NEWSLETTER No. 48June 2003

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

Editorial by David Salinger ......................................................................... 3

Introducing the Committee, part 2 ................................................................ 5

Popularising mathematics: from eight to infinity ......................................... 7

Brought to book: the curious story of Guglielmo Libri ................................ 12

Interview with D V Anosov, part 2 .............................................................. 15

Latvian Mathematical Society .................................................................... 21

Slovenian Mathematical Society: further comments ................................... 25

Forthcoming Conferences ............................................................................ 26

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

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NOTICE FOR MATHEMATICAL SOCIETIESLabels for the next issue will be prepared during the second half of August 2003.Please send your updated lists before then to Ms Tuulikki Mäkeläinen, Department ofMathematics, P.O. Box 4, FIN-00014 University of Helsinki, Finland; e-mail:makelain@cc.helsinki.fi

INSTITUTIONAL SUBSCRIPTIONS FOR THE EMS NEWSLETTERInstitutes and libraries can order the EMS Newsletter by mail from the EMS Secretariat,Department of Mathematics, P. O. Box 4, FI-00014 University of Helsinki, Finland, or by e-mail: (makelain@cc.helsinki.fi). Please include the name and full address (with postal code), tele-phone and fax number (with country code) and e-mail address. The annual subscription fee(including mailing) is 80 euros; an invoice will be sent with a sample copy of the Newsletter.

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

ASSOCIATE EDITORSVASILE BERINDEDepartment of Mathematics,University of Baia Mare, Romania e-mail: vberinde@univer.ubm.ro KRZYSZTOF CIESIELSKIMathematics Institute Jagiellonian UniversityReymonta 4 30-059 Kraków, Polande-mail: ciesiels@im.uj.edu.plSTEEN MARKVORSENDepartment of Mathematics Technical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: s.markvorsen@mat.dtu.dk

SPECIALIST EDITORSINTERVIEWSSteen Markvorsen [address as above]

SOCIETIESKrzysztof Ciesielski [address as above]

EDUCATIONTony GardinerUniversity of BirminghamBirmingham B15 2TT, UKe-mail: a.d.gardiner@bham.ac.uk

MATHEMATICAL PROBLEMSPaul JaintaWerkvolkstr. 10D-91126 Schwabach, Germanye-mail: PaulJainta@aol.com

ANNIVERSARIESJune Barrow-Green and Jeremy GrayOpen University [address as above]e-mail: j.e.barrow-green@open.ac.ukand j.j.gray@open.ac.uk

CONFERENCESVasile Berinde [address as above]

RECENT BOOKSIvan Netuka and Vladimir Sou³ekMathematical InstituteCharles UniversitySokolovská 8318600 Prague, Czech Republice-mail: netuka@karlin.mff.cuni.czand soucek@karlin.mff.cuni.cz

ADVERTISING OFFICERVivette GiraultLaboratoire d’Analyse NumériqueBoite Courrier 187, Université Pierre et Marie Curie, 4 Place Jussieu75252 Paris Cedex 05, Francee-mail: girault@ann.jussieu.fr

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CONTENTS

EMS March 2003

EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

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

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

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

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

ORDINARY MEMBERSProf. VICTOR BUCHSTABER (2001–04)Steklov Mathematical InstituteRussian Academy of SciencesGubkina St. 8, Moscow 117966, Russiae-mail: buchstab@mendeleevo.ru Prof. DOINA CIORANESCU (2003–06)Laboratoire d’Analyse NumériqueUniversité Paris VI4 Place Jussieu75252 Paris Cedex 05, Francee-mail: cioran@ann.jussieu.frProf. PAVEL EXNER (2003-06)Mathematical InstituteCharles UniversitySokoloská 8318600 Prague, Czech Republice-mail: exner@ujf.cas.czProf. MARTA SANZ-SOLÉ (2001-04)Facultat de MatematiquesUniversitat de BarcelonaGran Via 585E-08007 Barcelona, Spaine-mail: sanz@cerber.mat.ub.esProf. MINA TEICHER (2001–04)Department of Mathematics and Computer ScienceBar-Ilan UniversityRamat-Gan 52900, Israele-mail: teicher@macs.biu.ac.il

EMS SECRETARIATMs. T. MÄKELÄINENDepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlandtel: (+358)-9-1912-2883fax: (+358)-9-1912-3213telex: 124690e-mail: makelain@cc.helsinki.fiwebsite: http://www.emis.de

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EMS NEWS

EMS June 2003

2003

30 June-4 July EMS Summer School at Porto (Portugal) Dynamical Systems Organiser: Maria Pires de Carvalho, e-mail: mpcarval@fc.up.ptwebpage: http://www.fc.up.pt/cmup/sds

6-13 July CIME-EMS Summer School at Bressanone/Brixen (Italy)Stochastic Methods in FinanceOrganisers: Marco Frittelli and Wolfgang J. Runggaldier e-mail: runggal@math.eunipd.itwebpage: www.math.unifi.it/~cime

15 August Deadline for submission of material for the September issue of the EMSNewsletterContact: Robin Wilson, e-mail: r.j.wilson@open.ac.uk

12-14 SeptemberSPM-EMS Weekend Meeting at the Calouste Gulbenkian Foundation,Lisbon (Portugal)Organiser: Rui Loja Fernandes, e-mail: rfern@math.ist.utl.pt

14-15 September EMS Executive Committee meeting at Lisbon (Portugal) Contact: Helge Holden, e-mail: holden@math.ntnu.no

15 November Deadline for submission of material for the December issue of the EMSNewsletterContact: Robin Wilson, e-mail: r.j.wilson@open.ac.uk

2004

31 JanuaryClosing date for nominations for delegates to the EMS Council to representthe individual membership.Contact: Tuulikki Mäkeläinen, e-mail: makelain@kantti.helsinki.fi

25-27 June EMS Council Meeting, Stockholm (Sweden)

27 June-2 July4th European Congress of Mathematics, Stockholmwebpage: http://www.math.kth.se/4ecm

2-6 September EMS Summer School at Universidad de Cantabria, Santander (Spain) Empirical processes: theory and statistical applications

EMS AgendaEMS Committee

Cost of advertisements and inserts in the EMS Newsletter, 2003(all prices in British pounds)

AdvertisementsCommercial rates: Full page: £210; half-page: £110; quarter-page: £66 Academic rates: Full page: £110; half-page: £66; quarter-page: £40 Intermediate rates: Full page: £160; half-page: £89; quarter-page: £54

Inserts Postage cost: £13 per gram plus Insertion cost: £52

The statutes of the EMS state:The purpose of the Society is to promote thedevelopment of all aspects of mathematics in thecountries of Europe, with particular emphasison those which are best handled on an interna-tional level.

The Society will concentrate on those activitieswhich transcend national frontiers and it will inno way seek to interfere with the national activ-ities of the member societies.

In particular, the Society will, in theEuropean context, aim to promote mathematicalresearch (pure and applied), assist and adviseon problems of mathematical education, concernitself with the broader relations of mathematicsto society, foster the interaction between mathe-maticians of different countries, establish asense of identity amongst European mathe-maticians, and represent the mathematicalcommunity in supra-national institutions.

I have emphasised one aim, because itseems to me that it is different from theothers, in that it is overtly political. Whyshould we foster a sense of identity amongEuropean mathematicians? If we acceptthis as a purpose, how can we tell whetherwhat we do achieves this purpose?

Mathematics, it can be argued, is inter-national and we should regard highly goodmathematics, whatever its provenance. Soany attempt at creating a ‘European iden-tity’ is, in principle, divisive and should beavoided.

I think this is a valid argument and thatthere is a danger of such division: but theSociety is reaching out to non-EuropeanSocieties (in particular, the AMS andSIAM) to facilitate European mathemati-cians’ membership of those societies (andvice versa). As a fledgling society, somemembers fear that we may be overwhelmedby more established and richer partners,but, as we grow, this will become less of aproblem (if it is one at all).

But there are positive reasons for foster-ing a European sense of identity. It is theacceptance that, if one European countryhas problems, then it is incumbent on all ofus to help. A small example is the responseto the flood damage of Prague’s mathe-matical library. A more routine example isthe way we devote funds regularly to giveconference grants to mathematicians fromthe poorer European countries – we alsodo this indirectly through applications toUNESCO and the European Frameworkprogrammes.

The European prizes are another way inwhich we try to foster a European identity.The Executive committee has had livelydebates in trying to frame an exact defini-tion of ‘European’ in this context. The twindemands of nationality and place of workare difficult to reconcile in a sensible way. There is a further problem with the word

‘European’: which countries does itinclude? The Society has interpreted thisterm very broadly when it comes to mem-bership: we do not limit ourselves to mem-bers and potential members of theEuropean Union. Again, we are seeking tounite, not divide, the mathematical com-munity.

The European congresses of mathemat-ics are occasions when mathematiciansfrom around the world, but particularlyfrom Europe, meet. There we discuss notonly mathematics, but common problemsof all kinds, informally in the corridors,

formally at round-table discussions. TheStockholm congress in 2004 will bring theFramework 5 networks into the proceed-ings.

The mathematical community cannotavoid politics, if it wishes to prosper, inthat it cannot afford to be absent from theEuropean arena, either for funding, or inmatters of mathematical education. Wehave to remind officials from time to timethat mathematics is basic to all science, andthat it has its own special needs which theFramework programmes should accommo-date. Often, the needs of mathematics areoverlooked (as in the thematic part ofFramework 6) and would be even more dis-regarded if it were not for the tirelessefforts of our colleagues. This is an area inwhich the Society can claim some success.

In mathematical education, the Bolognaprogramme will impinge on all of us, but,so far, the Society has not been able toarrive at a position representative of themathematical community, partly becauseopinion is divided in our community, andpartly because key individuals on theExecutive Committee have been over-loaded with crucial applications to theFramework 6 programme on behalf of theSociety. But in many countries, mathemati-cians are already having to come to termswith degree programmes curtailed in thename of a common European educationalarea.

On a small scale, by its efforts as a team,the Executive Committee finds a commonpurpose. In working together, it sees whatcommon problems need solving and isaware of the diverse strengths comingfrom different traditions which can bebrought to bear upon on those problems,whether totally solvable or not. In short,the EMS Executive has a European identi-ty. By participating in EMS events, mem-bers can share that sense of identity.

EDITORIAL

EMS June 2003 3

Editorial: A EurEditorial: A European Senseopean Senseof Identity?of Identity?

David Salinger (Leeds, UK)

EurEuropean Mathematical Society opean Mathematical Society PPrizes, 2004rizes, 2004

David Salinger (EMS Publicity Officer)

A call for nominations for the 2004 EMS prizes has been issued and is on theEuropean Mathematical Society’s webpages (www.emis.de) and in the March 2003 issueof the EMS Newsletter.

Ten prizes (covering all areas of mathematics) were awarded by the city of Pariswhen the inaugural European Congress of Mathematics was held there in 1992, andthe same number of EMS prizes have been awarded at each European Congress sincethen. These prizes are for young mathematicians, but, in response to comments atthe last Congress, the age limit has now been increased to 35 and a further allowanceof up to three years can be made for career breaks. The prizes are awarded toEuropean mathematicians, meaning those whose nationality is European or whosenormal place of work is in Europe. (When discussing the rules, some members of theExecutive Committee felt unhappy about restricting the prizes to Europe – even ifthat was taken in the widest possible sense – but the Committee as a whole felt thatthe time was not yet ripe for removing this restriction.)

Several of the EMS prizewinners have become Fields medallists. The names of allprizewinners can be found in the information about past congresses on the EMIS web-site.

Nominations have to be made (to the 4ecm Organising Committee, Prof. AriLaptev, Department of Mathematics, Royal Institute of Technology, SE-100 44Stockholm, Sweden) by 1 February 2004, and the nominees will be judged on thestrength of work published by December 2003.

SUMMER SCHOOL of the Centro Internazionale Matematico Estivo and the European Mathematical Society

Supported by UNESCO-Roste Cusanus Akademie Bressanone (Bolzano)

6th-13th July 2003The aim of the SUMMER SCHOOL is to provide a broad and accurate knowledge of some of the most up todate and relevant topics in Mathematical Finance. Particular attention will be devoted to the investigationof innovative methods from stochastic analysis that play a fundamental role in the mathematical modelling

of finance or insurance.

The school will provide an occasion to facilitate and stimulate discussion between all participants, inparticular, young researchers and lecturers, for whom limited financial support will be available.

Courses* Partial and asymmetric information. (Kerry Back, University of St.Louis, USA)* Stochastic Methods in Credit Risk Modeling, Valuation and Hedging

(Tomasz Bielecki, Northeastern Illinois University, USA)* Finance and Insurance (Christian Hipp, University of Karlsruhe, Germany)* Nonlinear expectations and risk measures (Shige Peng, Shandong

University, China)* Utility Maximization in Incomplete Markets (Walter Schachermayer,

Technical University of Vienna, Austria)

Course directorsProf. Marco Frittelli (Università di Firenze, Italy) marco.frittelli@dmd.unifi.it

Prof. Wolfgang Runggaldier (Università di Padova, Italy) runggal@math.unipd.it

Webpage: http://www.math.unifi.it/cime

EuropeanMathematical Society

Stochastic Methods

in Finance

Victor Buchstaber graduated from the Moscow State University (MSU) in 1969 and went on topostgraduate study there, with advisors Sergei P. Novikov and Dmitri B. Fukhs. He received hisPhD in 1970 and a DSc in 1984.

He has been Research Leader of the topology division of the Steklov Mathematical Institute ofthe Russian Academy of Science, Professor in Higher Geometry and Topology at the MSU, andHead of the Mathematical Modelling Division at the National Scientific and Research Institute forPhysical, Technical and Radio-technical Measurements.

He has been on the Council of the Moscow Mathematical Society, Deputy Editor-in-Chief ofUspekhi Mat. Nauk, and Head of the Expert Committee in Mathematics in the Russian Foundationfor Basic Research.

Doina Cioranescu has been Directeur du Recherche at the Centre National de la RechercheScientifique (CNRS), Laboratoire d’Analyse Numérique Paris VI, since 1981. She graduated fromthe University of Bucharest in partial differential equations in 1966, and became associateresearcher at the CNRS in 1971, receiving her Doctorat d’État in 1978 at Université Paris VI.

Her research interests are mainly concentrated in the modelling of non-Newtonian fluids,homogenisation theory (in particular, applied to perforated domains and reticulated structures),and control problems in heterogeneous media. She has been involved in several international sci-entific projects, including being scientific coordinator of the CE Eurohomogenisation Science pro-gramme. Since 1997 she has been a member of the management committee of the French Societyof Applied Mathematics. In January she was joint organiser of the EMS-SMAI-SMF meetingMathématiques Appliquées – Applications des Mathématiques in Nice.

Pavel Exner graduated from Charles University, Prague, in 1969, and has worked in severalplaces, including a dozen years at the Laboratory of Theoretical Physics in Dubna. He currentlyheads a mathematical-physics research group at the Czech Academy of Sciences.

His scientific interests are primarily concerned with mathematical problems and methods ofquantum theory – in particular, unstable systems, scattering theory, functional integration andquantum mechanics on graphs, surfaces, quantum waveguides, and has written about 130 researchpapers and three books on these subjects. He is a member of IAMP and secretary of the IUPAPcommission for mathematical physics.

Marta Sanz-Solé is Professor of Mathematics at the University of Barcelona. She received her PhDthere in 1978. Her research interests lie in probability theory, in particular in stochastic analysis– stochastic partial differential equations, analysis on theWiener space, and Malliavin calculus, andshe has been active in the creation of a research group in stochastic analysis in Barcelona. She hasbeen a visiting professor at the University of North Carolina at Chapel Hill, the MathematicalSciences Research Institute in Berkeley, Université Pierre et Marie Curie in Paris, Universitéd’Angers, Clermont-Ferrand II, Messina, among others.

From 1993 to 1996 she was Dean of the Faculty of Mathematics, and from 2000 to 2003 Vice-President of the Division of Sciences, at the University of Barcelona. She has been a member ofthe EMS Executive Committee since 1997, and was organisational secretary of 3ecm, the ThirdEuropean Congress of Mathematics, in Barcelona in 2000.

Mina Teicher is Professor of Mathematics and Computer Science at Bar-Ilan University, Israel,specialising in algebraic geometry and topology, with applications to theoretical physics (shape ofthe universe), computer vision, and mathematical models in brain research. She is a world leaderin braid techniques and fundamental groups.

She obtained her PhD from Tel-Aviv University and did postdoctoral studies in Princeton. Shehas received many awards and grants, including the prestigious Batsheva de Rothschild award,Excellency Center of the Israel National Academy of Sciences, and a European Commission fund-ed network which was rated first (out of 430) in all fields of science and the humanities.

She has been Chair of the Department of Mathematics and Computer Science at Bar-IlanUniversity, Director of the Emmy Noether Research Institute for Mathematics, and a member ofthe Israeli Council for Higher Education, the National Forum for Brain Research, the IsraeliNational Forum ‘Academia-Industry’ and the council of the Wolf Foundation, and Advisor for theGerman Science Foundation, the European Union and the Italian Science Foundation. She isVice-president for Research at Bar-Ilan University and has created bilateral programmes betweenBar-Ilan and leading centres in Germany, Italy, Poland and India.

EMS NEWS

EMS June 2003 5

IntrIntroducing the Committeeoducing the Committee(part 2)

2003 Abel P2003 Abel Prize awarrize awarded to Jean-Pded to Jean-Pierrierre Serre SerreeThe Niels Henrik Abel Memorial Fund was established on 1 January 2002, to award the Abel Prizefor outstanding scientific work in the field of mathematics. The prize amount is 6 million Norwegiankroner (about 750,000 Euro) and the name of the first Abel laureate was announced at theNorwegian Academy of Science and Letters on 3 April 2003.

The Prize committee, appointed for a period of two years, consisted of: Erling Stoermer (Chair),University of Oslo (Norway), John Macleod Ball, University of Oxford (UK), Friedrich Ernst PeterHirzebruch, Max Planck Institute for Mathematics (Germany), David Mumford, Brown University(USA), and Jacob Palis, Instituto Nacional de Matemática Pura e Aplicada, IMPA (Brazil).

The Norwegian Academy of Science and Letters has awarded the Abel Prize for 2003 to Jean-Pierre Serre,Collège de France, Paris (France), ‘for playing a key role in shaping the modern form of many parts of mathe-matics, including topology, algebraic geometry and number theory’. The Prize was presented by HM KingHarald in a special ceremony at the University of Oslo on 3 June 2003.

Jean-Pierre Serre was born in 1926 in Bages, France. He studied at the École Normale Supérieureand received his DSc in 1951 from the Sorbonne in Paris. After holding a position at the CentreNational de la Recherche Scientifique, he became an associate professor at the Université de Nancy.In 1956 he was appointed a professor at the Collège de France. He was made a Commander Légiond’Honneur and High Officer Ordre National du Mérite. He has been elected to many national academies, in particular, the acad-emies of France, Sweden, the United States and the Netherlands. He was awarded the Fields Medal in 1954 (the youngest recipi-ent ever), the Prix Gaston Julia in 1970, the Balzan Prize in 1985, the Steele Prize in 1995, and the Wolf Prize in 2000. He hasbeen awarded honorary degrees from many universities, most recently from the University of Oslo in 2002 in connection with theAbel Bicentennial.

Serre developed revolutionary algebraic methods for studying topology, and in particular studied the transformations betweenspheres of higher dimensions. He is responsible for a spectacular clarification of the work of the Italian algebraic geometers, byintroducing and developing the right algebraic machinery for determining when their geometric construction works. Serre’s pow-erful technique, with its new language and viewpoint, ushered in a golden age for algebraic geometry.

For the past four decades Serre’s magnificent work and vision of number theory have been instrumental in bringing that subjectto its current glory. This work connects and extends in many ways the mathematical ideas introduced by Abel, in particular hisproof of the impossibility of solving the fifth-degree equation by radicals, and his analytic techniques for the study of polynomialequations in two variables. Serre’s research has been vital in setting the stage for many of the most celebrated recent breakthroughs,including the proof by Wiles of Fermat’s last theorem.

Although Serre’s effort has been directed to more conceptual mathematics, his contributions have connections with importantapplications. The practical issues of finding efficient error-correcting codes and of public-key cryptography make use of solutionsof polynomial equations (specifically over finite fields) and Serre’s work has substantially deepened our understanding of this topic.

Many events took place in ‘Abel week 2003’, including:• an Abel lecture by Jean-Pierre Serre entitled ‘Prime numbers, equations and modular forms’ • a wreath-laying by Serre at Gustav Vigeland’s Abel monument in the Palace grounds• a one-day Abel symposium at the University of Oslo• a special banquet at Akershus Castle, attended by the King and Queen• a Maths Carnival on Universitetsplassen in Central Oslo, involving children of all ages and the awarding of prizes to Year 9 and

upper secondary school pupils for the mathematical contests ‘KappAbel’ and ‘The Abel Competition’

More details of Serre’s work and of the events of Abel week 2003 can be found on the website http://www.abelprisen.no

EMS NEWS

EMS June 20036

FFourth Eurourth European Congropean Congress of Mathematicsess of MathematicsDavid Salinger (EMS Publicity Officer)

In November 2000 in De Morgan House (London), the Executive Committee of the European Mathematical Society accepted onbehalf of Council the offer of holding 4ecm in Stockholm. It is now just over a year to the Congress – 27 June to 2 July 2004 – andmuch of the framework is already in place. There are a poster and a website: http://www.math.kth.se/4ecm/

We are honoured that Lennart Carleson has agreed to be President of the Scientific Committee. The President of the OrganisingCommittee is Ari Laptev, whose good-humoured reports have reassured the Executive Committee that the Congress is in very capa-ble hands.

The Congress has a subtitle: Mathematics in Science and Technology. Although it will try to cover all of mathematics as usual (a daunt-ing task), there will be a special place this time for applications of mathematics; this is in keeping with the Society’s recent efforts toput applied mathematics high on its agenda. Since Stockholm is the home of the Nobel Prizes, a number of Nobel prizewinnershave been asked to speak on the role of mathematics in their discipline. Creating room for this has meant that there will be noround-table discussions at 4ecm, but this is a decision affecting 4ecm alone: it will be open to the organisers of 5ecm to reinstate theround tables.

Another feature of 4ecm is that the current European networks have been invited to hold their annual meetings in associationwith the Congress: some of them may present their work there.

There will be satellite conferences: so far there have been six expressions of interest.4ecm will be the occasion for the presentation of the European Mathematics Prizes. Nina Uralt’seva (St. Petersburg) has beenappointed Chair of the Prize Committee and a call for nominations for prizes is on the website. The Felix Klein Prize will also bepresented at the Congress, and the committee for that is now complete.

Everything is shaping up for a really stimulating Congress. All it needs is for the mathematical community to turn up in droves,as it did in Barcelona.

It is rare to succeed in getting mathematicsinto ordinary conversation without meeting allkinds of reservations. In order to raise publicawareness of mathematics effectively, it is nec-essary to modify such attitudes. Here we pointto some possible topics for general mathemati-cal conversation.

Introduction“One side will make you grow taller, andthe other side will make you grow short-er”, says the caterpillar to Alice in thefantasy Alice’s Adventures in Wonderland asit gets down and crawls away from a mar-vellous mushroom upon which it has beensitting. In an earlier episode, Alice hasbeen reduced to a height of three inchesand she would like to regain her height.Therefore she breaks off a piece of eachof the two sides of the mushroom, inci-dentally something she has difficultiesidentifying, since the mushroom is entire-ly round. First Alice takes a bit of one ofthe pieces and gets a shock when her chinslams down on her foot. In a hurry sheeats a bit of the other piece and shoots upto become taller than the trees. Alice nowdiscovers that she can get exactly theheight she desires by carefully eatingsoon from one piece of the mushroomand soon from the other, alternatingbetween getting taller and shorter, andfinally regaining her normal height.

Few people link this to mathematics,but it reflects a result obtained in 1837 bythe German mathematician Dirichlet, onwhat is nowadays known as conditionallyconvergent infinite series – namely, that onecan assign an arbitrary value to infinitesums with alternating signs of magni-tudes tending to zero, by changing theorder in which the magnitudes areadded. Yes, mathematics can indeed be

fanciful, and as a matter of fact, Alice’sAdventures in Wonderland was written,under the pseudonym Lewis Carroll, byan English mathematician at theUniversity of Oxford in 1865. The fanta-sy about Alice is not the only place in lit-erature where you can find mathematics.And recently, mathematics has evenentered into stage plays [18] and movies[19].

On the special position of mathematics When the opportunity arises, it can befruitful to incorporate extracts from liter-ature or examples from the arts in theteaching and dissemination of mathemat-ics. Only in this way can mathematicseventually find a place in relaxed conver-sations among laymen without immedi-ately being rejected as incomprehensibleand relegated to strictly mathematicalsocial contexts.

Mathematics occupies a special positionamong the sciences and in the education-al system. This position is determined bythe fact that mathematics is an a priori sci-ence building on ideal elements abstract-ed from sensory experiences, and at thesame time mathematics is intimately con-nected to the experimental sciences, tra-ditionally not least the natural sciencesand the engineering sciences.Mathematics can be decisive when formu-lating theories giving insight intoobserved phenomena, and often formsthe basis for further conquests in thesesciences because of its power for deduc-tion and calculation. The revolution inthe natural sciences in the 1600s and thesubsequent technological conquests wereto an overwhelming degree based onmathematics. The unsurpassed strengthof mathematics in the description of phe-nomena from the outside world lies in thefascinating interplay between the con-

crete and the abstract. In the teaching ofmathematics, and when explaining theessence of mathematics to the public, it isimportant to get the abstract structures inmathematics linked to concrete manifes-tations of mathematical relations in theoutside world. Maybe the impression canthen be avoided that abstraction in math-ematics is falsely identified with puremathematics, and concretisation in math-ematics just as falsely with applied math-ematics. The booklet [3] is a contributionin that spirit.

