contents editorial team european mathematical …emis.de/newsletter/newsletter53.pdf · 2004. 10....

NEWSLETTER No. 53September 2004

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

Editorial - Manuel Castellet .................................................................................. 3

EMS meetings at Uppsala ........................................................................................ 4

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

Report on the 4th ECM - Ulf Persson ................................................................... 7

Visions of Young Mathematicians - Jan Larsson and Mikael Johansson.......... 12

Reflections on ICME-10 - Vagn Lundsgaard Hansen ...................................... 14

25 years proof of the Kneser conjecture - Mark de Longueville ......................... 16

100th Birthday of Henri Cartan ............................................................................ 20

Interview with Nuno Crato .................................................................................... 22

Interview with Michael Atiyah and Isadore Singer ............................................. 24

Abel Prize Celebration 2004 .................................................................................. 31

Greek Mathematical Society ................................................................................. 34

French-Polish cooperation in Approximation Theory ........................................ 36

ERCOM: CRM Barcelona ..................................................................................... 37

Forthcoming Conferences ..................................................................................... 40

Recent Books .......................................................................................................... 42

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Published by European Mathematical SocietyISSN 1027 - 488X

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

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EDITOR-IN-CHIEFMARTIN RAUSSENDepartment of Mathematical Sciences, Aalborg UniversityFredrik Bajers Vej 7GDK-9220 Aalborg, Denmarke-mail: [email protected] EDITORSVASILE BERINDEDepartment of Mathematics,University of Baia Mare, Romania e-mail: [email protected] KRZYSZTOF CIESIELSKIMathematics InstituteJagiellonian UniversityReymonta 4, 30-059 Kraków, Polande-mail: [email protected] MARKVORSENDepartment of Mathematics, TechnicalUniversity of Denmark, Building 303DK-2800 Kgs. Lyngby, Denmarke-mail: [email protected] WILSONDepartment of Pure MathematicsThe Open UniversityMilton Keynes MK7 6AA, UKe-mail: [email protected] EDITOR:KELLY NEILSchool of MathematicsUniversity of SouthamptonHighfield SO17 1BJ, UKe-mail: [email protected] EDITORSINTERVIEWSSteen Markvorsen [address as above]SOCIETIESKrzysztof Ciesielski [address as above]EDUCATIONTony GardinerUniversity of BirminghamBirmingham B15 2TT, UKe-mail: [email protected] PROBLEMSPaul JaintaWerkvolkstr. 10D-91126 Schwabach, Germanye-mail: [email protected] ANNIVERSARIESJune Barrow-Green and Jeremy GrayOpen University [address as above]e-mail: [email protected] [email protected] Berinde [address as above]RECENT BOOKSIvan Netuka and Vladimír SouèekMathematical InstituteCharles University, Sokolovská 8318600 Prague, Czech Republice-mail: [email protected] [email protected] OFFICERVivette GiraultLaboratoire d’Analyse NumériqueBoite Courrier 187, Université Pierre et Marie Curie, 4 Place Jussieu75252 Paris Cedex 05, Francee-mail: [email protected]

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CONTENTS

EMS September 2004

EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

EXECUTIVE COMMITTEEPRESIDENTProf. Sir JOHN KINGMAN (2003-06)Isaac Newton Institute20 Clarkson RoadCambridge CB3 0EH, UKe-mail: [email protected]

VICE-PRESIDENTSProf. LUC LEMAIRE (2003-06)Department of Mathematics Université Libre de BruxellesC.P. 218 – Campus PlaineBld du TriompheB-1050 Bruxelles, Belgiume-mail: [email protected]. BODIL BRANNER (2001–04)Department of MathematicsTechnical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: [email protected]

SECRETARY Prof. HELGE HOLDEN (2003-06)Department of Mathematical SciencesNorwegian University of Science and TechnologyAlfred Getz vei 1NO-7491 Trondheim, Norwaye-mail: [email protected]

TREASURERProf. OLLI MARTIO (2003-06)Department of MathematicsP.O. Box 4, FIN-00014 University of Helsinki, Finlande-mail: [email protected]

ORDINARY MEMBERSProf. VICTOR BUCHSTABER (2001–04)Steklov Mathematical InstituteRussian Academy of SciencesGubkina St. 8, Moscow 117966, Russiae-mail: [email protected] Prof. DOINA CIORANESCU (2003–06)Laboratoire d’Analyse NumériqueUniversité Paris VI4 Place Jussieu75252 Paris Cedex 05, Francee-mail: [email protected]. PAVEL EXNER (2003-06)Department of Theoretical Physics, NPIAcademy of Sciences25068 Rez – Prague, Czech Republice-mail: [email protected]. MARTA SANZ-SOLÉ (2001-04)Facultat de MatematiquesUniversitat de BarcelonaGran Via 585E-08007 Barcelona, Spaine-mail: [email protected]. MINA TEICHER (2001–04)Department of Mathematics and Computer ScienceBar-Ilan UniversityRamat-Gan 52900, Israele-mail: [email protected]

EMS SECRETARIATMs. T. MÄKELÄINENDepartment of Mathematics and StatisticsP.O. Box 68 (Gustav Hällströmintie 2B)FI-00014 University of HelsinkiFinlandtel: (+358)-9-1915-1426fax: (+358)-9-1915-1400telex: 124690e-mail: [email protected] Web site: http://www.emis.de

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EMS NEWS

EMS September 2004

200415 NovemberDeadline for submission of material for the December issue of the EMS NewsletterContact: Martin Raussen, e-mail: [email protected]

12-19 DecemberEMS Summer School and Séminaire Européen de Statistique at Munich (Germany)The Statistics of Spatio-Temporal SystemsWeb site: http://www.stat.uni-muenchen.de/semstat2004/

20055-13 FebruaryEMS Summer School at Eilat (Israel)Applications of braid groups and braid monodromy

16-17 AprilEMS Executive Committee meeting at Capri (Italy)

25 June – 2 JulyEMS Summer School at Pontignano (Italy)Subdivision schemes in geometric modelling, theory and applications

17-23 JulyEMS Summer School and European young statisticians’ training camp at Oslo(Norway)

13-23 SeptemberEMS Summer School at Barcelona (Catalunya, Spain)Recent trends of Combinatorics in the mathematical context

16-18 SeptemberEMS-SCM Joint Mathematical Weekend at Barcelona (Catalunya, Spain)

25 September – 2 OctoberEMS Summer School and Séminaire Européen de Statistique at Warwick (UK)Statistics in Genetics and Molecular Biology

12-16 DecemberEMS-SIAM-UMALCA joint meeting in applied mathematicsVenue: the CMM (Centre for Mathematical Modelling), Santiago de Chile

200622-30 AugustInternational Congress of Mathematicians in Madrid (Spain)Web site: www.icm2006.org

200716-20 JulyICIAM 2007, Zurich (Switzerland)

2008July5th European Mathematical Congress, Amsterdam (The Netherlands)

EMS AgendaEMS Committee

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

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

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

ERCOM is a committee of the EuropeanMathematical Society created in 1996 con-sisting of Scientific Directors of EuropeanResearch Centres in the MathematicalSciences, or their chosen representatives.Only centres in which the number of visitingstaff substantially exceeds the number of per-manent and long-term staff and that broadlycover Mathematical Sciences are eligible forrepresentation in ERCOM. The eligibility ofcentres is decided by the EMS ExecutiveCommittee.

The aims of ERCOM are to contribute tothe unity of Mathematics, from fundamentalto applications, with the purpose of constitut-ing a forum for communication andexchange of information and fostering col-laboration and co-ordination between thecentres themselves and the EMS, to fosteradvanced research training on a Europeanlevel, to advise the Executive Committee ofthe EMS on matters relating to activities ofthe centres, to contribute to the visibility ofthe EMS and to cultivate contacts with simi-lar research centres within and outsideEurope.

The 24 centres at present represented inERCOM have different working and organi-sational structures, they are also different insize and scope, but they can be grouped inthree different classes, as centres mainly formeetings, centres without any permanentresearchers, only visitors and post-docs, andcentres with a small permanent staff and alarge number of visitors. They broadly coverthe European Area, from Russia and Swedento Portugal and Israel. A Chairman and aVice-chairman, at present myself, as Directorof the Centre de Recerca Matemàtica, andKjell-Ove Widman, Director of the InstitutMittag-Leffler, co-ordinate the annualERCOM meeting and the other initiativesproposed by the ERCOM members. Sir JohnKingman, Director of the Isaac NewtonInstitute, acts as link between ERCOM andthe EMS Executive Committee.

The Editor-in-Chief of the EMSNewsletter, Martin Raussen, recently offeredto the ERCOM institutions the opportunity topresent themselves in order to give the read-ers an idea of how they might use the possi-bilities offered by the centres. EURAN-DOM, Eindhoven, presented itself in theJune issue of the Newsletter and the Centrede Recerca Matemàtica (CRM), Barcelona,is doing so in the present issue.

As Peter Hilton already wrote twenty yearsago “A research institute is at least two things

at the same time: it is a building and it is anorganisation of people working together anddedicated to the pursuit and support ofresearch. But at its best it is more. It starts lifeas an idea in the mind of one or more per-sons, of insight and imagination, and thenlives and grows by spreading the spirit,imbued by their founders, through the heartsand minds of all those benefiting from itspresence and contributing to its future”.

In mathematical research, exchange ofideas plays a central role and needs a deephuman contact. A contact which is the truemathematical laboratory, and, as Friedrichvon Siemens said at the end of the nineteenthcentury, “Laboratories are the fundamentalbasis of knowledge and power”. The highdegree of abstraction of mathematics and thecompact way it is presented need direct per-sonal communication, since mathematics isan attempt to develop tools that can be usedto achieve a better understanding of theworld’s measurable aspects and to identifyprocesses that appear in very different natur-al situations but that are essentially analo-gous.

But mathematics, apart from this, is a trulyinternational science, perhaps the most inter-national of sciences, since, compared withother disciplines, it is based less on the use ofinstruments and more on a strong humancontact. This is where the research institutesplay a crucial role, allowing not only theexchange of ideas between specialists in thesame field, but also profound and sometimessurprising links between different lines ofresearch.

In order to foster collaboration and co-ordination between the European researchinstitutes and to provide communication andexchange of information, ERCOM meetsyearly, usually by March, in one of the mem-ber institutes. Since the 1999 meeting inCambridge the Administrators have beeninvited to discuss separately and jointly withthe Directors matters of their competence.

Several topics are considered for discus-sion in the meetings: The European ResearchArea and the presence of Mathematics in theFramework Programmes of the EuropeanUnion, the annual call for post-doctoralgrants by the European Post-doctoralInstitute for the Mathematical Sciences(EPDI), the situation of Mathematics and themathematicians in Eastern Europe, theINTAS programme, the Digital MathLibrary, issues arisen from the EMSExecutive Committee, etc.

In particular, ERCOM has established anexchange programme for short-term visits ofpost-doctoral fellows from one ERCOMcentre to another, flexible and capable ofadapting to different centres and situations,the main goal being to stimulate theexchange and the increase of knowledge andto create synergies enriching research.

The Administrators have prepared a reporton the percentage of women doing researchin the ERCOM centres. There are about 14%of female researchers conducting long termvisits and an average of 16% femaleresearchers conducting short term visits inthe ERCOM centres, while, on the otherhand, the administrative staff is mostlyfemale and the directorate is mostly male.Last March in the Aarhus meeting the fol-lowing resolution was adopted: “ERCOMencourages its members to take actions tofacilitate the presence of women in their sci-entific activities, and to collect data regardingthe number of applications from femaleresearchers received and approved”.

ERCOM, as a committee of the EuropeanMathematical Society, wishes and has thecapacity to act as one of the EMS means ofscientific outreach, fostering the presence ofmathematicians in the new emerging multi-disciplinary research domains. With this aimwe are preparing a proposal to be submittedto the European Commission as a NESTSupport Action through a Co-ordinationAction instrument, with the main objectiveof designing curricula and finding ways totrain researchers into emerging specialities ofMathematics, linked especially with SystemBiology, Neuroscience, Risk Assessmentand Data Security, and identifying futureresearch opportunities on the interfacebetween Mathematics and suitable areas ofMedicine, Industry and Social Sciences.

A home page (http://www.ercom.org),from which you can reach that of eachERCOM centre, provides information on theERCOM initiatives and, in particular onopen positions offered by the centres andscheduled conferences and training courses.

EDITORIAL

EMS September 2004 3

EditorialEditorialManuel Castellet (Barcelona)

Chairman of ERCOM

The Council of the EuropeanMathematical Society (its “parliament”)met in the weekend before the EuropeanCongress in Stockholm on the premisesof Uppsala University, about 60 kilome-tres north of the Swedish capital. Therewere 65 delegates representing 43Mathematical Societies from all overEurope, 14 delegates representing indi-vidual members, and institutions wererepresented by 9 delegates. The EMSexecutive committee and several inviteesmade up the rest of the assembly.

Membership issues, budget and elec-tionsCouncil had its share of routine business,but finished with several lively discus-sions. The routine business was, howev-er, important, starting with the approvalof the membership applications from theMathematical Societies of Malta, Cyprusand Romania and the StatisticalSocieties of Belgium, Sweden, Norwayand Spain.

The President underlined two impor-tant facets of the Society’s many activi-ties: its lobbying for mathematics at aEuropean level, in which Luc Lemaireplayed an invaluable role; and the suc-cessful establishment of its own publish-ing house, for which Rolf Jeltsch hadbeen the driving force. All activitiesdepended on a small group of dedicatedpeople, in particular he warmly thankedSecretary Tuulikki Mäkeläinen, but theSociety needed more volunteers as itswork expanded. The President com-mended the willingness of mathemati-cians to serve on the recently createdScientific Advisory Panel. Councilagreed to send congratulations to HenriCartan on the occasion of his 100thbirthday.

The financial statements, presented byOlli Martio, were accepted. The financeswere in good health, but a slight rise inthe corporate membership fee was need-ed, which Council approved.

Pavel Exner was elected as Vice-President; Victor Buchstaber, Olga Gil-Medrano, Carlo Sbordone, and KlausSchmidt as ordinary members of the

Committee. The President thanked theoutgoing members of the Committee:Bodil Branner, Marta Sanz-Solé, andMina Teicher.

Reports and DebatesMartin Raussen, editor of the Newsletterreported on a smooth handover fromRobin Wilson. Much of the Newsletterwould be made available on EMIS, andit was hoped that the Book Reviews andConferences could be made available ina searchable form. Council thankedRobin Wilson and Martin Raussen fortheir work for the Newsletter.

There had been some excellent workon behalf of the applied mathematicscommunity, including the AMAM con-ference in Nice and the focus on applica-tions of mathematics in the forthcoming4ecm, but the applied mathematics com-mittee had been unable to decidewhether to recommend a separate seriesof JEMS for applied mathematics andthere had been a loss of momentum in itsactivities.

Mina Teicher was the new chair of theEducation Committee. She would do

some work on the Pisa report and workclosely with the education committees ofnational societies.

Vagn Lundsgaard Hansen talked aboutthe activities of the Committee forRaising Public Awareness and the neces-sity for mathematicians to engage withpoliticians, journalists and the generalpublic. This stimulated several sugges-tions and examples from delegates, suchas the forthcoming public activities inCopenhagen during the ICME meeting,links on EMIS and a website for teachers(like that of the DMV).

There was a report from the SpecialEvents Committee and from the EasternEurope committee, whose principalfunction remains the allocating of travelgrants.

Luc Lemaire reported on the successof the General Meetings Committee inattracting a grant of half a million eurosfor EMS Summer Schools. A new roundof applications was due. TheMathematical Weekends had startedwith a successful meeting in Lisbon in2003, with another due in Prague thisSeptember and a further one planned for

EMS NEWS

EMS September 20044

EMS Council MeetingEMS Council MeetingUppsala, Sweden, 26th-27th June 2004

David Salinger, EMS Publicity Officer

Host Sten Kaijser, Uppsala Univ., and EMS-President Sir John Kingman

Barcelona in 2005. On the other hand,there had been no suggestions receivedfor EMS lecturers.

A short report on the state of theBologna process was followed by a stim-ulating, and occasionally agonised, dis-cussion. The President summed up bysaying that the Society should be willingto back the mathematical community inparticular countries rather than formulat-ing a general policy.

Thomas Hintermann, managing direc-tor of the EMS Publishing House, spokeof its progress. From this year, EMSphwas publishing JEMS, Interfaces andFree Boundaries, Oberwolfach Reportsand, from 2005, Commentarii Helveticaeand Elemente der Mathematik wouldjoin the list. Book series had also start-ed. A delegate suggested reduced pricesfor developing countries.

The Committee on DevelopingCountries continued to be very active.Herbert Fleischner said the books projectwas continuing, and the Committee wastrying to increase the geographicalspread of its work. It had a working fundwith contributions from EMS andZentralblatt-MATH. John Ball spoke ofthe work the IMU was doing.

Helge Holden talked about the DigitalMathematical Library project. This wasbeing pursued at the national level; theEMS might have a co-ordinating role,but it could not fill it without additionalvolunteers. John Ball spoke of theimportance with which the IMU regardsthis project, which was moving well.

Council unanimously agreed to holdthe fifth European Congress ofMathematics in Amsterdam in July

2008.The President spoke of the importance

of having a viable Zentralblatt-MATH,as a counterweight to MathematicalReviews-MathSciNet. There were cur-rently some difficulties, in that the finan-cial base (mainly German and French)needed to be broadened and that the pro-ject rested heavily on the shoulders oftoo few people. There was a need for ascientific advisory board, not to manage,but to provide scientific input. An EUfunding application (two had failed)should be a secondary activity to support

a strategy, but should not be allowed todistort or dominate. Laurent Guillopéspoke of the French government’sreview of Zentralblatt-MATH.

Reporting on the work of theCommittee on Women and Mathematics,Emilia Mezzetti spoke of the limitedsuccess of the questionnaire, so that itwas not possible to base action on it. Butthe Committee was collecting data andthis did show some improvement in theposition of women, particularly at start-ing levels. The Committee was alsoinvolved with mentoring schemes.

Council concluded with a discussionon the position of mathematics withinEuropean research programmes. Thethemed networks were not very good formathematics. In the Marie Curie andother schemes, the success rate, thoughlow, was fixed to be same in each sub-ject. A delegate spoke passionatelyabout the time-consuming and demean-ing bureaucratic hurdles encountered ifan application was successful. Though areport had come out in favour of aEuropean Research Council with signifi-cant funding, the political will did notappear to be there; nor did this appear tobe a high priority for the accession coun-tries.

In his closing remarks, the Presidentappealed to delegates to bore peopleabout the importance of mathematics,particularly the development of abstractmathematics as ‘tomorrow’s tools’.

EMS NEWS


Council agreed that the search for a new EMS-president shouldbegin now. The terms of office is four years starting with January2007. Members of the Society are invited to send suggestions toPresident Sir John Kingman at [email protected]

Delegates and members of the EC during a break

The Newsletter's Editorial Board held a meeting at Uppsala, too. From left to right: MartinRaussen, Steen Markvorsen, Krzysztof Ciesielski, Robin Wilson.

Following the success of its first E.U.application (which allows the funding ofeight Summer Schools or Conferences in2004-2005, see the EMS agenda page 2 orhttp://www.emis.de/etc/ems-summerschools.html), the European Mathematical Societyis now launching a new call for proposalsfor such Schools and Conferences for 2006,2007 and 2008. The deadline for this call isJanuary 12, 2005, by e-mail to theaddress: llemaire@ ulb.ac.be.

The deadline will allow the EMS to pre-sent a coherent proposal of activities forEU funding, thereby allowing organisers ofsingle meetings to be part of a series ofevents. EMS direct support being limited,the result of this application will make amajor difference to the funding for themeetings selected by EMS. There will besimilar calls every two or three years in thefuture.

This call for proposals concerns allSummer schools or Conferences that anygroup of mathematicians – pure or applied- would like to run in 2006, 07 or 08 in theEU or associated states.

The EU guidelines for these events mustbe followed strictly in order to have a rea-sonable hope for funding (the success ratefor EU-applications being extremely low).

Thus, there must be a very strong com-ponent of training of young researchers (inthe first 10 years of their career) by meansof integrated courses and lectures at anadvanced level. These can be supplement-ed by conference type research lectures, butthe training component is needed.

Each course must aim at an internationalaudience (no more than 30% of participantsshould come from a single state; otherwisethe event in not eligible for funding).

The EMS will make a selection amongstthe proposals received, taking into consid-eration the criteria applied by the EU, i.e.:- Scientific quality of the project (high

quality, cutting edge subject, speakersof high calibre…)

- Quality of the research training activity(training provided to the youngresearchers, level of interactionbetween the main speakers and theyoung ones…)

- Quality of the host (quality of infra-structure, experience in running confer-

ences, contact making opportunities…)- Management and feasibility (organisa-

tion and management, plans for public-ity, dissemination of the results by pub-lications, web sites, any other means…)

- Relevance to the objectives of the EUaction and added value to the EU (rele-vance of the themes in relation withEuropean interests and achievements,measures taken in favour of gender bal-ance, use for integration of less devel-oped regions, plans to favour publicunderstanding of science…).

Concerning the last two criteria, part of thevalue comes from the EMS organisation,and this will be added in the final applica-tion. However, any little point will count,and each organiser can help by stressingqualities.

Proposers are asked to present a projectas detailed as possible, with (provisional!)lists of speakers, subjects, and addressingall questions above. They should alsoinclude an estimate of the number of eligi-ble participants for each of the three cate-gories (EU or associated states and lessthan 4 years research experience; less than10 years; no time limit but a researcherfrom EU or associated states living outsidethat zone). For these, no financial estimatesshould be given. An estimate of “organisa-tional” expenses is also needed, including

all expenses of key speakers which are noneligible.

The proposers should be aware of therules of funding by the EU: the travel andliving expenses of all eligible researcherscan be covered, up to a rather generousmaximum determined by the EU. If a fee ischarged to all participants, the EU willreimburse the fees of eligible ones but thensubtract this amount from the expensesbelow. All other expenses, includingexpenses of non-eligible key speakers, willbe covered at a proportion equal to the ratioeligible participants / all participants.The EU will pay a maximum of 80 % inadvance, the last 20 % only when allreports are in, and possibly a year later.

Thus a statement from an organisation(university, conference centre…) mustaccompany proposals, agreeing that it willcover the complement of organisationalexpenses, and also be ready to advance 20% of the funding until payment from theEU has arrived.

To give an idea, for the eight schools inthe present project, we had a total of 140participants of category 1, 155 of category2, 20 of category 3. Their funding amountsto 366 000 euros, the organisationalexpenses of each event varies between12000 and 17 000 euros, of which the EUcovers the prescribed percentage.

Note that by a new rule, EU will not fundseries of events in a single specific subject.In case that several proposals submit a par-ticular theme, the EMS will have to make achoice or have to determine if a mergerbetween these proposals makes sense.

Luc LEMAIREPhone: 00-32-2-6505837

e-mail: [email protected]

EMS NEWS

EMS September 20046

CALLCALL FOR PROPOSALSFOR PROPOSALSEMS SUMMER SCHOOLS IN FUNDAMENTAL AND

INTERDISCIPLINARY MATHEMATICS

EURYI Call for ProposalsEuropean Young Investigators Awards

The European Heads of Research Councils (EUROHORCS) in collaborationwith the European Science Foundation (ESF) have created the European YoungInvestigator (EURYI) Awards, launched for the first time in 2003. Awards willlast for a 5 year period with a total value normally no less than 750 kEuro. InJuly 2004, 25 awards were granted after the first call.

Young researchers from anywhere in the world with between 2 and 10 yearspostdoctoral experience – taking into account career breaks – are eligible. Theselection criteria for EURYI Awards are the research quality and potential of theapplicant and the originality as well as the groundbreaking character of theresearch proposal and its feasibility.

Applications must be submitted through a national EUROHORCS organisa-tion no later than November 30, 2004. They will then be assessed first at thenational level and later at the European level. Awards are expected to beannounced at the end of July 2005. Detailed information can be obtained fromwww.esf.org/euryi .

How it came aboutIn spite of a difficult start, the fourthEuropean Congress in Mathematics (ECM)took place after all. As its President AriLaptev revealed in an earlier issue of thisnewsletter, its eventual location inStockholm was not initially planned butwas the result of a crisis perceived in thesummer of 2000. Despite what many out-siders may have assumed, finding sufficientfunding from Swedish sources was not astraightforward matter (as the first bidmade painfully clear). It was only due to thedaring and ‘savoire faire’ of Ari Laptev andhis colleague at Kungliga TekniskaHögskolan (KTH), Anders Lindquist, thatsufficiently promising leads were extractedfrom some of the major Swedish fundinginstitutions, thus enabling the SwedishMathematical Society to send a small dele-gation to the executive meeting in Londonlater that fall for negotiations. However, afinal commitment to hold the congress wasnot made until just a week before Christmas- after a rather tense encounter with the thenPresident of the European MathematicalSociety (EMS), Rolf Jeltsch. The meetingonly came to a desired conclusion after afinal short interview with the President ofthe KTH, Anders Flodström, in which thelatter agreed to the necessary financialunderwriting. The upshot was that the plan-ning and organization of the congress wasshortened by at least a year from what hasbeen customary for this kind of event.

Why big congresses? Was it worth it? This is obviously not the place to delvedeeper into that philosophical question, yeta few reflections may not be amiss. In thegood old days, which meant up into thefifties, conferences were few and small, andin particular the International Congresses ofMathematicians (ICM) were rather selectand as a consequence not bigger thanallowing group portraits to be taken. Butwhen the ICM was held in Stockholm backin 1962, it was the largest scientific con-gress that had ever been held in Sweden.Since then the ICM have turned from ratherexclusive meetings to large affairs attendedby the ‘mathematical masses’. This fact,

together with the recent acceleration in thenumber of specialized conferences, oftenreferred to as workshops to emphasize theirfocused and businesslike character, hassown doubts in the minds of many as to thescientific relevance of such gargantuanmeetings, which, cynics protest, are morethe occasion for tourism than serious scien-tific interchange. In such a perspective, theestablishment of a European Congress mayseem redundant. One may point out thoughthat the situation of mathematics is notunique, but is shared by most academic dis-ciplines, and that it is very important,among other things, to counter the frag-mentation of a discipline with opportunitiesfor the contemplation of it as a whole. Forthis to be fully successful there is both aneed for the lectures to be directed to a gen-eral mathematical audience and for thataudience to seek out lectures not primarilywithin their own speciality. Popular sur-veys do not rank highly among the priori-ties of research mathematicians. They arenotoriously difficult to do and the rewardsare marginal. A scientific committee selectstheir choices primarily on the basis of sci-entific excellence and topicality of the sub-ject matter, not on expository skills; andnotwithstanding the desirability of the latterthis is basically a sound principle, tamper-ing with which would court disaster andseriously undercut the legitimacy of thewhole enterprise. Thus selection as a speak-er is seen primarily as recognition of scien-tific merit and not as an opportunity, as wellas an obligation, to communicate effective-ly.

What is needed is a change of culture,something that cannot be decreed but has toevolve slowly. On what constitutes a goodlecture, one can of course argue, there obvi-ously being no specific rules on whicheveryone can agree. And besides, in allkinds of human interaction rules are simplythere to be broken. It suffices to point outthat a scientific lecture, and especially amathematical one, is not necessarily mademore accessible by the simple process of‘watering down’ (i.e. removing technicali-ties and making unwarranted simplifica-tions). What is required of a lecture is the

conveying of an idea (usually one isenough) and some motivation (whichshould not be confused with justificationoften of the type: ‘this has applications tophysics’) and placing the subject matter ina wider mathematical context. Apart fromthat, a good lecture can contain highly tech-nical material, and there is no reason whythe audience should understand most of itas long as they have gotten something totake home with them. It would be very dan-gerous if mathematicians were to abandontheir tradition of honesty and aim for mere-ly the ‘flashy’. To reflect on why your workis interesting and to try to truly motivate it,should not be seen as a concession to igno-rance, but rather as an additional source ofinspiration for your own research.

New featuresObviously this is not the place to com-

ment upon how successfully the variousspeakers performed their task of communi-cating to a general mathematical audience.Fortunately, although few lectures can beexpected to please everyone, the lecture israre indeed that does not bring rewards to atleast a token number in the audience, andone may argue that this is indeed all that isneeded to make it worthwhile. However,there was a feeling among the organizersthat something extra was needed to justifyyet another general conference in mathe-matics, as well as serving as a rejuvenationof the concept. Two new features wereadded. One was to invite scientists in natur-al science, not only in chemistry andphysics, but also mathematical biologists,to give lectures. The other was to latch onto the pre-existing structure of theEuropean Networks and thus give to theECM a natural European anchoring. Thefirst may be seen as somewhat controver-sial, wedding mathematics and its ultimate

EMS NEWS


SStockholm 2004tockholm 2004The FourThe Fourth Eurth European Congropean Congress ofess of

MathematicsMathematicsUlf Persson (Göteborg, Sweden)

justification too tightly to applications. Yet,whether we like it or not, practical mathe-matical applications are what brings inmaterial resources, and for public relationsthe importance of the willingness of mathe-maticians to interact externally, and in theprocess maybe also acquiring greater pub-lic visibility, should not be underestimated.From a less opportunistic standpoint,applications should not merely be seen asnecessary justifications, but also as sourcesof inspiration. The second new feature mayhopefully ensure that the new tradition ofECM’s continues. The funding and organi-zation of a big international congress isindeed a major undertaking and the diffi-culties involved may be expected toincrease with time rather than decrease.The European Union has already invested aformidable apparatus of networks repletewith their own special conferences. Whatwould be more natural than a unifying one,which should bring with it a large body ofactive participants, and hopefully alsochannels for necessary funding? AtStockholm, admittedly the network lecturesplayed a rather marginal role, but if the ideais accepted and developed in future ECM’sthey may provide the core activity ontowhich various extras may be attached asembellishments.

