved had to fall back on improvisa-tion time and again and sorely felt theneed of a national mathematics body inSwitzerland: a mathematical organisa-tion was needed to coordinate andrealise an undertaking of these dimen-sions and to secure funding. This is whythe publication of Eulers works led indi-rectly to the founding of the SwissMathematical Society. On 4 September1910 the SMG/SMS (SchweizerMathematische Gesellschaft, its Germanname, and Société Mathématique Suisse, itsFrench one) was founded as a section ofthe Schweizerische NaturforschendeGesellschaft (today, the Swiss Academy ofNatural Science). The advancement ofpure mathematics should stand in theforeground and its promotion within anational and international framework iswhat Rudolf Fueter, a founding memberand the first president of the society,wrote in an article in the Neue ZürcherZeitung on 26 June 1960, on the occasionof the 50th anniversary of the SMG.

The founding of the society in 1910was greeted with great interest by Swissmathematicians: in its first year the soci-ety numbered 100 members. The annu-al meetings, which took place jointlywith the Schweizerische NaturforschendeGesellschaft at different venues inSwitzerland, were very well attended andundoubtedly helped to create and

strengthen the contact and communica-tion between mathematicians from thedifferent (language) regions. That thishad been one of the main aims at thefounding of the society can be seen bythe fact that, from the start, the choice ofthe president took the various universi-ties of the German and French-speakingregions into consideration, taking carethat neither region should dominate theother; the first president, R. Fueter, wasat the University of Basle, the second, H.Fehr at the University of Geneva and thethird, M. Grossmann at the ETH Zurich.It was, incidentally, mainly due to theinitiative of these three men that thesociety came to be founded. Havingmathematics embedded in a trulynational society became especiallyimportant during the first World War asthe belligerent actions betweenGermany and France threatened to poi-son the atmosphere between the French-speaking Swiss and their German-speak-ing counterparts.

Commentarii Mathematici HelveticiScientific demands on mathematics

continued to grow, and with these grow-ing expectations the lack of a Swiss peri-odical for the publication of scientificmathematical works became painfullyapparent. A plan was devised during the1920s to establish such a journal. Thepreparation was carried out by H. Fehr,together with some of the former presi-dents of the society. Faced with themulti-lingualarity of Switzerland, it wasclear from the beginning that neither aFrench or German title would serve; forthis reason a Latin name was decided on,and Commentarii Mathematici Helvetici waschosen from three suggestions.

The journal was formally founded atthe SMG meeting of 20 May 1928, andR. Fueter was elected as its first editor-in-chief. In those early years, practicallyonly articles from mathematicians work-ing in Switzerland were published, inorder, as Fueter said in the above-men-tioned NZZ article, to give as completea picture as possible of what our countryhas to offer in the way of mathematics.In the following years the contributionspublished in the Commentarii MathematiciHelvetici became more and more inter-national. Even though the journal filleda void and sold well from the beginning- it had 140 subscribers in its first year ofpublication - it could not be financed bysales of subscriptions alone. Furtherfunds were needed. In 1929 the SMGtherefore decided to create a Stiftung zurFörderung der mathematischenWissenschaften in order to attract moneyfrom the private sector. Eventually, thisfoundation was also entrusted with othertasks, and today it still supports specifi-cally targeted projects in teaching andresearch.


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The 1994 International Congress logo and a Swiss postage stamp issued to commemorate theCongress. The stamp features a portrait of Jakob Bernoulli, painted by his brother Nicholas,

together with his law of large numbers.

The frontispiece of the Proceedings of the FirstInternational Congress of Mathematicians,

Zurich 1897; the mathematicians featured are(top) Daniel, Jakob and Johann Bernoulli;(bottom) Leonhard Euler and Jakob Steiner

International Congress ofMathematicians, 1932 In 1932 the ICM took place in Zurich forthe second time. R. Fueter from theUniversity of Zurich and M. Plancherelfrom the ETH had received a request inBologna that the next ICM take place inZurich. The preparations were very dif-ferent from those for the 1897 congress.Care was taken to carry them out underthe patronage of the SMG and that theorganising committee should includepeople from universities from otherparts of Switzerland. In a far greatermeasure than in 1897, this congress wasorganised with the mutual participationof the entire Swiss mathematical com-munity. R. Fueter from the University ofZurich served as president and therewere two vice-presidents, M. Plancherelfrom the ETH and H. Fehr from theUniversity of Geneva.

The Steiner and Schläfli Archives As early as 1930 the SMG had laid thefoundation for a Steiner archive andformed a responsible committee. Thearchive was to preserve and manage J.Steiners scientific legacy; his CompleteWorks had been published by Weierstrassin 1881-82. In 1937 the task of the com-mittee was extended to include siftingthrough the legacies of the two greatSwiss mathematicians Steiner andSchläfli and making their work accessi-ble. Schläflis work was eventually pub-lished in 1956 in three volumes.

International Contacts The meetings of the SMG brought manyopportunities to invite renowned foreignmathematicians to Switzerland to givelectures. For the smaller Swiss universi-ties, these invitations provided amongthe few opportunities until well intothe 1940s for making internationalcontacts. Furthering international rela-tions thus remained an especially impor-tant task of the SMG. With this in mind,the society created numerous honorarymemberships for foreign mathemati-cians with whom it wished to foster rela-tions. These included, amongst others,R. Dedekind, D. Hilbert, H. Weyl, R.Nevanlinna, G. Pólya, H. Whitney and J.Leray, as well as worthy Swiss mathe-maticians (we name only deceased col-leagues) R. Fueter, C. F. Geiser, H.Hopf, A. M. Ostrowski, A. Pfluger, M.Plancherel, G. de Rham, W. Saxer andB. L. van der Waerden.

The ever stronger relationshipbetween Switzerland and theInternational Mathematical Union mustbe mentioned in this context. H. Hopfwas president from 1955-58, while B.Eckmann served as secretary. Laterpresidents of the IMU from Swiss uni-versities were G. de Rham from the uni-versities of Geneva and Lausanne, andK. Chandrasekharan and J. Moser fromthe ETH Zurich. Recently, the relation-

ship with the European MathematicalSociety has included a prominent posi-tion: the current president, Rolf Jeltschfrom the ETH, is also a member of theSMG board.

Elemente der Mathematik In 1946 L. Locher-Ernst started publica-tion of the journal Elemente derMathematik, quite independent of theSMG. It was aimed at a readership con-sisting mostly of teachers of higher edu-cation, although it dealt primarily withmathematics as science and not didac-tics. In this, Elemente der Mathematik wasfollowing aims similar to those ofLEnseignement Mathématique, the journalset up in 1899 by H. Fehr and Ch.Laisant at Geneva. In 1975 ownership ofElemente der Mathematik was transferredto the SMG and the Society has beencharged with its publication ever since.

International Congress ofMathematicians, 1994

Searching for a venue for the 1994International Congress ofMathematicians, the IMU again turnedto the SMG with the request that it takeplace in Switzerland. After a short loca-tion assessment, Zurich was chosen, forthe third time, as the most suitableplace. Once again, as in 1932, a com-mittee was put together under thepatronage of the SMG, made up ofmathematicians from all overSwitzerland. H. Carnal from theUniversity of Berne served as presidentand Chr. Blatter from the ETH Zurichwas secretary. This congress provided awelcome opportunity to steer the inter-ests of the general public towards math-ematics and elucidate its role in todaystechnological society. At the openingaddress, the Federal Minister of HomeAffairs, Ruth Dreifuss, picked up on thisvery point: It is the task of the scientificcommunity to tell the public why sciencematters. It is your task and mine. Therole of our scientific society towards soci-ety as a whole cannot be betterexpressed.

Urs Stammbach is Professor of Mathematicsat the ETH Zurich. He wishes to thank Dr.F. Lanini from the WissenschaftshistorischeSammlungen der ETH Bibliothek for hercritical proof reading of this manuscript andfor many useful suggestions.

Philip Heywood

The Edinburgh Mathematical Society wasformed at a meeting in the mathematicsclassroom of Edinburgh University on

EdinburghMathematical

Society

Friday 2 February 1883, following the cir-culation of a letter from A. J. G. Barclay, A.Y. Fraser and C. G. Knott to Gentlemen inEdinburgh, in Cambridge, and throughoutScotland generally whom they deemedlikely to take an interest in such a Society.The letter proposed the establishment, inconnection with the University, of a societyfor the mutual improvement of its mem-bers in the mathematical sciences, pureand applied. Methods suggested for theattainment of this object were reviews ofwork, both British and foreign, historicalnotes, discussion of new problems or newsolutions, and comparisons of the varioussystems of teaching in different countries.

It is interesting that schoolteachers tooka leading part in the foundation of theSociety. Mr Barclay and Mr Fraser wereteachers at George Watsons College inEdinburgh, while Cargill Knott, who tookthe chair at the inaugural meeting, wasassistant to the professor of natural philos-ophy in the University of Edinburgh, PeterGuthrie Tait. At the first meeting,Professor Tait and George Chrystal, theprofessor of mathematics at Edinburgh,were elected honorary members.Professor Chrystal gave an address onPresent fields of mathematical research,and 51 people joined as ordinary mem-bers. J. S. Mackay, chief mathematics mas-ter at the Edinburgh Academy, was electedas first president of the Society, with DrKnott as both secretary and treasurer. MrFraser succeeded Dr Knott in these officesfrom the autumn of 1883, when the latterleft to become professor of physics at theImperial University of Japan; Cargill Knottwas an authority on magnetism and seis-mology, and was responsible for conduct-ing the magnetic survey of Japan. After hisreturn to a lectureship in mathematics atEdinburgh in 1891, he served twice as theSocietys president, in 1893-94 and in1918-19. His name is known to manythousands, including the present writer,


Professor Ian Sneddon of Glasgow Universityand Professor Emeritus W. L. Edge of

Edinburgh University enjoy a joke at theSocietys centenary dinner in February 1983.

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Meetings are normally on Friday after-noons, and are open to all who are inter-ested. Speakers are chosen to cover a widerange of topics in pure and applied math-ematics. From time to time, the Societyorganises joint meetings with the LondonMathematical Society, and regularly hostsin Scotland that Societys PopularLectures, intended for senior school pupilsand other non-professional mathemati-cians.

Each year the Society holds a meeting ofpostgraduate students at the Burn atEdzell, and every four years it holds the StAndrews Colloquium, a major conferenceheld for a week in July, at which three orfour distinguished international speakersgive short courses of lectures; the next suchcolloquium will be in 2001.

The Proceedings of the EdinburghMathematical Society have been publishedby the Society since 1884. Three issues arepublished each year, containing researchpapers covering a wide range of topics inpure and applied mathematics. TheProceedings has an international board ofconsulting editors and has a world-widecirculation.

The Society is conscious that itsdeclared objective, the promotion andextension of the Mathematical Sciences, issometimes best achieved through the sup-port of activities planned and instigated byothers. It therefore has schemes forawarding grants for mathematical activi-ties, and it joins with other Societies in sup-porting various initiatives at national andinternational level through subscriptions,donations and the sending of representa-tives.

Membership of the Society is open to allwho are interested. Prospective membersare nominated at one of the Societys meet-ings and elected at the following meeting.Currently membership costs 11 poundssterling per year, or 20 pounds includingan individual subscription to theProceedings. Further information is avail-able from the Honorary Secretary,

Department of Mathematics and Statistics,James Clerk Maxwell Building, MayfieldRoad, Edinburgh EH9 3JZ, Scotland, or onthe internet at http://www.maths.ed.ac.uk/~chris/ems. The Society has reciprocityagreements with a number of overseasmathematical societies, through whichmembership of one society enables one tojoin others at a reduced subscription.Postgraduate students at Scottish universi-ties are offered free membership of theSociety for up to three years to encouragethem to take part in the Societys activities.

The Societys library is housed in theJames Clerk Maxwell Building at theUniversity of Edinburgh, and consistsmainly of periodicals obtained byexchange with other learned societiesthroughout the world. Members may con-sult or borrow books from the Societyslibrary, and are also entitled to use theEdinburgh University Library.

Each year since 1983, the Society has setaside a sum of money, known as theCentenary Fund, to give financial supportto a variety of mathematical activities,including research visits, conferences andpublications. Any member of the Societymay apply for a grant or guarantee fromthis fund. Awards are normally paid to anapplicants institution rather than to theapplicant personally, and a key feature inthe consideration of an application is theperceived benefit to the wider mathemati-cal community. Part of the CentenaryFund each year is earmarked for applica-tions from members resident overseas.The Society also sets aside each year a sumof money known as the Education Fund, togive financial support to educational activ-ities of a mathematical nature. As with theCentenary Fund, applications may bemade by members of the Society, but addi-tionally the Education Committee itselfactively seeks out projects worthy of sup-port. In addition to the Centenary andEducation Fund schemes, the GeneralCommittee of the Society from time totime makes special grants. Sometimesthese are major grants for the support oflarge international conferences, but equal-ly they may be small grants for worthymathematical purposes that do not fallwithin the remits of the two schemes.

Acknowledgement I have drawn freely on thehistory of the Society, The First HundredYears 1883-1983, Proc. Edinburgh Math.Soc. 26 (1983), 135-150, by Robert Rankin,formerly professor of mathematics atGlasgow University, President of theSociety from 1957-58, and currently anHonorary Member, and also on theAnnual Report for 1997-98 by the presentHonorary Treasurer, Neil Dickson ofGlasgow University.

Philip Heywood teaches in the Department ofMathematics and Statistics at the University ofEdinburgh.


who were school pupils before the age ofcalculators, through the publication of hisbooklet of four-figure mathematical tables.

Prominent and faithful early membersof the Society included George AlexanderGibson, president of the Society in 1888-89, professor of mathematics at theGlasgow and West of Scotland TechnicalCollege from 1895-1909 and at GlasgowUniversity from 1909-27, and RobertFranklin Muirhead, president in 1899-1900.

With the growth in size of universitydepartments, the Society gradually camemore under their influence and paid moreattention to mathematical research. SirEdmund Whittaker, professor of mathe-matics at the University of Edinburgh from1912-46, was responsible for the firstmathematical colloquium sponsored by theSociety, in 1913. A second colloquium washeld in Edinburgh in 1914, before the out-break of war. After the war, colloquia wereresumed at St Andrews in 1926, largelythrough the enthusiasm of H. W. Turnbull,who succeeded to the Regius Chair ofmathematics there in 1921, and colloquiahave been held in St Andrews regularlyever since. The Society is now firmly estab-lished as the principal mathematical soci-ety for the university community inScotland. Its membership, which exceeds400, is drawn from the Scottish universitiesand other educational institutions, as wellas from mathematicians in industry andcommerce both at home and overseas.

