contents editorial team european mathematical … · newsletter no. 39 march 2001 ems agenda ........

NEWSLETTER No. 39

March 2001

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

Editorial - Bodil Branner .............................................................................. 3

EMS Summer School 2000 ............................................................................ 5

Statutes of the European Mathematical Foundation .................................... 6

Raising Public Awareness of Mathematics ................................................... 7

Maths Quiz 2000 ............................................................................................7

Interview with Bernhard Neumann .............................................................. 9

Interview with Sir Roger Penrose, part 2 ................................................... 12

Problem Corner ........................................................................................... 18

SIAM-EMS Conference ................................................................................ 21

Fondazione CIME Courses ........................................................................... 22

Forthcoming Conferences ............................................................................ 23

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

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

ISSN 1027 - 488X

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

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

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

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

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

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CONTENTS

EMS March 2001

EDITORIAL TEAM EUROPEAN MATHEMATICAL SOCIETY

EXECUTIVE COMMITTEEPRESIDENT (1999�2002)Prof. ROLF JELTSCHSeminar for Applied MathematicsETH, CH-8092 Zürich, Switzerlande-mail: [email protected]. LUC LEMAIRE (1999�2002)Department of Mathematics Université Libre de BruxellesC.P. 218 � Campus PlaineBld du TriompheB-1050 Bruxelles, Belgiume-mail: [email protected]. BODIL BRANNER (2001�2004)Department of MathematicsTechnical University of DenmarkBuilding 303DK-2800 Kgs. Lyngby, Denmarke-mail: [email protected] (1999�2002)Prof. DAVID BRANNANDepartment of Pure Mathematics The Open UniversityWalton HallMilton Keynes MK7 6AA, UKe-mail: [email protected] (1999�2002)Prof. OLLI MARTIODepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlande-mail: [email protected] ORDINARY MEMBERSProf. VICTOR BUCHSTABER (2001�2004)Department of Mathematics and MechanicsMoscow State University119899 Moscow, Russiae-mail: [email protected] Prof. DOINA CIORANESCU (1999�2002)Laboratoire d�Analyse NumériqueUniversité Paris VI4 Place Jussieu75252 Paris Cedex 05, Francee-mail: [email protected]. RENZO PICCININI (1999�2002)Dipartimento di Matematica e ApplicazioniUniversità di Milano-BicoccaVia Bicocca degli Arcimboldi, 820126 Milano, Italye-mail: [email protected]. MARTA SANZ-SOLÉ (1997�2000)Facultat de MatematiquesUniversitat de BarcelonaGran Via 585E-08007 Barcelona, Spaine-mail: [email protected]. MINA TEICHER (2001�2004)Department of Mathematics and ComputerScienceBar-Ilan UniversityRamat-Gan 52900, Israele-mail: [email protected] EMS SECRETARIATMs. T. MÄKELÄINENDepartment of MathematicsP.O. Box 4FIN-00014 University of HelsinkiFinlandtel: (+358)-9-1912-2883fax: (+358)-9-1912-3213telex: 124690e-mail: [email protected]: http://www.emis.de

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EMS NEWS

EMS March 2001

20014-6 MayEMS Workshop on Applied mathematics in Europe, Berlingen (Switzerland)Contact: Rolf Jeltsch, e-mail: [email protected]

11-12 May EMS working group on Reference Levels in MathematicsConference on Mathematics at Age 16 in EuropeVenue: LuxembourgContact: V. Villani, A. Bodin or Jean-Paul Piere-mail: [email protected], [email protected] or [email protected]

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

19-21 JuneEMS lectures at the University of Heraklion, Crete (Greece)Lecturer: Prof. George Papanicolaou (Stanford, USA)Title: Time Reversed AcousticsContact: George N. Makrakis, e-mail: [email protected]

9-25 JulyEMS Summer School at St Petersburg (Russia)Title: Asymptotic combinatorics with applications to mathematical physicsOrganiser: Anatoly Vershik, e-mail: [email protected]

15 AugustDeadline for proposals for 2003 EMS Summer SchoolsContact: Renzo Piccinini, e-mail: [email protected] for submission of material for the August issue of the EMSNewsletterContact: Robin Wilson, e-mail: [email protected]

19-31 August EMS Summer School at Prague (Czech Republic)Title: Simulation of fluid and structure interactionOrganiser: Miloslav Feistauer, e-mail: [email protected]

24-30 AugustEMS lectures in Malta, as part of the 10th International Meeting of EuropeanWomen in MathematicsLecturer: Michèle Vergne (Ecole Polytechnique, Palaiseau, France)Title: Convex polytopesContact: Dr. Tsou Sheung Tsun, e-mail: [email protected]

1-2 SeptemberEMS Executive meeting in Berlin (Germany)

3-6 September1st EMS-SIAM conference, Berlin (Germany)Organiser: Peter Deuflhard, e-mail: [email protected]

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

22-23 NovemberDiderot Mathematical Forum 5Title: Mathematics and telecommunicationsVenues: Eindhoven (Netherlands), Helsinki (Finland) and Lausanne(Switzerland)Contact: Jean-Pierre Bourgignon, e-mail: [email protected]

20021-2 June EMS Council Meeting in Oslo (Norway)

EMS AgendaEMS Committee

Better interaction During the EMS Council meeting inBarcelona last July, some of the corporatemember societies called for better interac-tion between the EMS and its member soci-eties. There is indeed a need for moredirect interaction in between the biannualCouncil meetings � the question is how thiscan be set up most effectively.

The best way to be informed about EMSactivities is through the EMS Newsletter,which is mailed to all member societies andall individual members. In addition, sincetaking office the EMS president RolfJeltsch has started a tradition of sendingout � by the end of each calendar year � aletter to each individual member andanother letter to all corporate membersocieties, describing the highlights of theprevious year and the most importantfuture undertakings. Also, more e-mailsthan ever before have been sent to mem-ber societies to inform, to ask for opinionson certain questions, and to ask for help inspreading information of general interestreceived by the EMS. I think we shoulddevelop such activities even more in thefuture.

For the EMS to be successful, it is highlydependent on collaborations with membersocieties and work done by individuals.Mathematicians working for the EMS arehowever all appointed as individuals, notrepresenting any society in particular.This gives a freedom which is of greatvalue, and I have no doubt that this is theway it should be. Only for Council meet-ings do all member societies directlyappoint their delegates. Together with theelected delegates of individual members,they elect the president, vice-presidentsand other members of the executive com-mittee, and can express their opinions andhave influence on the broader aspects ofthe work and the initiatives taken by thevarious committees of the EMS.

The best idea would be to interact morewith the delegates between Council meet-ings, or with other appointed representa-tives of the member societies. In the past,the role of a society�s delegate has beenmore-or-less restricted to participation inCouncil meetings. Indeed, some delegateswere mainly chosen because they were alsospeakers at the conference associated withthe Council meeting. To make a morelively and valuable exchange of ideasbetween and during Council meetings, itcould be important for the EMS and itsmember societies to rethink the role of thedelegates and increase their responsibili-ties. The EMS Newsletter too wishes to havea person appointed by each society,responsible for transferring local news to

the Newsletter. Some societies haveappointed such a person, others not yet.

What can member societies do to makethe EMS more visible? Many societies publish a newsletter andhave a homepage. We wish to encourageall societies to publicise its EMS member-ship, both in its newsletter and on the fronthomepage of the society, and to advocateindividual membership.

Individual membership of the EMS isusually obtained and paid through mem-ber societies, but this may not be obvious tothe society�s members, and neither may itbe obvious how to distinguish between thesociety being a member and a personbeing a member. We encourage societiesto make it easy for their members to jointhe EMS. However, the way in which eachsociety collects its EMS fees should be aseasy as possible for that society, and maydiffer from one society to another.

The most visible sign of EMS member-ship is the receipt of the EMS Newsletter.Societies can add new EMS members fourtimes per year, in mid-February, May,August and November, by mailing theadditional list to Tuulikki Mäkeläinen.The individual benefits, besides theNewsletter, include reduced conference feesfor EMS conferences and reduced prices ofcertain books and journals (see the list atEMIS under �How to join the EMS�).

Societies should also encourage mathe-matical departments to subscribe to theEMS Newsletter: it is very cheap (60 eurosannually). As explained above, it is mainlythrough the Newsletter that the mathemati-cal community can be informed about thework of the EMS. Many mathematicaldepartments subscribe to the AMSNewsletter, the Notices, and it should bejust as natural to subscribe to the EMSNewsletter. Individual members of the EMScan of course also encourage their depart-

ment to subscribe. I believe that all the above can be han-

dled by the individual societies without toomuch difficulty.

What does the EMS offer? The purpose of the EMS is to �promote thedevelopment of all aspects of mathematicsin the countries of Europe, with particularemphasis on those which are best handledon the international level�.

Some of the society�s activities are of apurely mathematical nature � for instance,summer schools, conferences, and lecturesare organised under the EMS umbrella,mainly by different volunteers. The weak-ness is that the EMS, besides moral sup-port, can offer at most symbolic economicsupport, and so the work involved is notvery different from organising any othersuch international mathematical activity;still, we hope that many will find it worthdoing so under the name of the EMS. TheDiderot Mathematical Forums are of a differ-ent nature. These are workshops that takeplace in three different cities simultane-ously, featuring a special topic � the lastone was Mathematics and Music and the nextis Mathematics and Telecommunications � withthree different themes, and with a round-table discussion in which the three citiesare linked (when possible) by electronicmeans.

Other activities need local support to beof a truly European dimension. Oneexample is the announcement of vacantjobs in mathematics in Europe. Some ofthese are collected by a number of membersocieties and can be found under Euro-Math-Job at EMIS; however, to be reallyuseful, the list should be not only morecomplete, but also searchable. The EMS isalso a partner of Zentralblatt MATH.Through the LIMES (Large Infrastructuresin Mathematics � Enhanced Services) pro-ject, funded by the EU, the database isbeing improved in many ways.

One of the goals of the several distrib-uted editorial units is to ensure a bettercoverage of the European mathematicalliterature. One example where membersocieties can be helpful is to ensure that allPhD theses are recorded in the database �this is done systematically in Germany, andought to be followed up throughout therest of Europe. Another activity is EULER(EUropean Libraries and ElectronicResources in mathematical sciences), a for-mer EU project that will be continued insome way. Institutions other than the coreparticipants can join by making theirlibrary resources available (for more detailssee EMS Newsletter 38, December 2000).The newest added activity is the EMF (theEuropean Mathematical Foundation),founded in March 2001. Its main purposeis to run the EMS publishing house, whichwill start to function this year. Again,interaction with member societies is highlyneeded.

In order to expand such activities as theabove, the EMS will have to describe moreprecisely how member societies or otherpartners may contribute. The list of activ-ities is by no means exhaustive.

EDITORIAL

EMS March 2001 3

EditorialEditorialBodil Branner (Lyngby, Denmark)

EMS Vice-President (2001�2004)President of the Danish Mathematical Society (1998�2002)

Asymptotic combinatorics with application to mathematical physics.Asymptotic combinatorics with application to mathematical physics.European Mathematical Society SUMMER SCHOOL

Euler International Mathematical Institute, St. Petersburg, Russia.9th July � 25th July 2001

The SUMMER SCHOOL aims to discuss recent progress in the asymptotic theory of Young tableaux and random matrices from the point of viewof combinatorics, representation theory and the theory of integrable systems. This subject belongs both to mathematics and theoretical physics.Systematic courses on the subjects and current investigations will be presented. The SUMMER SCHOOL is aimed at physicists as well as math-ematicians. Graduate students are encouraged to apply.

Main SpeakersP. Biane ENS, Paris, France E. Brezin ENS, Paris, France P. Deift UPenn, USAK. Johansson KTH, Stockholm, Sweden V. Kazakov ENS, Paris, France R. Kenyon University Paris-Sud, Orsay, FranceM. Kontsevich IHES, France A. Lascouix University Marne-la-Valiee, France A. Okoun�kov UCB, USAG. Ol�shansky IPPI, Moscow, Russia L. Pastur University Paris-7, France R. Speicher University of Heidelberg, GermanyR. Stanley MIT, USA C. Tracy UCD, USA H. Widom UCS Cruse, USA

The lectures will be devoted to asymptotic combinatorics and applications to the theory of integrable systems, random matrices, free probability,quantum field theory etc. They will also cover topics in low-dimensional topology, QFT, a new approach to the Riemann-Hilbert problem, asymp-totics of orthogonal polynomials, symmetric functions, representation theory, and random Young diagrams. Lately there has been great progress;new links and problems have appeared.

The following long-standing problems have recently been solved: fluctuation of random Young diagrams and of the maximal eigenvalues of the ran-dom Hermitian matrix. A new approach to the Riemann-Hilbert problem and integrable systems has been developed by Deift-Baik-Johansson (basedon the Korepin-Izergin-Its approach, papers by Tracy-Widom and others). Alternative methods come from the asymptotic theory of representationsand in particular from studying the Plancherel measure. These ideas let us calculate the correlation functions of the corresponding point processes(Ol�shansky-Borodin and previous results by Kerov-Vershik) and also to apply the boson-fermion correspondence (Okoun�kov). The explicit distri-bution of the fluctuations of the characteristics of Young diagrams is one of the main results as is the precise distribution of the fluctuations of theeigenvalues of the random matrix.

Such progress has initiated great activity in many related topics, namely � the growth of polymers, ASEP, random walks on groups and semigroups.Prospective problems and applications of the results will be discussed during the SUMMER SCHOOL in seminars and round-table discussions.

Scientific CommitteeE. Brezin, O. Bohigos, P. Deift, L. Faddeev, V. Malyshev, A. Vershik (chair).

Local Organizing CommitteeA.Vershik, Ju. Neretin, K. Kokhas, E. Novikova

OrganizationThe SUMMER SCHOOL will take place 9th �25th July, 2001, at the International Euler Institute, St.Petersburg, Russia.The Summer school has financial support from the EMS and the RFBR. Additional information about the SUMMER SCHOOL can befound at http://www.pdmi.ras.ru/EIMI/2001/emschool/index.html E-mail: [email protected].

EuropeanMathematical Society

Around one hundred mathematiciansfrom twenty countries, mainly youngresearchers, took part in the Year 2000Summer School in Probability Theory at Saint-Flour. This was the thirtieth such summerschool; the series was founded in 1971 byPaul Louis Hennequin, and is well knownby the international community in proba-bility theory, statistics, and their applica-tions.

The three series of lectures were chosento have applications in different fields, andprovided advanced training in these areas.Each series consisted of ten 90-minute lec-tures.

The lectures by Sergio Albeverio,University of Bochum, were on Dirichletforms and infinite-dimensional processes, theirtheory and applications. These covered basictheory: analytic and probabilistic tools; dif-fusion processes; jump processes; connec-tions with systmes of stochastic ordinarydifferential equations and stochastic par-tial differential equations; applications tostochastic dynamics (statistical mechanics,quantum field theory and polymerphysics); and analysis and geometry onconfiguration spaces and applications.

Walter Schachmayer, University ofVienna, gave an updated overview of Themathematical probabilistic tools of finance andthe mathematics of arbitrage; the applicationshere concern banks and finance and insur-ance companies. This subject has beengreatly developed in the past ten years,

and this summer school provided theopportunity for a state-of-the-art review byone of the main specialists in Europe.

The course dealt with the applications ofstochastic analysis to mathematicalfinance. The basic economic conceptunderlying the modern approach to math-ematical finance is the �principle of arbi-trage�. The theory was developed by sys-tematically elaborating this concept and itsapplications. The mathematical model ofa financial market is a stochastic process Smodelling the discounted price of a finan-cial asset (a �stock�); the time index can beeither discrete or continuous. Trading onthe stock S is modelled by predictableprocesses H, describing the amount ofstock held by an investor at a given time.The basic object of study is the process ofgains and losses given by the stochasticintegral H.S.

The course was in two parts. In the first,the underlying probability space wasassumed to be finite. In this elementarysetting all the measure-theoretic difficul-ties disappear, and the basic themes andideas of mathematical finance can then bepresented without unnecessary technicali-ties: the basic relation between arbitrageand equivalent martingale measures (the�fundamental theorem of asset pricing�),pricing and hedging of derivative securi-ties in complete and incomplete markets,optimal investment and consumptionproblems, etc. In the second part, the

same programme was carried out in prop-er generality to develop the natural frame-work in continuous time. The tools usedare general semi-martingale theory andfunctional analysis.

Michael Talagrand, research director atthe CNRS (National centre of ScientificResearch) in France, taught us about Spinglasses. Under this name one denotes anumber of stochastic models introduced byphysicists. These models are purely math-ematical objects of a remarkably simpleand canonical character. They have beenstudied by physicists using non-rigorousmethods, and predict a number ofextremely interesting behaviours thatpotentially open new areas of probabilitytheory. The challenge is to say somethingrigorous about this topic, and the purposeof the course was to introduce the basicideas and tools that have been developedin this direction.

The young participants were all invitedto present their research work in 40-minute presentations, and forty of themdid so. The environment of the summerschool was very favourable for exchangesbetween young researchers and moreexperienced participants.

The help of the EMS was much appreci-ated, and I wish to thank all the sponsorsof the Summer School: the EuropeanCommission XII, Région Auvergne,Départment du Cantal, City of Saint-Flour,UNESCO, the Blaise Pascal University,and CNRS.

The next Saint-Flour Summer Schoolwill be held from 8-25 July 2001, and thethree lecturers will be Olivier Catoni(CNRS, Paris VI University): Statisticallearning theory and stochastic optimization;Simon Tavaré (University of SouthernCalifornia): Ancestral inference from molecu-lar data; Ofer Zeitouni (Technion�IsraelInstitute of Technology): Random walk inrandom environments: asymptotic results.

EMS NEWS

EMS March 2001 5

EMS Summer School inEMS Summer School inPPrrobability Theoryobability Theory

Saint-Flour, Cantal, France17 August � 2 September 2000

Pierre Bernard (Blaise Pascal University, Clermont-Ferrand, France)

Article 1: Name, seat and duration The European Mathematical Foundationis a Foundation in the sense of Article 80 ofthe Swiss Civil Code. It has its seat inZurich. The duration of the Foundation isnot limited.

Article 2: Purpose of the FoundationThe Foundation has the purpose of fur-thering Mathematics in all its aspects inEurope (including scientific, educationaland public awareness).

To this purpose the foundation canundertake the following activities:

a) the establishment and supervision of apublishing house, whose mode of opera-tion will be regulated by by-laws deter-mined by the Board of Trustees of theFoundation;

b) furtherance of the activities of theEuropean Mathematical Society [�EMS�];in particular, the furtherance, dissemina-tion and popularization of mathematicalknowledge, furtherance of exchange ofideas between mathematicians in Europeand between mathematicians in Europeand other mathematicians;

c) collaboration with the administrationof the EMS;

d) support of activities of corporatemember societies of the EMS.

The Foundation may, on the basis ofdecisions made by its Board of Trustees,extend all its activities towards furthertasks with similar objectives.

Article 3: Foundation capitalThe foundation capital is Euro 10000. Thefoundation capital may be increased any-time by additions from the founder orthird parties.

Article 4: Bodies of the FoundationThe operating bodies of the Foundationwill be:

a) its Board of Trustees;b) the Executive Committee of the Board

of Trustees;c) the managing director of the publish-

ing house;d) the Auditors of the Foundation.

Article 5: Board of TrusteesThe Board of Trustees will consist of atleast 4 members but not more than 12members.

Members of the Board of Trustees willinclude:

* the current President of EMS;* the previous President of EMS, unless

he/she renounces this office;* the current Secretary of EMS;* the current Treasurer of EMS;* a representative of the Swiss Federal

Institute of Technology in Zürich[�ETHZ�].

The Board of Trustees may appoint fur-ther members at its discretion, as specifiedin the by-laws of the Foundation.

The President of EMS will chair theBoard of Trustees.

The Board of Trustees will appoint aSecretary.

The Board of Trustees will supervise themanagement of the funds of theFoundation, and will ensure that the fundsare used according to the aims of theFoundation. The Board of Trustees willmeet at appropriate intervals, and at leastonce a year for an Annual Meeting. In thatAnnual Meeting the financial statementsand budget of the Foundation will beapproved. The Executive Committee ofthe Board may call a meeting of the Boardof Trustees at any time.

The board of trustees will establish by-laws on the details of the organization andthe management. These by-laws can bechanged by the board of trustees within thepurpose of the foundation any time andhave to be approved by the supervisingauthorities.

A member of the board of trustees will inprinciple receive no payment from theFoundation. Expenses will be coveredupon presentation of receipts. If sometasks are extraordinarily work intensivethese can be paid for in an isolated case.

Article 6: Agency for AuditingThe Board of Trustees elects an indepen-dent external Agency for Auditing whichaudits the books of the foundation on anannual basis and submits its result to theBoard of Trustees in a detailed report withthe motion for approval. In addition it hasto ensure that the Statutes and the purposeof the foundation are adhered to.

The Agency for Auditing has to informthe Board of Trustees on shortcomingsobserved during their investigation. Ifthese are not corrected in a reasonabletime period, the agency has to inform thesupervising body.

Article 7: Executive CommitteeThe Executive Committee of the Boardshall comprise the following:

* the Chairman of the Board of Trustees(or a nominee from amongst the Boardmembers);

* two nominees of the Board of Trustees,at least one of whom is also a member ofthe Board of Trustees.

The Executive Committee is responsiblefor the day-to-day running of theFoundation, the Annual Report of theBoard, the audited Annual Financial

Report of the Board, and a budget. TheBoard of Trustees will specify who is autho-rized to sign for the Foundation. TheExecutive Committee will supervise theinvestment of the funds of the Foundation.

Article 8: Changes to the Statutes of theFoundationThe Board of Trustees may, within the lim-itations of Article 85 and 86 of the SwissCivil Code, with a two-thirds majority ofthe present or represented members of theBoard of Trustees, decide on a total revi-sion or a partial revision of theFoundation�s Statutes.

The Board must submit its decision tothe supervising body, the Swiss Ministry ofthe Interior, with a request for an appro-priate changement decree.

Article 9: Termination of the FoundationShould the objectives of the Foundation nolonger be achievable for any reason, thenthe Board of Trustees may at its discretiondecide to dissolve the Foundation, subjectto the approval of the supervising body(the Swiss Ministry of the Interior). In theevent of a dissolution of the Foundation,its remaining funds, after payment of alldebts, may be given to organizations, foun-dations and institutions with the same orsimilar objectives to the presentFoundation.

In the name of the EuropeanMathematical Society

PresidentRolf Jeltsch

EMS NEWS

EMS March 20016

S T A T U T E SS T A T U T E Sof the European Mathematical Foundation

At the London meeting of the EMS Executive Committee in November 2000 it was decided that an EMS publishing house shouldbe created and that a foundation, the European Mathematical Foundation, should be the legal owner of such a publishing house.The statutes of this foundation appear below.

EURESCO Conferencesin Mathematics in 2001

18-23 August: Algebra andDiscrete Mathematics, Hattingen,GermanyEuroConference on Classification,Non-Classification and IndependenceResults for Modules, Groups and ModelTheoryChairs: Rüdiger Göbel, Essen, andManfred Droste, Dresden

1-6 September: Number Theoryand Arithmetical Geometry,Acquafredda di Maratea (nearNaples), ItalyEuroConference on ArithmeticAspects of Fundamental GroupsChair: Anthony J. Scholl, Durham,UK

EMS March 2001 7

Articles in many ways,math displays,

says,the EMS committee of RPA:A competition surely may,

inspire the way,in which to pay,

as we say,attention to public awareness!

During World Mathematical Year2000, several articles on mathematicsaddressing a general audience werepublished throughout the world, andmany valuable ideas for articles popu-larising mathematics have been gener-ated. The newly appointed committeefor Raising Public Awareness ofMathematics of the EuropeanMathematical Society (acronym RPA)believes that it is vital that such articlesbe written. In order to inspire futurearticles with a mathematical theme andto collect valuable contributions, whichdeserve translation into many lan-guages, the EMS wishes to encouragethe submissions of articles on mathe-matics for a general audience, in acompetition. The EMS is convincedthat such articles will contribute to rais-ing public awareness of mathematics.The RPA-committee of the EMS invitesmathematicians, or others, to submitmanuscripts for suitable articles onmathematics.

To be considered, an article must bepublished, or about to be published, ina daily newspaper, or some other gen-

eral magazine, in the country of theauthor, thereby providing some evi-dence that the article does catch theinterest of a general audience. Articlesfor the competition shall be submittedboth in the original language (the pub-lished version) and in an English trans-lation. The English version shall besubmitted both in paper and electroni-cally.

There will be prizes for the three bestarticles of 200, 150 and 100 Euros, andthe winning articles will be published inthe EMS Newsletter. Other articles fromthe competition may also be publishedif space permits. Furthermore, it isplanned to establish a web-site contain-ing English versions of all articles fromthe competition approved by the RPA-committee.

By submitting an article for the com-petition, it is assumed that the authorgives permission to translation of thearticle into other languages, and forpossible inclusion into a web-site.Translations into other languages willbe checked by persons appointed byrelevant local mathematical societiesand will be included on the web-site.