Aula MagnaAfter those general preambles it may beappropriate to describe the FourthEuropean Congress of Mathematicians asan actual event tied to a physical location ata given slot in time. The basic problem oforganizing a conference for one thousandodd expected participants is to find a lec-ture hall big enough. Unlike the case of theOlympics, the erection of new buildings isnot an option. Of course big conferencesare legion these days, but commerciallyavailable space often comes with price-tagsnot within the capabilities of mathematicalmeetings. The Royal Swedish Institute ofTechnology (Kungliga TekniskaHögskolan - KTH) was not able to providesuch a hall on its premises, making a directcollaboration with the University ofStockholm a necessity on this basis alone.The fact that there are two separate depart-ments of mathematics in Stockholm hasincidentally been a bone of contention forat least fifty years, and may continue to beso for another fifty years, but this is ofcourse only a matter of local interest. TheAula Magna is the official grand lecturehall of the University of Stockholm, andhas of course no direct connection, physicalor not, to its department of mathematics,nor to that of the KTH, being about equally

(physically) distant from both. It is of fairlyrecent vintage, located on the main campusof the University of Stockholm, easilyaccessible by the Stockholm Underground(‘T-banan’, ‘T’ as in tunnel). Shaped likean amphitheatre, with options of subdivi-sion, the Aula actually boasts a capacitywell in excess of the number of actual par-ticipants (around 900), which resulted inthe somewhat unfortunate impression ofnot only the lectures but also the opening

ceremony being sparsely attended. Asexpected, the Aula along with its adjacenthallways became the locus of the meeting.Walking along its perimeter you had imme-diate access to the young assistants donningyellow T-shirts, as well as to the variousbookstalls providing not only opportunitiesfor browsing but also seducing discounts.The Springer stall also supplied piles ofcopies of the Stockholm Intelligencer (stillsmelling of fresh print), featuring short arti-cles on Sweden and mathematics, includinga short but morbid list of distinguishedmathematicians who died here. Climbing afew stairs you could also inspect the vari-ous poster-sessions. Additional smaller lec-ture halls were available at the proverbialstone-throws distance, as well as what inrecent years has become an absolute neces-sity for wayward mathematicians - accessto e-mail. A fairly spacious room, accessi-ble only by pressing a code at the entrance,was filled with a sufficient number of com-puters to keep waiting lines rather than tem-pers short. Furthermore, a small staff ofknowledgeable yellow-shirts was always athand during opening hours. For those ableto resist the allure of the screen at

lunchtime, there was the temptation of theofficial cafeteria situated halfway betweenthe two locations (actually accessible fromthe Aula Magna remaining indoors theentire way, a godsend in the case ofinclement weather). It provided the stan-dard Swedish lunch fare to be expectedfrom that kind of self-serve establishment,confidently assured of a captive audience,not only by its isolated existence but alsodue to pre-paid lunch coupons.

The opening ceremonyThe conference got a head start on Sundayafternoon, on June 27, by providing regis-tration outside the Aula. This involved get-ting a handy black briefcase, doubling as arucksack, with the logo of the 4ECM suffi-ciently discreet to encourage post-confer-ence use. It would be tedious to list its con-tents of ‘goodies’, but I am sure that mostpeople appreciated the free public transportpasses intended to cover the entire period.One of its more trivial items was a couponto be exchanged for a single glass of wine(of optional colour) to be served at thehour-long reception starting at six o’clock.The next day, Monday June 28, involvedthe official opening of the meeting at ninethirty in the morning, preceded by anopportunity for last-minute registrations.Back at the ICM of 1962, the old SwedishKing Gustavus VI attended, giving out theFields Medals. Such a spectacle of royaltyat a mathematical meeting was, alas, notrepeated this time. The presence of theSwedish Majesty back then has been attrib-uted to the above mentioned fact that at thetime, it was the largest scientific meetingever to have taken place in Sweden. No

EMS NEWS

EMS September 20048

4ecm staff studying new books at the booth of the EMS publishing house

such distinction could be given to the4ECM, so neither the old King’s grandson,the present King Charles XVI, nor thegreat-granddaughter, the crown princessVictoria, were able to squeeze the eventsinto their busy schedules. (One should notoverestimate the love the general public,including that of royalty, feels for mathe-matics) Instead two academic notables,Bremer and Franke, provided the requiredofficial lustre. The latter, Sigbrit Franke,the chancellor of the university system inSweden, also served the function of award-ing the EMS prizes to the young mathe-maticians, to which we will return shortly.Both Bremer and Franke, not surprisingly,made a special point in referring to SofiaKovalevskaya in their short speeches.Kovalevskaya, as can have escaped thenotice of few mathematicians, was the firstever woman professor in mathematics,holding her position at the precursor ofStockholm University at the end of the 19thcentury, and ever since then being a mathe-matical role model for half the populationof the world. Kingman, in his capacity asthe President of the EMS, gave the mathe-matical speech with commendable aplomb,stressing the importance of mathematics,and urging young mathematicians to go forthe hard problems, irrespective of whetherpure or applied, because you can never tell.He also encouraged the new generation byreminding them that they are as great andimaginative as the great ones of the past,because what puzzled our ancestors doesnot necessarily puzzle us. The chair of thePrize Committee, Nina Uraltseva from St.Petersburg, who complained that she was abit too short for the microphone, explainedthat the work of the prize committee hadbeen very hard. There were fifty nominees,out of which ten had to be selected. Tothose who did not make it, there is only onething to say - work harder and try to outdothose who were selected. This is not theplace to give a list of the winners, alongwith short descriptions of their accomplish-ments. It suffices to add that in addition tothe traditional EMS prizes, a so called BITprize was awarded for work in numericalanalysis. No official ceremony is completewithout some kind of entertainment. Themusical interlude in this case was providedby a small Swedish ensemble specializingin Elizabethan music but also performing,fittingly, some classical Swedish songs.Dressed in period costumes, they did theirthing, singing with their own instrumentalsupport, concluding their act by struttingaround the stage pilfering the belongings ofthose unfortunate enough to have the privi-lege of sitting on the first row.

The week in reviewThe conference was kicked off by the firstplenary lecturer, Oded Shramm, associatedwith Microsoft Research. An appropriatebeginning in view of the fact that PresidentLaptev, earlier in the ceremony, got stuckon his power-point presentation, cursingmodern computer technology. Schramm,however, did not address such wordlyissues of practicality, but expounded onconformally invariant random processesinstead, albeit with many a computer visu-alization. And then there was time forlunch, and in spite of the customary denialof the existence of such entities, free to allparticipants. The afternoon was devoted toparallel sessions, four in fact, involvingtwelve lectures in total. The day wascapped off by an EMS reception at the oldlocation for the department of mathematicsat KTH, a building commonly referred to asSing-sing, due to its intimidating lay-out.The reception wisely took place on theground floor, the limited space of whichquickly got extremely crowded. One sur-mises that afterwards there were onlyempty wine-bottles among the left-overs.The next day started out with presentationsof the prize winners and their work (itshould be noted that some of them had also,independently one assumes, been invited asregular speakers as well.) Those were fol-lowed by invited lectures and then in theafternoon there were three Science lectures.The last one was that of R.Ernst, a NobelPrize winner in Chemistry, giving a surveyfrom Fourier to Medical imaging, constitut-ing no doubt a very instructive lecture tothe mathematicians, giving them, amongother things, a sense of the somewhat alienculture of big science. Tuesday was cappedoff by a visit to the Town Hall ofStockholm replete with a buffet courtesy ofthe city of Stockholm. The Town Hall,designed by the Swedish architect RagnarÖstberg and built in the twenties, is one ofthe most commonly pictoriallyreproduced landmarks of the city, located atthe edge of an island, and commanding apresence on the main waterway. Its designwas inspired by the palace of the dodges inVenice and its main feature is the great ban-quet hall inside, somewhat puzzlinglyknown as the Blue Hall (guides of thebuilding are just delighted to explain to youthe historical reason why), which everyDecember is host to the Nobel banquet fol-lowing the prize awards at the ConcertHall. This time however, the setting wassomewhat less sumptuous. Two grand buf-fets (presumably identical) were displayedon two long tables on one side. Food wasplentiful, but to savour it you needed to do

the customary juggle of balancing yourwineglass, your plate, and your knife andfork with just two hands, while standingfirmly on your feet and trying to makecoherent conversation. AfterwardsKingman rose to the occasion thanking thehosts referring to the great success of theconference (so far). He concluded by deliv-ering a splendid celebration of the impor-tance of mathematics, reminding everyonethat while back in 1900, some of the scien-tific accomplishments honoured at this veryplace involved no mathematics, nowadaysthis would be almost impossible, and thatall scientists should acknowledge this fact.In fact, he reminded us all that the entirehuman race will benefit from the develop-ment of mathematics. In order to gentlyusher out the mathematical crowd from thepremises, a guided tour was offered at theend providing a natural conclusion. Forthose of us who afterwards lingered on bythe waterfront, we may late forget the glo-rious view made sublime (as one used tosay) by the slowly setting sun. The tradi-tional association of Stockholm withVenice, made particularly palpable by theTown Hall, is not plucked out of thin air,but rests on water. Wednesday was takenover by plenary talks in the morning andthree additional science lectures in theafternoon. The Austrian Nowak expoundedon mathematical biology with specialemphasis on evolutionary processes, andBerry presented a cascade of computer gen-erated pictures relating to optics and con-comitant singularities. Thursday was onlyhalf a day, with invited lectures in themorning and scheduled excursions in theafternoon. In the evening, the Frenchambassador gave a reception for the nota-bles of the EMS, the organizers and last butnot least the young prize-winners. Franceas a country and culture should be com-mended for the official respect it accordsmathematics and for always recognisingmathematical achievement. I fear that the4ECM may very well have received morepublicity in France than in Sweden. Fridaywas the closing up, with parallel networklectures in the morning and a series of ple-nary talks in the afternoon, the last fittinglydelivered by a local speaker, Johan Håstadat KTH, talking on the difficulty of provingthe generally believed NP≠P.

Wrapping upThe concluding ceremony was, as such

things tend to be, rather short. Kingmanexpressed the pride of the EMS to be asso-ciated with the ECM and thanked the orga-nizers, and especially its President. Therewas a lot of applause. Then there was a ref-

EMS NEWS


erence to the upcoming centenary birthdayof Henri Cartan, who had an honorary rolein the first ECM which took place in Paris1992, and to whom the entire congressrelayed its congratulations. Laptev thenreferred to the statistics of the event, withthe final tally of 930 members, over threehundred posters, and sixty three scheduledtalks, of which only one had to be can-celled. As expected he could not refrainfrom congratulating himself on havingarranged such nice weather for the entireduration. No mean feet indeed, and a defi-nite contribution to the general pleasure.Finally the congress was treated to an invi-tation to the fifth ECM to take place inAmsterdam in 2008. After the usual prob-lems with an unresponsive machine, theaudience was eventually exposed to apower-point show by the representative ofthe next ECM, promising future delights.By the time the audience emerged out ofthe Aula Magna for the last time, bookstallswere being dismantled, posters taken down,and the last remaining conference briefcas-es sold off at very reasonable prices.Clearly within hours, no physical traceswould be left of the congress. On the otherhand, one would hope that mental traces,preferably of the positive and edifyingkind, will remain for a very long time.

Ulf Persson [[email protected]] hasbeen professor at Chalmers University ofTechnology in Gothenburg (Göteborg)since 1989. He earned his Ph.D at Harvard1975 as a student of David Mumford.Works in Algebraic Geometry, especiallycompact complex surfaces. Hobbiesinclude mathematical pictures pro-grammed directly in PostScript. Has servedas the President of the SwedishMathematical Society and has been active-ly engaged in public debate on mathemati-cal education. Publishes regularly reviewson scientific-philosophical matters in theSwedish press and is the editor of thenewsletter of the aforementioned SwedishMathematical Society.

Starting the race towards 5ecm atAmsterdam in 2008. From left to right:H.J.J. te Riele,J.O.O. Wiegerinck, C.M. Ran(http://sta.science.uva.nl/ brandts/5ECM/)

The EMS prizes are awarded by theEuropean Mathematical Society in recog-nition of distinguished contributions inMathematics by young researchers notolder than 35 years. The prizes are pre-sented every four years at the EuropeanCongresses of Mathematics.

The Prize Committee is appointed bythe EMS and consists of a number of rec-ognized mathematicians from a wide vari-ety of fields. The prizes were first award-ed in Paris in 1992 and then in Budapest in1996 and in Barcelona in 2000.

Each prize winner 2004 received 5,000Euro.

Prize committeeEnrico Arbarello, - RomeVictor Buchstaber, - MoscowJohn Coates,- Cambridge, UKJacek Graczyk, - OrsayBertil Gustafsson, - UppsalaStefan Hildebrandt, - BonnJean-François Le Gall, - ParisVladimir Lin, - HaifaLeonid Polterovich, - Tel AvivDomokos Szasz, - BudapestDimitri Yafaev, - RennesEduard Zehnder, - Zürich

EMS Prize winners 2004 with the presi-dent of the Prize committee, NinaUraltseva. Sitting from the left: XavierTolsa, Paul Biràn, Sylvia Serfaty,Stefano Bianchini, Otmar Venjakob.Standing from the left: Franck Barthe,Warwick Tucker, Nina Uraltseva(President of the Prize committee), ElonLindenstrauss, Andrei Oukonkov,Stanislav Smirnov.

EMS Prizes 2004 with citations

Franck Barthe, Institut deMathématiques: Laboratoire deStatistique et Probabilités, Toulouse,FranceBarthe pioneered the use of measure-transportation techniques (due toKantorovich, Brenier, Caffarelli, McCann and others) in geometric inequali-ties of harmonic and functional analysiswith striking applications to geometry ofconvex bodies. His major achievement isan inverse form of classical Brascamp-Lieb inequalities. Further contributionsinclude discovery of a functional form ofisoperimetric inequalities and a recentsolution (with Artstein, Ball and Naor) ofa long-standing Shannon’s problem onentropy production in random systems.

Stefano Bianchini, Instituto per leApplicazioni del Calcolo “M. Picone”,Rome, ItalyStefano Bianchini has introduced anentirely new perspective to the theory ofdiscontinuous solutions of one-dimen-sional hyperbolic conservation laws, rep-resenting solutions as local superpositionof travelling waves and introducing inno-vative Glimm functionals. His ideas haveled to the solution of the long standingproblem of stability and convergence ofvanishing viscosity approximations. Inhis best individual achievement, pub-lished in 2003 in Arch.Ration. Mech.Anal., he shows convergence of semidis-crete upwind schemes for general hyper-bolic systems. In the technically demand-ing proof the travelling waves are con-structed as solutions of a functional equa-tion, appling center manifold theory in aninfinite dimensional space.

Paul Biràn, School of MathematicalSciences, Tel-Aviv University, IsraelPaul Biràn has made fundamental andinfluential contributions to symplectictopology as well as to algebraic geometryand Hamiltonian systems. His work ischaracterised by new depths in the inter-actions between complex algebraic geom-etry and symplectic topology. One of theearlier contributions is his surprisingsolution of the symplectic packing prob-lem, completing work of Gromov,McDuff and Polterovich, showing that

EMS NEWS

EMS September 200410

EMS Prizes EMS Prizes

compact symplectic manifolds can bepacked by symplectic images of equallysized Euclidean balls without wastingvolume if the number of balls is not toosmall. Among the corollaries of his proof,Biràn obtains new estimates in the Nagataproblem. A powerful tool in symplectictopology is Biràn’s decomposition ofsymplectic manifolds into a disc bundleover a symplectic submanifold and aLagrangian skeleton. Applications of thisdiscovery range from the phenomenon ofLagrangian barriers to surprising novelresults on topology of Lagrangian sub-manifolds. Paul Biràn not only provesdeep results, he also discovers new phe-nomena and invents powerful techniquesimportant for the future development ofthe field of symplectic geometry.

Elon Lindenstrauss, Clay MathematicsInstitute, Massachusetts and CourantInstitute of Mathematical Sciences, NewYork, USAElon Lindestrauss has done deep andhighly original work at the interface ofergodic theory and number theory.Although he has worked widely in ergod-ic theory, his recent proof of the quantumunique ergodicity conjecture for arith-metic hyperbolic surfaces breaks fertilenew ground, with great promise for futureapplications to number theory.

Already, in joint work with Katok andEinsiedler, he has used some of the ideasin this work to prove the celebrated con-jecture of Littlewood on simultaneousdiophantine approximation for all pairs ofreal numbers lying outside a set ofHausdorff dimension zero. This goes farbeyond what was known earlier aboutLittlewood’s conjecture, and spectacular-ly confirms the high promise of themeth-ods of ergodic theory in studying previ-ously intractable problems of diophantineapproximation.

Andrei Okounkov, PrincetonUniversity, USAAndrei Okounkov contributed greatly tothe field of asymptotic combinatorics. Anextremely versatile mathematician, hefound a wide array of applications of hismethods. His early results include a proofof a conjecture of Olshanski on the repre-sentations theory of groups with infinite-dimensional duals. Okounkov gave thefirst proof of the celebrated Baik-Deift-Johansson conjecture, which states thatthe asymptotics of random partitions dis-tributed according to the Plancherel mea-sure coincides with that of the eigenval-ues of large Hermitian matrices. An

important and influential result ofOkounkov is a formula he found in jointwork with Borodin, which expresses ageneral Toeplitz determinant as theFredholm determinant of the product oftwo associated Hankel operators. Thenew techniques of working with randompartitions invented and successfullydeveloped by Okounkov lead to a strikingarray of applications in a wide variety offields: topology of module spaces, ergod-ic theory, the theory of random surfacesand algebraic geometry.

Sylvia Serfaty, Courant Institute ofMathematical Sciences, New York, USASylvia Serfaty was the first to make a sys-tematic and impressive asymptotic analy-sis for the case of large parameters inTheory of Ginzburg-Landau equation.She established precisely the values of thefirst, second and third (with E.Sandier)critical fields for nucleation of one stablevortex, vortex fluids and surface super-conductivity. In micromagnetics, herwork with F. Alouges and T. Rivièrebreaks new ground on singularly per-turbed variational problems and providesthe first explanation for the internal struc-ture of cross-tie walls.

Stanislav Smirnov, KTH, Sweden andGeneva University, SwitzerlandStanislav Smirnov’s most striking resultis the proof of existence and conformalinvariance of the scaling limit of crossingprobabilities for critical percolation onthe triangular lattice. This gives a formu-la for the limiting values of crossing prob-abilities, breakthrough in the field, whichhas allowed for the verification of manyconjectures of physicists, concerningpower laws and critical values of expo-nents. Stanislav Smirnov also made sev-eral essential contributions to complexdynamics, around the geometry of Juliasets and the thermodynamic formalism.

Xavier Tolsa, ICREA and UniversitatAutònoma de Barcelona, SpainXavier Tolsa has made fundamental con-tributions to Harmonic and ComplexAnalysis. His most outstanding worksolves Vitushkin’s problem about semi-additivity of analytic capacity. The prob-lem was raised in 1967 by Vitushkin inhis famous paper on rational approxima-tion in the plane. Tolsa’s result has impor-tant consequences for a classical (100years old) problem of Painlevé about ageometric characterization of planar com-pact sets are removable in the class ofbounded analytic functions. Answering

affirmatively Melnikov’s conjecture,Tolsa provides a solution of the Painlevéproblem in terms of the Menger curva-ture. Xavier Tolsa has also publishedmany important and influential resultsre-lated to Calderón-Zygmund theory andrational approximation in the plane.

Warwick Tucker, Uppsala University,SwedenWarwick Tucker has given a rigorousproof that the Lorenz attractor exists forthe parameter values provided by Lorenz.This was a long standing challenge to thedynamical system community, and wasincluded by Smale in his list of problemsfor the new millennium. The proof usescomputer estimates with rigorous boundsbased on higher dimensional intervalarithmetics. In later work, WarwickTucker has made further significant con-tributions to the development and appli-cation of this area.

Otmar Venjakob, MathematischesInstitut: Universität Heidelberg, GermanyOtmar Venjakob has made a number ofimportant discoveries in both the algebra-ic and arithmetic aspects of non-commu-tative Iwasawa theory, especially onproblems which appeared intractablefrom the point of view of the classicalcommutative theory. In arithmetic geom-etry, Iwasawa theory is the only generaltechnique known for studying the myste-rious relations between exact arithmeticformulae and special values of L-func-tions, as typified by the conjecture ofBirch and Swinnerton-Dyer. Venjakob’swork applies quite generally to towers ofnumber fields whose Galois group is anarbitrary compact p-adic Lie group(which is not, in general, commutative),and has done much to show that a richtheory is waiting to be developed. Hismost important results include the proofof a good dimension theory for modulesover Iwasawa algebras, and the proof ofthe first case of a structure theory formodules over these algebras. WithHachimori he discovered the first exam-ples of arithmetic Iwasawa moduleswhich are completely faithful, as well asproving a remarkable asymptotic upperbound for the rank of the Mordell Weilgroup of elliptic curves in certain towersof number fields over Q whose Galoisgroup is a p-adic Lie group of dimension2. Very recently, he found the key to theproblem of defining, in non-commutativeIwasawa theory, the analogue of the char-acteristic series of modules over Iwasawaalgebras.

EMS NEWS


During the week immediately prior to theEuropean Congress of Mathematics, oneof the congress’s many satellites tookplace at the Royal Institute ofTechnology (KTH) in Stockholm. Thisparticular one differed slightly from mostsatellite events in that the participants ofthis conference were aged 14-20 and notyet active research mathematicians.

The Junior Mathematical CongressThe first Junior Mathematical Congresswas held in Paris, France, in conjunctionwith the first ECM. Several hundredyoung secondary school students, mainlyfrom France but also from several otherEuropean countries, gathered and lis-tened to seminars from participants andfrom invited lecturers. As the time camefor the second ECM in Budapest, theJMC was this time held in Miskolc(Northeast Hungary), again with a largeinternational contingent present.

After the Miskolc event, some of the

organisers realised that a periodicity offour years is much too sparse for theintended age group, as a student wouldbe able to both start and finish secondarystudies - or for that matter higher studies– between congresses and without anycontact with the congress activities.Thus, the third JMC was held inPotsdam, in conjunction with the ICM inBerlin. The fourth was supposed to beheld in Spain, together with theBarcelona ECM, but the intensity ofarranging different activities was toohigh and thus in 2000, the JMC was heldin Miskolc. 2002 again saw a small JMCin Miskolc, this time with mostlyHungarian participants.

In Sweden, home of the authors of thisarticle, the JMC was unheard of until1998, when Mikael got involved in thepreparations for the Potsdam congressduring an exchange year in Germany.Two years later, the two of us bothattended the fourth JMC in Miskolc.

Thrilled and excited after the experience,we were determined to seek out similarevents back in Sweden, or - if need be -bring them there.

Stockholm 2004When news that the 4ECM was to be heldin Stockholm reached us three years ago,it was clear that the JMC of 2004 mustalso take place here - on our home turf.With the help and guidance mainly of DrPéter Körtesi of the University ofMiskolc, and of the Swedish NationalCentre for Mathematics Education, theStockholm district of the SwedishFederation of Young Scientists throughus started the preparations for the con-gress.

By the end of June 2004, after havingraised EUR 30,000 in funding and gath-ered 40 participants from seven Europeancountries, it was finally time. A staff oftwelve, all just a little older than the par-ticipants, got everyone through the week-

NEWS


VVisions ofisions ofYYoungoung

MathematicsMathematicsin Eurin Europeope

Jon Larsson and Mikael Johansson

(Stockholm, Sweden)Jon Larsson Mikael Johansson

Congress participants at work ... and at dinner

long event which, according to the ques-tionnaire at the end, was a great success.

The sixth JMC was indeed a success,but it was also a lot of hard work, uncer-tainty and confusion. During the courseof preparations and the congress itself,we experienced in a very profound waythe lack of continuity, established coop-eration and well-functioning communica-tion and information networks in theyoung mathematics movements inEurope. It had been a difficult task at bestto reach out to like-minded youngstersthroughout the continent and informthem about the JMC’s existence and thefact that we wanted them to attend it.Something needed to be done.

Founding a Mathematical SocietyIn 2000, Péter Körtesi had already for-mulated the idea of cooperation acrossnational borders on a more general level,and as the work progressed, this ideamatured in our discussions. A EuropeanJunior Mathematical Society would bewell fitted to supply the much neededcontinuity between congresses by actingas formal principal of the JMC’s, keepingrecords from earlier events and facilitat-ing personal contact between organisersin different countries. On a more generallevel, the EJMS can act as a commonforum and a coordinating body for sever-al national or regional organisationswithin the field of young mathematics.

The EJMS would not only be an assetfor congress organisers and assortedNGO’s, but also, perhaps even moreimportantly, for the individual youngmathematicians on the continent. For theteenager interested in mathematics, twoof the main problems are often findingothers who share their interest, and find-ing further mathematical stimulation andguidance. These are both problems thatcould hopefully be alleviated by the exis-tence of a Pan-European network ofyoung mathematicians. That networkwould also have a natural place in thegreater quest to further and increaseinterest in mathematics among youngEuropeans in general.

These discussions were brought to alarger audience at the JMC this summer.Many good ideas were brought forth, byyoungsters and adults alike. The culmi-nation was reached during the closingceremony of the congress, when the par-ticipants - young mathematicians fromseven countries - unanimously declaredthat it was the wish of the sixth JMC thata European Junior Mathematical Societybe founded as soon as possible.

Here in Stockholm, the work has nowbegun. The formation of a Swedish soci-ety for young mathematicians is under-way in close cooperation with theSwedish Federation of Young Scientists.With a national organisation in place, wewill have a starting point from which wecan gather the contacts, support andfunding necessary to launch the EJMS.

In closing, we would like to make a callfor support to you, the mathematicians ofEurope. If you feel that you can help usin any way, be it by founding a JuniorMathematical Society in your country,providing us with funds or simply shar-ing valuable experience with us, we urgeyou to contact us by e-mail at

[email protected]. We believe the EJMShas great potential and hope that you willshare that belief.

Jon Larsson [[email protected]] is a stu-dent of theoretical computer science atthe Royal Institute of Technology (KTH)in Stockholm, Sweden, and Chairman ofthe Stockholm District of the SwedishFederation of Young Scientists.

Mikael Johansson[[email protected]] has recently fin-ished his Master thesis at the Departmentof Mathematics of Stockholm University,Sweden.

Both authors were members of the4ECM organising committee.

NEWS


Max-Planck-Institut

für Mathematik in den Naturwissenschaften Max Planck Institute for Mathematics in the Sciences

The Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, is seeking a

Coordinating Scientist

for the

International Max Planck Research School

Mathematics in the Sciences at the University of Leipzig

The Max Planck Research School Mathematics in the Sciences is a cooperation of the Max Planck Institute for

Mathematics in the Sciences and the Department of Mathematics and Computer Sciences and the Department

of Physics of the University of Leipzig.

It will lead interested graduate students towards research problems in the physical and life sciences. This

involves a broad range of mathematical fields, including geometry, partial differential equations and functional

analysis, stochastics, and discrete mathematics. Specific subject areas are:

- Partial Differential Equations and Material Sciences

- Numerical Analysis and Scientific Computing

- Riemannian and Symplectic Geometry and Hamiltonian Systems

- Quantum Field Theory, Particle Physics

- Algebraic Geometry, String Theory

- Geometric and Functional Analytic Methods in Mathematical Physics

- Stochastic Processes, Many Particle Systems

- Complex Systems in Evolutionary Processes and Neurobiology

It is expected that candidates have a Ph.D. in one of the areas covered by the research programm and they are

encouraged to pursue their research interest in parallel to the duties as coordinator of the research school.

Ideally this research interest is in nonlinear pde or many particle systems. The coordinator will work closely

with the speaker of the research school, Prof. Dr. Stephan Luckhaus.

His/her duties include the organization of the selection procedure of candidates, organization of lecture series

and summer schools, and the presentation of the IMPRS. He/she is responsible for the general management of

the IMPRS and the support of its graduate students.

We offer a position with diverse activities in an excellent research environment. The financial conditions

depend on the qualifications with a maximum salary base of BAT IIa/Ib (including Social Security benefits)

according to German public regulations. The position is temporary, with an initial appointment of three years

and possible extension for another three years. The Max Planck Society is an equal opportunity employer and

encourages disabled persons to apply. It also aims at increasing the number of female staff members in fields

where they are underrepresented, and therefore encourages women to apply.

The successful applicant should, if possible, start the offered position not later than January 1, 2005. In order to

ensure full consideration, a curriculum vitae, publication list, statement of professional goals, and the name of

three references should be directed by October 15, 2004 to:

Prof. Dr. Stephan Luckhaus

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

D-04103 Leipzig

Germany

e-mail: [email protected]

Later applications will be considered as long as the evaluation process is going on.

In the second week of July 2004, the bigquadrennial event in mathematical educa-tion, or didactics of mathematics as we sayin Europe, took place on the campus of theTechnical University of Denmark situatedat the outskirts of Copenhagen. We aretalking about the “10th InternationalCongress on Mathematical Education”,abbreviated as ICME-10.

About 2300 researchers in mathematics,researchers in mathematics education andteachers from all levels in the educationalsystem from primary school to university,discussed new developments in relation toquality in mathematical education. Therewere participants from 119 different coun-tries. Registration took place on SundayJuly 4 in the magnificent celebration hallof the University of Copenhagen, close tothe hotels of the participants. The registra-tion day was a beautiful sunny day.During the conference, participantsenjoyed rich variations in the Danishweather! At registration, the participantsreceived cards for the public transporta-tion system in the Copenhagen area,which in the following days quickly and

easily brought them to the site of the con-gress.

Researchers and PractitionersThe international congress on mathemati-cal education is the event that bringstogether active research mathematicians,deeply devoted to develop and dissemi-nate their subject, and practitioners,strongly devoted to create a well-function-ing and rich teaching environment inmathematics in their countries of origin,under the guidance of researchers in math-ematical education, who have contributedsignificantly to structuring thinking aboutproblems related to the teaching and learn-ing of mathematics. During the congress, Icame to realise how important it is thatresearchers in mathematics andresearchers in mathematics education donot loose contact with each other to theconfusion of the practitioners in the class-room, who face concrete problems withspecific topics from mathematics. Morethan ever, I now believe that the voices ofboth groups of researchers are necessaryin the debate about the contents, the meth-

ods, the presentation and the assessmentof mathematics in schools.

Ceremonies and AwardsMany participants will remember theopening ceremony since two politiciansmade very good speeches and stayed forthe whole ceremony. The Danish ministerof education, Ulla Tørnæs, faced the prob-lems for mathematics in schools withoutgetting into general political polemic, andthe Mayor of Lyngby Municipality, RolfAagaard-Svendsen, who holds a Ph.D. inmathematical statistics, made a wittyspeech where he even showed a page fromhis thesis. Other speeches were made byprofessor Hyman Bass, the president ofICMI (International Commission onMathematical Instruction), professorMogens Niss, the chair of the internation-al programme committee (IPC), professorMorten Blomhøj, the chair of the localorganizing committee (LOC), professorChristian Stubkjær, the dean of research atthe Technical University of Denmark, andprofessor Bernard Hodgson, the generalsecretary of ICMI. The ceremony was fur-ther enlightened by a musical performanceby Royal Danish Brass.

The opening ceremony also included theawarding of the first Felix Klein Medal toprofessor Guy Brousseau, France, for hislifetime contributions to didactics ofmathematics, and the first HansFreudenthal Medal to professor CeliaHoyles, England, for outstanding contri-butions to research in the domain of tech-nology and mathematics education1. Withthe choices of the first recipients, a high

ICME-10


From the left; Morten Blomhøj (chair of the LOC), Elin Emborg (administrativesecretary for LOC and IPC), Mogens Niss (chair of the IPC), all three IMFUFA,Roskilde University, Denmark and Gerd Brandell (chair of the Nordic ContactCommittee for ICME-10), Lund University, Sweden.

SOME PERSONALSOME PERSONALREFLECTIONS ON ICME-10REFLECTIONS ON ICME-10

Vagn Lundsgaard Hansen (Lyngby, Denmark)

1 The citations can be found in issue 52 ofthe Newsletter.

standard has been set for these awards,making them especially prestigious forresearch in mathematics education.