The first meeting outside Edinburghwas held in Glasgow in March 1900 duringMuirheads Presidency. Regular meetingsin Glasgow followed, and the Society beganto meet annually in St Andrews in 1922.The first meetings in Dundee andAberdeen were held in 1930 and 1937,respectively. At present, the Society meetsregularly in ten different universities.Eight ordinary meetings are held in eachSession, from October to June, three at theUniversity of Edinburgh, and the remain-der in other Scottish universities.

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George Chrystal and Peter Guthrie Tait, the first honorary members of the Society.

Please e-mail announcements of European con-ferences, workshops and mathematical meetingsof interest to EMS members, [email protected]. Announcementsshould be written in a style similar to thosebelow, and sent as text files (but not as TeXinput files). Space permitting, each announce-ment will appear in detail in the next issue of theNewsletter to go to press, and thereafter will bebriefly noted in each new issue until the meetingtakes place, with a reference to the issue in whichthe detailed announcement appeared. The pre-sent issue includes conferences up to September2000.

4-8: International Workshop on GeneralTopological Algebras, Tartu, EstoniaInformation:e-mail: [email protected]

4-8: ParaOpt VI, 6th InternationalConference on Parametric Optimizationand Related Topics, Dubrovnik, CroatiaInformation:URL:http://www.math.hr/dubrovnik/index.htm

4-15: Isaac Newton Institute Workshop,Defect Mechanics and Non-locality,Cambridge, UK Information:URL: http://www.newton.cam.ac.uk/programs/

5-9: Géométrie des équations differen-tielles, Luminy, FranceInformation: contact J.-P. Francois,Equipe «Géométrie différentielle,Systèmes dynamiques, applications, UFR920, Mathématiques, B.C. 172, Tour 46,5ème étage, Université P.-M. Curie, ParisVI 75252, Paris, France; fax: +33-1-442-5345 e-mail: [email protected][For details, see EMS Newsletter 32]

6-9: New Trends in the Calculus ofVariations, Lisbon, PortugalInformation: Contact CMAF/Univ.Lisboa, Av. Prof. Gama Pinto 2, 1649-003Lisboa, Portugal e-mail: [email protected]: http://www.math.ist.utl.pt/~ntcv99/[For details, see EMS Newsletter 32]

15-16: Two-day LMS meeting, NewApplications of Twistor Theory, London,UKSpeakers: K.P. Tod (Oxford), M.G.Eastwood (Adelaide), N.J. Hitchin(Oxford), S. Merkulov (Glasgow), L.J.Mason (Oxford), Sir Roger Penrose(Oxford)Grants: apply to [email protected]: to book reserved accommoda-tion, 1 October 1999Information:e-mail: [email protected]

October 1999

21-26: Foundation Workshop onStochastics and Quantum Physics,Aarhus, Denmark Aim: to foster fruitful discussions and col-laboration on the role and use of stochas-tics in quantum physics, by bringingtogether leading physicists and mathe-maticians having an active interest in thethemes of the workshopMain themes: (confirmed key-note speak-ers in parentheses) laser physics/quantumoptics (F. Bardou, H. Carmichael, G.Mahler), quantum stochastic processes (A.Barchielli, V.P. Belavkin, G. Lindblad),Wick products, white noise analysis andMalliavin calculus (B. Øksendal), the roleof generalised measurements and quan-tum statistical inference (R. Gill, S.Massar, H. Wiseman), quantum informa-tion (A.S. Holevo, P. Høyer, A. Peres, S.Popescu)Programme: during the first two days, B.Øksendal will give a short course on Wickproducts, Malliavin calculus and theirapplications in physicsOrganisers: O.E. Barndorff-Nielsen(Aarhus), K. Molmer (Aarhus)Fee: the registration fee is DKK600, tocover lunches, coffee during breaks andthe conference dinner. Those who partici-pate only in the short course need not paythe registration feeSite: University of Aarhus Deadlines: for registration, already passedInformation: URL:http://www.maphysto.dk/events/QuantumStoc99/index.html

1-12: Isaac Newton Institute Workshop,Models of Fracture, Cambridge, UKInformation:URL: http://www.newton.cam.ac.uk/programs/

2-5: Workshop on Hilberts 10th prob-lem, relations to arithmetic and algebra-ic geometryInformation: contact Jan Van Geel orKarim Zahidi, University of Gent,Department of Pure Mathematics,Galglaan 2, B-9000 Gent, Belgium; fax.+32-9-264-49 93e-mail: [email protected]:http://cage.rug.ac.be/~hilbrt10/hilbert10.html[For details, see EMS Newsletter 32]

25-26: XXème Rencontre Franco-Belgede Statisticiens, Brussels, Belgium Topic: factor models. In a factor model,one tries to explain the correlationbetween several random variables using alimited number of factors. This confer-ence aims to present the latest develop-ments as well as the numerous applica-tions of factor models

November 1999

Programme committee: ChristopheCroux, Alois Kneip, Lucrezia Reichlin,Eric Renault, Bas Werker Organisers: Institut de Statistique et deRecherche Opérationelle, EuropeanCenter for Advanced Research inEconomics and StatisticsSite: Universite Libre de Bruxelles,Brussels Deadline: for submission of papers,already passedInformation: contact RFBS99, Institut deStatistique, CP210, ULB, B-1050Bruxelles, Belgium; tel/fax: +32-2-650-5899 e-mail: [email protected] URL: http://isro.ulb.ac.be

2-4: Conference on Mathematical andComputational Methods in Music,Vienna [Part of the EMS Diderot Forum series]Aim: to bring together mathematiciansinterested in or working on subjects ofrelevance to music, and musicians whomake use of computational methods; toraise public awareness of the role of math-ematics in our society, in particular byshowing the multiple connections andmutual influences which exist betweenmusic and mathematics Topics: synthesis of musical sounds,analysis of musical sounds (transcriptiondigital data - MIDI, time-frequency meth-ods, quantitative analysis of instrumentsand musical interpretation), restorationand improvement of old recordings andmathematical algorithms, instrumentoptimisation (instrumental parameteridentification), coding strategies for musi-cal information, mathematical models formusical sound or rhythm Invited speakers: (confirmed) W.Fitzgerald (Cambridge), G. DePoli(Padua), X. Serra (Barcelona), G.Wakefield (Ann Arbor, Michigan) Scientific committee: Werner A. Deutsch(Austrian Acadademy of Science,Kommission für Schallforschung), HansG. Feichtinger (Institut für Mathematik,University of Vienna, main organizer),Christian Krattenthaler (Institut fürMathematik, University of Vienna), ErichNeuwirth (Institut für Statistik, Universityof Vienna), Gregor Widholm (Institut fürWiener Klangstil, University of Music andDramatic Arts, Vienna), Gerhard Widmer(Austrian Research Institute for ArtificialIntelligence, Vienna) Local organising committee: MonikaDörfler, Hans G. Feichtinger, ChristianKrattenthaler, Stefan Thurner Supporting institutions: UniversitätWien, EMS (European MathematicalSociety), OEMG (Austrian MathematicalSociety), Mathematischer Zirkel derUniversität Wien, OECG (AustrianComputer Society), Bank Austria Proceedings: to be published in a specialissue of the OECG series (AustrianComputer Society)Site: Kleiner Festsaal der UniversitätWien, main building of the University ofVienna

December 1999

CONFERENCES


FForthcoming conferorthcoming conferencesencescompiled by

Kathleen A S Quinn

Accommodation: participants shouldmake their own arrangements; the organ-isers may be consulted Social event: a concert with contributionsfrom participants is plannedDeadline: for submission, already passedInformation: contact Hans G.Feichtinger, Institut für Mathematik,Universität Wien, Strudlhofgasse 4, A-1090 Vienna tel: +43-1-4277-50696; fax: +43-1-4277-50620 e-mail: [email protected]: http://tyche.mat.univie.ac.at/~diderot/

20-22: Seventh IMA InternationalConference on Cryptography andCoding, Cirencester, UK Information: e-mail: [email protected]: http://www.ima.org.uk/mathematics/conferences.htm[For details, see EMS Newsletter 32]

17-22: Workshop on ComputationalStochastics, Aarhus Information: contact Eva B. VedelJensen, Department of MathematicalSciences, University of Aarhus, NyMunkegade, DK-8000 Aarhus C e-mail: [email protected]:http://www.maphysto.dk/events/CompStoc2000/[For details, see EMS Newsletter 32]

3-5: Mathematics Today, Trondheim,NorwayNote: primarily intended for aScandinavian audienceInformation: URL: http://www.math.ntnu.no/talltiltusen/

28-3 March: Eighth InternationalConference on Hyperbolic Problems,Magdeburg, Germany Topic: theory, numerics and applicationsof hyperbolic conservation laws and relat-ed fieldsPlenary speakers: Yann Brenier (Paris),Thomas Hou (Pasadena), ShuichiKawashima (Fukuoka), Ingo Mueller(Berlin), Alfio Quarteroni (Lausanne),Phil Roe (Ann Arbor), Giovanni Russo(Laquila), Steve Schochet (Tel Aviv), JoelSmoller (Ann Arbor), Michael Struwe(Zuerich), Kevin Zumbrun (Bloomington) Invited Speakers: Francois Bouchut(Orleans), Suncica Canic (Houston),Pierre Degond (Toulouse), EduardFeireisl (Prague), Emmanuel Grenier(Lyon), David Hoff (Bloomington), Shi Jin(Atlanta), Kenneth H. Karlsen (Bergen),Smadar Karni (New York), Claus-DieterMunz (Stuttgart), Benedetto Piccoli(Salerno), Bradley Plohr (Stony Brook),Ed Seidel (Potsdam), Tao Tang (HongKong), Eleuterio Toro (Manchester),Cheng-Chin Wu (Los Angeles), TongYang (Hong Kong), Shi-Hsien Yu (Osaka)Scientific committee: J. Ballmann, A.Bressan, C. Dafermos, B. Engquist, M.Feistauer, H. Freistühler, J. Glimm, L.Hsiao, R. Jeltsch, P. Lax, T.-P. Liu, P.

February 2000

January 2000

Marcati, D. Marchesin, K.W. Morton, B.Perthame, D. Serre, E. Tadmor, A. Tveito,and G. Warnecke. Organisers: G. Warnecke, H.Freistueühler. Call for papers: abstracts for contributedpapers may be submitted immediatelySite: Otto-von-Guericke University,MagdeburgInformation: contact HYP-2000 c/oInstitut für Analysis und Numerik Otto-von-Guericke-Universität Magdeburg PSF4120, D-39016 Magdeburg, Germany; fax: HYP-2000 at +49-391-67-18073e-mail: [email protected]: http://rubens.math.uni-magdeburg.de/ ~hyp2000

11-12: School Mathematics 2000,Helsinki, FinlandInformation:e-mail: [email protected]

11-14: Workshop on Harmonic Mapsand Curvature Properties ofSubmanifolds 2, Leeds, UK Aim: to stimulate cooperation betweenresearchers working in harmonic mapsand those working in submanifold theorySpeakers: (expected plenary speakers) F.Burstall, T.E. Cecil, M.A. Guest, F.Helein, Y. Ohnita, F. Pedit, U. Pinkall, G.Thorbergsson Programme committee: S. Carter, J.C.Wood. Information: contact J.C. Wood, Schoolof Mathematics, University of Leeds,Leeds LS2 9JT, UK e-mail: [email protected] URL: http://www.amsta.leeds.ac.uk/pure/geometry/leeds2000.html

17-20: 52nd British MathematicalColloquium, Leeds, UKScope: the annual meeting of (pure)mathematicians in the UK; all mathemati-cians are welcomePlenary speakers: Sir Michael Atiyah(Edinburgh), Simon Donaldson (ImperialCollege), Vaughan Jones (Berkeley),Harvey Friedman (Ohio), Jens Jantzen(Aarhus); also 14 main speakers, mostlybased in the UKSpecial sessions: harmonic maps andminimal surfaces; operator algebrasSplinter groups: number theory, mathe-matical logic, algebraic topology, mathe-matical education, functional analysis,algebra, and integrable systemsOrganisers: H.G. Dales and H.D.Macpherson, LeedsSite: University of LeedsRegistration: forms will be available inearly 2000; they will be circulated with theLondon Mathematical Society Newsletter,and sent to all UK Departments ofMathematics; to obtain a form, pleasecontact H.D. Macpherson at the e-mailaddress belowAccommodation: available on the campusof the University of Leeds Satellite conferences: 11-14 April,

April 2000

March 2000

Harmonic maps and curvature propertiesof submanifolds, Leeds; 14-17 April,Mathematical methods of regular dynam-ics - dedicated to the 150th anniversary ofSonja Kowalevski, Leeds; 16-17 April,British topology meeting, Sheffield; 17April, Model theory, LeedsInformation: e-mail: [email protected]: http://www.amsta.leeds.ac.uk/bmc/

26-28: Mathematical Education ofEngineers, Loughborough, UKAim: to reflect the progress and experi-ences of initiatives within the teaching ofmathematics to engineers in recent years,to debate areas of known concern and tolearn together from current best practice;to examine collectively as mathematicians,academic engineers and engineers inindustry the engineering mathematicsprovision for the futureProgramme: invited speakers, contributedpapers (or posters), workshop sessionsand a forumOrganising committee: Leslie Mustoe(Loughborough), Stephen Hibberd(Nottingham), Trevor Easingwood (IMA),Duncan Lawson (Coventry); HeatherLiddell (London), John McWhirter(DERA), Stephen Reid (UMIST), RalphSmith (Jaguar)Proceedings: to be published by theInstitute of MathematicsSponsors: The Institution of CivilEngineers and The Institution ofChemical EngineersSite: Loughborough UniversityDeadline: for abstracts, 3 DecemberInformation:URL: http://www.ima.org.uk/mathematics/conferences.htm

29-9 June: Foliations: Geometry andDynamics Revisited, Banach Centre,Warsaw, PolandAim: the exchange of scientific informa-tion among specialists in the theory offoliation and related topics, in particularin the area of relations between this theo-ry, differential geometry, dynamical sys-tems and ergodic theory