Articles should be sent before 31December 2001, to the chairman of theRPA-committee of the EMS:Professor Vagn Lundsgaard Hansen,Department of Mathematics, Technical University of Denmark,Building 303, DK-2800 Kongens Lyngby, Denmark.e-mail: [email protected]

EMS NEWS

Maths Quiz 2000The international competition Maths Quiz 2000took place on 4-5 December. There were 378teams from around the world registered to play,including 108 from the USA, 27 from Russia,and teams from Mongolia, Azerbaijan, thePhilipines and Thailand, to name a few. As thecompetitors and organisers can testify, it was anemotional experience and an excellent way tocelebrate World Mathematical Year 2000.

It is now time to announce the winners andexplain the background to this unique event.The competition was conceived and designed atthe Centre de Recerca Màtematica in Barcelonaby Manuel Castellet, Rafel Serra and JaumeAguadé. Designed as a real-time question-and-answer competition to be played over the inter-net, with successive levels of scoring followingthe Fibonacci sequence, it was played in onecontinuous 24-hour session to guarantee fair-ness to players from all time zones. In this wayit was also a test of endurance!

When Sun Microsystems offered to sponsor thegame with five magnificent prizes, as well asdonating the server needed to host it, Maths Quiz2000 was born. Now that the game is over, theserver will be presented to a mathematicsdepartment in a less-favoured region.Birkhäuser publishers and Wolfram Researchalso showed interest, offering prizes of bookvouchers and Mathematica licences. The Centrede Recerca Matemàtica offered a special prize tothe team from the Catalan countries whichachieved the best result in the competition.

To make the game fun to play it was necessaryto think carefully about the design (and theprizes!) and to have good questions. A commit-tee of five mathematicians (Jaume Aguadé, JoanJ. Carmona, Enric Nart, Pere Puig and JaumeSoler) worked for months on the task of produc-ing a large database of questions that weresimultaneously hard, enjoyable and correct. Atthe technical end, the game was programmed byRafel Serra and Waldo Mateo, with a group fromUniversitat Oberta de Catalunya, implementingthe hardware and providing and configuring theinternet connection for the server.

The competition started at noon, GreenwichMean Time, on Monday 4 December and endedexactly 24 hours later. The organisers, followingits progress from the UOC, were consistentlysurprised by the high level of play sustained bythe competitors during this period and theirextraordinary skill on really hard questions!

The leading team was a group of youngresearchers from the Universitat Autònoma deBarcelona, led by Raul Fernandez, who achieveda final score of 925 points. In second place camean Italian team led by Andrea Rizzi from Pisa.Third was a Brazilian team from IMPA in Rio deJaneiro, lead by Krerley Oliveira, while thefourth position went to Martin Kollar and hiscollaborators from the Comenius University inSlovakia. Positions 5, 6, 7 and 8 were taken byAndrei Moroianu (currently in USA), Juan PabloRossetti (Dartmouth College, USA), GregMartin (University of Toronto, Canada) andRicardo Perez-Marco (UCLA, USA).

The CRM is now considering ways to make thegame MQ2000 available to the public.

Raising PRaising Public Aublic Awarwarenessenessof Mathematicsof MathematicsAn ARTICLE COMPETITION

Vagn Lundsgaard Hansen

JEMSJEMSThe Journal of the European Mathematical Society (JEMS) is a joint publication of theEuropean Mathematical Society and Springer-Verlag. There is one volume per year, con-sisting of four quarterly issues; Volume 1 was published in 1999. JEMS includes researchpapers in all areas of pure and applied mathematics. The Editor-in-Chief is Dr. JürgenJost.

JEMS is one component of the continuing effort of the European MathematicalSociety to promote joint scientific activities among the many diverse structures that char-acterise European mathematics.To look at the journal, use the website:

http://link.springer.de/link/service/journals/10097/index.htmTo see the contents of the first two volumes, use the website:

http://link.springer.de/link/service/journals/10097/tocs.htmIndividual members of the EMS can purchase JEMSat a special price of DM 52 in Centraland Eastern European, and DM 80 outside Central and Eastern European (plus carriagecharges). The corresponding rates for institutional subscribers are DM 180 and DM 396.These subscription rates include both the printed and electronic forms of the journal. Tosubscribe, please contact Springer-Verlag on the website:

http://link.springer.de/link/service/journals/10097/subs.htmThe contents of the most recent issue (Volume 3, No. 1) are as follows:D. Mucci, A characterization of graphs which can be approximated in area by smooth graphsM. Harris, A note on trilinear forms for reducible representations and Beilinson�s conjecturesL. Ambrosio, V. Caselles and J.-M. Morel, Connected components of sets of finite perimeter andapplications to image

Simulation of Fluid and Structure Interaction

European Mathematical Society SUMMER SCHOOLCampus of the Faculty of Mathematics and Physics, Charles University, Prague

19th-31st August 2001

The SUMMER SCHOOL will be concerned with mathe-matical and numerical methods in fluid dynamics, structur-al mechanics and, particularly, the interaction of fluids andstructures. This last is a relatively new, but extensivelydeveloping, area having great importance from the stand-point of applications in science and technology. There willbe a comprehensive series of lectures on the above subjects.

The level will be appropriate for graduate students andyoung researchers. Limited financial support will also beavailable for graduate students.

SpeakersJ. Ballmann, TH Aachen, GermanyD. Causon, University of Manchester, UKM. Feistauer, Charles University, Prague, CzechRepublicJ. Felcman, Charles University, Prague, Czech RepublicP. Hemon, Institut Aerotechnique, FranceP.Leyland, EPFL, SwitzerlandA. Quarteroni, EPFL, SwitzerlandM. Schäfer, TH Darmstadt, GermanyL. Tobiska, University of Magdeburg, GermanyJ.Vierendeels, University of Gent, Belgium

OrganizersP. Wesseling, Delft University of Technology,NetherlandsL. Tobiska, Otto-von-Guericke-University, Magdeburg,GermanyM. Feistauer, J. Felcman, Charles University, Prague,Czech Republic

Co-OrganisersP. Chocholatý, Commenius University, Bratislava,SlovakiaG. Stoyan, ELTE University, Budapest, HungaryJ. Rokicki, Warsaw University of Technology, Poland

Contact addressMiloslav Feistauer and Jiøï Felcman, Faculty of Mathematics and Physics, Charles University Prague, Sokolovská 83, 186 75 Praha 8, Czech Republice-mail: [email protected]: (+420 2) 2191 3392, 2191 3388fax: (+420 2) 24 81 10 36

EuropeanMathematical SocietyEuropeanMathematical Society

Bernhard Neumann was born in Berlin in1909, and took his Dr. phil from the Universityof Berlin in 1932. He moved to England in1933, taking a PhD at the University ofCambridge, and taught at the Universities ofWales, Hull and Manchester before moving toAustralia in 1962, to create the mathematicsdepartment within the Institute for AdvancedStudies of the Australian National University.This interview focuses on his life, work andexperiences in Europe up to 1962.

Professor Neumann, I wanted to ask youfirst about the influence of your family onyour mathematical interests and develop-ment.My home environment was a very stronginfluence. My father, Richard Neumann,was an engineer who worked for the elec-tricity company AEG (AllgemeineElektrizitäts Gesellschaft) in Berlin, and welived in a prosperous Berlin suburb whereI went to school. It was one of his studenttextbooks that first introduced me to thefascination of mathematics, at the age ofeleven or twelve: it was Stegemann andKiepert�s textbook on differential and inte-gral calculus. Studying differential andintegral calculus was in those days not gen-erally done until university�it had notmigrated down to high school yet. Myfather showed this textbook to me, to showme the shape of some curve, I think, and itfired my imagination. I worked throughthe differential calculus from cover tocover, did all the exercises, and enjoyed itimmensely. The second volume, on inte-gral calculus, I think I got stuck on after awhile. I must have been eleven, but thatcertainly was a great influence.

At school I was thoroughly lazy, but verygood at mathematics, and that saved me,except on one occasion when my formmaster sent for my mother and told her Iwas very close to not being moved up at theend of the year. That would have been aterrible thing, for me and for the family,and my sister, who was four years olderthan I, was delegated to see to it that Iworked. I worked really hard under herguidance for something like six weeks, andby then I had got into the habit of doingenough work to get by.

And how did you enjoy your school days?I spent three years in primary school andnine years in high school. All in all I tooknine years of French, eight years of Latin,and three years of English. The Frenchwas uninspiring. The Latin I didn�t like�I didn�t like the teacher because he insist-ed on our working�until one day, I was inthe fifth form by then, when he said wehave covered what we have to do, let usread an author who is not on the syllabus.He chose a minor writer from the secondcentury AD, Minucius Felix, an earlyChristian apologetic, and suddenly this was

interesting. Caesar was somewhat spoiledfor us by having to translate word for word,but here was something that was alive, andsuddenly I got interested in Latin. MyLatin improved tremendously and I didrather well at the examination.

It was when I went to university that Ibegan to read more books in Latin, read-ing the authors we hadn�t got to in eightyears� study of Latin at school. I wouldread them on the train journey, using atiny little dictionary in which I looked upthe words I didn�t know. After a while Icould read Latin with reasonable fluency.Among other things, I read engineeringtexts of Frontinus, from the end of the firstcentury, about the Roman water supply. Itsounds a very dry subject but I was fasci-nated by the author�s tables of pipes, theirdiameter, circumference & carrying capac-ity, which he calculated by a simple formu-la. I taught myself calculating with Romanfractions and checked all the tables, whichhad been rather badly transmitted.

Where did you go to university?I spent my first two semesters (1928-29) atthe University of Freiburg, in the BlackForest, but returned to the Friedrich-Wilhelms University in Berlin for the rest

of my studies (1929-32). There were fourprofessors in Berlin: Issai Schur, Richardvon Mises, Ludwig Bieberbach and ErhardSchmidt. They made an excellent team,each completely different in style, in lec-turing style. Issai Schur was myDoktorvater, a distant but kindly and sup-portive man.

Were you always primarily interested inalgebra?I had really intended to become a topolo-gist, influenced by Heinz Hopf, and myconversion to algebra was rather bychance. In my sixth semester, towards theend of my third year, there was a studentseminar and I was the understudy to theperson reporting on a paper about auto-morphism groups by the Danish mathe-matician Jacob Nielsen. I realised that Icould reduce the number of generators heused from four to two, in general; so Iwrote it up and showed it to Heinz Hopf.He immediately suggested I might like totake my doctorate with this work, to whichI replied that the paper was too slight andI was too young. Nonetheless, Hopfshowed the work to Issai Schur, who senthis assistant Alfred Brauer with the samesuggestion, to which I gave the same

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Interview with BernharInterview with Bernhard Neumannd Neumanninterviewer: John Fauvel, Open University, UK

response. In the next semester Schur sentBrauer again to say �But Professor Schurwants you to take your doctorate with thiswork.� So I agreed. A little later Schurhimself spoke to me and said that perhapsI was right and it was rather thin, so wouldI like to use the same methods to explorean additional topic in which he hadbecome interested, that is now called thewreath product of groups. My suspicion isthat Schur was thinking in that directionbecause of a recent paper by Loewy inwhich the idea had been adumbrated. Idid the work in a fortnight, which doubledthe length of my thesis, and submitted itsecretly, as a surprise for my parents.

I took my oral examination for the doc-torate in November 1931�my examinerswere Issai Schur and Erhard Schmidt inmathematics, and for my subsidiary sub-jects Peter Pringsheim in physics, andWolfgang Köhler in psychology. My exam-ination with Schmidt was intended to takeforty minutes but we found the conversa-tion so interesting that it took twice aslong. I received the degree some monthslater in July 1932, since I had to get copiesof my dissertation. I expected Issai Schurto publish my dissertation in MathematischeZeitschrift, of which he was founding editor,but he didn�t, and instead OttoBlumenthal published it in MathematischeAnnalen. Then Jacob Nielsen, on whosework it was based, reviewed it inZentralblatt. I was then 23, which was a veryyoung age to take a doctorate in mathe-matics in Berlin.

Of the four full professors at Berlin, LudwigBieberbach became notorious later for hisNazi sympathies. Were there signs of thiswhen you knew him?One always had the feeling that this was adefensive move, because Bieberbach hadbeen too friendly before with wives ofJewish colleagues�not in Berlin, but else-where�at least, that was the rumour.Perhaps he was concerned for his ownfuture, and that is why he became not justNazi but ultra-Nazi. He was in fact quitean inspirational teacher, whose effect wasrather great in a number of ways onHanna, my first wife.

What happened after you took your doctor-ate?I continued to go to lectures and seminars,and also worked as an unpaid assistant inexperimental physics. Some time, it musthave been in January 1933, a friend intro-duced me to a young student, Hanna vonCaemmerer. I thought her name soundedrather right-wing (and indeed her ances-tors were Prussian officers), so thought lit-tle more about it.

But then I met her again at Mapha, theM a t h e m a t i s c h - P h y s i k a l i s c h eArbeitsgemeinschaft, which was a wonder-ful institution within the university, set upby Alfred Brauer, one of Professor Schur�sassistants, and Hans Rohrbach. It had,uniquely, its own rooms and its ownlibrary, and there we would play chess, playgo, and discuss mathematics and otherthings. Hanna was in her second semester

then, and when we talked more I discov-ered that she too was anti-Nazi�Hitlerhad come to power at the end of January,and it was now the end of February. So webecame friendlier and at Easter I invitedher to come for a walk with me in one ofthe forests on the outskirts of Berlin. Afterthe walk we came back and played tabletennis. Suddenly she called me, by mis-take�by deliberate mistake, perhaps�itwas certainly a highly significant slip interms of German etiquette�the familiarDu instead of the impersonal Sie. That ledto my really falling in love with her.

A little later I took her home to meet myfamily. My mother immediately took toher, and that became very important forus.

1933 was an eventful year in Germany.Yes, in August of that year I moved toCambridge, in England. I knew I had nochance in Nazi Germany, but my arrival inCambridge was quite coincidental. Aschool friend of mine knew I wanted toemigrate somewhere, I didn�t mind partic-ularly where, and rang up one day to say�Meet me in Amsterdam tomorrow.� Iphoned home and my father, who was atwork, immediately arranged for me to geta passport, which took a day to obtain. Itook a train to Amsterdam, met my schoolfriend there, who said Cambridge was thebest place for a mathematician, and sawme on my way. So I arrived in Cambridge.Like so many of the emigrés at that time Ihad a doctorate already, but unlike some Iimmediately registered to do a second doc-torate in Cambridge. This was against theadvice of G. H. Hardy, who said that a doc-torate was unnecessary; what was impor-tant was to do good mathematics. Most ofus ignored his advice, except for HansHeilbronn, who indeed went on to do verywell.

What was your experience of research atCambridge?I was allocated to Philip Hall as myresearch supervisor�I had no idea whoPhilip Hall was, as the allocation was madeby someone else, I don�t know who. I start-ed my research working on a topic which atthe time was just too early, on rings of non-associative polynomials; this problem wasonly solved later, and not by me. I workedon it until Christmas 1934 and then Irealised that it would not make for a PhDthesis in the months remaining. Then Iremembered something I had started a fewyears earlier and had not really pursued,and looked at it again, and within a veryfew days I realised I had struck oil. Iworked on it and worked on it and one dayin May 1935 I said to myself �I must stopnow and write up�, otherwise it would justgo on and on and on.

I saw Philip Hall once a week, when Iwould look him up in King�s College afterdinner. He would offer me a cigarette. Wethen talked about everything under thesun. He was very well informed over awide range: about rhizomorphs in botany,about the political situation in Europeancountries, everything. At about ten o�clock

or so he might ask me a question or twoabout my research work, but never pointedme in any particular direction (indeed hedid not interfere in my false starts). In fact,at that time I knew a bit more about infi-nite groups than he did. This didn�t last;later he knew a tremendous amount aboutinfinite groups. Anyway, the experience ofbeing supervised by Philip Hall was verypleasant.

The oral examination for my thesis wasover lunch in my supervisor Philip Hall�srooms. He was one examiner, and MaxNewman was the other. They asked metwo questions: did I prefer beer or winewith my lunch, and would I like my coffeeblack or white. I must have answered thesequestions to their satisfaction, for Ireceived my PhD.

Meanwhile Hanna was in Germanythroughout these years?Hanna was in Germany over this period.We had become engaged in 1934, and cor-responded in secret through my parentsand her old geography teacher. In 1936we managed to spend a fortnight togetherin Denmark, where I stopped off on myway back from the ICM in Norway.

In that year, too, 1936, Hanna took herteachers examination, the Staatsexamen.She had been warned by a good friend thatthis would have involved her being ques-tioned in the oral examination by LudwigBieberbach on �political attitudes�, whichwas a new requirement. Because he sus-pected her of being friendly with Jews, hewould certainly have failed her. So the oralwas arranged for the long summer vaca-tion, when Bieberbach was away. In theevent it was conducted by Neiss, who wasmore sympathetic and reliable. But clear-ly it was impossible for Hanna to take herdoctorate at Berlin, as she wanted, so shemoved to Göttingen to continue herresearch under Helmut Hasse. She got onwell there, but after three semesters shesaw the way the wind was blowing, aban-doned her research (it was only completeda few years later, I think by André Weil),and came to join me in England in July1938 where we got married.

You had a job by this time?Yes, after completing my Cambridge doc-torate I stayed on there for a while. At thattime Olga Taussky was in Cambridge,teaching a course on algebraic numbertheory; this needed a preparatory coursewhich I was asked to give. At the end ofthese lectures the students who had attend-ed gave their judgement of the lecturer ina letter delivered to one of the members ofstaff who was a confidential go-between. Igot given a sealed letter with the results ofthe survey. I eagerly opened it, to find itsaid: �No comment�. I was paid £10 forthat course. The year after they offeredme the same course again, for which theywould pay £50. If I had stayed on givingthat course I would be rich now.

But by that time I was in Cardiff, inWales. I had applied for every job going,and was interviewed by University College,Cardiff, who wanted a temporary assistant

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for three years and had shortlisted onemathematician from the north of England,and myself. I got the job, I suspect becauseI was not English. It was temporarybecause they had a very bright student whohad just finished her first degree and hadgone off to Cambridge to take a PhD, andthey wanted to keep an opening for her forwhen she had finished. After those threeyears the war had started, and things werevery different. In fact, she didn�t comeback after three years but after four years,by which time I was no longer in Cardiff, Iwas in the army.

What happened to you during the War?At first I continued to teach in Cardiff, liv-ing there with Hanna and with my parents,who had come to join us, after much per-suasion, just before the outbreak of war. InMay 1940 there were a lot of scare storiesin the press about enemy aliens and wewere moved away from Cardiff, which wasa port, and went to live in Oxford, safelyinland. After a few days I was interned, atfirst in Southampton and then inLancashire. By autumn of that yearinternees were being released if their for-mer employer asked for them. My formeremployer, University College, Cardiff,made no move at all, so I decided to jointhe Army. When later Cardiff asked me tocome back I said �No�, since they did notsupport me and try to secure my release

when they could have done. I served for two-and-a-half or three years

in the Pioneer Corps, an interesting out-door life building Nissen huts and the like,and when we were allowed to volunteer forcombat service I transferred to the RoyalArtillery where I had a slightly more math-ematical role using surveying equipmentand doing basic numerical work. Later Iwas in the Intelligence Corps.

After the fighting had stopped I volun-teered to go with a unit of the IntelligenceCorps to Germany, as I hoped to makecontact with Hanna�s family near Lübeck.When I arrived with a kitbag full of foodtins, my mother-in-law became further rec-

onciled to our marriage which she had notat first been at all enthusiastic about. Ofcourse my having produced three grand-children for her, with a fourth on the way,was a great help in bringing her round.

So after the war you didn�t return to Wales?No, I turned down their invitation toreturn, since they did nothing for me dur-ing the War, and instead took a lectureshipat University College, Hull. Shortly after-wards Hull offered a job also to Hanna,who had her DPhil by now, supervised atOxford by Olga Taussky. She stayed therefor twelve years, but in 1948 I was recruit-ed by Max Newman to join the mathemat-ics department at Manchester.

Why did Max Newman lure you toManchester? Of course, you had longknown him, as he was your PhD examinerat Cambridge.It wasn�t just that. What happened wasthat Hanna and I decided to organise ameeting, a colloquium, of British mathe-maticians in Hull. We hoped the LondonMathematical Society would support it, butsome members were a bit lukewarm aboutthe idea. In particular, our own professorof mathematics in Hull, George Steward,was rather dismissive of the idea of holdinga colloquium of mathematicians in Hull.The problem was that he had beenappointed quite young to the Chair in

Hull, and in about 1936 he had died, butno-one told him he had died, so he carriedon living until his nineties, when in fact hehad been dead for many years before that.

So there was no mathematical colloqui-um in Hull, but meanwhile Max Newman,who was putting a lot of energy into build-ing up the department in Manchester,learned through his LMS contacts of ourproposal which he thought an excellentone. The upshot was that he invited me tojoin the faculty at Manchester, and the fol-lowing year, 1949, we were able to organ-ise the first British MathematicalColloquium, in Manchester.

Alan Turing was also on the staff atManchester. Was he someone you knewthere?I did not know him well, but he showed methe computer they were developing, whichwas a matter of pink string and sealingwax, down in the basement. In the cornerwas a big column which was the memory.It was a strange contraption. He was thebrains behind it. He gave a number oftalks, from which it was quite clear that hewas a genius. He was not easy to under-stand, as he would stumble over thewords�he thought so much faster than hecould speak�but it was very clear he was agenius. There is no doubt about that.

It was in Manchester too, I understand,that you developed your interest in the his-tory of mathematics?I had an engineer friend who worked forFerranti in Manchester, B. V. Bowden(later Lord Bowden) who was very con-cerned with computers and their history,and one day said to me that he had foundpapers belonging to Ada Lovelace whichhe had on loan from the family. Thesewere largely letters from Ada Lovelace toher mathematics tutor, Augustus DeMorgan, with some of his replies. I readthese papers and found them fascinating�they included also notebooks from herdaughter�s study with the first professor ofmathematics at Owen College,Manchester�and as they were in disarrayI put some effort into arranging them inchronological order, using watermarks,internal evidence, and so on. I read asmuch as I could about her and De Morganand their circles, including Augustus DeMorgan�s son William de Morgan, whobecame famous for his tiles and also forwriting novels. In fact, my father gave mea copy of William De Morgan�s first novel,Joseph Vance, to read on the journey when Iwent to Cambridge in 1933. Eventuallythe Lovelace papers went back to the fam-ily, but my son Peter made photocopies ofthem and I have kept up my interest in thehistory of that time. I later wrote a paperabout this which appeared in theMathematical Gazette.

Do you think history of mathematics canhelp to stimulate and inform students�interest in learning mathematics?Well, for example, Charles Curtis�s bookon the history of group representationswill do a lot to introduce students to thework of Frobenius, Schur, Burnside, and soon. And my son Peter has done a lot ofwork in history, including a forthcomingedition of the work of Burnside, whichshould help further in bringing mathemat-ics from the past alive to students today.

In 1962 Bernhard and Hanna Neumannmoved to Australia, where he took up theFoundation Chair in mathematics at theInstitute of Advanced Studies, AustralianNational University in Canberra. Hanna wasappointed a Professorial Fellow there, andshortly afterwards became head of the puremathematics department in the School ofGeneral Studies.

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Three octo- and nono-geraniums (as they called themselves) met again at the 2000 Christmas meet-ing of the British Society for the History of Mathematics, held in University College, London: thelate Robert Rankin (left) with two refugees from Nazi Germany, Bernhard Neumann and WalterLedermann.

In the early 1970s you discovered two�chickens� that can tile the plane in a waythat must be non-periodic. How did youfind these non-periodic � or perhaps Ishould say aperiodic � tilings?Yes, aperiodic tile sets, I suppose, but thetilings are non-periodic. Tiling problemshave always been a doodling side interestof mine, just for fun; if I got bored withwhat I was doing I�d try and fit shapestogether, for no particular scientific reason� although I supposed that there was someconnection with my interest in cosmology,in that there seem to be large structures inthe universe that are very complicated on alarge scale, whereas one believes that theyshould be governed by simple laws at root.So I tried to find a model where we havesimple structures that produce great com-plication in large areas; I had an interest intypes of hierarchical design.

So I played around with such hierarchi-cal tilings, where you form bigger shapesout of smaller ones; the bigger ones youproduce have the same character but areon a larger scale than what you just did. Ialso had an interest in Escher and his workand met him on one occasion: I had pro-duced single tile shapes that would tileonly in rather complicated ways, andEscher himself used one of these in his lastpicture.

What was the name of these tiles? Themagic something?That�s different: those are the impossibleobjects. The staircase and the tribars that

people now call the �impossible triangle�were things my father and I played aroundwith. Later, Maurits Escher incorporatedthem in some of his pictures: Ascending anddescending used the staircase and the water-fall used the triangle. And he actually usedours, because we sent him a copy of ourpaper.

I met Escher once, and left him a copy ofa puzzle I�d made which consisted of wood-en pieces which he had to try and assem-ble. Well, he managed to do this all right,and somewhat later when I explained thebasis on which it was constructed he pro-duced a picture called Ghosts � as far as I�maware it was his last picture, when he wasquite ill � and it�s based on this tile I�dshown him � twelve different orientationsof this shape.

But that was just a sideline, an amuse-ment really, and the way the tilings cameabout was in two stages. I�m sure I owe adebt to Kepler, although I didn�t realise itat the time, because my father owned abook showing the picture that Keplerdesigned which had a number of differenttilings that he played with. Some of thesewere of pentagons, and these tilings withpentagons are very close to the tilingshapes I produced later.