The ProgrammeAnd then we were ready for the confer-ence. The programme was huge: 8 plenaryactivities, some 80 regular lectures, 29topic study groups, 24 discussion groupsand - as a new thing - a thematic afternoonwith 5 choices. There were 5 national pre-sentations during one full afternoon:Korea, Mexico, Romania, Russia and theNordic countries (Denmark, Finland,Iceland, Norway, and Sweden). Eachnational presentation had a detailed andcomprehensive programme. Obviously,most activities were in parallel sessionswith many options. All in all, there weremore than half a million possibilities forcomposing your own programme. Toassist the individual planning, participantsgot an extra copy of the one page sched-ule, where they could fill in which activi-ties they wanted to attend and where tofind them.

For a programme of this size, no singleparticipant can capture it all. But everyonefelt the spirit and the energy and compas-sion with which the many contributors hadprepared their contributions. In all fairnessit is not possible to single out specific con-tributions in a general overview. But againyou realise that the listeners and the eager-ly engaged contributors to discussions arereally what make up a good congress. I dothink that ICME-10 was a success in thatrespect.

I attended a thematic afternoon on math-

ematics and mathematics education thatcaused a heated debate. Although the tonewas not exactly polite, it was very directand explicit, and it may actually lead toreflections on the various roles ofresearchers in mathematics andresearchers in mathematics education andtheir possibilities for the shaping of goodmathematics teaching in the classroom.

Circus MathematicusDuring the congress, a mathematical cir-cus “Circus Mathematicus” was arrangedat the congress site for the general public.The creativity shown was enormous,encompassing among other things: bowl-ing with fractions, a labyrinth, life size

puzzles, boomerangs, juggling, origami,wood carving, beautiful Christmas deco-rations, various rope tricks and construc-tion of kites. The mathematical circus wasa great success and helped attract theattention of the news media to the con-gress in general, including leading Danishnewspapers, TV and radio. Also a veryinteresting exhibition developed in co-operation between ICMI and UNESCO,“Why mathematics”, was shown duringICME-10.

Being one of the local organisers of thecongress, I must be careful with appraisal.But I do feel that the congress was suc-cessful in fulfilling its aim, namely stimu-lating the continual process towards goodteaching of mathematics. As professorJeremy Kilpatrick assured us under a veryinteresting plenary interview session withfour distinguished mathematical educa-tors: “The strive for good teaching willnever stop. There will always be new chal-lenges.”

The closing ceremony took place onSunday, July 11, and the organisers wererelieved after more than four interestingyears of planning in collaboration with ourextremely effective and helpful congressbureau: Congress Consultants.

ICME-11 takes place in Mexico in2008. You should join it. An ICME con-gress is worth the effort.

Vagn Lundsgaard Hansen[[email protected]] is a Professorof Mathematics at the TechnicalUniversity of Denmark, Lyngby,Denmark. He serves as chairman of theEMS-committee on Raising PublicAwareness of Mathematics.

ICME-10


The president of the International Commission on Mathematical Instruction,Hyman Bass, presents the first Felix Klein Medal to Professor Guy Brousseau (onthe right-hand side).

Children and congress participants engaged in mathematical activities in CircusMathematicus.

25 years proof of the Kneser conjectureThe advent of topological combinatorics

Mark de Longueville (Berlin)

Figure 1: The origin of the Kneser conjecture in the hand writing of Martin Kneser.

Laszlo Lovasz’s proof of the Kneser conjecture about 25 years ago marked the beginning of the history of topological combinatorics:applications of topological methods and theorems to problems of discrete mathematics which did not (seem to) have any connection totopology.

With this article I want to take the op-portunity to sketch the development oftopological proofs in discrete mathe-matics beginning with the proof of theKneser conjecture about 25 years ago,which eventually led to a new discipli-ne: topological combinatorics.

In the beginning of the twentiethcentury the discipline of combinatorialtopology already made use of combina-torial concepts in topology, finally lea-ding to the emergence of algebraic to-pology. Meanwhile, discrete mathema-tics did not make much use of tech-niques from (algebraic) topology untilthe seminal proof of the Kneser con-jecture. This situation was going tochange in an unexpected and fascina-ting way.

The essence of topological combina-

torics can be characterized by a sche-me that many proofs in this field pur-sue. If we want to solve a combinato-rial problem by topological means wecarry out the following steps.

(1) Associate a topologicalspace/continuous map to the gi-ven discrete structure such as agraph/graph homomorphism.

(2) Establish a relationship bet-ween suitable topological inva-riants of the space, e.g. dimensi-on, k-connectedness, homologygroups, etc. and the desired com-binatorial features of the originalstructure.

(3) Show that the associated spaceresp. the map has the desired to-

pological properties.

Thus we are concerned with a (func-torial) procedure, which is common-ly used in mathematics. However, thedifference to other fields is that theconstructions have to be invented andtailored anew for almost each indivi-dual problem. This calls for a certainamount of ingenuity and deeper in-sight, in particular in steps (1) and (2)of the above procedure.

The first proof of this kind is theproof of the Kneser conjecture byLaszlo Lovasz. Because of its relevancefor the emergence of topological com-binatorics, its elegance, and its 25th an-niversary, I will sketch this proof andreport on the development of topologi-cal combinatorics.

1

Figure 2: From “Jahresbericht der DMV” 1955. Figure 3: Martin Kneser. Figure 4: Laszlo Lovasz.

The Kneser conjecture and itsproof

The occupation with an article by Ir-ving Kaplansky on quadratic forms,from 1953, led Martin Kneser to que-stion the behaviour of partitions of thefamily of k-subsets of an n-set:

Consider the family of allk-subsets of an n-set. Itis easy to partition thisfamily into n − 2k + 2classes C1 ∪ · · · ∪ Cn−2k+2,such that no pair of k-setswithin one class is disjoint.Is it possible to partition thefamily into n − 2k + 1 clas-ses with the same property?

Kneser conjectured that this is notpossible! He presented his conjecturein the “Jahresbericht der DeutschenMathematiker–Vereinigung” in 1955 inthe form of an exercise [6] (cf. Figure 2).

We will translate Kneser’s conjectureinto graph theory language. For thatpurpose we define the Kneser graphKGn,k in a suggestive manner: the ver-tices are the k-subsets of an n-set andthe edges are given by pairs of disjointk-sets. Figure 5 shows this graph forthe parameters n = 5 and k = 2. Forexample, there is an edge between thesets {1, 2} and {3, 5} because they haveempty intersection.

The afore mentioned partitions ofthe k-subsets of an n-set now corre-spond to partitions of the vertex set ofthe graph in so called colour classes. Theproperty that no partition set containsa pair of disjoint k-sets now translatesinto the property that no colour classcontains two vertices that are adjacentvia an edge in the graph. Such a partiti-on is called a graph colouring. The chro-matic number of a graph is the smallestnumber of colour classes in a graph co-louring. As mentioned above it is easyto define a graph colouring of KGn,k

with n−2k+2 colour classes. The follo-wing example shows 5−2·2+2 = 3 pos-sible colour classes C1, C2, C3, whichdefine a colouring of KG5,2.

C1 = {{1, 2}, {1, 3}, {1, 4}, {1, 5}}

C2 = {{2, 3}, {2, 4}, {2, 5}}

C3 = {{3, 4}, {3, 5}, {4, 5}}

(The example already suggests the ge-neral principle for a partition into n −2k + 2 colour classes!) The Kneser con-jecture now states that it is impossibleto colour the graph with fewer colourclasses, i.e., a lower bound of n−2k+2 forthe chromatic number of the graph KGn,k.

Twenty three years after Kneser po-sed his problem Laszlo Lovasz initia-ted the development of topologicalcombinatorics with the publication ofhis proof of the Kneser conjecture [7].In the following sketch of his proof we

will make use of some topological no-tions that will not be further explained,but which can be found in almost anytextbook on topology, such as [11].

Lovasz’ proof of the Kneser conjec-ture pursues the scheme that we men-tioned in the introduction. Step (1) isbased on the invention of a simplicialcomplex associated with any graph G,the so called neighbourhood complexN (G). Simplices in the neighbourhoodcomplex are defined by sets of verticesthat have a common neighbour in thegraph.

As an example we consider thegraph G in Figure 6. It defines theneighbourhood complex N (G):

N (G) = {∅, {1}, {2}, {3}, {4}, {5},

{1, 2}, {1, 3}, {1, 4}, {1, 5},

{2, 3}, {2, 5}, {3, 4},

{1, 2, 5}, {1, 3, 4}}

The inclusion maximal simplices inN (G) are given by {1, 2, 5}, {1, 3, 4}and {2, 3}. In the graph G they cor-respond to the neighbourhoods of thevertices 3, 2, 1 respectively. Figure 7shows the topological space that reali-zes the simplicial complex geometrical-ly. The simplices {1, 2, 5} and {1, 3, 4}correspond to the shaded triangles, thesimplex {2, 3} corresponds to the linesegment from 2 to 3.

{3, 5}

{1, 2}

{1, 3}

{2, 5}

{1, 4}

{2, 4}

{3, 4}{4, 5}

{2, 3} {1, 5}

Figure 5: The familiar Petersen graphin the guise of the Kneser graph K5,2.

1

5 3

4 2

Figure 6: A graph G.

3

4

1

2

5

Figure 7: The neighbourhood complexN (G) associated to G.

In step (2) Lovasz applies the Borsuk–Ulam theorem from topology, aboutwhich Gian–Carlo Rota, in an essayabout Stan Ulam, once wrote:

“While chatting at the Scot-tish Cafe with Borsuk, anoutstanding Warsaw topo-logist, he [Ulam] saw ina flash the truth of whatis now called the Borsuk–Ulam theorem. Borsuk hadto commandeer all his tech-nical resources to prove it.”

One of the commonly used versionsof this theorem reads:

If there exists an anti-podal continuous mapf : S

n → Sm from the n-

sphere to the m-sphere, i.e.a continuous map that sa-tisfies f(−x) = −f(x) forall x ∈ S

n, then m ≥ n.

Lovasz now shows: if N (G) is to-pologically m-connected, then G is not(m + 2)-colourable, i.e., there is no co-louring of G with m + 2 colour classes.In the example of Figure 7, the neigh-bourhood complex is 0-connected, i.e.connected, but not 1-connected sincefor example the loop 1→2→3→1 can-not be contracted to a point. This re-flects the fact that G is not 2-colourablebut 3-colourable.

We want to sketch this essential partof the proof from a slightly more mo-dern point of view (see e.g. [2] andalso quite current is [10]). An (m + 2)-colouring of a graph G induces a graphhomomorphism G → Km+2 from G

to the complete graph Km+2 on m + 2vertices in which all pairs of verti-ces are connected by an edge. Sucha graph homomorphism induces acontinuous map N (G) → N (Km+2)of topological spaces. It is easyto verify that N (Km+2) is an m-dimensional sphere. Now, if N (G)is n-connected, then with the helpof the map N (G) → N (Km+2), onecan construct an antipodal continuousmap f : S

n+1 → Sm. Hence by the

Borsuk–Ulam theorem m ≥ n+1, resp.n ≤ m − 1 as desired.

In step (3) Lovasz completes hisproof by verifying that N (KGn,k) is in-deed (n − 2k − 1)-connected.

With his proof Lovasz had identifiedthe topological core of the problem:the Borsuk–Ulam theorem. In the sa-me year, 1978, a much shorter proof byImre Barany followed, which employsthe Borsuk–Ulam theorem in a mo-re direct fashion. As recently as 2002,another substantial simplification hasbeen found by the American mathe-matics student Joshua Greene [4]. Forhis proof, which can be considered as“Proof from the Book”, Greene has be-en awarded the 2003 AMS-MAA-SIAMMorgan prize.

Topological combinatorics

The role of the Borsuk–Ulam theoremis not restricted to the proof of the Kne-ser conjecture. General bounds for thechromatic number of a graph, partiti-on results, complexity bounds for algo-rithmic problems, and much more, ha-ve all been established with the help

of Borsuk–Ulam’s theorem and its ge-neralizations. Maybe one of the mostimportant generalizations is a theorempresented by Albrecht Dold [3] in 1983:

Let G be a non-trivial fi-nite group which acts free-ly on (well behaved) spacesX and Y . Suppose X is(n − 1)-connected and Y

has dimension m. If thereexists a G-equivariant mapfrom X to Y , then m ≥ n.

For X = Sn, Y = S

m, and G thetwo–element group acting via the an-tipodal map, we re-obtain the Borsuk–Ulam theorem. As a matter of fact,there is such a variety of applicationsof this theorem and its generalizationsthat Jirı Matousek dedicated an entire(and entirely wonderful) book to themwith the title “Using the Borsuk–UlamTheorem” [8].

While Borsuk–Ulam’s theorem so farplays the most prominent role in topo-logical combinatorics, most standardtools from algebraic topology havenow found their applications in combi-natorics, from homology- and cohomo-logy computations, characteristic clas-ses up to spectral sequences. A recentexample is the article “Complexes ofgraph homomorphisms” by Eric Bab-son and Dmitry Kozlov [1]. Even somemethods from differential topology ha-ve found a combinatorial analog, e.g. inthe invention of discrete Morse theory byRobin Forman.

I want to mention a few applicationsof these methods. Most notably thereare graph colouring problems. By now amultitude of bounds for the chroma-

tic number of graphs and hypergraphshave been established with topologicalmethods. Partition results of differentkinds were solved, such as the neck-lace problem, which was solved in itsfull generality in 1987 by Noga Alon.Moreover, complexity problems, such asthe complexity of linear decision treealgorithms and the complexity of mo-notone graph properties in connectionwith the evasiveness conjecture have be-en addressed with techniques from to-pological combinatorics. Another hu-ge topic is the topology of partially or-dered sets. In the early 80s the Swe-dish mathematician Anders Bjorner in-troduced the concept of shellability ofa partially ordered set. With a partial-ly ordered set one can associate a sim-plicial complex and thus a topologi-cal space. Shellability of a partial or-der as defined by Bjorner implies thatthe associated topological space is abouquet of spheres. This combinatori-al concept along with its topologicalconsequences found numerous appli-cations, such as in the theory of Bru-hat orders and questions in the areaof algebraic combinatorics. It shouldbe remarked that using the concept ofshellability, it is easy to see that theneighbourhood complex N (KGn,k), asit appears in Lovasz’ proof of the Kne-ser conjecture, is indeed (n − 2k − 1)-connected.

Back to combinatorics

Ever since combinatorial theoremswere proved with topological me-thods, the natural question followed asto whether the topological argumentcould be replaced by a combinatorialone. A first breakthrough in this direc-tion was made by Jirı Matousek in theyear 2000 with a combinatorial proofof the Kneser conjecture [9]. His proofrelies on a special case of a combina-torial lemma by A.W. Tucker which isessentially “equivalent” to the Borsuk–Ulam theorem. Combinatorial relativesof the Borsuk–Ulam theorem have alsobeen applied in the construction of ap-proximation algorithms in connectionwith fair division problems . Moreover,the concrete questions in discrete ma-thematics pointed out the need for ex-plicit methods for the computation of

homology groups and other invariantsof simplicial complexes which led to ef-ficient programs for the computation ofthese invariants.

As mentioned in the introduction,combinatorial topology led to the for-mation of algebraic topology. The term“algebraic topology” was apparentlyfirst used in a public lecture by Solo-mon Lefschetz at Duke University in1936 (cf. [5]):

“The assertion is often ma-de of late that all mathe-matics is composed of alge-bra and topology. It is notso widely realized that thetwo subjects interpenetrateso that we have an alge-braic topology as well as atopological algebra.”

The latter has now also become truefor combinatorics and topology.

Addendum: Prof. Dr. Martin Kneserdied on February 16, 2004 in Gottingen.

The EMS-Newsletter is grateful to theeditors of the Newsletter of DeutscheMathematiker–Vereinigung for their kindpermission to publish the author’s Englishtranslation of this article which original-ly appeared in Mitteilungen der DMV,4/2003 under the title “25 Jahre Beweisder Kneservermutung”.

Mark de Longueville [[email protected]] studied mathematics in Berlinand New Orleans. Supported by a grantfrom German Research Foundation (DFG),he received his Ph.D. from Technische Uni-versitat Berlin. He has been assistant pro-fessor at University of Minnesota in Min-neapolis and is now assistant professor atFreie Universitat Berlin. He is mainly in-terested in combinatorial problems that po-tentially have connections to other fields inmathematics.

Literature

[1] Eric Babson and Dmitry N.Kozlov. Complexes ofgraph homomorphisms.http://arXiv.org/abs/math/0310056, Preprint 2003.

[2] Anders Bjorner. Topologi-cal methods. In R. Graham,M. Grotschel, and L. Lovasz,Editors, Handbook of Combinato-rics, 1819–1872. North Holland,Amsterdam, 1994.

[3] Albrecht Dold. Simple proofs ofsome Borsuk–Ulam results. Con-temp. Math., 19:65–69, 1983.

[4] Joshua Greene. A new shortproof of Kneser’s conjecture.Amer.Math.Monthly, 109:918–920,2002.

[5] Ioan M. James. From Combinato-rial Topology to Algebraic Topolo-gy. In I. M. James, Editor, History ofTopology, 561–573. North Holland,Amsterdam, 1999.

[6] Martin Kneser. Aufgabe 360. Jah-resbericht der DMV, 58(2):27, 1955.

[7] Laszlo Lovasz. Kneser’s conjec-ture, chromatic number and ho-motopy. J. Combinatorial Theory,Ser. A, 25:319–324, 1978.

[8] Jirı Matousek. Using the Borsuk–Ulam Theorem. Lectures on topolo-gical methods in combinatorics andgeometry. Written in cooperationwith Anders Bjorner and Gunter M.Ziegler. Universitext, Springer,Heidelberg, 2003.

[9] Jirı Matousek. A combinatorialproof of Kneser’s conjecture. Pre-print 2000, Combinatorica, to appe-ar.

[10] Jirı Matousek and Gunter M.Ziegler. Topological lower boundsfor the chromatic number: A hier-archy. Jahresbericht der DMV, Pre-print 2003, to appear.

[11] James R. Munkres, Elements of al-gebraic topology, Addison Wesley,1984.

[12] Rade T. Zivaljevic. Topologicalmethods. In Jacob E. Goodmanet al., Editoren, Handbook of discreteand computational geometry, 305–329. CRC Press, Second edition,2004.

Three-quarters of a century with HenriCartan

Jean Cerf (CNRS, Université Paris-Sud, Orsay, France)

This article appeared (in French) in Gazettedes Mathématiciens 100, April 2004, p. 7-8.The Newsletter’s thanks go to the author andto the editor of Gazette for the permission toreproduce it and to Larry Siebenmann (Univ.Paris-Sud) for the translation into English.

In a special section in Le Monde of 3 March2004 entitled “Matière grise: la bataille mon-diale” (Grey Matter: the worldwide contest),one of the first observations by the biologistE.E. Baulieu reads: “French mathematiciansare among the very best in the world”. Aboveand beyond reservations concerning certainexcesses perpetrated under the Bourbakibanner, it is my conviction (hopefully, onewidely shared) that this high reputation is inlarge measure the heritage of that group ofyoung men who, in the thirties of the lastcentury, conceived the Bourbaki project towrite a treatise that would develop mathe-matics from the ground up. In this group, Isee, in the front rank, Henri Cartan flankedby his friends André Weil and JeanDieudonné.

I have had the privilege to frequently crossthe path of Henri Cartan. The first occasion(by his own account) was in Strasbourg in1930; admittedly, I have forgotten thisencounter – I was just two years old – but Ihave retained clear memories from succeed-ing years. Cartan was a colleague of myfather’s and a friend of the family, a friendwho was admired, but somewhat feared forhis caustic turn of mind and tongue; at thesame time, his fragile health was a cause for

concern! I saw him again in 1939 inClermont-Ferrand, where the University ofStrasbourg had been moved following theoutbreak of war, and later in La Bourboule,after the defeat of June 1940, of which he hadbeen one of the few to voice premonitions.Then again in Strasbourg in 1945 after theabyss of war from which my own family wasrising safe and sound, but not his. Next inParis at the École Normale (rue d’Ulm),when he was the professor and I a student:the course for second year students, in whichhe taught us what a differentiable manifoldwas, and where, from the back of the lectureroom, an unknown person (it was AlexanderGrothendieck) ventured to dialog with himas an equal; the first Cartan Seminar onTopology (1948), for which, under his guid-ance, I presented “exposé no. 3” (thereafterentirely written up by him alone); and fre-quently in his family gathering, BoulevardJourdan, on Sunday, where I was a little like

one more child. Then as research advisor,who did not propose a thesis topic, but who,one day, pointed out to me the article ofFeldbau concerning homeomorphisms ofspheres, which became the starting point ofmy thesis.

Even before this last period, I was suffi-ciently close to him to occasionally gleanconfidential remarks such as the followingthat I remember concerning René Thom,whom he ‘discovered’ before anyone else inspite of their quite different ways of thinking:“Thom is a boy brimming with ideas, buthow hard it is to make him write themdown!” Then, over a long period: his battles,mostly victorious, to, as he formulated it,“raise the level of the Sorbonne inMathematics”; his vain efforts to have a chaircreated at the Collège de France for AndréWeil. His precocious commitment in favourof Franco-German reconciliation – in a peri-od when that demanded lofty vision, a utopi-an vision whose realization is today one ofour grounds for hope in the face of seeming-ly insoluble conflicts. His militance in thedefence of Human Rights everywhere in theworld, and for the construction of Europe asa federal state, an aim to which he attachessuch high value.

Jean-Pierre Serre has written: “I believethat the Cartan style is the finest to be foundin mathematics”.

I would add that the Cartan style is thefinest to be found in life in general. Thankyou, Monsieur Cartan, for showing, by yourexample, that it is possible to become withage ever more humane.

Jean Cerf has been Professeur and Directeurde Recherche (CNRS) at Université Paris-Sud (Orsay), France.

ANNIVERSARY


On July 8, 2004, the outstanding French mathematician Henri Cartan turned onehundred years old. The French Mathematical Society held a seminar in honour ofHenri Cartan on June 28. Its programme, some of the contributions, a biography,several photos and documents can be found on the web sitehttp://smf.emath.fr/VieSociete/Rencontres/JourneeCartan. The 100th issue ofGazette des Mathématiciens, the news publication of Société Mathématique deFrance, carried some tributes to Cartan, two of which are reproduced in this issueof the Newsletter along with a resolution from the International MathematicalUnion (IMU). At the closing ceremony of the 4th European congress of mathe-maticians, EMS-president Sir John Kingman sent his cordial congratulations toProf. Cartan in the name of all the participants.

100th Bir100th Birthday of thday of Henri CarHenri Cartantan

Personal souvenirs of Henri CartanPierre Samuel (Université de Paris-

Sud, Orsay, France)

This article appeared (in French) inGazette des Mathématiciens 100, April2004, p. 13-14. The Newsletter thanks theeditor of gazette for the permission toreproduce it and the author for providingthe translation into English.

My first meeting with Henri Cartan tookplace in August 1940. I was a candidate atthe Ecole Normale Supérieure and he wasone of the examiners. Classmates, whohad already taken the examination, toldme that he was a “nice” examiner asopposed to some examiners at the EcolePolytechnique who tried to destabilize thecandidates. In fact, Henri Cartan wanted tomake the candidates reveal as much aspossible in order to find out which oneswere the most promising. I noticed thesame quality when, much later, we sattogether in doctorates juries. The goodquestions he put to the candidates enabledhim to predict from their answers whichcandidates would become first-rate mathe-maticians. He was never wrong.

I met Henri Cartan again in 1944-1945at the Ecole Normale Supérieure, when Iwas preparing for the Agrégation deMathématiques. At that time, the oral partof this competitive examination consistedin preparing (without documents) and giv-ing lectures on classical subjects (likeEuclidian Geometry, Analytic Geometry,Calculus) in front of the jury – just like infront of a high school class. We practicedlecturing in front of Henri Cartan. His crit-icisms were incisive and to the point andhis suggestions were very valuable. Evenat this elementary level, we learned a lot ofdeep Mathematics from him.

I had written to Bourbaki, pointing outmistakes in the exercises of the publishedvolumes, so he took an interest in me andinvited René Thom and myself as “guineapigs” to a Bourbaki congress, which tookplace in July 1945 at the Ecole NormaleSupérieure. This was one of the greatexperiences in my life.

From 1945 to 1947, I got a research fel-lowship at Princeton University and Ireturned with a research project on inter-section multiplicities in AlgebraicGeometry, mostly inspired by ClaudeChevalley. When this work was completedin 1949, I presented it as a thesis to theUniversité de Paris. I asked Henri Cartanto be in the jury by giving me the subjectof the so-called “seconde thèse” (in whichthe candidate studies and gives an outline

of a topic which has been the subject ofrecent research, but in a field distinct fromthe field of the thesis itself). Cartan chosethe relations between homology andhomotopy and spent a lot of time pointingout the articles to be read and making surethat I was mastering this topic that wasnew to me.

In the meantime, I became full-fledgedmember of Bourbaki. During theCongresses, I admired the acuteness ofCartan’s comments and his mastery ofmost branches of Mathematics. If one ofhis proposals was not accepted immediate-ly, he tried hard to convince other mem-bers during the recesses; either they wereor the proposals got improved. When trav-elling by train to the Congresses, he also

made us share his enthusiasm for recentdiscoveries or trends, e.g. the notion of afunctor and various topics from S.Eilenberg and his book “Homologicalalgebra”.

In spite of the fact that our domains ofresearch were quite distinct, our friendshipand our common concerns intensified.Gradually, and especially when I becamean environmental activist, he made meshare his enthusiasm for the unification ofEurope.

I also admired his ceaseless actions forpersecuted mathematicians. During theMay 1968 movement, we were both opento some demands of the students, e.g. theinclusion of personal studies in the cur-riculum. In 1970, we both successfullyrequested to be transferred to the newUniversité de Paris-Sud (Orsay). There heworked very hard to embody statutes pro-viding a good balance between teachingand research.

Other mathematicians, more competentthan me, will surely describe his results onfunctions of several complex variables,sheaves, algebraic topology, homologicalalgebra, etc. I just know that they are fun-damental and that the “SéminairesCartan”, the topics of which were chosenby him each year with a remarkable intu-ition of what would be important, awak-ened the vocations of many first-ratemathematicians.

Pierre Samuel is Professeur émérité atUniversité Paris-Sud (Orsay), France.

ANNIVERSARY


On the Occasion of the 100th Birthday ofHenri Cartan

It is a great honour for the International Mathematical Union to associate itself withthe hundredth birthday of Henri Cartan, on July 8, 2004.

The son of the great mathematician Elie Cartan, his contributions to mathematicshave been fundamental, from several complex variables to algebraic topology andhomological algebra. A member of the Bourbaki group, his participation in the reju-venation of the French mathematical school was essential, in particular through hisseminar held at the École Normale Supérieure. His roles as teacher and mentor werealso exceptional, and were felt well beyond national boundaries.

During the critical years after the second world war, Cartan's enduring friendshipwith the German mathematician Heinrich Benhke, and his own personal generosity,contributed greatly to the rebirth of German mathematics. He was made an honorarymember of the German Mathematical Society (DMV) in 1994.

His natural preoccupation with international cooperation led to his active involve-ment with the International Mathematical Union, of which he was President from1967 to 1970. As such he chaired the Fields Medal Committee for the NiceInternational Congress of Mathematicians in 1970.

He has been actively involved in the defense of mathematicians who were jailed ordiscriminated against in their countries, and is an ardent defender of European unity.

Mathematicians worldwide unite with respect and admiration in warmly congratu-lating Henri Cartan, great mathematician and great man, on this happy occasion.

International Mathematical Union

Nuno Crato is a mathematician and awell-known science writer in his nativePortugal. He is frequently present in pub-lic discussions related to research, educa-tion, and science popularization. He regu-larly writes for the press and oftenappears in radio and television programs.He was recently elected President of thePortuguese Mathematical Society (SPM),and took charge this September.

Last year, he won the First Prize of theEMS competition Raising PublicAwareness (RPA) on Mathematics. Thelast EMS Newsletter, issue 52, reproducedpart of the three-article series on cryptog-raphy he submitted to the competition.English and Portuguese versions of thesearticles are now available online athttp://pascal.iseg.utl.pt/~ncrato/EMS. Hun-garian and Italian versions will soon beavailable as well.

Chaired by Vagn Lundsgaard Hansenand formed by eight mathematicians froman equal number of countries, the RPAcommittee selected Crato’s articles among26 proposals from 14 different countries.

José Francisco Rodrigues, who has beencollaborating with the EMS since its foun-dation (at which he represented thePortuguese Mathematical Society), con-ducted the conversation that follows.

Let me first congratulate you, Nuno, onthe well-deserved prize. I have beenreading your articles in the Portuguesepress and one thing that surprises me isthe way you chose the topics. They arevery diverse and very often are notrelated to your research. Is this true?Thanks, José Francisco. My research is inStochastic Processes and applications,mainly now on long-memory time series, akind of generalization of fractal Brownianmotion. This is necessarily a restrictedarea and only once I wrote about it to thegeneral public. But I am usually interestedin many other subjects, as you are and aseverybody who likes mathematics and sci-ence usually is. What I find is that I can

read an article in, say, Science, or Annalsof Statistics, or even Scientific American,faster than most interested readers withouta scientific background. As I read journalsand magazines very often, just out of mysimple curiosity, I very often indeed comeacross simple ideas on topics. Then I think“I’d like to explain this to someone”. It’s anatural thing for a teacher. Unfortunately,I can’t go to class and say “today, insteadof power series, let me explain you howthese guys from Scotland are assessing thenumber of stars in our galaxy using anovel simulation algorithm”…

Then, you decide to write an article onit…More or less... But I have to do someresearch on the topic, think about the wayto explain it, restrain myself to a limitedspace, and wait for the right moment…

Couldn’t you write immediately?Not usually. It seems to be better to waitfor a moment when the topic gains actual-ity and becomes interesting for the largepublic. For example, going back to thearticles on the competition, I have beenseduced by the marvels of modern cryp-tography for a while. I have read DavidKahn’s and Simon Singh’s books and anumber of expository papers and articles.But I waited. Suddenly, there was a publicdebate in Portugal on the use of creditcards and electronic purchasing. Somebanks even created a special debit cardthat could be used as a credit card, but onlyfor electronic transactions. They were try-ing to dispel the fears on the public. It wasthe ideal moment to talk about cryptogra-phy. If you read my first article, you willnotice it starts precisely discussing thesefears. I wrote it this way to entice the read-

er. If he or she is hooked with the first sen-tences, there is a chance the article will beread. If not…

So you try to bring people into scienceusing ordinary subjects as a pretext. Doyou consider this work as science popu-larization or science vulgarization?Maybe it’s both. But I can’t give generalrules. Sometimes it works the way Idescribed. Sometimes it is different. Somepeople explain mathematics and science ina straight forward manner; other peopleuse pretexts, as I usually try to do; otherpeople write only chronicles, that is, com-mentaries on the public scene that aremade from a special point of view (in ourcase, science or math). Everything is use-ful, when it is done seriously and serves toexpand people awareness of science, Ithink.

Why do you think popularization ofmathematics is useful? Some people sayit is useless, since the public cannot beeducated with light articles that onlyscratch the surface of things.I’m sorry: I completely disagree with thisidea. We can’t confuse formal educationwith journalism, and science writing isakin to journalism. Popularization ofmathematics brings our discipline to pub-lic attention. That’s it! If some people areenticed by an article and read further,that’s fantastic. But if most people onlyread it and get a general idea, that’s alsofine. I have nothing against it.