13-16: First AMS-ScandinavianInternational Mathematics Meeting,XXIII Scandinavian Congress ofMathematicians, Odense, DenmarkSpeakers: Tobias Colding (New York),Nigel J. Hitchin (Oxford), Elliott Lieb(Princeton) Pertti Mattila (Jyväskylä),Curtis McMullen (Harvard), AlexeiRudakov (Trondheim), Dan-VirgilVoiculescu (Berkeley), Johan Hostad(Stockholm)Special sessions: (to run in parallel) alge-braic groups/representation theory, com-plex analysis in higher dimensions, differ-ential geometry, discrete mathematics,dynamical systems, geometricanalysis/PDE, K-theory and operator alge-bras, linear spaces of holomorphic func-tions, mathematical physics, mathematicseducation, stochastic DE and financial

June 2000

May 2000

CONFERENCES


mathematics, joint EWM and AWM ses-sion Information: contact Hans J. Munkholm,Odense University, Campusvej 55, DK5230 Odense M, Denmark tel: +45-6557-2309/+45-6593-2691e-mail: [email protected]:http://www.imada.ou.dk/~hjm/AMS.Scand.2000.html

14-17: International Workshop forOperator Theory and Applications(IWOTA), Bordeaux, FranceAim: to bring together mathematiciansand engineers interested in operator the-ory and its applications to related fieldsScope: operator theory and related topicsin mathematics and its applications: dif-ferential operators, reproducing kernelspaces, harmonic analysis, control theory,system theory and signal processingInvited speakers: (confirmed) V.M.Adamyan, D. Alpay, A. Böttcher, L deBranges, R.F. Curtain, K. Davidson, C.Foias, I. Gohberg, W. Helton, R.Kaashoek, A. Megretski, Y. Meyer, S.N.Naboko, J. Partington, G. Pisier, C.Sadosky, E.B. Saff, K. Seip, O. Staffans, S.Treil, S. Verduyn LunelOrganising committee: L. Baratchart(Sophia-Antipolis), A. Borichev(Bordeaux), G. Cassier (Lyon), J. Esterle(Bordeaux), N. Nikolski, chairman(Bordeaux and St.Petersburg), V.-H.Vasilescu (Lille)President of the steering committee: I.Gohberg (Tel Aviv)Site: University of Bordeaux-1 Information: e-mail: [email protected] URL: http://www.math.u-bordeaux.fr/~iwota/

18-21: International Conference onMonte Carlo Simulation, Monte Carlo,MonacoAim: to provide an opportunity for engi-neers, mathematicians and other profes-sionals who are interested in the theoreti-cal and practical aspects of Monte Carlosimulation to exchange ideas on the statusof MCS proceduresTopics: include algorithms for randomnumber generation, methods for solutions(spectral simulation, solving the PDEsdirectly, numerics of PDEs), algorithms(evolutionary, genetic), practical engineer-ing applicationsDeadline: for abstracts, 12 December Information: URL:http://www.uibk.ac.at/c/c8/c810/conf/mcs_2000.html

18-24: Perspectives of Mathematics,Goslar, Germany[World Mathematical Year event]Aim: to bring together senior andyounger mathematicians to discuss theperspectives of mathematics at the turn ofthe centurySpeakers: (confirmed) V. Arnold (Parisand Moscow), M. Gromov (Bures surYvette), R. James (Minnesota), J.P. Morel(Cachan), G.C. Papanicolaou (Stanford),A. Schrijver (Amsterdam), D. Zagier(Bonn) Scientific committee: J.-P. Bourguignon

(Paris), F. Hirzebruch (Bonn), K.-H.Hoffmann (Bonn), S. Müller (Leipzig). Local organiser: K. HulekNote: the number of junior participants islimited to 50. The workshop is aimed atyounger mathematicians, that is, thosewho have had a PhD for at least one yearand have done research beyond their the-sis, but do not yet hold a senior tenuredpositionDeadline: for applications, 31 January Information: contact K. Hulek, Institut fürMathematik, Universität Hannover,Postfach 6009, D-30060 Hannover,Germany e-mail: [email protected]: http://www-ifm.math.uni-hannover.de/info/perspectives.html

26-30: POISSON 2000, FranceInformation:e-mail: [email protected]

28-1 July: First World Congress of theBachelier Finance Society, Paris, FranceInformation:e-mail: [email protected]

29-3 July: International Workshop onNonlinear Spectral Theory, Würzburg,Germany Topic: state-of-the-art of spectral andeigenvalue theory for nonlinear operators Scope: to bring together experts on non-linear analysis to discuss recent develop-ments and open problems in the theory,methods, and applications of spectra ofnonlinear operators Programme: 15 invited one-hour lectures,short communications, informal discus-sion Lecturers: R. Chiappinelli (Siena), G.Conti (Florence), E. De Pascale (Cosenza),W. Feng (Toronto), M. Furi (Florence), M.Martelli (Fullerton), M.Z. Nashed(Newark), T. Riedrich (Dresden), P.Santucci (Rome), C.A. Stuart (Lausanne),V.A. Trenogin (Moscow), M. Vath(Wurzburg), A. Vignoli (Rome), J.R.L.Webb (Glasgow), P.P. Zabrejko (Minsk)Sponsor: DeutscheForschungsgemeinschaft (DFG), Bonn Site: Department of Mathematics,University of Würzburg (Franconia) Information: contact Jurgen Appell,Department of Mathematics, University ofWürzburg, Am Hubland, D-97074Würzburg, Germany tel: +49-931-8885017; fax: +49-931-8885599e-mail: [email protected]: www.mathematik.uni-wuerzburg.de/~appell/nlst.html

3-7: ALHAMBRA 2000, Granada, SpainInformation:e-mail: [email protected]

3-7: Functional Analysis Valencia 2000,Spain[satellite conference to the ThirdEuropean Congress of Mathematics inBarcelona, 10-14 July]Information: contact: K.D. Bierstedt or J.Bonet, Univ. Paderborn, FB 17, Math., D-

July 2000

33095 Paderborn, Germany orUniversidad Politècnica de Valencia,Departamento de Matemática Aplicada,E-46071 Valencia, Spain e-mail: [email protected]: http://math-www.uni-paderborn.de/VLC2000[For details, see EMS Newsletter 32]

3-7: Sixth International Conference onp-Adic Analysis, Ioannina, GreeceInformation:e-mail: [email protected]

4-7: Second International Conference onMathematical Methods in Reliability,Bordeaux, FranceInformation:e-mail: [email protected]

10-14: IUTAM Symposium on FreeSurface Flows, Birmingham, UKTopics: axisymmetric free surface flows,moving contact lines, non-linear waterwaves, collapsing bubblesScientific committee: J.R. Blake, J.B.Keller, A.C. King, W. Lauterborn, D.H.Peregrine, A. Prosperetti, E.O. Tuck, L.van Wijngaarden Local organising committee: J.R. Blake(co-chair), A.C. King (co-chair), J.Billingham, S.P. Decent, Y.D.Shikhmurzaev, J.R. King, J.R.A. Pearson,E.J. Hinch, J.M. Vanden-Broeck Site: University of BirminghamRelated meetings: 17-20 July, IUTAMSymposium 2000/10 Diffraction andScattering in Fluid Mechanics andElasticity, Manchester; 17 July-4 August,Free Boundary Problems in Industry,Isaac Newton Institute, Cambridge, UKDeadline: for abstracts, 1 JanuaryInformation:URL:http://www.mat.bham.ac.uk/research/iutam.htm

10-14: Third European Congress ofMathematics, Barcelona Information: contact Societat Catalana deMatemátiques, Carrer del Carme, 47, E-08001 Barcelonatel: +343-270-16-26; fax: +343-270-11-80 e-mail: [email protected]: http://www.iec.es/3ecm/[For details, including satellite confer-ences, see First Announcement in EMSNewsletter 31]

17-20: IUTAM Symposium 2000/10Diffraction and Scattering in FluidMechanics and Elasticity, Manchester,UKAim: to bring together researchers from arange of different subject disciplines andindustrial focuses, who employ commontechniques and approaches, to examinethe ways that the various subjects havedeveloped and to stimulate cross-fertilisa-tion of the theoretical ideas and method-ologies. Themes: elastic waves, acoustic phenome-na in stationary fluids, aeroacoustics, dif-fraction of free surface and other geo-physical waves Organisers: hosted jointly by theDepartments of Mathematics, Universityof Manchester and Keele UniversityLocal organising committee: I. David

CONFERENCES


Abrahams (Manchester, chair), C. JohnChapman (Keele), Paul A. Martin(Manchester), Mike J. Simon(Manchester). Graham Wilks (Keele),Andrew J. Willmott (Keele) Proceedings: will be published Site: University of ManchesterRelated conference: IUTAM Symposiumon Free Surface Flows, 10-14 July 2000 Deadline: for abstracts, 31 January Information: contact Professor DavidAbrahams, Department of Mathematics,University of Manchester, Oxford Road,Manchester M13 9PL, UKtel: +44-161-275-5901, fax: +44-161-275-5819 e-mail: [email protected]: http://www.keele.ac.uk/depts/ma/iutam/

17-22: Colloquium on Lie Theory andApplications, Vigo, Spain[Satellite Activity of the Third EuropeanCongress of Mathematics]Programme: three courses of three hourseach, eleven invited lectures and severalshort communicationsCourses: D.V. Alekseevsky, SemisimpleLie algebras, Dynkin diagramms andgeometry of flag manifolds; A.T.Fomenko, Lie groups and integrableHamiltonian systems; M. Scheunert Speakers: S. Benayadi, M. Bordemann, V.Cortés, A. González-López, Yu.B.Hakimjanov, E. Koelink, M. de León, E.Macías-Virgós, A. Medina, C. Moreno, K-H. Neeb Scientific committee: D.V. Alekseevsky(Germany), S. Benayadi (France), M.Bordemann (Freiburg), V. Cortés (Bonn),A.T. Fomenko (Moscow), A. González-López (Madrid), Yu.B. Hakimjanov(France), K.H. Hofmann (USA), E.Koelink (Netherlands), M. de León(Spain), E. Macías-Virgós (Santiago), A.Medina (France), C. Moreno (Bourgogneand Madrid), K-H. Neeb (Germany), M.Scheunert (Bonn) Organising committee: N. Alonso, I.Bajo, R. González, A. Martín and E.Sanmartín (Vigo) Languages: English, Spanish and French;English is recommended in abstracts andin the written version of the communica-tionsRegistration: possible from 1 February to30 April Deadline: for abstracts, 30 November Note: first of a series of conferencesdevoted to all aspects of lie theory to beheld in different locations biannuallyInformation: contact I Colloquium on LieTheory and Applications, E.T.S.I.Telecomunicación, Universidad de Vigo,36280 Vigo, Spain; tel: +86-81-21-52 // +86-81-24-45; fax:+86-81-21-16 // +86-81-24-01 e-mail: [email protected] URL: http://www.dma.uvigo.es/~clieta/

17-22: International Congress ofMathematical Physics, London, UKInformation:URL: http://icmp2000.ma.ic.ac.uk/

23-31: ASL European Summer Meeting(Logic Colloquium 2000), Paris, FranceMain themes: proof theory and logical

foundations of computer science, set theo-ry, model theory, computability and com-plexity theory, history of 20th-centurylogic, philosophy, applications of logic tocognitive sciencesProgramme: 24 plenary talks, four three-hour tutorials, parallel sessions of con-tributed talks.Programme committee: Daniel Andler(Paris), Chantal Berline (Paris), BarryCooper (Leeds), Akihiro Kanamori(Boston), Charles Parsons (Harvard),Alexander Razborov (Moscow), HelmutSchwichtenberg (Munich), John Steel(Berkeley), Stevo Todorcevic (Paris), Dirkvan Dalen (Utrecht), Alex Wilkie(Oxford), Carol Wood (Chairperson,Wesleyan University)Organising committee: Chantal Berline(Paris), Zoé Chatzidakis (Paris), René Cori(Chairman, Paris), Maximo Dickmann(Paris), Jacques Dubucs (Paris), Jean-Baptiste Joinet (Paris), Daniel Lascar(Paris), Yves Legrandgérard (Paris), JeanMosconi (Paris), Marie-Hélène Mourgues(IUFM de Créteil), Catherine Muhlrad-Greif (Paris), Leszek Pacholski (Wroclaw),Jean-Pierre Ressayre (Paris), BobanVelickovic (Paris), Francoise Ville (Paris)Site: the Sorbonne (Université Paris 1)Deadline: for abstracts, 31 MarchInformation: to receive the congressannouncements, send a request by e-mailto [email protected] withsubject get-announcement, or by fax to+33-1-44-27-61-48 (from abroad) and 01-44-27-61-48 (from France), or by letter toLC2000, UFR de Mathématiques, case7012, Université Paris 7-Denis Diderot, 2place Jussieu, 75251 Paris Cédex 05,France.e-mail: [email protected]: http://lc2000.logique.jussieu.fr

31-3 August: Third Conference ofBalkan Society of Geometers, Bucharest,RomaniaTopics: Riemannian geometry, symplecticgeometry, submanifolds theory, Cheninvariants, harmonic maps, spectralgeometry, Finsler-Lagrange-Hamiltongeometry, geometry of PDEs, criticalpoint theory and its applications, convexi-ty and optimisation on Riemannian mani-folds, electromagnetic dynamical systems,numerical integrator of dynamical systemsProgramme: 30-minute lectures and 15-minute papers; a workshop engagingMasters and Ph.D. students in geometry,and an open forum of the Balkan Societyof GeometersConference chairs: R. Miron (Romania),Gr. Tsagas (Greece), C. Udriste(Romania)Programme committee: M. Anastasiei(Romania), D. Andrica (Romania), P.L.Antonelli (Canada), Gh. Atanasiu(Romania), D. Blair (USA), N. Blazic(Yugoslavia), V. Boskoff (Romania), K.Buchner (Germany), B.Y. Chen (USA), V.Cruceanu (Romania), D. Hrimiuc(Canada), S. Ianus (Romania), L.Nicolescu (Romania), D. Opris (Romania),D. Papuc (Romania), Gh. Pitis (Romania),P. Popescu (Romania), M. Puta(Romania), H. Shimada (Japan), P.