On the element of surprise in mathe-maticsWithout a doubt, mathematics makes thelongest-lasting impression when it is usedto explain a counter-intuitive phenome-non. The element of surprise in mathe-matics therefore deserves particularattention.

As an example, most people – eventeachers of mathematics – find it close tounbelievable that a rope around theEarth along the equator has to be only 2ðmetres (i.e., a little more than 6 metres)longer, in order to hang in the air 1metre over the surface all the way aroundthe globe.

Another easy example of a surprisingfact in mathematics can be found in con-nection with figures of constant width. Atfirst one probably thinks that this proper-ty is restricted to the circle. But if youround off an equilateral triangle by sub-stituting each of its sides with the circulararc centred at the opposite corner, oneobtains a figure with constant width. Thisrounded triangle is called a Reuleaux tri-angle after its inventor, the German engi-neer Reuleaux. It has had several tech-nological applications and was amongothers exploited by the German engineerWankel in his construction of an internalcombustion engine in 1957.Corresponding figures with constantwidth can be constructed from regularpolygons with an arbitrary odd number ofedges. In the United Kingdom, the reg-ular heptagon has been the starting pointfor rounded heptagonal coins (50p and20p).

Figures of constant width were alsoused by the physicist Richard Feynmannto illustrate the dangers in adapting fig-ures to given measurements withoutknowledge of the shape of the figure.Feynmann was a member of the commis-sion appointed to investigate the possiblecauses that the space shuttle Challengeron its tenth mission, on 28 January 1986,

FEATURE

EMS June 2003 7

PPopularising mathematics: opularising mathematics: FFrrom eight to infinityom eight to infinity

V. L. Hansen

exploded shortly after take off. A prob-lematic adaptation of figures might havecaused a leaky assembly in one of the lift-ing rockets.

Mathematics in symbolsMathematics in symbols is a topic offeringgood possibilities for conversations with amathematical touch. Through symbols,mathematics may serve as a common lan-guage by which you can convey a messagein a world with many different ordinarylanguages. But only rarely do you findany mention of mathematics in art cata-logues or in other contexts where thesymbols appear. The octagon, as a sym-bol for eternity, is only one of manyexamples of this.

The eternal octagonOnce you notice it, you will find the octa-gon in very many places – in domes,cupolas and spires, often in religioussanctuaries. This representation is strongin Castel del Monte, built in the Bariprovince in Italy in the first half of the1200s for the holy Roman emperorFriedrich II. The castle has the shape ofan octagon, and at each of the eight cor-ners there is an octagonal tower [9]. Theoctagon is also found in the beautiful Al-Aqsa Mosque in Jerusalem from aroundthe year 700, considered to be one of thethree most important sites in Islam.

In Christian art and architecture, theoctagon is a symbol for the eighth day (inLatin, octave dies), which gives the daywhen the risen Christ appeared to his dis-ciples for the second time after his resur-rection on Easter Sunday. In the Jewishcounting of the week, in use at the time ofChrist, Sunday is the first day of the week,and hence the eighth day is the Sundayafter Easter Sunday. In a much favouredinterpretation by the Catholic Fathers ofthe Church, the Christian Sunday is boththe first day of the week and the eighthday of the week, and every Sunday is acelebration of the resurrection of Christ,which is combined with the hope of eter-nal life. Also in Islam, the number eightis a symbol of eternal life.

Such interpretations are rooted in oldtraditions associated with the numbereight in oriental and antique beliefs.According to Babylonian beliefs, the soulwanders after physical death through theseven heavens, which corresponds to thematerial heavenly bodies in theBabylonian picture of the universe, even-

tually to reach the eighth and highestheaven. In Christian adaptation of thesetraditions, the number eight (octave)becomes the symbol for the eternal salva-tion and fuses with eternity, or infinity.This is reflected in the mathematical sym-bol for infinity, a figure eight lying down,used for the first time in 1655 by theEnglish mathematician John Wallis.

The regular polyhedraPolyhedra fascinate many people. Adelightful and comprehensive study ofpolyhedra has been made by Cromwell[4]. Already Pythagoras in the 500s BCknew that there can exist only five typesof regular polyhedra, with respectively 4(tetra), 6 (hexa), 8 (octa), 12 (dodeca) and20 (icosa) polygonal lateral faces. In thedialogue Timaeus, Plato associated thetetrahedron, the octahedron, the hexahe-dron [cube] and the icosahedron with thefour elements in Greek philosophy, fire,air, earth, and water, respectively, whilein the dodecahedron, he saw an image ofthe universe itself. This has inspiredsome globe makers to represent the uni-verse in the shape of a dodecahedron.

It is not difficult to speculate as to

whether there exist more than the fiveregular polyhedra. Neither is it difficultto convince people that no kinds ofexperiments are sufficient to get a finalanswer to this question. It can only besettled by a mathematical proof. A proofcan be based on the theorem that thealternating sum of the number of ver-tices, edges and polygonal faces in thesurface of a convex polyhedron equals 2,stated by Euler in 1750; see e.g. [10].

The cuboctahedronThere are several other polyhedra thatare ascribed a symbolic meaning in vari-ous cultures. A single example has to suf-fice.

If the eight vertices are cut off a cube byplanes through the midpoints in thetwelve edges in the cube one gets a poly-hedron with eight equilateral trianglesand six squares as lateral faces. Thispolyhedron, which is easy to construct, isknown as the cuboctahedron, since dually itcan also be constructed from an octahe-dron by cutting off the six vertices in theoctahedron in a similar manner. Thecuboctahedron is one of the thirteensemiregular polyhedra that have beenknown since Archimedes. In Japancuboctahedra have been widely used asdecorations in furniture and buildings.Lamps in the shape of cuboctahedra wereused in Japan already in the 1200s, andthey are still used today in certain reli-

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The International Congress of Mathematics, Beijing, 2002

The five regular polyhedra

Cuboctahedron

The first appearance of ∞∞

gious ceremonies in memory of the dead[16].

The yin-yang symbolYin and yang are old principles inChinese cosmology and philosophy thatrepresent the dark and the light, nightand day, female and male. Originally yinindicated a northern hill side, where thesun does not shine, while yang is thesouth side of the hill. Everything in theworld is viewed as an interaction betweenyin and yang, as found among others inthe famous oracular book I Ching (Bookof Changes), central in the teaching ofConfucius (551-479 BC). In this ancientChinese system of divination, an oraclecan be cast by flipping three coins andthe oracle is one of sixty-four differenthexagrams each composed of two tri-grams. The three lines in a trigram areeither straight (yang) or broken (yin),thus giving eight different trigrams andsixty-four hexagrams. In Taoism, theeight trigrams are linked to immortality.Yin and Yang as philosophical notionsdate back to the 400s BC.

The mathematical symbol for yin-yangis a circle divided into two equal parts bya curve made up of two smaller semicir-cles with their centres on a diameter ofthe larger circle. The mathematics in thisbeautiful and very well known symbol issimple, but nevertheless it contains somebasic geometrical forms, and therebyoffers possibilities for mathematical con-versations.The Borromean linkBraids, knots and links form a good topicfor raising public awareness of mathemat-ics, as demonstrated in the CD-Rom [2].

Here we shall only mention theBorromean link. This fascinating linkconsists of a system of three interlockingrings, in which no pair of rings interlocks.In other words, the three rings in aBorromean link completely fall apart ifany one of the rings is removed from thesystem.

The Borromean link is named after theItalian noble family Borromeo, whogained a fortune by trade and banking inMilano in the beginning of the 1400s. Inthe link found in the coat of arms of theBorromeo family, the three rings appar-ently interlock in the way described, but acareful study reveals that the link oftenhas been changed so that it does not com-pletely fall apart if an arbitrary one of therings is removed from the system. The

link in the coat of arms of the Borromeofamily is a symbol of collaboration.

Mathematics in art and designArtists, architects and designers give lifeto abstract ideas in concrete works of art,buildings, furniture, jewellery, tools fordaily life, or, by presenting human activi-ties and phenomena from the real world,dynamically in films and television. Thevisual expressions are realised in theform of paintings, images, sculptures, etc.From a mathematical point of view, allthese expressions represent geometricalfigures.

For a mathematician, the emphasis ison finding the abstract forms behind theconcrete figures. In contrast, the empha-sis of an artist or a designer is to realisethe abstract forms in concrete figures. Atthe philosophical level many interestingconversations and discussions aboutmathematics and its manifestations in thevisual arts can take place following thisline of thought.

More down to earth, the mathematicsof perspective invites many explanationsof concrete works of art [8], and the beau-tiful patterns in Islamic art inspire discus-sions on geometry and symmetry [1]. Inrelation to the works of the Dutch artistEscher, it is possible to enter into conver-sations about mathematics at a relativelyadvanced level, such as the Poincaré discmodel of the hyperbolic plane [5], [7].The sculptures of the Australian-Britishartist John Robinson – such as his won-derful sculpture Immortality that exhibits aMöbius band shaped as a clover-leaf knot– offer good possibilities for conversa-

tions on knots [2].In his fascinating book [17], Wilson

tells many stories of mathematics in con-nection with mathematical motifs onpostage stamps.

Mathematics in natureThe old Greek thinkers – in particularPlato (427-347 BC) – were convinced thatnature follows mathematical laws. Thisbelief has dominated thinking about nat-ural phenomena ever since, as witnessedso strongly in major works by physicists,such as Galileo Galilei in the early 1600s,Isaac Newton in his work on gravitationin Principia Mathematica (1687), andMaxwell in his major work on electro-magnetism in 1865; cf. [10]. In hisremarkable book On Growth and Form(1917), the zoologist D’Arcy WentworthThompson concluded that wherever wecast our glance we find beautiful geomet-rical forms in nature. With this point ofdeparture, many conversations on mathe-matics at different levels are possible. Ishall discuss a few phenomena that can bedescribed in broad terms without assum-ing special mathematical knowledge, butnevertheless point to fairly advancedmathematics.

Spiral curves and the spider’s webTwo basic motions of a point in theEuclidean plane are motion in a line, andperiodic motion (rotation) around a cen-tral point. Combining these motions, onegets spiral motions along spiral curvesaround a spiral point. Constant velocity inboth the linear motion and the rotationgives an Archimedean spiral around the spi-ral point. Linear motion with exponen-tially growing velocity and rotation withconstant velocity gives a logarithmic spiral.

A logarithmic spiral is also known as anequiangular spiral, since it has the follow-ing characteristic property: the anglebetween the tangent to the spiral and the lineto the spiral point is constant.

Approximate Archimedean and loga-rithmic spirals enter in the spider’s con-struction of its web. First, the spider con-structs a Y-shaped figure of threads fas-tened to fixed positions in the surround-ings, and meeting at the centre of theweb. Next, it constructs a frame aroundthe centre of the web and then a system ofradial threads to the frame of the web,temporarily held together by a non-stickylogarithmic spiral, which it constructs byworking its way out from the centre to theframe of the web. Finally, the spiderworks its way back to the centre along asticky Archimedean spiral, while eatingthe logarithmic spiral used during theconstruction.

Helices, twining plants and an optical illu-sionA space curve on a cylinder, for which allthe tangent lines intersects the genera-tors of the cylinder in a constant angle, isa helix. This is the curve followed by twin-ing plants such as bindweed (right-hand-ed helix) and hops (left-handed helix).

As we look up a long circular cylinder

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Borromean rings

Immortality

in the wall of a circular tower enclosing aspiral staircase, all the generators of thecylinder appear to form a system of linesin the ceiling of the tower radiating fromthe central point, and the helix in thebanister of the spiral staircase appears asa logarithmic (equiangular) spiral. Amagnificent instance of this is found inthe spiral staircase in Museo do PopoGalergo, Santiago, Spain, where you havethree spiral staircases in the same tower.

Curvature and growth phenomena in natureMany growth phenomena in natureexhibit curvature. As an example, we dis-cuss the shape of the shell of the primi-tive cuttlefish nautilus.

The fastest way to introduce the notionof curvature at a particular point of aplane curve is by approximating thecurve as closely as possible in a neigh-bourhood of the point with a circle, calledthe circle of curvature, and then measuringthe curvature as the reciprocal of theradius in the approximating circle. If thecurve is flat in a neighbourhood of thepoint, or if the point is an inflectionpoint, the approximating circle degener-ates to a straight line, the tangent of thecurve; in such situations, the curvature isset to 0. Finally, the curvature of a curvehas a sign: positive if the curve turns tothe left (anticlockwise) in a neighbour-hood of the point in question, and nega-tive if it turns to the right (clockwise).

Now to the mathematics of a nautilusshell. We start out with an equiangularspiral. At each point of the spiral, takethe circle of curvature and replace it withthe similar circle centred at the spiral andorthogonal to both the spiral and theplane of the spiral. For exactly one anglein the equiangular spiral, the resultingsurface winds up into a solid with the out-

side of one layer exactly fitting againstthe inside of the next. The surface lookslike the shell of a nautilus.

In the Spring of 2001, I discovered thatthis unique angle is very close to half thegolden angle (the smaller of the twoangles obtained by dividing the circum-ference of the unit circle in golden ratio),so close in fact that it just allows for thick-ness of the shell. More details can befound on my home page [14].

Geometry of spaceQuestions about geometries alternative toEuclidean geometry easily arise in philo-sophical discussions on the nature ofspace and in popularisations of physics.It is difficult material to disseminate, butstill, one can come far by immediatelyincorporating a concrete model of a non-Euclidean geometry into the considera-tions. In this respect, I find the discmodel of the hyperbolic plane suggested byPoincaré (1887) particularly useful. Inaddition to its mathematical uses, it wasthe inspiration for Escher in his fourwoodcuts Circle Limit I-IV [7, p. 180].

In [13, Ch. 4], I gave a short account ofthe Poincaré disc model, which has suc-cessfully been introduced in Denmark atthe upper high-school level, and whereyou encounter some of the surprisingrelations in hyperbolic geometry. In par-ticular, the striking difference betweentilings with congruent regular n-gons in

the Euclidean plane, where you can tilewith such n-gons only for n = 3, 4, 6, andin the hyperbolic plane, where you cantile with such n-gons for each integer n ≥3.

At a more advanced level, one canintroduce a hyperbolic structure on sur-faces topologically equivalent to the sur-face of a sphere with p handles, for each

genus p = 2, by pairwise identification ofthe edges in a regular hyperbolic 4p-gon;cf. [11].

The appearance of non-Euclideangeometries raised the question of whichgeometry provides the best model of thephysical world. This question was illus-trated in an inspired poster designed byNadja Kutz; cf. [6].

Eternity and infinityEternity and, more generally, infinity arenotions both expressing the absence oflimits. Starting from the classical para-doxes of Zeno, which were designed toshow that the sensory world is an illusion,there are good possibilities for conversa-tions about mathematical notions associ-ated with infinity. And one can get fareven with fairly difficult topics such asinfinite sums; cf. [15].

The harmonic seriesA conversation about infinite sums(series) inevitably gets on to the harmonicseries:

1 + 1/2 + 1/3 + 1/4 + … + 1/n + ... .As you know, this series is divergent, i.e.,the Nth partial sum

SN = 1 + 1/2 + 1/3 + 1/4 + … + 1/Nincreases beyond any bound, for increas-ing N.

The following proof of this fact made avery deep impression on me when I firstmet it. First notice, that for each numberk, the part of the series from 1/k+1 to 1/2kcontains k terms, each greater than orequal to 1/2k so that

1/k+1 + 1/k+2 + … + 1/2k > 1/2k + 1/2k + … + 1/2k = 1/2.

With this fact at your disposal, it is notdifficult to divide the harmonic seriesinto infinitely many parts, each greaterthan or equal to ½, proving that the par-tial sums in the series grow beyond anybound when more and more terms areadded.

The alternating harmonic seriesThe alternating harmonic series

1 – 1/2 + 1/3 – 1/4 + … – (–1)n 1/n + ...offers great surprises. As shown in 1837by the German mathematician Dirichlet,this series can be rearranged – that is, theorder of the terms can be changed, sothat the rearrangement is a convergentseries with a sum arbitrarily prescribed inadvance. The proof goes by observing

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Nautilus shell

The Poincaré disc model

that the positive series (the series of termswith positive sign) increases beyond anyupper bound, and that the negative series(the series of terms with negative sign)decreases beyond any lower bound, whenmore and more terms are added to thepartial sums in the two series. Now, saythat we want to obtain the sum S in arearrangement of the alternating har-monic series. Then we first take as manyterms from the positive series as areneeded just to exceed S. Then take asmany of the terms from the negativeseries as needed just to come below S.Continue this way by taking as manyterms as needed from the positive series,from where we stopped earlier, just againto exceed S. Then take as many terms asneeded from the negative series, fromwhere we stopped earlier, just again tocome below S, and so on. Since {1/n}tends to 0 as n increases, the series justdescribed is convergent with sum S.

The alternating harmonic series is themathematical idea behind the episodefrom Alice’s Adventures in Wonderland,mentioned at the beginning of this paper.

Mathematics as a sixth senseIn 1982, Dirac’s string problem was pre-sented on a shopping bag from a Danishsupermarket chain [12]. This problemillustrates the property of half-spin of cer-tain elementary particles, mathematicallypredicted by the physicist P. A. M. Diracin the 1920s for elementary particles suchas the electron and the neutron. To con-vince colleagues sceptical of his theory,Dirac conceived a model to illustrate acorresponding phenomenon in themacroscopical world. This model consistsof a solid object (Dirac used a pair of scis-sors) attached to two posts by loose (orelastic) strings, say with one string fromone end of the object and two stringsfrom the other end to the two posts.Dirac then demonstrated that a doubletwisting of the strings could be removedby passing the strings over and round theobject, while he was not able to remove asingle twisting of the strings in this way.Rather deep mathematics from topology

is needed to explain the phenomenon,and in this case mathematics comes in asa ‘sixth sense’, by which we ‘sense’ (under-stand) counter-intuitive phenomena. Anapplication to the problem of transfer-ring electrical current to a rotating plate,without the wires getting tangled andbreaking, was patented in 1971.

Making mathematics visibleIn many countries there are increasingdemands to make the role and impor-tance of the subjects taught in the schoolsystem visible to the general public. Ifthe curriculum and the methods of teach-ing are not to stagnate, it is importantthat an informed debate takes place insociety about the individual subjects inschool.

For subjects strongly depending onmathematics, this is a great challenge.Where mathematicians (and scientists ingeneral) put emphasis on explanations(proofs), the public (including politicians)are mostly interested in results and con-sequences. It is doubtless necessary to gopart of the way to avoid technical lan-guage in presenting mathematics for awider audience.

On the initiative of the InternationalMathematical Union, the year 2000 wasdeclared World Mathematical Year. Duringthe year many efforts were made to reachthe general public through poster cam-paigns in metros, public lectures, articlesin general magazines, etc. and valuableexperience was gained. The mathemati-cal year clearly demonstrated the value ofan international exchange of ideas insuch matters, and the issue of raisingpublic awareness of mathematics is nowon the agenda all over the world fromeight to infinity.

Bibliography1. S. J. Abas and A. S. Salman, Symmetries of

Islamic Geometrical Patterns, WorldScientific, Singapore, 1994.

2. R. Brown and SUMit software, RaisingPublic Awareness of Mathematics throughKnots, CD-Rom (http://www.cpm.informat-ics.bangor.ac.uk/), 2001.

3. M. Chaleyat-Maurel et all, Mathematiquesdans la vie quotidienne, Booklet producedin France in connection with WorldMathematical Year 2000.

4. P. R. Cromwell, Polyhedra, CambridgeUniv. Press, 1997.

5. M. Emmer, The Fantastic World of M. C.Escher, Video (50 minutes), © 1980.

6. EMS-Gallery (design Kim ErnestAndersen), European MathematicalSociety, WMY 2000, Poster competition,http://www.mat.dtu.dk/ems-gallery/, 2000.

7. M. C. Escher, J.L. Locher (Introduction),W.F. Veldhuysen (Foreword), The Magic ofM. C. Escher, Thames & Hudson, London,2000.

8. J. V. Field, The Invention of Infinity –Mathematics and Art in the Renaissance,Oxford Univ. Press, 1997.

9. H. Götze, Friedrich II and the love ofgeometry, Math. Intelligencer 17 (4) (1995),48-57.

10. V. L. Hansen, Geometry in Nature, A K

Peters, Ltd., Wellesley, MA, 1993.11. V. L. Hansen, Jakob Nielsen (1890-1959),

Math. Intelligencer 15 (4) (1994), 44-53.12. V. L. Hansen, The story of a shopping

bag, Math. Intelligencer 19 (2) (1997), 50-52.

13. V. L. Hansen, Shadows of the Circle – ConicSections, Optimal Figures and Non-EuclideanGeometry, World Scientific, Singapore,1998.

14. V. L. Hansen, Mathematics of a NautilusShell, Techn. Univ. of Denmark, 2001.See:http://www.mat.dtu.dk/people/V.L.Hansen/.

15. V. L. Hansen, Matematikkens UendeligeUnivers, Den Private Ingeniørfond, Techn.Univ. of Denmark, 2002.

16. K. Miyazaki, The cuboctahedron in thepast of Japan, Math. Intelligencer 15 (3)(1993), 54-55.

17. R. J. Wilson, Stamping ThroughMathematics, Springer, New York, 2001.

18. Mathematics in plays: (a) David Auburn:Proof (Review Notices AMS (Oct. 2000),1082-1084); (b) Michael Frayn:Copenhagen; (c) Tom Stoppard: Hapgood.See also: Opinion, Notices AMS (Feb.2001), 161.

19. Mathematics in movies: (a) Good WillHunting (dir. Gus Van Sant), 1997. (b) Abeautiful mind (dir. Ron Howard), 2001;based in part on a biography of JohnForbes Nash by Sylvia Nasar.

A version of this article originallyappeared in the Proceedings of theInternational Congress of Mathematicians(ICM 2002, Beijing, China, 20-28 August2002), Vol. III (2002), 885-895, and is repro-duced with permission.V. L. Hansen, is at the Department ofMathematics, Technical University ofDenmark, Matematiktorvet, Bygning 303,DK-2800 Kgs. Lyngby, Denmark; e-mail:V.L.Hansen@mat.dtu.dk

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Solution to Dirac’s string problem

Thought for the DayThe following remarks appeared in theEditor’s column Internet: The Editor’sChoice in a recent issue of theMitteilungen der Deutschen Mathematiker-Vereinigung, in connection with theICM in Beijing.

It is all too familiar. You go to a collo-quium talk, supposedly meant for a “gener-al mathematical audience” and “graduatestudents”, and you don’t understand aword. OK, being clueless for one hour perweek is not the end of the world, but what ifyou go to a National Meeting?, or worse,THE BIG HAPPENING of mathematics,ICM, and try to go to ALL the plenarytalks?

I recently came back from ICM 2002, inBeijing, and once again was struck by thefact that the Tower of Babel hit us mathe-maticians particularly hard. In addition tothe intrinsic compartimization, overspecial-ization and splintering of math, most math-ematicians (even, or perhaps especially, themost prominent) have no idea how to give ageneral talk.

This year marks the 200th birthday of anintriguing and somewhat enigmatic math-ematician. For despite being one of themost colourful figures in the history ofmathematics, he is also one of the mostobscure. An Italian aristocrat of nobledescent, he spent most of his life in exile,first in France, and later in England. Hiscareer was scarred by turbulence and con-troversy, yet in his lifetime he managed toachieve fame as a mathematician, a histo-rian of mathematics and, most notorious-ly, a mathematical bibliophile. In thisarticle, we look at the work and fascinat-ing life of the Italian mathematicianGuglielmo Libri.

Guglielmo Bruto Icilio Timoleone LibriCarrucci Dalla Sommaia was born into anoble Tuscan family on 2 January 1803.After an excellent and wide-ranging ele-mentary education, he entered theUniversity of Pisa in 1816, initially study-ing law, but then concentrating on math-

ematics and the natural sciences.Academically precocious, his graduationcoincided with the publication of his firstpaper, Memoria … sopra la teoria dei numeriin 1820. This paper attracted muchpraise from scholars both at home andabroad, including Charles Babbage inEngland, Augustin-Louis Cauchy inFrance, and Carl Friedrich Gauss inGermany.

In 1823, aged only 20, he was appoint-ed professor of mathematical physics atthe University of Pisa. However, he wasnot a good teacher and, not relishing theposition, managed to obtain permanentleave after just one academic year. Yet,despite the brevity of his tenure, heretained the title (and salary) of the pro-fessorship for the rest of his life.

Travelling to Paris in 1824-25, hebecame personally acquainted with manyof the greatest scientists of the day,including Pierre-Simon Laplace, Siméon

Denis Poisson, André-Marie Ampère,Joseph Fourier and François Arago.These friendships proved valuable to himwhen, in 1830, he was forced into exilefrom Tuscany after being involved in aplot to impose a new liberal constitutionon the Grand Duke of Tuscany. Fleeing toFrance, Libri took refuge amongst his sci-entific friends in Paris, of whom Aragowas particularly generous, providing thepenniless refugee with funds for a time.

Meanwhile, his mathematical reputa-tion continued to grow. In March 1833,shortly after being granted French citi-zenship, Libri was elected a full memberof the Académie des Sciences to replace therecently deceased Adrien-MarieLegendre. He threw himself wholeheart-edly into the Académie’s proceedings, buthis ambitious and arrogant nature,together with the fact that he was not anative Frenchman, soon led to resent-ment amongst many of his colleagues.