Now, a completely different problem isthe rigor we put in the writing. We can’tsay mistakes and we are always treadingdangerous grounds. We have to writethings in such a way people are not boredand, at the same time, avoid making

INTERVIEW


BRINGING MATHTO THE PUBLIC

FRONTLINEInterview with Nuno Crato, winner ofthe 2003 ‘Raising Public Awareness ofMathematics’ article competition of the

EMS, conducted by José FranciscoRodrigues (Lisbon, Portugal)

N. Crato (left) and J.F. Rodrigues (right)

errors. How do we do it? I can’t give gen-eral rules. We just have to be careful andimaginative.

Let me just give an example. If you aretalking about something that grows fast,you can’t say: “it’s an exponentialgrowth”, unless it really is. A politician,though, could say it, since “exponentialgrowth” is now a common phrase. But ifyou are a mathematician or a scientist—people read you in a different way. If youdon’t want to go into details and explainthe difference between polynomial andexponential growth, you can just say: “it isa fast growth”.

Let me insist: do you think it is possibleto popularize mathematics and sciencewithout vulgarizing it?Many mathematicians and scientists areafraid of vulgarization, I know, becausethey identify it with oversimplification. Idon’t like to discuss words, but I havenothing against simplification, that’s whatwe are always doing in life. I condemnerrors and try to avoid mistakes, which isanother question.

But don’t you think that, simplifyingmatters, the popularization of sciencefavors a wrong idea about scientistswork?It may and may not. It all depends on theway things are done. You can say “on adark night, he suddenly had the idea…” oryou can say “after many discussions andmany days thinking about the problem, hecame to the idea…” My impression,though, is that the public has already a ter-ribly wrong idea about the way scienceworks. Everything you can do to clarifythings is positive.

You didn’t talk yet about one often citedbenefit of mathematics popularization,which is the appreciation of mathemat-ics by the public. How would you rate

this benefit?It’s always difficult to do ratings, but thisbenefit is certainly important. I would adda couple of others, putting in first place theappreciation of the essential ethic of mathand science: the intellectual honesty, thecritical rationalism, the respect for thereality, the international cooperation, theeffort to avoid prejudices.

In general, I would say mathematics andscience are an essential part of our cultureand they deserve to be on the forefront ofpublic life. This is important for the sup-port of our efforts. And, more importantly,for the creation of a general culture thatrespects mathematics and science and triesto educate the citizens accordingly.

You have recently been electedPresident of SPM—SociedadePortuguesa de Matemática. Do youthink our societies should give moreattention to scientific popularization?I can’t speak in general. I believe scientif-ic societies should promote research, edu-cation, and popularization. All these goalsare important, but not all societies can pro-mote them equally.

… as we all should?I believe scientific organizations shouldnot forget their role in society, whichincludes popularization of science. Butthis is a goal for organizations, not forindividuals. I think it would be a terriblemistake to try to involve everybody inpopularization. Some people may like todo it and may have some aptitude for it.Other people not. The basic thing isresearch. Teaching comes next.Popularization is at the end of the list.

Isn’t it possible to do all three?It’s very difficult, and one activity alwaysharms the other. Very few people are likeIan Stewart, who apparently can write agreat book a year, a research paper amonth, and still be a dedicated teacher. Formost of us, to do science writing necessar-ily harms research and teaching. It’s truethat these activities are also complemen-tary. Sometimes, research helps us withideas for writing to the public, and teach-ing can give us ideas on ways to explainthings. But I never noticed a good researchidea coming out a good effort on popular-ization.

So you think popularization should beleft to journalists?Popularization should be done by peoplewho know what they are talking about andwho know how to talk about it. Some jour-

nalists have a basic scientific backgroundand the professionalism necessary to doscience popularization well. Ideally, wewould have plenty of good science jour-nalists. But we do not have so. And it isalso refreshing that different worlds anddifferent people communicate and, occa-sionally and for a while, even trade places.

What do you mean?Think about research and teaching: peoplewho do research can bring to teaching aninsider’s view and an insight on mathproblems that other people usually can’t.

The same way, professional mathemati-cians can bring to math popularization arigor and a point of view no good journal-ist can. So, I believe it’s useful for sciencejournalism that some of us, once a while,practice science popularization. The morepeople talk about math and sciences on thenewspapers and the better they do it, thebetter math and science are appreciated bysociety. And the better society appreciatesscience, the better education is. Then,more resources and more people come tomathematics and sciences. And this isgood for everybody.

José Francisco Rodrigues [[email protected]] is professor ofMathematics at the Faculty of Science ofthe University of Lisbon, where he direct-ed the Centro de Matemática e AplicaçõesFundamentais for several years until2002. His research interests are in thefield of nonlinear PDEs and their applica-tions, in particular, in free boundary prob-lems. He is currently collaborating withthe Centro de Matemática of theUniversity of Coimbra, also in Portugal.He co-organized the 1999 DiderotMathematical Forum on “Mathematicsand Music” and is a member of the EMSRaising Public Awareness of MathematicsCommittee. He is a main editor of“Interface and Free Boundaries”, an EMSjournal.

INTERVIEW


Illustration from the prize-winning articles

Illustration from the prize-winning articles

The interview took place in Oslo, on the24th of May 2004, prior to the Abel prizecelebrations.

First, we congratulate both of you forhaving been awarded the Abel Prize 2004.This prize has been given to you for “thediscovery and the proof of the IndexTheorem connecting geometry and analy-sis in a surprising way and your outstand-ing role in building new bridges betweenmathematics and theoretical physics”.Both of you have an impressive list of fineachievements in mathematics. Is theIndex Theorem your most importantresult and the result you are most pleasedwith in your entire careers?ATIYAH First, I would like to say that Iprefer to call it a theory, not a theorem.Actually, we have worked on it for 25 yearsand if I include all the related topics, I haveprobably spent 30 years of my life workingon the area. So it is rather obvious that it isthe best thing I have done.SINGER I too, feel that the index theoremwas but the beginning of a high point thathas lasted to this very day. It’s as if weclimbed a mountain and found a plateauwe’ve been on ever since.

We would like you to give us some com-ments on the history of the discovery ofthe Index Theorem.1 Were there precur-sors, conjectures in this direction alreadybefore you started? Were there onlymathematical motivations or also physicalones? ATIYAH Mathematics is always a contin-uum, linked to its history, the past - nothingcomes out of zero. And certainly the IndexTheorem is simply a continuation of workthat, I would like to say, began with Abel.So of course there are precursors. A theo-rem is never arrived at in the way that log-ical thought would lead you to believe orthat posterity thinks. It is usually muchmore accidental, some chance discovery inanswer to some kind of question.Eventually you can rationalize it and saythat this is how it fits. Discoveries neverhappen as neatly as that. You can rewritehistory and make it look much more logi-

The Index Theorem

cal, but actually it happens quite different-ly.SINGER At the time we proved the IndexTheorem we saw how important it was inmathematics, but we had no inkling that itwould have such an effect on physics someyears down the road. That came as a com-plete surprise to us. Perhaps it should nothave been a surprise because it used a lot ofgeometry and also quantum mechanics in away, à la Dirac.

You worked out at least three differentproofs with different strategies for theIndex Theorem. Why did you keep onafter the first proof? What differentinsights did the proofs give?ATIYAH I think it is said that Gauss hadten different proofs for the law of quadrat-ic reciprocity. Any good theorem shouldhave several proofs, the more the better.For two reasons: usually, different proofshave different strengths and weaknesses,and they generalize in different directions -they are not just repetitions of each other.And that is certainly the case with theproofs that we came up with. There are dif-ferent reasons for the proofs, they have dif-ferent histories and backgrounds. Some ofthem are good for this application, someare good for that application. They all shedlight on the area. If you cannot look at aproblem from different directions, it isprobably not very interesting; the more per-spectives, the better!SINGER There isn’t just one theorem;

there are generalizations of the theorem.One is the families index theorem using K-theory; another is the heat equation proofwhich makes the formulas that are topolog-ical, more geometric and explicit. Eachtheorem and proof has merit and has differ-ent applications.

Both of you contributed to the index theo-rem with different expertise and visions –and other people had a share as well, Isuppose. Could you describe this collabo-ration and the establishment of the resulta little closer?SINGER Well, I came with a backgroundin analysis and differential geometry, andSir Michael’s expertise was in algebraicgeometry and topology. For the purposesof the Index Theorem, our areas of exper-tise fit together hand in glove. Moreover,in a way, our personalities fit together, inthat “anything goes”: Make a suggestion -and whatever it was, we would just put iton the blackboard and work with it; wewould both enthusiastically explore it; if itdidn’t work, it didn’t work. But oftenenough, some idea that seemed far-fetcheddid work. We both had the freedom to con-tinue without worrying about where itcame from or where it would lead. It wasexciting to work with Sir Michael all theseyears. And it is as true today as it waswhen we first met in ’55 - that sense ofexcitement and “anything goes” and “let’s

Collaboration

INTERVIEW


Interview withInterview withMichael Michael Atiyah and Isadore SingerAtiyah and Isadore Singer

Interviewers: Martin Raussen and Christian Skau

from left to right: I. Singer, M. Atiyah, M. Raussen, C. Skau

see what happens”.ATIYAH No doubt: Singer had a strongexpertise and background in analysis anddifferential geometry. And he knew cer-tainly more physics than I did; it turned outto be very useful later on. My backgroundwas in algebraic geometry and topology, soit all came together. But of course there area lot of people who contributed in the back-ground to the build-up of the IndexTheorem – going back to Abel, Riemann,much more recently Serre, who got theAbel prize last year, Hirzebruch,Grothendieck and Bott. There was lots ofwork from the algebraic geometry side andfrom topology that prepared the ground.And of course there are also a lot of peoplewho did fundamental work in analysis andthe study of differential equations:Hörmander, Nirenberg… In my lecture Iwill give a long list of names2; even thatone will be partial. It is an example ofinternational collaboration; you do notwork in isolation, neither in terms of timenor in terms of space – especially in thesedays. Mathematicians are linked so much,people travel around much more. We twomet at the Institute at Princeton. It was niceto go to the Arbeitstagung in Bonn everyyear, which Hirzebruch organised andwhere many of these other people came. Idid not realize that at the time, but lookingback, I am very surprised how quicklythese ideas moved…

Collaboration seems to play a bigger rolein mathematics than earlier. There are alot of conferences, we see more papersthat are written by two, three or even moreauthors – is that a necessary and com-mendable development or has it draw-backs as well?ATIYAH It is not like in physics or chem-istry where you have 15 authors becausethey need an enormous big machine. It isnot absolutely necessary or fundamental.But particularly if you are dealing withareas which have rather mixed and inter-disciplinary backgrounds, with people whohave different expertise, it is much easierand faster. It is also much more interestingfor the participants. To be a mathematicianon your own in your office can be a little bitdull, so interaction is stimulating, both psy-chologically and mathematically. It has tobe admitted that there are times when yougo solitary in your office, but not all thetime! It can also be a social activity withlots of interaction. You need a good mix ofboth, you can’t be talking all the time. Buttalking some of the time is very stimulat-ing. Summing up, I think that it is a gooddevelopment – I do not see any drawbacks.

SINGER Certainly computers have madecollaboration much easier. Many mathe-maticians collaborate by computer instant-ly; it’s as if they were talking to each other.I am unable to do that. A sobering coun-terexample to this whole trend isPerelman’s results on the Poincaré conjec-ture: He worked alone for ten to twelveyears, I think, before putting his preprintson the net. ATIYAH Fortunately, there are many dif-ferent kinds of mathematicians, they workon different subjects, they have differentapproaches and different personalities –and that is a good thing. We do not wantall mathematicians to be isomorphic, wewant variety: different mountains need dif-ferent kinds of techniques to climb.SINGER I support that. Flexibility isabsolutely essential in our society of math-ematicians.

Perelman’s work on the Poincaré conjec-ture seems to be another instance whereanalysis and geometry apparently getlinked very much together. It seems thatgeometry is profiting a lot from analyticperspectives. Is this linkage between dif-ferent disciplines a general trend – is ittrue, that important results rely on thisinterrelation between different disci-plines? And a much more specific ques-tion: What do you know about the statusof the proof of the Poincaré conjecture?SINGER To date, everything is workingout as Perelman says. So I learn fromLott’s seminar at the University ofMichigan and Tian’s seminar at Princeton.Although no one vouches for the finaldetails, it appears that Perelman’s proofwill be validated.

As to your first question: When any twosubjects use each other’s techniques in anew way, frequently, something specialhappens. In geometry, analysis is veryimportant; for existence theorems, themore the better. It is not surprising thatsome new [at least to me] analysis impliessomething interesting about the Poincaréconjecture.ATIYAH I prefer to go even further – Ireally do not believe in the division ofmathematics into specialities; already ifyou go back into the past, to Newton andGauss… Although there have been times,particularly post-Hilbert, with the axiomat-ic approach to mathematics in the first halfof the twentieth century, when peoplebegan to specialize, to divide up. TheBourbaki trend had its use for a particulartime. But this is not part of the general atti-tude to mathematics: Abel would not havedistinguished between algebra and analy-

sis. And I think the same goes for geome-try and analysis for people like Newton.

It is artificial to divide mathematics intoseparate chunks, and then to say that youbring them together as though this is a sur-prise. On the contrary, they are all part ofthe puzzle of mathematics. Sometimes youwould develop some things for their ownsake for a while e.g. if you develop grouptheory by itself. But that is just a sort oftemporary convenient division of labour.Fundamentally, mathematics should beused as a unity. I think the more exampleswe have of people showing that you canusefully apply analysis to geometry, thebetter. And not just analysis, I think thatsome physics came into it as well: Many ofthe ideas in geometry use physical insightas well – take the example of Riemann!This is all part of the broad mathematicaltradition, which sometimes is in danger ofbeing overlooked by modern, younger peo-ple who say “we have separate divisions”.We do not want to have any of that kind,really.SINGER The Index Theorem was in factinstrumental in breaking barriers betweenfields. When it first appeared, many old-timers in special fields were upset that newtechniques were entering their fields andachieving things they could not do in thefield by old methods. A younger genera-tion immediately felt freed from the barri-ers that we both view as artificial.ATIYAH Let me tell you a little storyabout Henry Whitehead, the topologist. Iremember that he told me that he enjoyedvery much being a topologist: He had somany friends within topology, and it wassuch a great community. “It would be atragedy if one day I would have a brilliantidea within functional analysis and wouldhave to leave all my topology friends andto go out and work with a different groupof people.” He regarded it to be his duty todo so, but he would be very reluctant.

Somehow, we have been very fortunate.Things have moved in such a way that wegot involved with functional analysts with-out losing our old friends; we could bringthem all with us. Alain Connes was infunctional analysis, and now we interactclosely. So we have been fortunate tomaintain our old links and move into newones – it has been great fun.

We would like to have your comments onthe interplay between physics and mathe-matics. There is Galilei’s famous dictumfrom the beginning of the scientific revo-

Mathematics and physics

INTERVIEW


lution, which says that the Laws of Natureare written in the language of mathemat-ics. Why is it that the objects of mathe-matical creation, satisfying the criteria ofbeauty and simplicity, are precisely theones that time and time again are found tobe essential for a correct description of theexternal world? Examples abound, let mejust mention group theory and, yes, yourIndex Theorem!SINGER There are several approaches inanswer to your questions; I will discusstwo. First, some parts of mathematics werecreated in order to describe the worldaround us. Calculus began by explainingthe motion of planets and other movingobjects. Calculus, differential equations,and integral equations are a natural part ofphysics because they were developed forphysics. Other parts of mathematics arealso natural for physics. I remember lec-turing in Feynman’s seminar, trying toexplain anomalies. His postdocs keptwanting to pick coordinates in order tocompute; he stopped them saying: “TheLaws of Physics are independent of a coor-dinate system. Listen to what Singer has tosay, because he is describing the situationwithout coordinates.” Coordinate-freemeans geometry. It is natural that geome-try appears in physics, whose laws areindependent of a coordinate system.

Symmetries are useful in physics formuch the same reason they’re useful inmathematics. Beauty aside, symmetriessimplify equations, in physics and in math-ematics. So physics and math have in com-mon geometry and group theory, creating aclose connection between parts of bothsubjects.

Secondly, there is a deeper reason if yourquestion is interpreted as in the title ofEugene Wigner’s essay “TheUnreasonable Effectiveness ofMathematics in the Natural Sciences”3.Mathematics studies coherent systemswhich I will not try to define. But it stud-ies coherent systems, the connectionsbetween such systems and the structure ofsuch systems. We should not be too sur-prised that mathematics has coherent sys-tems applicable to physics. It remains to beseen whether there is an already developedcoherent system in mathematics that willdescribe the structure of string theory. [Atpresent, we do not even know what thesymmetry group of string field theory is.]Witten has said that 21st century mathe-matics has to develop new mathematics,perhaps in conjunction with physics intu-ition, to describe the structure of string the-ory. ATIYAH I agree with Singer’s description

of mathematics having evolved out of thephysical world; it therefore is not a big sur-prise that it has a feedback into it. More fundamentally: to understand theoutside world as a human being is anattempt to reduce complexity to simplicity.What is a theory? A lot of things are hap-pening in the outside world, and the aim ofscientific inquiry is to reduce this to as sim-ple a number of principles as possible.That is the way the human mind works, theway the human mind wants to see theanswer.

If we were computers, which could tabu-late vast amounts of all sorts of informa-tion, we would never develop theory – wewould say, just press the button to get theanswer. We want to reduce this complexi-ty to a form that the human mind canunderstand, to a few simple principles.That’s the nature of scientific inquiry, andmathematics is a part of that. Mathematicsis an evolution from the human brain,which is responding to outside influences,creating the machinery with which it thenattacks the outside world. It is our way oftrying to reduce complexity into simplicity,beauty and elegance. It is really very fun-damental, simplicity is in the nature of sci-entific inquiry – we do not look for com-plicated things.

I tend to think that science and mathe-matics are ways the human mind looks andexperiences – you cannot divorce thehuman mind from it. Mathematics is partof the human mind. The question whetherthere is a reality independent of the humanmind, has no meaning – at least, we cannotanswer it.

Is it too strong to say that the mathemati-cal problems solved and the techniquesthat arose from physics have been thelifeblood of mathematics in the past; or atleast for the last 25 years?ATIYAH I think you could turn that intoan even stronger statement. Almost allmathematics originally arose from externalreality, even numbers and counting. Atsome point, mathematics then turned to askinternal questions, e.g. the theory of primenumbers, which is not directly related toexperience but evolved out of it.

There are parts of mathematics where thehuman mind asks internal questions justout of curiosity. Originally it may be phys-ical, but eventually it becomes somethingindependent. There are other parts thatrelate much closer to the outside worldwith much more interaction backwards andforward. In that part of it, physics has for along time been the lifeblood of mathemat-ics and inspiration for mathematical work.

There are times when this goes out of fash-ion or when parts of mathematics evolvepurely internally. Lots of abstract mathe-matics does not directly relate to the out-side world. It is one of the strengths of mathematicsthat it has these two and not a singlelifeblood: one external and one internal,one arising as response to external events,the other to internal reflection on what weare doing. SINGER Your statement is too strong. Iagree with Michael that mathematics isblessed with both an external and internalsource of inspiration. In the past severaldecades, high energy theoretical physicshas had a marked influence on mathemat-ics. Many mathematicians have beenshocked at this unexpected development:new ideas from outside mathematics soeffective in mathematics. We are delightedwith these new inputs, but the “shock”exaggerates their overall effect on mathe-matics.

Can we move to newer developments withimpact from the Atiyah-Singer IndexTheorem? I.e., String Theory andEdward Witten on the one hand and onthe other hand Non-commutativeGeometry represented by Alain Connes.Could you describe the approaches tomathematical physics epitomized by thesetwo protagonists?ATIYAH I tried once in a talk to describethe different approaches to progress inphysics like different religions. You haveprophets, you have followers – eachprophet and his followers think that theyhave the sole possession of the truth. If youtake the strict point of view that there areseveral different religions, and that theintersection of all these theories is empty,then they are all talking nonsense. Or youcan take the view of the mystic, who thinksthat they are all talking of different aspectsof reality, and so all of them are correct. Itend to take the second point of view. Themain “orthodox” view among physicists iscertainly represented by a very large groupof people working with string theory likeEdward Witten. There are a small numberof people who have different philosophies,one of them is Alain Connes, and the otheris Roger Penrose. Each of them has a veryspecific point of view; each of them hasvery interesting ideas. Within the last fewyears, there has been non-trivial interactionbetween all of these.

They may all represent different aspectsof reality and eventually, when we under

Newer developments

INTERVIEW


stand it all, we may say “Ah, yes, they areall part of the truth”. I think that that willhappen. It is difficult to say which will bedominant, when we finally understand thepicture – we don’t know. But I tend to beopen-minded. The problem with a lot ofphysicists is that they have a tendency to“follow the leader”: as soon as a new ideacomes up, ten people write ten or morepapers on it and the effect is that everythingcan move very fast in a technical direction.But big progress may come from a differ-ent direction; you do need people who areexploring different avenues. And it is verygood that we have people like Connes andPenrose with their own independent linefrom different origins. I am in favour ofdiversity. I prefer not to close the door orto say “they are just talking nonsense”.SINGER String Theory is in a very specialsituation at the present time. Physicistshave found new solutions on their land-scape - so many that you cannot expect tomake predictions from String Theory. Itsoriginal promise has not been fulfilled.Nevertheless, I am an enthusiastic support-er of Super String Theory, not just becauseof what it has done in mathematics, but alsobecause as a coherent whole, it is a marvel-lous subject. Every few years new devel-opments in the theory give additionalinsight. When that happens, you realizehow little one understood about StringTheory previously. The theory of D-branesis a recent example. Often there is mathe-matics closely associated with these newinsights. Through D-branes, K-theoryentered String Theory naturally andreshaped it. We just have to wait and seewhat will happen. I am quite confident thatphysics will come up with some new ideas

in String Theory that will give us greaterinsight into the structure of the subject, andalong with that will come new uses ofmathematics.

Alain Connes’ program is very natural –if you want to combine geometry withquantum mechanics, then you really wantto quantize geometry, and that is what non-commutative geometry means. Non-com-mutative Geometry has been used effec-tively in various parts of String Theoryexplaining what happens at certain singu-larities, for example. I think it may be aninteresting way of trying to describe blackholes and to explain the Big Bang. I wouldencourage young physicists to understandnon-commutative geometry more deeplythan they presently do. Physicists use onlyparts of non-commutative geometry; thetheory has much more to offer. I do notknow whether it is going to lead anywhereor not. But one of my projects is to try andredo some known results using non-com-mutative geometry more fully.

If you should venture a guess, whichmathematical areas do you think aregoing to witness the most important devel-opments in the coming years?ATIYAH One quick answer is that themost exciting developments are the oneswhich you cannot predict. If you can pre-dict them, they are not so exciting. So, bydefinition, your question has no answer.

Ideas from physics, e.g. QuantumTheory, have had an enormous impact sofar, in geometry, some parts of algebra, andin topology. The impact on number theoryhas still been quite small, but there aresome examples. I would like to make arash prediction that it will have a big

impact on number theory as the ideas flowacross mathematics – on one extreme num-ber theory, on the other physics, and in themiddle geometry: the wind is blowing, andit will eventually reach to the farthestextremities of number theory and give us anew point of view. Many problems that areworked upon today with old-fashionedideas will be done with new ideas. I wouldlike to see this happen: it could be theRiemann hypothesis, it could be theLanglands program or a lot of other relatedthings. I had an argument with AndrewWiles where I claimed that physics willhave an impact on his kind of number the-ory; he thinks this is nonsense but we had agood argument.

I would also like to make another predic-tion, namely that fundamental progress onthe physics/mathematics front, StringTheory questions etc., will emerge from amuch more thorough understanding ofclassical four-dimensional geometry, ofEinstein’s Equations etc. The hard part ofphysics in some sense is the non-linearityof Einstein’s Equations. Everything thathas been done at the moment is circum-venting this problem in lots of ways. Theyhaven’t really got to grips with the hardestpart. Big progress will come when peopleby some new techniques or new ideas real-ly settle that. Whether you call that geom-etry, differential equations or physicsdepends on what is going to happen, but itcould be one of the big breakthroughs.

These are of course just my speculations.SINGER I will be speculative in a slightlydifferent way, though I do agree with thenumber theory comments that Sir Michaelmentioned, particularly theta functionsentering from physics in new ways. I thinkother fields of physics will affect mathe-matics - like statistical mechanics and con-densed matter physics. For example, I pre-dict a new subject of statistical topology.Rather than count the number of holes,Betti-numbers, etc., one will be more inter-ested in the distribution of such objects onnoncompact manifolds as one goes out toinfinity. We already have precursors in thenumber of zeros and poles for holomorphicfunctions. The theory that we have forholomorphic functions will be generalized,and insights will come from condensedmatter physics as to what, statistically, thetopology might look like as one approach-es infinity.

Mathematics has become so specialized, itseems, that one may fear that the subjectwill break up into separate areas. Is there

Continuity of mathematics

INTERVIEW


Isadore Singer and Sir Michael Atiyah receive theAbelprize from King Harald. (Photo: Scanpix)

a core holding things together?ATIYAH I like to think there is a coreholding things together, and that the core israther what I look at myself; but we tend tobe rather egocentric. The traditional partsof mathematics, which evolved - geometry,calculus and algebra - all centre on certainnotions. As mathematics develops, thereare new ideas, which appear to be far fromthe centre going off in different directions,which I perhaps do not know much about.Sometimes they become rather importantfor the whole nature of the mathematicalenterprise. It is a bit dangerous to restrictthe definition to just whatever you happento understand yourself or think about. Forexample, there are parts of mathematicsthat are very combinatorial. Sometimesthey are very closely related to the continu-ous setting, and that is very good: we haveinteresting links between combinatoricsand algebraic geometry and so on. Theymay also be related to e.g. statistics. I thinkthat mathematics is very difficult to con-strain; there are also all sorts of new appli-cations in different directions.

It is nice to think of mathematics havinga unity; however, you do not want it to bea straitjacket. The centre of gravity maychange with time. It is not necessarily afixed rigid object in that sense, I think itshould develop and grow. I like to think ofmathematics having a core, but I do notwant it to be rigidly defined so that itexcludes things which might be interesting.You do not want to exclude somebody whohas made a discovery saying: “You are out-side, you are not doing mathematics, youare playing around”. You never know!That particular discovery might be themathematics of the next century; you havegot to be careful. Very often, when newideas come in, they are regarded as being abit odd, not really central, because theylook too abstract.SINGER Countries differ in their attitudesabout the degree of specialization in math-ematics and how to treat the problem of toomuch specialization. In the United States Iobserve a trend towards early specializa-tion driven by economic considerations.You must show early promise to get goodletters of recommendations to get good firstjobs. You can’t afford to branch out untilyou have established yourself and have asecure position. The realities of life force anarrowness in perspective that is not inher-ent to mathematics. We can counter toomuch specialization with new resourcesthat would give young people more free-dom than they presently have, freedom toexplore mathematics more broadly, or toexplore connections with other subjects,

like biology these days where there is lotsto be discovered.

When I was young the job market wasgood. It was important to be at a major uni-versity but you could still prosper at asmaller one. I am distressed by the coer-cive effect of today’s job market. Youngmathematicians should have the freedomof choice we had when we were young.

The next question concerns the continuityof mathematics. Rephrasing slightly aquestion that you, Prof. Atiyah are the ori-gin of, let us make the followinggedanken experiment: If, say, Newton orGauss or Abel were to reappear in ourmidst, do you think they would under-stand the problems being tackled by thepresent generation of mathematicians –after they had been given a short refresh-er course? Or is present day mathematicstoo far removed from traditional mathe-matics?ATIYAH The point that I was trying tomake there was that really importantprogress in mathematics is somewhat inde-pendent of technical jargon. Importantideas can be explained to a really goodmathematician, like Newton or Gauss orAbel, in conceptual terms. They are in factcoordinate-free, more than that, technolo-gy-free and in a sense jargon-free. Youdon’t have to talk of ideals, modules orwhatever – you can talk in the commonlanguage of scientists and mathematicians.The really important progress mathematicshas made within 200 years could easily beunderstood by people like Gauss andNewton and Abel. Only a small refreshercourse where they were told a few terms –and then they would immediately under-stand.

Actually, my pet aversion is that manymathematicians use too many technicalterms when they write and talk. They weretrained in a way that if you do not say it 100percent correctly, like lawyers, you will betaken to court. Every statement has to befully precise and correct. When talking toother people or scientists, I like to usewords that are common to the scientificcommunity, not necessarily just to mathe-maticians. And that is very often possible.If you explain ideas without a vast amountof technical jargon and formalism, I amsure it would not take Newton, Gauss andAbel long – they were bright guys, actual-ly!SINGER One of my teachers at Chicagowas André Weil, and I remember his say-ing: “If Riemann were here, I would puthim in the library for a week, and when hecame out he would tell us what to do next.”

Next topic: Communication of mathe-matics: Hilbert, in his famous speech atthe International Congress in 1900, inorder to make a point about mathemati-cal communication, cited a Frenchmathematician who said: “A mathemati-cal theory is not to be considered com-plete until you have made it so clear thatyou can explain it to the first man whomyou meet on the street”. In order to passon to new generations of mathemati-cians the collective knowledge of the pre-vious generation, how important is itthat the results have simple and elegantproofs?ATIYAH The passing of mathematics onto subsequent generations is essential forthe future, and this is only possible ifevery generation of mathematiciansunderstands what they are doing and dis-tils it out in such a form that it is easilyunderstood by the next generation. Manycomplicated things get simple when youhave the right point of view. The firstproof of something may be very compli-cated, but when you understand it well,you readdress it, and eventually you canpresent it in a way that makes it lookmuch more understandable – and that’sthe way you pass it on to the next genera-tion! Without that, we could never makeprogress - we would have all this messystuff. Mathematics does depend on a suf-ficiently good grasp, on understanding ofthe fundamentals so that we can pass it onin as simple a way as possible to our suc-cessors. That has been done remarkablysuccessfully for centuries. Otherwise,how could we possibly be where we are?In the 19th century, people said: “There isso much mathematics, how could anyonemake any progress?” Well, we have - wedo it by various devices, we generalize,we put all things together, we unify bynew ideas, we simplify lots of the con-structions – we are very successful inmathematics and have been so for severalhundred years. There is no evidence thatthis has stopped: in every new generation,there are mathematicians who make enor-mous progress. How do they learn it all?It must be because we have been success-ful communicating it.SINGER I find it disconcerting speakingto some of my young colleagues, becausethey have absorbed, reorganized, and sim-plified a great deal of known material intoa new language, much of which I don’tunderstand. Often I’ll finally say, “Oh; isthat all you meant?” Their new conceptu-

Communication of mathematics

INTERVIEW


al framework allows them to encompasssuccinctly considerably more than I canexpress with mine. Though impressedwith the progress, I must confess impa-tience because it takes me so long tounderstand what is really being said.