Now I was aware of these things becauseI�d seen them, but they were not what Ithought of when I was producing my own.They just coloured my way of thinking,which must be rather similar to what hap-pened to Shechtman when he discoveredquasi-crystals. He didn�t think about my

tilings, but when I spoke to him later hesaid he was aware of them. I suspect thatit�s the kind of thing that puts you in a kindof frame of mind, so that when you seesomething, you�re more receptive to it thanyou would have been otherwise. So yes,I�m sure it�s true of me with Kepler that Iwas more receptive of his kind of design .

These three-dimensional forms of your tileshave appeared in recent years, as quasi-crystals. Did you ever anticipate such appli-cations of your non-periodic tilings?Well I did, but I was overcautious I sup-pose, because I certainly knew this was atheoretical possibility. But what worriedme was that if you ever tried to assemblethese them you�d find it very hard, andwithout kind of foresight it�s difficult not tomake mistakes. I sometimes gave lectureson these tilings, and people asked me �doesthis mean that there�s a whole new area ofcrystallography� � and my response wouldbe �yes, that�s true � however, how wouldNature produce things like this, becausethey would require this non-local assem-bly?�. And it seemed to me that maybe youcould synthesise such objects with great dif-ficulty in the laboratory, but I didn�t seehow nature would produce them sponta-neously.

Now I think, although people nowunderstand them better, the situation ismuch the same. I still don�t think we knowhow they�re produced spontaneously, andthere are different theories about how theymight come about � maybe there wassomething a little bit non-local, somethingbasically quantum-mechanical, about thoseassemblies which I came to think is proba-bly true, but it�s not an area that people areagreed about � in fact, it�s not totallyagreed that quasi-crystals are this kind ofpattern, although I think its getting prettywell accepted now.

I was first shown the physical objects, thediffraction patterns, by Paul Steinhardt at aconference in Jerusalem to do with cosmol-ogy. I was talking about general relativityand energy and he was talking about infla-tionary cosmology, and he came up to meand said �look, I want to talk to you aboutsomething nothing to do with this confer-ence�. He showed me these diffraction pic-tures that he�d produced, and it was quitestartling but very gratifying � in fact, curi-ously enough, I wasn�t completely sur-prised. I suppose I felt that it must be rightand nature is doing it somehow. Natureseems to have a way of achieving things inways which may seem miraculous; this wasjust another example of that.

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Interview with Sir RInterview with Sir Roger Poger Penrenrose ose part 2

Interviewer : Oscar Garcia-Prada

In part 1 of this interview (in the previous issue), Roger Penrose described his work in twistor theory and cosmology. He now talksabout his work on tilings and impossible objects, and about his popular books on mathematics.

A Penrose tiling

Have other examples of your recreationalmathematics appeared in nature orphysics?One thing I got from my father is that younever draw a line between the two. He waslike that � he did things for fun, whichmight be making children�s toys out ofwood � puzzles, or gadgets connected withhis work. He�d make things like compli-cated slide rules that were supposed to testsome statistical results. He made a bust ofone of his patients, I seem to remember,and then he spent his later years produc-ing wooden models that reproduce them-selves. There was no line � you couldn�tsay what was recreational and what wasprofessional work.

In 1989 you published a best-selling book,�The Emperor�s New Mind�, where you�reconcerned with computers and artificialintelligence, the mind, the laws of physics,and many other things. What is the centralquestion you address in this book?I think I mentioned earlier that I had for-mulated a certain view while I was a gradu-ate student. Before that, I�d been quitesympathetic to the idea that we were allcomputers, but it seemed to me thatGodel�s theorem tells us that there areaspects of our understanding which youcannot encompass in a computational pic-ture. Nevertheless, I still maintain a scien-tific viewpoint that something in the lawsof physics allows us to behave the way wedo, but that the laws of physics are muchdeeper. My view is that we know much lessabout them than many people would main-tain.

So I was quite prepared to believe thatthere was something outside computation.I�ve been interested in mathematical logicfor a long time, and I�d known for a long

time that there are things of a mathemati-cal nature that are outside computation;that didn�t frighten me � it just seemed tome �all right, why not?� Then I saw a tele-vision programme where Marvin Minsky,Edward Fredkin and others were makingoutrageous statements about how comput-ers were going to exceed everything wecould do. What they were saying was logi-cal if we were indeed computers � but sinceI didn�t believe that, it seemed to me thatthis was something they�d completelymissed � and not only they had missed it,I�d never seen it anywhere else. So as I�dpreviously considered writing a popular orsemi-popular book on physics � somethingI thought I�d do at some stage in life, butperhaps not until I retired � this providedan opportunity: �All right, let�s explainthings about physics and what the world islike, as far as we know, but with a differentfocus: let�s explore the laws of physics tosee whether there�s scope for something ofa non-computational character�. I�d neverseen anybody put this forward as a seriousviewpoint. Since it seemed to me that itneeded to be put forward, that�s what I did.

So this was an attack on artificial intelli-gence; but you also questioned the fact thatas quantum mechanics is incomplete, theunderstanding of physical laws has to pre-cede understanding the functioning of themind?Yes, I�d felt that if there was somethingnon-computational it needed to be outsidethe laws that we presently understand inphysics, because they seem to have thiscomputational character. And it alsoseemed to me that the biggest gap in ourunderstanding is when quantum theoryrelates to large-scale objects, where therules of quantum mechanics give us non-

sense � they tell us that cats canbe alive and dead at the sametime, and so on, which is non-sense; we don�t see our worldlike that. Yet quantum theorywas supposed to be so absolutelyaccurate, so why are we notaware of the manifestations ofthat theory on a large scale? Itseemed to me that the theorycan�t be quite accurate and thatthere must be some changes thattake place when it gets involvedwith large-scale objects.

I think I�d already thought thiswas to do with when gravitation-al phenomena start to becomeentangled with quantum effects� that�s when the changes startto appear, and there are goodreasons for believing that. Sothat is what I believed at thetime, and still believe; but whenwriting The Emperor�s New Mind Ididn�t really have any clear ideawhere in brain function quan-tum effects could start tobecome important and have aninfluence on large-scale effects,and where these new physicalprocesses that should be non-computable could come in.

So I started writing the book, expectingthat by the time I�d finished I�d have someclearer ideas on this. This happened tosome degree � I was very ignorant aboutlots of things to do with the brain when Istarted, and I had to study to write thechapters specifically on them, and in par-ticular the idea of �brain plasticity�, that theconnections between neurons can changeand that these changes can take placequickly � it seemed to me that this was veryimportant, and that the new physics comesin to regulate these changes.

So has this given you any clue as to whatchanges are needed in quantum mechanics?A little, but the physical motivations arelargely independent. I did, however, thinkabout the needed changes in quantummechanics a little more seriously thanbefore and I changed my views betweenwriting the two books.

You�re now referring to your second book�Shadows of the Mind�. Do you pursue thesame problems in this book?Yes, but Shadows of the Mind had slightlyambivalent purposes. Originally I startedto write it to address some of the pointsthat arose out of people�s criticisms andmisunderstandings of The Emperor�s NewMind, and in particular my treatment ofGödel�s theorem. I just didn�t expect thekind of vehement response that I got. Iwas very naïve, perhaps. I didn�t realisethat people would feel attacked in the waythey did and would therefore respond byattacking me, while misunderstanding a lotof what I was trying to say.

The main point that I was making aboutGödel�s theorem was that if you have a sys-tem which you believe in and which youbelieve might be usable as mathematicalproof, then you can produce a statementwhich lies beyond the scope of the system,but which you must also believe in. Nowthere is an assumption here that the systemis consistent, something which I didn�tbother to stress particularly, because itseemed to me quite obvious that there�s nopoint in using the system if you don�t thinkit�s consistent: the proof is no proof if it�s ina system you don�t thoroughly believe inand therefore don�t trust its consistency.That seemed to me to be obvious, but Ididn�t make those points strongly and sothere are loopholes that people couldpoint to, which of course they did. So I feltit necessary to address these issues with agreat deal more care in Shadows. It was notmeant to be a particularly long or popularbook; it was just addressing these pointsand was quite technical and complicated inplaces, but in the process of writing thisbook, two things happened.

One of them was that I received a letterfrom Stuart Hameroff telling me about thecytoskeleton whose structures inside cells Iwas totally ignorant of. But most peoplewho work in artificial intelligence didn�tseem to know about them either; MarvinMinsky didn�t, as he told me afterwards.But it seemed to me that here was a com-pletely new area for which it was muchmore plausible that quantum effects could

INTERVIEW

EMS March 2001 13

be important. They are much smallerstructures than neurons and are muchmore tightly organised structures. Themost relevant of these were microtubules,where one has a much more believableplace for coherent quantum phenomena totake place. It is still hard to see how this

takes place, because it is dif-ficult to maintain quantumcoherence on the large scalethat one needs in order forthese ideas to work. Oneneeds to go beyond what canbe done in any physics labtoday, and there is no physi-cal experiment performedtoday that can achieve thekind of quantum coherenceat the level that I wouldneed in order for the kind ofphenomena to take placethat I suspect are takingplace all the time in ourbrains. So nature has been alot cleverer than physics hasbeen able to be so far, butwhy not? It seems to mequite plausible that this isthe case. The cytoskeleton,and in particular micro-tubules, seem to me to bestructures where quantumcoherence is much moreplausible. So I changed thenature of this book. I want-ed to put in microtubulesand the cytoskeleton, and soI needed to learn a little bitmore about it and to expresswhy I think that�s importantin brain action.

The other thing whichhappened was that I somewhat shifted myviewpoint on quantum state reduction, inrelation to gravity. The viewpoint I�d heldfor quite a long time, and which isexpressed in The Emperor�s New Mind, ismore or less that if you have too big a dis-crepancy between two states, then they

don�t superpose and the statereduces. This discrepancy is to bemeasured in terms of space-timegeometry; I called it the one gravi-ton criterion. It is to do with howmany gravitons come into this dif-ference between the two states. Now in work that I did subse-quently, and also in work that hadbeen done by others (particularlyDiósi, a Hungarian, and Ghirardiin Italy) who had developed dif-ferent ideas in connection withquantum state reduction, itseemed to me that I needed tomodify the view that I had before.I think it�s quite a significant mod-ification. Basically, when youhave two states that are signifi-cantly different from each other,then their superposition becomesunstable, and there is a calculabletime scale involved in how long ittakes for the superposition to�decay� to one state or the other.The details are probably a bit tootechnical for here, but there is atime scale, rather than an instan-taneous reduction, and this timescale produces figures that aremuch more plausible � also, it iseasier to use and it may be muchmore relevant to brain action.

One can see how to use it, and these ideasI developed with Stuart Hameroff. Weproduced a number of suggestions abouthow this idea could be carried forward, butagain we found a great deal of opposition.A lot of these ideas are clearly speculative,but I don�t think it is that speculative thatsomething like this is going on. It seems tome it has got to be. Consciousness seemsto be such a different phenomenon fromthe other things we see in the physicalworld that it�s got to be something veryspecial, in physical organisation. I can�tsee that it�s really just the same old physicsput together in more complicated systems.It�s got to be something of quite a differentcharacter from other things that areimportant in the way the world operates.This new physics would be only occasional-ly employed in a useful way, and you haveto have a very careful organisation thattakes advantage of whatever is going on instate reduction and channels it in a direc-tion which makes us operate. But it is veryrarely actually taken advantage of in phys-ical systems, where most things don�t usethis phenomenon in any useful way at all.

In these books you reveal yourself to be aphilosopher. How have you come to termswith the Great Mystery?Well, there are lots of hugely unansweredquestions � there�s no doubt about that.I�d certainly want to emphasize that even ifeverything that Stuart Hameroff and I sayturns out to be absolutely correct, it wouldnot answer these questions. I hope we aremoving a little in the right directiontowards answering those questions.However, I think there�s very littleprogress towards answering the deep ques-tions of what is going on in mentality, whowe are, what is consciousness, why are wehere and why does the universe allowbeings who can be aware, or is there lifeafter death? Any questions of that natureseem to me to require us to know moreabout what the world is like � we reallyknow very little.

People say, and physicists often say: wenearly have the solution to the grand theo-ry of the universe which is just around thecorner, the theory of everything. I simplyjust don�t believe that. I think there aremajor areas of which we have almost nounderstanding at all, and it�s quite curiousthat one can have a view which seems toencompass (at least in principle) most ofthe things that you see around you, whythey behave this way and that way.

One of the major things that isn�texplained is simply ignored and just sweptunder the carpet by physicists generally,which is quantum state reduction. Theysay: quantum theory is a beautiful theory;it works perfectly well and describes howtiny particles behave. But, to put it blunt-ly, it gives you the wrong answer. What thetheory tells us is that, for example, if youhad Schrödinger�s famous cat, the catcould very easily be put into a state ofbeing alive and dead simultaneously.That�s simply wrong; it doesn�t do that. Sowhat is it? I mean, there is something bigmissing from our view of the world, it�s

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INTERVIEW

huge and it�s not just a tiny phenomenonwhich we haven�t quite got because we�vegot to get the last decimal place right andthe coupling constant or something: it�s ahuge aspect of the way the world behaveswhich we simply do not understand, andunderstanding it is, in my view, one smallstep towards understanding what mentali-ty is. I think it must be part of it, but it isnot going to answer the question of men-tality. We may well know what state reduc-tion is (and I expect we will know if wedon�t destroy ourselves first); then that the-ory will have as part of its nature somecompletely different way of looking at theworld from the way we have now. We�vealready seen that happen in Einstein�s gen-eral relativity, because before that we hadNewton�s theory which told us how bodiesattracted each other with forces and movedaround and so on. It�s beautifully accurate:Newton�s theory is incredibly precise. Ittells you how the planets move around intheir orbits to almost complete precision �not quite, but almost. You might think itonly requires a little tiny modification to

make it completely right, but that�s notwhat happened. Einstein produced a the-ory that is completely different. Its struc-ture is utterly different from that ofNewton�s theory, but it gives almost exact-ly the same answers. The philosophicalframework of that theory is quite different.The very nature of space is warped, and

that changes our whole outlook. Now whatI am saying is that when we see how quan-tum theory is to be changed to accommo-date state reduction as a phenomenon,with the measurement process as a phe-nomenon, we shall have to come to termswith a completely different way of lookingat what matter is and what the world is like.

Are you saying that this is a step that willhelp us to address questions that are typi-cally addressed, by some people at least, byreligion?Well it will, I think so. But you see I�mmore� �tolerant� isn�t quite the right word� I think I�m a little bit more supportiveof religious viewpoints than many totallyscientific people would be. It�s not that Ibelieve in the dogma that is attached toany of the established religions, because Idon�t. On the other hand, religions aretrying to address questions which are notaddressed by science, particularly moralissues. I regard morality as something withan absolute �platonic� component which isoutside us. Is there really a platonic

absolute notion of�the good�? I haveto say I�m inclinedsomewhat in thatdirection. I thinkmorality is not justman made, butthere are thingsoutside us whichwe have nothing inthe way of scientif-ic understandingof at the moment,but neverthelessthey are part of abig overall picturewhich maybesomeday will cometogether. So Imean these ques-tions are nowadaysconsidered not tobe part of science,but they�re part ofreligion, you see.Well, as I say, Idon�t believe in thedogma of religion,but I do think thatreligion is gropingfor somethingwhich we don�tknow the answersto yet, and which isoutside traditionalscience. So I sup-pose what I think isthat the whole sci-entific enterprisemust broaden itsscope and eventu-

ally change its character.

We have touched on only a fraction of yourwork but what is the result, theorem or the-ory that you are most happy with?Well, I�d have to say twistor theory. Thenon-periodic tilings are nice, and they�resomething to show to somebody. But this

is not something which is as deep, and it�snot something I�ve devoted so much of mylife to. Even when I call it �twistor�, this isslightly inappropriate, because we don�tknow what the theory is yet, the real theo-ry. But if I can answer your question inthat way, I think I�d say that the whole pro-gramme of representing Einstein�s theoryin terms of twistors is what I�m proudest of.I don�t know if you�d call that an answer toyour question, because it�s not a theorem,it�s not one result, it�s a body of ideas. Isuppose in amongst those I would thinkthe �non-linear graviton construction� isprobably the thing which I feel mostpleased about so far, but that is part of theprogramme, and I don�t regard that as anend-point; it�s something on the way. Ihope that when one really understandshow the Einstein equations can be incorpo-rated into twistor theory, we�ll see some-thing, a much broader picture of whichthat is just part. But we don�t quite havethat fully yet.

How do you actually work? How do youselect a problem? I�ve seen you many timesin your lectures and seminars making pic-tures � do you visualise things that way?A lot is very visual, but not entirely visual.Certainly some things I�ve done are notentirely visual; for example, algebraicthings aren�t. But I do find visual imageryabsolutely essential in my work, and that�sthe way I do it usually. I often start to writesomething down, but it doesn�t help verymuch. I may see a picture sometimes, buteven drawing it on a piece of paper doesn�thelp because it�s not something you canexpress very well that way. It�s an imagewhich is hard to put down, but I do findexpressing things by drawing pictures rea-sonably accurately sometimes helps, butsometimes it�s an accuracy that�s very mis-leading. You�ve got to know what it means.I mean, the pictures are not really accuratebecause I might draw a picture of some-thing which has got the wrong number ofdimensions and it�s really in a complexspace rather than a real space, but you geta feeling for knowing what�s reliable andwhat isn�t in a picture. In some ways it�s arelief when one can do a calculation. If youcan reduce something to a calculation youcan deal with it, but so few of the things Ido actually find their way into calculations,because that�s not the problem. The prob-lem is a conceptual or a geometrical prob-lem � often conceptual: you have to look atthings in the right way. And that can bequite hard sometimes. But I suppose whatI am probably best at, when it comes tomathematics, is the geometrical, where Ican visualise things well. I don�t find itvery easy to work with complicated formu-lae or analytic notions. It�s an uphill jobsometimes. To a certain extent, things inanalysis can be looked at geometrically.

What are you working on right now?Well, what I think about most of the timewhen I�m not doing something else liketrying to write up endless things one afterthe other, is this problem about theEinstein equations and twistors, which I

EMS March 2001 15

INTERVIEW

Roger Penrose and Jean-Pierre Bourguignon, Durham 1982

think is almost there or is very close tobeing resolved; but it�s not completelyclear. I regard that as the major problem.I also try to make progress with the quan-tum state-reduction problem � even withsome possible experiment.

You�ve mentioned a number of mathemati-cians and physicists, but who is your mostadmired mathematician or physicist?That�s a tough one � there are a lot ofthem. I�m not sure there�s a single one asa mathematician. I always had a tremen-dous admiration for Riemann because hewas a miracle man. As a mathematician Ithink it probably would be him, I guess, butit�s not quite clear because on the physicsside I�ve always thought of Galileo,Newton, Maxwell and Einstein as being thefour most major figures. Of those, I sup-pose one would have to say that Newtonwas a more impressive figure thanEinstein, no matter how impressiveEinstein is. I mean, he�s had absolutelywonderful ideas. General relativity is a fan-tastic theory � of any physical theory thatI�ve seen, I�d say put general relativity atthe top because it�s amazing. But Newtonwas such a powerful mathematician, andEinstein was less so. Einstein had thesetremendous physical insights, but Newtonhad this great power as a mathematician as

well as deep physical insights. And I�vealways had a soft spot for Maxwell, but alsoGalileo � I think I�ve always admired himfrom a long time back. I don�t know � I�mnot trying to single out any one in particu-lar. Riemann as a mathematician I sup-pose, possibly more than Gauss I wouldsay. Archimedes also, I suppose, was a veryimpressive character; also Euler.

What is your opinion about the direction inwhich research in mathematics and physicsis going nowadays?

Something I�ve found slightly discouragingabout the way things go these days, whichis not the fault of the mathematicians orthe physicists, is a problem of technology.Although technology is a wonderful thing,it has the effect of magnifying fashions. It�sso easy to communicate ideas from oneend of the world to the other instanta-neously, and this means that the fashionshave global control over what goes on. Itmeans that a lot of progress can be made infashionable areas, but there�s somethingwhich is lost in there, I feel. It�s a bit sillyto hark back or think about things in thisway, but when communications were muchmore difficult there were these little pock-ets of people working on different things.Maybe I have this starry eyed view of whatit was like; it probably wasn�t like that.Fashions were still important in the olddays too, but somehow there�s somethingabout the global nature of the fashionsthese days which I find a bit disturbing.But it�s worse in physics than it is in math-ematics, I think. I always used to think thatmathematics was immune from this kind ofthing. You would have these people work-ing away and developing their wonderfulideas in relative isolation, and occasionallygetting together, and some things wouldspring from the coming together of differ-ent viewpoints. But now it�s much more as

if you can go anywhere and they�re allworking on the same thing, which is ratherdiscouraging. In physics it is particularlylike that. This applies to highly theoreticalwork where there is no stabilising inputfrom experiment.

Communication is obviously beneficial, butthe aspect you�re remarking on is actually anegative one.It is a negative one, I think. And there�sanother thing that goes along with this.It�s not relevant in mathematics, but in

physics it�s to do with the expense of bigexperiments, which means that you have tobuild bigger and bigger machines to lookat higher energies, and so on. I can seewhy they do that, but since the experi-ments are so expensive, they require a lotof money and government backing andhuge organisations, and therefore youmust have committees to decide which tosupport, and so on. Committees have todecide where the money�s going, so theyconsult people who are considered to bethe experts in the relevant areas, andtherefore the things tend to get locked incertain directions because the experts havegot there, as they were the ones who wereimportant in the development of the cur-rent theories. It�s hard to break away fromthis.

It becomes a political thing as well.It does become political as well, becausemoney is involved. One doesn�t have thefree-ranging way of thinking about thingsthat was there before. So I�m only express-ing the negative points, because maybe thepositive ones are more obvious. Obviouslythere�s huge activity � all over the worldyou find people who work in areas and whohave never before had a chance to thinkseriously about science. Now the internetallows them to get involved. That�s allpositive and I agree with that, but I�m justpointing out that there�s a downside also,which I find disturbing.

Are you questioning the way in whichwhat�s important and what�s not are decid-ed?It is hard to advise them, you see, becausethey�re caught up in the system, and ifyoung researchers want to get jobs afterdoing research, they must work in an areawhich is going to be recognised by the peo-ple who employ them. If I were talkingfrom the point of view of science I�d saythat in quantum state reduction there aresome important problems: you see whyquantum mechanics needs modification.But if they�re working on that, it�s regard-ed as marginal at best, and crackpot atworst. I could even say that with twistortheory: in mathematics it seems to havecaught on, but as a physical theory � if youwork in twistor theory you�ll find it hard toget a job. There are very few people inphysics departments who know about thissubject and consider it important.

I remember reading in one of your papersthat twistor theory was an esoteric subjectfrom that point of view. Do you still main-tain this view?Yes, it�s not much studied as a physical the-ory. I�m not unhappy from my own per-sonal point of view, because when it getsstudied by lots of people it becomes toocomplicated to find out what they�re doing.I�d have to learn and understand theirnotation, which would probably be differ-ent from mine � and that�s hard work. Butif I know that nobody�s working on it, thenthat�s fine � I�m not rushed, and I don�thave to think I�ve got to get in there beforesomeone else does!

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Durham 1982

That�s right. But what recent developmentin science at large has made the greatestimpact on you?Gosh, I�d say cosmology is one of the areas.Gravitational lensing I worked on for a lit-tle bit at one point; it�s an amazing thing.I�d say that astrophysics and cosmology areexciting areas.

What aspects do you find most interesting?What I was talking about, gravitationallensing. I find it interesting because it�s aneffect of Einstein�s general relativity, whichI think people thought was very hard tomeasure. It was the first thing that con-vincingly suggested that Einstein�s theorymight be right. The Eddington experi-ment showed the deflection of light of thestars by the sun, but to see this effect on acosmological scale, to see a galaxy focussedby another galaxy, would have seemedabsolutely ridiculous. Nowadays it�s anobservational tool, people use it all thetime � it�s a way you can tell the mass of anobject by how much focussing it exerts onthe image behind it, and it�s wonderful.It�s not just that it�s a powerful tool in cos-mology, but because it uses somethingwhich is close to my heart I suppose; that�swhy I like that one so much.

But that�s just one area; there are lots ofother things. I suppose the experimentson quantum entanglements (non-locality):you can get two ends of the room, 12metres apart � well, nowadays it�s longerthan that (10 kilometres) � by these quan-tum entanglement effects the two are con-nected through quantum mechanics, it�samazing. One knew it had to be there inquantum theory, but it�s very impressive tosee that it�s real.

Quasi-crystals are remarkable. Hightemperature superconductors are amaz-ing, and so are developments in molecularbiology. Some of the things that peopleare now learning about cells and aboutcytoskeletons and microtubules I find fan-tastic. Partly this is because I didn�t knowwhat was going on in biology, and havinggot slightly involved in this subject I seesome remarkable things.

I came across Schrödinger�s book �What islife�, for which you wrote a preface.That�s right, and it mentions the aperiodiccrystal ideas that he believes are at theheart of life.