Nevertheless, with Arago’s help, heobtained a teaching position at the presti-gious Collège de France in 1833, and wasalso elected as professeur suppléant (assis-tant professor) in the calculus of probabil-ity at the Sorbonne in 1834. But, for rea-sons unknown, his friendship with Aragodid not last, and by 1835 they were bitterenemies. Unfortunately for Libri, Aragowas an extremely powerful figure withinthe Académie, and opposing him meantopposing those who were under his influ-ence or patronage. One of these was themathematician Joseph Liouville, who setabout undermining Libri’s mathematicalreputation within French scientific soci-ety. Indeed, the feuding betweenLiouville and Libri was to become a char-acteristic feature of the French academicworld throughout the 1830s and 1840s.

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BrBrought to book: ought to book: the curious story of Guglielmo Libri (1803-69)the curious story of Guglielmo Libri (1803-69)

ADRIAN RICE

François Arago (1786-1853)

Guglielmo Libri (1803-69)

Joseph Liouville (1809-82)

Libri’s mathematical research coveredseveral areas, but chiefly included num-ber theory and mathematical physics. Hewas also extremely interested in the theo-ry of equations although, in the words ofJesper Lützen, it is now generally agreed that there is at mostone good idea in Libri’s mathematics: theattempts to transfer theorems from the solutionof algebraic equations (Galois theory) to thefield of differential equations, but Libri nevergot far with his idea. [1]

Indeed, it was found that, although hiswork was non-trivial, its originality wasoften questionable and his methods ofproof were not always the most efficientor elegant. Liouville, in particular, whoproduced important and innovative workin several of the same areas, was able toprove many of Libri’s results in a far lessconvoluted fashion.

The two mathematicians clashedrepeatedly during stormy sessions of theAcadémie, bombarding each other withabstruse mathematical arguments. A lit-tle-known consequence of these disputesis that Liouville made his famousannouncement of Evariste Galois’s impor-tant work on theory of equations inresponse to an attack by Libri in 1843. [2]Somewhat unsettled by the prolongedassaults, not just from Liouville, but alsofrom other mathematical scientistsincluding Arago, Charles-François Sturmand Peter Lejeune Dirichlet, Libri wasgradually to abandon mathematics and tolook for new research areas.

He had always been particularly inter-ested in the history of mathematics, and itwas here that he produced the work forwhich he is probably best remembered.His four-volume Histoire des SciencesMathématiques en Italie was published

between 1838 and 1841. A wide-rangingand provocative study of the mathemati-cal produce of the Italian peninsula fromantiquity until the seventeenth century,the work was never completely finished.With a strong emphasis on originalsources, Libri leads us from the Etruscansto Galileo via the invention of spectacles,the travels of Marco Polo and the geniusof Leonardo da Vinci (whom he wronglycredits with the invention of the + and –signs and the discovery of blood circula-tion). Along the way, he revealed for thefirst time the originality of Fibonacci’sgeometry, and that the Italians GiovanniPorta and Branca had envisaged steampower before James Watt invented thesteam engine.

The title page of Libri’s Histoire

However, in his aim to raise foreignawareness of the richness of Italian sci-ence and the extent to which Europeansavans were indebted to their Italian fore-bears, Libri was only moderately success-ful, and because of his tendency to praiseItalian achievements at the expense ofothers, the Histoire must today be readwith caution. As Benedetto Croce wrote in1947, the Histoire was excellent as a rich collection of uncommonknowledge and as a work of versatile learningand vivid mind, [but] not, however, as anexample of what a history of science should be.

As Libri’s fascination with the history ofmathematics grew, so too did his interestand expertise in rare books and mathe-matical manuscripts. Throughout the1830s and 1840s he rapidly acquired asubstantial collection, including somevaluable finds, such as previously lostpapers of Fermat, Descartes, Euler,D’Alembert and Arbogast, found in abookshop in Metz. Further manuscripts

by Galileo, Leibniz, Mersenne, Gassendi,and even Napoleon were acquired, untilby 1841 he estimated his library to con-tain around 1800 manuscripts, rangingfrom a 9th-century copy of Boethius topapers of Lagrange and Laplace. By1847, his collection of printed books andincunabula amounted to around 40,000volumes. As he said in 1846, I have books everywhere, even behind the bed.

Thanks to the influence of his goodfriend François Guizot, a minister in theFrench government, Libri was appointedInspector of French Public Libraries in1841. But it was this post that was to leadto his downfall. When rumours began tocirculate that he had used the position tosteal large quantities of valuable booksand manuscripts for his own collection,his enemies began to compile evidenceagainst him, and a police investigationwas begun. Libri had nothing to fear as long as hisfriend was in power, but in February 1848came the revolution that toppled theregime of Louis-Philippe. Fearing immi-nent arrest, Libri fled to London withintwenty-four hours, arranging for around30,000 of his most precious books andmanuscripts to be packed and sent afterhim.

Once again, he was a refugee in a for-eign land. However, on his arrival inLondon, he was immediately welcomedby fellow Italian ex-patriot AntonioPanizzi, who occupied the position ofDirector of the Library of the BritishMuseum. Panizzi was well connected inLondon’s literary circles, having brieflyheld the Chair of Italian at UniversityCollege. It was through this connectionthat Libri was introduced to the Britishmathematician and logician Augustus DeMorgan, who became his most vigoroussupporter.

Augustus De Morgan (1806-71)

British public opinion in general,although divided, was largely favourable

EMS June 2003 13

towards Libri at first. Although The Timeswas opposed, articles in Libri’s defenceappeared in the Daily News, The Guardianand The Examiner, as well as a host of arti-cles by De Morgan in The Athenæum,where Libri was portrayed as a patriotwhose belief in Italian precedence hadaroused the animosity of French dogma-tists such as Arago. His real crime, DeMorgan claimed, was that in science he would not be a Frenchman, butremained an Italian. One of his great objectswas to place Italian discovery, which theFrench historians had not treated fairly, in itsproper rank.

He continued, We suspect that political animosity generatedthis slander, and a real belief in the minds ofbad men that collectors always steal, and thatthe charge was therefore sure to be true.

Perhaps ironically, in order to supporthimself financially while in England, Libriwas forced to sell a large proportion of hiscollection at a series of auctions. Many ofthese items found their way into DeMorgan’s library! Not surprisingly, thesesales attracted further attention and criti-cism from the French government, andfor the next few years, Libri and DeMorgan published a barrage of articlesand pamphlets attacking the Frenchauthorities and protesting Libri’s inno-cence of the charges made against him.As De Morgan dryly put it, the accusationhad created a new form of syllogism: Jack lost a dog; Tom sold a cat; therefore Tomstole Jack’s dog.

If the French prosecutors ever read thestream of publications issued by Libri andDe Morgan on the affair, it made no dif-ference. On 22 June 1850, a French courtfound Libri guilty in absentia, and sen-tenced him to ten years imprisonment.With De Morgan’s unwavering support,Libri continued to argue his case fromacross the English Channel, but he neversucceeded in clearing his name. Heremained in England for the next eigh-teen years, finally returning to his nativeTuscany (by now part of a unified Italy) in1868. In failing health, he moved to avilla in Fiesole, where on 28 September1869, his eventful life finally came to anend.

But the question remained: was Librireally a thief? Up to this point, all the evi-dence that had been compiled againsthim had been purely circumstantial. Buthis death initiated a resurgence of inter-est in the affair. As De Morgan observedwhen reporting Libri’s death to theastronomer John Herschel, of course, he being removed, the accusationswhich he put down [now] begin to revive.

De Morgan did not live to see the even-tual outcome of this revival, as he himselfdied in 1871. But in a detailed investiga-tion that stretched through the 1870s and1880s, the French archivist LéopoldDelisle finally managed to piece togetherthe missing strands of evidence, theabsence of which had previously prevent-ed any adequate confirmation of the alle-gations, to prove conclusively the truth ofthe charges. Complex negotiations

between the British, French and Italiangovernments eventually resulted in thereturn of the stolen manuscripts toFrance in 1888, finally bringing an end tothe regrettable saga of L’affaire Libri.

How then can one sum up the life andwork of Guglielmo Libri? So wide-rang-ing were his interests and abilities, yet sobizarre and inexplicable were his actions,it is difficult to draw any definitive con-clusions about him. Over the years therehave been several who have attemptedbiographical appraisals, but perhaps themost succinct was one of the earliest,made shortly after Libri’s death by theFrench critic Philarète Chasles, whooffered the following assessment:Admirable in the salons and incomparablyfriendly, flexible, with gentle epigrams of sweethumour, elegant flattery, a good writer in bothFrench and Italian, a profound mathemati-cian, geometer, physicist, knowing historythrough and through, a very analytical andcomparative mind…; more expert than anauctioneer or a bookseller in the science ofbooks, this man had only one misfortune: hewas essentially a thief.

Further ReadingA. De Morgan, Arithmetical books from the inven-

tion of printing to the present time, Taylor andWalton, London, 1847.

S. E. De Morgan, Memoir of Augustus DeMorgan, Longmans, Green and Co.,London, 1882.

I. Grattan-Guinness, Note sur les manuscriptsde Libri conservés à Florence, Revue d’histoiredes sciences 37 (1984), 75-76.

G. Libri, Histoire des Sciences Mathématiques enItalie, 4 vols., Jules Renouard, Paris, 1838-41.

J. Lützen, Joseph Liouville 1809-1882: Master ofPure and Applied Mathematics, Springer-Verlag, New York, 1990.

P. A. Maccioni, Guglielmo Libri and the BritishMuseum: a Case of Scandal Averted, BritishLibrary Journal 17 (1991), 36-60.

P. A. Maccioni Ruju and M. Mostert, The Lifeand Times of Guglielmo Libri: Scientist, Patriot,Scholar, Journalist, and Thief, Hilversum,1995.

Adrian Rice [arice4@rmc.edu] is AssistantProfessor of Mathematics at Randolph-MaconCollege, Virginia, USA.

NOTES[1] In 1839, Libri claimed to have shown

that all properties of algebraic equa-tions could be transferred to linearordinary differential equations. Theonly result he mentioned explicitlywas: if a homogeneous linear differen-tial equation of order m + n is given,and if we know another homogeneouslinear differential equation of order mall of whose solutions are also solu-tions of the first equation, then it ispossible to find a third homogeneouslinear differential equation of order nsuch that the solutions of the two lat-ter span the solutions of the first.

[2] See J. Liouville, Réplique à M. Libri,Comptes rendus 17 (1843), 445-449.

CompositioCompositioMathematicaMathematica

Gerard van der Geer (Managing editor)

From January 2004, the journalCompositio Mathematica will be publishedby the London Mathematical Society onbehalf of its owners, the FoundationCompositio Mathematica. TheFoundation is based in the Netherlandsand its Board members are all mathe-maticians. The journal goes back to1934 when L. E. J. Brouwer, expelledfrom the editorial board of MathematischeAnnalen by Hilbert, founded his ownjournal.

The aim of the journal is to publishfirst-class mathematical research papers,and during the last 25 years it has earnedan excellent name. By tradition, thejournal focuses on papers in the main-stream of pure mathematics and includesthe fields of algebra, number theory,topology, algebraic and analytic geome-try, and geometric analysis.

Through the years the journal hasbeen published by Dutch publishers, andin recent years Kluwer AcademicPublishers took care of this although theownership of the journal remained withthe Foundation. Alarmed by risingprices, the Foundation has changed pol-icy and now the London MathematicalSociety will be managing the journal ontheir behalf, but without taking on own-ership of the journal. There will be nochange to the editorial policy or to theextent of the mathematics published inthe journal, but there will be a change offormat to a small A4 size with about 256pp per issue. It will be published in onevolume of six issues per year.

Prices for subscriptions to the 2004volume are to be a third less than thosefor the 2003 volumes. The prices are1200 euros, US$1200 or £750 for the fullsubscription to print plus electronic ver-sions. The intention is that this pricereduction is not just a one-off introducto-ry offer but a permanent change to alower pricing policy. The journal will bedistributed via Cambridge UniversityPress.

With this notice, we hope to persuadeyou to pass on the information to yourlibrary committees and to support thismove to lower prices and learned-societypublishing by taking up new subscrip-tions. We have agreed to buy the elec-tronic archive of papers dating back to1997 from Kluwer and these will be madeavailable to subscribers from January2005. This will provide continuity forlibraries that have cancelled their sub-scriptions in recent years.

Mathematicians are encouraged tosubmit good papers to the journal.Information about the aims and scope ofthe journal, the Editorial Board and theprocedure for submission of papers canall be found via the Foundation’s websiteat http://www.compositio.nl

EMS June 200314

This is the second part of R. I. Grigorchuk’sinterview with Professor Anosov; the first partappeared in issue 47.

Stephen Smale played an important role inyour scientific career. How did this hap-pen?First, I need to explain what preceded ouracquaintance.

Having found out from the ‘Theory ofoscillations’ structurally stable flows in aplanar region, and having some under-standing of closed smooth manifolds, Inaturally started thinking of structurallystable systems on n-dimensional smoothclosed manifolds. These thoughts werenaive. I myself came to the two conjecturesthat S. Smale dared to publish, which wereundoubtedly known to many, that struc-turally stable systems are dense in thespace of all Cn dynamical systems, and that

structurally stable systems are Morse-Smale systems (using a later term). Theseconjectures were immediate generalisa-tions of the Andronov-Pontryagin theoremon structurally stable systems in a boundedplanar region, and of the analogous theo-rem by Peixoto on structurally stable flowson surfaces which appeared around 1960.Note that Smale proved the Morse-Smaleinequalities, which were quite meaningfuland required a non-trivial argument forMorse-Smale systems. When publishingthem, Smale also published his conjec-tures. I (and probably many others) did notprove anything about these systems, whichsaved us from publishing wrong conjec-tures.

While attempting to show the genericityof Morse-Smale systems, I proved what isnow called the Kupka-Smale theorem: ormaybe I thought I proved it. The proof is

based on the method of ‘bringing to gen-eral position by small perturbations’, whichI had learned by that time, and I found theproof not difficult. Later, in 1964, whentalking to Peixoto I mentioned it but hedid not support me, which was not surpris-ing — he had just published a simplifiedproof of it and was probably aware of someconcealed underwater rocks. So he waspartly right. I’m not sure now that mysketches contained a complete proof, but Iwas also partly right – if, under threat ofdeath (or at least expulsion from graduateschool) I had been ordered to write a com-plete proof of the Kupka-Smale theorem, Iwould have done it, and would have foundand dealt with all those problems. Later Iused my earlier considerations in order toprove Abraham’s theorem on bumpy met-rics, which hung in the air for fifteen years.This was not much of an achievement, buteven so, during those years an incorrectargument was published by two well-knownauthors.

I must confess one of my misunder-standings. In 1959 I constructed the fol-lowing example of a map f of the interval[0,1] to itself which is not invertible andnot smooth: f(x) = 3x (0 ≤ x ≤ 1/3); 2 – 3x (1/3

≤ x ≤ 2/3); 3x – 2 (2/3 ≤ x ≤ 1). I realised thatthe cascade {fn} (the dynamical system withdiscrete time n) is topologically transitive,has countably many periodic trajectories,and that it did not look as though it couldbe approximated by something like aMorse-Smale cascade – ‘something like’,because the map is not bijective; also, thegraph has ‘corners’, so we cannot speak ofa C1 approximation. But I attributed this tof being not smooth and not invertible. Whydidn’t I (who thought in manifolds) thinkof the expanding map of the circle R/Zinto itself, sending x to 3x (mod Z)? Theissue of non-differentiability would disap-pear, and the role of orbit instability wouldprobably become obvious. Then one couldaddress the non-invertibility, and in high-er dimensions – in dimension 1, there is noinvertible example. In fact, why didn’t Ibecome Smale for a couple of hours? Thatis too much, but it seems that I could get tocircle expandings, and to their connectionsto the one-sided Bernoulli shift and to themap x →{3x} (where {.} denotes the frac-tional part), so loved by number theorists.However, I did not think of it.

This was all about reflections, whichstayed ‘reflections for the soul’. In parallel,I worked on the theory of differentialequations – without much spark, but withsome success. The latter is due to the factthat Pontryagin and Mishchenko fortu-nately gave me several problems to workon, gradually increasing their difficultyand leaving me more room for initiative.Besides, by 1961 I had become ratherknowledgeable about a number of ques-

INTERVIEW

EMS June 2003 15

InterviewInterview with D. Vwith D. V. Anosov. Anosov (part 2)conducted by R. I. Grigorchuk

Dmitry Anosov

tions in the classical theory of differentialequations. Some things I used in mypapers, but usually I learned not only what‘worked’ for me but also many relatedthings. In particular, I knew the works of A.M. Lyapunov and O. Perron on stabilityand conditional stability, and not onlythose parts related to equilibria and peri-odic trajectories. I also knew N. N.Bogolyubov’s papers on averaging andintegral manifolds; the former were relat-ed to my PhD thesis, while the latter werenot in my immediate area but also influ-enced me.

For some reason, as well as the areas oftopology and differential equations, I wasattracted to Riemannian geometry, and Iread (at least to a first approximation) thefamous book of E. Cartan, whereas mostpeople read other books.

So this is approximately what my statuswas by September 1961 when theInternational Symposium on Non-linearOscillations took place in Kiev. At first Ihesitated whether to go to it. I had report-ed on my last work (on averaging in multi-frequency systems) in the spring of 1961 atthe fourth (and last) All-unionMathematical Congress in Leningrad, andthat time I defended my PhD thesis on thattopic. I did not want to repeat myself byspeaking on the same subject. However,Smale, who was already famous in topolo-gy, but not yet in dynamical systems, wroteto Novikov that he was coming to Kiev andwanted to stop in Moscow and meet him,and maybe other young Moscow mathe-maticians. But not myself – he did notknow about me, and there was little toknow – I’d published some papers, but notin topics of interest to Smale and not of ahigh enough level to attract his attention. Iwent to Kiev mainly to assure him thatNovikov and other advanced young peoplewould be in Moscow.

The organisers had published theabstracts as booklets by the beginning ofthe conference (it required quite an effortthen), and the abstracts of the foreignerswere translated into Russian. While stand-ing in the registration line and lookingover someone’s shoulders ahead, I read

the title of one of them: ‘S. Smale. A struc-turally stable, differentiable homeomor-phism with infinitely many periodicpoints’. At this moment the world turnedover for me and a new life started.

In addition to his formal lecture, Smalewas kind enough to explain his discovery(‘Smale’s horseshoe’) in more detail to agroup of interested participants. Therewere several people from Gorky; fromMoscow I remember only myself and M.M. Postnikov. Then he went to Moscow,met Novikov, Arnold, Sinai and myself atthe Steklov Mathematical Institute andgave us still more details. He alsoremarked that a hyperbolic automorphismof the 2-torus disproves his naive conjec-ture of the density of Morse-Smale systemsand stated two conjectures on the structur-al stability of this automorphism and of thegeodesic flow on a closed surface (or on aclosed n-dimensional manifold?) of nega-tive (constant or variable?) curvature. Weknow now that Smale’s horseshoe, and thetorus (n-dimensional) with a hyperbolicautomorphism, and the phase space of thegeodesic flow are hyperbolic sets, butSmale did not know that – such a generalnotion did not exist then. (Even later, in1966, after Smale had already introducedthis general notion, he was doubtful – orrather, careful – about the existence ofinvariant stable and unstable manifolds forall trajectories from a hyperbolic set, notonly for periodic ones.) But I am sure thatintuitively he felt that these three exampleshad something in common.

I must tell you about the path that ledSmale to the horseshoe. As he explainedlater, after he had stated his unfortunateconjectures, several people told him thathe was wrong. He named only two: R.Thom and N. Levinson, who had differentexplanations.

Thom suggested that Smale should con-sider a hyperbolic toral automorphism f. Itis trivial that the Lefschetz number |L(f n)|→ ∞ as n → ∞; therefore it holds for any gclose to f. This allows one to expect that thenumber of periodic points of g of period ≤n also tends to ∞ as n ≤ ∞. We must excludethe possibility that there are finitely manyperiodic points, but the local (Kronecker-Poincaré) indices of f n at these points areunbounded. This is impossible for Morse-Smale diffeomorphisms; it is easy to showthat this is impossible for g which is C1 closeto f; this argument was obvious to Smale.We immediately deduce that f is notapproximated by Morse-Smale diffeomor-phisms.

On the other hand, Levinson’s lettermade Smale think hard. Levinsoninformed Smale of papers by M.Cartwright and J. E. Littlewood, and alsoby himself, which proved the existence ofinfinitely many periodic solutions for cer-tain second-order differential equationswith periodic perturbing force (which leadsto a flow in 3-space), and, Levinsonemphasised, this property and all majorproperties of the ‘phase portrait’ are pre-served under small perturbations.According to him, this more-or-less fol-lowed from Cartwright and Littlewood’s

argument, but Levinson even proved itdirectly (for a slightly different equation).What would a mathematician do afterreceiving Levinson’s message? He wouldprobably start reading the correspondingpapers. This is not easy, especially withCartwright and Littlewood’s papers. If oneis interested only in the existence of a cer-tain example, it is enough to read the lessformidable paper by Levinson, but eventhat paper is hardly a gift to the reader.

Smale chose a different road. He feltthat he was a god who had to create a worldwith certain phenomena: how could he doit? He extracted from the papers of thethree authors (or maybe only fromLevinson’s letter?) only that strong frictionand a large perturbing force are present inthe corresponding examples, and at thesame time, the trajectories stay in a bound-ed region in the phase space. The first canbe modelled geometrically by assumingthat trajectories approach fast in one direc-tion and diverge fast in another. We canconsider not a continuous motion, butrather iterations of a map (in this case acertain natural map – time shift by theperiod – is present), so that we can imaginea square becoming a long and narrow rec-tangle. But since the trajectories stay in abounded region in the phase space, weshould bend the rectangle so that it doesnot leave the region. Smale might haveknown the ‘baker’s transformation’ ofabstract ergodic theory: it is the followingmap f from the unit square S = [0, 1]2 toitself: first the square is stretched horizon-tally and contracted vertically into the rec-tangle R = [0, 2] × [0, ½]; then R is cutinto two rectangles, R1 = [0, 1] × [0, ½]and R2 = [1, 2] × [0, ½], and without mov-ing R1, R2 is mapped onto S \ R1 by (x, y) →(x – 1, y + ½).

The iterates f n of this map perfectly imi-tate the random process consisting of aninfinite sequence of coin tossings, so thatits trajectories have strikingly apparent‘quasi-regular’ behaviour. In abstractergodic theory, the discontinuities of f arenot troublesome. Contraction and expan-sion suited Smale, while the cutting of Rdid not – he needed a smooth map. But inergodic theory, they want preservation ofarea (the corresponding measure). Smaledid not need that (I think he knew that theLebesgue measure is decreased under thetime shift by the period in the Littlewood-Cartwright-Levinson case). He couldtherefore contract the rectangle verticallyeven more, bend the obtained strip, andtry to put back it inside S. Experimentationwith pictures – a friend of mine said thatSmale ‘is great at drawing pictures’ –quickly led to the horseshoe.

Although Smale did not intend to godeeply into the theory of ODE, he some-how managed to learn about homoclinicpoints. It was not trivial then. I rememberthat in 1956, while speaking at the III All-union Mathematical congress, E. A.Leontovich-Andronova (the widow and co-worker of A. A. Andronov and a sister ofanother famous physicist, M. A.Leontovich) said that difficulties in theinvestigation of multi-dimensional systems

INTERVIEW

EMS June 200316

Stephen Smale

are related to homoclinic points, and L. S.Pontryagin asked publicly: ‘And what isthat?’ Strangely I do not remember theanswer – maybe E. A. L.-A. refrained fromanswering. Of course, L. S. P. was a new-comer in the theory of ODE, but he hadstill been studying it for several years. Andone more vivid demonstration – there areno homoclinic points in the book ofNemytsky-Stepanov!

Smale realised that a horseshoe appearsnear a transverse homoclinic point. Hemight have mentioned it during our dis-cussions in 1961. At that time it was only aplausible picture; later he wrote a paper onit. But the paper imposed certain condi-tions on the corresponding periodic pointthat actually were not necessary. L. P.Shilnikov (of Nizhni Novgorod) found atreatment of this question that made theconditions unnecessary: it can be easilyachieved using the general theory ofhyperbolic sets (in particular, the theoremon families of ε-orbits). Shilnikov laterstudied later non-transverse homoclinicpoints, where this does not work. He wasthe first to give them special attention, andalthough the most famous discoveries inthis area belong to Newhouse, Shilnikov’scontributions around that time were alsosignificant.

By the way, I have always had good con-tact with mathematicians in Gorky, andsome of my interests were close to theirs.Although Smale saw me only a few times,he understood that, and even counted meas a member of the ‘Gorky school’ (found-ed by Andronov), even though he knewthat I lived in Moscow.

Finally, Smale described the coding ofthe horseshoe trajectories by a topologicalBernoulli shift. (He might have known thecoding for the baker’s transformation witha similar shift with the standard invariantmeasure.) Later, a similar coding was wide-ly used in more complicated situationsarising in the hyperbolic theory, startingwith the works of R. Adler-B. Weiss andSinai. There are other studies in this direc-tion that deal with ‘genuine’ hyperbolicsets, as well as with sets with weaker hyper-bolic properties. In the general case thesymbolic description is somewhat worse,

since the phase space of a Bernoulli shift is0-dimensional, whereas a hyperbolic setmay have positive dimension. Certain‘glueing’ must be performed when passingfrom the symbolic system to the originalone. The usefulness of this depends onwhether it is possible to get a good descrip-tion of the glueing, but even without goinginto details, it is possible to have the glue-ing affect a very small (‘negligible’) portionof symbolic sequences.

As far as I know, Smale did not describethe details of his considerations. I think,though, that in this case they can be recon-structed as above. But how he guessed thestructural stability of the other two systems(the hyperbolic toral automorphism andthe geodesic flow on a surface of negativecurvature) remains a mystery to me.