Has the time passed when deep andimportant theorems in mathematics canbe given short proofs? In the past, thereare many such examples, e.g., Abel’sone-page proof of the addition theoremof algebraic differentials or Goursat’sproof of Cauchy’s integral theorem.ATIYAH I do not think that at all! Ofcourse, that depends on what foundationsyou are allowed to start from. If we haveto start from the axioms of mathematics,then every proof will be very long. Thecommon framework at any given time isconstantly advancing; we are already at ahigh platform. If we are allowed to startwithin that framework, then at every stagethere are short proofs.

One example from my own life is thisfamous problem about vector fields onspheres solved by Frank Adams where theproof took many hundreds of pages. Oneday I discovered how to write a proof ona postcard. I sent it over to Frank Adamsand we wrote a little paper which thenwould fit on a bigger postcard. But ofcourse that used some K-theory; not thatcomplicated in itself. You are alwaysbuilding on a higher platform; you havealways got more tools at your disposalthat are part of the lingua franca whichyou can use. In the old days you had asmaller base: If you make a simple proofnowadays, then you are allowed toassume that people know what group the-ory is, you are allowed to talk aboutHilbert space. Hilbert space took a longtime to develop, so we have got a muchbigger vocabulary, and with that we canwrite more poetry.SINGER Often enough one can distil theideas in a complicated proof and makethat part of a new language. The newproof becomes simpler and more illumi-nating. For clarity and logic, parts of theoriginal proof have been set aside and dis-cussed separately.ATIYAH Take your example of Abel’sParis memoir: His contemporaries did notfind it at all easy. It laid the foundation ofthe theory. Only later on, in the light ofthat theory, we can all say: “Ah, what abeautifully simple proof!” At the time, allthe ideas had to be developed, and theywere hidden, and most people could notread that paper. It was very, very far fromappearing easy for his contemporaries.

I heard you, Prof. Atiyah, mention that onereason for your choice of mathematics foryour career was that it is not necessary toremember a lot of facts by heart.Nevertheless, a lot of threads have to bewoven together when new ideas are devel-oped. Could you tell us how you work best,how do new ideas arrive? ATIYAH My fundamental approach todoing research is always to ask questions.You ask “Why is this true?” when there issomething mysterious or if a proof seemsvery complicated. I used to say – as a kindof joke – that the best ideas come to you dur-ing a bad lecture. If somebody gives a terri-ble lecture, it may be a beautiful result butwith terrible proofs, you spend your timetrying to find better ones, you do not listento the lecture. It is all about asking ques-tions – you simply have to have an inquisi-tive mind! Out of ten questions, nine willlead nowhere, and one leads to somethingproductive. You constantly have to beinquisitive and be prepared to go in anydirection. If you go in new directions, thenyou have to learn new material.

Usually, if you ask a question or decide tosolve a problem, it has a background. If youunderstand where a problem comes fromthen it makes it easy for you to understandthe tools that have to be used on it. Youimmediately interpret them in terms of yourown context. When I was a student, Ilearned things by going to lectures and read-ing books – after that I read very few books.I would talk with people; I would learn theessence of analysis by talking to Hörmanderor other people. I would be asking ques-tions because I was interested in a particularproblem. So you learn new things becauseyou connect them and relate them to oldones, and in that way you can start to spreadaround.

If you come with a problem, and you needto move to a new area for its solution, thenyou have an introduction – you have alreadya point of view. Interacting with other peo-ple is of course essential: if you move into anew field, you have to learn the language,you talk with experts; they will distil theessentials out of their experience. I did notlearn all the things from the bottomupwards; I went to the top and got theinsight into how you think about analysis orwhatever.SINGER I seem to have some built-in senseof how things should be in mathematics. Ata lecture, or reading a paper, or during a dis-cussion, I frequently think, “that’s not theway it is supposed to be.” But when I try

Individual work styleout my ideas, I’m wrong 99% of the time. Ilearn from that and from studying the ideas,techniques, and procedures of successfulmethods. My stubbornness wastes lots oftime and energy. But on the rare occasionwhen my internal sense of mathematics isright, I’ve done something different.

Both of you have passed ordinary retire-ment age several years ago. But you arestill very active mathematicians, and youhave even chosen retirement or visitingpositions remote from your original workplaces. What are the driving forces forkeeping up your work? Is it wrong thatmathematics is a “young man’s game” asHardy put it?ATIYAH It is no doubt true that mathemat-ics is a young man’s game in the sense thatyou peak in your twenties or thirties in termsof intellectual concentration and in original-ity. But later you compensate that by expe-rience and other factors. It is also true thatif you haven’t done anything significant bythe time you are forty, you will not do sosuddenly. But it is wrong that you have todecline, you can carry on, and if you man-age to diversify in different fields this givesyou a broad coverage. The kind of mathe-matician who has difficulty maintaining themomentum all his life is a person whodecides to work in a very narrow field withgreat depths, who e.g. spends all his life try-ing to solve the Poincaré conjecture –whether you succeed or not, after 10-15years in this field you exhaust your mind;and then, it may be too late to diversify. Ifyou are the sort of person that chooses tomake restrictions to yourself, to specialize ina field, you will find it harder and harder –because the only things that are left are hard-er and harder technical problems in yourown area, and then the younger people arebetter than you.

You need a broad base, from which youcan evolve. When this area dries out, thenyou go to that area – or when the field as awhole, internationally, changes gear, youcan change too. The length of the time youcan go on being active within mathematicsvery much depends on the width of yourcoverage. You might have contributions tomake in terms of perspective, breadth, inter-actions. A broad coverage is the secret of ahappy and successful long life in mathemat-ical terms. I cannot think of any counterexample. SINGER I became a graduate student at theUniversity of Chicago after three years inthe US army during World War II. I wasolder and far behind in mathematics. So Iwas shocked when my fellow graduate stu-dents said, “If you haven’t proved the

INTERVIEW


Riemann Hypothesis by age thirty, youmight as well commit suicide.” How infan-tile! Age means little to me. What keeps megoing is the excitement of what I’m doingand its possibilities. I constantly check [andcollaborate!] with younger colleagues to besure that I’m not deluding myself – thatwhat we are doing is interesting. So I’mhappily active in mathematics. Another rea-son is, in a way, a joke. String Theory needsus! String Theory needs new ideas. Wherewill they come from, if not from Sir Michaeland me? ATIYAH Well, we have some students…SINGER Anyway, I am very excited aboutthe interface of geometry and physics, anddelighted to be able to work at that frontier.

You, Prof. Atiyah, have been very muchinvolved in the establishment of theEuropean Mathematical Society around1990. Are you satisfied with its develop-ment since then?ATIYAH Let me just comment a little onmy involvement. It started an awful longtime ago, probably about 30 years ago.When I started trying to get people interest-ed in forming a European MathematicalSociety in the same spirit as the EuropeanPhysical Society, I thought it would be easy.I got mathematicians from different coun-tries together and it was like a mini-UN: theFrench and the Germans wouldn’t agree; wespent years arguing about differences, and –unlike in the real UN – where eventually atthe end of the day you are dealing with realproblems of the world and you have to cometo an agreement sometime; in mathematics,it was not absolutely essential. We went onfor probably 15 years, before we foundedthe EMS.

On the one hand, mathematicians havemuch more in common than politicians, weare international in our mathematical life, itis easy to talk to colleagues from other

History of the EMS

countries; on the other hand, mathemati-cians are much more argumentative. Whenit comes to the fine details of a constitution,then they are terrible; they are worse thanlawyers. But eventually – in principle – thegood will was there for collaboration.

Fortunately, the timing was right. In themeantime, Europe had solved some of itsother problems: the Berlin Wall had comedown – so suddenly there was a new Europeto be involved in the EMS. This very factmade it possible to get a lot more peopleinterested in it. It gave an opportunity for abroader base of the EMS with more oppor-tunities and also relations to the EuropeanCommission and so on.

Having been involved with the set-up, Iwithdrew and left it to others to carry on. Ihave not followed in detail what has beenhappening except that it seems to be active.I get my Newsletter, and I see what is goingon.

Roughly at the same time as the collapseof the Berlin Wall, mathematicians in gen-eral – both in Europe and in the USA –began to be more aware of their need to besocially involved and that mathematics hadan important role to play in society. Insteadof being shut up in their universities doingjust their mathematics, they felt that therewas some pressure to get out and getinvolved in education, etc. The EMS tookon this role at a European level, and theEMS congresses – I was involved in the onein Barcelona – definitely made an attempt tointeract with the public. I think that theseare additional opportunities over and abovethe old-fashioned role of learned societies.There are a lot of opportunities both in termsof the geography of Europe and in terms ofthe broader reach.

Europe is getting ever larger: when westarted we had discussions about wherewere the borders of Europe. We met peoplefrom Georgia, who told us very clearly, thatthe boundary of Europe is this river on theother side of Georgia; they were very keen

to make sure that Georgia is part of Europe.Now, the politicians have to decide wherethe borders of Europe are.

It is good that the EMS exists; but youshould think rather broadly about how it isevolving as Europe evolves, as the worldevolves, as mathematics evolves. Whatshould its function be? How should it relateto national societies? How should it relateto the AMS? How should it relate to thegovernmental bodies? It is an opportunity!It has a role to play!

Could you tell us in a few words about yourmain interests besides mathematics?SINGER I love to play tennis, and I try todo so 2-3 times a week. That refreshes meand I think that it has helped me work hardin mathematics all these years.ATIYAH Well, I do not have his energy! Ilike to walk in the hills, the Scottish hills – Ihave retired partly to Scotland. InCambridge, where I was before, the highesthill was about this (gesture) big. Of courseyou have got even bigger ones in Norway. Ispent a lot of my time outdoors and I like toplant trees, I like nature. I believe that if youdo mathematics, you need a good relaxationwhich is not intellectual – being outside inthe open air, climbing a mountain, workingin your garden. But you actually do mathe-matics meanwhile. While you go for a longwalk in the hills or you work in your garden– the ideas can still carry on. My wife com-plains, because when I walk she knows I amthinking of mathematics.SINGER I can assure you, tennis does notallow that!

Thank you very much on behalf of theNorwegian, the Danish, and the EuropeanMathematical Societies!

The interviewers were Martin Raussen,Aalborg University, Denmark, andChristian Skau, Norwegian University ofScience and Technology, Trondheim,Norway.

1 More details were given in the laureates’lectures.

2 Among those: Newton, Gauss, Cauchy,Laplace, Abel, Jacobi, Riemann,Weierstrass, Lie, Picard, Poincaré,Castelnuovo, Enriques, Severi, Hilbert,Lefschetz, Hodge, Todd, Leray, Cartan,Serre, Kodaira, Spencer, Dirac,Pontrjagin, Chern, Weil, Borel,Hirzebruch, Bott, Eilenberg,Grothendieck, Hörmander, Nirenberg.

3 Comm. Pure App. Math. 13(1), 1960.

Apart from mathematics…

INTERVIEW


The Abel lectures were given by I. Singer, E. Witten, M. Atiyah and J.- M. Bismut

On 25 March, the Norwegian Academy ofScience and Letters announced that theAbel Prize for 2004 was to be awarded toSir Michael F. Atiyah of the University ofEdinburgh and Isadore M. Singer of MIT.

This is the second Abel Prize awardedfollowing the Norwegian Government’sdecision in 2001 to allocate NOK 200million to the creation of the AbelFoundation, with the intention of award-ing an international prize for outstandingresearch in mathematics. The prize,amounting to NOK 6 million, was insti-tuted to make up for the fact that there isno Nobel Prize for mathematics. In addi-tion to awarding the international prize,the Foundation shall contribute part of itsearnings to measures for increasing inter-est in, and stimulating recruitment to,mathematical and scientific fields.

The first Abel Prize was awarded in2003 to the French mathematician Jean-Pierre Serre for playing a key role in shap-ing the modern form of many parts ofmathematics. In 2004, the AbelCommittee decided that Michael F.Atiyah and Isadore M. Singer shouldshare the prize for:

their discovery and proof of the indextheorem, bringing together topology,geometry and analysis, and their out-standing role in building new bridgesbetween mathematics and theoreticalphysics.

This year’s committee consisted of ErlingStørmer (Oslo, Leader), David Mumford(Brown University), Jacob Palis (IMPA,Brazil), Gilbert Strang (MIT) and DonZagier (Max-Planck-Institut fürMathematik, Germany).

The Abel Prize for 2004 was presentedon 25 May, the occasion being marked bya number of associated events in Oslo.

Café ScientifiqueOn Sunday 23 May, Sir Michael Atiyahparticipated in the first event in connec-tion with the award of this year’s AbelPrize. In collaboration with theNorwegian Association of YoungScientists, the British Council arranged aCafé Scientifique at the Kafé Rust inOslo, in which Atiyah gave an informallecture on his chosen subject: Man versus

machine – the brain and the computer,with the subtitle “Will a computer ever beawarded the Abel Prize?” QuentinCooper, one of the BBC’s most popularradio presenters, chaired the meeting, inwhich Sir Michael spoke for an hour to anaudience of about 50 people. He pointedout that while computers are extremelyadept at following pre-determined rulesand that he himself is not surprised that,for example, very good chess programshave been developed, what is surprising isthat people are still able to play chess onan equal footing with machines. In otherwords, computers are good at followingrules, but what they are not able to do is tobreak the rules in a creative manner. Asan example, he cited Niels Henrik Abel’sproof of the impossibility of solving thegeneral quintic equation. A computerwould have continued to search for thesolution, and would never have been ableto break the rules, as Abel did, and look atthe inverse of the problem. As a mathe-matician you have to know the rules, butto create something new you have to

break those rules creatively, just like anartist or a musical composer.

After a brief interval, Quentin Cooperinvited questions from the audience and anumber of points were brought up thatAtiyah addressed thoroughly and profes-sionally.

After a highly successful meeting last-ing almost two hours, Atiyah answeredhis own question, “Will a computer everbe awarded the Abel Prize?”: – Only if theAbel Prize Committee is replaced by com-puters.

Youth and maths in the celebration ofNiels Henrik AbelMonday 24 April was Youth Day, onwhich the Abel Committee invited thewinners of various mathematical competi-tions for young people to Oslo. In addi-tion to the winners of the AbelCompetition for Norwegian upper sec-ondary schools and of KappAbel, themathematics competition for schools, thewinners of mathematics competitions inBerlin and France were also invited. Themain event took place at Oslo CathedralSchool – the school at which Abel himselfwas a pupil. An audience of 200 wasassembled, mostly consisting of theschool’s own mathematics pupils. After abrief introduction by Paul Jasper, head-master of the school, one of the pupils,Jon Strand, took over and led the pro-ceedings with capable hands. First theFrench students presented their winning

NEWS


Oslo 2004: Oslo 2004: The The Abel Prize celebrationsAbel Prize celebrationsNils Voje Johansen and Yngvar Reichelt (Oslo, Norway)

Nils Voje Johansen Yngvar Reichelt

Café Scientique: Quentin Cooper(BBC) and Sir Michael Atiyah

entry to the competition, “Niels HenrikAbel in the French tradition”. The audi-ence were then given an introduction tothe winning project in this year’sKappAbel competition, and Sir MichaelAtiyah and Professor Isadore Singerawarded a book prize to the young partic-ipants. After this, Per Manne, the leaderof the Norwegian Mathematics Council,took the floor and presented informationabout a recently established prize forgood mathematics teaching, in memory ofBernt Michael Holmboe. Holmboe wasthe teacher who discovered Niels HenrikAbel’s exceptional talent and who was hisearly tutor. The Norwegian MathematicsCouncil arranges the Holmboe Prize, withfinancial support from the AbelFoundation. The prize is to be awardedannually to one or more teachers whohave distinguished themselves throughhigh quality and inspiring mathematicsteaching. In conjunction with the award ofthe prize, a symposium on mathematicsand mathematics teaching will also beheld.

Wreath laying at the Abel monumentAt 5 p.m. there was a wreath laying cere-mony at Gustav Vigeland’s monument toAbel, which stands in front of the RoyalPalace in Oslo. The ceremony began witha display by a troop from His Majesty’sCorps of Signals. Then the leader of theAbel Committee, Jens Erik Fenstad, helda short speech in which he explainedabout the origin of the Abel monument.The ceremony culminated in the wreathlaying by Atiyah and Singer. Afterwards,the younger contingent retired to a restau-rant, while the two prize winners and themembers of the Norwegian Academy ofScience and Letters were invited to dinnerat the Academy.

The prize ceremonyAfter an audience with Their MajestiesKing Harald and Queen Sonja earlier inthe day, it was time for the prize ceremo-ny in the Great Hall of Oslo University.The main street of Oslo, Karl Johans gate,was decorated for the occasion with

colourful Abel Prize banners. This year’sprize winners arrived at the packed hall tothe sound of Klaus Sanvik’s recently-composed Abel Fanfare, performed bySidsel Walstad on electric harp, followedby the arrival of the King and Queen.

Lars Walløe, the President of theNorwegian Academy of Science andLetters, welcomed those present to theceremony. Before the official presenta-tion, the audience were given a surprise inthe form of a new arrangement of MichaelJackson’s Billy Jean, performed by SidselWalstad (electric harp), Mocci Ryen(vocals) and Børre Flyen (percussion).The lively and youthful performance wasappreciated, at least by Isadore Singer,who tapped his foot enthusiastically intime with the music.

The leader of the Abel Committee,Erling Størmer, briefly explained the rea-sons for the selection of Atiyah andSinger as this year’s prize winners: “TheAtiyah-Singer index theorem is one of themost important mathematical results ofthe twentieth century. It has had an enor-mous impact on the further developmentof topology, differential geometry andtheoretical physics. The theorem also pro-vides us with a glimpse of the beauty ofmathematical theory in that it explicitlydemonstrates a deep connection betweenmathematical disciplines that appear tobe completely separate.”

After the address, His Majesty KingHarald presented the Abel Prize to the twowinners.

Sir Michael Atiyah commenced hisacceptance speech by thanking colleagueswho had made important contributions tothe work, mentioning in particular Fritz

Hirzebruch, Raoul Bott, Graeme Segaland Nigel Hitchin. He went on to explainthat right from the time of Newton to thatof Einstein there has been a close rela-tionship between mathematics and theexploration of the natural world. “One ofthe unexpected joys of my partnershipwith Is Singer has been that these linkswith physics have been reinforced duringour time” In conclusion, Atiyah said thatin his opinion, “Abel was really the first

modern mathematician. His wholeapproach, with its generality, its insightand its elegance, set the tone for the nexttwo centuries. If Abel had lived longer, hewould have been the natural successor tothe great Gauss: a statement with which Ifully concur except for the qualificationthat Abel was a much nicer man, modest,friendly and likeable. I am proud to havea prize that bears his name.”

After that it was the turn of IsadoreSinger, who started with a confession.“Outside of the university environment itis difficult to be a pure mathematician. Noone in my family understands what I do.At parties, when someone learns I am amathematician, they frown and say; “Oh,I never could understand calculus”, andthey turn away.” After this description,all too familiar to many, Singer describedhow mathematicians are fascinated by thebeauty, logic and power of mathematics.The index theorem itself had “providednew insight in such fields as Gauge theo-ry and String theory. Breakthroughs inphysics needed new mathematics, and theindex theory frequently supplied what wasneeded. Mathematicians and physicistsbegan talking to each other again. Nowwe take for granted this new discipline ofmathematical physics.” Finally, Singerstated that the establishment of the AbelPrize attracts the attention of the worldand emphasises the fundamental rolewhich mathematics plays in modern liv-ing.

The ceremony was concluded withEdvard Grieg’s Halling before the Kingand Queen and the Abel Prize winners leftthe hall.

Press conferenceFollowing the prize ceremony, a pressconference was held in the “Annen Etage”restaurant at the Hotel Continental. JensErik Fenstad and Jacob Palis commencedby providing information about theinvolvement of the Abel Foundation andthe IMU with regard to the developingcountries, after which the floor was opento the Press to put their questions to theprize winners. There was also an opportu-nity to taste something new. MortenHallan, the hotel’s Chef, had created acompletely new Abel cake, which was onsale in the legendary Theatercafeen dur-ing “Abel Week”. The recipe is of coursesecret, but the different layers of the cakeare as follows: First a base of chocolatecake soaked in blackcurrant liqueur, fol-lowed by a layer of chocolate truffle andthen a layer of blackcurrant preserve, cov-ered by nut meringue, and topped off with

NEWS


Awards to the winners ofthe Abel competition

Abel Fanfare (Photo: Anne Lise Flavik)

Italian meringue. The cake is decoratedwith white and dark chocolate andcaramel cornets filled with blackcurrants.Bon appétit! The cake will also be servedin connection with future prize cere-monies.

BanquetAt 7 p.m. the same day, the NorwegianGovernment held an Abel banquet atOslo’s historic Akershus Fortress, hostedby Kristin Clemet, the Minister ofEducation and Research. The banquet wasattended by the King and Queen,Norwegian and foreign mathematicians,eminent politicians and members ofNorwegian society. Many people recog-nised the mathematician John Donaldson,father of Danish Crown Princess MaryDonaldson, who was specially invited tothe Abel Prize ceremony by King Haraldat the wedding of the Crown Prince ofDenmark some weeks before. After a wel-coming cocktail in Christian IV’s Hall,the party proceeded to Skriverstuen (theScribes’ Hall), where all were welcomedby the Minister of Education andResearch, Sir Michael and ProfessorSinger. Dinner was then served in theRomerike Hall. The menu consisted ofNorwegian trout and turbot, veal fillet andcaramel mousse.

In her address, the Minister, KristinClemet, mentioned that there was onewinner (Jean-Pierre Serre) at the firstAbel Prize ceremony, while at the secondthere were two winners. Based on this,one could perhaps draw the conclusionthat the number of prize winners would bedetermined by the equation x = n, where nis the number of years the prize has exist-ed. Having reminded the audience of thestory of Descartes, who died of pneumo-nia when he visited Scandinavia, shetouched on the effect of the Abel Prize onthe recruitment of future mathematicians.The Holmboe Prize has now been createdto honour the best mathematics teachers –those with the ability to impart under-

standing of the beauty of mathematics totheir students. In connection with this shequoted Sir Michael: By exploring thewhole country of mathematics you get tothe top of Mount Everest and look around.It’s a long route, and when you get to thetop, it’s a big scene you can see. TheMinister went on to express her delightwith the international response to theAbel Prize: it had been extremely posi-tive, and she noted that the selection ofwinners had gained widespread support.

She then gave the floor to the presidentof the European Mathematical Society,Sir John Kingman.

In his speech, Sir John touched on thecriteria for the successful establishment ofthe Abel Prize: “The great name of NielsHenrik Abel is important. Of course theactual value of the prize is important. It isalso important that those who select theprize winners select people that futureprize winners will be proud to follow.”

He went on to point out that such a prizecould have a positive effect on recruit-ment, which is of considerable impor-tance since we need “new mathematiciansdoing new mathematics. Whether it iswhat we call pure mathematics, mathe-matics for its own sake, or whether it ispursuing interesting new applications,applying the techniques which the so-called pure mathematicians have invent-ed.”

ExhibitionIn connection with the award of thisyear’s Abel Prize, an exhibition wasstaged with the objective of informinglaypersons about the work of the prizewinners. The exhibition was based onsimple concepts in the fields of topology,geometry and analysis, and demonstratedhow the links between the disciplines hadbeen discovered. An example of such a

link is illustrated by the Gauss-Bonnettheorem. By means of the Atiyah-Singerindex theorem, an overall unification ofthe disciplines was acquired. In turn, thisnew insight formed the origin of countlessapplications in the field of theoreticalphysics. The exhibition paid special atten-tion to applications in the fields of gaugetheory and string theory. The exhibitionwas located in the foyer outside the audi-torium in which the Abel Lectures wereheld and was the result of collaborationbetween the Departments of Mathematicsand Physics at the University of Oslo.

The mathematicians’ own partyOn the evening of 26 May, it was time todrop the formalities. The Abel days inOslo were wound up with a party at theNorwegian Academy of Science andLetters. Mathematicians from near and farwere invited, in addition to other peoplewho had worked with this year’s events.The atmosphere was pleasant and every-body had an opportunity to meet oldfriends and make new acquaintances. As acontribution to the convivial atmosphere,refreshments were served and throughoutthe venerable building musical treats ofvarious types could be enjoyed.

Photos in this article without a credit are© Dept of Mathematics: UoO

Nils Voje Johansen [[email protected]] and Yngvar Reichelt [[email protected]] are associated to the Departmentof Mathematics at the University of Oslo.Both were members of a group of mathe-maticians that formulated the suggestionto the Norwegian government to installthe Abel Prize in commemoration of NielsHenrik Abel’s bicentenary, which wasfinally adopted by the Norwegian parlia-ment.

NEWS


The Abel Prize 2005King Harald of Norway will present the Abel Prize for 2005 to the winneron May 24th in the Aula of the University of Oslo. The deadline for nom-inating candidates is the 15th of November 2004.

Nominations letters should contain a CV and a description of the can-didate's work, together with names of distinguished specialists in the fieldof the nominee who can be contacted for independent opinion. The lettershould be marked "Abel Prize Nomination" and addressed to:

The Norwegian Academy of Science and LettersDrammensveien 78

NO-0271 OSLONorway

Detailed information is obtainable from the web site www.abelprize.no .

The Abel cake with the logo of the AbelPrize (Photo: Knut Falch/Scanpix)

The beginningsThe history of “modern” Greek mathemat-ics begins with Nicholas Nicolaidis (1826-1889). Born during the Greek Revolutionin the very center of Arcadia, officer-engi-neer in the Greek Army, he went with ascholarship to Paris and became a Docteur

d’Etat es Sc. Math., with O. Bonnet in hisCommittee. He published several originalworks in the Editions of Gauthier-Villars.When he returned to Greece he becameprofessor at the University of Athens. Hisresearch work continues to enjoy someinterest even in the present time (he isknown, in particular, for the term“Nicolaidis’ theorem” in Kinematics).With him begins our story.

The next to appear in Greek mathemat-ics were the professors CyparissosStéphanos (1857-1917) from the island ofKea, and John Hadzidakis (1844-1921)from Crete. Stéphanos was a Docteurd’Etat es Sc. Math. from Paris, withCharles Hermite (1822-1901) as a memberof his Committee. He was a geometer andalgebraist, with references made to hiswork by a number of mathematiciansincluding the young (at that time) DavidHilbert (1862-1943). He also served as a

member of a number of HonoraryCommittees of International Congresses.J. Hadzidakis is known for his work inGeometry with the notion “Hadzidakistransformation”, that is cited in M.Spivak’s “Differential Geometry” and in J.McCleary’s “Geometry from a

Differential Viewpoint”. He was a pupil ofthe Paris school and of Karl Weierstrass(1815-1897) in Berlin. He is also wellknown in Greece for writing good booksfor all levels of mathematical education.

J. Hadzidakis and C. Stéphanos estab-lished the mathematical tradition of theUniversity of Athens and were the teach-ers of the founders of the GreekMathematical Society (G.M.S.). Thesefounders, all three professors at theUniversity of Athens, were NicholasHadzidakis (1872-1942), a son ofJ. Hadzidakis, George Remoundos (1878-1928) from Athens, and Panayotis Zervos(1878-1952) from Cephalonia. All of themhad been pupils of the Paris School, in par-ticular N. Hadzidakis of David Hilbert.Following the geometric approach of JeanGaston Darboux (1842-1917),N. Hadzidakis introduced Kinetic Geome-try, as well as the study of complexes of

lines and surfaces, to Greece. Remoundoscontributed a lot in function theory and inparticular with his generalization ofPicard’s theorem to algebraic functions.In the year 1905, Jacques Hadamard(1865-1963) proposed to P. Zervos thestudy of Monge’s conjecture (sometimescalled Monge’s problem) in the field ofPartial Differential Equations, which wasposed in 1784 and has not been solvedsince. In 1905, P. Zervos was the firstmathematician to notice the falsity ofMonge’s conjecture. In the year 1912,after some intensive work by P. Zervos,E. Goursat, and O. Botasso among others,D. Hilbert solved a major part of Monge’sproblem. Elie Cartan’s seminal paper ofthe year 1914 on Pfaff’s problem, beginsas follows: “In a recent paper (1913),P. Zervos generalizes a theorem ofD. Hilbert…” (Bull. de la Soc. Math. deFrance, 1914).

Foundation of the G.M.S.In the year 1918, N. Hadzidakis,G. Remoundos and P. Zervos founded theG.M.S. as well as the Research Seminarsin Mathematics at the University ofAthens. In addition, the year after, in 1919,they started the international journal:Bulletin of the G.M.S. The first issue of theBulletin of the G.M.S. publishedP. Zervos’ inaugural lesson entitled“Relation of Mathematics to the otherSciences and to Philosophy” as well assome “technical” articles in mathematicsby the above three mathematicians andothers. Zervos had been in the audience ofPoincaré’s lectures on CelestialMechanics in Paris during the period1903-1905, and he was also the translatorinto Greek of Poincaré’s celebrated book“Science and Hypothesis”. In 1935, theBulletin of the G.M.S. published a paperby Nicholas Criticos (1894-1985) that wasdevoted to a new proof of the Jordan

SOCIETIES


THE GREEK THE GREEK MAMATHEMATHEMATICALTICAL SOCIETYSOCIETYIts predecessors, its founders, and some highlights from its life

(Themistocles M. Rassias, Athens, Greece)

Cyparissos Stephanos John Hadzidakis

Curve Theorem. In the context of two lec-tures given by him in the G.M.S., a kind offirst advanced text in “Point Set Topology”was provided, written in Greek.

Papakyriakopoulos and CarathéodoryA rigorous presentation of MathematicalAnalysis on the basis of Dedekind Cutsand Cantor’s Set Theory had been madeby P. Zervos in his masterpiece:“Infinitesimal Calculus”. Both of theabove mentioned texts, by N. Criticos andP. Zervos, contributed greatly to the orien-tation in mathematical studies of the well-known topologist Christos D. Papakyria-kopoulos (Athens, 1913 - Princeton,1976). The doctoral thesis ofPapakyriakopoulos, entitled “A new proofof the invariance of the homology groupsof a complex”, was published in the year1943 in the Bulletin of the G.M.S. The154-page thesis of Papakyriakopoulos thatwas published in the Bulletin also consti-tutes the first text written in Greek onAlgebraic Topology. The (unofficial) ref-eree for the Thesis was C. P. Papaioannou(1899-1979), who was Professor ofMechanics at the University of Athens.Papaioannou had the ability to immediate-ly foresee Papakyriakopoulos’ brilliantresearch career. Another prophetic visionof Papaioannou appears in his inauguralspeech at the Academy of Athens in theyear 1965, in which he sees the universe asPlaton’s dodecahedron. This is somethingthat was later very beautifully presented ina modern mathematical language byWilliam P. Thurston (Fields Medal inTopology, 1982) in the ScientificAmerican. This seems to be a major sub-ject of research by satellites (cf. Notices ofthe A.M.S., June-July 2004).