Stavrinos (Greece), L. Tamassy (Hungary),K. Trencevski (Macedonia), I. Vaisman(Israel), L. Vanhecke (Belgium), L.Verstraelen (Belgium), E. Vassiliou(Greece)Organiser: C. UdristeProceedings: selected papers will be pub-lished in the Balkan Journal of Geometryand its ApplicationsSite: University Politehnica of Bucharest,Splaiul Independentei Street 313,Bucharest, RO-77206Deadline: for registration, 15 AprilInformation: contact V. Balan, UniversityPolitehnica of Bucharest, Department ofMathematics I, Splaiul Independentei313, RO-77206, Bucharest, Romania fax: +401-411-53-65e-mail: [email protected]

8-12: XVIII Nevanlinna Colloquium,Helsinki, Finland[World Mathematical Year event]Scope: the emphasis will be on subjects insome way connected to analysis, especiallygeometric aspectsSpeakers: include A. Baernstein (St.Louis), J. Cheeger (New York), W.Bergweiler (Kiel), P. Mattila (Jyväskylä),C. Bishop (Stony Brook), C. McMullen(Berkeley), B. Bowditch (Southampton),Yu. Reshetnyak (Novosibirsk), L. Carleson(Stockholm) Programme: about 45 invited talksOrganising committee: Peter Buser(Lausanne), Seppo Rickman (Helsinki),Ilpo Laine (Joensuu), Kurt Strebel(Zürich), Olli Lehto (Helsinki),.PekkaTukia (Helsinki), Olli Martio (Helsinki),Matti Vuorinen (Helsinki)Site: University of HelsinkiInformation: to receive the secondannouncement, send e-mail [email protected] or writeto Riitta Ulmanen, Department ofMathematics, P.O. Box 4 (Yliopistonkatu5), FIN-00014 University of Helsinki,Finlande-mail: [email protected]: http://www.math.helsinki.fi/~analysis/NevanlinnaColloquium/

21-25: IMACS 2000, Lausanne,Switzerland[International Association forMathematics and Computers WorldCongress]Information: contact Prof. Robert Owens,IMACS Congress 2000, DGM-IMHEF-LMF, Swiss Federal Institute ofTechnology, CH-1015 Lausanne,Switzerlandtel: +41-21-693-35-89; fax: +41-21-693-36-46 e-mail: [email protected]: http://imacs2000.epfl.ch[For details, see EMS Newsletter 32]

30-2 September: Innovations in HigherEducation 2000, Helsinki, FinlandTheme: higher education in general (notjust mathematics)Information:e-mail: [email protected]: http://www.helsinki.fi/inno2000

August 2000

CONFERENCES


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

S. K. Berberian, Fundamentals of Analysis,Universitext, Springer, New York, 1999, 479pp., DM99, ISBN 0-387-98480-1This book is essentially a record of a courseon functions of a real variable for first-yeargraduate students in mathematics, offeredat the University of Texas at Austin.

The foundational material includes con-struction of the reals, cardinal and ordinalnumbers, Zorns lemma and transfiniteinduction. Chapter 2 is devoted toLebesgue measure on R introduced viaCarathéodory outer measure, and Chapter3 presents a short introduction to metricand topological spaces. Chapter 4 dealswith abstract Lebesgue integral, includingconvergence theorems, finite signed mea-sures and the Radon-Nikodym theorem.Chapter 5, Differentiation, includesabsolutely continuous functions, functionsof bounded variation, F. Rieszs Rising SunLemma, indefinite integrals, LebesguesFundamental theorem of calculus,Lebesgue decomposition of a function ofbounded variation and a criterion forRiemann-integrability. The Stone-Weierstrass approximation theorem, Lp-spaces and real and complex measures arestudied in Chapter 6. Product measuresand Fubini-Tonelli theorem are covered inChapter 7. Chapter 8 deals with Picardsand Peanos existence theorems for y =f(x, y). Additional topics in measure andintegration (Jordan-Hahn decompositionof a signed measure, the Radon-Nikodymtheorem for σ-finite measures, Lebesguedecomposition of measures, convolution)are included in Chapter 9.

The material is well chosen, the presen-tation nice, the balance between specialand general is reasonable, and each sectionis accompanied by exercises. This text-book will surely find many readers amongboth students and teachers interested inmathematical analysis. (in)

C. Blatter, Wavelets. A Primer, A. K. PetersLtd., Natick, 1998, 202 pp., £24, ISBN 1-56881-095-4This excellent book is intended as an intro-duction to the wavelet transform for stu-dents in mathematics. It provides a solid,yet accessible, mathematical foundationfor those interested in learning aboutwavelets and pursuing the broad range ofapplications for which the wavelet trans-form has proved successful.

The book is divided into six chapters.The introductory Chapter 1 presents a tourdhorizon over various ways of signal repre-sentation. Chapter 2 serves primarily as atutorial on Fourier analysis. The continu-

ous wavelet transform is treated in Chapter3, while Chapter 4, Frames, decribes a gen-eral framework that allows one to handlethe continuous and the discrete wavelettransforms in a uniform way. Multi-resolu-tion analysis with its fast algorithms is pre-sented in Chapter 5. The construction oforthonormal wavelets with compact sup-port is given in Chapter 6, and the bookends with a brief treatment of splinewavelets in Section 6.4.

Numerous illustrations and fullyworked-out examples further enhance thevalue of this exemplary introduction to thefield. (kn)

J. Bochnak, M. Coste and M.-F. Roy, RealAlgebraic Geometry, Series of Modern Surveysin Mathematics 36, Springer, Berlin, 1998,430 pp., ISBN 3-540-64663-9This is a substantially enlarged and updat-ed edition of Géométrie Algébrique Réelle bythe same authors published in the samepublishing house and in the same series asVol. 12. It is a highly competent mono-graph written by the leading specialists inthe field. Without any doubt this is anindispensable book for mathematiciansworking in real algebraic geometry, but itis so well written that it can be used as atextbook for postgraduate students. Ofcourse, a preliminary knowledge of com-plex algebraic geometry is very helpful forunderstanding the differences between alittle classical complex algebraic geometryand real algebraic geometry. But thedevelopment of mathematics is very quick,and I can well imagine students who startto study algebraic geometry via the realalgebraic geometry. On the other hand,mathematicians who are at least a littlefamiliar with the complex algebraic geom-etry will appreciate the new real ideas andmethods. Some chapters of the book, likethe Nash Functions, will be interesting formathematicians working in analysis, otherparts like Topology of real algebraic varieties,Algebraic vector bundles, Polynomial or regularmappings with values in spheres will be inter-esting for topologists, the chapter Algebraicmodels of C∞ manifolds will attract the atten-tion of differential geometers, while alge-braists will find here a plenty of interestingmaterial.

The prerequisities for this book arequite modest. Moreover, the authors haveincluded a chapter on ordered fields andreal closed fields, where a lot of preparato-ry material is gathered. (We remark thatthe book contains much material that ismore general than the title of the booksuggests; in particular, we find here alge-braic geometry over an arbitrary realclosed field.) It is not surprising that theauthors were not able to cover the wholearea of real algebraic geometry.Nevertheless they cover a lot both from

classical and modern algebraic geome-try.

Where they have not enough space, theysometimes omit proofs, but they alwaysproperly explain all notions and assertionsand give references where the correspond-ing proofs can be found. Moreover, at theend of each chapter there are bibliograph-ical notes where we find hints for furtherreading. The bibliography is large andincludes 350 items. (jiva)

B. Bollobás, Linear Analysis: AnIntroductory Course, CambridgeMathematical Textbooks, Cambridge UniversityPress, Cambridge, 1999, 240 pp., £16.95,ISBN 0-521-65577-3This is a well-written concise introductionto functional analysis, intended foradvanced undergraduate students. Thecontents include a nice exposition of stan-dard material (normed spaces and bound-ed linear operators, Hahn-Banach,Banach-Steinhaus, closed-graph theorem,Stone-Weierstrass, contraction-mappingtheorem, Hilbert spaces, orthonormal sys-tems, adjoint operators, compact opera-tors, compact normal operators, weaktopologies and duality), as well as unusualtopics like invariant subspaces, fixed-pointtheorems (Brouwer, Schauder), theBishop-Phelps theorem and geometry offinite dimensional spaces. This text,although concentrating on abstract spaces,shows the relevance of functional analysisto other areas of mathematics. A nice fea-ture of the book is a large collection ofexercises and notes following each chapterwhere the most important references areprovided and historical comments present-ed. The book will surely be appreciated bystudents as well as teachers of mathemati-cal analysis. (in)

J. Borwein, P. Borwein, L. Jörgenson andR. Corless, eds., Organic Mathematics,CMS Conference Proceedings 20, AmericanMathematical Society, Providence, 1997, 412pp., ISBN 0-8218-0668-8This book is the hardcopy version of theelectronic Proceedings of the OrganicMathematics Workshop held at SimonFraser University. The least commondenominator of the conference was themutual affects of mathematics and moderntechnology. We are seeing a steady growthin the use of new technology, not only incomputation but also in research, teachingand communication (including publica-tion). Last but not least, a large portion ofthe current mathematical knowledge isencoded in computer algebra systems andpackages for scientific computation. Theorganisers of the conference were interest-ed in the benefits of all these facets of thismodern development, and want the infor-mation of the Proceedings to form exam-ples of living documents, connected totheir references, connected to each other,connected to algorithms for live mathe-matical work on the part of the reader.They want them to be organic.

The proceedings contain fifteen invitedpapers and two associated articles. Manyof the underlying papers appeared else-

RECENT BOOKS


Recent booksRecent booksedited by Ivan Netuka and Vladimír Souèek

where, and each paper has been selectedby author reputation and with the aim toselect papers with a good potential foractivation. Since not every activity dur-ing the conference can be archived in textform, the editors have attempted at least topersonalise the Proceedings by adding ashort biography of each speaker, with indi-vidual pictures. (spor)

J.-P. Bourguignon, P. de Bartolomeis andM. Giaquinta (eds.), Geometric Theory ofSingular Phenomena in PartialDifferential Equations, Instituto Nazionale diAlta Matematica Francesco Severi.Symp.Mathematica vol. XXXVIII, CambridgeUniversity Press, Cambridge, 1998, 182 pp.,£40, ISBN 0-521-63246-3In May 1995 a workshop on the geometrictheory of singular phenomena of PDEsoccuring in real and complex differentialgeometry was held in Cortona. The bookcontaining contributions of participantsand related articles is dedicated to FrancoTricerri who was a member of the organis-ing committee of the workshop before histragic death in a plane crash in China.

The longest contribution (70 pages) isan introduction to twistor techniques by P.de Bartolomeis and A. Nannicini. Thispaper contains a very nice systematic anddetailed description of geometrical prop-erties of twistor spaces for even-dimension-al compact connected manifolds with agiven conformal structure. The paperends with a characterisation of manifoldsfor which the corresponding twistor spaceis Kählerian. Glueing procedures for con-structions of solutions of various geometri-cal problems (complete immersed minimalsurfaces in 3-dimensional Euclidean space,complete embedded surfaces of constantmean curvature and complete conformalmetrics of constant positive scalar curva-ture on subsets of compact Riemannianmanifolds) and a discussion of modulispaces of such solutions are discussed in apaper by R. Mazzeo and D. Pollack.

The conribution by L. Habermann andJ. Jost contains a discussion of propertiesof the metric induced on the Teichmüllerspace of (marked) Riemann surfaces Σ of agiven genus by a choice of a Hermitianmetric gΣ on Σ. The paper by M.Giaquinta, G. Modica and J. Souèekdescribes weak solutions of the variationalproblem for the Dirichlet integral on thespace of maps from a bounded domain inR³ into the sphere S² with prescribedboundary values. Two different approach-es are discussed, one in the setting ofSobolev maps and the other in the settingof Cartesian currents. The orbifold funda-mental group of the Persson-Noether-Horikawa surfaces is computed in the con-tirbution by F. Catanese and S.Manfredini.

The book also contains two short papersby F. Labourie (on solutions of the Monge-Ampère equation and its relations to pseu-do-holomorphic curves) and by A. M.Nadel (existence problem for Kähler-Einstein metrics on a given Fano mani-fold). (vs)

G. Buttazzo, M. Giaquinta and S.Hildebrandt, One-dimensional VariationalProblems, Oxford Lecture Series inMathematics and its Applications 15,Clarendon Press, Oxford, 1998, 262 pp., £35,ISBN 0-19-850465-9The authors, all of them well-known spe-cialists in the field, demonstrate the ideasof modern variational calculus in the one-dimensional case. It enables them to avoidthe difficulties connected with multiplevariational integrals and to explain mainideas of the calculus of variations (empha-sising direct methods) in extremely clearmanner. This project has been realisedvery successfully. The book represents anexcellent tool for those who wish to spe-cialise in the calculus of variations and togo on to study more complicated theoriesof variational problems for multiple inte-grals. At the same time, it is an interestingtextbook for analysts working in differentbranches who wish to complete their edu-cation. The text can be inspiring for grad-uate students beginning their scientificactivities. I much enjoyed the historicalnotes and the vivid style of exposition.

In Chapter 1, the classical indirectmethods based on necessary and sufficientconditions for optimality are treated.Chapter 2 gives a framework of functionspaces indispensable for applying thedirect methods (absolutely continuousfunctions, BV-functions and Sobolevspaces). Lower semicontinuity methods arediscussed in Chapter 3. In Chapter 4, reg-ularity of minimizers is treated; here alsothe Lavrentiev phenomenon is studied indetail. Chapter 5 is dedicated to applica-tions, such as the Sturm-Liouville eigenval-ue problem, the vibrating string, variation-al problems with obstacles, periodic solu-tions of variational problems, periodicsoloutions of Hamiltonian systems, non-coercive variational problems, an existenceproblem in optimal-control theory andparametric variational problems. InChapter 6 (Scholia) some ramifications ofcalculus of variation and various connec-tions to pertinent problems for multiplevariational integrals are pointed out. (oj)

P. J. Cameron, Permutation Groups,London Mathematical Society Student Texts 45,Cambridge University Press, Cambridge, 1999,220 pp., £15.95, ISBN 0-521-65302-9,ISBN 0-521-65378-9This book is intended as a course thatrequires only rudimentary knowledge ofgroup theory. It has seven chapters, thelast comprising tables of simple groups,affine 2-transitive groups and almost sim-ple 2-transitive groups. The sixth chapteraddresses several stand-alone topics (suchas Blichtfelds or Jordans theorem) and sothe bulk of the content rests with the firstfive chapters. The first two are standard(regular primitive and multiply transitivegroups, wreath products, orbitals, cen-traliser algebra, and characters) and thefourth is dedicated to the ONan-ScottTheorem, with a short sketch of the proofand with applications (orders and degreesof primitive groups, distance-transitivegraphs and some others).

The third chapter connects permuta-tion groups to combinatorial regular struc-tures (such as association schemes, coher-ent configurations, strongly regular anddistance-transitive graphs).

The fifth chapter is somewhat differentfrom the first four, since it concerns infi-nite permutation groups, and in particularthe so-called oligomorphic groups, i.e.,such permutation groups on an infinite setΩ for which the number fn(G) of orbits aris-ing from G when it acts on n-element sub-sets of Ω is finite for all n. Connections withrandom infinite graphs, with the theory ofmodels (automorphism groups of count-ably categorical structures) and with grad-ed algebras are presented with varyinglevel of detail. Attention is also paid to thegrowth rate of fn(G) and to preservations oflinear and circular orders.