Here is how things developed further.Arnold and Sinai came up with a proof ofthe structural stability of a hyperbolic auto-morphism of the 2-torus – unfortunately,wrong. Before discovering their mistake,and taking their theorem at face value (thetheorem is actually correct), I could nothelp but notice a certain similarity with therecent D. M. Grobman-Ph. Hartman theo-rem on the local structural stability of amulti-dimensional saddle. (Bogolyubov’sintegral manifolds were somewhere at theedge of peripheral vision.) Finally mythoughts went in the following direction:let a diffeomorphism g of the torus be C1-close to a hyperbolic automorphism f, andassume that the latter is structurally stable;then for each point x, there must be anorbit {gny} somewhere near the orbit {f nx}.How can one be certain of its existencewithout assuming that f is structurally sta-ble? How many such orbits {gny} are there?As soon as I thought about it, everythingbecame clear.

Clear to me, but I might have been theonly one for whom it was obvious then. Iheard that English conservatives claim thatgood laws and institutions are not soimportant, the right person in the rightplace is important (the English version of‘the cadre decides everything’). Of course,conservatives meant politics and econom-ics, and in my case this formula has workedin the world of science. Through my back-ground I turned out to be the best personfor this question – I knew manifolds (with-in ‘no-man’s land’) and classical stuff aboutdifferential equations. Finally, the hyper-bolicity of geodesics in a space of constantnegative curvature (at least in the two-dimensional case) was already known toevery educated mathematician, but Ilearned from Appendix III to Cartan’sbook that hyperbolicity is also present invariable curvature.

I was not allowed to go to theInternational Congress of Mathematiciansin Stockholm in 1962, although the pow-ers-that-be were generally not as suspiciousof young people then as later, and severalof my contemporaries went. Probably thequestion of my trip arose too late, and Icould not get an invitation. However,Arnold and Sinai informed the communityof my work.

On the other hand, I was able to go to

the USA in the autumn of 1964. I accom-panied Pontryagin and his wife, but theyspent part of the time with Gamkrelidzeand Mishchenko who went for a muchlonger time, so I could fly to California andspend some time there. At Berkeley I, ofcourse, spent time with Smale, who waskind enough to give my name to thosedynamical systems for which the wholephase space is a hyperbolic set, and bydoing so moved me to one of the top slotsamong mathematicians by the number of‘Math. Reviews’ citations. (Later W.Thurston contributed to that by his ‘pseu-do-Anosov maps’. By doing this, Smaleand Thurston created a discrepancy withthe ‘scientifically based’ citation indexwhich, as far as I know, is not very high forme. So, am I famous in the mathematicalworld or not?) I also met Ch. Pugh, whowas then finishing the proof of the closinglemma, but later he turned seriously tohyperbolicity and co-discovered ‘stableergodicity’. Anosov systems are stablyergodic, but Pugh and Shub discoveredthat there are other such systems.

I repeat that, as I recollect, Smale forsome time could not express in rigorousterms what he felt intuitively. My mainmathematical achievement was thus that Iexplained to Smale what his achievementwas.

Next year Smale formulated the generalnotion of a hyperbolic set, which becameone of the central notions in the theory ofdynamical systems. Together with the pre-viously known hyperbolic equilibria andclosed orbits, locally maximal hyperbolicsets acquired the role of principal structur-al elements that must be studied first whendescribing the phase portrait. Smalehoped initially that, with this expansion ofthe list of structural elements, the corre-sponding modifications of his conjectureswould become correct. But as soon as heand his co-workers started analysing howthese elements are connected to each otherby trajectories travelling from one elementto another, the conjectures were again dis-proved, as well as new hasty attempts tomodify them.

Currently, few people, except J. Palis,risk making conjectures about the struc-ture of ‘typical’ dynamical systems. (Palis’sconjectures are continuously changing.)Hyperbolicity still plays an important rolein the above-mentioned examples, but it isweakened hyperbolicity by comparisonwith the version of the 1960s. So the grandhopes of the 1960s did not come true.(Only the conjecture on the ‘phase por-trait’ of structurally stable systems wasproved: see the references below.)However, the horizon widened tremen-dously, although not all systems (by far) fallinside the wider field of vision. Smale’sfamous survey, which to a large extent isdevoted to the hyperbolic theory andwhich for a time became the main sourceof inspiration for many in dynamical sys-tems, is a bright monument to that era.

That is more or less it about my contactswith Smale and the emergence of the gen-eral concept of hyperbolicity as it hap-pened in pure mathematics. I have not

INTERVIEW

EMS June 2003 17

L. S. Pontryagin

described the long prehistory that startswith the discovery of homoclinic points inPoincare’s memoir ‘On the three bodyproblem and equations of dynamics’, andwith Hadamard’s paper on surfaces of neg-ative curvature: enough is said about themin the literature. However, the literatureconnected with applications tells the histo-ry differently. The discovery of homoclinicpoints by Poincaré is still mentioned, butthere is not a word about geodesic flows,and the main development starts with E.Lorenz’s paper ‘Deterministic non-period-ic flow’, published in 1963 in the J.Atmospheric Sciences, a periodical that is notvery popular among pure mathematicians.As a result of numerical experiments with aspecific 3rd-order system motivated byhydrodynamic considerations, Lorenz dis-covered several interesting phenomena.But his discovery became of wide interestonly about ten years later (after hyperbol-icity and related effects had become knownto theorists), and applied and theoreticalmathematicians became interested in it fordifferent reasons. It proved to the appliedpeople the existence of strange (or chaotic)attractors: hyperbolic strange attractorsdiscovered by mathematicians wereunknown to them, because such examplesdid not appear in the problems of naturalsciences. (Not then and not now: theimpression is that the Lord prefers toaccept a weakened hyperbolicity ratherthan labour with the limitations on thetopology of the genuine uniformly hyper-bolic attractor of the 1960s.) Mathe-maticians, on the other hand, becameinterested in the Lorenz attractor – not asone more manifestation of known possiblebehaviours of trajectories, but quite theopposite, as the object which althoughclose to hyperbolic attractors in many ways(those that impressed the applied people),is different from them in other ways: math-ematicians are interested in such nuances.

Tell us more about your first works beforethe hyperbolic period.We have departed from our chronologicalexposition. But for a person connected tomixing, etc., this comes naturally.

I have already mentioned one of myearly student papers. There were others,roughly of the same level; I will not talkabout them. I will describe my Candidatethesis. (There were two scientific degreesin the USSR – a Candidate and a Doctor:the latter is roughly equivalent to anAmerican full professor, a Candidate islower.) Usually, a topic for a Candidatethesis is offered by the scientific supervisor.This happened to me as well, and was thelast time that L.S.P. offered me a topic.M. M. Postnikov wrote that L.S.P. gave stu-dents questions for which he already knewthe answers; this guaranteed the timelycompletion of the thesis. Postnikov him-self, having realised that L.S.P. alreadyknew everything, rebelled and wrote a the-sis on a different topic – close to L.S.P.’sinterests, of course, but the results of whichL. S. P. did not know in advance.

I also became an exception to this usualpractice. In graduate school it was pro-

posed that I study multi-frequency averag-ing with slow motions influencing the fastones. These again were singular perturba-tions, but in a completely different situa-tion. The task was general and not welldefined – my teachers obviously had noidea what the answer could be (conver-gence in initial conditions by measure). Asto the time of my defence, it took place sev-eral months before the official end of thegraduate school, after which to my regret Ihad to be expelled. Graduate student lifeat MIAN (during the second and thirdyears when there is no philosophy or for-eign language) is so good! – no require-ments at all – but now I had to go to work,and in the morning (!) – and I am an ‘owl’who works well at night. For a while I camealone in the morning and took a nap on asofa. Fortunately, the MIAN leadershipwas never strict on issues of attendance andwithin reason muted the memoranda fromabove. So, in the winter of 1961-62 I couldwork in a more suitable regime – in theevening and at home (where, by Sovietstandards, I had good conditions – a sepa-rate room).

My first paper on hyperbolicity andstructural stability was written at home andat night. Later and previously, useful ideascame to me at a different time of the dayand not at home, but nothing ever came tome at my work place in MIAN. On theother hand, I could do other things there,useful and even necessary. (The main oneis to discuss scientific questions with col-leagues. I could also edit texts, writereports, have a preliminary look at the lit-erature from the library and decidewhether to take it home for careful study,and now there is also a computer.)

I thought through many issues for mythesis while attending with many othergraduate students (from MIAN and fromother institutes of the AS) the mandatorylectures on dialectic and historical materi-alism that were given in a large hall at theInstitute of Philosophy on Volhonka street,and also on the way home from those lec-tures. This once again shows the role of theonly true Marxist-Leninist philosophy inthe development of science.

Later, A. I. Neishtadt brilliantly extend-ed my work, and in 2001 we were awardedthe A. M. Lyapunov prize of the RussianAcademy of Sciences for our investigations.It is curious that I received that prize formy Candidate thesis (and partly forNeishtadt as well) and that 40 years (25 forNeishtadt) had passed since its publica-tion!

What can you say about your students andcolleagues?Formally I worked at the MSU in 1968-73and from 1996. My main achievement isthe joint seminar with A. B. Katok ondynamical systems, which was a study sem-inar at first and later became a scientificseminar. After Katok emigrated, A. M.Stepin became the leader of the seminar,and for a while R. I. Grigorchuk. The sem-inar continued to function when I did notformally work at the MSU. It has existedfor 30 years, although this is not a record

for the MSU.About the students: I have never super-

vised anybody that started from their earlyundergraduate student years. M. I.Monastyrsky was my student during thefirst period of my pedagogical career. I wasnot quite fortunate with him – I always hada strong analytical component, and he wasnot inclined that way. Later he became astudent of N. N. Meiman, a pure analyst.Nevertheless, Monastyrsky ‘survived’ andmanaged to find his ‘ecological niche’,actually in physical questions. Here he gotlucky – some issues in physics were discov-ered (with his participation) that requiredmore-or-less pure topological considera-tions. He is also widely known to mathe-maticians and theoretical physicists as theauthor of historical books, not on ancienthistory but on the Fields medals andRiemann (his biography as well as his her-itage).

Then I had a couple of talented studentswho wrote one or two papers under mysupervision, but later quit science for onereason or another. But around the sametime several people appeared who wereinitially students of others but later, with-out quitting their advisors, became someway or another connected with me – some-times formally, sometimes not. Some ofthese not only stayed in science, but alsoearned considerable recognition. The firstwas Katok, who cannot be considered mystudent – we collaborated as equals, but Iwas older and this showed at first. Thencame M. I. Brin, Ya. B. Pesin, E. A. Satayevand A. V. Kochergin, who initially wereKatok’s students. A bit later (it seems that Ialready did not work at the MSU)Grigorchuk came to our seminar, who hadinitially been Stepin’s student. The secondperiod of my pedagogical career has beenless successful in terms of immediate stu-dents.

I already mentioned my colleagues. Atthat time I communicated a lot with thelate V. M. Alekseyev. From the end of the1980s I had close ties with A. A. Bolibruch.

What interfered, and what helped, withyour work?Let me add two things.When visiting MIAN, the famous Lefschetz

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S. Lefschetz

said that the institute was good becauseone could do nothing there. Imagine howawful this sounds to the bureaucrats!Lefschetz was not only an outstanding sci-entist but also an outstanding scienceadministrator. Princeton has not only theInstitute for Advanced Study but also FineHall, a period of whose history is closelyconnected with Lefschetz. So, he under-stood these issues better than bureaucrats.This was what helped – now about thingsthat have significantly interfered lately.The situation of the first post-war years (asI learned from older people) has beenrepeated during peace time. There areolder scientists: some of them left, butmany didn’t, and most of those who diddon’t break their ties and try to compen-sate for their absence by spending part ofthe year in Russia. There are also youngpeople (fewer than before, and definitelyfewer than we wish), but there is a definitelack of people in the middle – both by ageand rank – and this affects everything. Thisstarted even before ‘democracy’, from theend of the 1970s when few were hired bythe MSU or MIAN, and those very peoplewould carry the main load now. It is point-less to discuss how everything has deterio-rated during the past decade.

Tell us about the department that you head.In his best years Pontryagin undoubtedlydominated all of his subordinates, and theODE department was thus a ‘theatre of oneactor’. He had no objection to somebodyplaying his role, and this is what I did aftergraduate school. Usually, though, the restof the department worked on topics closeto him and stayed in his shadow. This sys-tem remained in place even when he grewold and could not be the leader as before.And when, in a theatre of one actor, hisperformance lost former brilliance, thequality of the shows went down. They evenstarted talking of the department’s degen-eration.

Curiously, an American mathematicianwho spent several months at the institute,wrote that everyone in the ODE depart-ment was a KGB agent exceptGamkrelidze and me! I do not know wherehe obtained this information about theKGB. If it were not simply silly I would addthat only a man of the KGB, and of highrank, could know such things. But it is truethat our department did not look goodthen, and that Gamkrelidze and I were theexceptions, at least with respect to ourresearch interests. (Unlike me,Gamkrelidze worked in the theory of opti-mal processes, but his interests in this widefield lay off the mainstream of the maingroup of the department.) Gamkrelidzebecame head of the department afterL.S.P.’s death. In essence we have a differ-ent organisation now — everyone plays hisown role. Also, we hired new people. Themain changes took place duringGamkrelidzee’s tenure, but I helped himas much as I could. (For some time beforeI became the Chief, the department had alaboratory for dynamical systems which Iheaded. Thus, even formally, I had someadministrative freedom and could do some

things myself.

How and why was the Chair of dynamicalsystems created at the Moscow StateUniversity?This was Bolibruch’s, Mishchenko’s andmy initiative. We reckoned that the theoryof dynamical systems, which historicallyappeared in ODE, had actually become amajor independent scientific field. Thiswas formally expressed by the fact thatthere was an ‘ODE and dynamical systems’section at several recent congresses ofmathematicians – as you see, dynamicalsystems play the role of an equal partnerand not a part of ODE. Indeed, the theoryof dynamical systems has many connec-tions with functional analysis, probabilitytheory, algebra, complex analysis, geome-try and topology. The seminar that Ifounded with Katok at the MSU about 30years ago has mainly been (and is now)affiliated with the Chair of the theory offunctions and functional analysis (TFFA).

The leadership of the RAS (Yu. S.Osipov and A. A. Gonchar) supported ourinitiative. The decision had to be made bythe MSU rector V. A. Sadovnichy, who alsounderstood us. At its start it was under-stood that the Chair would be quite small(it is probably the smallest one in thedepartment of mechanics and mathemat-ics), but would grow later. We hope thatthis will happen. We cooperate with the oldChairs of differential equations and TFFA,and also with several university researcherswho deal with analytical mechanics andprobability. We have only begun working,so it is too early to talk about the results.

Many are interested in your work at theschool reform commission.There is no such commission. There is acommission of the Division ofMathematical Sciences of the RAS onschool mathematics, of which I am theChair. And, to believe official statements,

there is no reform, there is modernisation.However, this modernisation touches onall sides of school (and not only school)education, so it is a reform, and a verydeep one. Its realisation would destroywhatever was best in our education. Someeven think that financial support fromabroad for the reformers is connected withthe hopes for precisely this result, afterwhich Russia will quickly lose the remnantsof past greatness and can be practically dis-counted. I do not share this point of view,since a similar process started earlier andhas already gone pretty far in several west-ern countries and first of all in the USA –well, it could be in the interests of Soviet orChinese communists, but did they have ahand in this?

Rather, a malignant tumour has grownin education. It did not appear here, butcould spread here metastatically, whichwould be even more dangerous to us thanthe original cancer to the place where itappeared.

Somebody may object: has the presentstate of American education led to any lossof American greatness? Answer: the draw-backs of the American educational systemare partly compensated by other factorsthat Russia lacks; one of them is the importof specialists from abroad. Recall, however,what happened after the launch of the firstSoviet sputnik. And quite recently, theGlenn commission issued a report whosetitle speaks for itself: ‘Before it’s too late’.Right now, the main US problems are con-nected with something else – terrorism –but it will not remain like that.

Our proposed reform-modernisationhas aspects that one cannot disagree with.First is the branching of the final grade’seducation into several directions or, asthey say now, into ‘profiles’. This was sounder the Czar, but in the Soviet periodthe idea of profiling somehow turned outto be ideologically bad. Further, only aproportion of children (in my time, at most

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Moscow State University

a quarter) received high-school educationearlier, those who aimed at colleges, pri-marily technically oriented; now it is dif-ferent, and it clearly requires a change.The main mistake of the reformers of the1960s was that they wanted to increase thecontent of the school material while thestudent population grew, and a reasonablesimplification and reduction of that mater-ial was in order. Much was said, and newprogrammes and textbooks were preparedwithout sufficient thought. But even if ithad been done well, the reform was unfor-tunate because of the principal mistake Imentioned above – they meant some idealeducational model realisable with non-existent populations of students and teach-ers. And finally, when preparing peoplefor a life in society, the school must famil-iarise them with the organisation of thesociety.

Professionally our commission deals onlywith the contents of school mathematicalprogramme: the curriculum, schedule,textbooks. But we also have to discussother sides of the reform. Speaking of theeducational curriculum, unlike previouscommissions, this one has no right to man-date anything. We can only express ouropinion on issues, and this is what we do.Our statements have so far had little effect,but I have witnessed many epochal reforms‘for ages’ (not only in education) that werefollowed by other reforms. I believe thesame thing will happen here.

What is the role of the MoscowMathematical Society (MMS), of whose gov-erning board you are a member?Presently, the leading roles in the mathe-matical life in Moscow are played byMIAN, the MSU and MMS; theIndependent Moscow University (seebelow) has become more visible recently.They all differ in their objectives, adminis-tration, and type of activities: this combi-nation is some sort of pluralism.Rephrasing a quote from a famous acade-mician A. Yu. Ishlinsky (his area ismechanics), who spoke of the role of thescientist in industry, we can say that goodconsequences of pluralism are not appar-ent if it exists, but become apparent whenpluralism does not exist.

What about the Independent University?The Independent Moscow University(IMU) is a part of the conglomerate calledthe ‘Moscow centre for continuing mathe-matical education’ (MCCME). The ‘conti-nuity’ in the name reflects the fact that thecentre works with schoolchildren, as well aswith college students. This work has manysides to it: I will give only one example.

In 2001 and 2002 a summer school washeld near Dubna for high-school juniorsand seniors and for college freshmen andsophomores. In between swimming andsports they had a number of lectures,accompanied by seminar-type classes.These lectures were quite different in topicand style; for example, I was into educa-tion (vector fields and related notions,mainly topological, and complex analysis),V. I. Arnold talked about new things (for

example, in 2001 it was asteroidal geome-try ‘and all that’). Notes were taken, and aconsiderable portion has already beenpublished as booklets. In general, I praisethe publishing aspects of MCCME (whichis wide with respect to the range and levelof books and booklets – from quite popularto the newest).

Now the IMU. For a number of reasonsit is not an ordinary university, but supple-ments courses for MSU students and forstudents from other colleges (includingsome high-school students). I gave severalcourses there. No matter what one calls it,these courses are useful and important.One can find out only from there about anumber of new developments in mathe-matics (and about some older things aswell). Several of our scientists, and foreignvisitors, have given lectures and mini-courses there, and in this respect the IMUis definitely the leader. (The MSU leads inlong-term topics courses – so to say, sprint-ers and long distance runners.)

The IMU consists of two colleges: math-ematics and mathematical physics; the for-mer is larger, while the second uses MIANas a base. When it was created, it was hopedto cover other fields of natural sciences,but this has not yet happened.

BibliographyMy book [1] contains my doctoral thesis.Smale’s famous survey is [2]. We both men-tion works on geodesic flows on manifoldsof negative curvature, beginning withHadamard, and give other references, forexample on structural stability; later I pub-lished a historical paper on the latter sub-ject [3]. More new information on this sub-ject and hyperbolic theory can be found in[4], which also contains some informationon the Lorenz attractor (including refer-ences for my early publications). MyCandidate thesis is [5]. M. M. Postnikov’sarticle is [6].

1. D. V. Anosov, Geodesic flows on closedRiemannian manifolds with negative curva-ture, Trudy MIAN 90, Nauka, Moscow, 1967;English translation: Proc. Steklov Inst. ofMath. 90 (1967), Amer. Math. Soc. (1969).

2. S. Smale, Differentiable dynamical systems,Bull. Amer. Math. Soc. 73 (1967), 747- 817.

3. D. V. Anosov, Structurally stable systems, in:Topology, ordinary differential equations, dynam-ical systems: Collection of survey papers on the50th anniversary of the Inst. Trudy MIAN 169,Nauka, Moscow, 1985, pp. 59-93: Englishtranslation: Proc. Steklov Inst. of Math., issue4 (1986), 61-95.

4. D. V. Anosov (ed.), Dynamical systems-9, Itoginauki i techn., Fundam. napravleniya, VINI-TI, Moscow 66 (1991), 5-81, 85-99: Englishtranslation: Dynamical systems-9, Encyclo-paedia of Math. Sciences 66, Springer,1995.

5. D.V. Anosov, Averaging in the systems ofordinary differential equations with rapidlyoscillating solutions, Izv. Akad. Nauk SSSR,ser. math. 24(5) (1960), 721-745 (Russian).

6. M.M. Postnikov, Pages from a mathematicalautobiography (1942-1953), Research in theHistory of Math. (2nd ser.) 2(37) (1997), 78-104 (Russian).

MathematicalMathematicalSociety Society

of of Japan Japan

PPrizes, 2003rizes, 2003The Spring Prize and the Algebra

Prize of the Mathematical Society ofJapan (MSJ) were awarded at itsAnnual Meeting in Tokyo in March2003.

The Spring Prize is awarded eachyear to a mathematician who is notolder than forty years old and who hasmade an outstanding contribution tomathematics. The 2003 Spring Prizewas awarded to Tomotada Ohtsuki ofTokyo University for his distinguishedcontributions to the study of quantuminvariants for 3-manifolds.

After the discovery of the Jones poly-nomial, much work has been devotedto topological invariants of knots and3-manifolds. Tomotada Ohtsukientered this field in the beginning ofthe 1990s with his notion of finite typeinvariants for 3-manifolds. He definedan arithmetic perturbative expansionof 3-manifold invariants due to Wittenand Reshstikhin-Turaev for homology3-spheres, and showed that each coeffi-cient of this expansion is of finite typein his sense. Later, in collaborationwith Le Tu Quoc Thang and JunMurakami, Ohtsuki established univer-sal finite type invariants for 3-mani-folds with values in the algebra ofJacobi diagrams, based on theVassiliev-Kontsevich invariants forframed links. Ohtsuki’s achievementsclarified greatly combinatorial aspectsof quantum invariants for 3-manifoldsand shed new light on our understand-ing of the class of homology 3-spheres.

The Algebra Prize is awarded everyyear to one or two algebraists in recog-nition of outstanding contributions inalgebra. This prize was established in1998 with funds donated to the MSJ,and awarded to Takeshi Saito andHiroshi Umemura in 1998, toKazuhiro Fujiwara and MasahikoMiyamoto in 1999, to Koichiro Haradain 2000, to Tamotsu Ikeda andToshiaki Shoji in 2001, and to MasatoKurihara in 2002. The 2003 Algebraprize was awarded to Kei-ichiWatanabe of Nihon University for hisoutstanding contributions to commuta-tive ring theory and its applications tosingularity theory.

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In January 2003 the Latvian MathematicalSociety (LMS) marked its 10th anniver-sary. Compared with such bodies as theLondon Mathematical Society (establishedin 1865) or our neighbour the EstonianMathematical Society (which celebrated its75th anniversary in 2001), we are still inour infancy, being one of the youngestmathematical societies in Europe.However, mathematics and mathemati-cians existed in Latvia long before theLMS was founded, and therefore I wouldlike to say some words about the prehisto-ry of the LMS, mentioning some factsabout mathematics and mathematiciansrelated to Latvia before 1993 when theLMS was founded.

Piers Bohl Probably, the first outstanding mathemati-cian whose life and work was closely relat-ed to Latvia was Piers Bohl. He was born in1865 in Valka (a town on the borderbetween Latvia and Estonia) in the familyof a poor German merchant. In 1884, aftergraduating from a German school inViljandi (Estonia) he entered the faculty ofphysics and mathematics at the Universityof Tartu 1. Two years later, in 1886, Pierspresented his first research work‘Invariants of linear differential equationsand their applications’, for which he wasawarded the ‘Golden Medal’. Bohl gradu-ated from Tartu University in 1887, earn-ing a Master’s degree at the same universi-ty in 1893. In his Master’s thesis, Über dieDarstellung von Functionen einer Variabelndurch trigonometrische Reichen mit mehrereneines Variabeln proportionalen Argumenten,he introduced a class of functions which,ten years later, were rediscovered by theFrench scientist E. Esklangon under thename quasiperiodic functions.

In the period 1887-95 Piers Bohlworked as a teacher of mathematics incountryside schools, but in 1895 he wasinvited to be the Head of the Departmentof Mathematics at the Riga PolytechnicInstitute – at that time the only institute ofhigher education in the territory of Latvia.In 1900 he was awarded the degree ofDoctor in Applied Mathematics fromTartu University for his research work inthe applications of differential equationsto mechanics. He continued intense workin different areas of mathematics – in par-ticular, in the field of differential equa-

tions. In his paper ‘Über die Bewegungeines mechanischen Systems in die Näheeiner Gleichgewichtslage’ (J. reine, angew.Math. 127 (1904), 179-276), he studied theexistence and smoothness problems of sta-ble and unstable manifolds for a quasilin-ear system of differential equations. In thecourse of studying these problems, as aux-iliary results he established that a sphere isnot a retract of a ball and proved that a con-tinuous mapping of a ball into itself has a fixedpoint, a famous result which is often calledthe ‘Brouwer fixed-point theorem’,although Brouwer published the resultseven years later (see L.E.J. Brower, ‘Überdie Abbildung von Mannigfaltigkeiten’,Math. Ann. 71 (1911), 97-115. In his paper,‘Über ein Dreikörperproblem’, Z. Math.Phys. 54 (1906), 381-418, Bohl proved afamous theorem about quasiperiodic func-tions, as well as some important resultsabout differential equations with quasi-periodic coefficients. Of course, here wehave been able to mention only a fewresults of this insightful mathematician.