The 1943 issue of the Bulletin of theG.M.S. was, naturally, dedicated toConstantin Carathéodory at the occasionof his 70-th birthday. Carathéodory wasborn in Berlin, of Greek parents, on 13September 1873, and he died in Munichon 2 February 1950. His work coveredseveral subjects of Mathematics, includingthe calculus of variations, function theory,measure and integration, as well as appliedmathematics. Carathéodory has also donevery fundamental work in mechanics,thermodynamics, geometrical optics andrelativity theory. An example ofCarathéodory’s wide ranging influence inthe international mathematical communitywas seen during the first Fields Medalsawards at the International Congress ofMathematicians, Oslo, 1936. The selectioncommittee consisted of GeorgeD. Birkhoff, Elie Cartan, C. Carathéodory,

F. Severi, and T. Takagi. Two medalswere awarded, one to Lars V. Ahlfors(Harvard University) and one to JesseDouglas (M.I.T). It was C. Carathéodorywho presented the work of both medalistsduring the opening of the InternationalCongress (see Constantin Carathéodory:An International Tribute, Vols I & II,World Scientific Publ. Co., 1991, edited byTh. M. Rassias).

Now let us come back to our story.Remoundos’ pupil Theodore Varopoulos(1894-1957), professor of the Universityof Thessaloniki (father of the well-knownParis Analyst Nicholas Th. Varopoulos),the closest pupil of Paul Montel, had a spe-cial interest in the Geometry ofPolynomials. From time to time one couldfind articles in the Bulletin of the G.M.S.devoted to polynomials (from both analgebraic and a geometric approach) influ-enced directly or indirectly byTh. Varopoulos. For instance, in the thir-ties one can find articles by ConstantineYannopoulos and by John Papademetriou.Around 1950, one has to mention the the-sis on polynomials written in Athens byTh. Varopoulos’ research studentDionysios Vythoulcas. Moving furtherand with more essential results in the con-text of polynomials and especially theirlocalization of roots (zeros), SpyrosP. Zervos (a son of P. Zervos) wrote hisThèse d’Etat in Paris (Ann. Sc. de l’EcoleNormale Sup., 1960) in the spirit ofModern Mathematics. Furthermore, someinteresting work in the classical version ofpolynomials and related subjects wasobtained by the late I. Ferentinou-Nicolacopoulou and was published in theBulletin of the G.M.S.

Other activitiesUntil now, we have been devoted to the

scientific side of the subject and have notcovered the general activity of the G.M.S.In this connection, we recall thePanhellenic Competition in Mathematicsfor High School Students, inaugurated bythe G.M.S. in the year 1931. The first win-ner was George Tsamis from Patras, atthat time a schoolboy, who was to becomea very influential High School Teacher inmathematics. In the sixties, theMathematical Competition got a new leaseof life by Aristide Pallas, a man who wasentirely devoted to the G.M.S and at thattime its president.

In 1975, the new Council of the G.M.S.,under the presidency of ProfessorS.P.Zervos, enlarged the domain of theactivities and publications of the G.M.S.In that year, S.P. Zervos conceived theidea for the foundation of a “University ofAegean”, with a Pythagorean Departmentof Mathematics, on the island of Samos,and other Departments (Schools) invarious subjects on other islands. This ideawas materialized later on. Some distin-guished mathematicians delivered lecturesduring the “Pythagorean Days”, in Athensand in Samos. Among the mathematicianswho delivered such lectures are the follo-wing: K. Borsuk (Warszawa), L. Iliev(Sofia), A. Kawaguchi (Japan),M. Krasner (Paris), K. Kuratowski(Warszawa), Dj. Kurepa (Beograd),M. Loi (Paris), O. Onicescu (Bucharest).

A subsequent article will deal with the‘discovery’ of the ‘Olympiads’ for HighSchools in Greece, as well as to activitiesof the G.M.S. with emphasis to the lastthree decades.

Themistocles M. Rassias [[email protected]] studied mathematics at theUniversity of California at Berkeley,where he received his Ph.D with StevenSmale. He has published more than 170papers and six research books and he hasedited 24 volumes on different subjects inMathematical Analysis, Geometry/Topology and their applications. Hisresearch work has found more than 2000citations. His work is known in the field ofMathematical Analysis with the term“Hyers-Ulam-Rassias Stability” and inGeometry with the term “Alexandrov -Rassias Problem”. He has lectured exten-sively and conducted research in variousacademic institutions in Europe and NorthAmerica. More than ten mathematicaljournals count him as a member of theirEditorial Board. He is Professor at theNational Technical University of Athens,Greece. He is married and has two chil-dren.

SOCIETIES


C.Carathéodory

There has always been a great tradition ofacademic exchange and contact betweenFrance and Kraków in the field of mathemat-ics. The origin of the formal French-Polishcooperation between the Institute ofMathematics at the Jagiellonian Universityand French universities in the domain ofapproximation theory and real and complexanalysis is almost anecdotic. During his stayat the Paul Sabatier University (Toulouse) in1985, Prof. Wies³aw Pleœniak learned thatProf. Jacek Gilewicz (Université de Toulonet du Var and Centre de Physique Théorique,Marseille-Luminy) and the physicists fromthe University of Warsaw had been intendingto organize an international conference onrational approximation at the Castle of£añcut, the home town of Pleœniak. Thisinspired Prof. Pleœniak to go to Marseille andmeet Prof. Gilewicz. As a consequence ofthat meeting, Prof. Jacek Gilewicz was invit-ed to the Institute of Mathematics at the

Jagiellonian University and spent twomonths there in 1986. During this visit, Prof.Gilewicz and Prof. Pleœniak had succeeded inattracting interest from the French Embassyin Warsaw. It ended in the cooperation thatyielded a formal cooperation project underthe ATP Programme (Action ThématiqueProgrammée) supported by the FrenchMinistry of Foreign Affairs and the PolishState Committee for Scientific Research, TheFrench universities were represented by theUniversite de Toulon et du Var, UniversitePaul Sabatier (Toulouse), Université de ParisXI (Orsay), Université des Sciences etTechnologies de Lille, and then the CentreInternational des Rencontres Mathématiques(Marseille-Luminy). The project was coordi-nated by Professor Jacek Gilewicz and thenby Professor Jean-Paul Brasselet (CentreInternational des Rencontres Mathém-atiques) and the undersigned Wies³aw

Pleœniak (Institute of Mathematics of theJagiellonian University).

Thanks to the financial support from theFrench Embassy in Poland and the Instituteof Mathematics of the JagiellonianUniversity, the coordinators had succeeded inorganising a series of seven French-Polishmeetings in Poland (Karniowice nearKraków, the Niedzica Castle, and thenKrynica) under the common theme: Lefrançais des mathématiciens. These meetingsserved as the permanent forum of the presen-tation of the progress of both Polish andFrench participants of the ATP Project inresearch concerning the subject of coopera-tion. The meetings were also opportunities offirst contacts with French scientific languageand French culture for young Polish mathe-maticians that were planning for invited staysin French universities.

First, we should make special note of animportant French-Polish meeting at theCentre of the Polish Academy of Sciences inParis in November of 1995 that was orga-nized by Prof. Jacek Gilewicz and Prof.Wies³aw Pleœniak, At this meeting, the pro-gramme of future co-operation in the abovementioned subject was scheduled. The nextimportant step was the common “Réseau-Formation-Recherche” project that wasimplemented 1995-1998 under the coordina-tion of Prof. Anne-Marie Chollet (Universitédes Sciences et Technologies de Lille) andProf. Wies³aw Pleœniak. Thanks to this pro-ject, Polish partners from the JagiellonianUniversity and the Adam MickiewiczUniversity of Poznañ obtained 56 months ofFrench Government fellowships for Polishdoctorate students and globally, more thanfive hundred thousand French Francs for theexchange programme.

Other common projects included the work-

shop on Differential Analysis at theInternational Banach Center of Warsaw(September 2000), (organized by Prof. Anne-Marie Chollet, Prof. Krzysztof Kurdyka(Université de Savoie a Chambery), and Prof.Wies³aw Pleœniak. Moreover, the workshopon Approximation Theory (January 2002)was organised at the same InternationalBanach Center (organised by Wies³awPleœniak with the cooperation of Prof. Anne-Marie Chollet, Vincent Thilliez (Universitédes Sciences et Technologies de Lille), andJacques Chaumat (Université de Paris-Sud).

Since 1998, the cooperation continuesunder the “Polonium” Project “Real andComplex Analysis” coordinated by Anne-Marie Chollet and Wies³aw Pleœniak whichhas been accepted by the French-PolishGovernment Commission for implementa-tion in 2004.

Wies³aw Pleœniak [[email protected]. pl],born in 1944, is a professor of theJagiellonian University at Kraków, Poland,where he holds the Chair in ApproximationTheory at the Department of Mathematics.His main mathematical interest is approxi-mation theory and complex analysis. He is amember of Editorial Boards of three jour-nals: Annales Polonici Mathematici,Commentationes Mathematicae and EastJournal on Approximation. For some yearshe served as the President of the Krakówbranch of the Polish Mathematical Society;moreover, he was the Vice-Dean of theFaculty. Currently he is the Vice-President ofthe Mathematical Committee of the PolishAcademy of Sciences.

Wies³aw Pleœniak has received a doctoratehonoris causa from the University of Toulonin France. He is a member of Société Royaledes Sciences de Liège and of the PolishAcademy of Arts and Sciences.

Among other distinctions, he was awardedthe Mazurkiewicz Great Prize and theZaremba Great Prize of the PolishMathematical Society (1977, 1989), and hehas been awarded scientific prizes from thePolish Academy of Science three times.

SOCIETIES


FRENCH-POLISH COOPERAFRENCH-POLISH COOPERATIONTIONIN IN APPROXIMAAPPROXIMATION TION THEORTHEORYY

AND REALAND REAL AND COMPLEX AND COMPLEX ANALANALYSISYSISWies³aw Pleœniak (Jagiellonian University, Kraków, Poland)

The participants of the 1992 conference inNiedzica Castle.

Ceremony at Toulon, 23 October 2003.Wies³aw Pleœniak, Jacek Gilewicz, JózefSiciak, Miros³aw Baran.

Institutional frameworkThe CRM was created in 1984 by the Institutd’Estudis Catalans (Institute for CatalanStudies), an academic, scientific and culturalbody whose aim it is to promote research inscience, technology and humanities, whilesupporting all aspects of Catalan culture.Since 2001, the CRM has been a consortium,with its own legal status, integrated by theInstitut d’Estudis Catalans and the CatalanGovernment. It is a research institute asso-ciated with the Universitat Autònoma deBarcelona and located on the Bellaterra cam-pus, about 15 km from Barcelona.

The Governing Board of the CRM haseight members and is chaired by the Ministerof Universities, Research and theInformation Society of the CatalanGovernment. In 2002, the Governing Boardre-elected Manuel Castellet as Director andappointed a new Scientific Advisory Board,with the following members: Joan Bagaria,Àngel Calsina, Carles Casacuberta, VicentCaselles, Alberto Facchini, Evarist Giné,Joan Girbau, Antoni Huerta, Jaume Llibre,Xavier Massaneda, M. Pilar Muñoz, JoanC. Naranjo, David Nualart, Pere Pascual,Joan Porti, Jordi Quer, Oriol Serra, and JuanL. Vázquez. Since 2004, Carles Casacubertaand Jordi Quer have acted as Vice-Directors.

Aims and scopeThe CRM gives support to local researchgroups in all areas of mathematics and fos-

ters emerging research directions by invitingoutstanding mathematicians and organisingspecialised research programmes, conferen-ces, workshops, and advanced courses.

It hosts post-doctoral fellows (thirteen in2003), including several Marie Curie grantholders, and it has also hosted doctoral stu-dents as a Marie Curie Training Site of theEuropean Union from 2000 to 2004.Between 2000 and 2003, the CRM was anode of an EC Training Network entitledModern Homotopy Theory, jointly withteams in Aarhus, Aberdeen, Lille, Louvain-la-Neuve, Paris, and Sheffield.

Research programmesIn 2002, the Governing Board of the CRMapproved a quadrennial plan that includestwo research programmes every year, toget-her with complementary activities. Eachresearch programme offers the followingpositions during one academic year: one full-time local researcher, one full-time visitingresearcher, two post-doctoral fellows, and 24months of visiting researchers for periods ofone to three months. Activities include aweekly seminar, a conference or a workshopand an advanced course at a doctoral or post-doctoral level. Partial funding for visitors andactivities is provided by the Department ofUniversities, Research and the InformationSociety (DURSI) of the CatalanGovernment, under a contract programmethat is revised every year, and by means of

competitive calls. Other sources of funding,through competitive applications, are theSpanish Ministry of Education and Scienceand the European Commission.

An open call for research programmes ismade every year. Each programme has to beapproved by the CRM Governing Board.Proposals are presented by the Director onthe grounds of evaluation reports prepared bythe Scientific Advisory Board.

During the academic year 2003-2004, aresearch programme on Set Theory wascarried out. The organisers were JoanBagaria (ICREA and Universitat deBarcelona) and Stevo Todorcevic (CNRSand Université Paris 7). For the academicyear 2004-2005, a research programme onthe Geometry of the Word Problem is plan-ned, under the direction of Josep Burillo andEnric Ventura (Universitat Politècnica deCatalunya), and Noel Brady (University ofOklahoma). Shorter specialised programmeswill be held on Control, Geometry andEngineering, co-ordinated by MiguelC. Muñoz (UPC), and on ContemporaryCryptology, co-ordinated by Jorge Villar andCarles Padró (UPC). The Scientific AdvisoryBoard selected two programmes for the aca-demic year 2005-2006: one on ArakelovGeometry and Shimura Varieties, co-ordina-ted by José I. Burgos (Universitat deBarcelona) and Jörg Wildeshaus (UniversitéParis 13), and another one on Hilbert’s 16thProblem, co-ordinated by Armengol Gasulland Jaume Llibre (Universitat Autònoma deBarcelona), together with Chengzhi Li andJiazhong Yang (Beijing University).

CRM Research Programmes2003-2004Set Theory2004-2005

Geometry of the Word Problem2005-2006

Arakelov Geometry and ShimuraVarieties

On Hilbert’s 16th Problem

ERCOM


CRMCRMCentre de Recerca Matemàtica, Barcelona

Excursion to Cabrera 1991

Scientific meetingsThe following conferences and courses werescheduled for 2004 and 2005: AdvancedCourse on Ramsey Methods in Analysis(January 19 to 28, 2004); Workshop on theFoundations of Set Theory (June 9 to 12,2004); Advanced Course on ContemporaryCryptology (February 2 to 13, 2004);Conference on Mathematical Foundationsof Learning Theory (June 18 to 23, 2004);Workshop on Non-Linear DifferentialGalois Theory (June 28 to July 2, 2004);Advanced Course on Automata Groups(July 5 to 16, 2004); HYKE Conference onComplex Flows (October 6 to 9, 2004); 4thCongress of the European Society forResearch in Mathematics Education(February 17 to 21, 2005); BarcelonaConference on Geometric Group Theory(June 8 to July 2, 2005); Advanced Courseon the Geometry of the Word Problem forFinitely Generated Groups (July 5 to 15,2005); Advanced Course on Recent Trendson Combinatorics in the MathematicalContext (September 12 to 23, 2005).

Around the International Congress ofMathematicians (ICM2006) in Madrid, theCRM will organise a three-month researchprogramme on Analysis and an advancedcourse on Computational Geometry, bothco-ordinated jointly by researchers fromBarcelona and Madrid. An ICM satelliteconference is also planned.

Links with European entitiesSince 2003, the CRM has been an institutio-nal member of the EMS. It is also a memberof ERCOM (European Research Centres onMathematics), a committee of the EMS con-sisting of scientific directors of Europeanresearch centres in mathematics. In fact, thecurrent CRM Director, Manuel Castellet,has been the chair of ERCOM since 2002.

In a similar vein to other ERCOM centres,the CRM undertakes actions to reinforce therole of mathematics in the thematic priori-ties of the 6th Framework Programme of theEuropean Commission. Funding is offeredto young mathematicians in order to fosterthe development of the following topics,which were selected on the basis of reportsprepared by local teams: Life sciences,

genomics, and biotechnology for health;nanotechnologies and nanosciences; infor-mation society technologies; sustainabledevelopment, global change and ecosys-tems. Several activities, including doctoraltraining and workshops, are planned on neu-roscience and cryptology for the years 2004,2005 and 2006.

The CRM is a member of the EPDI(European Post-Doctoral Institute for

Mathematics), a network of nine Europeanresearch institutes that jointly offer post-doctoral fellowships in mathematics andmathematical physics every year.

PublicationsBesides its twenty-year old preprint collec-tion the CRM has published, since 2001, amonograph series entitled Advanced

Courses in Mathematics CRM Barcelona,which is produced and distributed byBirkhäuser Verlag (Basel). The series isespecially addressed to doctoral and post-doctoral students. Volumes contain care-fully edited notes written by lecturers atCRM advanced courses. The followingvolumes were published in 2003 and 2004:Symplectic Geometry and IntegrableHamiltonian Systems, by M. Audin,

ERCOM


Lecture room

Congress at Institut d'Estudis Catalans. Andrew Barron, Steve Smale, and José M. Matías

A. Cannas da Silva and E. Lerman; GlobalRiemannian Geometry: Curvature andTopology, by S. Markvorsen and M. Min-Oo; Proper Group Actions and the Baum-Connes Conjecture, by G. Mislin andA. Valette; Polynomial Identity Rings, byV. Drensky and E. Formanek.

Facilities and infrastructureThe CRM occupies 940 square metres in theScience Building of the UniversitatAutònoma de Barcelona. Office spaceallows the allocation of up to fifteen guests.Two lecture rooms are used for seminarsand meetings. All offices and rooms arefully equipped and air conditioned. TheCRM has a LAN Ethernet with twenty-fiveworking stations and four printers. Access tomajor bibliographic data bases is provided.

In addition, visitors have free access to thescientific infrastructure of the Catalan uni-versities, including the use of the UABScience and Engineering Library. The CRMhas several furnished apartments at its dis-posal in the Vila Universitària of theBellaterra campus and in the nearby town ofSant Cugat.

20th anniversaryIn the Autumn of 2004, the CRM will cele-brate its 20th anniversary. On this occasion,Jean-Pierre Serre will deliver a lectureduring an academic event on November 9,which will be presided by the foremost aca-demic and political authorities of Catalonia.The history of the CRM during its firsttwenty years of existence will be published

in a commemorative volume and a CD. Inthis period, the CRM hosted 969 visitorsfrom 58 different countries, including 44post-doctoral fellows. Twenty-four congres-ses were organised, together with 19 works-hops and 23 advanced courses. These eventswere attended by a total number of 3,480participants, coming from 72 countries.Many of them, hopefully all, enjoyedBarcelona, the Catalan countryside, or thehospitality of their Catalan colleagues, andkeep pleasant memories from their stay.

Web siteAdditional information about the CRM,including full lists of visitors and activities,can be found at the internet addresswww.crm.es.

20th Anniversary of the CRMJean-Pierre Serre, Collège de France

Groupes finis: Choix de théorèmesNovember 9, 2004

Institut d’Estudis Catalans, Barcelona

ERCOM


Centres at present represented in ERCOM, ordered by decreasing geographical latitude

Euler International Mathematical Institute, St. Petersburg (59º55')Institut Mittag-Leffler, Stockholm (59º20')Network in Mathematical Physics and Stochastics, Aarhus (56º10')International Centre for Mathematical Sciences, Edinburg (55º57')Isaac Newton Institute for Mathematical Sciences, Cambridge (52º21')Centrum voor Wiskunde en Informatica, Amsterdam (52º21')Stefan Banach International Mathematical Center, Warszawa (52º15')Thomas Stieltjes Institute for Mathematics, Leiden (52º10')Lorentz Center, Leiden (52º10')Mathematical Research Institute, Nijmegen (51º50')EURANDOM, Eindhoven (51º26')Max Planck Institute for Mathematics in the Sciences, Leipzig (51º20')Max-Planck Institut für Mathematik, Bonn (50º44')Institut Henri Poincaré, Centre Emile Borel, Paris (48º52')Institut des Hautes Études Scientifiques, Bures-sur-Ivette (48º52')Mathematisches Forschungsinstitut Oberwolfach (48º18')Erwin Schrödinger International Institute for Mathematical Physics, Wien (48º13')The Abdus Salam International Center for Theoretical Physics, Trieste (45º39')Centro di Recerca Matematica Ennio Di Giorgi, Pisa (43º43')Centre International de Rencontres Mathématiques, Luminy (43º18')Instituto Nazionale di Alta Matemática Francesco Severi, Roma (41º53')Centre de Recerca Matemática, Barcelona (41º25')Centro Internacional de Matemática, Lisboa (38º44')Emmy Noether Research Institute for Mathematics, Ramat-Gan (32º35')

Facilities at CRM

Please e-mail announcements ofEuropean conferences, workshops andmathematical meetings of interest toEMS members, to one of the followingaddresses [email protected] [email protected] should be written in astyle similar to those here, and sent asMicrosoft Word files or as text files (butnot as TeX input files). Space permit-ting, each announcement will appear indetail in the next issue of the Newsletterto go to press, and thereafter will bebriefly noted in each new issue until themeeting takes place, with a reference tothe issue in which the detailedannouncement appeared.

8–11: Dixièmes journées montoisesd’informatique théorique, à Liège(Tenth Mons theoretical computer sci-ence days, in Liège)Information: e-mail: [email protected] Web site : www.jm2004.ulg.ac.be[For details, see EMS Newsletter 50]

13–18: 3rd CIME Course - StochasticGeometry, Martina Franca, Taranto,ItalyInformation:e-mail: [email protected] site: http://www.math.unifi.it/CIME Address: Fondazione C.I.M.E. c/oDipart.di Matematica “U. Dini”Viale Morgagni, 67/A - 50134 FIREN-ZE (ITALY)Phone: +39-55-434975 / +39-55-4237111 Fax: +39-55-434975 / +39-55-4222695[For details, see EMS Newsletter 51]

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

23–26: 4th International Conferenceon Applied Mathematics (ICAM-4),

September 2004

Baia Mare, Romania (previous edi-tions in 1998, 2000 and 2002)Information: e-mail:[email protected]; [email protected] site: http://www.ubm.ro/site-ro/ facultati/departament/manifestari/icam4/index.html[For details, see EMS Newsletter 49]

26–October 1: Potential theory andrelated topics, Hejnice, CzechRepublicInformation:e-mail: [email protected] site:http://www.karlin.mff.cuni.cz/PTRT04

[For details, see EMS Newsletter 52]

29–October 2: XII Annual Congressof the Portuguese Statistical Society,Evora, PortugalInformation:e-mail: [email protected] site: www.eventos.uevora.pt/spe2004Address: Comissao Organizadora XIICongresso SPE, Departamento deMatematica, Universidade de Evora,Rua Romao Ramalho 59, 7000-671EVORA, PORTUGALPhone: +351-266745370Fax: +351-266745393 [For details, see EMS Newsletter 52]

6–9: HYKE Conference on ComplexFlows, Centre de RecercaMatematica, Bellaterra, SpainInformation:e-mail: [email protected] Web site: http://www.crm.es/ComplexFlows[For details, see EMS Newsletter 52]

24–30: Partial Differential Equationsin Mathematical Physics (in memoryof Olga A. Ladyzhenskaya), Trento,ItalyOrganizer: CIRM, Centro Internazio-nale per la Ricerca Matematica, Trento,ItalyScientific Committee: H. Beirao daVeiga (Pisa), G. Seregin (St. Petersburg),V. Solonnikov (Ferrara), N. Uraltseva

October 2004

(St. Petersburg), A. Valli (Trento) Location: Grand Hotel Bellavista, Levi-co Terme, Trento, ItalyInformation:e-mail: [email protected] sites: www.science.unitn.it/cirm/ ,http://www.science.unitn.it/cirm/Ladylecture.html

6–9: 24th Nordic and 1st Franco-Nordic Congress of Mathematicians,Reykjavik, IcelandAim: The main goal of the congress is tobring together mathematicians to pre-sent recent results in several importantareas of research in the Nordic countriesand France. Around half of the pro-gramme will be devoted to 3 mainthemes: Algebraic Geometry, GeometricAnalysis, and Probability. Satellitemeetings will be organized in the daysbefore the congress.Scientific committee: H. Thorisson(chair) (Iceland), J. Kr. Arason(Iceland), G. Grubb (Denmark),Ch. Kiselman (Sweden), J.-F. Le Gall(France), P. Koskela (Finland), MireilleMartin-Deschamps (France), R. Piene(Norway), and M. Waldschmidt(France)Plenary speakers: S. Helgason(Iceland/USA), M. Yor (France),M. Audin (France), S. Neshveyev(Norway), H. Schlichtkrull (Denmark),C. Villani (France), F. Loeser (France),A. Buch (Denmark), M. Passare(Sweden), Xiao Zhong (Finland), OlavKallenberg (Sweden/USA)Sessions and their organizers: 1. Arithmetic theory of differentialequations and q-difference equations,Y. André (France); 2. Combinatorics,E. Steingrimsson (Iceland/Sweden);3. Geometric topology, J. Dupont(Denmark); 4. Group theory,R.G. Moller (Iceland); 5. Homologicalmethods in algebra and topology,B. Dundas and I. Reiten (Norway);6. Lie groups and harmonic analysis,F. Rouviere (France); 7. Mathematicalphysics, A. Kupiainen (Finland);8. Nonlinear partial differential equa-tions, K. H. Karlsen (Norway); 9. Realalgebraic geometry and applications(theme: Algebraic Geometry), Marie-Francoise Coste-Roy (France);10. Singularities (theme: AlgebraicGeometry), B. Teissier (France);11. Foliations in algebraic and arith-metical geometry (theme: AlgebraicGeometry), T. Ekedahl (Sweden);12. Moduli spaces (theme: AlgebraicGeometry), C. Faber (Sweden);13. Complex geometry (theme:

January 2005

CONFERENCES


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

Geometric Analysis), A. Tsikh(Russia/Sweden); 14. Quasiconformaltechniques in analysis (theme:Geometric Analysis), M. Zinsmeister(France); 15. Spectral geometry (theme:Geometric Analysis), R. Nest(Denmark); 16. Analysis on metricspaces (theme: Geometric Analysis),Juha Kinnunen (Finland).17. Percolation and spatial interaction(theme: Probability), O. Haggstrom(Sweden); 18. Applied probability andstochastic networks (theme:Probability), F. Baccelli (France);19. Branching processes (theme:Probability), P. Jagers (Sweden);20. Large random matrices (theme:Probability), Alice Guionnet (France).Information: e-mail: [email protected] Web site: http://www.raunvis.hi.is/1FrancoNordic Congress/

28–30: 4th Mediterranean Conferenceon Mathematics Education, Palermo-ItalyDescription: The Fourth MediterraneanConference on Mathematics Educationwith International participation followsa successful series of three such confer-ences since 1997. At this stage educatorsand researchers who may be interestedin attending the conference with or with-out a paper should send a message withfull address, fax and e-mail to the orga-nizing committee.Themes: Mathematics in the ModernWorld; Mathematics and Didactics;Mathematics and Society; Mathematicsand Talent; Mathematics and Sciences;Mathematics and Technology;Mathematics and Motivation;Mathematics and Statistics; History ofMathematicsDeadline: for full papers is 15 October2004Organizers: Cyprus MathematicalSociety and University of PalermoFormat: The conference will consist ofseveral sessions covering multiplethemes but all under the general aim ofMathematics Education.Grants: The conference will offer a twoday free accommodation to one presen-ter of each paper provided that theirpapers are accepted by the scientificcommittee. All papers will be reviewedby at least two referees.Local Organising Committee: Chair:F. Spagnolo, University of Palermo;Gianna Manno; Claudia Sortino;A. Scimone; Teresa Marino (all GRIM,Department of Mathematics, Universityof Palermo)International Organizing Committee:Chair: G. Makrides, Vice-Chair:F. Spagnolo; Secretary A: Gianna

Manno; Secretary B: Katerina Nicolaou;Treasurer A: Claudia Sortino; TreasurerB: A. Philippou; Members: MATHEUpartners; S. AntoniouScientific/Program Committee:A. Gagatsis; F. Spagnolo; B. D’Amore;N. Alexandris; S. Grozdev; G. Makrides;C. Christou; P. Damianou; DemetraPitta; A. Brigaglia; U. BottazziniInformation:e-mail: [email protected]; Web sites: http://math.unipa.it/~grim/mediterranean_05.htm; www.cms.org.cy

5–13: Applications of braid groupsand braid monodromy, EMS SummerSchool, Eilat, IsraelInformation: Web site: www.emis.de/etc/ems-summer-schools.html

9–April 1, 2005: 14th INTERNA-TIONAL WORKSHOP ON MATRI-CES AND STATISTICS (IWMS-2005), Massey University, AlbanyCampus, Auckland, New ZealandAim: to stimulate research and, in aninformal setting, to foster the interactionof researchers in the interface betweenstatistics and matrix theory. TheWorkshop will provide a forum throughwhich statisticians may be betterinformed of the latest developments andnewest techniques in linear algebra andmatrix theory and may exchange ideaswith researchers from a wide variety ofcountries. The International Workshopsin Matrices and Statistics are held annu-ally in different locations.

This Workshop will be the fourteenthin the series, with the previousWorkshops held at the following loca-tions: 1) Tampere, Finland, August1990; 2) Auckland, New Zealand,December 1992; 3) Tartu, Estonia, May1994; 4) Montreal, Quebec, Canada,July 1995; 5) Shrewsbury, England, July1996; 6) Istanbul, Turkey, August 1997;7) Fort Lauderdale, Florida, USA,December 1998; 8) Tampere, Finland,August 1999; 9) Hyderabad, India,December 2000; 10) Voorburg, TheNetherlands, August 2001; 11) Lyngby,Denmark, August 2002; 12) Dortmund,Germany, August 2003 (http://www.statistik.uni-dortmund.de/IWMS/main.html);13) Bedlêwo, Poland, August 2004(http:// matrix04.amu.edu.pl/). IWMS-2005 is a Satellite Conference tothe 55th Biennial Session of theInternational Statistical Institute to beheld in Sydney, April 5 - 12, 2005.