Students should find this book verystimulating because of the many differentconnections it mentions, and this is partic-ularly true about the fifth chapter. Thecombinatorial interests of the author obvi-ously influenced the choice of topics andone should not expect an introduction toall aspects of permutation groups.

Another feature of the book is its inclu-sion of computational group theory. TheSchreier-Sims algorithm and Jerrums fil-ter are included in the first chapter andthere is even a short introduction to GAP.The book contains information aboutresources available on the World WideWeb. (ad)

C. M. Campbell, E. F. Robertson, N.Ruskuc and G. C. Smith, eds., Groups StAndrews 1997 in Bath, I, II, LondonMathematical Society, Lecture Note Series260/261, Cambridge University Press,Cambridge, 1999, 737 pp., £29.95/£29.95,ISBN 0-521-65588-9/0-521-65576-5Two volumes of conference proceedingscontain 64 papers that cover many aspectsof group theory. The main lectures arereflected in the proceedings by five surveypapers, each 20-50 pages long, concentrat-ing on probabilistic algorithms and geo-metric group theory. Similar topicsappear also in many of other longer papers(those with around 20 pages).

The common theme of the probabilisticpapers is the exploitation of Aschbachersclassification of maximal subgroups of thefinite classical groups for the purposes ofprobabilistic recognition in these groups.

Babai and Beals present the black-boxgroup concept and give a number of theo-rems concerning constructions, recogni-tion and description in Monte-Carlo poly-nomial time. The paper by Shalev starts byciting some classical probabilistic results onsymmetric groups and the author com-ments on the proof of Dixons conjecturefor simple finite groups. He also presentssome applications to free groups and themodular group, and points out connec-tions with profinite groups and Kac-Moodyalgebras, finishing with a sketch of proof ofCamerons conjecture on the base size ofalmost simple primitive permutationgroups. The essay of Praeger is aboutprimitive prime divisor elements that play

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an essential role for recognising algo-rithms in GL(d, q).

The longest article is an introductorysurvey on non-positive curvature in grouptheory by Bridson. He carefully explainsthe notion of the non-positive curvatureand its connection to hyperbolicity, anddevelops then the theory of CAT(0) spaces.First he describes properties of groups thatact properly and cocompactly by isome-tries on CAT(0) spaces, and then gives var-ious examples of an interaction betweeninteresting groups and these spaces.

The lecture of Brookes blends filtra-tions of group rings with non-commutativetoric geometry. The developed techniquesaim at groups with no non-Abelian freesubgroups. Nilpotent groups receive spe-cial attention.

Among other papers let us mention thefollowing three: Bogomolov and Katzarkovpresent a base change construction whichallows to prove that fundamental groups ofprojective surfaces are rather densely dis-tributed among all finitely presentedgroups. Olshanskii shows that underrather weak assumptions, one can find 2-generated 2-groups (p-groups in somecases) with a prescribed growth function.McCammond introduces a generalisationof small cancellation theory which he hasapplied to the word problem of freeBurnside groups. (ad)

M. Capiñski and E. Kopp, Measure,Integral and Probability, SpringerUndergraduate Mathematics Series, Springer,London, 1999, 227 pp., DM59, ISBN 3-540-76260-4This book aims to present Lebesgue inte-gral in a way accessible to undergraduatestudents as a background to probabilitytheory. This is considered as the mainarea of application for the theory and thechoice of material for the book is influ-enced by this aim.

A probabilistic interpretation of dis-cussed notions is introduced at the begin-ning, and leads naturally to the choice of ameasure-theoretical approach to theLebesgue integral. Among the topics cov-ered, we mention the law of the large num-bers and the central limit theorem. Thecourse is kept at a level accessible toAmerican students at the third year of theirstudies. Certain notions from functionalanalysis are also included. The book con-tains exercises with solutions. The finalChapter explains the axiom of choice as atool used to prove the existence of non-measurable sets.

Readers of the book, especially begin-ners, will appreciate another strategy ofthe authors: instead of the most elegant orthe shortest approach, they prefer whatthey consider to be more direct or natural.Some assertions are first formulated andleft to be proved by readers using the hintsprovided; their proofs are presented lateron. (jiva)

R. W. Carter and M. Geck (eds.),Representations of Reductive Groups,Publications of the Newton Institute, CambridgeUniversity Press, Cambridge, 1998, 191 pp.,

£35, ISBN 0-521-64325-2This book serves as an introduction to asystematic study of the topic and consists ofnine articles by different authors. The bestdescription of the content of the book isgiven by listing the articles and theirauthors: Introduction to algebraic groups andLie algebras (R. W. Carter), Weyl groups,affine Weyl groups and reflection groups (R.Rouquier), Introduction to abelian andderived categories (B. Keller), Finite groups ofLie type (M. Geck), Generalized Harish-Chandra theory (M. Broué and G. Malle),Introduction to quantum groups (J. C.Jantzen), Introduction to the subgroup struc-ture of algebraic groups (M. W. Liebeck),Introduction to intersection cohomology (J.Rickard), An introduction to the LusztigConjecture (S. Donkin). (lbi)

C. Constatinescu, W. Filter and K.Weber, Advanced Integration Theory,Mathematics and its Applications 454, KluwerAcademic Publishers, Dordrecht, 1998, 861pp., £237, ISBN 0-7923-5234-3This volume is devoted to a detailed andsystematic development of the abstract the-ory of integration. The aim is to build alogically consistent and advantageous con-struction, suitable for integration in gener-al topological spaces (without local com-pactness or countability assumptions). Thecost is that the terminology in the book dif-fers from the most widely accepted one.Among the major features of the authorsapproach, let us mention the following: theintegral is defined via the Daniell construc-tion as a functional on a Riesz lattice; theclass of integrable functions is wide due toa wide family of negligible sets: null sets forintegration are those which are locallyinsignificant; measures are finite-valuedand defined on δ-rings rather than on σ-rings.

The volume is divided into chapters onthe following topics: vector lattices, defini-tion of the integral, Lp-spaces, real mea-sures, the Radon-Nikodym theorem andduality, integration of real functions. Then-dimensional integration is not consid-ered separately (of course, it falls withinthe general theory).

The text is organised in a definition-theorem-proof format and each chapter isendowed with exercises, some of thembeing important theoretical excursions.To end with, a brief historical chapter isincluded.

There is another book that uses thisapproach to integration theory: IntegrationTheory: Measure and Integral, by C.Constantinescu and K. Weber (in collabo-ration with A. Sontag), John Wiley & Sons,1985, where the old and new definitionsare compared and the advantage of thenew approach is explained. The presentbook contains more material, but for moti-vation and intuitive remarks the reader isreferred to the older book.

The book presents a possibility how toteach measure and integration theory forstudents whose interest consists in abstractmathematics. It also provides also a theo-retical background for various directionsin abstract analysis. It is an important doc-

ument on this way of developing integra-tion theory. (jama)

D. Cox, J. Little and D. OShea, UsingAlgebraic Geometry, Graduate Texts inMathematics 185, Springer, New York, 1998,499 pp., DM78, ISBN 0-387-98487-9 and 0-387-98492-5Many applications of algebraic geometrymethods are demonstrated in this book,which covers a variety of topics related tothe algorithmic theory of polynomials. Inthe first chapter basic results and notionsabout basic algebraic structures, polynomi-als, Gröbner basis algorithms and affinevarieties can be found. The second andthird chapters discuss several approachesto solving polynomial equations, such aselimination theory or resultants. Chapters4, 5 and 6 are devoted to topics of classicalcommutative algebra; the reader can findhere basic facts about local rings, Milnornumbers, syzygies, Hilbert functions, etc.Chapter 7 deals with geometry polytopes,toric varieties, Minkowski sums andBernsteins theorem, thereby coveringsome connections between polynomialsand convex polytopes. Chapter 8 illus-trates some applications of Gröbner basesto problems in integer programming,combinatorial enumeration problems,spline functions. The last chapter discuss-es applications from computational alge-bra and algebraic geometry to problemsfrom coding theory.

Reading the book does not requiremore than standard undergraduate knowl-edge. The book can be used in a variety ofcourses involving solving equations, com-mutative algebra, and their various appli-cations. It contains a lot of exercises withindications for further development, andincludes pointers to the abilities of variouscomputer algebra packages. It would beuseful (but not necessary) to read this bookin conjunction with Ideals, Varieties andAlgorithms by the same authors. (spr)

L. Debnath, Nonlinear Partial DifferentialEquations for Scientists and Engineers,Birkhäuser, Boston, 1998, 593 pp., DM138,ISBN 0-8176-3902-0 and 3-7643-3902-0One of the major goals of the book is toprovide an accessible working knowledgeof some of the current analytical methodsrequired in modern mathematics, physicsand engineering. From the immensewealth of the world of non-linear PDEs, thevolume emphasises the part dedicated tonon-linear wave propagation problems. Itcontains many new examples of applica-tions in fluid dynamics, plasma physics,non-linear optics, gas dynamics, analyticaldynamics and acoustics. With twenty-onepages of bibliography, the work can alsoserve as a reference book for those with amore detailed interest in some of the con-sidered subjects.

Chapter 1 is an introduction to linearPDEs (method of characteristics, Fouriermethod, the use of integral transforms andthe method of Greens functions). Chapter2 is dedicated (among other things) to vari-ational principles and the Euler-Lagrangeequations. Chapters 3-6 are devoted to

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non-linear first-order PDEs. Several sec-tions of Chapter 6 discuss properties ofsolutions of real-world non-linear models,including traffic flow, flood waves, chro-matographic models, etc.

In Chapter 7, various aspects ofWhithams equation (non-linear dispersivewaves) are studied. Non-linear diffusion-reaction phenomena are considered inChapter 8. Chapter 9 develops the theoryof solitons and the inverse scattering trans-form; many recent results on the basicproperties of the KdV and Boussinesqequations are discussed. The non-linearSchrödinger equation and solitary wavesare the main focus of Chapter 10. Chapter11 is concerned with the theory of non-lin-ear Klein-Gordon and sine-Gordon equa-tions with applications. Non-linear evolu-tion equations and asymptotic methods aretreated in the last chapter.

In a way, the authors style of expositionfollows the idea of R. P. Feynman, quotedin the preface: However, the emphasisshould be somewhat more on how to dothe mathematics quickly and easily, andwhat formulas are true, rather than themathematicians interest in methods of rig-orous proof. A great advantage of thisbook lies in the striking balance main-tained between the mathematical andphysical aspects of the subject. (oj)

R. W. Easton, Geometric Methods forDiscrete Dynamical Systems, OxfordEngineering Science Series 50, OxfordUniversity Press, New York, 1998, 157 pp.,£50, ISBN 0-195-08545-0This small book is an essay on discretedynamical systems and it is very pleasant toread it. There are no long computationsor technically difficult considerations, andthe sampling ideas clearly demonstrate theview of topological dynamics which wasclose to C. Conley. Several open problemsare also formulated. The main subject ofthe book is the long-time behaviour oforbits.

After illustrative examples and basicdefinitions, chain-recurrent sets and theConley decomposition theorem are pre-sented in Chapter 2. Chapter 3 is devotedto local analysis near hyperbolic invariantsets (stable manifolds, Hartman-Grobmantheorem, Smales horseshoe and the recentnotion of resonance zone). The heart ofthe book is in Chapters 4 and 5, where iso-lated invariant sets and the Conley indexof an isolated block are explained. A dis-crete analogue of Hamiltonian systems,symplectic maps, is studied in Chapter 6and the existence of the Birkhoff normalform is proved. The book ends with thenotion of an invariant measure and thePoincaré recurrence theorem. (jmil)

J. Elias, J. M. Giral, R. M. Miró-Roig andS. Zarzuela, eds., Six Lectures onCommutative Algebra, Progress inMathematics 166, Birkhäuser, Basel, 1998,398 pp., DM128, ISBN 3-7643-5951-X and0-8176-5951-XThis volume arises from the followingseries of lectures given at the BarcelonaSummer School on Commutative Algebra

in July 1996: Infinite free resolutions (L. L.Avramov), Generic initial ideals (M. L.Green), Tight closure, parameter ideals, dualgeometry (C. Huneke), On the use of localcohomology in algebra and geometry (P.Schenzel), Problems and results on Hilbertfunctions on graded algebra (W. V.Vasconcelos), Cohomological degree of gradedmodules (W. V.Vasconcelos). Each series isclose to a monograph and is a good intro-duction to its subject. (tk)

G. Friedlander and M. Joshi, Introductionto the Theory of Distributions, 2nd edition,Cambridge University Press, Cambridge, 1998,175 pp., £42.50, ISBN 0-521-64015-6 and0-521-64971-4In this book the basic notions of the theoryof Schwartz distributions are explained. Itis especially convenient for those readerswho wish to learn the theory of distribu-tions without a knowledge of functionalanalysis, namely of the theory of topologi-cal (locally convex) linear spaces, which isincluded as an appendix. The book doesnot even require a knowledge of the theo-ry of the Lebesgue integral. This approachshows that some very deep theorems canbe proved in an elementary way; for exam-ple, the Schwartz kernel theorem is provedwithout using the theory of nuclear spaces,referring to the theory of Fréchet spacesonly. There are a lot of exercises supple-menting the theory. They contain applica-tions mostly to those ordinary or partialdifferential equations to which this theoryis applicable, linear differential equationswith C∞ coefficients. The basic notions ofthe theory of distributions are not gener-alised except for a few pages devoted toSobolev spaces. No more general multipli-cation is introduced than that of a distrib-ution with a C∞ function; for example, themultiplication of a distribution of order nwith a Cn function is not introduced. Onlythe last chapter of the book is an excep-tion: it contains harder notions that cannotbe found in other textbooks on distribu-tions. In this chapter, multiplication ofdistributions is generalised using the wave-front sets defined by the growth of theirFourier transforms in a given direction.Roughly speaking, the product of two dis-tributions is defined if, in each direction,one distribution is C∞. (jjel)

W. Fulton, Intersection Theory, 2ndEdition, Ergebnisse der Mathematik und ihrerGrenzgebiete, Modern Surveys in Mathematics,Springer-Verlag, Berlin, 1998, 470 pp.,DM178, ISBN 3-540-62046-XThis is the second and unchanged editionof a famous and fundamental monograph,the first edition of which appeared in1984. The author has added some refer-ences that appeared shortly after the firstedition, and refers the reader to the sec-ond edition of his book Intersection Theoryin Algebraic Geometry, CBMS 57, Amer.Math. Soc., 1996, for the more recent liter-ature.