Can you tell us something about this?My father was very taken with the idea, Irecall, and we had these mechanicaldevices I mentioned earlier which repro-duce themselves, and he then developedmuch more elaborate devices which hemade of wood, little things with leverswhich copied themselves. He sometimesreferred to these things in Schrödinger�sterminology as �aperiodic crystals� � a crys-tal that went on for a while and thenstopped growing, because that was how itcame to an end. Life was to be thought ofin that way. So it had some influence onhim, and indirectly on me. But I don�tknow about the quasi-periodic tilings,whether that has much connection histori-cally.

What books are you reading at the moment?Gosh, I never get a chance, though some-times I have to review a book. I�m notreading one right now. It�s one of myregrets that I find myself so busy that I

have hardly any time to read anything forfun, which I like to do.

If the day had 36 hours, what would youlike to do?If it had 36 hours, there should be a rulethat the extra twelve were only allowed tobe used for things that were not directly todo with one�s work. I enjoy reading whenI get the chance, and also going to the the-atre, to films, and listening to concerts � allthe things I�d love to do more of. Butthere are films that I never get the chanceto see, including some wonderful ones thatare going around now. I used to read quitea bit of science fiction, but I hardly have achance to now. I sometimes read MichaelFrayn; I find him tremendously funny.

What about music? Do you have afavourite composer?On the whole I prefer classical music, but Ienjoy jazz too � my wife�s influence! Myparents each had a favourite composer: formy father it was Mozart, who was his God,and for my mother it was Bach. I eventu-ally realised that Bach is really myfavourite by quite a long way; I�ve alwaysregarded those two as a cut above the othercomposers. Bach I think I see much morein, there�s something you can always goback to, and there�s more and more of it.But I think it�s the perfection in Mozartthat I find somewhat magical. I�ve alwaysrated him above Beethoven, who neverhad quite the same magic for me. I can seehe had the power and originality, butsomehow there�s not quite the magic there.Maybe even with Schubert there�s a bit ofmagic that I don�t quite see in Beethoven.To go back further, I like Vivaldi andPurcell and others. But I also quite likesome modern composers, such asStravinsky, Prokofiev and Shostakovitch.

What do you consider the most profounddevelopment of the last century?Einstein�s general theory of relativity. Imight have said quantum mechanics, but Ithink that that theory isn�t finished yet,because of the measurement paradox. Idiscuss these things in my new book, TheRoad to Reality, which should be publishedsometime early in 2002. It�s almost com-pleted, but still there�s work to be done onit. It�s about mathematics and physics andthe profound relation between the two. Init I try to give my views of some of the pop-ular developments in modern theoreticalphysics � I may get myself into more trou-ble!

Finally, what will keep you busy in thefuture?I have many more books in mind, so I�llkeep writing. Also, I�ll keep working ontwistor theory, as there�s a great deal to doin that subject. I have new ideas to do withquantum state reduction � even an experi-ment (FELIX) which I hope will be per-formed in space while I�m still alive! Inaddition to all this, I have a new son(Maxwell Sebastian � Max, for short) whowas born on 26 May 2000 and will keep meeven busier!

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EMS March 2001 17

Roger Penrose is the current Gresham Professor of Geometry, in London. Last year he met theearliest Gresham Professor, Henry Briggs.

122 Let A1 A2 A3 A4 A5 A6 A7 be a regular heptagon. Prove that 1/A1A2 = 1/A1A3 + 1/A1A4. (Romanian high school textbook)

123 The sequence {xn} satisfies √(xn+2+ 2) ≤ xn ≤ 2, for all n ≥ 1. Find all possible values of x1986. (Romanian IMO selection test, 1986)

124 Find the real numbers x1, x2, � , xn satisfying √(x1 � 12) + 2√(x2 � 22) + �+ n√(xn � n2) = (x1 + x2 + �+ xn)/2 (Gazeta Matematica)

125 Prove that there is a perfect cube between n and 3n, for any integer n ≥ 10. (Gazeta Matematica)

126 Show that if √7 � m/n > 0 for the positive integers m, n, then √7 � m/n > 1/mn. (Romanian Mathematical Olympiad, 1978)

127 Find all positive integers x, y, z satisfying 1+ 2x3y = z2. (Romanian IMO selection test, 1984)

PROBLEM CORNER

EMS March 200118

PPrroblem Corneroblem CornerContests from Romania, part 3

Paul Jainta (Schwabach, Germany)

Competing against teams representing 81 countries, a teamof six Romanian high-school students won six medals at theFortieth International Mathematical Olympiad (IMO), heldin their capital Bucharest in summer 1999. While the com-petition is between individuals, unofficial rankings placedthe Romanian team fourth, after China, Russia andVietnam.

It�s no accident that Romania is exceptional in identifyingmathematically able young persons. As Mark Saul,Bronxville School, N.Y., wrote in his foreword to a new col-lection of problems consisting mainly of difficult problemsfrom Eastern Europe: �Preparation for participation in achallenging contest demands a relatively brief period ofintense concentration, asks for quick insight on specificoccasions, and requires a concentrated but isolated effort�.

The rich Romanian Olympic ambience can serve as amodel for studying the conditions that go to make greatproblem-solvers. For these young people an Olympiadproblem is an introduction, a glimpse into a world of math-ematics not afforded by the usual classroom situation, for agood Olympiad problem captures in miniature the processof creating mathematics: immersion in the situation, exam-ination of possible approaches, and the pursuit of variouspaths to the solution. But struggling with tough questionsoften leads to fruitless dead ends. Grappling with a goodproblem provides practice in dealing with the frustration ofworking at material that refuses to yield, until a lucky solvercatches sight of a silver lining that heralds the start of a suc-cessful solution. Like a well-crafted work of fiction, a goodOlympiad problem tells the story of mathematical ability tocreate which apprehends a good part of the real experience,and leaves the participant wanting still more.

The Grigore Moisil Contest In Romania, thanks to VasileBerinde, North University of Baia Mare, a student canchoose between nine intercounty competitions. The biggestand most famous is the contest named after the greatRomanian mathematician Grigore Moisil (1906-73). It was

initiated in 1986 between the four counties of Maramures,Bistrita-Nasaud, Satu-Mare and Salaj, in the north-west ofRomania, and alternates between the capital towns of thecounties involved. The Moisil contest is open for pupilsfrom grades 5-12, and is run as a team-contest with teams of3 or 4 individuals for each grade, selected from the contes-tants who completed the county round of the NationalMathematics Olympiad. Thus, the Moisil competitionserves as additional screening for the national event. It ismainly supported by the Ministry of Education and also bythe regional branch of the Romanian Society forMathematical Sciences and private sponsors. Diplomas andprizes are awarded. Normally, the problems are chosen byan independent selection committee of experts outside thecatchment area of the contestants.

Another tricky competition, similar to the Moisil contest,is named after the Romanian mathematician, Traian Lalescu(1882-1929). The Lalescu contest is for students fromgrades 5-12 in one of the four south-west counties Arad,Caras-Severin, Hunedoara and Timis. The Lalescu contestis organised by the Faculty of Mathematics, West Universityof Timisoara. Here the problems are first-hand, or emergefrom prestigious journals such as Gazeta Matematica orRevista Matematica din Timisoara. The Lalescu competitionhas been running for fifteen years.

The following contests form a small portion of what isavailable to youngsters hungry for mathematics: the SpiruHaret � Gh. Vrânceanu contest takes place in the Eastern partsof Romania; the L. Duican competition is established inBrasov, and the N. Paun event occurs in and aroundBucharest; the Gh. Mihoc tournament takes place in thesouth-east of the country; the D. V. Ionescu � Gh. Lazar con-test occurs in the central region of Transylvania, the homeof Count Dracula.

From this contest scene I�ve compiled a cross-section ofquestions; readers are invited to compete with the leadingjunior minds of Romania.

Solutions to some earlier problems

110 The quadrilateral ABCD has two parallel sides. Let M and N be the midpoints of DC and BC, and let P be the common pointof the lines AM and DN. If PM/AP = ¼, prove that ABCD is a parallelogram.

Solution (Partially) solved by Niels Bejlegaard, Stavanger (Norway), Oddvar Iden, Bergen (Norway), J. N. Lillington, Dorchester (UK), Mr &Mrs Mudge, Pwl �Trap, St. Clears (Wales), Dr Z. Reut, London. No solver considered the two cases: AD parallel to BC and AB parallel to DC.Two submitters used plane coordinate geometry. Mr & Mrs Mudge even affixed to their submission:�The most inelegant proof of problem 110�.I shall outline an elementary geometrical solution. Case 1: AD is parallel to BC (see the left-hand figure). Let Q be the midpoint of DN. Then QM, BC and AD are all parallel, and QM = ½NC = ¼BC and QM = PM/PA·AD = ¼AD. Hence AD = BC and ABCD is a parallelogram.Case 2: AB is parallel to DC (see the right-hand figure). Let R be the midpoint of AM. In the trapezium ABCM we then have RN = ½(AB + MC) = ½AB + ¼DC. On the other hand, from similar proportions we have RN = RP/PM·DM = (3/2)·DC/2 = ¾CD (for PM = (1/5)AM, AB = (4/5)AM and RP = (1/2)AM(1/5)AM = (3/10)AM, giving RP/PM = 3/2). It follows that ½AB+¼DC = ¾DC, which simplifies to AB = DC.

111 Let k be an integer and let p(x) be the polynomial p(x) = x1997 � x1995 + x2 � 3kx + 3k + 1. Prove that the polynomial has nointeger roots, and that the numbers p(n) and p(n) + 3 are relatively prime, for each integer n.

Solution by Pierre Bornsztein, Courdimanche (France); also solved by Niels Bejlegaard, Oren Kolman, Jerusalem (Israel), Maohua Le,Guangdong (P.R. China) and Dr Z. ReutMore precisely we first prove that for each integer n, p(n) ≠ 0 (mod 3). Indeed, rewrite p(x) as p(x) = x1994 (x3 � x) + 3(k � kx) + x2 + 1.Let n be an integer. It is well known that n2 = 0 or 1 (mod 3), and thus n2 + 1 ≠ 0 (mod 3). From Fermat�s Little Theorem, we have n3 � n = 0 (mod 3). It follows that p(n) = n2 + 1 (mod 3) ≠ 0 (mod 3), as claimed. We deduce easily that for each integer n, p(n) ≠ 0 and thus p(x) has no integer root.Now suppose, for a contradiction, that q is a prime number dividing both p(n) and p(n) + 3. Then q divides p(n) + 3 � p(n) = 3, and hence q = 3. Thus p(n) = 0 (mod3), which is impossible from the above. It follows that, for each integer n, the numbers p(n) and p(n) + 3 are relatively prime.

112 Find the image of the function f(x) = (3 + 2sin x) / [√(1 + cos x) + √(1 � cos x)].

Solution by Gerald A. Heuer, Moorhead, MN (USA); also solved by Pierre Bornsztein, J. N. Lillington, Dr Z Reut.The image is the interval [0.5, 2.5]. Since f is continuous and f(x + 2) = f(x) for all x, it suffices to show that the minimum and maximum values of f(x) on [�π,π] are 0.5and 2.5, respectively. We shall show that f(x) is decreasing on [�π,�π/2], increasing on [�π/2, π/2] and decreasing on [π/2,π]. Since f(�π/2) = 0.5, f(π/2) =2.5 and f(�π) = f(π) = 3/√2, the stated result then follows.For 0 ≤ x ≤ π,cos (x/2)= cos (x/2) andsin (x/2)= sin (x/2), so with u = x/2, and using the half-angle formulas √(1 + cos x) = √2 cos (x/2) and √(1 � cos x) = √2 sin (x/2), we get √2 f(x) = g(u) = (3 + 4 sin u cos u) / (cos u + sin u), 0 ≤ u ≤ π/2.Then g (u) = [(cos u sin u)(1 + 4 sin u cos u)] / (cos u + sin u)2, which has the same sign as cos u � sin u. Thus g (u) > 0 when 0 < u < /4, and g (u) < 0 when π/4 < u < π/2. It follows that f(x) is increasing on [0, π/2] and decreasing on [π/2,π].For �π ≤ x ≤ 0,cos(x/2)= cos(x/2) andsin(x/2)= �sin(x/2), so with u = x/2 again,

√2f(x) = h(u) = (3 + 4 sin u cos u) / (cos u � sin u).Then h (u) = [(cos u + sin u)(7 � 4 sin u cos u)] / (cos u � sin u)2, which has the same sign as cos u + sin u. Thus h (u) > 0 for �π/2 < u < �π/4 and h (u) < 0 for �π/4 < u < 0. It follows that f(x) is increasing on [�π,�π/2] and decreasing on [�π/2,0], establishing the claim.

PROBLEM CORNER

EMS March 2001 19

113 Let f(x) = ax3 + bx2 + cx + d be such that f(2) + f(5) < 7 < f(3) + f(4). Prove that there exist u, v such that u + v = 7 and f(u) + f(v) = 7.

Solution by Oren Kolman, Jerusalem; also solved by Niels Bejlegaard, Pierre Bornsztein, Gerald A. Heuer, J. N. Lillington, Claude Portenier,Marburg (Germany) and Dr Z Reut.Since the function G(x) = f(x) + f(7 � x) � 7 is continuous, and G(2) < 0, G(3) > 0, it follows from the Intermediate Value Theoremthat there exists u0 in (2, 3) such that G(u0) = 0. Now u0 and v0 = 7 � u0 are as required.

114 Let f be a real function that satisfies the following conditions:(i) f(0, y) = y + 1, for all y; (ii) f(x + 1, 0) = f(x, 1) for all x; (iii) f(x + 1, y + 1) = f(x, f(x + 1, y)) for all (x, y).Compute f(3, 1997).

Solution by Dr Ranjeet Kaur Sehmi, Chandigarh (India); also solved by Niels Bejlegaard, Pierre Bornsztein, Nira Chamberlain, Derby (UK), J.N. Lillington, Dr Z Reut.We have f(0, 1) = 2, by (i), f(1, 0) = f(0, 1) = 2 by (ii), and by (iii), f(1, 1) = f(0, f(1, 0)) = f(1, 0) + 1 = 3.Now suppose that f(1, x) = x + 2 for some x; then f(1, x + 1) = f(0, f(1, x)) = 1 + f(1, x) = 1 + x + 2 = (x + 1) + 2. Thus, by induction, f(1, x) = x + 2 for all x. (1)Next, f(2, 0) = f(1, 1) = 3, by (ii), f(2, 1) = f(1, f(2, 0)) = f(1, 3) = 3 + 2 = 5, by (1).Now suppose that f(2, x) = 2x + 3 for some x; then f(2, x + 1) = f(1, f(2, x)) = f(1, 2x + 3) = 2x + 3 + 2 = 2(x + 1) + 3, by (1) .Thus, by induction, f(2, x) = 2x + 3 for all x. (2)Next, f(3, 0) = f(2, 1) = 5 = 23 � 3, by (ii) and (2), f(3, 1) = f(2, f(3, 0)) = f(2, 5) = 13 = 21+3 � 3, by (2).Suppose that f(3, x) = 2x+3 � 3 for some x; then f(3, x + 1) = f(2, f(3, x)) = 2f(3, x) + 3 = 2[2x+3 � 3] + 3 = 2x+4 � 3 = 2(x+1)+3 � 3, by (2).Thus, by induction, f(3, x) = 2x+3 � 3 for all x. So f(3, 1997) = 21997+3 � 3 = 22000 � 3. [Pierre Bornsztein indicates that this function is better known as Ackermann�s function.]

115 Let n ≥ 3 be an integer and x be a real number such that the numbers x, x2 and xn have the same fractional parts. Prove thatx is an integer.

Solution by Gerald A. Heuer; also solved by Niels Bejlegaard, Pierre Bornsztein, Erich N. Gulliver, Schwäbisch Hall (Germany), Oren Kolman ,J. N. Lillington, Maohua Le and Dr Z Reut.We have x2 � x = a and xn � x = b, where a and b are integers. We show first that if x is rational then it is an integer. From x2 � x � a = 0 we have x = [1 ± √(1 + 4a)]/2 . If this x is rational, then so is √(1 + 4a). Thus √(1 + 4a) is an odd integer, and x = [1 ± √(1 + 4a)]/2 is an integer.It therefore suffices to show that x is rational. Now, x2 = x + a, so x3 = x2 + ax = (x + a) + ax = (1 + a)x + a.Suppose that xk = rkx + sk where rk and sk are integers. Then xk+1 = rkx2 + skx = rk(x + a) + skx = rk+1x + sk+1 , where rk+1 = rk + sk and sk+1 = rka. It follows by induction that for each positive integer m, xm = rmx + sm for some integers rm and sm . In particular, xn = rnx + sn . This, together with xn = x + b implies that (rn � 1)x + (sn � b) = 0, and thus x is rational, provided that rn ≠ 1.Note that a ≥ 0 because x is real. If a = 0, then x = 0 or 1, so is an integer. In the remaining case a ≥ 1. Now r2 = 1 and s2 = a ≥ 1; r3 = a + 1 > 1 and s3 = a ≥ 1. Suppose that rk > 1 and sk ≥ 1.Then rk+1 = rk + sk > rk > 1 and sk+1 = rka ≥ rk > 1. It follows by induction that rk > 1 for all k > 2. Thus x is rational, and is therefore an integer.

That completes the Problem Corner for this issue. Please send me your Olympiad Contests and other suitable materials, as well as your solutions andsuggestions or comments about the Problem Corner.

PROBLEM CORNER

EMS March 200120

Applied Mathematics

in our

Changing World

September 2-6, 2001Berlin, Germany

First SIAM-EMS Conference

ORGANIZED BY: EMS, SIAM and ZIB

CHAIRS: R. Jeltsch, G. Strang and P. Deuflhard

INTERNET: http://www.zib.de/amcw01E-MAIL: [email protected]

First AnnouncementCall for papers and registration

PROGRAM COMMITTEE ❐❐ Vincenzo Capasso, Milano, Italy ❐❐ Peter Deuflhard, Berlin (chair)❐❐ Heinz Engl, Linz, Austria❐❐ Björn Engquist, Stockholm, Sweden❐❐ David Levermore, Maryland, USA❐❐ Volker Mehrmann, Berlin❐❐ Bill Morton, Oxford, UK❐❐ Stefan Müller, Leipzig, Germany

INVITED PLENARY SPEAKERS❐❐ Medicine: Alfio Quarteroni, I / CH❐❐ Biotechnology: Michael Waterman,

USA❐❐ Materials Science: Jon Chapman,

UK❐❐ Environmental Science: Andrew

Majda, USA❐❐ Nanoscale Technology: Michael

Griebel, D❐❐ Commumication: Martin Grötschel,

D❐❐ Traffic: Kai Nagel, CH❐❐ Market and Finance: Benoit

Mandelbrot, USA❐❐ Speech / Image Recognition: Pietro

Perona, USA❐❐ Engineering Design: Thomas Y.

Hou, USA

CONFERENCE OFFICESIAM/EMS Conference 2001

Erlinda C. Körnig ❐❐ Sigrid WackerKonrad-Zuse-Zentrum Berlin (ZIB)

Takustr. 7D-14195 Berlin-Dahlem

GermanyINTERNET: http://www.zib.de/amcw01

E-MAIL: [email protected]

DEADLINES AND IMPORTANTDATES

March 31 Deadline for early registrationApril 15 Deadline for submission of min-isymposia proposalsMay 31 Second Announcement Invitedspeakers and minisymposia June 30 Deadline for submission of abstracts and postersJuly 30 Program and collected abstracts

REGISTRATION FEESSIAM / EMS Non-Members MembersEarly Registration (prior to 04/01/01) US$ 150,� US$ 200,�DM 300,� DM 400,�EUR 150,� EUR 200,�Full Registration (after 03/31/01)US$ 210,� US$ 260,�DM 420,� DM 520,�EUR 210,� EUR 260,�

Meals and accommodation expenses arenot covered. SIAM and EMS are trying toraise special support for students andEastern European scientists.

Keep observing our website.

IMPORTANTThere will be invited talks, minisymposia,contributed talks and posters. Please con-sult our website for registration, accom-modation information and submissionguidelines. The conference office preferson-line registration and on-line submis-sion of abstracts.

TOPICS:1.Medicine

❐❐ medical imaging methods; ❐❐ computational assistence of surgery,

therapy planning; ❐❐ hospital information systems; ❐❐ pharmacokinetics, tumor growth

modelling; ❐❐ artificial organs, immune system

modelling; ❐❐ infectuous disease control,

epidemic spreading; ❐❐ physiology (e.g. dynamics of cardio-

vascular or of respiratory system).

2.Biotechnology ❐❐ biomolecular structural storage

schemes, patent recognition and cir-cumvention;

❐❐ conformational molecular dynamics,drug design, cell factory;

❐❐ mathematical modelling in biopoly-merization;

❐❐ sequence alignment, fuzzy reason-ing;

❐❐ density functional theory, ab-initiocomputation.

3.Materials science❐❐ realistic modelling and simulation of

composite materials, magnetic mate-rial, polymers, glass, and paper;

❐❐ crack propagation and further fail-ure mechanisms;

❐❐ phase transitions, crystal growth, superconductivity, and hysteresis;

❐❐ control of phase transitions andsolidification, modelling of ironmak-ing process;

❐❐ coupling of atomistic and continu-um models, quantum-classicalapproximation and calculation.

4.Environmental science ❐❐ climate and climate impact research,

stochastic climate modelling, inter-mediate complexity modelling;

❐❐ short and medium range meteorolo-gy and oceanography;

❐❐ pollution transport in air, water,and soil;

❐❐ atmospheric chemistry, ozone hole; ❐❐ computational hydrology.

5.Nanoscale technology ❐❐ integrated optics, optical networks; ❐❐ quantum electronics and optics,

general microwave technology; ❐❐ nanoscale techniques in medicine,

porous materials.

6.Communication ❐❐ telecommunication and optical net-

works: analysis, simulation, opti-mization;

❐❐ transmission rate optimization; ❐❐ survivable networks, network design; ❐❐ frequency assignment, channel allo-

cation, load balancing.

7.Traffic ❐❐ optimal periodic train scheduling,

network planning; ❐❐ schedule synchronization; ❐❐ discrete and continuous traffic flow

models; ❐❐ traffic on-line simulation and con-

trol, route guidance and planning; ❐❐ traffic assignment

8.Market and finance ❐❐ financial mathematics and statistics; ❐❐ option pricing; ❐❐ derivative trading, risk manage-

ment; ❐❐ economic time series.

9.Speech and image recognition ❐❐ signal analysis; ❐❐ pattern recognition.

10. Engineering design ❐❐ transport systems in air, in water, or

on land; ❐❐ energy conversion, distribution and

conservation; ❐❐ smart design of consumer products.

These include mathematical subjectslike:

❐❐ PDE analysis and modelling, ❐❐ complex coupled PDE systems, ❐❐ optimal control of PDEs and het-

erogenous systems, ❐❐ variational principles, ❐❐ inverse problems, ❐❐ stability and bifurcation analysis, ❐❐ PDE computational finite element

methods, ❐❐ spatial and temporal homogeniza-

tion, ❐❐ spatial statistics, ❐❐ stochastic geometry, ❐❐ interacting particle systems, ❐❐ stochastic analysis, ❐❐ multiscale analysis and algorithms, ❐❐ multigrid and domain decomposi-

tion, ❐❐ wavelets, ❐❐ turbulence modelling.

SIAM-EMS

EMS March 2001 21

ApplicationsIf you wish to attend any session youshould fill in an application to the C.I.M.EFoundation at the above address onemonth before the beginning of each course(not later than 20 July for courses startingin September). In your application youmust specify your field of current research.An important consideration in the accep-tance of applications is the scientific rele-vance of the session to your field of inter-est. Participation will be allowed only topersons who have applied in due time andhave had their application accepted.CIME will be able to provide partial sup-port to some of the youngest participants.Those who plan to apply for supportshould mention this on the applicationform.

Sites and lodgingMartina Franca is a charming, beautifullypreserved ancient city on the hills of Pugliain southern Italy. Participants are lodgedat the Park Hotel S. Michele; which has awell-kept garden and a large swimmingpool. The lectures will be at the PalazzoDucale (City Hall), within a short walkingdistance.

Cetraro is a beatiful location on theTirrenian coast of Calabria in southernItaly. The nearest train station is Paola, onthe Roma-Salerno-Reggio line.

CIME activities are made possible thanksto the generous support received from:Ministero degli Affari Esteri; TheEuropean Commission, Division XII, TMRProgramme �Summer Schools�; ConsiglioNazionale delle Ricerche; Ministerodell�Università e della Ricerca Scientifica eTecnologica; UNESCO-ROSTE, VeniceOffice.

Lecture notesThese are published as soon as possibleafter the session.

Course 1: 2-10 June in Cetraro (Cosenza):Topological Fluid Mechanics Scientific direction: Prof. Renzo L. Ricca(University College London, UK) [[email protected]] (a) Topological methods in astrophysics (5 lec-tures) Prof. Mitch A. Berger, University CollegeLondon (UK) [[email protected]](b) Applications of knot theory (5 lectures) Prof. De Witt Sumners, Florida StateUniversity (USA) [[email protected]](c) Elements of knot theory (5 lectures)Prof. Louis H. Kauffman, University ofIllinois at Chicago (USA) [[email protected]] (d) Topological methods in hydrodynamics (5lectures) Prof. Boris Khesin, University of Toronto(Canada) [[email protected]](e) Invariants and energy relaxation techniques(5 lectures)Prof. H. Keith Moffatt, Isaac NewtonInstitute for Mathematical Sciences (UK)[[email protected]] (f) From fluid knots to complex systems (5 lec-tures) Prof. Renzo L. Ricca, University CollegeLondon (UK) [[email protected]]

Course 2: 1-8 July in Martina Franca(Taranto): Spatial Stochastic Processes Scientific direction: Prof. Ely MERZBACH(Bar-Ilan University) [[email protected]](a) Mixing results for critical nearest particlesystems via spectral gap estimates (5 lectures) Prof. Tom Mountford, University ofCalifornia, Los Angeles.