It is worth mentioning that Piers Bohlwas known also as a very strong chess play-er. One of the chess openings discoveredby him is known as the ‘Riga version of theSpanish game’. During the First WorldWar, Riga Polytechnic Institute was evacu-ated to Moscow, and Bohl spent threeyears in the capital city of Russia. Hereturned to (now already independent)Latvia in 1919 and was elected Professor ofthe Engineering faculty of the recentlyfounded University of Latvia.Unfortunately, two years later inDecember 1921, Piers Bohl died from acerebral haemorrhage.

The first Society of Latvian Physicistsand Mathematicians. Of all the mathematicians who were activeduring the time known as the first republicof Latvia (1918-40) we shall mentionEdgars Lejnieks, Alfreds Meders, EiensLeimanis, N. Brazma (Brower), Kârlis Zaltsand J. Cizareviès. None of them is knownas a world-famous mathematician or as thefounder of a ‘mathematical school’, buteach played an enormous role in thedevelopment of the mathematical cultureand education in a newly founded state –the Republic of Latvia. The first time thatLatvia gained its independence was afterthe World War I in 1918, and this was alsothe first time that Latvian came into use asthe official language and that studentscould acquire education in their nativelanguage.

Edgars Lejnieks was born in Riga in1889. After graduating from MoscowUniversity in 1907, he worked for a time in

Göttingen University. In 1919 he returnedto Riga and was elected the first dean ofthe faculty of mathematics and natural sci-ences at the University of Latvia. In 1923he organised the first congress of Latvianmathematicians. Lejnieks is also recog-nised as the founder of the library of theUniversity of Latvia, which now containsthe largest collection of books and journalsin mathematics and physics in Latvia(more than 180,000 items).

On 10 March 1939 the Latvian Societyof Physicists and Mathematicians (LSPM)was founded. The first Chair of the Societywas the physicist F. Gulbis. The mathe-matician members of this society includedE. Leimanis, A. Lûsis, A. Meders, P.Putniòð, N. Brazma and E. Grinbergs: onthe whole, the physicists and astronomerswere better represented and more activein that society than the mathematicians.The Society held about ten scientific ses-sions. In December 1940 the activity of theSociety’s activities were stopped by theSoviet authorities, who aimed to excludethe appearance of national thought in allits possible forms. The Society had a briefrecovery in January 1943, but in May 1944it was again closed. The total time that theSociety existed was 26 months. The con-clusion of these not very cheerful glimpsesinto the history was that after Latviaregained its independence in 1991, thephysicists decided to establish the LatvianPhysical Society as the renewed LSPM,while the mathematicians founded theLatvian Mathematical Society as a newbody.

Mathematics in Latvia during the Sovietperiod (1945-91)In the years 1940-45 some mathematiciansemigrated to the West, while others foundthemselves in Stalin’s camps. Thus the faceof mathematics in Latvia started to pro-duce other persons. Some of them origi-nated from Latvia, but after graduatingfrom the University of Latvia they oftenwent to develop their theses at other uni-versities – usually Moscow (Âriòð, Klokov,Reiziòð et. al) and Leningrad (Detlovs, etal.), where the level of mathematical edu-cation and researches was very high. At thesame time, a considerable number ofmathematicians, some outstanding onesamong them, came to Latvia from otherparts of the USSR. This process was espe-cially intense during the first fifteen yearsafter the incorporation of Latvia into theUSSR.

There were two main stimuli for thisprocess. On the one hand, the living stan-dards in Latvia (and in the Baltic on thewhole) were relatively higher than in the‘old part’ of the Soviet Union, and hencethere was a natural interest by scientists(and others!) to come to live and to work inthe Baltic. On the other hand, the politicsof the Soviet government (with regard tothe incorporated states) was to assimilatethe local population, and it thereforeencouraged such a movement of people.In any case, this process was beneficial formathematics in Latvia, since the Latvianmathematical community was replenished

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1 We use here modern geographical names. Atthe time when Bohl studied in Tartu this citywas called by its German name Dorpat, butlater, by the time he defended his DoctoralThesis, it had been renamed by the Russiansas Yurjev.

The Latvian MathematicalThe Latvian MathematicalSociety after 10 yearsSociety after 10 years

Alexander Šostak

with such serious mathematicians as A.Myshkis, B. Plotkin, I. Rubinštein, M.Goldman, S. Krachkovskij, L.Ladyzhenskij and others. Some of themcreated strong schools here in differentareas of mathematics: we should especiallymention A. Myshkis, B. Plotkin, I.Rubinštein and M. Goldman, whose stronginfluence on mathematical life in Latvia isstill felt nowadays. Unfortunately, theirprofessional activity was also often ham-pered by the authorities.

We now mention briefly some of themathematicians who ‘defined the mathe-matical face of Latvia’ in this period andafterwards.

Eiþens Âriòð (1911-84), a Latvian, wasborn in Krasnojarsk, in Siberia, where hisfather was in exile. In 1920 the familyreturned to Riga. After having graduatedfrom the University of Latvia, Âriòð contin-ued his education at postgraduate level atthe Moscow State University. He devel-oped his Candidate (PhD) thesis under thesupervision of L. V. Keldysh, a famousRussian topologist. Âriòð was a mathe-matician with a diverse spectrum of inter-ests. He wrote papers in the descriptivetheory of functions (his thesis was devotedto partially continuous functions on prod-ucts of topological spaces), theoreticalcomputer science, and cybernetics.Besides, Ariòð had a great organisationaltalent: in particular, he is remembered asthe founder and the first director of thecomputer centre at the University ofLatvia, one of the first such institutions inthe Soviet Union.

Emanuels Grinbergs (1911-82) wasborn in St Petersburg into a family ofdiplomats and artists. In 1930 he enteredthe University of Latvia, and graduatedfrom it in 1934. Later he studied at theEcole Normale Superieure in Paris.Grinbergs was elected as docent at theUniversity of Latvia in 1940. In 1943 hedefended his doctoral thesis, ‘On oscilla-tions, superoscillations and characteristicpoints’, but unfortunately was then draftedinto the German Army. In 1944 the Sovietsagain entered Latvia, and his presence inthe German Army (over which he had nochoice) was regarded as an unforgivablesin. He was sent to Stalin’s camp in Kutaisiin the Caucasus.

Although permitted to return to Latviain 1947, he was only allowed menial workat a factory. Only after Stalin’s death in1954 was Grinbergs permitted to work atthe University. In 1956 he defended hisCandidate thesis, ‘Problems of analysisand synthesis in simple linear circuits’ (hisfirst thesis was not acknowledged by theSoviet authorities, since it had beendefended during the period of Germanoccupation), and was then appointed asHead of the section in the computer cen-tre of the University of Latvia. The maincontributions of Grinbergs and his collab-orators were in the theory of electrical cir-cuits, in research on non-linear circuit the-ory, in magnetic hydrodynamics, in tele-phony, and in the development of numer-ical methods in the analysis of Markovprocesses. A significant achievement of

Grinbergs’ group was the development ofanalytic methods for calculations of planarcontours of 3-dimensional components ofthe hulls of ocean-going ships. Thesemethods were highly acclaimed through-out the Soviet Union and were used bymany shipbuilding companies. However,Grinbergs’ main interest was in graph the-ory, where he had several publications andan archive of about 20,000 (!) pages ofunpublished papers.

The first mathematician who graduatedfrom the University of Latvia and spent allhis working life lecturing at that universitywas Arvids Lûsis (1900-69), whose brotherJânis became a well-known geneticist.Born to a family of peasants, he receivedhis first education at a village school, andthen entered the University of Latvia fromwhich he graduated in 1924. During thefollowing 45 years he lectured in differentcourses of theoretical mechanics andapplied mathematics. His main scientificinterests were in differential and integralequations, but he is mostly remembered asa the teacher of many Latvian mathemati-cians who started their research work inthe second half of the 20th century.

Anatoly Myshkis, at that time one of theleading USSR specialists in the field of dif-ferential equations, came to Riga andstarted to work at the University of Latviain 1948. He was soon promoted to the postof Head of the Department of HigherMathematics. His outstanding skills of lec-turing and competence drew to him manygood young mathematicians and students.As a result, a very strong group ofresearchers in the field of differentialequations was created, and can still be felt,more than 50 years later. It is unfortunatethat he soon had to relinquish the post ofhead of the department, due to politicalinsinuation, and then in 1956 was forcedto leave the University of Latvia and Latviaforever. Nevertheless, his influence on thedevelopment of mathematics there was(and still is) extremely strong. In particu-lar, most of the people researching intodifferential equations (H. Kalis, L. Reiziòð,A. Reinfelds, A. Lepins, et al.) can considerthemselves as the mathematical offspringof Myshkis.

One of the outstanding representativesof this school was Linards Reiziòð (1924-91). Linards’ studies at school were inter-rupted by the Second World War. Toescape being drafted into the GermanArmy, he went to the North of Latvia andthen to Estonia. Near the small town ofPaide he was arrested by German soldiersand taken as a prisoner to Riga. He wasreleased in 1942, and soon after, he passedhis high-school final exams. After the warhe entered the University of Latvia, andafter graduating in 1948 he continued hisstudies at postgraduate level under theguidance of Lûsis, specialising in the fieldof differential equations. In his Candidatethesis he studied the qualitative behaviourof homogeneous differential equationsand obtained results that were highlyregarded by specialists.

A large part of Reiziòð’ research energywas directed to the investigation of condi-

tions under which two systems of differen-tial equations are dynamically (topologi-cally) equivalent. Among other results hefound a formula giving the relationshipbetween the solutions of full and truncatedsystems of differential equations. Duringhis later years he worked on Pfaff’s equa-tions.

Reiziòð was the author of two mono-graphs. He was the scientific advisor ofseveral students who later became success-ful researchers and administrators –among them, Andrejs Reinfelds (the direc-tor of the Institute of Mathematics), OjârsJudrups (from 1991-2002 the dean of thefaculty of physics and mathematics at theUniversity of Latvia) and Kârlis Dobelis (aformer rector of the Liepâja PedagogicalUniversity).

Ernsts Fogels worked in number theory,and his main interest was the ζ-function.Trying to solve the famous problem on thelocation of the zeros of the ζ-function,Fogels obtained a number of importantresults as by-products. In particular, hediscovered new effective proofs of theGauss-Dirichlet formula on the number ofclasses of positively definite quadraticforms, as well as the de la Vallée-Poussinformula for the asymptotic location ofprime numbers in an arithmetic progres-sion.

In the 1960s and 1970s, much interestamong Latvian mathematicians wasfocused on functional analysis. Especially,noteworthy was the work of SergejKrachkovski and Michael Goldman,whose scientific interests includedFredholm-type functional equations andthe spectral theory of linear operators.Further work in functional analysis wasdone by Imants Kârkliòð, a student of D.Raikov, and Andris Liepiòð.

In 1960 a very talented algebraist, BorisPlotkin, came to Riga. His scientific com-petence, energy and enthusiasm attractedmany young and not-so-young mathemati-cians. Under his supervision several dozenof Candidate (= PhD) and doctoral theseswere worked out. (Recall that in the USSRthere were two levels of scientific degree –Candidate of Sciences and a Doctor ofScience; the demands for the second onewere usually very high.)

In 1967 a large algebraic conference wasorganised in Riga by Plotkin and his col-laborators. One of his important accom-plishments was the founding and guidingof the ‘Riga Algebraic Seminar’ – wellknown among specialists throughout thewhole Soviet Union and in other countries.Among active members of this seminarwere many of Plotkin’s former students: I.Strazdiòð, A. Tokarenko, L. Simonyan, Ja.Livchak, A. Kushkuley, L. Gringlass, J.Cîrulis, A. Pekelis, A. Bçrziòð, R. Lipyanskij,Je. Plotkin, V. Shteinbuk, L. Krop and I.Ripss. The last of these merits specialattention.

The most talented of Plotkin’s students,Ripss, was very sensitive and keenly felt theinjustice reigning around him. On 13April 1969 he attempted self-immolationin the centre of Riga, near the Monumentof Freedom, protesting the Soviet invasion

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of Czechoslovakia in August 1968. Themilitia beat him severely and he was latercommitted to a mental hospital; then, aftersome days, he was imprisoned. The condi-tions in the prison were terrible, but nev-ertheless Ripss found the strength andenergy to continue his work in mathemat-ics and to obtain some brilliant results. Inparticular, while in prison, the 20-year-oldmathematician solved the famous long-standing Magnus problem on dimensionalsubgroups, constructing a counter-exam-ple for the case n = 4. This was a majorevent in 20th-century mathematics.However, for political reasons, this workcould not be published in the Soviet Unionand only later was it published in Israel,creating a great deal of interest amongmathematicians.

One of the consequences of Ripss’ actionwas that Plotkin, the author of five mono-graphs and an outstanding scientist andlecturer, lost his job at the University ofLatvia: the ‘crime’ of a student was also the‘sin’ of the teacher, arising from ‘bad edu-cation’ – this was an unwritten rule of theSoviet system. Fortunately, Plotkin hadinfluential friends who helped him to finda job in another institution, the RigaInstitution of Aviation. As a result, the‘Algebraic centre’ in Latvia moved fromthe University of Latvia – its appropriatehome – to the Institute of Aviation.

Many of the members of the RigaAlgebraic Seminar have now emigratedfrom Latvia, mainly to the USA and Israel.Since 1996 Plotkin has been living andworking in Jerusalem. However, he stillkeeps close contact with colleagues andformer students in Latvia and regularlycomes to Riga and gives talks at the ses-sions and seminars of the LMS.

Mathematics in independent Latvia In the late 1980s in Latvia, as well as inother Baltic republics, much of the coun-try’s intellectual and artistic energies werefocused on the independence movement.This movement in Latvia was called Atmoda(Awakening) and showed itself in differentforms. In particular, organising nationalscientific and professional Societies wasamong the most vivid and popular ideas.(However, as the experience of Estonianmathematicians shows, the idea of organis-ing such societies during the Soviet Erawas usually put to an end by the stateauthorities, who were afraid of any nation-al movement.)

One of the first bodies of this kind wasthe Union of Scientists of Latvia, createdin 1988. The Latvian Mathematical Societywas founded on 15 January 1993 by thedecision of the Constituent Assembly ofmathematicians of Latvia. Before its foun-dation much preliminary work was doneby a group of enthusiasts, including UldisRaitums (the first Chair of the Society,from 1993-97), Andrejs Reinfelds (the pre-sent Chair), Jânis Cîrulis, and the author ofthese notes (the Chair from 1997-2000).The Society is governed by a Board ofseven members, one of which is the Chairof the Society. As its main aims, the societyconsiders the following activities:

• Consolidating the mathematical com-munity of Latvia, to make Latvian math-ematicians feel as a body and not as iso-lated individuals. In particular, theSociety aims to stimulate exchange ofideas between mathematicians repre-senting different areas and fields of sci-ence. This is especially important in acountry of the size of Latvia, where thenumber of mathematicians is small.Besides being interested in various fieldsof mathematics, they usually keep closerscientific contacts with foreign col-leagues and not with their compatriotcolleagues.

• Stimulating the development of Latvianmathematical terminology. This is espe-cially important since the overwhelmingmajority of mathematical publicationsare written in English, while our lecturesfor students are usually given in Latvian.

• Representing Latvian mathematiciansand mathematics developed in Latvia ininternational circles such as theEuropean Mathematical Society, andestablishing to keeping contacts withmathematical societies in other coun-tries.

• Supplying libraries with current mathe-matical publications. This was especiallyurgent during the first years of indepen-dence, when the financial situation ofour universities and our Academy wasextremely poor and they could afford nonew publications. During these years theLatvian Mathematical Society receivedbooks, journals and other kinds of infor-mation, through some agreements inthe form of exchange, as donations, orby greatly reduced prices. In particular,we have got such important materials forevery working mathematician asMathematical Reviews and Zentralblatt-MATH. (Here especially should be notedthe activities of the AMS by the FSU pro-gram, as well as support by the EMS andpersonally by Bernd Wegner, the Editor-in-chief of Zentralblatt-MATH.

As one of its main goals, the Society viewsthe organisation of seminars and confer-

ences. During the first ten years of its exis-tence, the LMS organised more than onehundred seminars, at which both localmathematicians and their foreign col-leagues have given talks. In particular, thefollowing speakers have given talks at ourseminar: Wies³aw elazko and TomaszKubiak (Poland), Wesley Kotzè (SA),Andrzhej Szymanski (ASV), Mati Abel,Arne Kokk and Heino Turnpu (Estonia),Ulrich Höhle, Hans Porst, Horst Herrlichand Helmut Neunzert (Germany), BorissPlotkin (Latvia-Israel), Juris Steprâns(Canada), and many others. The topicspresented at these seminars covered prac-tically all fields of mathematics.

The LMS organises biannual confer-ences. The guiding principle of these con-ferences is that they are not specialised:everyone working in mathematics inLatvia is welcome to give a talk. The work-ing language of the conference is Latvian –this has both positive effects (stimulatingthe development of mathematical termi-nology in Latvian) and negative ones:there are usually no foreign guests at theseconferences – the only exception was thethird conference held in 2000, whereabout 20 foreign guests were invited byAndþans to commemorate the 50thanniversary of the movement ofMathematical Olympiads in Latvia. Thefirst two conferences (in October 1995 and1997) were held in Riga, the capital ofLatvia, where about 80% of Latvian math-ematicians live and work, while later wedecided to go to other regions, in order topropagate mathematics more widely whilegaining a better conception about theproblems of mathematicians in the rest ofthe country. For example, our third con-ference in April 2000 was in Jelgava, atown known for its University ofAgriculture. The fourth conference inApril 2002 was in Ventspils, a town in thewest of the country, in the region calledKurzeme, an important harbour on theBaltic Sea. The main organisers of theseconferences were Andrejs Reinfelds andlocal organisers: Aivars Aboltiòð from theUniversity of Agriculture in Jelgava, and

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Participants of the 4th conference of the LMS in Ventspils. The castle is mid-14th century of the Livonian Order.

Jânis Vucâns, the rector of VentspilsUniversity. The next conference will beheld in 2004 in Daugavpils, a city in theSoutheast of Latvia in the region calledLatgale.

So far about the activities of the societyas a whole. However, any professionalsociety gains its strength through its mem-bers and their scientific and social activi-ties. So, here are the main fields in whichLatvian Mathematicians are working.

We first mention ordinary differentialequations, interest in which was essentiallyinfluenced by P. Bohl and later by A.Myshkis and L. Reiziòð (see above). Theleading people in this area are nowAndrejs Reinfelds, Yurij Klokov, ArnoldLepin and Felix Sadyrbajev. Reinfelds’main scientific interests are the qualitativetheory of differential equations, dynamicalsystems and impulse differential equa-tions; Klokov, Lepin and Sadyrbajev workin the qualitative theory of ordinary differ-ential equations, studying boundary-valueproblems (Klokov, Sadyrbajev) and thetopological properties of solutions of dif-ferential equations (Lepin).

Problems of optimal control for partial dif-ferential equations and calculus of variationsare investigated by Uldis Raitums, whoalso works on the problem of G-conver-gence of elliptic operators.

In recent decades one of the most rapid-ly developing areas has been mathematicalsimulation, represented by Andris Buiíis,Jânis Cepîtis, Andrejs Reinfelds, AivarsZemîtis and their students. One of theprincipal aims of this group is to developmathematics for the needs of industry,specifically for the wood industry that isamong the most important branches inLatvia. Evident confirmation of the successof the activities in this direction were pro-vided by the seminar ‘Baltic days:Mathematics for industry’ in June 1995with the participation of ECMI experts,and the 14th ECMI conference inSeptember 2002. Both of these events tookplace in Jurmala, near Riga, and wereorganised under close collaboration withthe LMS. For more information aboutthese and related activities of Latvianmathematicians, see ECMI Newsletter 29(March 2001), 20-23.

Research in partial differential equationsreceived a strong impulse from IsaakRubinshtein, who came to Riga in 1961and left for Israel in 1974. The main rep-resentatives of this field are now AndrisBuiíis, Harijs Kalis, MaksimilianAntimirov, Vladimir Gudkov and AndrejKolyshkin.

Important results in the theory of functionswere obtained by Eduards Riekstiòð (1919-92). In particular, he studied problems ofasymptotic expansion of functions: he wasthe author of five monographs related tothis subject. The fate of Riekstiòð duringthe war and the first years after it has muchin common with Reiziòð’ fate which we dis-cussed above. Interesting results in thefield of the theory of functions wereobtained by Georgs Engelis (special func-tions, operator theory), Teodors Cîrulisand Svetlana Asmuss (approximation and

expansion) and their collaborators.Researches in mathematical logic, and in

particular in the theory of recursive func-tions, started in the early 1950s by VilnisDetlovs who, after graduating from theUniversity of Latvia, completed hisCandidate thesis under the supervision ofA. A. Markov (Junior) at the University ofLeningrad. One of Detlovs’ first students,Jânis Bârzdiòð, developed his Candidatethesis in Novosibirsk (Russia) under thesupervision of a well-known Russian logi-cian B. A. Trahtenbrot: the thesis wasdevoted to the problem of universality inautomata theory. Bârzdiòð continued hisprofessional education at the MoscowState University where he wrote his Dr Scthesis, collaborating with the famousAndrej Kolmogoroff. The most active inthe field of mathematical logic is currentlyJânis Cîrulis, who has important results inalgebraic and many-valued logics as well asresearch papers in different areas ofabstract algebra.

The field of algebraic geometry is repre-sented by the talented mathematician

Aivars Bçrziòð, who, in particular, hasfound conditions under which two exten-sions of a given field have the same geom-etry.

Set-theoretic topology, the direction startedin Latvia by Michael Goldman in the1960s is now developed mainly by theauthor of these notes and his students.Our group is currently working in classicalset-theoretic and in categorical topology,as well as in many-valued (or fuzzy) topology.

The field of numerical analysis is repre-sented by Harijs Kalis, Andris Buiíis,Teodors Cîrulis and Ojârs Lietuvietis.

Stochastic aspects of Probability theory is thefield of research by Jevgenij Tsarkov andhis group including Kârlis Ðadurskis, at pre-sent the Minister of Education and Sciencein Latvia.

Mathematical statistics is represented byJevgenij Tsarkov, Aivars Lorencs, JânisLapiòð and their students and collabora-tors.

It should also be noted that a largeamount of work in the field of theoreticalcomputer science is carried out by a group of

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Jelgava Palace. At present the Latvian University of Agriculture is in this palace. The 3rd coference of the LMS took place here in April 2000.

The 3rd LMS conference in Jelgava, April 2000

Latvian scientists, headed by Jânis Bârzdiòðand Rusiòð Mârtiòð Freivalds. In particular,the results obtained by Freivalds and hiscollaborators in the theory of probabilisticautomata and of quantum computers arewell known and highly regarded by col-leagues all over the world. Problems ofstructural synthesis of probabilisticautomata are studied also by AivarsLorencs.

As a separate direction in mathematicsin Latvia, we should mention modern ele-mentary mathematics, whose founder andundoubted leader is Agnis Andþâns. Everyyear Andþâns and his group organise twoolympiads for school pupils interested inmathematics. In one of these olympiads allpupils can participate on their own initia-tive. To participate in the other one, apupil has to get good results in a series oflocal olympiads. Many advanced textbooksand problem books in elementary mathe-matics have been published. As evidenceof the immense work done by Andþâns andhis group (including Aivars Bçrziòð, AndrisCîbulis and Lîga Ramana) are the relativelyhigh places which the Latvian team usual-

ly earns at international mathematicalcompetitions. In 1998 the WorldFederation of National MathematicalCompetitions selected Andþâns as a recip-ient of a Paul Erdös Award to honour his‘significant contribution to the enrichmentof mathematical learning in Latvia’.Andþâns was the main organiser of the 2ndInternational Conference on ‘Creativity inMathematical Education and Education ofGifted Students’, which took place in Rigalast July.

Last, but not least, we wish to emphasisethe important role played by an outstand-ing teacher of mathematics, Jânis Mencis(Senior) in the development of mathemati-cal education in Latvia. His son, JânisMencis (Junior), has followed in hisfather’s footsteps and is also the author ofseveral mathematical textbooks forschools. For their enormous contributionsto the rise in level of the mathematical cul-ture of pupils in primary and secondaryschools in Latvia, Jäanis Mencis (Senior)and Agnis Andþâns were awarded the‘Three Star Order’, the highest nationalaward from the Latvian Government.

Acknowledgements When preparing this article we used materialswritten by L. Reiziòð, J. Dambîtis, I. Rabinoviès,I. Heniòa, D. Taimiòa. In particular, see theelectronic version of ‘Mathematics in Latviathrough the centuries’ by D. Taimiòa and I.Heniòa inwww.math.cornell.edu/~dtaimina/mathin.lv. htmlInformation about the first Society of LatvianPhysicists and Mathematicians was obtainedfrom a physicist Jânis Jansons. The authoracknowledges discussions with Jânis Cepîtis,Jânis Cîrulis, Boriss Plotkin, Uldis Raitumsand Andrejs Reinfelds, which helped to shapethese notes. We express special gratitude toJuris Steprâns, a Canadian mathematician ofLatvian descent, who helped to improve thelanguage style of these notes. In conclusionwe emphasise that, although the author hastried to be unbiased and has discussed thecontents of these notes with colleagues, theflavour of subjectivity is inevitable in such apaper.