For young mathematicians interested inalgebraic geometry this book is quite indis-pensable. It is also of interest to alge-braists, topologists, specialists in several

complex variables, and for mathematicianswho want to familiarise themselves with theprofound ideas of intersection theory. It isa modern treatise that includes many newresults, as well as simpler proofs of someolder results, and also takes into accountthe historical development of the subject.The text is designed so as to be accessibleto anybody equipped with a first course ofalgebraic geometry, assuming that he/sheconsults the appendices at the end of thebook. Moreover, the organisation of thebook enables one, after the first six chap-ters, to read the other chapters separately.An important role is played by theExamples, which appear in large quantitiesat the end of each section. These illustratethe general theorems and build bridgesbetween classical and modern approaches,present generalisations and counter-exam-ples, and serve to motivate later results. Atthe end of each chapter, the Notes andReferences contain historical remarks andput the material of the chapter into a moregeneral framework. Simply, it is a verygood book. (jiva)

L. Gårding, Mathematics andMathematicians. Mathematics in Swedenbefore 1950, History of Mathematics 13,American Mathematical Society, Providence,1998, 288 pp., ISBN 0-8218-0612-2This book is an extremely readable historyof mathematics in Sweden up to 1950, writ-ten by a leading personality from Swedishmathematics. This makes the book differ-ent from usual books on the history ofmathematics: the text brings an expertoverview of the development of mathemat-ics, presented in a master style that reflectsa deep insight into the subject. Thus thereaders learn important facts from the his-tory of mathematics and extend their pre-vious knowledge in fields that do not exact-ly overlap their own specialities.

It is natural that the history of mathe-matics in Sweden is more-or-less a story ofuniversity towns and their mathematicians.Thus the main body of the book includes adetailed description of mathematics as wellas university life in Uppsala, Lund andStockholm. The 18th century and up to1850 are briefly presented, and the time inUppsala and Lund from 1860 to 1880 isdescribed. There is a chapter devoted toalgebraic geometry in Lund before 1900,another chapter deals with the situation inUppsala during the period 1860 to 1900and whole chapters are devoted toBäcklund, Mittag-Leffler, and Mittag-Lefflers and Sonya Kovalevskis mathe-matical works. A further chapter describesthe development of astronomy and opticsin the 19th century. More than a half of thebook deals with the mathematics devel-oped in Stockholm (1880-1920 and 1925-50), Uppsala (1900-25 and 1930-50) andLund (1900-50). Here the important con-tributions of famous Swedish mathemati-cians are explained, in particular those ofBendixon, Phragmén, von Koch,Holmgren, Nörlund, Carleman, Pleijel,Carlson, M. Riesz, Frostman, Nagell,Beurling and Carleson. This book onmathematics and mathematicians is

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strongly recommended to anybody wholikes mathematics and its history. (in)

A. J. Hahn, Basic Calculus. FromArchimedes to Newton to its Role inScience, Springer, New York, 1998, 545 pp.,DM98, ISBN 0-387-94606-3This book is based on calculus coursesdelivered at the University of Notre Dame.As the author remarks, the title of the bookcould be The story of the calculus. He givesfive examples of different first-year coursesthat could be based on the book. Otherexamples are available on the web sitehttp://www.nd.edu:80/~hahn/ (the prefacecontains an obvious URL misprint: ahyphen instead of a tilde). This diversity ispartly due to the fact that a large portionof college mathematics is also included.

The first two chapters are devotedmainly to connections between trigonome-try and astronomy. The third chapterdescribes Archimedes area research andincludes the quadrature of the parabola,while the main subject of the fourth chap-ter is analytic geometry; this comprisesabout one fifth of the book. The next fourchapters deal with the mathematics ofLeibniz and Newton. The relation of therest of material to the history of mathe-matics is more complicated, but the math-ematics included is easier to describe: itconcerns properties of elementary func-tions and their use in simple problemsinvolving differentiation and integration.

The book contains a lot of material thatcan be used to relate mathematics to theevolution of human knowledge. It givesvarious ways, both for students and instruc-tors, for coping with the elements of calcu-lus. On the other hand, it is written fornon-specialists, such as for students ofinformatics, biology, economics, or thehumanities. A descriptive style of presenta-tion without formal proofs helps to illus-trate what is going on, but the approacheliminates a deeper understanding of thenature of mathematics. This is a strongfeature of the book as well as its majorweakness. The book is more useful forteaching the history of calculus than forteaching the basics of mathematics. Detailsof the books contents can be found on theabove URL. (jiva)

S. Hassani, Mathematical Physics. AModern Introduction to its Foundations,Springer, New York, 1999, 1025 pp., DM179,ISBN 0-387-98579-4This book is a condensed exposition of themathematics that is met in most parts ofphysics. The presentation attains a verygood balance between the formal introduc-tion of concepts, theorems and proofs onone hand, and the applied approach onthe other, with many examples, fully orpartially solved problems, and historicalremarks.

An impressive amount of mathematicsis covered. Starting from a set-theoreticalintroduction and linear algebra, throughfunctional analysis, complex analysis, anda lot of differential equations (both ODEsand PDEs, exact and numerical solutions),up to differential geometry and Lie theory.

All of this offers a chance to enjoy mathe-matics as a source of applicable tools thatare all related one to the other and whichcreate the beautiful abstract world of ideas,definitions and theorems. The author haswritten the book both for physics students,to show them the mathematics they use,and for mathematics students, in order tooffer a glimpse of many applications of theabstract ideas they know. The prerequi-sites comprise just a basic calculus courseand all advanced undergraduates shouldbe able to follow the exposition.

The headings of the eight parts of thebook are: finite-dimensional vector spaces(covering standard linear algebra), infi-nite-dimensional vector spaces (includingthe theory of orthogonal polynomials,Fourier series and Fourier transforms),complex analysis (complex series, calculusof residues, multivalued functions, andanalytic continuation), differential equa-tions (mainly second-order equations,including analytical and numerical meth-ods), operators on Hilbert spaces (spectraltheory, integral equations and Sturm-Liouville systems), Greens functions (forODEs and PDEs), groups and manifolds(representation theory and tensor analy-sis), Lie groups and their applications (dif-ferential geometry, symmetries of ODEsand PDEs, variational calculus and Lie the-ory).

This book can be warmly recommendedas a basic source for the study of mathe-matics for advanced undergraduates orbeginning graduate students in physicsand applied mathematics, and also as a ref-erence book for all working mathemati-cians and physicists. (jsl)

J. Hofbauer and K. Sigmund,Evolutionary Games and PopulationDynamics, Cambridge University Press,Cambridge, 1998, 323 pp., £16.95, ISBN 0-521-62570-X and 0-521-62365-0The same authors previous book TheTheory of Evolution and Dynamical Systems(Cambridge, 1988) has been used both as areference book and as a textbook. Theauthors have now restructured it andadded a lot of new material to make it clos-er to evolutionary game theory. This goalhas forced them to reduce the biologicalmotivation and ideas. The result is a math-ematical text that divides into four parts.

The first part, Dynamical systems andLotka-Volterra equations, can be under-stood either as an introduction to popula-tion dynamics and the basic ideas of V.Volterra or as an introduction to theasymptotic behaviour of low-dimensionalsmooth dynamical systems (mainly ofdimension 2 for the continuous case). Thesecond part is a concise course in thedynamics of non-cooperative games; theproperties of replicator equations are usedas a main tool. In fact, these equations areequivalent to Volterra-Lotka systems(Section 7.5). Global properties of replica-tor equations, mainly permanence andpersistence, and algebraic properties ofinteraction matrices (M,B,P-matrices) areinvestigated in the third part and areapplied to n-species communities. The last

part is devoted to connections between theselection, mutation and recombination ofgenetic information and evolutionarygame theory. Special gradients (withrespect to a metric on a simplex), so-calledShahshahani gradients, and correspond-ing gradient systems are continually usedhere.

A basic knowledge of ordinary differen-tial equations and linear algebra is suffi-cient for reading this book. There aremany exercises of very different types,some providing significant extensions ofthe material. The systematic explanation,exercises and extensive bibliography canstimulate new research. (jmil)

B. B. Hubbard, The World According toWavelets, A. K. Peters, Natick, 1998, 330 pp.,£28, ISBN 1-56881-072-5This is one of the best books in the popu-lar mathematical literature, remarkable forits fresh style that makes wavelet history,applications and technical advances inter-esting for non-mathematicians of any age.The author begins her examination ofwavelets with the story of the developmentof Fourier analysis, the essential underpin-ning for telephones, X-ray machines, andcomputers. Without any dizzying technicalor mathematical details, she describes themore recent meteoric rise of wavelet analy-sis and its many practical applications.Part II lucidly presents the mathematicalformulas and details of wavelet analysis forthose seeking a deeper understanding.

In this second edition she includes a dis-cussion of new medical and genetic appli-cations, such as mammography, heart dis-ease and fingerprints. The Part IIIAppendices contain proofs of theHeisenberg uncertainty principle and thesampling theorem. Readers who are lesssophisticated mathematically will also finda brief review of trigonometry, a list ofmathematical symbols, and a discussion ofintegrals and the Fourier transform of aperiodic function. The bibliographyincludes technical books and articles. Theauthor has also included a list of waveletsoftware and electronic recources.

This excellent book is a wonderfulintroduction to the world of wavelets. Itcan be strongly recommended to anybodyinteresting in using wavelets. (kn)

O. A. Ivanov, Easy as ππ?, Springer, NewYork, 1999, 187 pp., DM59, ISBN 0-387-98521-2When lecturing to future teachers of math-ematics, possible connections with sec-ondary school mathematics should beemphasised. Many times the chance islost, because of the lack of good referencebook where such things are treated in anaccessible way.

This book is aimed at filling the gap. Itoffers numerous elementary facts, forminga basis for further development of relatedmathematics going far beyond the scope ofthe book. In ten chapters the author dealswith a range of topics, including inductionand Peano axiomatics, combinatorics andgenerating functions, ornaments and geo-metrical transforms, quaternions, and

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functional equations. Each chapter con-tains (mostly solved) problems and relatedtheorems that extend the readers knowl-edge. Exercises help to test the level ofunderstanding. The book is based on acourse taught for several years at StPetersburg University, and will be of inter-est not only for people oriented towardseducation but as pleasant and interestingreading for professional mathematicians.It can be highly recommended for themathematical library of any university.(jiva)

J. Kollár and S. Mori, Birational Geometryof Algebraic Varieties, Cambridge Tracts inMathematics 134, Cambridge University Press,Cambridge, 1998, 254 pp., £30, ISBN 0-521-63277-3This is a very good introduction to the con-temporary research on algebraic geometrycalled minimal model programme orMoris programme. The general idea isto find in each birational equivalence classof algebraic varieties a variety that can beconsidered as the simplest one. For exam-ple, each irreducible curve is birationallyequivalent to a unique smooth curve. Asimilar investigation of surfaces was startedat the beginning of this century and result-ed in finding (with few exceptions) the sim-plest smooth surfaces in each class, calledminimal models. It is only in the last twodecades that efforts have been made toextend this programme to higher dimen-sions. There one meets many complica-tions; for example, as minimal models onemust allow not only smooth varieties butalso varieties with certain reasonable sin-gularities. Next, already in dimension 3,there have appeared quite new birationaltransformations called flips and flops.So far, many results have been obtainedthat are valid in arbitrary dimensions, butthe minimal model programme has beencompleted only in dimension 3, by Mori in1988.

This book gives a very good survey ofpresent research, including possible exten-sions of the minimal model programme.We mention that this programme has alsofound interesting applications in otherparts of algebraic geometry. The reader isassumed to be familiar with algebraicgeometry at the level of R. HartshornesAlgebraic Geometry. The authors have suc-ceeded in giving a very clear presentationof the subject. The bibliography has 154items and goes up to 1988. (jiva)

P. Koosis, The Logarithmic Integral I,Cambridge Studies in Advanced Mathematics12, Cambridge University Press, Cambridge,1998, 606 pp., £29.95, ISBN 0-521-30906-9 and 0-521-59672-6The theme of this book is the logarithmicintegral. The author shows how one canbuild up an investigation that explains andclarifies many different, seemingly unrelat-ed problems, from a few simple ideas.With this book one can begin a seriousstudy of real and complex analysis. It canbe read as a whole, and presents manyresults, some unpublished, some new, andsome available only in inaccessible jour-

nals.This first of two volumes is self-con-

tained, but assumes graduate real andcomplex variable theory with a bit of func-tional analysis (as contained in Rudinswell-known text). In this paperback edi-tion, a number of small errors and somemathematical mistakes in the originalhard-cover version have been corrected.The material is presented with useful pic-tures to help the reader understand thesubject.

Chapter I starts with the Jensens for-mula. Szegõs theorem is studied inChapter II with the pointwise approximateidentity property of the Poisson kernel.Chapter III deals with the entire functionsof exponential type (Hadamard factorisa-tion, Lindelöfs theorem, Phragmén-Lindelöf theorems, the Paley-Wiener theo-rem, representation of positive harmonicfunctions as Poisson integrals, Blaschkeproducts, Levinsons theorem on the den-sity of zeros). Quasi-analyticity is studiedin Chapter IV, using Carlemans criterion(including the theorem of Cartan andCorny). Chapter V contains a discussion ofthe moment problem on the real line(method based on moment sequences,Carlemans sufficient condition, a neces-sary condition, M. Rieszs general criterionfor indeterminacy). Weighted approxima-tion on the real line is the topic of ChapterVI (Mergelians treatment of weightedpolynomial approximation, Akhiezersmethod, Mergelians criterion, Pólyasmaximum density, the analogue ofPollards theorem, de Brangess descrip-tion of external unit measures, Kreinsfunctions). A question How small can theFourier transform of a rapidly decreasing non-zero function be? is the title of Chapter VII(Levinsons result, Beurlings theorems,Kargaevs example, Volbergs work). Thefinal chapter presents the persistence ofthe form dx/(1+ x²) with special cases (theset of positive lower uniform density, theset of integers, harmonic estimation in slitregions). Some discussion of special topicsappears in an Addendum.