(b) Level sets and excursions of the Browniansheet (5 lectures)Prof. Robert Dalang, Ecole PolytechniqueFederal de Lausanne [[email protected]] (c) Weak convergence of set-indexed martin-gales (5 lectures)Prof. Gail Ivanoff, University of Ottawa(d) Set-indexed martingales and point processes(5 lectures)Prof. Ely Merzbach, Bar-Ilan University[[email protected]] (e) Stochastic geometry of spatially structuredbirth and growth processes: applications to poly-mer crystallization processes (5 lectures) Prof. Vincenzo Capasso, Università diMilano [[email protected]] (f) Local time and sample path propertiesof n-parameter processes. (5 lectures)Prof. Marco Dozzi, Université de Nancy, E-mail [email protected] }}

Course 3: 2-8 September in MartinaFranca (Taranto): Optimal transportationand applicationsScientific direction: Prof. Luis Caffarelli,University of Texas, Austin (USA) [[email protected]]Prof. Sandro Salsa, Politecnico di Milano,Italy [[email protected]](a) The Monge-Ampere equation, optimal trans-portation and periodic media (6 lectures) Prof. Luis Caffarelli, University of Texas,Austin [[email protected]](b) title to be announced (6 lectures)Prof. L. C. Evans, University of CaliforniaBerkeley [[email protected]](c) Shape optimization problems through theMonge-Kantorovich equation (4 lectures)Prof.Giuseppe Buttazzo, Università di Pisa[[email protected]](d) Geometric PDEs related to fluids and plas-mas (4 lectures)Prof. Yann Brenier, LAN-UPMC, 4 PlaceJussieu, Paris VI [[email protected]](e) Mass transportation tools for dissipativePDEs (4 lectures)Prof. Cedric VILLANI, University ofTexas, Austin [[email protected]]

Course 4: 9-15 September in MartinaFranca (Taranto): Multiscale Problemsand Methods in Numerical Simulation Scientific direction: Prof. Claudio Canuto,Politecnico di Torino, Italy [[email protected]](a) Multilevel methods in finite elements (6 lec-tures)Prof. James H. Bramble, Texas A. & M.University(b) Nonlinear approximation and applications(6 lectures)Prof. Albert Cohen, Laboratoire d�AnalyseNumérique, Université Pierre et MarieCurie Paris VI(c) Wavelet methods for operator equations (6lectures) Prof. Wolfgang Dahmen, Institut fürGeometrie und Praktische Mathematik,RWTH Aachen(d) Variational multiscale methods, physicalmodels and numerical methods (6 lectures)Prof. Thomas J. R. Hughes, StanfordUniversity, USA

CIME COURSES

EMS March 200122

Fondazione CIMEInternational

Mathematical Summer Center2001 courses

The Fondazione CIME will present four courses in 2001: three will beheld at Martina Franca (Taranto, Italy); the other will be at Cetraro(Cosenza, Italy).

Information concerning these courses is presented below. Further infor-mation can be found on the Web server http://www.math.unifi.it/~cime/

Participants can also address their requests to:Fondazione C.I.M.E. c/o Dipartimento di Matematica �U. Dini�

Viale Morgagni, 67/A - 50134 Firenze, Italy tel: +39-55-434975 or +39-55-4237123 fax: +39-55-434975 or +39-55-4222695

e-mail: [email protected] del CIME: Prof. Arrigo Cellina [[email protected]]

Segretario del CIME: Prof. Vincenzo Vespri [[email protected]]

Please e-mail announcements of Europeanconferences, workshops and mathematicalmeetings of interest to EMS members, [email protected]. Announcementsshould be written in a style similar to thosehere, and sent as Microsoft Word files or astext files (but not as TeX input files). Spacepermitting, each announcement will appearin detail in the next issue of the Newsletter togo to press, and thereafter will be brieflynoted in each new issue until the meetingtakes place, with a reference to the issue inwhich the detailed announcement appeared

2-6: Lévy Processes and Stable Laws,Coventry, UKInformation:Web site:http://science.ntu.ac.uk/msor/conf/Levy/7-9: 16th British Topology Meeting,Edinburgh, UKInformation:Web site: ttp://www.ma.hw.ac.uk/icms/current/britopol9-12: British MathematicalColloquium, Glasgow, UKInformation:Web site: www.maths.gla.ac.uk/bmc200115-21: Spring School in Analysis,Paseky nad Jizerou, Czech Republic Information:e-mail: [email protected] Web site: http://www.karlin.mff.cuni.cz/katedry/kma/ss/apr01/ss.htm [For details,see EMS Newsletter 38]17-20: EuroConference: Applicationsof the Macdonald Polynomials,Cambridge, UK Organisers: B. Leclerc, M. Nazarov, M.Noumi, J.-Y. Thibon Topics: Macdonald polynomials andrelated special functions in mathemati-cal physics, harmonic analysis, and rep-resentation theory. In particular: clas-sical and quantum many-body prob-lems, integrable models in statisticalphysics and quantum field theory,quantum symmetric spaces, deforma-tions of Virasoro and W-algebras, quan-tum Knizhnik-Zamolodchikov equationsSpeakers: H. Awata, J.-F. van Diejen,C. Dunkl, J. Faraut, P. Forrester, K.Hasegawa, G. Heckman, N. Jing, T.Koornwinder, K. Mimachi, T. Miwa, S.Odake, E. Opdam, E. Rains, S.Ruijsenaars, J. Shiraishi, J. Stokman, V.Tarasov. Site: Isaac Newton Institute forMathematical Sciences Deadline: for receipt of applications,already passedInformation: Web site: http://www.newton.cam.ac.uk/programs/sfmw02.html

April 2001

18-20: Third Postgraduate GroupTheory Conference, Imperial CollegeLondon, UK Theme: group theory Aim: the conference is aimed at post-graduate students who are currentlyworking towards obtaining a PhD inmathematics in an area that is relatedto group theory. It is meant to bringstudents together in a less intimidatingatmosphere, and we encourage partici-pants to give a talkMain speakers: Roger Carter(Warwick), Dan Segal (Oxford) Organising committee: Jason Fairleyand Alain Reuter Sponsors: (provisional) LondonMathematical Society Grants: (provisional) subsidy of regis-tration fee Notes: participation is restricted topostgraduate students Deadlines: 16 March for registration Information:e-mail: [email protected] [email protected]: http://www.ma.ic.ac.uk/~jtf98pggtc2001/

May � July: Thematic Term onSemigroups, Algorithms, Automataand Languages, Coimbra, Portugal Theme: semigroups, algorithms,automata and languages Aim: to make Coimbra the gatheringpoint of researchers in the subjects ofsemigroup theory and automata theoryduring May, June and July, besides pro-viding a basepoint for the developmentof joint research projectsTopics: algorithmic aspects of the theo-ry of semigroups and its applications,automata and languages, model theory,profinite topology and semigroups,semigroups and applications, presenta-tions and geometryMain speakers: J. Almeida (Porto), C.Choffrut (Paris VII), J. Fountain (York),S. Margolis (Bar-Ilan), L. Ribes(Carleton), M. Sapir (Vanderbilt), M.Volkov (Ekaterinburg), T. Wilke (Kiel),M. Branco (Lisbon), V. Bruyère (Mons),O. Carton (Marne-la-Vallée), A. Restivo(Palermo), T. Coulbois (Paris VII), H.Straubing (Boston College), P. Trotter(Tasmania), P. Weil (Bordeaux), K.Auinger (Vienna), M. Lawson (Bangor),W. D. Munn (Glasgow), A. Pereira doLago (São Paulo), R. Gilman (StevensInst. of Tech.), D. McAlister (DeKalb),J. Meakin (Lincoln), S. Pride (Glasgow),N. Ruskuc (St. Andrews), B. Steinberg(Porto)Programme: 2-11 May, School on

May 2001

Algorithmic Aspects of the Theory ofSemigroups and its Applications; 4-8June, School on Automata andLanguages; 11-13 June, Workshop onModel Theory, Profinite Topology andSemigroups; 2-6 July, School onSemigroups and Applications; 9-11 JulyWorkshop on Presentations andGeometry Sessions: each school consists of several5-hour courses held by prominentresearchers. The workshops include 50-minute invited lectures and a limitednumber of 20-minute talks on the spe-cific topics of the workshop, proposedby the participantsLanguages: English and Portuguese Programme and organising commit-tee: Gracinda M. S. Gomes (Lisbon,CAUL), Jean-Eric Pin (Paris VII,CNRS), Pedro V. Silva (Porto, CMUP) Sponsors (confirmed): Centro de Álge-bra da Universidade de Lisboa, Centrode Matemática da Universidade doPorto, Centro Internacional deMatemática, Fundação CalousteGulbenkian, Fundação Luso-Americanapara o Desenvolvimento, Fundaçãopara a Ciência e a Tecnologia. Proceedings: to be published Site: Observatório da Universidade deCoimbra (Almas de Freire, Coimbra,Portugal) Grants: A limited number of scholar-ships will provide funding for thoseparticipants in need of financial sup-port, particularly postgraduate studentsDeadlines: already passedInformation:e-mail: [email protected] Web site: http://alf1.cii.fc.ul.pt/term2001 27-2 June: Spring School in Analysis:Function Spaces and Interpolation,Paseky nad Jizerou, Czech Republic Information:e-mail: [email protected] Web site: http://www.karlin.mff.cuni.cz/katedry/kma/ss/jun01/ss.htm [For details, see EMS Newsletter 38]28-1 June: Harmonic morphisms andharmonic maps, Marseille, FranceInformation: e-mail: [email protected] Web-site: http://beltrami.sc.unica.it/harmor/ [For details, see EMS Newsletter 38]

4-8: Conference on AlgebraicTopology, Gdansk, Poland Theme: algebraic topology Topics: homotopy theory, topology ofmanifolds, geometric group theory,algebraic K-theory and homotopy theo-ry in algebraic geometry Main speakers: Gregory Arone(University of Aberdeen), MatthewAndo (University of Illinois at Urbana-Champaign), Mladen Bestvina(University of Utah), AlexanderDranishnikov (University of Florida),Nicholas Kuhn (University of Virginia),Jack Morava (Johns Hopkins

June 2001

CONFERENCES

EMS March 2001 23

FForthcoming conferorthcoming conferencesencesCompiled by Kathleen Quinn

University), Fabien Morel (Universitede Paris VII,) Teimuraz Pirashvili(Georgian Academy of Sciences) Organising committee: StanislawBetley (Warszawa), Tadeusz Kozniewski(Warszawa), Stefan Jackowski(Warszawa), Adam Przezdziecki(Warszawa), Andrzej Szczepanski(Gdansk) Deadlines: for registration, 15 AprilInformation: e-mail: [email protected] Web site: www.mimuw.edu.pl/~cat01 4-9: Fractals in Graz 2001, Analysis-Dynamics-Geometry-Stochastics, Graz,Austria Information: e-mail: [email protected] Web site: http://finanz.math.tu-graz.ac.at/~fractal/[For details, see EMS Newsletter 38]18-22: Fourth European Conferenceon Elliptic and Parabolic Problems:Theory, Rolduc, NetherlandsInformation:e-mail: [email protected],[email protected] Web site: http://www.math.unizh.ch/rolducgaeta [For details, see EMS Newsletter 38]18-23: Tools for mathematical model-ling, St. Petersburg, Russia Theme: mathematical modelling Topics: mathematical modelling,applied mathematics, computer alge-bra, methods of approximation, dis-cretisation and computation, designtechniques, numerical methods, paralleland distributed algorithms, computermodelling in dynamical systems, shad-owing, adaptive methods and integralequations, mathematical models in biol-ogy, medicine etc., applications tophysics, electrotechniques and electron-ics, dynamic economic models, generalmacro-economic models, market mod-elsMain speakers: N. Ajabyan (Armenia),V. Aladjev (Estonia), A. Alawneh(Jordan), M. Alrefaei (Jordan), M.Andramonov (Russia), I. Andrianov(Germany), V. Bakhrushin (Ukraine), V.Chiroiu (Romania), A. B. Dragut(Romania), J. Esquivel-Avila (Mexico),S. Gavryushin (Russia), V. Ivanov(Russia), H. Ji-Huan (China), D.Khusainov (Ukraine), M. Kulshrestha(India), M. Lopez Garcia (Spain), A.Markov (Russia), R. Meyer-Spasche(Germany), A. Mikitinskiy (Russia), A.Prykarpatsky (Poland), K. Pucher(Austria), J. Purczynski (Poland), G. K.Raju (India), A. Rasulov (Uzbekistan),R. Ravindran (India), M. Razzaghi(USA), G. Sadovoi (Russia), V. Samovol(Russia), M. Schahz (German), M.Shamolin (Russia), M. Stojanovic(Yugoslavia), A. Stulov (Estonia), M.Veljovic (Yugoslavia), V. Vladimirov(Ukraine), R. Volosov (Russia), M.Yunusi (Tajikistan) Sessions: equations of mathematicalphysics, theory of control processes,group analysis of differential equations,statistic modelling and integral equa-

tions, economic models, stochastic mod-elling and data analysis, variationalmodel for non-linear problems, mathe-matical models in biology Languages: English and Russian Call for papers: an abstract of a com-plete typewritten page should includethe title of the paper, names of allauthors, complete affiliations, address-es, e-mail. A collection of abstracts willbe printed before the conference. Theparticipants must send their abstracts (2copies) in camera-ready format to theOrganising Committee by normal mail.We do NOT use electronic versions ofabstractsProgramme committee: S. M. Ustinov,P. A. Jilin, G. G. Malinechkii, V. M.Ivanov, G. Yu Riznichenko, A. G.Chenchov, V. N. Kozlov, A. V.Zinkovskii, A. F. Andreev, Yu. N.Bibikov, S. Yu Piliugin, N. N. Petrov, E.N. Rozenvasser, I. V. Miroshnik, A. V.Timofeev, V. N. Fomin, N. E.Barabanov, Yu.V. Churin, E. K. Ershov,A. S. Matveev, V. B. Smirnova, A. I.Shipiliavui, A. V. Flegontov (all Russia) Organising committee: conferencechair, George Osipenko, St PetersburgTechnical Univ.; Valentin Zaitsev,Russian State Pedagogical Univ.;Dmitrii Arseniev, Institute ofInternational Education Program.Secretary, Lidiya Linchuk, StPetersburg State Technical Univ. Sponsors: Russian foundation for BasicResearch, Laboratory of NonlinearAnalysis, St Petersburg Technical Univ.,Editorial Board of the ElectronicJournal �Differential Equations andControl Processes�, Institute ofInternational Education ProgramsProceedings: it is suggested that refer-eed Conference proceedings includingfull versions of selected papers acceptedfor presentation will be publishedSite: St Petersburg State TechnicalUniversityDeadlines: for abstracts and registra-tion forms, 15 April Information: contact Lidiya Linchuk,MATHTOOLS�2001, Dept. ofMathematics, State TechnicalUniversity, Polytechnicheskaya st., 29,St Petersburg, 195251, Russia, fax: +7-812-5343314, +7-812-5527770. e-mail: [email protected] Web page: http://www.neva.ru/journal 19-22: Computational Intelligence,Methods and Applications (CIMA2001), Bangor, UKInformation:e-mail: [email protected]; [email protected];[email protected] site: http://www.icsc.ab.ca/cima2001.htm[For details, see EMS Newsletter 37]25-28: Banach Algebras andCohomology, Newcastle, UKInformation:Web site: http://www.ncl.ac.uk/icbacn25-29: Cmft2001, ComputationalMethods and Function Theory, Aveiro,Portugal

Theme: function theory, scientific com-putation, complex variables (includinggeneralisations like quaternions, etc.),approximation theory and numericalanalysisMain speakers: · Alan Beardon (UK),Percy Deift (USA), Richard Delanghe(Belgium), Toby Driscoll (USA),Alexandre Eremenko (USA), KlausGürlebeck (Germany), Doron S.Lubinsky (South Africa), AndrewOdlyzko (USA), Lawrence Zalcman(Israel) Organising committee: Helmuth R.Malonek (Aveiro), Nicolas Papamichael(Cyprus), Stephan Ruscheweyh,(Würzburg), Edward B. Saff, (SouthFlorida)Sponsors: Fundação para a Ciência e aTecnologia, Sociedade Portuguesa deMatemática, COSTED, The BritishCouncil, Universidade de Aveiro UI&D�Matemática e Aplicações� (as of 31October 2000) Proceedings: as with past CMFT meet-ings, the conference proceedings ofCMFT2001 will appear in a carefullyrefereed volume distributed by a notedmathematics publisher. The style-filecmft.sty and the description-file cmft-manual.ps will be available for down-loading from the homepage of CMFT2001 by 1 April. Both files are alsoavailable from the local organizingcommittee upon requestSite: University of Aveiro, Campus deSantiago, Aveiro Deadlines: for final registration, pay-ment of conference fee (US$150)already passed; for payment of late con-ference fee (US$180), 25 June; for sub-mission of papers for the proceedings,15 OctoberInformation: contact: H. R. Malonek,Departamento de MatematicaUniversidade de Aveiro, Portugaltel./fax: +351-234-370359 /+351-234-382014 e-mail: [email protected] Web site: http://event.ua.pt/cmft2001/

1-6: Eighteenth British CombinatorialConference, Brighton, UnitedKingdomInformation:e-mail: [email protected] sites:http://www.maths.susx.ac.uk/Staff/JWPH/ http://hnadel.maps.susx.ac.uk/TAGG/Confs/BCC/index.html [For details, see EMS Newsletter 36]3-7: Barcelona 2001 EuroPhDTopology Conference, Bellaterra,Barcelona Invited speakers: O. Cornea(Université de Lille), N. P. Strickland(University of Sheffield), P. Salvatore(Università degli Studi di Roma), D. J.Green (Universität Wuppertal), S.Schwede (Universität Bielefeld), S.Whitehouse (Université d�Artois), W.Chachólski (Yale University), P. Turner

July 2001

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(Heriot-Watt University)Scientific committee: Antonio Viruel(Conference Coordinator) (Universidadde Málaga), Jérôme Scherer (Universitéde Lausanne), Natàlia Castellana(University of Aberdeen)Site: Centre de Recerca MatemàticaInformation: Web site: http://www.crm.es/b2001etc 4-6: MathFIT workshop: TheRepresentation and Management ofUncertainty in GeometricComputations, Sheffield, UKInformation:Web site:http://www.shef.ac.uk/~geom2001/5-7: British Congress of MathematicsEducation, Keele, UKInformation:Web site: http://www.bcme.org.uk8-13: Second Workshop on AlgebraicGraph Theory, Edinburgh, Scotland Information: e-mail: [email protected] Web site: http://www.ma.hw.ac.uk/icms/current/graph/index.html[For details, see EMS Newsletter 38]9-22: European summer school�Asymptotic combinatorics with appli-cation to mathematical physics�, StPetersburg, RussiaInformation:e-mail: [email protected] site: www.pdmi.ras.ru/EIMI/2001/emschool/index.html[For details, see EMS Newsletter 38]10-15: Advanced Course onSymplectic Geometry of IntegrableHamiltonian Systems, Bellaterra,Barcelona Speakers: Michèle Audin (Strasbourg),Lagrangian and special lagrangian sub-manifolds, Ana Cannas da Silva (Lisboa),Symplectic toric manifolds, EugeneLerman (Illinois), Symmetries of symplecticand contact manifolds Coordinators: Carlos Curràs-Bosch,Eva Miranda Site: Centre de Recerca Matemàtica Information: Web site: http://www.crm.es/sgihs 15-20: Algorithms for ApproximationIV International Symposium,Huddersfield, UKInformation: e-mail: [email protected] Web-site:Http://Helios.Hud.Ac.Uk/A4a4/ [For details, see EMS Newsletter 38]17-23: Advanced Course on GlobalRiemannian Geometry: Curvature andTopology Speakers: S. Markvorsen (Denmark),Distance geometric analysis on Riemannianmanifolds, M. Min-Oo (McMasterUniversity), K-area, mass and assymptoticgeometryCoordinator: Vicent Palmer Site: Universitat Jaume I, Castelló de laPlana Information: Web site: http://www.crm.es/geom2001 18-28: Advanced Course on ModularForms and p-adic Hodge Theory,Bellaterra, Barcelona

Speakers: Christophe Breuil(Université Paris-Sud), p-adic Hodge the-ory, deformations and local Langlands, BasEdixhoven (Université de Rennes 1),Modular forms, Galois representations andlocal LanglandsCoordinators: Xavier Xarles, AdolfoQuirós Site: Centre de Recerca Matemàtica, Information: Web site: http://www.crm.es/tn2001 23-27: 20th IFIP TC 7 Conference onSystem Modelling and Optimization,Trier, GermanyInformation:e-mail: [email protected] site: http://ifip2001.uni-trier.de [For details, see EMS Newsletter 38]

5-18: BALTICON 2001, BanachAlgebra Theory in Context, KrogerupHojskole, Humlebaek, Denmark Information:e-mail: [email protected] Web site:http://www.math.ku.dk/conf/balti-con2001/[For details, see EMS Newsletter 38]5-18: Groups St Andrews 2001,Oxford, UKInformation: Groups St Andrews 2001,Mathematical Institute, North Haugh,St Andrews, Fife KY16 9SS, Scotland e-mail: [email protected] Web site: http://www.bath.ac.uk/~masgcs/gps01/ [For details, see EMS Newsletter 36]12-18: 39th International Symposiumon Functional Equations, DenmarkOrganising committee: Peter Friis andHenrik Stetkaer (Aarhus University)Information: e-mail: [email protected] site: http://www.mi.aau.dk/isfe39/12-19: Summer School 2001Homological Conjectures for FiniteDimensional Algebras, Nordfjordeid,Norway Information: contact Øyvind Solberg,([email protected], NTNU,Trondheim) Web sites: http://www.mathematik.uni-bielefeld.de/~sek/summerseries.htmlhttp://www.math.ntnu.no/~oyvinso/Nordfjordeid/[For details, see EMS Newsletter 38]19-25: 9th Prague TopologicalSymposium, International Conferenceon General Topology and its Relationsto Analysis and Algebra, Prague,Czech Republic Topics: topology (mainly set-theoreticaltopology, continua, descriptive topolo-gy, categorical topology, geometricaltopology), topological groups and semi-groups, topology of Banach spaces,topology and computer science, topo-logical dynamics Main speakers: A. V. Arhangelskii(Moscow and Athens, OH), Z. Balogh(Oxford, OH), M. Bell (Winnipeg), A.Bouziad (Rouen), D. Dikranjan (Udine),

August 2001

A. Dow (Charlotte, NC), I. Farah(Staten Island, NY), D. Fremlin (Essex,UK), P. M. Gartside (Oxford, UK), G.Godefroy (Paris), G. Gruenhage(Auburn), N. Hindman (Washington,D.C.), I. Juhász (Budapest), H.-P.Kuenzi (Cape Town), K. Kunen(Madison, WI), D. Lutzer(Williamsburg, VA), W. M. Marciszewski(Warsaw), M. Megrelishvili (Ramat-Gan), J. van Mill (Amsterdam), P.Nyikos (Columbia, SC), J. Pelant(Prague), V. Pestov (Wellington), M.Reed (Oxford, UK), D. Repovs(Ljubljana), M. E. Rudin (Madison, WI),D. Shakhmatov (Matsuyama), L.Shapiro (Moscow), S. Solecki(Bloomington, IN), M. Tkachenko(Mexico), V. Tkachuk (Mexico), S.Todorcevic (Belgrade and Paris), V. V.Uspenskiy (Athens, OH), S. Watson(York), Y. Yajima (Yokohama)Programme committee: SimonDonaldson (representative of IMU), B.Balcar, M. Husek, J. Pelant, P. Simon,V. Trnková Organising committee: B. Balcar, M.Husek (Chairman), J. Pelant, P. Simon,V. Trnková, P. Holický, O. Kalenda, A.Klíc, J. Coufal, R. Rohácková Sponsors: International MathematicalUnion, Mathematical Institute of CzechAcademy of Sciences, Department ofMathematics of Prague Institute ofChemical Technology, Department ofMathematics of Economical Universityand Faculty of Mathematics and Physicsof Charles University. Deadlines: for abstracts and registra-tion, 31 May Information: Web site: http://www.karlin.mff.cuni.cz/~toposym/ e-mail: [email protected] 24-30: 10th International Meeting ofEuropean Women in Mathematics,Malta Information: contact Dr. Tsou SheungTsun (EWM01), Mathematical Institute,24-29 St. Giles, Oxford OX1 3LB, UK, fax: +44-1865-273583 Web site:http://www.maths.ox.ac.uk/~ewm01/27-31: Equadiff 10, CzechoslovakInternational Conference onDifferential Equations and theirApplications, Prague, Czech RepublicInformation:e-mail: [email protected] Web site: www.math.cas.cz/~equadiff/[For details, see EMS Newsletter 38]