The author [e-mail: sostaks@latnet.lv] is at theDepartment of Mathematics, University of Latvia,LV-1586 Riga.

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Slovenian Mathematical Society: some further commentsSlovenian Mathematical Society: some further comments

Peter Legisa

In my report on our Society, in the previous issue, I concluded with an unfortunate personal critical opinion of connections with ournorthern and western neighbours. It stirred dust and I should like to do some explaining.

It is a fact of life that undergraduates who study abroad tend to stay abroad. For obvious reasons, many undergraduates study atuniversities directly across the border. The concern of losing too much talent is one of the reasons why Slovenia is now establishingthe third university near our western border. (Language is the main obstruction to the arrival of foreign undergraduates in our coun-try.) Graduate students, who are more likely to return, disperse far and wide, and in countries to the north and west cooperationseems to be better with the more distant centres. With hindsight I connected these two facts, plus some political concerns, in a waythat strongly resembled the proverbial elephant in a china shop. No wonder that mathematicians who invested a lot of effort in ourcooperation have been upset.

I received a letter from Prof. Dr. Heinz W. Engl, President of the Austrian Mathematical Society. He presented counter-examplesto several of my conjectures. He also helped to dispel many of my concerns and offered more contacts, a gesture that I appreciatevery much. My colleagues also informed me that there was much more cooperation than I was aware of, especially with Austria. Theyeven warned me against including a list of names, since contacts were so numerous I would almost surely omit somebody. They alsotold me that personal relations with mathematicians across the border are excellent and that even in political discussions agreementwas quite common.

To correct the imbalance in my previous writing, I shall state some facts.• Prof. Josip Globevnik from Ljubljana gave an invited plenary lecture at the International Congress of the Austrian Mathematical

Society in Linz in 1993. We can consider this as a formal contact between the two societies.• The extremely fruitful cooperation with Montanuniversität in Leoben resulted in a joint book, several PhD degrees and exchanges

of graduate students. Prof. Wilfried Imrich (Leoben) recently held a colloquium talk in Ljubljana to a large audience. The LL[Leoben-Ljubljana] Seminar in Discrete Mathematics is an annual (or semi-annual) meeting with 30-40 participants and a historygoing back two decades.

• There is a long established cooperation between universities in Graz and Maribor (as well as Ljubljana), producing at least onePhD degree. We work with other Austrian universities as well.

• There were two joint regional conferences: one, concerning technology-supported mathematics teaching, was organised in 2000in Portoro on the Adriatic coast (by the universities in Maribor and Klagenfurt/Celovec).

• If we look towards Italy: Prof. Toma Pisanski from Ljubljana lectured at the University of Udine for three years. Exchanges ingraduate students produced at least one PhD degree in Italy and a MSc degree in Slovenia.

• Several of our colleagues have been guests in the Abdus Salam International Centre for Theoretical Physics in Trieste. There havebeen other joint ventures in numerical analysis, discrete mathematics, geometric topology, etc.

• There is good research co-operation with Hungary, another bordering country.

The title ‘Links in my report’ was inaccurate, since I limited myself to regional contacts. I never mentioned the prevailing connec-tions of the Slovenian mathematical community with other countries. (A high point was reached this year when Prof. Dana S. Scottof Carnegie-Mellon University (US) was awarded an Honorary PhD degree by the University of Ljubljana.)

Finally, I turn to the future. Slovenia voted overwhelmingly for membership in the EU and that means better opportunities forexchanges of graduate students and researchers in all directions. I am also glad to quote Prof. Engl of the Austrian MathematicalSociety:Our next international congress in Klagenfurt in 2005 will be specifically devoted to contacts to our neighbours in the south and in the east, just aswe do this year with our Italian colleagues in Bolzano.

Please e-mail announcements of European confer-ences, workshops and mathematical meetings ofinterest to EMS members, to one of the followingaddresses: vberinde@ubm.ro or vasile_berinde@yahoo.com. Announcements should be writtenin a style similar to those here, and sent asMicrosoft Word files or 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.

1-5: EuroConference VBAC 2003, Porto,PortugalInformation: ttp://www.math.ist.utl.pt/~cfloren/VBAC2003.html[For details, see EMS Newsletter 47]

1-10: PI-rings: structure and combinatorialaspects (summer course)Bellaterra (Barcelona), CataloniaInformation: e-mail: PI-rings@crm.es website: http://www.crm.es/PI-rings[For details, see EMS Newsletter 47]

3-8: Wavelets and Splines, St Petersburg,RussiaInformation: e-mail: ws@imi.ras.ruwebsite: http://www.pdmi.ras.ru/EIMI/2003/ws[For details, see EMS Newsletter 47]

6-12: Journées Arithmétiques XXIII, Graz,AustriaInformation: e-mail: ja03@tugraz.atwebsite: http://ja03.math.tugraz.at/[For details, see EMS Newsletter 47]

9-12: XV Italian Meeting on Game Theoryand Applications, Urbino, ItalyInformation: e-mail: XVimgta@uniurb.it website: www.econ.uniurb.it/imgta[For details, see EMS Newsletter 46]

13-28: VI Diffiety School in the geometryof Partial Differential Equations, SantoStefano del Sole (Avellino), Italy Information: e-mail: school@diffiety.org; website: http://www.diffiety.ac.ru orhttp://www.diffiety.org.[For details, see EMS Newsletter 47]

14-18: International Conference onAlgebras, Modules and Rings, Lisboa,Portugal [in memory of António AlmeidaCosta, on the centenary of his birth]Information: e-mail: : lisboa03@cii.fc.ul.pt;website: http://caul.cii.fc.ul.pt/lisboa2003/Subscribe to the conference mailing list athttp://caul.cii.fc.ul.pt/alg.announce.html[For details, see EMS Newsletter 46]

20-27: Hodge Theory in a New Century - AEuro Conference celebrating the Centenary

July 2003

of Sir William Hodge (1903-75),Edinburgh, UKInformation: e-mail: icms@maths.ed.ac.ukhttp://www.ma.hw.ac.uk/icms/meetings/2003/HODGE/index.html[For details, see EMS Newsletter 47]

21-25: The W³adys³aw Orlicz CentenaryConference and Function Spaces VII,Poznañ, PolandTopics: Banach space, geometry and topol-ogy of Banach spaces, operators and inter-polations in Banach spaces, Orlicz spacesand other function spaces, decomposition offunctions, approximation and related topics Main speakers:1. Conference in Honour of Orlicz, G. Bennet(USA), J. Diestel (USA), P. Domanski(Poland), F. Hernandez (Spain), N. Kalton(USA), B. Kashin (Russia), H. König(Germany), S. Konyagin (Russia), J. Kurzweil(Czech Republic), A. Pinkus (Israel), K.Urbanik (Poland), P. Wojtaszczyk (Poland); 2. Function Spaces VII, J. Appell (Germany),A. Kaminska (USA), D. Pallaschke(Germany), L. Persson (Sweden), A.Rozhdestvensky (Russia), B. Sims (Australia),F. Sukochev (Australia), H. Triebel(Germany) Programme Committee:

1. Conference in Honour of Orlicz, Z. Ciesielski(Poland), L. Drewnowski (Poland), W.Johnson (USA), F. Hernandez (Spain), J.Lindenstrauss (Israel), N. Kalton (USA), J.Musielak (Poland), G. Pissier (France), A.Pelczynski (Poland), P. Ulyanov (Russia); 2. Function Spaces VII (all from Poland), J.Musielak, H. Hudzik, L. Skrzypczak Organising Committee (all from Poland): Z.Palka (Chair), B. Bojarski (Vice-Chair), S.Janeczko (Vice-Chair), L. Skrzypczak(Secretary), P. Domanski, H. Hudzik, J.Kaczmarek, J. Kakol, I. Kubiaczyk, M.Mastylo, P. Pych-Tyberska, S. Szufla, R.Urbanski, J. Werbowski, A. Waszak, W.Wnuk Proceedings: to be publishedLocation: main building of the Faculty ofMathematics and Computer Science, AdamMickiewicz University, Poznañ, Poland Deadlines: for abstracts and registration,passedInformation: e-mail: orlicz@amu.edu.plwebsite: http://orlicz.amu.edu.pl

3-9: The Third International Conference‘Creativity in Mathematics Education andthe Education of Gifted Students’, Rousse,Bulgaria.Information: e-mail:conf_orgcom@ami.ru.acad.bg oremily@ami.ru.acad.bg website: www.cmeegs3.rousse.bg,www.ami.ru.acad.bg/conference2003/ www.nk-conference.ru.acad.bg[For details, see EMS Newsletter 47]

August 2003

5-10: Workshops Loops ‘03, Prague, CzechRepublicInformation: e-mail: loops03@karlin.mff.cuni.czwebsite: http://www.karlin.mff.cuni.cz/~loops/workshops.html[For details, see EMS Newsletter 47]

10-17: Loops ‘03, Prague, Czech RepublicInformation: e-mail: loops03@karlin.mff.cuni.czwebsite: http://www.karlin.mff.cuni.cz/~loops[For details, see EMS Newsletter 47]

11-15: Workshop on Finsler Geometry andIts Applications, Debrecen, HungaryInformation: e-mail: kozma@math.klte.huhttp://www.math.klte.hu/finsler/[For details, see EMS Newsletter 47]

18-22: ENUMATH 2003, EuropeanConference on Numerical Mathematicsand Advanced Applications, Prague, CzechRepublicInformation:e-mail: enumath@karlin.mff.cuni.czhttp://www.karlin.mff.cuni.cz/~enumath/ [For details, see EMS Newsletter 47]

18-22: 7th International Symposium onOrthogonal polynomials, Special Functionsand Applications, Copenhagen, DenmarkInformation: e-mail: opsfa@math.ku.dk,website:http://www.math.ku.dk/conf/opsfa2003[For details, see EMS Newsletter 46]

2-5: Symposium on ‘Cartesian Set Theory’,ParisInformation: e-mail: editions.europeenne@wanadoo.fr[For details, see EMS Newsletter 46]

2-6: Barcelona Conference on AsymptoticStatistics, Bellaterra (Barcelona), CataloniaInformation: e-mail: bas2003@crm.es web site: http://www.crm.es/bas2003[For details, see EMS Newsletter 46]

12-16: International Conference ofComputational Methods in Sciences andEngineering (iccmse 2003), Kastoria,Greece Information: iccmse@uop.grwebsite: http://www.uop.gr/~iccmse/ orhttp://kastoria.teikoz.gr/~iccmse/ [For details, see EMS Newsletter 47]

15-21: First MASSEE Congress, Borovetz,BulgariaInformation:http://www.math.bas.bg/massee2003/ [For details, see EMS Newsletter 47]

16-20: Barcelona Conference on SetTheory, Bellaterra (Barcelona), CataloniaInformation: e-mail: : set-theory@crm.es website: http://www.crm.es/set-theory[For details, see EMS Newsletter 46]

17-19: Conference on ComputationalModelling in Medicine, Edinburgh, UKInformation: e-mail: icms@maths.ed.ac.ukhttp://www.ma.hw.ac.uk/icms/current/index.html[For details, see EMS Newsletter 47]

September 2003

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FForthcoming conferorthcoming conferencesencescompiled by Vasile Berinde (Baia Mare, Romania)

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

The review of the book by J.-P. Pier (see below)appeared in EMS Newsletter 47 with an incorrecttitle. We apologise for this unfortunate mistake.

T. Andreescu and Z. Feng (eds.),Mathematical Olympiads 1998-1999:Problems and Solutions From Around theWorld, MAA Problem Books, MathematicalAssociation of America, Washington, 2000, 290pp., £19.95, ISBN 0-88385-803-7This is the authors’ second volume in theseries, and is a sequel to Mathematical Contests1997-1998: Olympiad problems and solutions fromaround the world. In the first part we find solu-tions to challenging problems from 1998 fea-tured in the previous book. In the secondpart, problems from 1999 are presented with-out answers. As the authors mention, differentnations have different mathematical cultures,and it is interesting for students to comparetheir knowledge of mathematics with that ofstudents from other countries. As a trainer ofmathematics competition teams, I often needto find problems related to some particulartopic. It is hard to find relevant ones in bookswith unsorted questions, so I was impressed bythe appendix containing a classification ofproblems. The glossary at the end of the bookwill be valuable for students.

The book will be particularly appreciated bystudents preparing for competitions, as well asby high-school teachers involved with thepreparation of students. If the series contin-ues, the material will be a valuable source for astudy of the development of mathematical cul-ture in various countries. From this viewpoint,it will also be of interest to advanced mathe-maticians, and a welcome addition to theirbookshelf. (mèi)

T. Andreescu and Z. Feng (eds.),Mathematical Olympiads 1999-2000:Problems and Solutions from Around theWorld, Mathematical Association of America,Washington, 2002, 323 pp., £21.95, ISBN 0-88358-805-3The editors of this book have collected morethan 500 problems from MathematicalOlympiads organised in various countries in1999-2000. The problems from 1999 appearwith solutions. The book is a sequel to theabove publication containing problems from1998-1999. The problems are arrangedaccording to individual countries and notaccording to topics. The first editor T.Andreescu is the coauthor of the successfulbook Mathematical Olympiad Challenges (seebelow). (lbo)

T. Andreescu and R. Gelca, MathematicalOlympiad Challenges, Birkhäuser, Boston,2002, 260 pp., 45 euro, ISBN 0-8176-4155-6and 3-7643-4155-6The book is the third of a successful collectionof mathematical problems. A review of thefirst edition (Math. Bohem. 127, p.504) reads:‘The authors of the book are two Romanianmathematicians living in the USA since 1991.Both of them had taken part successfully in a

number of mathematical competitions andafter completing their University studies haveengaged themselves in organisation of mathe-matical competitions for high school students,first in Romania and now in USA. They par-ticipate intensively in the preparation ofAmerican students talented for Mathematicsfor their participation in InternationalMathematical Olympiads’.

In this book they have collected about 400problems from various fields of mathematics,divided into three chapters: Geometry andtrigonometry, Algebra and analysis, andNumber theory and combinatorics. Eachchapter is divided into sections, (cyclic quadri-laterals, power of a point, periodicity, meanvalue theorem, Pell equations, etc.). Each sec-tion presents mathematical assertions usedsubsequently, several solved problems, andfurther unsolved problems: solutions of thelatter can be found in the other part of thebook. The problems come from mathematicaljournals and competitions in various coun-tries, as well as from InternationalMathematical Olympiads.

The book can be heartily recommended tostudents who wish to succeed in mathematicalcompetitions, and also to teachers preparingtheir students for demanding mathematicalcompetitions at both national and interna-tional levels. (lbo)

V. I. Arnold and S. P. Novikov (eds.),Dynamical Systems IV, 2nd edition,Encyclopaedia of Mathematical Sciences 4,Springer, Berlin, 2001, 335 pp., DM 159,ISBN 3-540-62635-2This is the second, expanded and revised edi-tion of the book. It consists of three contribu-tions: Symplectic geometry by V. I. Arnold and A.B. Givental’, Geometric quantisation by A. A.Kirillov, and Integrable systems I by B. A.Dubrovin, I. M. Krichever and S. P. Novikov.

In the first part, a survey of fundamentalconcepts of symplectic geometry is given.Starting with linear symplectic geometry inthe first chapter, the theory of symplecticmanifolds is developed in Chapter 2 andapplied to mechanics in Chapter 3. Anoverview of contact geometry is presented inChapter 4. The last two chapters are devotedto the study of Lagrangian and Legendre sin-gularities and Lagrangian and Legendrecobordisms. The whole contribution is verynicely written, with many figures appealing togeometric intuition. The second part offersan overview of the theory of geometric quanti-sation by one of its founders. There is a niceintroductory part containing mathematicalmodels of classical and quantum mechanics,the statement of quantisation problems andexplaining a connection with orbits in repre-sentation theory. Then prequantizations,polarisations and quantisation are introduced.This part ends with definition and propertiesof quantisation operators, and there is a sup-plement to this edition containing some topicsinvestigated and exploited in recent years.The last part surveys the modern theory ofintegrable systems, based on the inverse scat-tering method. In the first chapter, hamilton-ian systems and classical methods of integra-tion are presented, and Chapter 2 describesmodern ideas on integrability of evolution sys-

tems. There are four appendices devoted tospecial problems and to historical remarks onmethods from algebraic geometry,Hamiltonian systems and spectral theory. Asis standard in volumes of the Encyclopaedia, allparts bring an excellent description of thechosen topics, with many examples, mostlywithout proofs. They can be found in thebooks and papers cited in a comprehensivebibliography. (jbu)

B. C. Berndt and R. A. Rankin (eds.),Ramanujan: Essays and Surveys, History ofMathematics 22, American MathematicalSociety, Providence, 2001, 347 pp., US$79,ISBN 0-8218-2624-7During his short life Srinivasa Ramanujan(1887-1920) published 37 papers (not count-ing solutions of problems for the Journal of theIndian Mathematical Society). Nevertheless, hewas a mathematical genius who heavily influ-enced many fields of mathematics and stimu-lated their development for many years ahead.A description of his work can be found inHardy’s Collected Papers of S. Ramanujan (1927)and many further studies. We recall, forexample, the outstanding volumes publishedby B. C. Berndt (Ramanujan’s Notebooks I-V,Springer, 1985-98), containing a detailed dis-cussion of Ramanujan’s preserved notes.

This book contains 28 papers coveringRamanujan’s exceptional life and magnificentmathematical work from many angles, andincluding several documentary photos. Let usmention explicitly the papers by B. C. Berndt(The books studied by Ramanujan in India), S.Janaki Ammal (Mrs. Ramanujan), The note-books of S. Ramanujan), R. A. Rankin (TheRamanujans’ family record, Ramanujan as apatient, Ramanujan’s manuscripts and note-books I-II), A. Selberg (Reflections aroundRamanujan’s centenary), R. Askey(Ramanujan and hypergeometric and basichypergeometric series) and G. N. Watson (Anaccount of the mock theta functions).

The book can be heartily recommended tomathematicians working in number theory,special functions and mathematical analysis,as well as to those interested in history ofmathematics. (bn)

A. Borel, Essays in the History of Lie Groupsand Algebraic Groups, History of Mathematics21, American Mathematical Society,Providence, 2001, 184 pp., US$39, ISBN 0-8218-0288-7This book consists of essays on topics belong-ing mainly to the first century of the history ofLie groups and algebraic groups. Some ofthem have been published before. In the firstchapter, the author describes the historicaldevelopment of the theory of Lie groups andLie algebras, starting from the local theory ofSophus Lie. Chapter 2 is devoted to problemsof full reducibility and invariants of the groupSL2 (C). In Chapter 3, the author explains theprincipal role played by Hermann Weyl in thedevelopment of theory of Lie groups and theirrepresentations; an important impact of hispapers on the work of Élie Cartan is also men-tioned. Chapter 4 concentrates on ÉlieCartan’s work on symmetric spaces and Liegroups, semisimple groups and global theory,as well as the role of the theory of Lie groupsin the Cartan calculus of exterior forms; thetheory of bounded symmetric domains is alsopresented here. Chapter 5 contains a reviewof the theory of linear algebraic groups in the19th century, and Chapter 6 describes themain lines of a spectacular development of thetheory in the 20th century. Chevalley’s work

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Recent booksRecent booksedited by Ivan Netuka and Vladimír Souèek

in the theory of Lie groups and algebraicgroups is presented in Chapter 7. The lastchapter is based on E. R. Kolchin’s work onalgebraic groups and Galois theory.

The book is an interesting collection ofpapers describing ideas that were essential tothe development of the theory. (jbu)

B. Buffoni and J. Toland, Introduction à laThéorie Globale des Bifurcations, CahiersMathématiques de l’EPFL, PressesPolytechniques et Univesitaires Romandes,Lausanne, 2002, 130 pp., 33,88 euros, ISBN 2-88074-494-6This small book is based on a course at ÉcolePolytechnique Fédéral de Lausanne duringthe winter semester 1999-2000. It is a simplywritten introduction to bifurcation theory withapplications. The style is telegraphic and ele-gant, and the book is therefore strongly rec-ommended to students and other interestedreaders as a first reading in this field. It mayalso help with the design of a course for lessadvanced (undergraduate) students. Thereader is supposed to have a basic knowledgein linear functional analysis, while other tools,such as differential calculus in Banach spacesand the local theory of analytic sets, are ade-quately contained in the book (20 and 38pages from 122). Readers will learn thefamous theorem of Crandall-Rabinowitz inlocal bifurcation theory and how to apply it tothe Euler-Bernoulli beam model, how tomodel surface waves on an infinite ocean(Stokes’ waves) as a bifurcation problem, andthey will see demonstrations of results of glob-al bifurcation theory to Stokes’ waves. A fewresults in the last section are presented with-out proof. (ef)

D. Burago, Yu. Burago and S. Ivanov, ACourse in Metric Geometry, Graduate Studiesin Mathematics 33, American MathematicalSociety, Providence, 2001, 415 pp., US$44,ISBN 0-8218-2129-6This book contains ten chapters. The first isdevoted to the standard advanced introduc-tion to metric spaces. Hausdorff measure anddimension are explained, and the Banach-Hausdorff-Tarski paradox is also mentionedinformally. Chapter 2 deals with the very gen-eral ‘length structure’ defined on a topologicalspace and later studied in metric spaces:examples range widely from Euclidean geom-etry to the Finsler geometry. The famous the-orem of Hopf and Rinow is generalised fromthe Riemannian case to a length space.Chapter 3 considers the glueings of lengthspaces, the quotient metrics, polyhedralspaces, metric graphs (including so-calledCayley graphs), direct products and warpedproducts on length spaces, concluding withthe abstract study of angles. Chapter 4 isdevoted to spaces of bounded curvature(Alexandrov spaces): here the curvature is notdefined explicitly – instead, spaces of non-pos-itive curvature and non-negative curvature arecharacterised by the geometric behaviour oftheir distance functions, when compared withthe Euclidean case. The first variation formu-la (known from differential geometry) isderived for Alexandrov spaces. Chapter 5 ison Riemannian metrics, Finsler metrics, sub-Riemannian metrics and a detailed study ofhyperbolic plane; the Besicovitch inequality isproved at the end. Chapter 6 deals with thecurvature of Riemannian metrics and therelated comparison theorems. Chapter 7investigates the ‘space of metric spaces’(Lipschitz distance, Gromov-Hausdorff dis-tance and convergence). Chapter 8, ‘Large-

scale geometry’, deals mainly with Gromovhyperbolic spaces and with periodic metrics.Chapter 9 is concerned with the spaces of cur-vature bounded from above, mainlyHadamard spaces (again in the most generalsetting); as an ‘example’, the theory of semi-dispersing billiards is treated. The last sectiontreats spaces of curvature bounded frombelow: the main results are general versions ofthe Toponogov theorem and the Gromov-Bishop inequality.

The book is written in an informal precisestyle, with many exercises. This is a reallyexcellent book which should be read by everyexpert in geometry, and which can be warmlyrecommended even to undergraduate stu-dents. A tribute is paid to the fundamentalcontributions to many parts of geometry andtopology by mathematicians of Russian origin.(ok)

C. Chatfield, Time-Series Forecasting,Chapman & Hall/CRC, Boca Raton, 2000, 267pp., US$69.95, ISBN 1-58488-063-5This book deals with forecasting in time seriesand is aimed at readers with a statistical back-ground and basic knowledge of time series. Itstarts with a short introduction to time-seriesanalysis, giving simple descriptive techniquesand basic facts. It is followed by a survey ofunivariate time-series models. Then followARMA and ARIMA models, including long-memory models, state-space models, non-lin-ear models, GARCH models, and other mod-els with changing variance, as well as neuralnetworks and chaos. The next chaptersdescribe univariate and multivariate forecast-ing methods and explore relationshipsbetween different methods, providing manyup-to-date references. Criteria for comparingand evaluating forecasting methods are intro-duced, and there is a discussion of a problemon how to choose the ‘best’ method for a par-ticular situation. Prediction intervals are alsoconsidered, and the effect of model uncertain-ty on the accuracy of the forecast is treated;the results are summarised at the end of thefinal chapter.

This book can serve as a textbook on fore-cast model building, and is an outstanding ref-erence source for both researchers and practi-tioners, especially in economics, operationsresearch and industry. (zp)

S. S. Cheng, Partial Difference Equations,Advances in Discrete Mathematics andApplications 3, Taylor & Francis, London,2003, 267 pp., £50, ISBN 0-415-29884-9The main topic of this book is a class of func-tional relations with recursive structures,called partial difference equations. Theauthor describes mathematical methods thathelp in dealing with recurrence relations gov-erning the behaviour of variables such as pop-ulation size or stock price.

The book begins with introductory exam-ples of partial difference equations: most ofthese stem from familiar concepts of heat dif-fusion, heat control, temperature distribution,population growth and gambling. Basic defi-nitions and a classification of partial differenceequations are given in Chapter 2, while thefollowing chapter introduces operators andstudies their properties. Chapter 4 is con-cerned with univariate sequences or bivariatesequences that satisfy monotone or convexfunctional relations; maximum and discreteWirtinger’s inequalities are derived. InChapter 5, the author employs methods ofoperators, and a number of other methods,for constructing explicit solutions for a num-

ber of basic partial difference equations.Certain methods useful for deriving stabilitycriteria appear in Chapter 6. Chapter 7 pre-sents several criteria for the existence of solu-tions, mainly those that are positive, bounded,or for travelling waves. In the final chapter,the author derives a number of conditions forthe non-existence of solutions of certain types.Every chapter ends with notes and remarks.