The book is well written and can be rec-ommended to anyone interested in realand complex analysis. (pp)

O. H. Kropholler, G. A. Niblo and R.Stöhr (eds.), Geometry and Cohomology inGroup Theory, London Mathematical SocietyLecture Note Series 252, Cambridge UniversityPress, Cambridge, 1998, 316 pp., £24.95,ISBN 0-521-63556-XThis volume is the proceedings of theDurham Symposium on Geometry andCohomology in Group Theory, held inJuly 1994, and contains 18 articles thatprovide a mixture of new results and sur-veys suggesting a framework for futureresearch. The longest survey is by Linnellwho studies an analytical version of thezero divisor conjecture over C when α (≠ 0)is in CG and β (≠ 0) is in Lp(G). He showsthat for p > 2 one can construct many ele-ments α in CG for which there are zerodivisors β in Lp(G). However, he conjec-tures that no such α and β exist when p =2. Much of the discussion of this case is

concerned with the classical right quotientring U(G) of the von Neumann algebraW(G), as the above conjecture is proved tobe true in the case when there exists a divi-sion ring D between CG and U(G).

In one of the longer papersMikhajlovskii and Olshanskii first give suf-ficient and necessary conditions for anHNN-extension of a hyperbolic group G tobe hyperbolic, when A and B are isomor-phic infinite elementary subgroups of G(here, elementary means cyclic-by-finite).Using this theorem they show that everynon-elementary hyperbolic group has anon-trivial verbally complete quotient thatis a torsion group, and give a similar resultwith respect to torsion-free hyperbolicgroups.

Further papers include a report byCarlson on some recent developments inthe area of quotient categories of modulesfiltered by complexity, a survey paper byCornick on homological techniques forstrongly graded rings, a paper by F.Jonson on polysurface groups, a survey byJ. S. Wilson on finitely presented solublegroups, and a paper by R. I. Grigorchuckwhich starts a systematic investigation ofabstract Tychonoff groups. (ad)

V. S. Kulikov, Mixed Hodge Structures andSingularities, Cambridge Tracts inMathematics 132, Cambridge University Press,Cambridge, 1998, 186 pp., £30, ISBN 0-521-62060-0Let f : (Cn+1, 0) → (C, 0) be a germ of holo-morphic function with 0 as an isolated sin-gularity. The main aim of the book is tostudy this singularity by means of globalmethods of algebraic geometry or the the-ory of analytic spaces. We mention thatthe global appears in this local situationroughly as follows. Let B be an open ballin Cn+1 of radius ε, S be an open disc in Cof radius δ, and S = S 0. We let X = B∩ f -1(S) and obtain a map f : X → S.Finally, setting X = X f -1(0) and restrict-ing f, we obtain a mapping f : X → S. If εand δ << ε are sufficiently small, then f isa smooth locally trivial fibration, called aMilnor fibration. This is already a veryimportant global object which representsthe starting point for the investigation ofthe singularity; the Gauss-Manin connec-tion, mixed Hodge structures, and the the-ory of period maps play an important rolehere. The book is on the contemporaryresearch level and will be interesting forspecialists in singularity theory, algebraicand differential geometry. The authorassumes the knowledge and training usualin algebraic and analytic geometry; thisincludes knowledge of sheaf theory andthe technique of spectral sequences. Thebook is nicely written, and with the aboveprerequisites makes this interesting topicaccessible for postgraduate students. Thebibliography has 78 items. (jiva)

T. Y. Lam, Lectures on Modules and Rings,Graduate Texts in Mathematics 189, Springer,New York, 1999, 557 pp., DM119, ISBN 0-387-98428-3This textbook is devoted to the basic partsof the modern structural theory of rings

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and modules. The first chapter deals withfree, projective and injective modules andthe second is devoted to flat modules andhomological dimensions. Chapter 3 treatssome special questions, such as uniformdimensions, singular submodules and non-singular rings, as well as the dense sub-modules and rational hulls. This is fol-lowed by two chapters on various types ofrings of quotients. In Chapter 6,Frobenius and quasi-Frobenius rings areinvestigated, and the last chapter dealswith matrix rings, categories of modulesand Morita theory, including equivalencesand dualities. (lbi)

E. M. Landis, Second Order Equations ofElliptic and Parabolic Type, Translations ofMathematical Monographs 171, AmericanMathematical Society, Providence, 1998, 203pp., ISBN 0-8218-0857-5This book is based on the authors lecturesat Moscow State University.

The main tool in Chapter 1 (non-self-adjoint elliptic equations) and in Chapter 3(parabolic equations) is the use of sub- andsuper-fundamental solutions, constructedwith the help of Riesz potentials. Thisapproach is used for qualitative studies ofthe behaviour of solutions near boundarypoints (Wiener-type theorems) as well as atinfinity (Phragmén-Lindelöf and Liouvilletheorems). Adding the assumptions ofCordes type on the spreading of the eigen-values of the coefficient matrix, the authoruses the same method to obtain a prioriestimates for the Hölder norms of solu-tions. It implies the existence of solutionsto boundary value problems for quasi-lin-ear equations. In Chapter 2 (self-adjointelliptic operators), the estimates for theHölder norms of solutions are derived withhelp of technical tools different from thoseused by E. De Giorgi and J. Moser.

The Russian edition of the bookappeared 26 years ago. Since then, pro-found new results have been obtained,such as the well-known results of Krylovand Safonov on the estimate of Höldernorms of the solutions without the assump-tions of Cordes type. However, a signifi-cant part of the book presents a non-tradi-tional approach which is currently of inter-est to specialists and cannot be found inother monographs and textbooks.

There are probably few mathematicaltexts that present these deep results insuch a vivid and readable manner. Mostsections include bibliographic remarks andcomments, some of which were added bythe author to the 1997 English edition. Ienjoyed reading them - they reflect brieflythe exciting way in which the subject hasbeen developed.

The book can be highly recommendedto all mathematicians working in the theo-ry of PDEs of elliptic and parabolic types.The precise and clear exposition alsomakes it attractive for graduate students.(oj)

R. Laubenbacher and D. Pengelley,Mathematical Expeditions. Chronicles bythe Explorers, Undergraduate Texts inMathematics. Readings in Mathematics,

Springer, New York, 1998, 275 pp., DM69,ISBN 0-387-98433-8 and 0-387-98433-XThe book is designed for those who preferto go back to the masters. The organisa-tion of the book is clear from the titles ofits chapters: Geometry: the parallel postu-late; Set theory: taming the infinite;Analysis: calculating areas and volumes;Number theory: Fermats last theorem;Algebra: the search for an elusive formula.Besides the historical development ofthese themes, the reader will find the rele-vant parts (in English translation) of theworks of the most important contributorsto the subject.

The book provides exciting reading forthose who wish to see how the subjectsevolved in time, what notation or languagewas used, and how the various ideasunfolded. The book can be recommendedfor anybody interested in the history of theabove branches of mathematics. (spor)

J. Lefort, La saga des calendriers ou le fris-son millénariste, Pour la Science, Paris,1998, 191 pp., ISBN 2-9029-003-5The book deals with the origin and devel-opment of different lunar and solar calen-dars (Gregorian, Jewish, Chinese, etc.),compares them, and presents algorithmsfor passing from a date in one calendar toa date in another, and algorithms for cal-culating the date of Easter. The reader willfind interesting information and get anastronomical explanation for why it wasdifficult for our ancestors to measure time.The book contains pictures (photographsand reproductions), well-arranged tablesand mathematical formulas. It is nicelydesigned and will provide interesting read-ing for a wide public. (efas)

I. G. MacDonald, Symmetric Functionsand Hall Polynomials, 2nd Edition, OxfordMathematical Monographs, Clarendon Press,Oxford, 1998, 475 pp., £35, ISBN 0-198-50450-0 and 0-198-53489-2The first edition of this fundamentalmonograph on symmetric functions andHall polynomials appeared in 1979. ARussian translation was published in 1985.This translation was substantially extend-ed, partially in cooperation with the trans-lator A. Zelevinsky. The second editionappeared in 1995 as a hardback, and rep-resents a further large extension. (TheRussian translation has 222 pages!) Thebook under review, published in 1998, isan unchanged version of the second edi-tion, appearing this time as a paperback.

The main aim of the book is to presentthe theory of Hall polynomials and theirapplications. The Hall polynomials areclosely connected with symmetric func-tions, and thus the author starts with arather long chapter (178 pp.) on symmet-ric functions. Indeed, this first chaptermakes the book a standard reference onsymmetric functions. The Hall polynomi-als were discovered only in the 1950s, buthave turned out to play an important rolein many areas of mathematics. For thisreason, MacDonalds book is interestingfor mathematicians working in variousdirections. The prerequisities for the read-

ing are very modest, and the book maytherefore be accessible even to undergrad-uate students. A large part of the book isdevoted to Examples; these substantiallyextend the theory contained in the basictext, and serve also as exercises and prob-lems. Because the book naturally dealswith many formulas, we find at the end ahelpful index of notations (for each chap-ter separately). This book should be avail-able in every library. (jiva)

G. Micula and S. Micula, Handbook ofSplines, Mathematics and Its Applications 462,Kluwer Academic Publishers, Dordrecht, 1999,604 pp., £174, ISBN 0-7923-5503-2This excellent book covers the global theo-ry of spline functions and their applica-tions to various fields, from the introduc-tion of the word spline by I. J. Schoenbergin 1946 to the newest theories of spline-wavelets and spline-fractals.

The book is divided into eleven chap-ters. Chapter 1 introduces the polynomialspline functions and their fundamentalproperties. Chapters 2 and 3 are devotedto multivariate and non-linear sets ofspline functions. The most importantmethods for the numerical solution of inte-gral equations and ordinary differentialequations are treated in Chapters 4 and 5.Chapter 6 shows that the most naturalframework for using spline functions isthat of finite element methods. Chapter 7is a thorough presentation of the finite ele-ment method for the solution of boundaryvalue problems for partial differentialequations. Using suitable spaces of splinefunctions, finite element methods for ellip-tic Dirichlet and Neumann problems andsome non-linear partial differential equa-tions are developed. The spline colloca-tion methods for parabolic and hyperbolicproblems in two space variables are dis-cussed. Chapter 8 is devoted to splinecurves, spline surfaces and their B-splinerepresentation for computer-aided geo-metric design; the rational point of view isalso studied. Chapter 9 briefly describes amodel of shape that combines determinis-tic splines and stochastic fractals, inherit-ing their complementary features. Thenotions of box splines and multivariatetruncated powers are introduced inChapter 10. Chapter 11 is a brief intro-duction to wavelet analysis, and some newaspects of the numerical methods usingspline wavelets for the solution of evolutionpartial differential equations are discussed.The references section at the end of thisbook aims to be the most exhaustive possi-ble (218 pages). All publications known tothe authors up to August 1998 are listed,and subdivided into three sections: books,monographs and conference reports; orig-inal papers; and dissertations for a doctor-al degree or habilitation.

This book can be strongly recommend-ed to researchers and graduate studentsinvolved in numerical methods and com-putation, approximations, differentialequations, and integral equations. (kn)

J. C. Migliore, Introduction to LiaisonTheory and Deficiency Modulus, Progress in

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Mathematics 165, Birkhäuser, Boston, 1998,215 pp., DM98, ISBN 0-8176-4027-4, ISBN3-7643-4027-4This is a highly specialised monographthat provides a very good introduction tocontemporary research in the fields of liai-son theory and deficiency modules. It willbe of interest first to algebraic geometersand algebraists.

Liaison (or linkage) is an equivalencerelation among subschemes of givendimension in a projective space Pn over analgebraically closed field. Two subschemesare directly CI-linked if their union is acomplete intersection, and directly G-linked if their union is arithmeticallyGorenstein. While the first notion is rela-tively classical, the second is rather newand represents an area of current research.The deficiency modules are defined for aclosed subscheme V of Pn of dimension ras cohomology modules H*i(IV), 1 ≤ i ≤ r, ofthe ideal sheaf of V, and measure the fail-ure of V to be arithmetically Cohen-Macaulay. They appear as an importanttool for the study of the liaison. To readthis book the reader needs an introductoryknowledge of algebraic geometry and com-mutative algebra, but the author tries tohelp as much as possible. The first chapterBackground presents some less standardnotions and results needed in the book.

The author pays great attention to moti-vation and the geometric aspects of thetheory. There are many examples throughwhich the reader is introduced into thetheory, thereby stimulating research inter-est in this field; some of these are comput-ed using the program Macaulay. Thebook will be useful both for specialists andfor postgraduate students in these areas.The bibliography includes 203 items andgoes up to 1997. (jiva)

A. D. Osborne, Complex Variables andtheir Applications, International MathematicsSeries, Addison Wesley Longman, Harlow,1998, 454 pp., £19.95, ISBN 0-201-34290-1The authors intention was to write a versa-tile book suitable for both first and secondcourses in complex analysis both for math-ematicians and for those interested inapplications. The only assumed knowl-edge is calculus and basic real analysis. Allimportant techniques and applications arecovered. The book contains a range ofexercises from standard to challengingquestions; there is a manual providing fullsolutions to all questions, available to lec-turers from the publishers. In addition toclassical material, the book includesasymptotic series and elliptic functions aswell as applications to ODEs and integraltransforms. Everything is carefullyexplained with a great sense for clear andstraightforward approach, accompanied bynumerous short historical remarks. Manypictures and very good typographical pre-sentation make reading pleasant endenjoyable. (jiva)

K. H. Parshall, James Joseph Sylvester:Life and Work in Letters, Clarendon Press,Oxford, 1998, 321 pp., £55, ISBN 0-19-

850391-1This volume presents for the first time 140letters from Sylvesters correspondence,selected from some 1200 letters from manyarchives, libraries and private collectionsin the USA, Great Britain, Germany andFrance. These letters cover Sylvesters lifeand work and provide a detailed look at histhoughts and thought processes. Theyshow him in both personal and profession-al spheres during his 82 years. The lettersreflect his research activities, the range ofhis correspondents, and his interests.

The letters are divided into six chap-ters. Each chapter opens with a short pre-lude that gives a thematic overview of itssubject. An extensive historical and math-ematical commentary accompanies the let-ters, and references to pertinent secondaryliterature (mathematical and historical)and bibliography are included. Chapter 1contains letters from 1834 to 1849,Chapter 2 spans the years 1850 to 1854when Sylvester formulated his theory ofinvariants, the 22 years from 1855 to 1876are covered by Chapters 3 and 4, Chapter5 is on Sylvester at Johns HopkinsUniversity (1876 to 1883) and the finalchapter is devoted to the years 1884 to1897. (mnem)

A. Pasini, Elementi di Algebra e Geometria.I. Naziono di base, II. Elementi di Algebra,III. Algebra lineare e Geometria, LiguoriEditore s.r.l., Napoli, 1998, 179/183/524 pp.,22000L/22000L/ 49000L, ISBN 88-207-2739-0, 88-207-2740-4 and 88-207-2741-2This book is designed for students whohave begun to study university mathemat-ics; it is divided into three volumes.