2-6: First SIAM-EMS Conference,Berlin, GermanyTopic: Applied Mathematics in OurChanging World Information: contact Peter Deuflhard(Chair of the Local OrganizingCommittee), SIAM/EMS 2001Conference Office c/o Konrad-Zuse-Zentrum Berlin (ZIB), Takustr. 7,14195 Berlin-Dahlem, Germany

September 2001

CONFERENCES

EMS March 2001 25

fax: +49 (30) 841 85-107 e-mail: [email protected] Web site: http://www.zib.de/amcw01/ 3-8: Sixth International Conference onFunction Spaces, Wroclaw, PolandTopics: the topics are connected withfunctional analysis, e.g. operator theo-ry, interpolation, geometry, topology,approximation in function spaces. Programme: plenary lectures (about 45minutes) and short communications(not more than 20 minutes)Organiser: Institute of Mathematics,Wroclaw University of Technology Organising committee: M. Burnecki(secretary), R. Grzaslewicz (chairman),H. Hudzik, J. Musielak, Cz. Ryll-Nardzewski Site: Institute of Mathematics, WroclawUniversity of TechnologyInformation: Web site: www.im.pwr.wroc.pl/~fsp/ e-mail: [email protected] 5-7: LMS Workshop on DomainDecomposition Methods in FluidMechanics, Greenwich, UKAim: to disseminate amongst graduatestudents, researchers and industrialiststhe state-of-the-art domain decomposi-tion methods applied to various engi-neering problems related to fluidmechanics. The workshop may beserved as a short intense course forgraduate students who will be exposedto review and overview lectures and cur-rent research interests and resultsInivited speakers: C. Bailey(Greenwich), X.-C. Cai (Boulder), M.Garbey (Lyon), I. G. Graham (Bath), G.Lube (Göttingen), F. Nataf (Paris) Organisers: C.-H. Lai, M. Cross, K. A.Pericleous, A. K. ParrottGrants: some scholarships are availableto graduate students with priority givento those from UK mathematics depart-mentsInformation:Web site: http://cms1.gre.ac.uk/conferences 12-14: Seventh Meeting on ComputerAlgebra and Applications (EACA-2001)Topics: effective methods in algebra,analysis, geometry and topology; algo-rithmic complexity; scientific computa-tion via symbolic- numerical methods;development of symbolic-numericalsoftware; analysis, specification, designand implementation of symbolic com-putation systems; applications to sci-ence and technology Speakers: Hoon Hong (North CarolinaState University), Jacques Calmet(Universität Karlsruhe), Marc Chardin(Institut de Mathématiques, CNRS,Paris), Eladio Domínguez (Universidadde Zaragoza), y Luis Español(Universidad de La Rioja), LaureanoGonzález-Vega (Univesidad deCantabria), y Rafael Sendra(Universidad de Alcalá)Programme: invited speakers will deliv-er one-hour lectures on current andfuture trends in their research fields. Anumber of sessions for presentation of30-minute talks are planned

Local organisers: Julio Rubio,Departamento de Matemáticas yComputación, Universidad de La Rioja,Edificio Vives, Calle Luis de Ulloa s/n,E-26004 Logroño (La Rioja, España) Site: Ezcaray (La Rioja, Spain)Note: this is a Spanish forum withinternational participation Information: e-mail: [email protected] Web site: http://www.unirioja.es/dptos/dmc/eaca2001/ 12-15: EuroConference onCombinatorics, Graph Theory andApplications, Bellaterra, Barcelona Invited speakers: Noga Alon (Tel AvivUniversity), Peter Cameron (QueenMary and Westfield College), DavidEpstein (University of California),Levon Khachatrian (University ofBielefeld), Jiri Matousek (CharlesUniversity), Kevin Phelps (AuburnUniversity), Vera T. Sós (Alfréd RényiInstitute of Mathematics), CarstenThomassen (Technical University ofDenmark), Nick Wormald (University ofMelbourne)Local organising committee: GaborLugosi (Universitat Pompeu), FabraConrado Martínez (UniversitatPolitècnica de Catalunya), Marc Noy(Universitat Politècnica de Catalunya),Josep Rifà (Universitat Autònoma deBarcelona), Oriol Serra (localChairman) (Universitat Politècnica deCatalunya)Site: Centre de Recerca MatemàticaInformation: Web site: http://www.crm.es/comb01 14-18: International Conference:Function Spaces, Proximities andQuasi-uniformities, Italia, Caserta[on the occasion of Som Naimpally�s 70birthday]Information: e-mail: [email protected] site: http://www.unina2.it/topological.sun/homesun.html16-22: Conference of the AustrianMathematical Society and the GermanMathematical Society, Vienna, Austria[please note corrected dates]Plenary speakers: V. Capasso (Milano),M. H. A. Davis (London), I. Ekeland(Paris), H. P. Schlickewei (Marburg), M.Kreck (Heidelberg), N. J. Mauser(Vienna), V. L. Popov (Moskau), T.Ratiu (California), D. Salamon (Zürich),G. Teschl (Vienna), J.-C. Yoccoz (Paris),D. Zagier (Bonn), G. M. Ziegler (Berlin) Local organisation: Karl Sigmund(University of Vienna) Information: Karl Sigmund, Universityof Vienna, Institute of Mathematics,Strudlhofgasse 4, 1090 Vienna;tel: +43 1 4277 506 02 or 506 12, fax:+43 1 4277 9506 e-mail: [email protected] Web site: http://www.mat.univie.ac.at/~oemg/Tagungen/2001/index.html 17-26: Functional Analysis VII,Dubrovnik, Croatia Topics: operator algebras, probabilitytheory, harmonic analysis, representa-tion theory

Main speakers: Drazen Adamovic(Croatia), Anne-Marie Aubert (France),Damir Bakic (Croatia), ChongyingDong (USA), Detlef Gronau (Austria),Boris Guljas (Croatia), JorgenHoffmann-Jorgensen (Denmark), Jing-Song Huang (China), David R. Larson(USA), Bojan Magajna (Slovenia), AllenMoy (USA), Goran Muic (Croatia),Pavle Pandzic (Croatia), ManosPapadakis (USA), Goran Peskir(Denmark), Gilles Pisier (France),Zoran R. Pop-Stojanovic (USA), MirkoPrimc (Croatia), Murali Rao (USA),Ludwig Reich (Austria), David Renard(France), Paul J. Sally (USA), BorisSirola (Croatia), Renming Song (USA),David Soudry (Israel), Marko Tadic(Croatia), Mitchell H. Taibleson (USA),David A. Vogan (USA), Guido Weiss(USA) Programme: series of lectures, singlelectures and a limited number of shortcommunications Organising committee: HrvojeKraljevic (Croatia), Davor Butkovic(Croatia), Murali Rao (USA), DamirBakic (Croatia), Pavle Pandzic(Croatia), Hrvoje Sikic (Croatia), ZoranVondracek (Croatia) Site: Inter-University Center,Dubrovnik, Croatia Deadlines: to register and submitabstracts, 1 April 2001Information: e-mail: [email protected] Web page: http://www.math.hr/~congress/Dubrovnik01/ 18-22: Euro Summer School on ProperGroup Actions, Bellaterra, Barcelona Speakers: Guido Mislin (ETH Zürich),Equivariant K-homology of the classifyingspace for proper actions, Alain Valette(Université de Neuchâtel), TopologicalK-theory of group C*-algebrasCoordinator: Carles Casacuberta Site: Centre de Recerca Matemàtica Information:Web site: http://www.crm.es/group-actions24-28: Fourth European Conferenceon Elliptic and Parabolic Problems:Applications, Gaeta, ItalyInformation:e-mail: [email protected],[email protected] Web site: http://www.math.unizh.ch/rolducgaeta [For details, see EMS Newsletter 38]24-28: III International Conference,Analytic and Probabilistic Methods inNumber Theory, Palanga, Lithuania [in honour of J. Kubilius]Theme: analytic and probabilistic meth-ods in number theory Organizing committee: B. Juodka(Chairman), A. Laurincikas(Programme Director), E. Manstavicius,(Vice-chairman), V. Stakenas (Vice-chairman)Proceedings: to be published Information: e-mail: [email protected] Web site: http://www.mif.vu.lt/ttsk/palan-ga.htm

CONFERENCES

EMS March 200126

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

M. Aschbacher, Finite Group Theory,Cambridge Studies in Advanced Mathematics10, Cambridge University Press, Cambridge,2000, 304 pp., £19.95, ISBN 0-521-78145-0 and 0-521-78675-4The second edition of Aschbacher�s FiniteGroup Theory differs from the first editionin three aspects. The first edition con-tained many misprints: using it for anoccasional reference I was aware of abouttwenty misprints in those parts I read withgreater care (some sixty pages). Of thesetwenty misprints all but two have disap-peared, and the second edition seemsmuch more reliable. Some of the exercis-es are quite difficult. The second editionhas an additional thirty pages devoted tosolutions of the more important andmore difficult exercises. The authorfound his extension of Bender�s proof ofthe solvable signaliser functor theorem toodd primes too complicated, and hedecided to return to Bender�s originalresult which covers only abelian 2-sub-groups. The new proof is not much short-er, but is presented in more detail and iseasier to follow.

The book covers most of the standardbasic material (series, semi-direct, centraland wreath products, linear representa-tions and characters, extensions, presen-tations, classical groups, Coxeter groupsand root systems, extremal p-groups,transfer), and emphasises themes that arerelevant to the Classification of FiniteSimple Groups (e.g., extra-specialgroups). The more advanced topicsinclude Thompson factorisation, fusion,central extensions, and various theoremson complements. The book also has a

chapter on buildings, BN-pairs and Titssystems.

The exposition is often quite terse andthe development of structures outside thegroup theory is done only to a minimalextent, which makes applications withingroup theory possible. The purpose ofthe book was to lay foundations for theproof of the CFSG, and the book finisheswith a chapter on finite simple groups,including an introduction to CFSG and toits proof. (ad)

G. Bachman, L. Narici and E.Beckenstein, Fourier and WaveletAnalysis, Universitext Springer, New York,2000, 505 pp., DM119, ISBN 0-387-98899-8This excellent book is intended as anintroduction to classical Fourier analysis,Fourier series, Fourier transforms andwavelets, for students in mathematics,physics and engineering. The textincludes many historical notes to placethe material in a cultural and mathemati-cal context. The topics are developedslowly for a reader who has never seenthem before, with a preference for clarityof exposition in stating and provingresults.

The first three chapters present requi-site background material, and these couldbe read as a short course in functionalanalysis. The standard features ofFourier analysis � Fourier series, Fouriertransform, Fourier sine and cosine trans-forms � are developed in Chapters 4 and5. More recent developments, such as thediscrete and fast Fourier transforms andwavelets, are covered in the last two chap-ters. Each section concludes with a set ofexercises with hints for many of these.(knaj)

C. Bandt, S. Graf and M. Zähle (eds.),

Fractal Geometry and Stochastics II,Progress in Probability 46, Birkhäuser, Basel,2000, 292 pp., ISBN 3-7643-6215-4This book includes the main contribu-tions to the second conference on fractalgeometry and stochastics, held atGreifswald, Germany in 1998. The con-ference and its proceedings reflect recentrapid and interesting research develop-ments on the boundary between fractalgeometry (of objects that show consider-able irregularities of magnitude) andprobability theory.

The topics covered are fractal dimen-sions of measures and sets, self-similar,self-affine sets and measures generated byiterated function systems, constructions ofrandom fractals via contractions, randomcoverings, etc., dynamical systems andfractals, and harmonic analysis on frac-tals. The contributions range in stylefrom survey papers to standard mathe-matical articles and an open problemscollection. (jstep)

A. J. Berrick and M. E. Keating, AnIntroduction to Rings and Modules withK-theory in View, Cambridge Studies inAdvanced Mathematics 65, CambridgeUniversity Press, Cambridge, 2000, 265 pp.,£35, ISBN 0-521-63274-9This is an introductory text for beginninggraduate students who intend to move onto study algebraic K-theory. The funda-mentals of ring and module theory aredeveloped from scratch, accompanied bynumerous examples and exercises.

After presenting the basics, the authorsdevote a chapter to direct-sum decompo-sitions and short exact sequences (includ-ing a section on IBN-rings). The follow-ing two chapters on (non-commutative)noetherian and artinian rings include theclassical Wedderburn-Artin theory. Inthis part, an interesting construction ispresented of a stably free non-free rightideal over the two-variable polynomialring over any skew-field which is not afield. This is in sharp contrast with thevalidity of Serre�s conjecture which is wellknown in the field case.

The final two chapters deal withDedekind domains and modules overthem. The class group is introduced andthe primary decomposition theorem isproved for finitely generated torsionmodules. Besides general theorems,explicit computations are also presented;for example, in Section 5.3, the factorisa-tion type of pO is completely determinedwhen p is a prime and O is the ring ofintegers in the quadratic extension Q(√d).The class group of Q(√d) is explicitly com-puted for small values of d. (jtrl)

A. J. Berrick and M. E. Keating,Categories and Modules with K-theory inView, Cambridge Studies in AdvancedMathematics 67, Cambridge University Press,Cambridge, 2000, 361 pp., £35, ISBN 0-521-63276-5This book is an introduction to categorytheory, with particular emphasis on mod-ule categories. The exposition starts withbasic material on categories, functors and

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Recent booksRecent booksedited by Ivan Netuka and Vladimír Sou³ek

natural transformations; this includesfirst facts on functors, categories and uni-versal objects. Next, the authors dealwith additive and abelian categories, andintroduce the Grothendieck group of a G-exact category. There is also a chapter onthe tensor product of modules and thechange of ring functors. Extn and Tornare introduced (with homological detailsleft as exercises). Chapter 4 deals with afundamental application of category the-ory to modules: the Morita equivalencetheory. Another important construction,the direct and inverse limits, is explainedin Chapter 5. The next chapter dealswith the localisation of rings and mod-ules. Localisation is first treated in termsof rings and modules of fractions, andthen more generally, as a quotient functorwith the universal property. The finalchapter deals with completions and withlocal-global methods. Particular atten-tion is paid to applications to lattices overDedekind domains. The rings and mod-ules of adèles are introduced, and projec-tive modules over lattices are charac-terised. The book may be viewed as acontinuation of Vol. 65 (see above), but itis of independent interest as well. (jtrl)

F. Blanchard, A. Maass and A. Nogueira(eds.), Topics in Symbolic Dynamics andApplications, London Mathematical SocietyLecture Note Series 279, CambridgeUniversity Press, Cambridge, 2000, 245 pp.,£24.95, ISBN 0-521-79660-1Symbolic dynamics studies iterations ofcontinuous selfmaps of spaces of symbolicspaces. These spaces are equipped withthe Cantor (product) topology and arehomeomorphic to subspaces of theCantor middle-third set. Symbolicdynamics originated in the study of geo-desics on surfaces with negative curva-ture. It is used for coding smooth trans-formations and has important connec-tions with computer science.

This volume contains the courses givenat the CIMPA-UNESCO Summer Schoolon Symbolic Dynamics and itsApplications held in Temuco, Chile, inJanuary 1997. The volume is divided intoeight chapters, each corresponding to acourse given at Temuco and devoted toan active area of symbolic dynamics orapplications to ergodic theory or numbertheory. The authors and topics are V.Berthé, Sequences of low complexity: auto-matic and Sturmian sequences, B. Host,Substitution subshifts and Bratteli diagrams,M. Boyle, Algebraic aspects of symbolicdynamics, B. Kitchens, Dynamics of Zd

actions on Markov subgroups, Z. Coelho,Asymptotic laws for symbolic dynamical sys-tems, V. Bergelson, Ergodic theory andDiophantine problems, Ch. Frougny, Numberrepresentation and finite automata, R.Labarca, A note on the topological classifica-tion of Lorenz maps on the interval. (pkur)

J. M. Borwein and A. S. Lewis, ConvexAnalysis and Nonlinear Optimization.Theory and Examples, CMS Books inMathematics 3, Springer, New York, 2000,273 pp., DM139, ISBN 0-387-98940-4

The authors offer convexity as a toolwhich allows for transition from classicalto modern analysis, from smooth to non-smooth problems, from linear to non-lin-ear optimisation, and from functions tomulti-functions. They present a widespectrum of results on convex analysis infinite-dimensional spaces, their applica-tions and some extensions. The well-known results on duality theory and theclassical necessary and sufficient condi-tions for non-linear optimisation areincluded in a succinct way and their non-smooth versions are developed. Theroles of polyhedral sets and functions, offixed-point theorems, and of variationalinequalities are explained. The lucid dis-cussion of the place of the assumed finite-dimensionality indicates possible general-isations. Eigenvalue optimisation is anexample of a less frequent type of appli-cation. Algorithms for solving optimisa-tion problems are not the subject of thebook.

The text consists of short sections, eachaccompanied by an extensive set of exer-cises marked into three categories: easyillustrative examples, more advancedexamples and additional theory, demand-ing examples and peripheral theory. Thestyle is not strictly formal, with definitionsincluded in the text without numbering,and with proofs and technical detailsmostly omitted or left as exercises. (Theoverall ratio is 45:55 in favour of exercis-es.)

This is an up-to-date, well-writtenadvanced text which aims at a broadmathematical audience. The generalrequirements correspond to a master�slevel in pure or applied mathematics, andselected sections and exercises may serveas an underlying text for advanced mas-ters� courses on optimisation. With adetailed list of named results, the bookcan be used as a compendium and asource of reference. (jdup)

C.-P. Bruter, La construction des nom-bres, Histoire et épistémologie, Ellipses,Paris, 2000, 237 pp., ISBN 2-7298-7959-5 This interesting book is a combination ofmathematics and its history with history.The author carefully describes the longhistory of mathematical thinking on thecreation and extension of the concept of anumber, up to the nineteenth century.The author prefers to present the mostsignificant problem in the history of num-bers so that the reader can appreciate thehistory of mathematics at first hand. Thisbook helps the reader to understand thereason for the creation of the positiveintegers, integers, rational and irrationalnumbers, complex numbers, quaternions,octonions, and the so-called Cliffordnumbers (elements of Clifford algebras).

For each type of number, the authorexplains how they were discovered andconstructed, their symbolic and geometri-cal representations, their properties anduses in geometry and physics. The lastchapter contains a philosophical answerto the question of what the word �number�really means. At the end of the book,

there are three appendices with a bibliog-raphy and two indices.

The book is nicely written and providesan important source for university stu-dents interested in mathematics and itshistory. (mnem)

E. B. Burger and M. Starbird, The Heartof Mathematics. An Invitation to EffectiveThinking, Springer, New York, 2000, 646pp., DM130, ISBN 1-55953-407-9This book is a wonderful and fascinatingexcursion into the realm of mathematicsand the world of creative and effectivethinking. The authors explain the math-ematical ideas and results in such a waythat the reader can discover the basicstrategies of thought and use them in reallife. In spite of this, the excursion ismade as pleasant as possible, with manyreferences to history, philosophy and thefine arts, without a lot of computation.The book can be read by teachers, stu-dents and anyone who is not afraid ofthinking and who wants to discover whatmathematics really is. (ec)

R. P. Burn, Numbers and Functions. Stepsinto Analysis, Cambridge University Press,Cambridge, 2000, 356 pp., £19.95, ISBN 0-521-78836-6The main purpose of this textbook is tohelp students with the transition fromhigh-school calculus to mathematicalanalysis as it is taught at universities.Through several hundreds of exercises,students are led from practical examplesto the theoretical foundations of rigorousanalysis. Solutions of all the exercises aregiven at the end of each chapter.

The topics presented in this bookinclude mathematical induction,sequences and series of real numbers,functions, differentiation and integrationand sequences of functions. Since thebook is written in a very comprehensiblebut exact way, it may serve as an excellentguide through the basic course of mathe-matical analysis at university. Unlikestandard textbooks, most of the contentsare presented in the exercises, which pro-vide the student with enough motivationfor the definitions and theorems. Eachchapter contains a summary of the pre-ceding theory and a few historical notes.Each item in the bibliography is com-mented on in a few words, so that even abeginner can easily find his/her way to theappropriate materials. (jvy) (= J. Vybiral)

N. L. Carothers, Real Analysis, CambridgeUniversity Press, Cambridge, 2000, 401 pp.,£19.95, ISBN 0-521-49749-3 and 0-521-49756-6This book presents a course on real analy-sis. It consists of three parts: metric (andnormed) spaces, function spaces, andLebesgue measure and integration (onthe real line). It does not supposedetailed knowledge of a large amount ofmaterial: a basic course of analysis(including deeper facts such as theBolzano-Weierstrass theorem, complete-ness, a certain familiarity with ε-δ-tech-niques, the Riemann integral, etc.) is suf-

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ficient to read the book. The expositionis clear, easily understandable, and writ-ten in a way accessible to non-specialiststudents. Good examples treated in suffi-cient depth (Cantor function, Peanocurve, nowhere-differentiable continuousfunctions) and a great number of exercis-es (484, 231 and 280 in the three above-mentioned sections) will be appreciatedby those who use the book for the self-study. The author also includes some his-torical remarks and comments, more than15 pages of references, and symbol andtopic indices. The book will be enjoyedby students: in several hundred pages itcontains almost all that a non-specialistmathematician should know about realanalysis on R. (jive)

C. Cheverry, Systèmes de Lois deConservation et Stabilité BV, Mémoires dela Société de Mathématique de France 75,Société Mathématique de France, Paris, 1998,106 pp., Ffrs150, ISBN 2-85629-072-8This short book considers questions con-nected with the existence of weak solu-tions of the Cauchy problem for strictlyhyperbolic systems of conservation laws.While the classical results give the exis-tence in the large for data which are smallin both the supremum and the total vari-ation norms, this book tries to relax theassumption of the smallness of the totalvariation. It is shown that under someadditional assumptions on the growthand (genuine) non-linearity of the fluxfunction, the total variation on intervalsof fixed appropriate length is decreasing.As a consequence, the compactness prop-erties of the solution operator are proved,and the existence in the large for period-ical initial data is proved in the case whenthe BV-norm of these data is large butBV-norm by period remains small.

This book will be appreciated both byspecialists in the field and by (postgradu-ate) students interested in basic questionsconnected with the solvability of systemsof hyperbolic conservation laws. (mrok)

A. Darte, Y. Robert and F. Vivien,Scheduling and AutomaticParallelization, Birkhäuser, Boston, 2000,264 pp., DM158, ISBN 0-8176-4149-1 and3-7643-4149-1This book deals with a very specific sub-ject: an automatic parallelisation of nest-ed loops in a sequential computer pro-gram. The core of the book is a detaileddescription of both classical and recentstate-of-the-art techniques in this quitespecialised field. The book also coversthe necessary preliminaries that areessential for understanding the mathe-matical background of the techniquespresented.

The book consists of five chapters, ofwhich the first three deal with uni-dimen-sional problems (task graph systems anddecomposed cyclic scheduling) and thelast two with multi-dimensional systems(recurrent equations and loop nests).Chapter 1 covers classical results on taskgraph scheduling with no communicationcost under both unlimited and limited

resource constraints. Chapter 2 is thecounterpart of Chapter 1 when takingcommunication costs into account.Chapter 3 is devoted to cyclic scheduling,which constitutes a transition from taskgraph scheduling to loop nest scheduling,the main subject of this book. Chapter 4provides mathematical foundations forthe core chapter of the book, Chapter 5,which deals with loop nests and loop par-allelisation algorithms.

The readers are assumed to have onlybasic mathematical skills and some famil-iarity with standard graph and linear pro-gramming algorithms. However, due toits high specialisation, this book is orien-tated more towards graduate studentsand researchers interested in loop nestparallelisation than to undergraduate stu-dents or the general mathematical public.(jkrat)

W. A. de Graf, Lie Algebras: Theory andAlgorithms, North-Holland MathematicalLibrary 56, North-Holland, Amsterdam,2000, 393 pp., US$118, ISBN 0-444-50116-9The theory of Lie algebras is ample, withexplicit constructions and concrete algo-rithms (Levi decomposition, branchingrules, Hall-Shirshov and Grbner bases,etc.). The present book contains a stan-dard course in Lie algebras and practical-ly all existing algorithms of this theory.The approach taken by the author simpli-fies proofs of some important theoremsand makes them more transparent andclear. Moreover, since current researchin the theory of Lie algebras requiresusing computers, such an exposition facil-itates understanding and practical use ofcomputational methods for solving con-crete problems. The author of this bookis also the author of the sub-package Liealgebras in the programme package GAP.This circumstance has enabled him to cre-ate a book which will be useful for theexperts, as well as for interestedresearchers from other fields of mathe-matics and mathematical physics. (ae)

P. Dehornoy, Braids and Self-Distributivity, Progress in Mathematics 192,Birkhäuser, Basel, 2000, 623 pp., DM178,ISBN 3-7643-6343-6In 1986 Richard Laver published a shortremark that connected non-trivial ele-mentary embeddings in set theory to sys-tems with one binary operation, in whichthe left-distributive law x . (y . z) = (x . y) .(x . z) is satisfied (the LD-systems). AnLD-system induced by a non-trivial ele-mentary embedding exhibits a certainproperty (the acyclicity) which implies itsfreeness (with respect to left-distributivi-ty). However, for some time there was nodirect proof that a free monogeneratedLD-system is acyclic. This was a ratherstrange situation, since non-trivial ele-mentary embeddings imply the existenceof large cardinals.