March 2005

February 2005

Format: The Workshop will includeinvited and contributed talks. Proceedings: It is intended that refereedConference Proceedings will be pub-lished.Local Organising Committee: JeffHunter (Chair) <[email protected]> International Organizing Committee:George Styan (Chair) <[email protected]>; Hans Joachim Werner (Vice-Chair) <[email protected]. uni-bonn.de> and Simo Puntanen<[email protected]>Information: Web site: http://iwms2005.massey.ac.nz/

12–24: Foliations 2005, £odz, PolandDescription: The conference is thefourth in a series devoted to the theoryof foliations. The previous three tookplace in 1990 (£odz), 1995 and 2000(both Warszawa). The main purpose ofthe conference is to interchange newideas in all aspects of foliation theoryand related topics: contact and symplec-tic structures, confoliations, Engel andGoursat structures, groups andpseudogroups of action on manifolds,holonomy groups and pseudogroups offoliations etc.Format: 2 mini-courses, invited lec-tures and contributed talksInvited speakers: V. Kaimanovich(mini-course), S. Matsumoto (mini-course), J. Alvarez Lopez, M. Asaoka,S. Fenley, V. Grines, X. Gomes Mont,J. Heitsch, Y. Kordyukov, H. Minakawa,K. Richardson, E. Zhuzhoma.Confirmation of speakers is in progress,see the conference web site: http://fol2005.math.uni.lodz.pl Organizers: Katedra GeometriiUniwersytetu £odzkiego (£odz),Banach Centre (Warszawa)Organizing Committee: S. Hurder(Chicago), R. Langevin (Dijon),T. Tsuboi (Tokyo), P. Walczak (£odz)and M. Czarnecki, secretary, (£odz) Location: Uniwersytet £odzki, £odz,PolandGrants: Several EU grants for youngmathematicians will probably be avail-ableInformation:e-mail: [email protected] Web site: http://fol2005.math.uni.lodz.pl

25–July 2 : Subdivision schemes ingeometric modelling, theory andapplications, EMS Summer School,Pontignano, ItalyInformation: Web site: www.emis.de/etc/ems-summer-schools.html

June 2005

CONFERENCES


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

B. Beckman: Codebreakers: Arne Beurling andthe Swedish Crypto Program during World WarII, American Mathematical Society, Providence,2002, 259 pp., $39, ISBN 0-8218-2889-4All stories about cryptanalytical work prior andduring the World War II read like a thriller. Thebook under review is no exception. The author, theformer head of the cryptanalytical section of FRA(The Radio Agency of the Swedish DefenceForces), presents a well of information. Its high-light is the detailed description of Arne Beurling’ssolution of the German Siemens machine, one ofthe most magnificent achievements of cryptanaly-sis of that time. During the previous century, in thethirties and the first half of the forties, the cryptan-alytical work of cryptographic “superpowers”(Poland, England, and USA) became dominatedby mathematicians. The transition of secret ser-vices to the search of cryptanalytical talentsamong graduates of mathematics departmentsprobably started in Poland and people like MarianRejewski became the founders of modern mathe-matically based cryptology. Well known are thecontributions of Alan Turing to the war efforts inBletchley Park. Arne Beurling is another, thoughless known, example of a top class mathematicianserving his country as a cryptanalytician duringthe war times.

The book starts with an explanation of what acode and a cipher is. Then the author overviewsthe Swedish cryptanalytical history to the end ofthe World War II. Most of the book is based ondeclassified Swedish documents and it traces thework of Arne Beurling in solving various ciphersused by different countries during the war, includ-ing the top secret German ciphers. For the review-er it is particularly interesting to read about theBeurling solution of the cipher used by a Czechresident in Sweden named Vanìk to communicatewith the Czechoslovak Government exiled inLondon. The second part of the book is dedicatedto the life and mathematical work of ArneBeurling, a close friend and collaborator of LarsAhlfors. The book can be highly recommended toanyone interested in the role of mathematics inclassical cryptology and in the history of the 20thcentury. (jtu)

J. Borwein, D. Bailey: Mathematics byExperiment. Plausible Reasoning in the 21stCentury, A. K. Peters, Natick, 2003, 350 p., $45,ISBN 1-56881-211-6J. Borwein, D. Bailey, R. Girgensohn:Experimentation in Mathematics.Computational Path to Discovery, A. K. Peters,Natick, 2004, 350 p., $49, ISBN 1-56881-136-5Despite the slightly different titles, the books formtwo volumes of the same publication.Mathematical experiments performed on comput-ers are more and more important factors of furtherdevelopment in the mathematics. The text is care-fully written and the plentiful short remarks onrelated topics are pleasant to read. The material ismostly accessible without knowledge of advanced

parts of modern mathematics. A certain familiari-ty with computer algebra programs like Maple orMathematica is helpful for a reader willing to trythese things in practice. The first volume of thework contains a gentle introduction to experimen-tal mathematics in its historical context and to itsmethodology, using a series of well-chosen exam-ples. The first chapter shows it as a rapidly devel-oping field of mathematics, the second containsfurther numerous illustrative examples. Thesequite often provoke a reader to work or play withthe examples on his own PC. The third chapterdescribes the progress in computing π, where bothauthors made significant contributions (BBP for-mula for computing π). A chapter on normality ofnumbers deals with another fascinating problem inwhich experimental mathematics promises at leastsome path to future discoveries. In contrast to thefact that the book introduces to the reader a highlyup-to-date part of mathematics, the next chapteroffers another look at classical themes like the fun-damental theorem of algebra, the Gamma functionor Stirling’s formula. After an exposition on basictools, the book is closed by a chapter with the title‘Making sense of experimental mathematics’.

For those wanting to know more on the pro-gression from numerical experiments to hypothe-ses and finally to deep mathematical theoremsproven within the frame of “classical mathemat-ics”, the authors prepared the second volume ofthe work together with R. Girgensohn. In fact,both volumes can be read independently.

The second volume starts with a chapter onsequences, series, products and integrals, which isfollowed by the chapter on Fourier series and inte-grals. These chapters form one third of the bookand will be of interest to any open-minded personwith a deeper interest in mathematics, equippedwith a basic knowledge of undergraduate mathe-matics. The following chapters on zeta functionsand multi-zeta functions, partition, powers, primesand polynomials are more specialised. The finaltwo chapters invite a reader to explore deepermethods and tools of experimental mathematics.Each chapter in both volumes is closed by com-mentaries and additional examples. The amount ofcollected material is tremendous. It is impossibleto describe the content of the whole work in detailin just a few lines. These are very nice and pro-voking books showing that experiments both wereand are an important part of the development ofmathematics. New computer-based tools arebroadly used mainly “outside” mathematics andthe book shows “… how today, the use ofadvanced computing technology provides mathe-maticians with an amazing, previously unimagin-able ‘laboratory’, in which examples can beanalysed, new ideas tested, and patterns discov-ered.” I strongly recommend visiting e.g. web-siteon URL http://www.expmath.info, where someparts of the first volume are also available. Alsothe very good typesetting and nice graphicalappearance of the whole work are worth mention-ing. It can be recommended not only to librariesbut to all members of the mathematical communi-ty. (jive)

T. Brzezinski, R. Wisbauer: Corings andComodules, London Mathematical Society

Lecture Note Series 309, Cambridge UniversityPress, Cambridge, 2003, 476 pp., £37,95, ISBN0 521-53931-5This is a first comprehensive monograph on cor-ings and comodules. Corings are generalizationsof coalgebras: while a coalgebra is an R-moduleover a commutative associative unital ring R,equipped with an R-linear coproduct and a comul-tiplication, a coring is an (A,A)-bimodule over anassociative, but not necessarily commutative, uni-tal R-algebra A, equipped with a (A,A)-bilinearcoproduct and comultiplication. Corings can alsobe viewed as coalgebras in a particular monoidalcategory (the ‘tensor category’). As observed byTakeuchi, important examples of corings are theso-called entwining structures connecting R-alge-bras with R-coalgebras. Moreover, in the particu-lar case of bialgebra entwinings, entwined mod-ules are exactly the classical Hopf modules. So -and this is the point of the book - the theory of cor-ings and comodules is a natural common general-ization of several theories connecting algebra withcategory theory, noncommutative geometry, andquantum physics.

The book consists of six chapters and anAppendix. After developing coalgebra andcomodule theory from the module-theoretic pointof view in Chapter 1, the authors deal with bialge-bras, classical Hopf algebras and their modules inChapter 2. The module theoretic approach ofChapter 1 pays back in Chapter 3, where basics ofthe coring and comodule theory are developed.Chapter 4 deals with an important class of coringscoming from ring extensions (together with theBOCS’s of Rojter and Kleiner, these were the firstexamples of corings truly generalizing coalge-bras). Chapters 5 and 6 deal with the relation toentwinings and weak entwinings. The Appendixrecalls basics on the module category σ[M], whichis one of the main tools for the theory. The reasonis that any right comodule over a coring C is a leftmodule over the left dual ring *C, and, for exam-ple, the so called ‘left α-condition’ just says thatthe category of all right C-comodules coincideswith σ[*C C]. The book provides a unified andgeneral treatment of a theory whose pieces wereoriginally developed by people working in ratherdistinct areas of algebra, category theory, non-commutative geometry, and mathematicalphysics. The book is a welcome addition to the lit-erature on this young and rapidly developing sub-ject. (jtrl)

V. M. Buchstaber, T. E. Panov: Torus Actionsand Their Applications in Topology andCombinatorics, University Lecture Series, vol. 24,American Mathematical Society, Providence,2002, 144 pp., $29, ISBN 0-8218-3186-0This book is an outgrowth of an extensive paper“Torus actions, Combinatorial topology andhomological algebra”, written by the authors andpublished in Uspekhi Mat. Nauk (Russian Math.Surveys) in 2000. It is yet another contribution tothe recent rich interplay of combinatorics and con-vex geometry with algebraic geometry and topol-ogy. In Chapters 1-5, the authors review basicnotions of both the combinatorial and topologicalside with a slight preference for combinatorialnotions (which they consider less familiar to aprospective reader). The last three chapters aredevoted to their own contribution, which is cen-tred around moment angle complexes, their coho-mology and to subspace arrangements. This is auseful book that will be of interest to the (algebra-ic) combinatorics community as well as toresearchers in algebraic geometry and topology. Itis a carefully written state of the art book. (jneš)

BOOKS


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

B. Buffoni, J. Toland: Analytic Theory of GlobalBifurcation, Princeton Series in AppliedMathematics, Princeton University Press,Princeton, Oxford, 2003, 169 pp., £29,95, ISBN0 691-11298-3In the book, bifurcation problems for non-linearoperator equations in infinite dimensional spacesare studied. To read the book requires certainknowledge, hence the first three chapters aredevoted to a review (without proofs) of basicnotions and facts from linear functional analysis,nonlinear functional analysis (e.g., the implicitfunction theory) and from the theory of analyticoperators in Banach spaces. Main facts on holo-morphic functions of several complex variables,real analytic functions of several real variables andon (finite-dimensional) analytic varieties, can befound in the next three chapters respectively.Analytic sets (in the complex setting, or their realversion) have a nice, distinguished structure,which can be used in a study of analytic operatorequations in infinite dimension. A tool needed forsuch a relation is a suitable version of the implicitfunction theorem, reducing infinite dimensionalquestions to finite dimension. The last two chap-ters contain applications of previous methods tosteady periodic water waves. (vs)

H. Cabral, F. Diacu, Eds.: Classical andCelestial Mechanics: The Recife Lectures,Princeton University Press, Princeton, 2002,385 pp., £35, ISBN 0-691-05022-8This book has its origin in a lecture series hostedby the Federal University of Pernambuco inRecife, Brazil, between 1993 and 1999.Immediately after opening the book, one gets afeeling of the great enthusiasm of its editors, orga-nizers of the corresponding series of lectures. It isa really attractive book, presenting many impor-tant topics from Hamiltonian dynamics and celes-tial mechanics in an accessible way. The editorsand the mathematical centre in the equator, colo-nial city of Recife, deserve great admiration forproducing such an impressive output - a record oftheir long term seminar, devoted to classicalmechanics. (Several contributions to the seminarwere published independently, before the publica-tion of this book.)

The book contains the following contributions:Central configurations and relative equilibria forthe N-body problem (by D. Schmidt),Singularities of the N-body problem (by F. Diacu),Lectures on the two-body problem (byA. Albouy), Normal forms of Hamilton systemsand stability of equilibria (by H. E. Cabral), ThePoincaré compactification and applications tocelestial mechanics (by E. Pérez-Chavela), Themotion of the moon (by D. Schmidt), Lectures ongeometrical methods in mechanics (by M. Levi),Momentum maps and geometric phases:Overview, classical adiabatic angles, holonomyfor gyrostats, microswimming (by J. Koiller et al.),and Bifurcation from families of periodic solu-tions (J. K. Hale and P. Táboas). (mzah)

A. Candel, L. Conlon: Foliations II, GraduateStudies in Mathematics, vol. 60, AmericanMathematical Society, Providence, 2003, 545 pp.,$79, ISBN 0-8218-0809-5This is the second volume of the two-volumeseries with the title Foliations. It has three inde-pendent parts, describing three special topics in thetheory of foliations: Analysis on foliated spaces,Characteristic classes of foliations and Foliated 3-manifolds. Each part contains a description of atopic in foliation theory and its relation to anotherfield of contemporary mathematics. In the first

part, the C*- algebras of foliated spaces are studiedand some of the classical notions fromRiemannian geometry (heat flow and Brownianmotion) are generalized to foliated spaces.Necessary analytic background can be found inthree appendices. The second part is devoted tocharacteristic classes and foliations. Here the read-er can find constructions of exotic classes based onthe Chern-Weil theorem, vanishing theorem forGodbillon-Vey classes and a discussion onobstructions to existence of a foliation transverseto the fibres of circle bundles over surfaces. In thethird part, compact 3-manifolds foliated by sur-faces are studied. Special methods of 3-manifoldstopology yield existence theorems and furtherresults unique for dimension three. There is anappendix with a proof and further discussion ofPalmeiras theorem, which says that the only sim-ply connected n-manifold foliated by leaves dif-feomorphic to Rn-1 is Rn. The book contains a lot ofinteresting results and can be recommended toanybody interested in the topic. (jbu)

X. Chen, K. Guo: Analytic Hilbert Modules,CRC Research Notes in Mathematics 433,Chapman & Hall/CRC, Boca Raton, 2003,201 pp., $99,95, ISBN 1-58488-399-5The main theme of the book is the structure ofmodules over function algebras of holomorphicfunctions in several complex variables. The bookconcentrates mainly on topics developed by bothauthors. The book starts with a nice introductionsummarizing main results described in the book.Chapter 2 describes the technique of characteristicspace theory for analytic Hilbert modules, whichwas developed by K. Guo. Chapter 3 shows thatanalytic Hilbert modules in several variables havea much more rigid structure than in one variable.The equivalence problem for Hardy submodulesin the cases of the polydisk or the unit ball is treat-ed in Chapter 4. The next chapter describes thestructure of the Fock space, or more generally,reproducing function spaces on Cn. Chapter 6 con-tains a discussion of modules over the Arvesonspace of square integrable functions on the unitball in Cn. In the last chapter, the authors describethe extension theory of Hilbert modules over func-tion algebras. The book is written in a clear andsystematic way and it describes an interesting partof the recent development in the field. Each chap-ter ends with useful bibliographic comments. (vs)

I. Dolgachev: Lectures on Invariant Theory,London Mathematical Society Lecture Note Series296, Cambridge University Press, Cambridge,2003, 220 pp., £29,95, ISBN 0-521-52548-9In many problems in mathematics, the structure ofthe set of all orbits of a group acting on a suitablespace is described by means of invariants of thegroup action. In particular, the case of the action ofa linear algebraic group G on an algebraic varietyX is the case studied in classical invariant theory.The present book is intended for beginners as afirst introduction to the theory. Knowledge of fun-damental notions and basic facts from algebraicgeometry is expected. The purpose of the book isto offer a short description of main ideas of theclassical invariant theory illustrated by many spe-cific examples. Every chapter ends with a set ofexercises and with hints for further reading.

The first two chapters treat the classical exam-ple of the action of the group GL(V) on homoge-neous polynomials of degree m on the space E,where E itself is the space of homogeneous poly-nomials of degree d on V. In the next chapter, theNagata theorem on the algebra of invariant poly-nomials on the space of a linear rational represen-

tation of a reductive algebraic group is proved.The next chapter is devoted to linear rational rep-resentations of a non-reductive algebraic group,including the Nagata counterexample to Hilbert’s14th problem. Covariants of the action are treatedin Chapter 5. Categorical and geometric quotientsand linearization of the action, a notion of stabili-ty of an algebraic action together with numericalcriterion of stability are described next. The lastthree chapters treat some special cases (hypersur-faces in projective space, ordered sets of linearsubspaces in projective space, and toric varieties).(vs)

R. M. Dudley: Real Analysis and Probability,Cambridge Studies in Advanced Mathematics 74,Cambridge University Press, Cambridge, 2002,555 pp., £32,95, ISBN 0-521-80972-XThis book serves as a clear, rigorous, and completeintroduction to modern probability theory usingmethods of mathematical analysis, and a descrip-tion of relations between the two fields. The firsthalf of the book is devoted to an exposition of realanalysis. Starting with basic facts of set theory, thebook treats e.g. the real number system, transfiniteinduction, and problems of cardinality, touchingboth the continuum hypothesis and axiom ofchoice together with its equivalences. Generaltopology is discussed, including compactness andcompactification, completion and completeness,and metric. Measure theory and integration istreated carefully, because it serves as the mostimportant tool for probability theory. Amongmore advanced topics of real and functional analy-sis, we can find here an introduction to functionalanalysis on Banach and Hilbert spaces, convexfunctions, convex sets and dualities, and measureson topological spaces. The second half of the bookcontains a description of modern probability theo-ry, including convergence laws, central limit theo-rems and laws of large numbers. Ergodic theory,as well as martingales, is studied. More advancedtopics include convergence laws on separable met-ric spaces, stochastic processes and Brownianmotion. The book can be considered as a textbook.It is a self-contained text and all relevant facts areproved. The appendices show that the author care-fully filled all gaps in mathematical backgroundneeded later. The book contains a number of exer-cises helping to understand the contents. It couldbe very useful for students interested in learningboth topics, it can also serve as complementaryreading to standard lectures. Teachers preparingtheir graduate level courses can use the book as anexcellent, rigorously written and complete source.(mrok)

J. Faraut, F. Rouviére, M. Vergne: Analyse surles groupes de Lie et théorie des représentations,Séminaires & Congrés 7, Société Mathématiquede France, Paris, 2003, 177 pp., €40, ISBN2-85629-142-2The book is based on lectures presented at thesummer school on analysis on Lie groups and rep-resentation theory, held in 1999 in Kénitra,Morocco. It contains a written form of three seriesof lectures. The first one (given by M. Vergne,recorded by S. Paycha) contains a description ofthe equivariant cohomology of a manifold, whichwas introduced independently by N. Berline andM. Vergne, E. Witten and M. F. Atiyah andR. Bott in the 80’s. In the special case of S1-actionwith isolated fixed points, the Paradan formula isused for a description of the equivariant cohomol-ogy in terms of fixed points of the action. As animportant application, it is shown how to obtainthe Duistermaat-Heckman stationary phase for-

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mula for a U(1)-action on a symplectic manifold. The second series of lectures (given by

F. Rouviére) was devoted to the Damek-Riccispaces. A manifold is harmonic if the mean valuetheorem holds for harmonic functions. TheDamek-Ricci spaces are harmonic manifolds butthey are not symmetric spaces. Basic facts fromgeometry and harmonic analysis on the Damek-Ricci spaces are described in the course.

The third series of lectures (given by J. Faraut)is devoted to Hilbert spaces of holomorphic func-tions invariant under the action of a group of auto-morphisms of a complex manifold. The set of suchHilbert spaces forms a convex cone and it is pos-sible to use methods of the Choquet theory for adescription of its structure and for an integral rep-resentation of their reproducing kernels. The lastparts are devoted to the case of invariant domainsin the complexification of a compact symmetricspace. A part of the school’s program wasreserved for lectures on the research work of par-ticipants, the list of lectures and their abstracts canbe found in the book. The book brings nicereviews of very interesting areas of the field. (vs)

M. Feistauer, J. Felcman, I. Straškraba:Mathematical and Computational Methods forCompressible Flow, Numerical Mathematics andScientific Computation, Clarendon Press, Oxford,2003, 535 pp., £59,95, ISBN 0-19-850588-4The book deals with the numerical solution ofequations describing motion of compressible flu-ids. Classical as well as modern numericalschemes are summarized here. At the beginning,the governing equations are determined and themathematical problems are defined. Some theoret-ical results, such as existence and uniqueness ofsolutions, are mentioned. The main part of thebook is concerned with the numerical solution ofinviscid as well as viscous compressible flows.Applications of various numerical methods (finitevolume method, finite element method, discontin-uous Galerkin method and their variants) for theEuler and Navier-Stokes equations are discussed.Convergence properties of numerical schemes aremostly derived for a scalar nonlinear convection-diffusion equation. Several types of mesh adapta-tion techniques are presented as well. In the bookthe reader can find a lot of numerical examplesdemonstrating the efficiency of described meth-ods. The book is suitable for researchers as well asfor students dealing with the numerical solution ofcompressible flows. (jdol)

W. Fenchel, J. Nielsen: Discontinuous Groupsof Isometries in the Hyperbolic Plane, de GruyterStudies in Mathematics 29, Walter de Gruyter,Berlin, 2003, 364 pp., €78,50, ISBN 3-1-017526-6In 1920’s and 1930’s, J. Nielsen published threelong papers in Acta Mathematica on discontinu-ous groups of isometries of the non-euclideanplane. His further research in the field led him tothe idea to describe the theory of discontinuousgroups of isometries systematically and in fullgenerality. After the Second World War, he start-ed a project along these lines together withW. Fenchel and they prepared the first version ofthe manuscript. W. Fenchel was able to continue(with his collaborators) the work on the projectafter J. Nielsen’s death and the typewritten versionof the manuscript was ready at the end of the 80’s.

The book under review is the final version ofthe manuscript, which was prepared for publica-tion by A. L. Schmidt after W. Fenchel’s death in1988. It offers a systematic geometric treatment ofdiscontinuous groups of isometries. It starts with acareful description of Möbius transformations of

the Riemann sphere and their use in non-euclideangeometry. The second chapter contains a detaileddescription of discontinuous groups of motions ofthe unit discs with its hyperbolic metric. In thethird chapter, associated surfaces and invariantsneeded for a classification are discussed.Elementary groups and elementary surfaces, thedecompositions of the discontinuous groups andtheir normal forms, are all treated in the fourthchapter. The final chapter is devoted to isomor-phisms of discontinuous groups, homeomor-phisms of the corresponding discs and their exten-sions to the boundary. A specific feature of thebook is that it is based entirely on geometric argu-ments. The Fenchel-Nielsen manuscript has beenfamous for a long time already and its final publi-cation is a valuable edition to mathematical litera-ture. (vs)

M. Georgiadou: Constantin Carathéodory,Mathematics and Politics in Turbulent times,Springer, Heidelberg, 2004, 651 pp., €89,95,ISBN 3-540-44258-8This is an excellent biography of a man whobelongs to the most renowned mathematicians ofthe first half of the 20th century. More than 50years after his death – much later than his contri-bution to the science deserves – this book bringshis personality and work to life again. It depictsthem in full context with that period, so rich on farreaching revolutionary events, in a very lively andsuggestive way. Being a German mathematicianborn in an outstanding Greek family of diplomats,Constantin Carathéodory was attached to Germanintellectual tradition by his German education, aswell as to Greece by his emotions. Moreover, aman of his intellectual influence must have alsobeen a man of political significance. Because ofthese facts and in connection with the dramaticperiod in which he lived, he is very suitable sub-ject for a biography. The author has gathered withexceptional care almost all accessible material thathas a bearing to the life of this scientist, and hasbuilt a narrative that will fascinate everyone whoopens the book. (jdr)

M. Gidea, C. P. Niculescu: Chaotic DynamicalSystems: An Introduction, Centre for NonlinearAnalysis and its Applications 3, UniversitariaPress, Craiova, 2003, 239 pp., €10, ISBN973-8043159-9 The book aims to be an introduction to the theoryof dynamical systems on the graduate level. It cov-ers all classical topics of the subject, includingstructural stability, Lyapunov exponents, thehorse-shoe dynamics, hyperbolic and symbolicdynamics, chaotic attractors, entropy, ergodicityand invariant measures. The particular emphasis isgiven to the concept of chaos and the variousmathematical tools for its understanding. Theexposition is clear and concise, yet perfectly rigor-ous. Many simple examples and a number of exer-cises serve excellently to the purpose of the bookto be an elementary introduction. The only minorsetback seems to be a somewhat lesser quality ofthe print. (dpr)

M. Gross, D. Huybrechts, D. Joyce: Calabi-YauManifolds and Related Geometries, Universitext,Springer, Berlin, 2003, 239 pp., €49,95, ISBN3-540-44059-3Summer schools in Nordfjordeid, Norway, havebeen organized regularly since 1996. Topics varyeach year but there are always three series of lec-tures by invited experts with evening exercises.The school held in June 2001 was devoted torecent interaction between differential and alge-

braic geometry. The book consists of notes writtenby lecturers of the corresponding three series oflectures. The first contribution (by D. Joyce) isdevoted to the introduction and study ofRiemannian properties. The last contribution (byD. Huybrechts) describes compact hyperkählermanifolds. The class of compact hyperkähler man-ifolds form an interesting class of Ricci-flat mani-folds, playing an important role in differentbranches of mathematics and mathematicalphysics. In this part, many interesting topics arepresented (including theory of holomorphic sym-plectic manifolds, deformation theory of complexstructures, cohomology and other properties ofcompact hyperkähler manifolds, twistor space andmoduli space of a hyperkähler manifold and a dis-cussion of the projectivity of hyperkähler metrics).The themes of all three contributions are interre-lated and together they give a nice introductioninto a very interesting field of research on the bor-der between mathematics and physics. I wouldlike to strongly recommend the book to anybodyinterested in the topic. (jbu)

B. C. Hall: Lie groups, Lie algebras, and repre-sentations: An Elementary Introduction,Springer, Heidelberg, 2003, 351 pp., €59,95,ISBN 0-387-40122-9To present a circle of ideas around Lie groups, Liealgebras and their representations, it is necessaryto make a few principal choices. The first questionis how to describe a relation between Lie groupsand Lie algebras. To make the book accessible toa broader audience, the author does not supposeknowledge of the theory of manifolds. He restrictsthe attention to matrix groups. Lie algebra of G isthen defined using simple properties of the expo-nential map. As for the correspondence betweenLie group homomorphisms and Lie algebra homo-morphisms, the author is using the Baker-Campbell-Hausdorff theorem for its description.

In the main part of the book, finite dimensionalrepresentations of classical semisimple Lie groupsare classified by their highest weights. Their con-struction is given in three different ways (as quo-tients of Verma modules, or by the Peter-Weyltheorem, or by the Borel-Weil realization). Theproof of the complete reducibility is based onproperties of representations of compact groups.The Weyl character formula and the classificationof complex semisimple Lie algebras end the mainpart of the book. To keep prerequisites minimal,the author also offers a few appendices. The bookis written in a systematic and clear way, eachchapter ends with a set of exercises. The bookcould be valuable for students of mathematics andphysics as well as for teachers, for the preparationof courses. It is a nice addition to the existing lit-erature. (vs)

U. Hertrich-Jeromin: Introduction to MöbiusDifferential Geometry, London MathematicalSociety Lecture Note Series 300, CambridgeUniversity Press, Cambridge, 2003, 413 pp.,£29,95, ISBN 0-521-53569-7The book is an introduction to the geometry ofsubmanifolds of the conformal n-sphere. AMöbius transformation is a conformal (i.e., anglepreserving) transformation of the sphere. Thesphere can be considered as a homogeneous spaceof the group of Möbius transforms, hence it is amodel of Möbius geometry from F. Klein’s pointof view. There are several other models of Möbiusgeometry (projective model, quaternionic modeland Clifford algebra model). Classically, a confor-mal structure is represented by a Riemannian met-ric (modulo a multiplication by a positive func-

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tion). The change of Riemannian properties underconformal change is described and the Weyl andSchouten tensors are introduced. Conformal flat-ness is discussed in details. The projective modeland related objects (congruencies, etc.) are intro-duced in the first chapter. The Cartan method ofmoving frames is often used for computations. Asan application of projective model constructions,conformally flat hypersurfaces, isothermic andWillmore surfaces are discussed. A quaternionicmodel introduced in the next chapter is used for astudy of isothermic surfaces in the four-dimen-sional case, while in higher dimensions, theClifford algebra model is used. At the end of thebook, the reader can find a discussion of triplyorthogonal systems, their Ribaucour transforma-tions and isothermic surfaces of arbitrary codi-mension. The book is a well-written survey ofclassical results from a new point of view and anice textbook for a study of the subject. (jbu)

J. Ize, A. Vignoli: Equivariant Degree Theory,de Gruyter Series in Nonlinear Analysis andApplications 8, Walter de Gruyter, Berlin, 2003,361 pp., €98, ISBN 3-11-017550-9The aim of the book is the development and appli-cations of the degree theory in the context of equi-variant maps. (Equivariant simply means that themapping has certain symmetries, e.g., beingeven/odd, periodic, rotational invariant, etc.). Thetheory is developed both in finite and infinitedimension. The first chapter gives necessary pre-liminaries. The second chapter brings the defini-tion of the degree and studies its basic properties.As the definition is somewhat abstract (the degreeis defined as an element of the group of equivari-ant homotopy classes of maps between twospheres), it is useful to compute the degree in var-ious particular cases. This is accomplished inChapter 3. The last and also the longest chapter,deals with applications to particular ODE’s and tobifurcation theory. The aim of the authors was towrite a book that would be easily accessible evento non-specialists, thus the exposition is accompa-nied by a number of examples and the use ofabstract special tools is limited. It is also worthnoting that each chapter is accompanied bydetailed bibliographical remarks. (dpr)

C. U. Jensen, A. Ledet, N. Yui: GenericPolynomials: Constructive Aspects of the InverseGalois Problem, Mathematical Sciences ResearchInstitute Publications 45, Cambridge UniversityPress, Cambridge, 2003, 258 pp., £45, ISBN0-521-81998-9The inverse Galois problem is to determine, for agiven field K and a given finite group G, whetherthere exists a Galois extension of K, whose Galoisgroup is isomorphic to G. And if there is such anextension, to find an explicit polynomial over K,whose Galois group is the prescribed group G. Theauthors present a family of “generic” polynomialsfor certain finite groups, which give all Galoisextensions having the required group as theirGalois group. The existence of such generic poly-nomials is discussed and a detailed treatment oftheir construction is given in those cases, whenthey exist. (jtu)

D. S. Jones, B. D. Sleeman: DifferentialEquations and Mathematical Biology, Chapman& Hall/CRC Mathematical Biology and MedicineSeries, Chapman & Hall/CRC, Boca Raton, 2003,390 pp., $71,96, ISBN 1-58488-296-4The book is written with two aims: firstly, to be anintroduction both to ordinary and partial differen-tial equations; secondly, to present main ideas on

how to model deterministic (and mostly continu-ous) processes in biology, physiology and ecolo-gy. The style of writing is subordinated to thesepurposes. It is remarkable that without the classi-cal scheme (definition, theorem and proof) it ispossible to explain rather deep results like proper-ties of the Fitz-Hugh-Nagumo model of nerveimpulse transmission or the Turing model of pat-tern formation. This feature makes the reading ofthis text pleasant business for mathematiciansalso. There exists a similar book written byJ. D. Murray (Mathematical Biology), which con-tains more biological models. In comparison withit, the book under review is also a textbook on dif-ferential equations. It can be recommended forstudents of mathematics who like to see applica-tions, because it introduces them to problems onhow to model processes in biology, and also fortheoretically oriented students of biology, becauseit presents constructions of mathematical modelsand the steps needed for their investigations in aclear way and without references to other books.(jmil)

J. W. Kammeyer, D. J. Rudolph: RestrictedOrbit Equivalence for Actions of DiscreteAmenable Groups, Cambridge Tracts inMathematics 146, Cambridge University Press,Cambridge, 2002, 201 pp., £35, ISBN0-521-80795-6The main topic of the book belongs to ergodic the-ory and measurable dynamics. It studies suitablenotions of similarity of two dynamical systemsusing structure of orbits of corresponding systems.The authors discuss free and ergodic actions ofcountable discrete amenable groups. The keynotions in the book are an orbit relationO={x,Tg(x)}g∈G ⊂ X x X generated by the action,an orbit equivalence (a measure preserving mapcarrying one orbit relation to the other), andarrangements and rearrangements of orbits. Basicdefinitions and examples can be found in Chapter2 (the vocabulary of arrangements and rearrange-ments, and m-equivalence classes of an arrange-ment). Fundamental results by Ornstein and Weisson ergodic theory of actions of amenable groupsare reviewed in Chapter 3. The next two chaptersinclude key technical lemmas and entropy theoryfor restricted orbit equivalences. Chapter 6 con-tains a construction of a topological model forarrangements and rearrangements using a notionof Polish spaces and Polish actions. The last chap-ter contains a formulation and a proof of the equiv-alence theorem. Similar questions were studiedcarefully in the last decades for Z-actions as wellas for Zd -actions (d≥ 1). Relations of the resultsdescribed in the book to results obtained in thesespecial cases can be found in the Appendix. (vs)

V. V. Kravchenko: Applied QuaternionicAnalysis, Research and Expositions inMathematics, vol. 28, Heldermann, Lemgo, 2003,127 pp., €24, ISBN 3-88538-228-8Function theory for the Dirac equation has grownsubstantially during the last several decades. Thecase of dimension four is special in two ways – itis a physical dimension and spinor fields can beidentified with quaternionic functions of a quater-nionic variable. Obtained results were applied to awide spectrum of problems in mathematicalphysics. The first part contains a summary of basicfacts from quaternionic analysis, including inte-gral formulae for unbounded domains. The selec-tion of topics is guided by their possible applica-tions. Boundary problems for Maxwell equationsin homogeneous media and the Dirac equation fora free particle are treated in the second part. The

last part contains a discussion of Maxwell equa-tions for inhomogeneous media, the Dirac equa-tion with potentials, and a generalization of theRiccati equation to a nonlinear quaternionic equa-tion. The book can be useful not only for mathe-maticians interested in the field but also for engi-neers (the book is based on a course given by theauthor to future engineers). (vs)

T. Lawson: Topology: A Geometric Approach,Oxford Graduate Texts in Mathematics 9, OxfordUniversity Press, Oxford, 2003, 388 pp., £45,ISBN 0-19-851597-9The book is a nice introduction to topology withan emphasis on its geometric aspects. It is writtenfor two purposes. In the first part, consisting ofthree chapters, there is material suitable for a onesemester basic course of topology. It starts withbasic point set topology with special attention tothe topology of Rn, followed by a description of theclassification of surfaces. The last topic introducedand studied in this part is the fundamental group ofa space, including its application to surfaces andthe vector field problem in the plane and on sur-faces. To compute fundamental groups, theSeifert-van Kampen theorem is introduced andproved.