The first volume contains basic notionsof mathematics (sets, maps, relations,logic, finite sets, binomial coefficients,mathematical induction, infinite sets,countable sets, axiom of choice, cardinalnumbers). The second volume covers thenecessary algebraic background (fields,rings, groups, semigroups); substantialparts of this volume are devoted to the realand complex fields, and to polynomials inone and several variables.

The third volume is devoted to linearalgebra and geometry. It begins with clas-sical material on vector spaces and linearmaps (subspaces, linear independence,bases, dimension, direct sums, etc.) and onbasic applications, such as systems of linearequations. Also included are inner prod-uct vector spaces (orthogonal bases,orthogonalisation, etc.). The next threechapters present standard material onaffine and Euclidean geometry. In thefinal chapters, linear algebra and geometryare presented as closely linked subjects; acentral idea is the matrix representation oflinear maps (algebra of linear maps, alge-bra of matrices, eigenvalues and eigenvec-tors, orthogonal and unitary transforma-tions and matrices, affine transformations,transposed and conjugate complex matri-ces, bilinear and quadratic forms, conicsections and quadrics).

Each volume includes exercises and anindex. They are well written and can berecommended for basic study. (jbe)

J. J. Risler (directeur de la publication),Matériaux pour lhistoire des mathéma-tiques au XXe siècle, Séminaires & Congrès3, Société Mathématique de France, Paris,1998, 282 pp., ISBN 2-85629-065-5This book is the third volume of materialsfor the history of mathematics in the XXthcentury, dedicated to the memory of JeanDieudonné (1906-92). The volume con-tains a short introduction that describesthe history of the book. In January 1996,on the occasion of the inauguration of thenew pavilion of Laboratoire J. A.Dieudonné at the Université de Nice -Sophia Antipolis, there was a conferenceorganised by the University, lAcadémie deSciences, dInstitut des Hautes ÉtudesScientifiques et du Centre National de laRecherche Scientifique. The book containsthe conference programme, with elevencontributions in French and English deliv-ered at the conference. They are devotedto fundamental ideas of 20th-centurymathematics, such as hyperbolic equations,functional analysis, homotopy theory offibre spaces, group representations, sheaftheory, the mathematical theory ofBrownian motion, and Hilberts twelfthproblem. At the end, an alphabetical indexis included.

This book can be recommended to allmathematicians interested in the history ofmodern mathematics. (mnem)

P. C. Roberts, Multiplicities and ChernClasses in Local Algebra, Cambridge Tractsin Mathematics 133, Cambridge UniversityPress, Cambridge, 1998, 303 pp., £37.50,ISBN 0-521-47316-0It is well known that the development ofalgebraic geometry greatly influencedprogress in algebra. In recent decades, thereverse trend appeared and proved to bevery fruitful; namely, methods of algebraicgeometry started to be applied within theframework of pure algebra. The main ideaconsists in substituting objects of algebraicgeometry by more general objects in purealgebra. This book represents an exposi-tion of some parts of this modern andrapidly developing branch of algebra.

An important notion in algebraic geom-etry is that of multiplicity of intersection.The author uses two notions of multiplici-ty. The first generalises the multiplicity ofintersection of a plane curve with itself andis called Samuel multiplicity. The secondgeneralises the multiplicity of intersectionof two subvarieties of a variety, and itsrequires the use of homological algebra; itis called Serre multiplicity. Most of thisbook is devoted to the introduction andinvestigation of these (and other) multi-plicities. For these purposes a lot of tech-niques are developed. We mention in par-ticular the Chern classes of locally freesheaves, Chern character, and local Cherncharacters. All these notions were inspiredby the corresponding notions of algebraictopology, but they are defined in purelyalgebraic terms.

The book is nicely written, and serveswell as an introduction to this interestingbranch of algebra, as well as containing

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very recent results. The necessary prereq-uisities from algebra and algebraic geome-try are rather modest, but a good back-ground in algebraic geometry gives thereader better motivation. There are manyexamples to make the exposition moreattractive, and there are exercises at theend of each chapter. The theory present-ed here is a beautiful interplay of severalbranches of mathematics and providesgood reading for algebraists, geometers,and topologists. (jiva)

K. A. Ross, J. M. Anderson, G. L.Litvinov, A. I. Singh, V. S. Sunder and N.J. Wildberger (eds.), Harmonic Analysisand Hypergroups, Trends in Mathematics,Birkhäuser, Boston, 1998, 249 pp., DM208,ISBN 3-764-33943-8, ISBN 0-817-63943-8This book contains the proceedings of aconference on harmonic analysis on hyper-groups held in December 1995 in Delhi.

Much of this book is concerned with thenotion of hypergroups. Let M(H) be thespace of all bounded Borel measures on alocally compact topological space H. If His a group, then there is a canonical prod-uct * defined on M(H) by δx * δy = δxy. Fora general H, if * is a product on M(H) sat-isfying a certain set of axioms (generalisingproperties of M(H) from the group case),the couple (H,*) is called a hypergroup.Many results in classical harmonic analysis(the Fourier transform, the Plancherel andinversion formulas, the Plancherel theo-rem) can be extended to hypergroups.

Signed hypergroups and relations toMarkov chains are discussed in the contri-bution by K. A. Ross. Connections amongcharacters of hypergroups, families oforthogonal polynomials (such as Legendrepolynomials) and the Sturm-Liouvilleproblem appear in the three lectures by A.L. Schwartz. An extension of the wavelettransform to hypergroups is discussed inthe paper by K. Trimeche. An approach tonon-commutative harmonic analysis on aLie group, based on an old idea ofFrobenius, and its relation to hypergroupsis described in the contribution of N. J.Wildberger.

Other topics include De Branges mod-ules (S. Agrawal and D. Singh), momentfunctions on hypergroups (L. Gallardo),behaviour of the Plancherel measure andmultiplier theorems (M.-O. Gebuhrer),disintegration of measures (H. Helson),multipliers of de Branges-Rovnyak spaces(B. A. Lotto and D. Sarason), applicationsto measures on compact spaces (R. Nair),applications to functional equations (H.Stetkaer), actions of finite hypergroups onfinite sets (V. S. Sunder and N. J.Wildberger), positivity of Turán determi-nants (R. Szwarc), relations to semigroupsof positive definite functions (M. E. Walter)and limit theorems for random walks (H.Zeuner).

The book contains nice survey paperson properties of hypergroups, as well aspapers presenting applications and rela-tionships with many different parts ofmathematics. (vs)

R. P. Stanley, Enumerative Combinatorics,

Vol. 2, Cambridge Studies in AdvancedMathematics 62, Cambridge University Press,Cambridge, 1999, 581 pp., £45, ISBN 0-521-56069-1The book is a continuation of Volume 1(1986), starting with Chapter 5, Trees andthe composition of generating functions, andcontinuing with Chapter 6, Algebraic, D-finite, and noncommutative generating func-tions, and Chapter 7, Symetric functions. Thelast chapter has two appendices: KnuthEquivalence, Jeu de taquin, and theLittlewood-Richardson rule (by S. Fomin)and The characters of GL(n, C). Eachchapter has its own set of exercises withsolutions, 261 in all but many more whensub-exercises are counted. The referencesto them and the bibliographies to thechapters constitute a giant survey of litera-ture on enumerative combinatorics.

What else is to be added to our com-ments on this excellent book? Perhaps aquotation from G.-C. Rotas foreword:Every once in a long while, a textbookworthy of the name comes along; ...Weber, Bertini, van der Waerden, Feller,Dunford and Schwartz, Ahlfors, Stanley.(mkl)

T.-T. Tay, I. Mareels and J. B. Moore,High Performance Control, Systems &Control: Foundations & Applications,Birkhäuser, Basel, 1998, 344 pp., sFr148,ISBN 3-764-34004-5 and 0-817-64004-5The main theme of this book is to deter-mine whether high performance can beachieved in the face of uncertainity. Thefirst three chapters are introductory, withattention paid to the description of all sta-bilising controllers for which the factorisa-

tion approach is developed; various normsare used as performance measures.Chapters 4 and 5 are devoted to an opti-mal design for tracking some signals orrejecting disturbances. It is supposed thatthe nominal plant G and the nominal con-troller K are augmented by a plant G and acontroller Q. Optimisation techniques aredeveloped for both types of designs: off-line and iterated or recursive (Q, S)-designs (for S-parametrisation). The nexttwo chapters describe direct and indirectadaptive-Q controls; in the latter, con-troller designs are based on on-line identi-fied models. As the adaptation proceedsslowly compared to the plant dynamics,averaging techniques are used. In Chapter8 it is shown how to apply the direct adap-tive-Q algorithm to non-linear systems bymeans of linearisation. The concludingtwo chapters are closer to real applications:the real-time implementation is discussedand three examples (a disc driver controlsystem, control of a heat exchanger and aflight control system) are studied in detail.

The book is directed to graduate stu-dents in system theory; beginners are firstrecommended to read the book FeedbackControl Theory by J. Doyle, B. Francisand A. Tannenbaum (Macmillan, NewYork, 1992). The proofs are sometimessketchy and more attention is paid to moti-vation and to system philosophy. Theextensive bibliography with comments onthe references is also very useful. Thisbook, together with (e.g.) I. Meerels and J.Poldermans Adaptive Control Systems(Birkhäuser, Basel, 1996), provides a goodaccount of the research that has been doneduring the last decade. (jmil)

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Centre de Recerca Matematica (Barcelona)List of visitors, September - December 1999

J. L. Balcazar, Barcelona, 5 Sep - 10 Dec 99, Computational theoryK. Baranski, Warszawa, 1 Dec 98 - 31 Oct 99, AnalysisM. Brunella, Dijon, 7 - 30 Nov 99, GeometryA. Candel, California, 22 Nov - 20 Dec 99, GeometryW. Chacholski, Connecticut, 1 - 31 Oct 99, Algebraic topologyJ. A. Crespo, Barcelona, 1 Sep 99 - 31 Jul 00, Algebraic topologyM. Cruz, Mexico, 10 Nov - 10 Dec 99, GeometryF. X. Dehon, Paris, 1 Oct 99 - 30 Sep 01, Algebraic topologyS. Dumitrescu, Lyon, 1 - 20 Nov 99, GeometryK. Faure, Toulouse, 1 Sep 99 - 30 Jun 00, Geometry J. M. Gambaudo, Dijon, 7 - 20 Nov 99, GeometryF. Gautero, Valbonne, 14 Sep 98 - 30 Sep 00, Dynamical systemsE. Ghys, Lyon, 1 - 21 Nov 99, GeometryA. Guillot, Lyon, 1 - 20 Nov 99, GeometryP. Koskela, Jyvaskyla, 18 Jan - 31 Dec 99, AnalysisM. Lagrange, Dijon, 8 - 20 Nov 99, GeometryS. Lamy, Prest, 1 Sep 99 - 30 Jun 00, GeometryJ. J. Loeb, Angers, 1 - 15 Nov 99, GeometryF. Loray, Villeneuve dAscq, 24 Oct - 20 Nov 99, GeometryM. McQuillan, Oxford, 7 Nov - 3 Dec 99, GeometryM. Mimura, Okayama, 1 Oct - 30 Nov 99, Algebraic topology I. Morrison, New York, 1 Jan - 31 Dec 99, Applied mathematicsH. R. Morton, Liverpool, 1 - 31 Oct 99, Applied mathematicsF. Sanchez-Bringas, Mexico, 1 - 15 Dec 99, GeometryJ. Seade, Mexico, 30 Nov - 14 Dec 99, GeometryC. Tarquini, Rennes, 1 Sep 99 - 30 Jun 00, GeometryA. Verjovsky, Mexico, 10 Nov - 15 Dec 99, GeometryX. Zhang, Beijing, 1 Mar 99 - 28 Feb 00, Dynamical systems

Journal of the European Mathematical JournalThe first issue of this EMS journal appeared in January 1999. The contents of the first two issues were givenin the June Newsletter. Further information can be obtained by e-mail: [email protected]

Volume 1, Number 3F. Lin and T. Rivière, Complex Ginzburg-Landau equations in high dimensions and codimensiontwo area minimizing currents J. Kollár, Effective Nullstellensatz for arbitrary idealsErratum to M. Burger and N. Monod, Bounded cohomology of lattices in higher rank Lie groups

Call for MonographsPortuguese Mathematical Society (SPM)

Editor-in-Chief: A. B. CruzeiroEditorial board: I. Fonseca, E. Marques de Sà, M. Pollicott, J. Rezende

This new series will publish research monographs of high scientific level in all domains of mathematics. Each mono-graph should be around 100 pages at least and, as far as possible, present a self-contained exposition of the subject.These can be written in English, French or Portuguese, but preferably in English.Proposals should be addressed to A. B. Cruzeiro, Grupo de Fsica Matemàtica da Universidade de Lisboa, Av. Prof.Gama Pinto 2, P-1649-003 Lisboa, Portugal.tel: +351-1-790-4726; fax: +351-1-795-4288e-mail: [email protected]

Post-doctoral Positions in Stochastic AnalysisWe wish to draw the attention of strong young researchers in stochastic analysis to the possibili-ty of research positions funded by the TMR (Training-Mobility-Research) contract in StochasticAnalysis ERB-FMRX-CT96-0075 of the European Union.

There are currently up to four one-year positions available. These positions are to carry outresearch within one of the teams of the contract, and the researcher would be based at one of thefollowing laboratories:(Prof. T. J. Lyons) Mathematics Dept, Imperial College, Huxley Building, 180 Queens Gate,London SW7 2BZ, UK ([email protected])(Prof. D. Elworthy) Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK([email protected])(Prof. Y. Le-Jan) Mathematiques, Université de Paris-Sud, Centre dOrsay, 91405 Orsay, Cedex,France ([email protected])Each institution has a top-quality mathematics department, and within it, has strong researchactivity in probability and stochastic analysis. Anyone who is well qualified and interested in oneof the positions should get in immediate contact with the relevant laboratory mentioned aboveand also send a note recording their interest to Prof. T.J. Lyons at Imperial College.

Please note that the European Union imposes strong nationality, and age restrictions on thesepositions. Researchers must be a national of a member State or an associated State of theCommunity (Austria, Belgium, Denmark, Finland, France, Germany, Greece, Iceland, Ireland,Italy, Liechtenstein, Luxembourg, Netherlands, Norway, Portugal, Spain, Israel, Sweden and theUK) and not from the country where the position is to be held. Details can be found onhttp://sagwww.ma.ic.ac.uk/.


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