Patrick Dehornoy started to contributeto the theory of LD-systems in the late1980s, and in 1994 he published a proofof acyclicity that avoids large cardinals.

His result was preceded by a systematicinvestigation of a (partial) monoid whichis generated by such transformations ofbinary terms that correspond to an LD-expansion at a certain term position. Theidentities that are fulfilled within thismonoid led Dehornoy to consider a con-nection with the braid group, and thisconnection led to many further discover-ies, some of which are concerned just withthe braid group.

This book presents the fruits ofDehornoy�s research up to 2000. It isdivided into three parts. Part A establish-es the connection to braids, and thenstudies divisibility in a wide class ofmonoids that includes the braid monoid(and which is later shown to contain also amonoid that is induced by LD-expan-sions). The applications to braids includenew linear orderings and a new normalform. Part B is devoted to a study of freeLD-systems. It starts by showing that anytwo LD-equivalent terms can be expand-ed to the same term. This observation isthen refined in various ways. First, a(right) normal form for free LD-systems isgiven, and then various results concern-ing the monoid that is induced by LD-expansions are obtained. These resultsthen point to an extension of the braidgroup that is so important for Dehornoy�sproof of acyclicity. Part C is concernedwith connections with set theory and withcertain finite LD-systems that appear nat-urally in this context and that yield anumber of formidable open problems.

The book is written very carefully andrequires almost no preliminary knowl-edge. Nevertheless, many readers willprobably regard it as quite difficult, sinceit penetrates deeply into structures thatare far from conventional mathematicalinterests. The book seems to me to be amajor achievement: its only weakness isperhaps a rather restricted attention toresults of other authors active in the field.Readers who have already come acrosssome LD-systems (such as symmetricspaces) should be warned that the book ismainly about non-idempotent LD-sys-tems, in contrast to various idempotentLD-systems that are induced by reflectionand knot colouring. (ad)

A. M. Etheridge, An Introduction toSuperprocesses, University Lecture Series20, American Mathematical Society,Providence, 2000, 187 pp., US$33, ISBN 0-8218-2706-5Measure-valued processes (also known assuperprocesses), as treated by this mono-graph, constitute an important and fre-quently visited topic of modern probabil-ity theory. The basic example and asource of motivation for the underlyingmathematical research, the Dawson-Watanabe superprocess (also calledsuper-Brownian motion) is known as alimit of branching Brownian motions andrecently became a principal tool in math-ematical population dynamics with therandom size of the population governedby the Feller diffusion. Among the top-ics in this book are branching Brownian

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motion, the DW superprocess as a martin-gale problem, the duality theorem for DWprocesses, the Fleming-Viot superprocessand its relation to the DW superprocess,discrete approximations, superprocessesin partial differential equation theory,and catalytic and competing species mod-els.

This monograph is aimed at graduatestudents and probabilists interested instochastic analysis who, the reviewerbelieves, will appreciate it as a deep andtechnically accessible summary of thedevelopments in the field during the lastdecade. (jstep)

M. V. Fedoryuk (ed.), Partial DifferentialEquations V, Encyclopaedia of MathematicalSciences 34, Springer, Berlin, 1999, 247 pp.,DM158, ISBN 3-540-53371-0This book is a collection of six papers byV. M. Babich, N. S. Bakhvalov, M. V.Fedoryuk, A. M. Il�in, V. F. Lazutkin, G.P. Panasenko, A. L. Shtaras, and B. L.Vainberg, published originally in 1988 byVINITI, Moscow. This edition presentsan English translation of these papers byJ. S. Joel and S. A. Wolf.

Each article considers contemporarytechniques in the study of properties ofpartial differential equations � in particu-lar, the asymptotic behaviour of solutions;for example, one of the articles studiesthe equations with rapidly oscillatingsolutions, using the Maslov canonicaloperator technique. Another article dealswith the asymptotic expansion of solu-tions of exterior boundary problems forhyperbolic equations, as the timeapproaches infinity. The remaining arti-cles consider problems connected withsemi-classical asymptotics of eigenfunc-tions, the higher-dimensional WKBmethod, boundary-layer problems, andaveraging methods (the methods ofhomogenisation).

As a whole, the book presents a collec-tion of papers dealing with particular top-ics in the theory of asymptotic and bound-ary behaviour of solutions of partial dif-ferential equations. It cannot be consid-ered to be a textbook on PDEs but will bevalued mostly by specialists working inthe field. (mrok)

C. Goffman, T. Nishiura and D.Waterman, Homeomorphisms in Analysis,Mathematical Surveys and Monographs 54,American Mathematical Society, Providence,1997, 216 pp., ISBN 0-8218-0614-9This book describes the interesting roleplayed by homeomorphisms in real func-tion and measure theories. More precise-ly, consider a function f from a certainfamily F. What can be said about the classf ° h: h is a homeomorphism on thedomain of f?. Here there is a typicalexample of the Maximoff theorem: let fbe a real function on the interval (0, 1);then there is a homeomorphism h: (0, 1)to (0, 1) such that f ° h is a derivative ifand only if f belongs to the Baire class 1and has the Darboux property.

The first part of the book deals with theone-dimensional case and classes of Baire

1-functions, absolutely measurable func-tions, continuous functions of boundedvariation, continuously differentiablefunctions and approximate derivatives.Chapters on Lebesgue equivalence, densi-ty topology and the Zahorski classes areincluded. The second part, Mappings andmeasures on Rn, is concerned with one-to-one transformations defined on intervalsin Euclidean space Rn. There are chap-ters on bi-lipschitzian homeomorphisms,lengths of non-parametric curves, exten-sions of homeomorphisms and construc-tions of deformations. The final chapterin this part contains a discussion of theconnection between Blumberg�s theorem,Baire spaces and self-homeomorphismsof dense subsets. The last part of thebook is devoted to Fourier series, andincludes such topics as the behaviour ofFourier series, uniform and absolute con-vergence, Bohr theorem, convergence ofFourier series after change of variable,and absolutely measurable functions.The appendix contains a supplementarymaterial including Baire, Borel andLebesgue sets and functions, Hausdorffdimension, non-parametric length andarea, and approximately continuousmaps. Brief discussions of the historicalbackground and other relevant mattersare contained in remarks to some chap-ters.

The monograph contains many resultsof the authors. To read it with under-standing, an advanced knowledge of realanalysis and topology is needed. (jl)

G. Grätzer, Math into LaTeX, Birkhäuser,Boston, 2000, 584 pp., DM108, ISBN 0-8176-4131-9 and 3-7643-4131-9This book is written by a mathematicianfor mathematicians and/or for other sci-entists (physicists, engineers,...) who usemany mathematical formulas in theirtexts. The core of the book consists oftypesetting such formulas, and all thepractical aspects of LaTeX are discussed.For typesetting mathematics, the use ofthe AMS packages is strongly recom-mended by the author. In 1999 theAmerican Mathematical Society releasedversion 2.0 of the AMS packages; thethird edition of the book covers thechanges made in this release. In addi-tion, this third edition contains three newchapters: Chapter 12 Books in LaTeX isdevoted to typesetting large documents inLaTeX; the other new chapters are TeX,LaTeX, and the Internet and Putting LaTeXon the Web. The latter explains how topublish LaTeX articles, reports and bookson the world wide web. (mdont)

R. L. Harris, Information Graphics: AComprehensive Illustrated Reference,Oxford University Press, Oxford, 2000, 448pp., £32.50, ISBN 0-19-513532-6This is an excellent reference for thosewishing to describe data using graphs (ofany nature). The book covers the fullspectrum of charts, graphs, maps, dia-grams, and tables, ranging from the mostbasic to very specialised ones. I was sur-prised how many types of graphs have

been suggested in the literature; this bookoffers more than 1000 different types.The approximately 4000 illustrations,prepared specially for this book, comple-ment the text. The book is arrangedalphabetically for easy reference andextensive cross-referencing is usedthroughout.

In addition to a detailed description ofeach type of information graphics, thebook includes: construction details suchas grids, symbols, text, lines, axes, leg-ends, etc.; features that might misleadviewers such as scale breaks and perspec-tives; specific applications such as break-even graphs, population pyramids, can-dlestick charts, and quality control charts;and general terminology, such as vari-able, class interval, cell, stub, coordinate,etc.

The audience for this book will not befrom those who specialise in pure mathe-matics. However, I can imagine manyapplied statisticians, engineers, peoplefrom business and/or newspapers etc., forwhom this reference could represent animportant addition to the library. (jant)

H. Hida, Modular Forms and GaloisCohomology, Cambridge Studies in AdvancedMathematics 69, Cambridge University Press,Cambridge, 2000, 343 pp., £42.50, ISBN 0-521-77036-XThe groundbreaking work of Wiles (com-pleted by Taylor-Wiles) on the modulari-ty of Galois representations and ellipticcurves was one of the most significantevents in number theory in the 1990s.The main result can be summarised as anisomorphism R = T between a Galois-the-oretic object R (a universal deformationring) and a modular object T (a suitableHecke algebra). The range of technicali-ties involved in this work is enormous,which is why it is not easy to give a com-prehensive introduction to the subject.This book is based on a series of graduatecourses given by the author at UCLA. Itdoes not aim at maximum generality, butconcentrates on the proof of R = T in theordinary case with trivial tame level.

The book consists of five chapters.Chapter 1 gives a brief introduction tothe subject, an overview of the book and areinterpretation of class field theory interms of the Hecke algebra for GL(1).Chapter 2 is devoted to basic results inthe representation theory of profinitegroups, including deformation theory ofrepresentations and pseudo-representa-tions. Chapter 3 is the heart of the book.It covers the following: an adelic descrip-tion of classical modular forms and Heckealgebras; p-adic Hecke algebras; Galoisrepresentations with values in Heckealgebras; deformations of unramifiedlocal Galois representations; Taylor-Wilessystems and the proof of R = T in a spe-cial case.

The key part of the proof is a construc-tion of a Taylor-Wiles system of Heckealgebras, which is in turn based on calcu-lations involving duality theorems forGalois cohomology. The latter is the sub-ject matter of Chapter 4; the proofs close-

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ly follow Milne�s book Arithmetic DualityTheorems. Chapter 5 gives, first, a reinter-pretation of the equality R = T in termsof a �generalized class number formula�relating the special value of the L-func-tion L(1, Ad(f)) and the order of a suitableSelmer group (still in the ordinary case).The final sections give a few new resultson p-adic Hecke algebras and deforma-tion rings.

This book will be useful to graduate stu-dents and researchers in number theory.Having gone through Chapters 1-3(backed up, if necessary, by Chapter 4),the reader can proceed to the originalwork of Wiles or to its detailed expositionin the book Modular Forms and Fermat�sLast Theorem (eds. G. Cornell, J.Silverman and G. Stevens). (jnek)

A. A. Ivanov, Geometry of SporadicGroups I: Petersen and Tilde Geometries,Encyclopedia of Mathematics and ItsApplications 76, Cambridge University Press,Cambridge, 1999, 408 pp., £45, ISBN 0-521-41362-1This is the first volume of a two-volumeset on Petersen and tilde geometries.There is an infinite family of tilde geome-tries associated with non-split extensionsof symplectic groups over the two-ele-ment field. Besides them, there are 12exceptional Petersen and tilde geome-tries. These geometries are related tosome sporadic simple groups, includingthe Monster group. The first volume pre-sents a construction of each of the excep-tional geometries; these constructionsalso provide an independent proof of theexistence of the corresponding automor-phism groups. Various applications, inparticular to the representation theory ofthe Monster, are given. The book isintended for researchers in finite groups,geometries and algebraic combinatorics.(jtu)

A. M. Kamchatnov, Nonlinear PeriodicWaves and Their Modulations, WorldScientific, Singapore, 2000, 383 pp., £33,ISBN 981-02-4407-XThe book presents mathematical treat-ment of non-linear waves. Althoughthere has lately been great progress in themathematical approach to these phenom-ena, the techniques involved are quite dif-ficult. It is the aim of the author to pre-sent some of the theory clearly throughapplications to physics.

The first part of the book consists of athorough treatment of a few importantexamples of equations, where the essen-tial features of the theory are shown. Thetheory is backed up by appropriate phys-ical background and motivation. Themost important examples in this sectionare the Burgers and Kortweg-de Vriesequations, which describe waves on shal-low water, wave breaking, and non-lineareffects of viscosity. The non-linearSchrödinger equation, which describeselectromagnetic pulses in dispersivemedia, is also discussed in detail in thissection. The author shows that theKortweg-de Vries and Schrödinger equa-

tions have more universal validity in thestudy of non-linear waves.

The second part of the book presentsthe basic concepts of the Whitham theoryof modulations, which describes the evo-lution of the parameters of periodic solu-tions. This theory serves as a basis for thedevelopment of some modern techniquesof the treatment of non-linear periodicwaves. The concepts are then againapplied to the Kortweg-de Vries andSchrödinger equations and some moreapplications are added.

At the end of each chapter a few exer-cises are provided to help the readerunderstand the topic. Solutions to theexercises are given in appropriate detailat the end of the book. Some of the math-ematics used throughout the book isexplained in the appendix. Each chapteris followed by bibliographic remarks withuseful comments. The book is especiallyvaluable for the author�s attempt to inter-connect the mathematical approach withits applications to physics, and it willundoubtedly be of great interest to bothmathematicians and physicists. (tf) (= T.Furst)

R. Korn and E. Korn, Option Pricing andPortfolio Optimization, Graduate Studies inMathematics 31, American MathematicalSociety, Providence, 2001, 253 pp., US$39,ISBN 0-8218-2123-7Modelling financial markets by proba-bilistic diffusion models has received aconsiderable acceleration since the Nobelprize Black-Scholes formula for the fairpricing of the (European) options provedto be applicable to real trading processes.As a result, many other types of very com-plex derivative have been used by finan-cial markets, each of them calling for anoriginal application of the stochastic Itocalculus. This monograph offers a mathe-matically sophisticated treatment of thefollowing subjects: modelling securityprices, trading strategies and wealthprocess, principal properties of the con-tinuous-time market model, option pric-ing based on the replication principle,arbitrage bounds, and the construction ofoptimal portfolios by the martingale andstochastic control methods. The pricingmethods are developed for European,American and exotic options.

The treatment of financial topics isaccompanied by mathematically rigorousexcursions into stochastic analysis. Theauthors explain and, where possible,prove results such as the Ito formula, theGirzanov theorem and the Bellman prin-ciple. The relevant mathematics is insert-ed to the text when the results are firstneeded. Theory is presented as far aspossible without an application of theDoob-Meyer theorem.

The book is meant as lecture notes andthe authors claim it to be accessible to allthose familiar with a basic course in prob-ability. The reviewer does not completelyaccept this claim, but has no reservationsabout the exceptional quality and useful-ness of the text. (jstep)

S. G. Krantz and H. R. Parks, TheGeometry of Domains in Space, BirkhäuserAdvanced Texts, Birkhäuser, Boston, 1999,308 pp., DM108, ISBN 0-8176-4097-5 and3-7643-4097-4This book serves as a comprehensivetreatment of domains in space andemphasises the interaction betweenanalysis and geometry. Usually, analysison Euclidean spaces is based on a study ofclassical groups (rotations, dilations andtranslations) acting on the space. Herethe authors focus on domains in spaceand on properties that arise from theintrinsic geometry of the domain.

The first chapter contains a brief pre-sentation of basic facts on smooth func-tions, defining functions and measuretheory. The next chapters are devoted torudiments of differential geometry andmeasure theory (rectifiability, Minkowskicontent, a space-filling curve, coveringtheorems, domains with finite perimeter,the area and co-area formulas andRademacher�s theorem). A short chapteron Sobolev spaces deals with restrictionand trace theorems and domain exten-sion theorems. The following chaptersare devoted to smooth mappings, Sard�stheorem, Stokes� theorem, the Whitneytheorem, various kinds of convexity, theKrein-Milman theorem, Steiner sym-metrisation, the isoperimetric inequality,quasi-conformal mappings and Weyl�stheorem on eigenvalue asymptotics.

The book is understandable with agood knowledge of multi-variableadvanced calculus and linear algebra, andcan be used as a text for graduate stu-dents. The monograph also offerspleasent reading to anybody who enjoysthe interplay between analysis and geom-etry. (jl)

G. R. Krause and T. H. Lenagan, Growthof Algebras and Gelfand-KirillovDimension, Graduate Studies in Mathematics22, American Mathematical Society,Providence, 1999, 212 pp., US$39, ISBN 0-8218-0859-1This book is based on invited lectures atthe Euroconference held under the sametitle at Bielefeld university in September1998. The main topic is the role of infi-nite length modules in the representationtheory of algebras. The book consists of23 survey papers written by leadingexperts in the field.

The scene is set in a survey by Ringelcontaining many illuminating examples.There follow papers on algebraically com-pact modules (by Huisgen-Zimmermannand Prest), decomposition theory (Eklof,Facchini, Göbel and Pimenov-Yakovlev),dimension theory (Bavula, Lenagan andSchröer), functor categories (Kuhn andPowell), homological methods(Martsinkovsky, Schwartz and Smalo),localisation and moduli spaces (Rickardand Schofield), modular representationsof finite groups (Benson and Carlson),and tameness (Bautista, Krause, Lenzingand Zwara).

This book is indispensable for anyoneinterested in current trends and methods

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EMS March 2001 31

of representation theory. (jtrl)

S. Levy, The Eightfold Way: The Beauty ofKlein�s Quartic Curve, MSRI Publications35, Cambridge University Press, Cambridge,1999, 331 pp., £35, ISBN 0-521-66066-1The Klein quartic is the complex curve inthe complex projective plane given by thesimple equation x3y +y3z+z3x = 0 and ithas many remarkable and amazing prop-erties. Felix Klein discovered it in 1879,and since then it has appeared in manybranches of mathematics (complex analy-sis, geometry, algebraic geometry, num-ber theory and topology). It has anexceptionally high degree of symmetry.The Hurwitz theorem asserts that theautomorphism group of a curve of genusg has at most 84(g - 1) elements; the Kleinquartic is one of two curves where thisupper bound is reached.

In this book, the reader can find anEnglish translation of Klein�s originalpaper, together with several reviewpapers on the role of the Klein quartic inthe further development of mathematics.The article by H. Karcher and M. Weberdescribes its geometry, derives its classicalequations and computes its Jacobian. Itsrole in number theory is treated in thepaper by N. D. Elkies. A short paper byA. M. Macbeath reviews automorphismgroups of Riemann surfaces. Two papersby A. Adler treat more general questionsconcerning rings of invariants of repre-sentations of certain finite groups, andprove Hirzebruch�s earlier results for cer-tain symmetric Hilbert modular surfaces.A historical review by J. Gray (first pub-lished in the Mathematical Intelligencer) isreprinted here.

A sculpture inspired by properties ofKlein�s quartic was created by H.Ferguson and since 1983 has attractedvisitors at the Mathematical SciencesResearch Institute in Berkeley. The bookstarts with W. Thurston�s speech at theoccasion of its inauguration and includesan essay written by the sculptor. (vs)

N. Limnios and M. Nikulin (eds.),Recent Advances in Reliability Theory.Methodology, Practice, and Inference,Statistics for Industry and Technology,Birkhäuser, Boston, 2000, 514 pp., DM178,ISBN 0-8176-4135-1 and 3-7643-4135-1This book discusses reliability, asymptotictheory of processes (with or withoutswitching), semi-Markov processes, mix-ing processes, asymptotic consolidation ofstate spaces, bootstrap, accelerated lifemodels, Bayesian models, software relia-bility.

Conceiving reliable systems is a strate-gic issue for any industrial society � andthis has been also the theme of the mostimportant international conference in thepast couple of years, MMR�2000, held inBordeaux in July 2000, in which morethan 350 people from all over the worldparticipated.

The main topics covered are stochasticmodels and methods useful for reliability,including the asymptotic theory ofprocesses (with or without switching), the

theories of semi-Markov processes andmixing processes, the asymptotic consoli-dation of state spaces, the statistical meth-ods of bootstrap, of accelerated life mod-els, of point processes, Bayesian modelsand software models, and so forth.

This volume contains a selection ofinvited and contributed papers in theabove topics that describe the state-of-the-art in their disciplines. The 31 chap-ters are grouped into eight parts: Generalapproach, Probability models and relatedissues, Asymptotic analysis, Statisticalmodels and data analysis, Methods com-mon to reliability and survival analysis,Software reliability, Statistical inferenceand Asymptotic methods in statistics. Formore details, see http://www.mass.u-bordeaux2.fr/MI2S/MMR2000 (jant)

H. Lüneburg, Die euklidische Ebene undihre Verwandten, Birkhäuser, Basel, 1999,207 pp., DM49,80, ISBN 3-7643-5685-5This book is a nice introduction to thetheory of affine and projective planes thatcan be described by two- and three-dimensional vector spaces over somefields. Special attention is paid to con-nections between properties of the planes(including finite ones) and the fields, e.g.Desargues� or Pappus planes and the cor-responding fields. The book also con-tains a description of the theory of conicsections, collineations and metric proper-ties of conics. In the final chapter, a geo-metrical characterisation of the real planeamong all affine planes is given. Thebook deals with basic questions of ele-mentary geometry and can be especiallyappreciated by teachers. (lbic)

A. Mader, Almost CompletelyDecomposable Groups, Algebra, Logic andApplications Series 13, Gordon and BreachPublishers, Singapore, 2000, 354 pp.,US$90, ISBN 90-5699-225-2 The structure of rank one torsion-freeabelian groups � the subgroups of theadditive group of rationals � is well knownand completely described in the terms oftypes. Arbitrary direct sums (finite orinfinite) of rank one groups are calledcompletely decomposable groups, andtheir structure is also well known. A tor-sion-free group is called almost complete-ly decomposable if it is a finite extensionof a completely decomposable group.Using regulating subgroups, regulatorsand near-isomorphisms and collectingknown results, the monograph presents asystematic study of the structural proper-ties of this class of groups. (lbic)

V. Maillot, Géométrie d�Arakelov des var-iétés toriques et fibrés en droites inté-grables, Mémoires de la SMF 80, SociétéMathématique de France, Paris, 2000, 129pp., FRF150, ISBN 2-85629-088-4Arakelov geometry combines algebro-geometric invariants of an arithmeticvariety X, a regular flat scheme of finitetype over Z with hermitian geometry ofthe complex manifold X(C).

In this monograph the author studiesArakelov geometry of smooth projective

toric varieties over Z � a natural generali-sation of projective spaces. Such varietiesand line bundles on them can bedescribed in purely combinatorial terms.The corresponding complex analytic linebundles admit natural hermitian metrics;however, these are not C8 in general. Asa result, the author first develops a gener-alisation of arithmetic intersection theoryof Gillet-Soulé, in which one can makesense of the first arithmetic Chern classfor line bundles with �integrable� hermit-ian metrics. Applications of this theory totoric varieties include a formula for theheight of a hypersurface in terms of ageneralised Mahler measure and an arith-metic version of a theorem of Bernstein-Konshnirenko. (jnek)

V. P. Maslov and P. P. Mosolov,Nonlinear Wave Equations Perturbed byViscous Terms, de Gruyter Expositions inMathematics 31, Walter de Gruyter, Berlin,2000, 329 pp., DM298, ISBN 3-11-015282-7This book is an extended and revised ver-sion of the Russian original publication,translated from Russian to English by M.A. Shishkova. It deals with the classicalsystem of equations describing the motionof a one-dimensional compressible gas �the one-dimensional Navier-Stokes equa-tions, the continuity equation and stateequation, and the equations similar tothese. The authors obtain sharper a prioriestimates than is usual in this context,and using these estimates, they are able togive necessary and sufficient conditionsfor the existence of smooth solutions ofthe classical system for vanishing viscosi-ty. Various possibilities on the form ofmodelling equations and their solutionare discussed, including the form of theviscosity perturbation, boundedness ofnon-linearities, etc.

This book provides a skilled insightinto the contemporary problems connect-ed with the solvability of the specifiedclass of partial differential equations, andcan be recommended both to specialistsin the field and to young mathematiciansbeginning their mathematical career inPDEs. (mrok)

A. S. Matveev and A. V. Savkin,Qualitative Theory of Hybrid DynamicalSystems, Control Engineering, Birkhäuser,Boston, 2000, 348 pp., DM148, ISBN 0-8176-4141-6 and 3-7643-4141-6Hybrid dynamical systems combine con-tinuous and discrete dynamics andinvolve both continuous and discrete statevariables. A well-known instance of ahybrid system is a dynamical systemdescribed by a set of ordinary differentialequations with discontinuous or multi-val-ued right-hand sides. Such mathematicalmodels can be used to describe variousengineering systems with relays, switches,and hysteresis. In the linear control area,a typical example of a hybrid system isthat which is created when a continuous-time plant described by differential equa-tions is controlled by a digital regulatordescribed by difference equations. These

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EMS March 200132

types of systems are studied in moderncontrol engineering courses under thename of computer-controlled systems, orsampled-data systems.