The book offers a concise introduction totools and techniques that were used to obtainresults in the theory of partial difference equa-tions. It will be useful for anyone who hasmastered the usual sophomore mathematicalconcepts. (kn)

E. N. Chukwu, Optimal Control of the Growthof Wealth of Nations, Stability and Control:Theory, Methods and Applications 17, Taylor& Francis, London, 2003, 384 pp., £75, ISBN0-415-26966-0The aim of this research monograph is toidentify realistic dynamic models for thegrowth of wealth of nations. For this purpose,the text derives the dynamics of the key eco-nomic variables (gross national product,employment, interest rates, capital stock,inflation, balance of payment), using rationalexpectations and market principles. Themodels are validated using economic timeseries of several countries, enabling one todeal with the growth of wealth of nations asexamples of mathematical optimal controlproblems. The corresponding governmentcontrol variables include taxation, govern-ment outlay, money supply and tariffs. Thecontrol variables of private firms includeautonomous consumption, investment, netexport, money holding, wages and productivi-ty; linear models using MATLAB are includedfor this purpose. The controllability and otherproperties of the systems are discussed as asimple explanation of an economic situation.The text compares the extent of governmentintervention with the activity of private firmsin the framework of controllabilty of the econ-omy.

The monograph is suitable for graduate stu-dents and researchers in applied mathematicsand economics dealing with problems of theglobal economy. (tc)

L. Conlon, Differentiable Manifolds, 2nd edi-tion, Birkhäuser Advanced Texts, Birkhäuser,Boston, 2001, 418 pp., DM 130, ISBN 0-8176-4134-3 and 3-7643-4134-3This is a well-organised and nicely written textcontaining the basic topics needed for furtherwork in differential geometry and globalanalysis. This extended second edition startswith topological manifolds and their proper-ties, including covering spaces and the funda-mental group. Smooth manifolds and smoothmaps are introduced, after a review of thelocal theory of smooth functions. The localtheory of local flows is then used to treat folia-tions and flows on manifolds. The next chap-ter, on Lie groups and Lie algebras, intro-duces an important tool for future use, andincludes key examples of Lie groups and theirhomogeneous spaces. The following threechapters introduce differential forms andtheir integration on manifolds, including deRham cohomology, Mayer-Vietoris sequences,Poincaré duality, the de Rham theorem anddegree theory; the Frobenius theorem is thenreformulated in terms of differential forms.The last two chapters present basic facts fromRiemannian geometry and introduce princi-pal fibre bundles. Appendices contain basicfacts on universal coverings, inverse function

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theorem, ODEs, and proofs of the de Rhamtheorem for Èech and singular cohomology.This book is very suitable for students wishingto learn the subject, and interested teacherscan find well chosen and nicely presentedmaterial for their courses. (vs)

N. J. Cutland, Loeb Measures in Practice:Recent Advances, Lecture Notes inMathematics 1751, Springer, Berlin, 2000,111 pp., DM 44, ISBN 3-540-41384-7This book presents an expanded version ofthe European Mathematical Society lecturesgiven by the author at Helsinki in 1997. Itdescribes important and interesting applica-tions of the Robinson non-standard analysis inmeasure theory, (stochastic) PDEs, Malliavincalculus and financial mathematics.

It starts with a condensed review of non-standard analysis, the Loeb construction of aLoeb measure from a non-standard measure(including simple constructions of theLebesgue, Haar and Wiener measures fromthe Loeb measure), Loeb integration and sim-ple applications. The three following chapterscontain applications of the principles inChapter 1 to three different fields: fluidmechanics (deterministic and stochasticNavier-Stokes equations), stochastic calculusof variations (infinite-dimensional Ornstein-Uhlenbeck process, Malliavin calculus) andfinancial mathematics (Cox-Ross-Rubinsteinand Black-Scholes models, convergence ofmarket models). These applications showclearly that methods of non-standard analysiscan be used very efficiently in many fields of‘ordinary’ mathematics. The basic principlesof possible applications can be well under-stood from the interesting material presentedhere, and the book will inspire mathemati-cians and mathematical physicists. (vs)

G. Da Prato and J. Zabczyk, Second OrderPartial Differential Equations in HilbertSpaces, London Mathematical Society LectureNotes Series 293, Cambridge University Press,Cambridge, 2002, 379 pp., £29.95, ISBN 0-521-77729-1This book is devoted to parabolic and ellipticequations for functions defined on infinite-dimensional Hilbert spaces. The first attemptto generalise the theory of partial differentialequations to functions defined on infinite-dimensional spaces was made by R. Gateauxand P. Lévy around 1920. A differentapproach, developed by L. Gross and Yu.Daleckij about 30 years ago, is used here; itsmain tools are probability measures in Banachspaces, semigroups of linear operators, sto-chastic evolution equations, and interpolationspaces.

The book is divided into three parts. Part 1presents the existence, uniqueness and regu-larity of solutions in the space of bounded anduniformly continuous functions. The theoryof Sobolev spaces with respect to a Gaussianmeasure is developed in Part 2, enabling astudy of equations with very irregular coeffi-cients arising in different applications, such asreaction-diffusion systems and stochasticquantisation. The final part contains someapplications to control theory.

This book presents the state of the art of thetheory. To make the presentation as self-con-tained as possible, it includes essential back-ground material. The reader will find numer-ous comments and references to more spe-cialised results not included here. It can bewarmly recommended to anyone interested inthe field. (rl)

A. Del Centina, The Abel’s ParisianManuscript, Cultura e Memoria 25, Leo S.Olschki Publisher, Firenze, 2002, 185 pp.,ISBN 88-222-5090-7This bilingual (Italian-English) book containsa reproduction of the famous 1826 Paris man-uscript by the Norwegian mathematician NielsHenrik Abel (1802-29). The manuscriptMemoire sur une propriété d’une classe très étenduede fonctions transcendantes is deposited in theMoreniana Library in Florence, and has beenrepublished for the bicentenary of Abel’sbirth. In its time, the paper had a greatimpact on mathematical research, openingnew research directions in the field.

The book contains the following chapters:The enhancement of the documents of theMoreniana library, Niels Henrik Abel, a bio-graphical portrait, Genius and orderliness,Abel in Berlin and Paris, The mathematicalsignificance of Abel’s Paris memoir (A verysuperficial tale), Abel’s manuscripts in theLibri collection: their history and their fate,The papers of Guglielmo Libri in theMoreniana library of Florence. They containnew contributions concerning Abel’s life andwork, written by mathematicians and histori-ans from Norway and Italy. A short history ofthe Moreniana Library is also included. Thefinal part of the book contains a reprint ofAbel’s manuscript. (mnem)

J. F. van Diejen and L. Vinet (eds.), Calogero-Moser-Sutherland Models, CRM Series inMathematical Physics, Springer, New York,2000, 561 pp., DM 189, ISBN 0-387-98968-4Thirty years ago, Calogero and Sutherland(on a quantum level) and Moser (on a classicallevel) showed the integrability of certain N-particle models. Perelomov and Olshanetskythen constructed similar integrable modelsconnected with root systems of simple Liegroups. Soon after, a relativistic deformation(Ruijsenaars and Schneider) and a long-rangespin model (Haldane and Shastry) were dis-covered. These origins were followed by abroad and intensive activity in the field inrecent years, leading to some very interestingconnections with other parts of mathematicaland theoretical physics, as well as of puremathematics.

The meeting organised in 1997 at CRM(Montréal) brought together people workingin many different areas connected with theoriginal models. Its proceedings contains 39contributions, covering a broad range of top-ics. There are papers by Calogero andSutherland, as well as longer papers on solu-tions of the quantised KZ difference equationby multidimensional q-hypergeometric inte-grals (E. Mukhin, A. Varchenko), relativisticLamé functions (S. Ruijsenaars), combinatori-al theory of the Jack and Macdonald symmet-ric polynomials (L. Lapointe, L. Vinet), gener-alisations of the elliptic Calogero-Moser andRuijsenaars-Schneider systems based on a spe-cial inverse problem (I. Krichever), a connec-tion between a Painlevé VI equation and theelliptic Calogero system (A. M. Levin, M. A.Olshanetsky), generalisations of R-matricesfor Calogero-Moser systems (J. Avan) andmany other interesting papers. The bookoffers a good overview of this modern andquickly developing area between mathematicsand physics, and will be of great interest to awide range of mathematicians and theoreticalphysicists. (vs)

G. Ellingsrud, W. Fulton and A. Vistoli (eds),Recent Progress in Intersection Theory, Trendsin Mathematics, Birkhäuser, Boston-Basel-

Berlin, 2000, Hardback, CHF 148.00, ISBN 0-8176-4122-X, This collection of articles originated in theInternational Conference in IntersectionTheory, organised in Bologna in December1997. The main aim of the conference was tobring postgraduate students to the borders ofcontemporary research in intersection theory.Four preparatory courses were prepared, byM. Brion (Equivariant Chow groups andapplications), H. Flenner (Joins and intersec-tions), E. M. Friedlander (Intersection prod-ucts for spaces of algebraic cycles), and R.Laterveer (Bigraded Chow (co)homology).With exception of the first one, the texts ofthese courses (in slightly modified form)appear here. In addition, the collection con-tains eight more research articles from inter-section theory.

Anyone reading this collection will need agood knowledge of algebraic geometry andcommutative algebra. On the other hand, theauthors were well chosen: excellent specialistswho are also very good writers. Because of thehigh mathematical quality of the articles, thiscollection should be attractive to specialists inalgebraic geometry and related fields. (jiva)

G. Fischer and J. Piontkowski, RuledVarieties. An Introduction to AlgebraicDifferential Geometry, Friedrich Vieweg &Sohn, Wiesbaden, 2001,141 pp., DM 68, ISBN3-528-03138-7Ruled varieties form a natural generalisationof ruled surfaces. A ruled variety is a variety ofarbitrary dimension which is swept out bymoving linear subspaces of ambient space; theruling is an extrinsic property of a varietyrelated to a chosen embedding into ambientaffine or projective space.

In this book, complex projective algebraicvarieties are mainly considered: theHermitian Fubini-Study metric and relativecurvature are not necessary here. It is suffi-cient to consider the second fundamentalform, which is the differential of the Gaussmap. In comparison with the book by P.Griffiths and J. Harris, Principles of algebraicgeometry, this book offers a more detailed andelementary explanation of related results, andalso brings a report on recent progress in thearea. An introductory chapter contains areview of facts needed from classical differen-tial and projective geometry. Chapter 1 isdevoted to Grassmanianns, and an elementarytheory of ruled surfaces can be found inChapter 2. In the final chapter, tangent andsecant varieties are studied, and the third andhigher fundamental forms are used todescribe their properties. The book is verywell written, and I can recommend it to any-one interested in these topics. (jbu)

L. C. Grove, Classical Groups and GeometricAlgebra, Graduate Studies in Mathematics 39,American Mathematical Society, Providence,2001, 169 pp., US$35, ISBN 0-8218-2019-2This book contains a detailed study of classicallinear groups over arbitrary fields, both finiteand infinite. The first part describes the prop-erties of classical groups over fields with char-acteristic different from 2. The discussionstarts with symplectic groups, Clifford alge-bras are treated together with orthogonalgroups, and the discussion ends with unitarygroups. The final chapters are then devotedto orthogonal groups and Clifford algebras incharacteristic 2. The author concentrates onindividual classes of classical groups, and gen-eral tools of Lie theory are not used here.Relationships with Chevalley groups are indi-

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cated. This text is written for a course at graduate

level, and contains many exercises. The read-er will need a good knowledge of linear alge-bra and some parts of general algebra. (vs)

M. Günter, Customer-based IP ServiceMonitoring with Mobile Software Agents,Whitestein Series in Software AgentTechnologies, Birkhäuser, Basel, 2002, 160pp., EUR 27,10, ISBN 3-7643-6917-5Software agents are software artefacts withspecific properties, which usually include:autonomy (the agent is a software applicationable to perform autonomous activities), mobil-ity (the agents are mobile agents (MA) that canautonomously move through a computer net-work) and cloning (the agent can produce itsnew instances). Further properties can beassociated with the agent intelligence in thesense of artificial intelligence, and a basic pre-condition for this is the ability of agents tobuild autonomously their communication, butsuch properties are not the topics of this book.The author describes in detail the implemen-tation of MA able to monitor the behaviourand effectiveness of services of nodes of thenetwork, related to requirements of a particu-lar user of the network. The implementationof MA and the supporting infrastructure,including the security issues, are also discussedin detail. The implemented mobile agents canmonitor aspects of the network behaviour thatinterest a user.

The book contains results of experimentsmeasuring various network properties (such asone-way delay, delay variation, round-triptimes and bottleneck bandwidth) and theoverhead connected with the agent activities.In the final chapter, the author mentions sev-eral similar related projects, such as Internet2Initiative, Qbone, and related IP measure-ment methodologies, and proposes ways ofapplying his MA here. He also discusses theapplications of his systems in network man-agement and formulates several open prob-lems. The book does not indicate whether thedeveloped system has been used in real-lifelong-term mass service practice. The bibliog-raphy contains more than 150 items.

The book is an excerpt from the author’sPh.D. thesis Management of Multi-ProviderInternet Services with Software Agents, publishedin July 2001. It is available, with the devel-oped source code, at http://www.iam.unibe.ch/~mguenter/phd.html. (jk)

S. Helgason, Differential Geometry, LieGroups, and Symmetric Spaces, GraduateStudies in Mathematics 34, AmericanMathematical Society, Providence, 2001, 641pp., US$69, ISBN 0-8218-2848-7This is a new edition of a volume published byAcademic Press under the same title in 1978.The author has corrected some inaccuraciesand has added a number of footnotes thatappear in the final section. It is useful to recallthat the author’s second book Groups and geo-metric analysis (Academic Press, 1984) wasreprinted by the AMS in 2000, as Volume 83in the series Mathematical Surveys andMonographs. An essentially different version ofthis book appeared in 1994 as Volume 39 ofthe same series, under the title Geometric analy-sis on symmetric spaces. (ok)

N. Higson and J. Roe, Analytic K-Homology,Oxford Mathematical Monographs, OxfordUniversity Press, 2000, 405 pp., £65, ISBN 0-19-851176-0K-homology can be formally described as the

homology theory dual to Atiyah-Hirzebruch’sK-theory (even though this fact does not seemto be extremely useful). It turns out, especial-ly in recent years, that K-homology finds manyvery interesting applications, both in topologyand geometry. Using K-homology techniquesit has been possible to attack problems thathad been considered rather hopeless, and topresent shorter and more conceptual proofs ofearlier theorems. (We mention that K-homol-ogy enables us to write a very short proof ofthe Atiyah-Singer index theorem.)

This book combines two interpretations ofK-homology, both closely related to functionalanalysis: the well-known Brown-Douglas-Fillmore theorem, and Kasparov’s K-homolo-gy. Here we underline that the Kasparovproduct plays an especially important role.The most interesting applications are concen-trated around the index theory, and we findhere index theory for hypersurfaces, the indextheorem for Spinc-manifolds, Toeplitz indextheorems, and index theory on stronglypseudoconvex domains. The last chapter isdevoted to the higher index theory, withapplications in Riemannian geometry.Because the whole book deals with complex K-homology, the authors include an appendixsuggesting a possible passage to real K-homol-ogy. However, they have decided not todescribe systematically Kasparov’s bivariantKK-theory.

Concerning prerequisites, the reader needsto be familiar with basic functional analysisand algebraic topology (or homological alge-bra), but this is not sufficient if the reader isinterested in applications: here more spe-cialised knowledge is required. But the bookis carefully written, and the authors includemany notes and commentaries that enable thereader to understand the main lines of theexposition. At the end of each chapter areexercises that enhance the main text and areoften rather difficult. In summary, this is avery nice book that will interest specialistsfrom several branches of mathematics and beattractive for postgraduate students. (jiva)

A. Holme, Geometry. Our Cultural Heritage,Springer, Berlin, 2002, 378 pp., 34,95 euros,ISBN 3-540-41949-7This book discusses central themes from thehistory of classical and modern geometry, andis divided into two parts.

Part 1 (A Cultural Heritage) contains sixchapters on the roots of geometric ideas, theirdevelopment and their historical context.Many interesting themes are included: geom-etry in prehistory, Egypt and Babylonia,Greek geometry (the Pythagoreans, Platonicsolids, classical problems and their solutions),the Hellenic Era (Euclid, Archimedes,Apollonius of Perga, Heron of Alexandria,etc.), and European geometry from theMiddle Ages to modern times. We find heresome problems that ancient mathematiciansthought about and how they solved them. Wealso find legends, stories and tales (aboutPlato, Pythagoras, Euclid, Archimedes, etc.).

Part 2 (Introduction to Geometry) containstwelve chapters on the development of someaspects of modern geometry (for example, thecreation of axiomatic geometry, projectivegeometry, models and properties of non-Euclidean geometry, the main properties ofalgebraic curves, solutions of classical prob-lems, the beginning of geometrical algebraand fractal geometry, and an introduction tocatastrophe theory). This part is a very inter-esting introduction to modern geometry andits historical aspects.

The book can be recommended as an excel-lent textbook for several possible courses (suchas Historical topics in geometry, orIntroduction to modern geometry). It can berecommended to historians of mathematics,and to professional mathematicians, teachersand students wanting to understand the ori-gins of modern geometry. (mnem)

R. Iorio and V. Iorio, Fourier Analysis andPartial Differential Equations, CambridgeStudies in Advanced Mathematics 70,Cambridge University Press, 2001, 411 pp.,£45, ISBN 0-521-62116-XThis book arises from several courses andseminar talks over the years, and is a greatlymodified version of the authors’ previous work(Equacoes diferenciais parciais, 1978). Its aim isto introduce students to the basic concepts ofFourier analysis and illustrate this theory inthe study of linear and non-linear evolutionequations.

After a preliminary chapter, the basic theo-ry of Fourier series is presented. Chapter 3develops the theory of periodic distributions.Since it is done in the context of (continuous)functions and series, the reader is not assumedto know the Lebesgue integral; a separate sec-tion presents material from topology, neces-sary for characterising the topology of distrib-utions.

The second part of the book is devoted toapplications of the previous chapters.Chapter 4 provides applications to linear evo-lution equations with periodic boundary con-ditions in Sobolev spaces, and includes a sec-tion on (unbounded) operator theory andsemigroup theory. The following two chaptersdeal with non-linear equations. In Sobolevspaces as phase spaces, the local and globalexistence and uniqueness of solutions areproved by means of fixed-point theorems anda priori estimates. In particular, theSchrödinger and Korteweg-de Vries equationsare treated and their Hamiltonian structure isexplained.

The third part deals with partial differentialequations on Rn. Chapter 7 presents the the-ory of distributions, Fourier transforms andSobolev spaces, simplified with references tosome analogous periodic cases developed inChapter 3. As applications, linear heat andSchrödinger equations, and the non-linearKorteweg-de Vries and Benjamin-Ono equa-tions are studied in greater detail.

The reader should have a basic knowledgeof functional analysis (Banach and Hilbertspaces, bounded operators), ordinary differ-ential equations, and, for the third part of thebook, the Lebesgue integral. Developing theperiodic distribution theory based onsequences first is an interesting feature of thebook. The exercises in each section form anintegral part of the text. It would have beenhelpful to have a list of symbols and a longerindex. Students and those beginning theiracademic career should find this text interest-ing and stimulating. (ef)

H. Iwaniec, Spectral Methods of AutomorphicForms, 2nd edition, Graduate Studies inMathematics 53, American MathematicalSociety, Providence, 2002, 220 pp., US$49,ISBN 0-8218-3160-7This is a very readable introduction to thespectral theory of automorphic forms onSL2(R), based on the author’s higher graduatecourses. The book begins with harmonicanalysis on the upper half-plane H, Fuchsiangroups and Eisenstein series, and proceeds todevelop spectral theory on Ã\H, beginning

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with cusp forms and continuing with analytictheory of Eisenstein series and their residues.Establishing the Kuznetsov formula and theSelberg trace formula, the author estimatesFourier coefficients of Maass forms, eigenval-ues of the hyperbolic Laplacian, and the cuspforms themselves. An appendix covers thenecessary analytic background. This is aremarkable book, written in a clear style byone of the world leaders in number-theoreticalapplications of the spectral theory of automor-phic forms. (jnek)

M. Jarnicki and P. Pflug, Extension ofHolomorphic Functions, de GruyterExpositions in Mathematics 34, Walter deGruyter, Berlin, 2000, pp., DM 248, ISBN 3-11-015363-7A domain of holomorphy for holomorphicfunctions of several complex variables is adomain G in Cn on which there exists a holo-morphic function that cannot be continuedacross any boundary point of G. Every domainis a domain of holomorphy for n = 1, but thisis not true in several complex variables.Moreover, studying the continuation of holo-morphic functions leads directly to multival-ued holomorphic functions and envelopes ofholomorphy.

A classical method for treating them in onevariable is to study holomorphic functions onRiemann surfaces. This book contains a studyof extensions of holomorphic functions of sev-eral complex variables in the framework ofRiemann domains over Cn. The first chapterintroduces the domain of holomorphy for afamily of holomorphic functions, and its char-acterisation in terms of holomorphic convexi-ty. Chapter 2 discusses pseudoconvexdomains and a solution of the Levi problemfor Riemann domains over Cn, while Chapter3 studies envelopes of holomorphy for specialtypes of domains. The final chapter describesthe structure of envelopes of holomorphy.

The book contains many open problems,and can be recommended to those interestedin the theory of several complex variables. (vs)

S. Kantorovitz, Introduction to ModernAnalysis, Oxford Graduate Texts inMathematics 8, Oxford University Press, 2003,434 pp., £45, ISBN 0-19-852656-3This book in the Oxford Graduate Texts inMathematics series is based on lectures in dif-ferent related subjects given since 1964 atYale, the University of Illinois and Bar IlanUniversity. The series is aimed at presentingtexts of high mathematical quality, especiallyin areas that are not sufficiently covered in thecurrent literature.

The reader will find here a carefully writtenexposition of classical topics (Lebesgue inte-gration, Hilbert spaces, and elements of func-tional analysis), and results less frequently cov-ered such as Haar measure, the Gelfand-Naimark-Segal representation theorem andthe von Neumann double commutant theo-rem. Many results appear in exercises (almost130 in total). One third of the book is devot-ed to two long applications: probability anddistributions (Appl. II). The book can thus beviewed as a multi-purpose text for advancedundergraduate and graduate students, andcan be recommended for mathematicallibraries. (jive)

W. Kierat and U. Sztaba, Distributions,Integral Transforms and Applications,Analytical Methods and Special Functions 7,Taylor & Francis, London, 2003, 148 pp., £45,ISBN 0-415-26958-X

This book is an approachable introduction tothe theory of distributions and integral trans-forms. The principal intention is to empha-sise the remarkable connections between dis-tribution theory, classical analysis and the the-ory of differential equations. The theory isdeveloped from its beginnings to the pointwhere fundamental and deep results areproved, such as the Schwartz kernel theoremand the Malgrange-Ehrenpreis theorem.Answers are given to natural questions arisingfrom the topics presented in the text. In caseswhere classical analysis is insufficient, practicalhints are given on using the theory of distrib-utions.

The book is divided into seven chapters. InChapter 1 the authors define distributions andintroduce some fundamental notions.Chapters 2 and 3 are devoted to local proper-ties of distributions, tensor products and con-volution products of distributions. In Chapter4 the authors give some applications to differ-ential equations. The next two chapters dealwith integral transforms, such as the Cauchy,Fourier and Hilbert transforms. In the finalchapter the author present some aspects of thetheory of orthogonal expansions of someclasses of distributions. An appendix containsbasic facts from functional analysis.

The book will be understandable to a stu-dent familiar with advanced calculus, and willbe of interest to mathematicians, physicistsand researchers concerned with distributiontheory and its many applications to theapplied sciences. (kn)

W. Kühnel, Differential Geometry.Curves–Surfaces–Manifolds, StudentMathematical Library 16, AmericanMathematical Society, Providence, 2002, 358pp., US$49, ISBN 0-8218-2656-5This book (a translation of Differential-geome-trie, Vieweg & Sohn Verlag, 1999) is a niceintroduction to classical differential geometryof curves and surfaces, containing also thetheory of curves and surfaces in Minkowskispace and basic results in the theory ofRiemannian manifolds, the Riemannian con-nections, examples of tensors, spaces of con-stant Riemannian curvature and Einsteinspaces. The text is illustrated with manyexamples, and several unsolved exercisesappear in each chapter. The whole exposi-tion is clear and the book can be recommend-ed to all students in the latter part of theirundergraduate curriculum. (lbo)

V. Lakshmikantham and S. Köksal, MonotoneFlows and Rapid Convergence for NonlinearPartial Differential Equations, Series inMathematical Analysis and Applications 7,Taylor & Francis, London, 2003, 318 pp., £55,ISBN 0-415-30528-4This book deals in a unified way with monoto-ne iterative techniques and generalised quasi-linearisations. Starting from a sub- (or super-) solution, this may create a sequence ofapproximate solutions converging pointwisemonotonically to the exact solution.Modification of this process to the non-monot-one case is also possible, leading to a moresophisticated iterative scheme. Sometimes,quadratic convergence can be achieved, whichhas possible numerical importance.