The second part contains an extension of thematerial of the first part to a full-year course. Itstarts with the description of covering spaces, cov-ering transformations and universal coveringspace, followed by a study of CW-complexes andtheir properties from a homotopy point of view.Simplicial complexes and ∆-complexes forCW-complexes are described as well. The lastchapter is devoted to the homology theory withspecial attention to its relations to homotopy.There are about 750 exercises of different levels,which can attract students to a more active studyof the subject. Solutions of selected exercises areincluded as an appendix (solutions to all exercisesare available to the instructor in electronic form onapplication to Oxford University Press). The bookis an excellent introduction to the subject and becan recommended to anybody interested in geom-etry and topology as his/her first reading. (jbu)

J. Lewin: An Interactive Introduction toMathematical Analysis, Cambridge UniversityPress, Cambridge, 2003, 492 pp., £27,95, ISBN0-521-01718-1The book under review provides an introduction tomathematical analysis using possibilities of com-puters. The book starts with some backgroundmaterial concerning quantifiers, sets, logic formu-las, and methods of proofs. The second part dealswith the set of real numbers, limits of sequencesand functions, differentiation of functions, theRiemann integral, and infinite series. The nextchapters are devoted to improper integrals,sequences and series of functions and integrationof functions of two variables. The enclosed CDforms an important on-screen part of the book.The CD contains a lot of exercises with solutions.Some parts of the book are explained here in amore detailed way and some chapters are added(e.g., calculus of several variables or complexvariable calculus) or are treated in an alternative(advanced) way. This book together with the CDwill be useful to many students. (mzel)

E. L. Lima: Fundamental Groups and CoveringSpaces, A. K. Peters, Natick, 2003, 210 pp., $49,ISBN 1-56881-131-4The book is an introduction to a part of algebraictopology. It concentrates on the circle of ideasaround homotopy theory, fundamental groups and

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covering spaces. The first part of the book intro-duces a general concept of a homotopy, togetherwith a particular case of path homotopy. The fun-damental group is defined and computed for manyexamples, including real and complex projectivespaces and classical matrix groups. The funda-mental group of the circle is related to windingnumbers of plane curves. The second part of thebook is devoted to basic properties of (differen-tiable) covering spaces and their relations to fun-damental groups. Each chapter ends with exercis-es. The book starts from the beginning and itsreading requires only a basic knowledge. Thebook is a pleasure to read, it contains a lot of illus-trations and presentation is clear and systematic.(vs)

M. Métivier: Semimartingales. A Course onStochastic Processes, de Gruyter Studies inMathematics 2, Walter de Gruyter, Berlin, 1982,287 pp., €68, ISBN 3-11-008674-3The presented monograph is an advanced book ongeneral martingale theory. A reader with a goodknowledge of probability and discrete timeprocesses will appreciate the deep results con-tained in the book. Theorems are formulated forquasimartingales, some parts of the book aredevoted to Hilbert space valued processes, and thetheory is not restricted to continuous square inte-grable martingales as is usually the case. Due tothese facts, the book is a necessity for allresearchers working with general stochasticprocesses.

The book consists of two parts. In the first partthe general theory of martingales is given. It startswith basic definitions and facts from stochasticprocesses; filtration, measurability, adapted andpredictable processes, stopping time, and decom-position. The next chapters continue with the mar-tingale property. We find many classical results ina quite general context like Doob’s inequalities orconvergence theorems. The first part is concludedby a chapter devoted to square integrable semi-martingales, quadratic variation and Meyer’sprocess. Here we also find the theory of Hilbertspace valued martingales and stochastic integralswith respect to them. In the second part, stochasticcalculus is the focus. First, the stochastic integralis built and a semimartingale version of Itô trans-formation theorem is proved. Then, as an applica-tion, the Brownian and Poisson processes are con-sidered and changes of probability, as well asGirsanov formula, are presented. The book culmi-nates with a chapter on SDE. This is not a bookwhich I would recommend as an introduction tostochastic processes and martingales. But for anyrigorous work in the theory of martingales, name-ly non-continuous processes, semimartingales andmultidimensional martingales, it is the book thatshould be consulted first. It is not easy to write aconcise book on general martingale theory but inmy opinion Michel Métivier has done great job.(dh)

I. Moerdijk, J. Mrèun: Introduction toFoliations and Lie Groupoids, CambridgeStudies in Advanced Mathematics 91, CambridgeUniversity Press, Cambridge, 2003, 173 pp., £30,ISBN 0-521-83197-0This is just a small book nicely covering principalnotions of the theory of foliations and its relationsto recently introduced notions of Lie groupoidsand Lie algebroids. After the first chapter, con-taining a definition of a foliation and main exam-ples and constructions, the authors introduce thekey notion of holonomy of a leaf, a definition of anorbifold and they prove the Reeb and the Thurston

stability theorems. Chapter 3 contains theHaefliger theorem (there are no analytic foliationsof codimension 1 on S3) and the Novikov theorem(concerning existence of compact leaves in a codi-mension 1 transversely oriented foliation of acompact three-dimensional manifold). TheMolino structure theorem for foliations defined bynonsingular Maurer-Cartan forms is treated inChapter 4. A holonomy groupoid of a foliation isa basic example of so called Lie groupoids. Thelast two chapters describe properties of Liegroupoids, a notion of weak equivalence betweenLie groupoids, a special class of étale groupoids,and a Lie algebroid as an infinitesimal version of aLie groupoid. The book is based on course lecturenotes and it still keeps its qualities and nice pre-sentation. (vs)

T. Mora: Solving Polynomial Equation SystemsI: The Kronecker-Duval Philosophy,Encyclopedia of Mathematics and its Applications88, Cambridge University Press, Cambridge,2003, 423 pp., £60, ISBN 0-521-81154-6This is an excellent book for readers interested inalgebraic methods. A part of the content will notbe new to most of them; its usefulness lies in thefact that so much is brought together in one book.In the first part of the book, the author describesthe Kronecker-Duval approach to the solution ofsystems of polynomial equations. The second partcontains a discussion of factorisation of polynomi-als. The author says: “It is my firm belief that thebest way of understanding a theory and an algo-rithm is to verify it through computation …”Accordingly, the book contains 52 numericalexamples provided with solutions and 27 pro-grams. I enjoyed the author’s language enormous-ly. The author’s words from the preface “the num-ber of hidden mistakes in a draft is always largerthan the number of the found ones” are true. I spot-ted a few, e.g., in Theorem 5.2.3 as well as inTheorem 5.5.6, one has to add the assumption thatthe degree of considered polynomials is at leastone. This criticism of the text is a little more thana quibble and, in any case, is greatly outweighedby its virtues.(lber)

J. Nestruev: Smooth Manifolds andObservables, Graduate Texts in Mathematics, vol.220, Springer, New York, 2003, 222 pp., €64,95,ISBN 0-387-95543-7Main themes of the book are manifolds, fibre bun-dles and differential operators acting on sectionsof vector bundles. A classical treatment of thesetopics starts with a coordinate description of amanifold M; the algebra of smooth functions on Mis a derived object in this approach. The presentbook is based on an alternative point of view,where calculus on manifolds is treated as a part ofcommutative algebra. In particular, the initialobject is a commutative associative unital algebraF with certain additional properties. The corre-sponding smooth manifold is reconstructed as thespectrum of F. (A generalization to the non-com-mutative case, which is usually called non-com-mutative geometry, is based on this point of view.This generalization, however, is not treated in thebook.)

The first few chapters describe properties ofalgebras that correspond to smooth manifolds,introduce a notion of charts and atlases and definesmooth maps between such manifolds. Theauthors (J. Nestruev is an invented name hiding agroup of authors) then show equivalence of thealgebraic definition with the usual one. In the sec-ond part of the book, the authors define tangent

and cotangent fibre bundles of a manifold, jet bun-dles and they introduce linear differential opera-tors in this algebraic setting. As explainedthroughout the book, and in particular in theappendix (written by A. M. Vinogradov), it is pos-sible to give a motivation coming from classicalmechanics for basic notions treated in the book.The commutative algebra F is related to the labo-ratory itself, elements of F to measuring devicesand points in the spectrum of F to states of anobserved physical system. The book containsquite a few exercises and many useful illustra-tions. (vs)

W. K. Nicholson, M. F. Yousif: Quasi-Frobenius Rings, Cambridge Tracts inMathematics 158, Cambridge University Press,Cambridge, 2003, 307 pp., £55, ISBN0-521-81593-2Quasi-Frobenius algebras provide the basic settingfor modular representation theory of finite groups.Indeed, the group algebra of any finite group isquasi-Frobenius. The presented book is a veryaccessible introduction to basic properties ofquasi-Frobenius rings and the modules over them.Rather than dealing with classical representationtheory, the authors consider a more general settingof mininjective rings and show that basics of theclassical theory can be developed using only ele-mentary module theory in a more general setting.(A ring is right mininjective, if any isomorphismbetween minimal right ideals is induced by a leftmultiplication. By a theorem of Ikeda, quasi-Frobenius rings are exactly the right and left min-injective, right and left artinian rings).

While basic notions and results on (weak) self-injectivity, CS- and C2-conditions, AB5*, anddualities are developed through Chapters 2-7, thereader is gradually introduced into three challeng-ing open problems: the Faith Conjecture (whetherevery left or right perfect right self-injective ring isquasi-Frobenius), the FGF-Conjecture (whetherthe condition that every finitely generated moduleembeds in a free module implies that the ring isquasi-Frobenius), and the Faith-Menal Conjecture(asking whether any right strongly Johns ring isquasi-Frobenius. A ring R is right Johns, if R isright noetherian and each right ideal of R is anannihilator; R is right strongly Johns, if all fullmatrix rings Mn(R), n ≥ 1 are Johns). The latterconjecture is investigated in Chapter 8, where anexample is given that the conjecture fails if theterm ‘strongly’ is omitted. Chapter 9 deals with theFaith Conjecture, providing a generic constructionof examples using particular 3x3-upper triangularmatrix rings with coefficients in bimodules overdivision rings.

The book concludes with three Appendices: onMorita theory of equivalence, on Bass’ theory ofperfect rings, and on the Camps-Dicks Theorem(proving that the endomorphism ring of any artin-ian module is semilocal). The authors haveachieved two seemingly incompatible goals: toprovide an elementary introduction to the classicaltheory of quasi-Frobenius rings, and to bring thereader up to the current research in the field. Thismakes the book interesting both for graduate stu-dents and researchers in contemporary moduletheory. (jtrl)

L. I. Nicolaescu: The Reidemeister Torsion of 3-Manifolds, de Gruyter Studies in Mathematics,vol. 30, Walter de Gruyter, Berlin, 2003, 249 pp.,€84, ISBN 3-11-017383-2This is a book on the Reidemeister torsion and its(mostly Turaev’s) generalizations. When compar-ing it with Turaev’s book (Introduction to

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Combinatorial Torsions, Lectures in Mathematics,ETH Zürich, Birkhäuser, 2001), which appearedrecently, we can immediately see that it has a dif-ferent character. While Turaev’s book can alsoserve as a kind of introduction to the subject (aswell as an introduction to the contemporaryresearch in the field), the book under review isdevoted to a wide range of applications of the tor-sion, and its reading requires certain prerequisites.

In the first chapter, which makes the readerfamiliar with the necessary algebraic notions, thereader is supposed to have some topological (andalso algebraic) background. The author modestlystates in the introduction that this is a computa-tionally oriented little book. But let us note that the“computations” we find here are very clever com-putations, and the wide variety of applications pre-sented here do not support the description of a lit-tle book. The book will be indispensable for spe-cialists in the field, and I think that it is very goodthat it exists together with Turaev’s book. It is verywell written, with many examples and also manyexercises. The wide range of applications will beinteresting not only for topologists but also for dif-ferential geometers. (jiva)

S. P. Novikov, V. A. Rohlin (Eds.): Topology II,Homotopy and Homology, Classical Manifolds,Encyclopedia of Mathematical Sciences, Springer,Berlin, 2004, 257 pp., ISBN 3-540-51996-3The volume under review consists of three chap-ters: Introduction to Homotopy Theory (byO. Ya. Viro and D. B. Fuchs), Homology andCohomology (again by O. Ya. Viro andD. B. Fuchs), and Classical Manifolds (byD. B. Fuchs). The original Russian text was trans-lated by C. J. Shaddock. The presentation ofnotions and results in all three chapters is reallyvery nice. The first two chapters contain quitedetailed exposition, while the third chapter has amore encyclopedia like character. This means thatthe first two chapters can also serve very well as atextbook. I would even recommend them for thispurpose, because the presentation is on one hand adetailed one (as already mentioned), and on theother hand it is not too long. The choice of mater-ial is very good, the text is saturated with manyexamples, and we find here information about fur-ther developments and recommendations for fur-ther study. Naturally, because this is an encyclo-pedia, we find no exercises. The last chapter con-tains information about the topology of classicalmanifolds, and I do not think that information ofthis type, in such a compact form and to such anextent, can be found elsewhere. Generally, thewhole volume makes a very good impression, andI would say that it is very clearly written. (jiva)

H. Pajot: Analytic Capacity, Rectifiability,Menger Curvature and the Cauchy Integral,Lecture Notes in Mathematics 1799, Springer,Berlin, 2002, 118 pp., €22,95, ISBN3-540-00001-1The question of characterization of removable setsfor bounded analytic functions is known as thePainlevé problem and has attracted the interest ofmathematicians for more than 115 years. In areformulation due to L. Ahlfors, the point is todescribe for which sets the analytic capacity van-ishes. In 1977, A. P. Calderón proved the DenjoyConjecture that one-dimensional rectifiable curvesare removable if and only if their Hausdorff lengthis zero. The proof was based on estimates of sin-gular integrals, namely of the Cauchy operator.The version of the Vitushkin conjecture that apurely unrectifiable set of finite Hausdorff lengthis removable has been proved in 1998 by G.David,

after significant steps forward by M. Melnikov,V. Verdera, P. Mattila, M. Christ and others. Inthis research the Menger curvature appeared to berelevant. In 2003, X. Tolsa characterized remov-able sets in terms of the Menger curvature. In thisvolume, the author presents the above mentionedresults and related developments.

The first two chapters are devoted to theHausdorff measure and rectifiability, includingbeta numbers and uniform rectifiability. TheMenger curvature is explained in Chapter 3. InChapter 4, the theory of singular integrals isapplied to the Cauchy operator. The Painlevéproblem and the analytic capacity are discussed inChapter 5. In Chapter 6, the proofs of the Denjoyconjecture and of the Vitushkin conjecture as wellas related results are presented. Some very recentresults including the Tolsa theorem are given inChapter 7. The book excellently explains thisbeautiful theory, which is a subtle mix of complexanalysis, harmonic analysis and geometric mea-sure theory. The text is almost self-contained. Thehistory of this development is nicely reviewed. Atthe end, main open problems of the theory are list-ed. (jama)

J. Roe: Lectures on Coarse Geometry, UniversityLecture Series, vol. 31, American MathematicalSociety, Providence, 2003, 175 pp., $39, ISBN0-8218-3332-4The book is based on lectures given by the authorat Penn State University. The notion of ‘coarsegeometry’ was invented to keep track of largescale properties of metric spaces. The first part ofthe book introduces an abstract notion of a coarsestructure, a notion of a bounded geometry coarsespace, its growth and a notion of an amenable met-ric space, and discusses coarse algebraic topology.The main topic in the middle part is the Mostowrigidity theorem saying that if two compact hyper-bolic manifolds of dimensions at least 3 are homo-topy equivalent, they are isometric. The last part ofthe book contains a discussion of a notion ofasymptotic dimensions and uniform embeddingsinto Hilbert spaces, together with relations to theKazhdan property of discrete groups. The bookoffers a very readable description of a circle ofideas around the notion of coarse geometry. (vs)

C. Sabot: Spectral Properties of Self-SimilarLattices and Iteration of Rational Maps,Mémoires de la SMF 92, Société Mathématique deFrance, Paris, 2003, 104 pp., €25, ISBN2-85629-133-3The method of a renormalization group, widelyused in physics, produces some considerablemathematical difficulties, when applied to systemson usual, regular lattices. Hierarchical, self-similarlattices are much better suited for a detailed math-ematical development of this method. The presentbook gives a detailed treatment of the problems,which were first studied by Rammal and Toulouseon the Sierpiñski gasket, and puts them in a moregeneral perspective. The goal of the book is tostudy systematically, in a mathematically rigorousway, the spectral properties of Laplace operatorson self-similar sets. A new renormalization map,which is a rational map defined on a smooth pro-jective variety, is introduced. Then, the character-istics of the spectrum of these operators are relat-ed with the characteristics of the dynamics of iter-ates of such a renormalization map. An explicitformula for the density of states is given, and it isshown that the spectral properties of the operatordepend substantially on the asymptotic degree ofthe renormalization map. The contents (a slightlyshortened list of the chapters of the book) are:

Definitions and basic results (self-similarLaplacian, density of states); Preliminaries(Grassmann algebra, trace of a Dirichlet form);The renormalization map (construction, the maintheorem in the lattice case); Analysis of the pshfunction G (the dichotomy theorem, asymptoticdegree, regularity of the density of states, somerelated rational maps); Examples; Remarks, ques-tions and conjecture; and Appendix (plurisubhar-monic functions and positive currents, dynamicsof rational maps on projective space, iteration ofmeromorphic maps on compact manifolds, G -Lagrangian Grassmanian). (mz)

I. Stewart: Galois Theory, third edition,Chapman & Hall/CRC Mathematics, Chapman &Hall/CRC, Boca Raton, 2003, 288 pp., $44,96,ISBN 1-58488-393-6This is the third edition of the classic textbook ofGalois theory, first published in 1972. But it is notjust a reprint of earlier editions. Those who knowthe first two editions will be surprised by a radicalchange of presentation. The author reversed theoriginal Bourbakiste approach expressed by a slo-gan “from general to concrete” and now presentsthe theory in the direction “from concrete to gen-eral”.

Thus after a historical chapter, he starts withsolutions in radicals of polynomial equations ofdegree 2, 3, 4, and presents a quintic equationsolvable in radicals. Factorization of complexpolynomials is developed from theory of polyno-mials with complex coefficients and the funda-mental theorem of algebra. Field extensions ofrational numbers follow including the definition ofrational expressions and the degree of an exten-sion. As a digression, the author proves non-exis-tence of ruler-and-compass solutions of classicalgeometric problems of squaring the cube, trisect-ing the angle and squaring the circle. Then theGalois theory starts. After a short explanation ofGalois groups according to Galois, he presentsmodern definitions of the Galois correspondence,splitting fields, normal and separable extensions,and field automorphisms. The fundamental Galoiscorrespondence between the subfields of a fieldextension and subgroups of the automorphismgroup of the extension is proved. An example ofthe correspondence resulting from a quartic equa-tion is also given. Solvable and simple groups areintroduced and the Galois theorem about solvabil-ity of equations in radicals is proved.

After all this, abstract rings and fields are intro-duced and the abstract theory of field extensions isdeveloped. The last part of the book contains someapplications, e.g., the construction of finite fields,constructions of regular polygons, circle divisions(including cyclotomic polynomials) and an algo-rithm on how to calculate the Galois group of apolynomial equation. The penultimate chapter isabout algebraically closed fields and the last chap-ter, on transcendental numbers, contains “what-every-mathematician-should-see-at-least-once”,the proof of transcendence of π. The book isdesigned for the second and third year undergrad-uate courses. I will certainly use it. (jtu)

D. Stirzaker: Elementary probability, Secondedition, Cambridge University Press, Cambridge,2003, 536 pp., £65, ISBN 0-521-83344-2This book provides an introduction to elementaryprobability and some of its applications. The wordelementary means that the book does not need theabstract Lebesgue measure and integration theory,only elementary calculus is used. The strong fea-ture of the textbook is a choice of good examples.Each theoretical explanation is accompanied by a

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large number of examples and followed byworked examples incorporating a cluster of exer-cises. The examples and exercises illustrate thetreated topics and help the student solve the kindof problems typical of examinations. Each chapterconcludes with problems. Solutions to many ofthese appear in an appendix, together with solu-tions to most of exercises.

The second edition of the book contains somenew sections. A new section provides a first intro-duction to elementary properties of martingales,which is now occupying a central position in mod-ern probability. Another section provides an ele-mentary introduction to Brownian motion, diffu-sion, and the Wiener process, which has under-pinned much of classical financial mathematics,such as the Black-Scholes formula for pricingoptions. Optional stopping and its applications areintroduced in the context of these important sto-chastic models, together with several associatednew examples and exercises. The list of chapters:Introduction, Probability, Conditional Probabilityand Independence, Counting, Random Variables:Distribution and Expectation, Random Vectors:Independence and Dependence, GeneratingFunctions and Their Applications, ContinuousRandom Variables, Jointly Continuous RandomVariables, and Markov Chains. Some sections canbe omitted at a first reading (e.g. GeneratingFunctions, Markov Chains). The book is suitablefor a first university course in probability and veryuseful for self-study. (br)

E. B. Vinberg: A Course in Algebra, GraduateStudies in Mathematics, vol. 56, AmericanMathematical Society, Providence, 2003, 511 pp.,$89, ISBN 0-8218-3318-9The book covers all topics, which are usuallyincluded in basic courses on linear algebra andalgebra in the first two years of study. In addition,it also includes a lot of non-standard and interest-ing material going into several different directions.The book is based on the long time teaching expe-rience of the author. At the beginning, main alge-braic structures are introduced (groups, rings,fields, algebras, vector spaces). Polynomial alge-bra is studied in detail and basic facts of group the-ory are covered. Linear algebra is covered inChapter 2, Chapter 5 and Chapter 6, together withbilinear and quadratic functionals. Chapter 8 con-tains a description of general multilinear algebra.The last four chapters are devoted to commutativealgebra (principal ideal domains, Noetherianrings, algebraic extensions, and affine algebraicvarieties), groups (Sylow theorems, simple groupsGalois extensions and Galois theory), linear repre-sentations of associative algebras (completereducibility, invariants, division algebras) and lin-ear Lie groups (the exponential map, the adjointrepresentation, basic facts on linear representa-tions). The book is beautifully written, the choiceof topics and their order is excellent and the bookis very carefully produced. It contains a huge num-ber of exercises and it appeals to geometric intu-ition whenever possible. It can be highly recom-mended for independent reading or as material forpreparation of courses. (vs)

C. Voisin: Hodge Theory and ComplexAlgebraic Geometry I, Cambridge Studies inAdvanced Mathematics 76, Cambridge UniversityPress, Cambridge, 2002, 322 pp., £55, ISBN0-521-80260-1As the title clearly implies, the book deals with thebasics of Hodge theory and its relation with com-plex algebraic geometry. The first four chapters

are devoted to preliminaries. The main result ofthe first chapter is the Riemann and HartogsTheorems on extension of holomorphic functionsand existence of local solutions to the Cauchy-Riemann equations. The second chapter containsan introduction to (holomorphic) vector bundlesand the Dolbeault complex on differentiable man-ifolds with complex structure (determined from analmost complex structure via Newlander-Nirenberg theorem). The third chapter contains aself-contained introduction to Kähler geometry,Kähler metrics and Chern connections on a holo-morphic vector bundle equipped with a Kählermetric. The introductory part ends with the fourthchapter describing theory of sheaves, cohomologyof a topological space with values in a sheaf andtheory of acyclic resolutions leading to a proof ofthe de Rham theorem.

The second part of the exposition is devoted toa proof of the Hodge and Lefschetz decompositiontheorems of cohomology of a complex manifold.The first two chapters of this part summarize anecessary background from analysis on Hilbertspaces used to define (formal) adjoints of ellipticoperators, their Laplace operators and finally, asan application, harmonic forms and their relationto cohomology. The following two chapters giveconceptual applications of these results. Thenotion of a polarized Hodge structure is explainedand applied to the Kodaira embedding theorem,which states that a complex manifold is projectiveif it admits an integral polarization. The next chap-ter is devoted to the holomorphic de Rham com-plex and the interpretation of the Hodge Theoremin terms of degeneracy of the Frölicher spectralsequence. The chapter ends with an introduction toholomorphic de Rham complex with logarithmicsingularities for quasi-projective smooth varietieswith mixed Hodge structure on their cohomology.The third part is devoted to a study of variations ofHodge structures. The notion of a family of com-plex differentiable manifolds leads to the con-struction of the Kodaira-Spencer map and Gauss-Manin connection associated to the local systemof cohomology of fibers of this family. Using theKodaira-Spencer map and the cup product inDolbeault cohomology, the period map and its dif-ferential are described. In the last chapter, various“cycle classes” are studied. In particular, Hodgeclasses arising in the study of morphisms ofHodge structures are described in more detail. TheDeligne cohomology and the Abel-Jacobi map areintroduced, although they are studied in moredetail in the next volume of the series. (pso)

C. Voisin: Hodge Theory and ComplexAlgebraic Geometry II, Cambridge Studies inAdvanced Mathematics 77, Cambridge UniversityPress, Cambridge, 2003, 351 pp., £60, ISBN 0-521-80283-0The book is the second volume of a two-volumetreatise on Hodge theory and its applications. Thevolume consists of three parts. The first part illus-trates various aspects of the qualitative influenceof Hodge theory on the topology of algebraic vari-eties. In particular, the Deligne’s theorem on thedegeneration of Leray spectral sequence of ratio-nal cohomology of a projective fibration at E2 andits consequences, e.g. surjectivity of the map fromrational cohomology of the total space to (the basegenerated) monodromy invariant subspace ofrational cohomology of the fiber, are discussed.

The second part is devoted to the study of infin-itesimal variations of Hodge structure for a familyof smooth projective varieties and its applications,in particular those concerning the case of completefamilies of hypersurfaces of complete intersec-

tions of a given variety. The main explicit result inthis part is Nori’s connectivity theorem.

The third (and final) part of the volume is devot-ed to relations between Hodge theory and algebra-ic cycles. Using infinitesimal techniques from thesecond part, certain cycle class maps and equiva-lence relations (rational, homological, algebraicand Abel-Jacobi equivalence) are established. Inparticular, variations on the theme of the relationof Chow groups and Hodge theory of smoothcomplex varieties are reviewed. (pso)

V. A. Zorich: Mathematical analysis I,Universitext, Springer, Berlin, 2004, 574 p.,€53,45, ISBN 3-540-40386-8V. A. Zorich: Mathematical analysis II,Universitext, Springer, Berlin, 2004, 681 p.,€53,45 , ISBN 3-540-40633-6This is a very nice textbook on mathematicalanalysis, which will be useful to both the studentsand the lecturers. The list of chapters is as follows:Vol. I. 1) Some General Mathematical Conceptsand Notation, 2) The Real Numbers, 3) Limits,4) Continuous Functions, 5) Differential Calculus,6) Integration, 7) Functions of Several Variables,8) Differential Calculus in Several Variables,Some problems from the Midterm Examinationsand Examination Topics. Vol. II. 9) ContinuousMappings (General Theory), 10) DifferentialCalculus from a General Viewpoint, 11) Multipleintegrals, 12) Surfaces and Differential forms inRn, 13) Line and Surface Integrals, 14) Elementsof Vector Analysis and Field Theory,15) Integration of Differential Forms onManifolds, 16) Uniform Convergence and BasicOperations of Analysis, 17) Integrals Dependingon the Parameter, 18) Fourier Series and FourierTransform, 19) Asymptotic Expansions, Topicsand Questions for Midterm Examinations andExamination Topics.

About style of explanation one can say that thedefinitions are motivated and precisely formulat-ed. The proofs of theorems are in appropriate gen-erality, presented in detail and without logicalgaps. This is illustrated in many examples (manyof them arise in applications) and each sectionends with a list of problems and exercises, whichextend and supplement the basic text. Finally, onecan make several remarks on the approaches used.Real numbers are introduced axiomatically, thegeneral concept of limits with respect to (filter)base is explained and used, e.g. in the definition ofRiemann integral, and real powers of a positivenumber are introduced as limits of rational pow-ers. Trigonometric functions are firstly introducedintuitively using the unit circle, then as sums ofsome power series and finally as inverse functionsto functions arcsine and arccosine, which aredefined after definition of the length of a curve.The multiple integral is firstly introduced as aRiemann integral over an n-dimensional interval(analogous to the 1-dimensional case). Then forthe bounded set, D is defined as the integral overan interval containing D of the product of thegiven function and the characteristic function ofthe set D. The Lebesgue necessary and sufficientcriterion for integrability is proved and frequentlyused. Line and surface integrals are introduced asintegrals of differential forms over surfaces.Smooth and piecewise smooth surfaces are con-sidered. Fundamental integral formulas (includingthe general Stokes formula) are proved. This mate-rial is explained in more detail in Chapter 15. InChapter 14, vector versions of fundamental inte-gral formulas are stated and vector fields having apotential are also studied. (jkop)

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