The book is primarily a research mono-graph that presents some recent researchin a unified fashion. The book consistsmainly of the authors� original results andis essentially self-contained. The mainconcepts are differential automata, cycliclinear differential automata and switchedsingle server flow networks. The authorsstudy asymptotic behaviour of these mod-els � in particular, the existence of limitcycles. (pkurk)

V. Maz�ya, S. Nazarov and B.Plamenevskij, Asymptotic Theory ofElliptic Boundary Value Problems inSingularly Perturbed Domains, I, OperatorTheory Advances and Applications 111,Birkhäuser, Basel, 2000, 435 pp., DM268,ISBN 3-7643-6397-5 and 3-7643-2964-5This book is devoted to the developmentand applications of asymptotic methodsto boundary value problems for ellipticequations in singularly perturbeddomains. The first volume contains PartsI-IV, in which boundary value problemswith perturbations near isolated singular-ities of the boundary of the domain arestudied. The second volume containsParts V-VII, which deal with other kindsof perturbations (problems with perturba-tions of the boundary of singular mani-folds, problems in thin domains, andproblems with rapid oscillations of theboundary of domain or coefficients of dif-ferential operators). In Part I the authorsdiscuss boundary value problems for theLaplace operator. Part II is devoted tothe study of general elliptic boundaryvalue problems. Parts III and IV dealwith expansion of functionals over solu-tions of boundary value problems andeigenvalues in the asymptotic series. InPart V the authors study boundary valueproblems in domains perturbed nearmulti-dimensional singularities of theboundary. The behaviour of solutions ofboundary value problems in thin domainsis investigated in Part VI. Part VII dealswith elliptic boundary value problemswith oscillating coefficients or boundaryof domain. (dmed)

N. Mukhopadhyay, Probability andStatistical Interference, Statistics: Textbooksand Monographs 162, Marcel Dekker, NewYork, 2000, 665 pp., US$95, ISBN 0-8247-0379-0This book covers standard topics belong-ing to a two-semester introductory courseof mathematical statistics. The first fourchapters introduce basic concepts such asprobability and conditional probability,expectation, variance, moments, andmoment generating functions of the com-mon discrete and continuous distribu-tions, multivariate random variables andtheir characteristics, some standard prob-ability inequalities, and functions of ran-dom variables, with a brief discussion onχ2, t, and F-distributions. Chapter 5 isdevoted to concepts of stochastic conver-

gence. The remainder of the book devel-ops concepts of statistical inference. Theauthor deals with sufficiency, complete-ness and ancillarity (including Basu�s the-orem), point estimation (method ofmoments and of maximal likelihood,Rao-Blackwell theorem, Lehmann-Scheff,theorems), tests of hypotheses, confi-dence interval estimation (including mul-tiple comparisons), Bayesian methods,likelihood ratio tests, large-sample infer-ence, and two-stage procedures for sam-ple size determination.

The only prerequisite is a one-yearcourse of calculus. In individual chaptersthe reader finds numerous examples, andat the end of each chapter a long list ofexercises is presented. The text containsinteresting historical remarks and theauthor gives selected biographical noteson some exceptional contributors to thedevelopment of statistics in an appendixto the book.

This is a nice textbook. The text is writ-ten in a very understandable way and theexplanation is illustrated with many care-fully selected examples. The book canalso be helpful as a supplementary text incourses for graduate students. It is a plea-sure to read and I can recommend it tostudents and teachers of mathematicalstatistics. (jand)

L. C. G. Rogers and D. Williams,Diffusions, Markov Processes andMartingales, Vols. 1, 2, CambridgeUniversity Press, Cambridge, 2000, 386 and480 pp., £22.95 and £24.95, ISBN 0-521-77594-9 and 0-521-77593-0; First and sec-ond editions published by John Wiley & Sons1979, 1994, reissued in 2000 by CambridgeUniversity Press.These books present a systematic treat-ment of topics on the title of the mono-graph and differ significantly from theirfirst edition.

Volume 1, Foundations, is divided intothree chapters, each adopting its ownindividuality as far as the purpose and thestyle of presentation are concerned.Chapter I (Brownian motion, Gaussianand Lévy processes) is designed as anintroduction that the reader both theo-rems and heuristics and reasons why thetopics are worth studying. Chapter II hasbeen transformed to a consistent treat-ment of what could be called a �handbookof probabilistic measure theory for youngprobabilists�. Its major subjects are theelements of measure and probability the-ory, measure and probability theory onLusin spaces, the Daniell-Kolmogorovtheorem, and discrete- and continuous-time martingales. The authors treat thesesubjects in a highly up-to-date manner;for example, the continuous-time martin-gale theory is presented for the processeswith trajectories that are continuous onthe right with limits on the left, and thesophistications as augmented filtrationsare discussed with rigorous mathematicaldetails. Chapter III (mostly identical withthe first edition) is a semi-advanced treat-ment of Markov processes. It presents aneat treatment of Feller, Lévy and Ray

processes, allowing sufficient space forresults on additive functionals and Martinboundary.

Volume 2, Itô Calculus, presents per-haps the most attractive topic in modernprobability (stochastic analysis) to inter-ested and mathematically cultured audi-ences, in a manner based on both intu-ition and formal mathematical reasoning.The text differs from its previous editionsin having many new examples and proofs.The authors justify the volume�s title byproviding many examples that help read-ers to develop their calculation abilities.Chapter IV covers the standard theory ofthe stochastic integral (of a previsibleprocess with respect to a continuous semi-martingale) up to the level of the semi-martingale local time as an occupationdensity. Chapter V is devoted to thestrong and weak theory of stochastic dif-ferential equations, and such other topicsas pathwise uniqueness and uniqueness indistribution, and martingale problemsare treated with an additional emphasison one-dimensional SDEs. The subject ofstochastic differential geometry receiveshere an original and clear exposition.Chapter VI extends the stochastic inte-gration theory to local martingales withtrajectories that are right continuous withlimits on the left, the general Meyer theo-rem and Itô excursion theory being thetopics whose presentation the reviewerconsiders the highlights of the volume.

The monograph as a whole is warmlyrecommended to post-PhD students ofprobability and will be welcomed as agood and reliable reference. (jstep)

J.-E. Rombaldi, Thémes pour l�agrégationde mathématique, EDP Sciences, Les Ulis,1999, 205 pp., Ffrs160, ISBN 2-86883-407-8This is a thorough and mathematicallyneat textbook on matrix theory. Thealgebraic and topological properties ofthe matrix vector algebra (over both thereal and complex numbers) are presentedin a self-contained manner, starting witha condensed treatment of the normedvector spaces that can be easily followedby any reader familiar with the elementsof linear algebra. Standard results of clas-sical matrix theory are proved comfort-ably via the endomorphism interpreta-tion, and applications to systems of lineardifferential equations are delivered care-fully with respect to possible numericalproblems. Each chapter is accompaniedby a collection of relevant solved exercis-es.

The textbook is meant as a teachingtext at first- or second-year graduate levelto provide a background for a course onnumerical analysis, for example. Somereaders might miss applications to linearprogramming or Markov chains. (jstep)

D. B. Shapiro, Compositions of QuadraticForms, De Gruyter Expositions inMathematics 33, Walter de Gruyter, Berlin,2000, 417 pp., DM248, ISBN 3-11-012629-XA composition formula of size [r, s, n] is a

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EMS March 2001 33

sum-of-squares formula of the type (x1

2+x22+ � + xr

2).(y12+y2

2+ � +ys2)

= z12+z2

2+ � +zn2,

where X = (x1, x2, � , xr) and Y = (y1, y2,� , ys) are systems of indeterminates andeach zk = zk(X, Y) is a bilinear form in Xand Y. The first part of the book,�Classical compositions and quadraticforms�, is devoted to classical sizes [r, n, n]and the second one, �Compositions of size[r, s, n]�, gives some partial answers to thepossible sizes [r, s, n] and the dependenceon the field of coefficients using themethods of algebraic topology, combina-torics and linear algebra and geometry.(lbic)

A. Schinzel, Polynomials with SpecialRegard to Reducibility, Encyclopedia ofMathematics and Its Applications 77,Cambridge University Press, Cambridge,2000, 558 pp., £60, ISBN 0-521-66225-7As the title indicates, the book deals withreducibility aspects of polynomials. Itcovers mainly those reducibility resultsthat hold over large classes of fields; inparticular, results valid only over therational field, or over finite and localfields, are not included. The book alsocontains many related results (e.g. Ritt�stheorems, or the Mahler measure).

The titles of the chapters are: Arbitrarypolynomials over an arbitrary filed (80pp.), Lacunary polynomials over an arbi-trary field (109 pp.), Polynomials over analgebraically closed field (62 pp.),Polynomials over a finitely generatedfield (52 pp.), Polynomials over a numberfield (75 pp.), Polynomials over aKroneckerian field (91 pp.).

Since polynomials touch on a huge vari-ety of topics, it is hard to expect that sucha book could be completely self-containedand simultaneously cover a broad spec-trum of results. Nevertheless, the bookcan be read fluently, and the eventualdefinitions and results from the theory ofpolynomials needed to follow the exposi-tion are collected in ten appendices. Forother standard results the reader isreferred to standard textbooks on realand complex analysis or the theory ofalgebraic numbers.

The book is written in a lucid style andcontains many notes and cross-referencesto recent improvements on stated resultsthat makes the book very versatile. Forinstance, the last appendix (written by U.Zannier) contains Bombieri and Zannier�sproof of one of the author�s conjectures.Also very helpful are the indices of defin-itions and theorems. The bibliographyconsists of over 300 items. The book canalso be recommended as a resource foranybody interested in reducibility prob-lems or in related facets of the theory ofpolynomials. (�p)

E. Ya. Smirnov, Stabilization ofProgrammed Motion, Stability and Control:Theory, Methods and Applications 12, Gordonand Breach Publishers, 2000, 265 pp.,US$110, ISBN 90-5699-256-2This monograph develops stabilisationand optimal stabilisation of so-called pro-

grammed motion controlled systems; thisis a controlled system governed by the ini-tial-value problem for a system of, in gen-eral non-linear, ordinary differentialequations dy/dt = f(t, y, w). The informa-tion for synthesising a stabilising controlw can be extracted from (some compo-nents of) the response y, either continu-ously, or in specific time instances only,considering possibly a measurementerror.

Chapter 1 treats the case of continuous-ly receiving information, for both so-called stationary and non-stationary sys-tems (usually called autonomous and non-autonomous systems). Reduction to acanonical form, Lyapunov�s function,relay stabilisation, or feedback control,are the methods used. Information abouty received in discrete-time instances(combined possibly with continuouslyreceived information) is analysed inChapter 2. Chapter 3 deals with variousvariants of the linear/quadratic optimalcontrol problem. Finally, Chapter 4 isdevoted to the so-called orbit stabilisationof variable-structure systems.

The book is based mostly on resultsfrom the Russian school (about 85% ofthe references are published in Russian),a large part (about 30%) being theauthor�s original research. The book,requiring only a three-year universitycourse in mathematics, is designed bothfor researchers in control theory and forgraduate students. (trou)

M. Stern, Semimodular Lattices: Theoryand Applications, Encyclopedia ofMathematics and its Applications 73,Cambridge University Press, Cambridge,1999, 370 pp., £50, ISBN 0-521-46105-7The theory of semimodular lattices is thepart of lattice theory with most connec-tions to other areas of mathematics � inparticular, to discrete mathematics andcombinatorics, and especially to matroidtheory. Other parts have relations togroup theory � namely, to the structure ofsubgroup lattices. Other connections areto the foundations of geometry. Theauthor develops the theory of semimodu-lar lattices and their generalisations, andpresents applications to the areas ofmathematics mentioned above.

The book will be valuable toresearchers, not only in lattice theory butalso in discrete mathematics, combina-torics and general algebraic systems. (jtu)

P. Taylor, Practical Foundations ofMathematics, Cambridge Studies inAdvanced Mathematics 59, CambridgeUniversity Press, Cambridge, 1999, 572 pp.,£50, ISBN 0-521-63107-6�Our system conforms very closely to theway mathematical constructions haveactually been formulated in the twentiethcentury. The claim that set theory pro-vides the foundations of mathematics isonly justified via an encoding of this sys-tem, and not directly�, says the author inparagraph 2.2 (page 72), after Chapter Ientitled �First order reasoning� and afterthe first paragraph of Chapter II entitled

�Constructing the Number System�. Hedoes not use �the encoding� and he workswith sets quite freely (using the axiom ofcomprehension: if X is a set and Φ[x] is apredicate on X, then {x: X Φ[x]} is also aset). From Chapter IV, his basic mathe-matical tool is category theory. The useof categories as a base of mathematics andthe application of categorical notions andmethods in logic and in computer science(more precisely: the developing of logicand of computer science on the basis ofcategorical notions and categorical meth-ods) has been widely and intensivelyexamined recently. The book presents asystematic exposition of this topic,explaining its ideas and summarising thecorresponding results. The author alsoaims to show how these modern ideasdevelop the classical ones and he presentsmany very interesting historical facts.The book is intended for programmersand computer scientists, rather than formathematicians and logicians, but it canbe useful for both groups of potentialreaders. (vt)

A. Terras, Fourier Analysis on FiniteGroups and Applications, LondonMathematical Society Student Texts 43,Cambridge University Press, Cambridge,1999, 442 pp., £18.95, ISBN 0-521-45108-6 and 0-521-45718-1The main goal of this book is to considerfinite analogues of the symmetric spacessuch as Rn and the Poincaré upper half-plane. The author describes finite ana-logues of all the basic theorems in Fourieranalysis, both commutative and non-com-mutative, including the Poisson summa-tion formula and the Selberg trace formu-la. One motivation for this study is toprepare the ground for understandingthe continuous theory by developing itsfinite model. The book is written in sucha way that it can be enjoyed by non-experts such as advanced undergradu-ates, beginning graduate students, andscientists outside of mathematics. Severalapplications are included, such as theconstruction of graphs that are goodexpanders, reciprocity laws in numbertheory, the Ehrenfest model of diffusionrandom walks on graphs, and vibratingsystems and chemistry of molecules. (ae)

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EMS March 200134

WORLD MATHEMATICAL YEAR 2000Last October an Italian WMY2000 stampwas issued. It features the WMY2000 logo,a fractal pattern and Archimedes� sphereinside a cylinder

AustraliaKnopfmacher, John

AustriaSchmidt, KlausTeichmann, Josef

BelgiumBellemans, M. J. Bergh, MichelBorrey, Sabine Botterman, Stefaan Kalinde, Ibert Kuijken, Elisabeth Luyckx, D. Mielants, Wim Plastria, Frank Quarta, Lucas Sebille, Michel Toint, Philippe Warrinnier, Alfred

BeninHouehougbe, Antoine

BotswanaCowen, R.

CanadaBrunner, H.Guastavino, Marc

Czech RepublicBohac, ZdenekFrancu, Jan

DenmarkFuglede, BentHoholdt, TomLind Jensen, AllanNilsson, Dennis

FinlandAstola, LauraGranlund, SeppoHeili, MattiHuuskonen, TaneliHyry, EeroJussila, TapaniLaine, MarkoNevanlinna, VeikkoPeltonen, PetriPirinen, AulisPohjolainen, SeppoSalli, ArtoStenberg, RolfSuuntala, SauliVuorinen, Matti

FranceAngeniol, BernardPerthame, Benoit

Bertrand, PierreChaudouard, Pierre-HenriChipot, MichelClement, EmmanueleCordier, StephaneCoudrais, JacquesCroisille, Jean-PierreDelaruelle, ChristianDi Maio, PierreDias, FredericDiener, FrancineDonato, PatriziaDe Reffye, JeromeDi Menza, LaurentDiener, MarcEl Badia, AbdellatifFoucher, FrancoiseFournier, YvanGallot, SylvestreGilquin, HerveGiovangigli, VincentGuillaume, AnneGuillas, SergeIntissar, AbdelkaderIooss, GerardJacob, ChristineJaffard, StephaneJoly, PierreKassel, ChristianKeller, BernardKold Hansen, SoerenLasry, Jean MichelLaunay, GenevieveLaurencot, PhilippeLe Roux, Alain YvesLemay, RobertLeruste, ChristianMaisonobe, PhilippeMassot, MarcMerindol, Jean-YvesPelletier, MichelePenot, Jean-PaulPicavet, GabrielRichard, DenisSablonniere, PaulSchuman, Bertrand MarcShort, HamishSimonot, FrancoisTerpolilli, PeppinoThieullen, MicheleTollu, ChristopheTreimany, CatherineWerner, WendelinWirtz, Bruno

GermanyAnsorge, RainerBallmann, JosefBenner, PeterCarstensen, CarstenCzech, Ingo SigurdForster, Brigitte

Fritz, GregorGagelmann, UrsulaGrötschel, MartinHatala, MichaelHerber, T.Jäger, WilliKorte, UlrikeLügger, JoachimMorgenstern, BrigitteRack, Heinz-JoachimSchmidt, WolfgangTrapp, MonikaUeberberg, JohannesWeber, Gerhard-W.Weber, HorstWinkler, Jörg

IndiaSharma, A. K.

IsraelBelenkiiy, AriBerg, Y.Braun, AmiramRudnick, Zeev

ItalyAmadori, DeboraArgiolas, RobertoBarletta, ElisabettaBernardi, ClaudioBischi, Gian ItaloBonanzinga, VittoriaBotteri Fattore, Maria LuisaBrenti, FrancescoCalabri, AlbertoCapparelli, StefanoCeccherini Silberstein, TullioCiarabelli, FedericoCiliberto, CiroCorti, BeatriceDedo, ErnestoDe Mari, FilippoDi Carpegna, RanieriDiomede, SabrinaEmmer, MicheleFedeli, AlessandroFerrara, MassimilianoFerrari, FaustoFrosini, PatrizioGabelli, StefaniaGalletto, DionigiGermano Crosa, BrunaGrimaldi, RenataIsopi, MarcoLanteri, AntonioLuciano, Claudia Ines ErnestaMantegazza, CarloManzi, GilbertoMarino, Maria CorinnaMennella, F.Miranda, Annamaria

EMS March 2001 35

Peccati, LorenzoPetronio, CarloRegazzini, EugenioRicca, RenzoSchiavi, V. A.Sernesi, EdoardoSimonetti, MauraTurrini, CristinaVaio, FrancoVisintin, Augusto

JapanKajiwara, Joji

LatviaSwylan, E.

NetherlandsLenstra, H. W.Lübke, MartinVenema, R.

New ZealandGlynn, David

PolandKubarski, Jan

PortugalBigotte Almeida, Maria EmiliaCastro, Luis Clemente, Maria Florencio Fidalgo, Carla IsabelMarado Torres, Delfim

FernandoMarques Cerejeiras, Paula

CristinaMelo Margalho, Luis Manuel Mendes Lopes, Margarida

MariaMonteiro Paixao, Miguel JorgePinto, Joao Silva, Jorge Nuno

RussiaBorichev, Andrey Rappoport, Juri

SpainAguilera Venegas, GabrielAlias, Luis J.Andradas Heranz, CarlosBerenguer Maldonado, Maria

IsabelBermejo Diaz, IsabelBerriochoa, EliasCasanovas Pujadas, XavierCastells Oliveres, EugeniCastro Jimenez, Francisco

JesusChavarriga Soriano, JavierCubedo Cullere, Marta

New MembersNew MembersWe welcome the following new members who have joined in the past year.

de Prada Vicente, MariaAngeles

Diez Fernandez, HonoratoEscoriza Lopez, JoseFagella, NuriaFortes Ruiz, InmaculadaGago Couso, FelipeGaray Bengoechea, Oscar

JesusGarcia Garcia, GloriaGarcia Hernandez, Jose LuisGarcia Rodriguez, DomingoGimenez, PhilippeGirela Alvarez, DanielGodoy Malvar, EduardoGomez-Perez, JavierGonzalez Enriquez, CristobalGonzales Jimenez, SantosGracian Rodriguez, EnriqueGual Arnau, Jose JoaquinIndurain Eraso, EstebanJimenez Alcon, FranciscoLaliena Clemente, JesusLlovet, JuanLobillo, Francisco JavierMaestre Vera, ManuelMartin Cabrera, FranciscoMartinez, JuanMartinez Ansemil, Jose MariaMata Hernandez, AguedaMaureso Sanchez, MontserratMiana, Pedro Jose

Milan Lopez, FranciscoMira Ros, Jose ManuelMontes Lozano, AntonioMorales Bueno, RafaelMunoz Martinez, Jose VicenteNarvaez Macarro, LuisNunez Gomez, MarianNuno Ballestros, JuanPerez-Fernandez, F. JavierPerez Fernandez, JavierPerez Gonzalez, FernandoPerez-Martinez, EugeniaRamos Alonso, Pedro AntonioSalvador Alcaide, AdelaSanchez-Pedreno Guillen,

SalvadorSanmartin Carbon, Maria

EsperanzaSanz Gil, JavierTorres Cabrera, Maria

BegonyaVelez Melon, Maria Piar

SwedenAndersson, Mats ErikBrzezinski, JuliuszDahlgren-Johansson, OlgaFainsilber, LauraGoos, ThomasPersson, LeifRoos, Jan-ErikSjögren, Peter

SwitzerlandAeschbach, FritzBaur, KarinBurger, MarcBuser, PeterGrote, MarcusGutknecht, MartinKappeler, ThomasKaup, BurchardLang, UrsLuethi, Hans-JakobMazzola, GuerinoMislin, GuidoPerret, St�phaneQuarteroni, AlfioRatiu, TudorRutishauser, ThereseStrebel, RalphStuart, Ch.Wuergler, UrsZehnder, Eduard

TurkeyAlpay, S.O.Zhang, P.

U. S. A.Fairweather, GraemeFossum, Robert Guidotti, PatrickLawvere, William Lee Keyfitz, Barbara

Maltese, GeorgeMichaelis, WalterMilea, ConstantinRobson, RobbySagle, A.A.Saxton, RalphSchötzau, DominikVogtmann, Karen

United KingdomBaesens, C. N.Darbyshire, P. M.Ferguson, W. A.Fokas, A. S.Grant-Ross, PeterHannabuss, KeithHou, ZhanyuanKelly, CliveKendall, WilfridKisil, VladimirMaenhaut, BarbaraNijhoff, FrankPickup, D. A.Pouroutidou, S.Robertson, E. F.Testermann, DonnaWellington-Spurr, GabrielleWilliams, David Wilson, RobinWotherspoon, C. I.Wright, H.C.

EMS March 200136

The Institut des Hautes Études Scientifiques, located in Bures-sur-Yvette (France),

hosts each year 200 to 250 mathematicians and theoretical physicists from all over the world and for various peri-ods (2 or 3 days up to 1 or 2 years).

Created in 1958, the IHÉS is a Private Foundation of international standing with the purpose of �enhancing the-oretical research in Pure Mathematics, Theoretical Physics, Human Sciences Methodology and every other disci-plines which could be linked to those�. The IHÉS is financed by the French Ministére de la Recherche, someEuropean research agencies, the US National Science Foundation, and several French and foreign foundationsand companies.

EUROPEAN PROGRAMMEIn February 2000, the European Commission acknowledged the IHÉS as a Large European ResearchInfrastructure centre. A contract has thus been signed with the Commission aiming to develop and promote accessto the Institute for European researchers (European Union and associated countries). This contract, valid through2002, endows the Institute with financial support of 450,000 Euros over a 3 year period. These funds will be usedto support approximately 50 visitors doing research in mathematics or theoretical physics and originating fromcountries of the European Union or associated countries. More information about conditions of applicationthrough the European programme are available on http://www.ihes.fr/APPLICATION/ProgEC/europe2A.htmlApplications are reviewed and selected by the IHÉS Scientific Committee twice a year: in January and inSeptember.

WILLIAM HODGE FELLOWSHIPSThe Engineering and Physical Sciences Research Council has been supporting the IHÉS for a number of years andrecently decided to foster closer links between British Institutions and French research centres of excellence.British mathematicians and theoretical physicists are invited to apply to the IHÉS to visit and additionally perhapsto use the opportunity to visit research groups in the Paris region. More information is given on the IHÉS website.In addition, the EPSRC and the IHÉS are offering annualy two 1-year fellowships under the name of the eminentBritish mathematician, Sir William Hodge, whose main interests were algebraic and differential geometry. The fel-lowships will enable outstanding young mathematicians and theoretical physicists to spend time at the IHÉS.

Conditions for application for the fellowship 2002 / 2003PhD in Mathematics or Theoretical Physics obtained in 1998 or later.

One of the two grants will be exclusively awarded to an applicant who has received his/her PhD from a UKUniversity or has spent the last year in a UK university or institution.

Selection of ApplicantsApplications will be reviewed and selection made based only on the criterion of excellence by the IHÉS ScientificCommittee in January 2002. This Committee consists of the permanent professors, the Director, and some exter-nal members.

Starting date of the fellowshipsSeptember or October 2002.

How to applyThe application file should be sent under the name of �Hodge Fellowships� through the IHÉS website(www.ihes.fr) and should include: a CV, a publication list, a research project and two or three letters or recom-mendation.

Deadline for applications: December 31, 2001.

InformationIHÉS - 35, route de Chartres, F-91440 Bures-sur-Yvette (FRANCE)

tel: +33 1 6092 6600 o fax: +33 1 6092 6669 o e-mail: [email protected] or website: www.ihes.fr

Institut des Hautes Études Scientifiques35, route de Chartres - 91440 BURES-SUR-YVETTE (FRANCE)

TEL: +33 1 6092 6667 FAX: +33 1 6092 6669 WEB SITE: www.ihes.fr

contents editorial team european mathematical … · newsletter no. 39 march 2001 ems agenda ........

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