The first part of this book addresses theseissues in the classical framework, while the sec-ond part uses variational techniques relyingon the weak solution. In each case, second-order elliptic, parabolic and hyperbolic equa-tions or systems are scrutinised (possibly onunbounded domains), and impulsive parabol-

ic equations are addressed. The first partallows semilinear equations or systems in non-divergence form, while the second part treatsequations or systems (possibly quasilinear) inthe divergence form. Reading this bookrequires a certain basic knowledge of partialdifferential equations and a knowledge of thestandard notation in this subject is assumed, asthere is no list of symbols. A fresh compressedstyle of explanation arises from reducing thenumber of definitions and from rathersketched assertions whose more detailed con-tent can be read from the proofs. This bookwill be a useful reference for applied scientistsand mathematically oriented engineers.(trou)

S. Lang, Introduction to DifferentiableManifolds, 2nd edition, Universitext, Springer,New York, 2002, 250 pp., 59,95 euros, ISBN 0-387-95477-5This book has ten chapters with the followingtitles: Differential calculus, Manifolds, Vectorbundles, Vector fields and differential equa-tions, Operations on vector fields and differ-ential forms, The theorem of Frobenius,Metrics, Integration of differential forms,Stokes’ theorem, and Applications of Stokes’theorem. Among the topics not mentioned inthe titles are a short introduction to Liegroups, Darboux’ theorem, elements of pseu-do-Riemannian geometry, the Morse lemmafor non-degenerate critical points, Sard’s the-orem, de Rham cohomology, and the residuetheorem for functions of complex variables asa non-standard application of Stokes’ theo-rem. The author recommends his text to ‘thefirst year graduate level or advanced under-graduate level’; it is not intended for the firstencounter with this topic for an average stu-dent. The author offers a rich bibliography ofeasier texts for beginners or those who prefermore visualisation and less abstraction.

Loyal to the Bourbaki style, the author startswith a paragraph on categories. Thus, hisexplanation is very precise, with rich formal-ism and with maximum generality; for exam-ple, he always deals with differentiable mani-folds of class Cp for sufficiently large positiveintegers p, and not only with those of class C∞,as is usual in other texts. Naturally, he mustpay a price for this generality, even in suchbasic concepts as a tangent vector. He alsodoes not exclude a consideration of singulari-ties, for example when treating various ver-sions of Stokes’ Theorem, but he does makeone concession to the reader: he does notspeak about infinite dimensions! In summary,this is an ideal text for people who like a moregeneral and abstract approach to the topic.(ok)

C. Lauro, J. Antoch, V. E. Vinzi and G.Saporta (eds.), Multivariate Total QualityControl. Foundation and Recent Advances,Contributions to Statistics, Physica-Verlag(Springer), Heidelberg, 2002, 236 pp., ISBN 3-7908-1383-4This very interesting book represents a homo-geneous collection of major contributionsfrom three European schools (Naples, Parisand Prague), whose researchers have facednew problems in the field of multivariate sta-tistics for total quality. The discussion coversboth theoretical and practical computationalaspects of the problem, as well as the necessarymathematical background and several appli-cations.

The first part on off-line control, by J.Antoch, M. Hušková and D. Jarušková, dealswith tests on the stability of statistical models.

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The problem is formulated in terms of testinghypotheses, and this chapter is the mostimportant part of the book. The next fourcontributions are closely connected with theon-line process control: J. Antoch and D.Jarušková offer a critical overview of classicalmethods for detection of a change in asequence of observations; two contributions ofL. Jaupi and N. Niang to multivariate processcontrol refer to the theory of robustness but indifferent frameworks; and G. Scepi considerssituations where control charts for monitoringindividual quality characteristics may be notadequate for detecting changes in the overallquality of products. The next two contribu-tions are closely connected with problems ofcustomer satisfaction: V. Esposito Vinzi pro-poses different strategies for comparativeanalysis, based on a joint use of non-symmet-rical multidimensional data analysis and pro-crustean rotations, while G. Giordano focuseson multi-attribute preference data that referto preference judgements expressed withrespect to a set of stimuli described by any rel-evant attributes, in order to perform a con-joint analysis.

This book can be used in courses related toquality control and industrial statistics, and isalso a reference book for real-world practi-tioners in both engineering and business dur-ing the phases of planning and checking forquality. (tc)

J. M. Lee, Introduction to Smooth Manifolds,Graduate texts in Mathematics 218, Springer,New York, 2002, 628 pp., 84,95 euros, ISBN 0-387-95495-3This text provides an elementary introductionto smooth manifolds which can be understoodby junior undergraduates. It consists of twen-ty chapters: Smooth manifolds; Smooth maps;Tangent vectors; Vector fields; Vector bun-dles; The cotangent bundle; Submersions;Immersions and embeddings; Submanifolds;Lie group actions; Embedding and approxi-mation theorems; Tensors; Differential forms;Orientations; Integration on manifolds; deRham cohomology; the de Rham theorem;Integral curves and flows; Lie derivatives;Integral manifolds and foliations; Lie groupsand their Lie algebras. The book ends with asixty-page appendix called ‘Review of prereq-uisites’ which is a short introduction to gener-al topology, linear algebra and calculus, so thebook is self-contained. There are 157 illustra-tions, which bring much visualisation, and thevolume contains many examples and easyexercises, as well as almost 300 ‘problems’ thatare more demanding. The subject index con-tains more than 2700 items!

It is obvious that the ‘skeleton’ of basic factshas been opulently enhanced by the ‘flesh’ ofcomments and explanations, bringing answersto any potential reader’s questions before theyare asked. The pedagogic mastery, the long-life experience with teaching, and the deepattention to students’ demands make this booka real masterpiece that everyone should havein their library. The author’s dedication “Tomy students” could easily be understood as“To my students with love”. (ok)

T. Lewinski and J. J. Telega, Plates,Laminates and Shells: Asymptotic Analysis andHomogenization, Series on Advances inMathematics for Applied Sciences 52, WorldScientific, Singapore, 2000, 739 pp., US$149,ISBN 981-02-3206-3 For computations in mechanics, it is impor-tant to know which simpler structures providesuitable replacements for the fine microstruc-

ture of the material. This can be done bymethods of homogenisation. The authorspresent a thorough study of homogenisationand related methods, with an emphasis onapplications to plates, laminates and shells.

Chapter 1 summarises the mathematicalbackground, including quasiconvexity, Γ con-vergence, G convergence and H convergence.Chapter 2 studies elastic plates constructed bya periodic repeating of three-dimensionalcells in two independent directions. The for-mal asymptotic expansion of the equilibriumsolution is justified by Γ convergence methods,and the effect of bending and stretching ofplates is investigated in Kirchhoff’s model.The non-linear model governed by the VonKármán equation is studied, including bifur-cation phenomena. The Reissner-Henckymodel is suitable for moderately thick plates;Hoff’s approach can be applied to sandwichplates with soft core. In Chapter 3, the authorsstudy elastic plates with cracks, and computestiffness loss of cracked laminates. In Chapter4, elastic perfectly plastic material which issubject to a loading is considered. The corre-sponding functional has linear growth andthus its analysis leads to non-reflexive spaces,such as the space of functions with boundedvariation. Chapter 5 is devoted to elastic andperfectly plastic shells. In Chapter 6,homogenisation methods are applied to theoptimum design of plates and shells. Thisrequires further mathematical methods, suchas Y-transformation and its Fourier analysis,quasiconvexity or quasiaffinity of some con-crete expressions, and Young measures. Theproblems of optimal design studied here havethe feature that optimising sequences increas-es the fineness of partitioning. The minimumcompliance problem of thin elastic plates is arepresentative of issues of this chapter.

This book contains a great amount of mate-rial. As always in mathematical modelling, theproblems studied here are fairly complicated.Despite this, the level of mathematical accura-cy is very high. The authors present a repre-sentative selection of known results, includingsome of their extensive research, and expertsin the field will find a lot of information.However, the methods used here are of broad-er significance and thus may provide inspira-tion for readers interested in quite distantfields of applied mathematics. (jama)

A. L. Lewis, Option Valuation underStochastic Volatility with Mathematica Code,Finance Press, Newport Beach, 2000, 350 pp.,USD$97,50, ISBN 0-9676372-0-1In this book option pricing is studied in thecase when the price of a security and its volatil-ity both follow diffusion processes. The theo-ry and methods presented here provide solu-tions for many option valuation problems thatare not captured by the Black and Scholes the-ory, such as smile and term structure ofimplied volatility. A generalised Fouriertransform approach is used to solve PDEs thatdetermine the option price; this approachplays an important role throughout the book,and the author presents new and generalresults. Volatility series expansions are alsoused to solve the corresponding PDE, and thistechnique is applied to obtain explicit formu-las for the smile. The term structure ofimplied volatility is considered, and methodsfor computing asymptotic implied volatilityare presented. Various types of stochasticprocesses for volatility are treated, especially,GARCH diffusion models of stochastic volatil-ity, and diffusion limits of well-known discretetime GARCH models are considered whose

parameters are easily estimated. Many exam-ples are given that illustrate the theory andmany Mathematica routines for evaluationformulas and solving the PDE are included.This makes the book very interesting and use-ful for both financial academics and traders.(zp)

H. Li and F. van Oystaeyen, A Primer ofAlgebraic Geometry. Constructive Comput-ational Methods, Pure and Applied Mathe-matics 227, Marcel Dekker, New York, 2000,370 pp., US$165, ISBN 0-8247-0374-X This book can be used as a first course in alge-braic geometry for students and researcherswho are not primarily pure mathematicians.It is also useful for applications in computeralgebra, robotics and computational geometryand mathematical methods in technology.Only a basic knowledge of linear algebra, basicalgebraic structures and elementary topologyare needed as prerequisites.

Chapter 1 describes affine algebraic setsand presents the Nullstellensatz, together withits applications. Chapter 2 is devoted to astudy of polynomial and rational functions onalgebraic sets, and the structure of coordinaterings of algebraic sets. Chapter 3 introducesprojective algebraic sets in projective spacesand studies their structure. Relations betweenaffine and corresponding projective notionsare discussed, and there are a definition andcharacterisations of multiprojective spaces,Segre spaces and the Veronese variety.Chapter 4 introduces the Gröbner basis for anideal in K = k[x1, …, xn] as a combination of anordered structure of ideals in K and the divi-sion algorithm; some applications of Gröbnerbases to basic questions in algebraic geometryare given. The dimension of an algebraic setand its geometric and algebraic role in projec-tive geometry are explained in Chapter 5,while Chapter 6 presents basic facts from thetheory of quasi-varieties (local theory). Thefinal two chapters develop the theory of planealgebraic curves (including intersection theo-ry), the theory of divisors and the Riemann-Roch theorem, and the class of elliptic curvesis studied in detail. Some additional resultsfrom algebra are given in two appendices, andthe book contains many examples. I wish torecommend this well-written book to anyoneinterested in applied algebraic geometry.(jbu)

H. Okamoto and M. Shoji, The MathematicalTheory of Permanent Progressive Water-Waves, Advanced Series in NonlinearDynamics 20, World Scientific Publishing Co.,New Jersey, 2001, 229 pp., £29, ISBN 981-02-4449-5 and 981-02-4450-9This small book is an analytical and numericaloverview of solutions to equations represent-ing water-waves in a two-dimensional regionof finite or infinite depth, following theresearch interests and results of the authors.It contains a brief mathematical derivation ofvarious forms of the governing equations: thewaves under consideration move with constantspeed and do not change their profile duringthe motion. Different kind of waves, such aspure capillary (Crapper’s), pure gravity(Stokes’ highest wave), rotational (Gerstner’strochoidal wave) and solitary waves are treat-ed, and further capillary-gravity waves, inter-facial progressive waves and waves with nega-tive surface tension are considered. The exis-tence of solutions is formulated as a bifurca-tion problem. The authors consider variousbifurcation parameters (gravity, surface ten-sion), and a normal form analysis is used for

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describing the bifurcation modes. There is anaccount of results from numerical experi-ments.

The book is aimed at readers desiring a firstinsight to water-wave theory, but with previousexperience in bifurcation theory and numeri-cal analysis. Looking for more sophisticatedresults and verifications, they will find manyreferences here. (ef)

J.-P. Otal, The Hyperbolization Theorem forFibered 3-Manifolds, SMF/AMS Texts andMonographs 7, American MathematicalSociety, Providence, 2001, 126 pp., US$39,ISBN 0-8218-2153-9This book aims to present a full and systemat-ic proof of the Thurston hyperbolisation theo-rem in the special case of fibred 3-manifolds.The 3-dimensional manifolds studied here aremanifolds that are fibre bundles over the cir-cle with a compact surface S as a model fibre:such manifolds M are described by the isotopyclass of their monodromy – that is, by a dif-feomorphism of S. It is proved here that acomplete hyperbolic metric of finite volumecan be constructed on such manifolds: themain ingredients for the proof are geodesiclaminations, the double limit theorem and theSullivan theorem. The proof is based onThurston’s original (unpublished) paper witha new approach to the double limit theorem.This very nice book is the English translationof the French original, published by theSociété Mathématique de France. It can berecommended to all mathematicians interest-ed in the fascinating field of low-dimensionalgeometry. (jbu)

J.-P. Pier (ed.), Development of Mathematics1950-2000, Birkhäuser, Basel, 2000, 1372 pp.,DM 298, ISBN 3-7643-6280-4To write comments on the history of mathe-matics in the second half of the 20th century isa difficult task, due to the enormous growth inthe number of working mathematicians. Theaim of this book is to contribute to such a pro-ject with a collection of essays on evolution incertain fields.

A few contributions are devoted to the foun-dation of mathematics, set theory, logic andtopology and their applications in variousparts of mathematics (J.-Y. Girard, J.-P.Ressayre, B. Poizat, F. W. Lawvere and S.Sorin). Papers by E. Fouvry, M. Waldschmidt,J.-L. Nicolas and C. Ciliberto discuss topicsfrom number theory and its applications.Graphs and their applications in other fieldsare discussed by C. Berge. Riemannian geom-etry is the topic of a paper by M. Berger. Apaper by M.-F. Roy surveys results in real alge-braic geometry. Various versions of K-theoryare discussed by M. Karoubi. Two papers byV. Arnold are devoted to the local theory ofcritical points of functions and to dynamicalsystems. Relations between thermodynamicsand chaotic dynamical systems are studied byV. Baladi. B. Mandelbrot discusses questionsin fractal geometry. A review by J. Dieudonnéis devoted to mathematics during the last 50years in France, while one by V. M.Tikhomirov discusses Moscow mathematics inthis period. A paper by M. Smorodinsky treatsquestions in information theory, and I.Chalendar and J. Esterle discuss some ques-tions in functional analysis. Fourier analysis istreated by P. L. Butzer, J. R. Higgins and R. L.Sten, and wavelets by S. Jaffard. PDEs are dis-cussed by J. Sjöstrand and R. Temam(microlocal analysis, Navier-Stokes equations).The calculus of variation is the topic of a paperby F. H. Clarke. Several complex variable the-

ory and plurisubharmonic functions are dis-cussed by P. Dolbeault and C. Kiselman.Random processes, branching processes,Brownian motion, random walks, statistics andstochastic analysis are treated by P.-A. Meyer,M. Yor, J.-F. Le Gall, Y. Guivarc’h, K. D.Elworthy, G. R. Grimmett, L. Le Cam and B.Prum. The book concludes with interviewswith A. Douady, M. Gromov and F.Hirzebruch, and various bibliographic andother data. It is important to know our ownhistory, and this book contributes very well tothis aim. (vs)

C. C. Pugh, Real Mathematical Analysis,Undergraduate Texts in Mathematics,Springer, New York, 2002, 437 pp., EUR59,95, ISBN 0-387-95297-7This introduction to undergraduate realanalysis is based on a course taught manytimes by the author over the last thirty-fiveyears at the University of California, Berkeley.It starts with the definition of real numbers bycuts and continues with basic facts about met-ric and topological spaces. A clear expositionof differentiation and Riemann integration ofa function of real variable appears in Chapter3, while Chapter 4 presents basic facts aboutthe space of continuous functions.Multivariable calculus, including a generalStokes’ formula, is studied in Chapter 5 andthe book ends with Lebesgue integration the-ory. The exposition is informal and relaxed,with a number of pictures. The emphasis is onunderstanding the theory rather than on for-mal proofs. The text is accompanied by verymany exercises, and the students are stronglyencouraged to try them. (sh)

W. Rossmann, Lie Groups: An IntroductionThrough Linear Groups, Oxford GraduateTexts in Mathematics 5, Oxford UniversityPress, 2002, 265 pp., £35, ISBN 0-10-859683-9A general theory of simple Lie groups andtheir finite-dimensional representations isusually left for courses at graduate level. Topresent the basic ideas of the theory to under-graduates is much easier and transparent if itis done for Lie groups of matrices. This is theapproach chosen in this book. Basic facts ofthe Lie theory for linear groups (exponentialmap, Campbell-Baker-Hausdorff series, Liealgebras of linear groups, homomorphismsand coverings, roots and weights, topology ofclassical groups) are explained in the firstthree chapters. The next short chapter intro-duces basic properties of manifolds, Liegroups and their homogeneous spaces.Chapter 5 describes integration on lineargroups and their homogeneous spaces, andthe Weyl integration formula. The final chap-ter describes finite-dimensional representa-tions of classical groups (including Schur’slemma, the Peter-Weyl theorem, characters,Weyl character and Weyl dimension formulasand the Borel-Weil theorem). Each chaptercontains many useful exercises, very suitablefor problem sets.

The book is very well written, coveringessential facts needed for understanding thetheory, and is easy to use when preparingcourses on Lie groups. The chosen approachfollows Weyl’s original ideas and helps onedevelop a suitable intuitive background for thegeneral theory. Historical remarks scatteredthrough the book are interesting and useful.This is an excellent addition to the existing lit-erature and should be useful for teachers andstudents, and for mathematicians and mathe-matical physicists. (vs)

A. Shen and N. K. Vereschagin, Basic SetTheory, Student Mathematical Library 17,American Mathematical Society, Providence,2002, 116 pp., US$21, ISBN 0-8218-2731-6The contents of this book are based on lec-tures given in the authors’ undergraduatecourses at the Moscow State University. Itgives the reader a survey of ‘naïve’ set theory,set theory without an axiomatic approach.The text is full of interesting challenges to thereader – exercises with a wide range of diffi-culty.

Chapter 1 introduces the notions of a setand its cardinality, and present the Cantor-Bernstein theorem, Cantor Theorem, etc.The final two paragraphs give the set-theoret-ical definition of a function and operations oncardinal numbers, with emphasis on injec-tions, surjections and bijections and their rolein comparing the cardinalities of sets. Thestyle of problems to be solved is rather combi-natorial. Chapter 2 focuses on ordered setsand different types of orderings. Under trans-finite induction and recursive definition,Zorn’s lemma and Zermelo’s theorem appear,followed by an application to the existence ofthe Hamel basis. Ordinals and ordinal arith-metics are presented here, followed by anapplication to Borel sets. The book concludeswith a short glossary and a bibliographicalindex. The book should be understandable toundergraduate students, while some partsshould also be of interest to others. (lpo)

H. Sohr, The Navier-Stokes Equations: AnElementary Functional Analytic Approach,Birkhäuser Advanced Texts, Birkhäuser, Basel,2001, 367 pp., 104 euros, ISBN 3-7643-6545-5This book is devoted to the system of Navier-Stokes equations for a velocity vector field uwith pressure π. The first n equations of thesystem express the law of balance of momen-tum, while the final equation is the continuityequation for incompressible fluid with con-stant density and viscosity 1. As a special case,we get the system describing Newtonian flow.The author uses a functional-analyticapproach to study the existence of a weaksolution u, the existence of the associatedpressure π and additional regularity of u andits uniqueness. The method is based on refor-mulating these problems in terms of theStokes operator A, its fractional powers, andthe semigroup generated by A. Although thisapproach is well known, various applicationsto the system of Navier-Stokes equations aredispersed throughout the literature.

The author assumes a basic knowledge offunctional analysis tools in Hilbert andBanach spaces, but most of the importantproperties of Sobolev spaces, distributions,operators, etc., are collected in the first twochapters of the book. Properties of the Stokesoperator are studied in detail in Chapter III,where the stationary variant of the system isalso considered. The system is non-linear dueto the convective term, which appears fromthe ‘convective derivative of momentum’ andis responsible for most of the troubles withexistence, regularity and uniqueness of solu-tion. Thus, the author drops the convectiveterm first and investigates the linearised vari-ant of the system in Chapter IV. It is interest-ing that problems with regularity of pressureappear already. The full non-linear system isconsidered in the last chapter.

The reader will find a very interesting,clear, concise and self-contained approach tothe mathematical theory of the Navier-Stokes

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equations. The book will be of interest toapplied and pure mathematicians, as well as topostgraduate students working in the field offluid mechanics. (pka)

F. G. Timmesfeld, Abstract Root Subgroupsand Simple Groups of Lie-Type, Monographsin Mathematics 95, Birkhäuser, Basel, 2001,389 pp., DM 220, ISBN 3-7643-6532-3The main topic treated in this book is the the-ory of groups generated by abstract root sub-groups. Their description (starting with thespecial case of rank 1 groups) can be found inthe first two chapters. The main part of thebook is Chapter 3, describing the classificationof groups generated by a class of abstract rootsubgroups. Chapter 4 describes a reduction ofthe classification of finite groups generated bya class of root involutions to the classificationof groups generated by abstract root sub-groups. The final chapter contains someapplications, and in this part proofs are mere-ly sketched. The book will be of interest tomathematicians interested in finite group the-ory. (vs)

G. Wüstholz, A Panorama of Number Theoryor The Ìiew from Baker’s Garden, CambridgeUniversity Press, 2002, 356 pp., £55, ISBN 0-521-80799-9These conference proceedings commemorat-ing Alan Baker’s 60th birthday have a two-foldpurpose: to document the continuing influ-ence of Baker’s seminal work on transcen-dence theory and diophantine approxima-tions from the 1960s and 1970s, and to takestock of the most recent developments in these(and related) areas. The contributors includeE. Bombieri, J.-H. Evertse, G. Faltings, D.Goldfeld, G. Margulis, D. W. Masser, Yu. V.Nesterenko, P. Sarnak and G. Wüstholz.(jnek)

A. Zygmund, Trigonometric Series, Vols. I, II,3rd edition, Cambridge University Press, 2003,364 pp., £39.95, ISBN 0-521-89053-5The third edition of this well-known and pop-ular book appears with a foreword by RobertFefferman as a paperback in the CambridgeMathematical Library. Both volumes are

included here. Seventeen chapters contain allthe classical related results: while Volume Ideals with basic results on Fourier series (withthe exception of Carleson’s and Hunt’s resultson the convergence almost everywhere ofFourier series of functions from Lp, p > 1),Volume II is devoted to such topics as trigono-metric interpolation, differentiation of series,generalised derivatives and the Fourier inte-gral. Instead of listing the missing results,which would probably be simpler than makingan account of those which are included, I willborrow the last few lines from Fefferman’sforeword: ‘In fact, what is surprising about thecurrent volume is not what is missing. What issurprising is that a single person could writesuch an extraordinary comprehensive andmasterful presentation of such a vast field.This volume is a text of historic proportion,having influenced several generations of someof the greatest analysts of the twentieth. Itholds every promise to do the same in thetwenty-first.’ The text can be recommendedfor libraries, students and anyone wanting agood reference book on the field. (jive)

RECENT BOOKS

EMS June 200334

InterInterdisciplinary rdisciplinary researesearch opportunity for mathematiciansch opportunity for mathematiciansunder the 6th Funder the 6th Framework Pramework Prrogrammeogramme

The European Commission recently published the first call for proposals for research on ‘new and emerging science and technol-ogy’ (NEST), under the 6th EU Research Framework Programme (2003-2006). The 215 million euro NEST action opens the pos-sibility for the Commission to finance research at the frontiers of knowledge in areas suggested by researchers themselves, with anemphasis on multi-disciplinary research. Research linking advanced mathematics with other disciplines is eligible for fundingunder NEST

The present call for proposals involves an indicative budget of 28 million euro, but the level of funding for NEST will increasesubstantially over the four years of the programme. It concerns two action lines:• ADVENTURE projects will feature novel and interdisciplinary research areas, ambitious and tangible objectives, with a potential

for high impact, but also imply a high risk of failure. • INSIGHT projects will investigate discoveries that might hamper our health and quality of life, such as for instance a new

pathogen or environmental contaminant, a potential failure in financial and other markets, new forms of crime etc.

Further information can be obtained from the European Commission at www. cordis.lu/nestNEST mailbox: rtd-nest@cec.eu.int

ŒUVRES COMPLÉTES DE NIELS HENRIK ABEL, 1881 ed.

As early as 1839, a collection of Niels Henrik Abel’s works was published under the editorship of Abel’s teacher, Bernt Holmboe. Thiscollection however, was incomplete. For example, the Paris memoir with the famous addition theorem was not included because themanuscript had disappeared. Also, some minor papers by Abel had been omitted. The Paris Memoir was found and published in 1841, and when the Holmboe edition of Abel’s œuvres went out of print, it was feltthat a mere reprint would be unsatisfactory. In 1872 the Norwegian Academy of Sciences and Letters commissioned Ludvig Sylow andSophus Lie to prepare a new and more complete edition. Sylow and Lie spent eight years on this task, searching for previously unpub-lished manuscripts and notebooks in French and German archives as well as those in Norway. According to Lie, most of this workwas done by Sylow. The two-volume edition was published in 1881. Volume I contains papers which had been published by Abel, most of them in Journal für die reine und angewandte Mathematik (Crelle’sJournal). Volume II contains papers found after Abel’s death and passages on the subject of mathematics that had been excerptedfrom his letters. The 1881 edition was composed of a print-run of 2000. The remaining 200 copies of the 1881 edition are in the possession of theNorwegian Mathematical Society. To commemorate the 200th anniversary of Abel’s birth, these have now been leather-bound in onevolume, and numbered.

It is now possible to order a copy of the limited edition. Price: NOK 5.000

Orders must be sent by mail to:Department of MathematicsUniversity of OsloP.O.Box 1053, BlindernNO-0316 OSLONORWAY

http://www.math.uio.no/abel/oeuvres.html

EMS June 200335